problem solving techniques in reasoning

35 problem-solving techniques and methods for solving complex problems

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How do you identify problems?

How do you identify the right solution.

Tips for more effective problem-solving

Complete problem-solving methods

Problem-solving techniques to identify and analyze problems
Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve.

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward.

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.

Try different approaches

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats
Lightning Decision Jam
Problem Definition Process
Discovery & Action Dialogue

Design Sprint 2.0

Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.

World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

6. Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

The Creativity Dice
Fishbone Analysis
Problem Tree
SWOT Analysis
Agreement-Certainty Matrix
The Journalistic Six
LEGO Challenge
What, So What, Now What?
Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.

SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!

Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!

LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

Improved Solutions
Four-Step Sketch
15% Solutions
How-Now-Wow matrix
Impact Effort Matrix

21. Mindspin

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.

MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.

Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.

How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.

Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in/Check-out
Doodling Together
Show and Tell
Constellations
Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.

Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!

Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.

Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback
Who What When Matrix
Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space.

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.

One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.

Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.

Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

Concepts and inferences (7.1)
Procedural knowledge (7.1)
Metacognition (7.1)
Characteristics of critical thinking: skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills (7.1)
Reasoning: deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic (7.2)
Fixation: functional fixedness, mental set (7.3)
Algorithms, heuristics, and the role of confirmation bias (7.3)
Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

Identify which type of knowledge a piece of information is (7.1)
Recognize examples of deductive and inductive reasoning (7.2)
Recognize judgments that have probably been influenced by the availability heuristic (7.2)
Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

Use the principles of critical thinking to evaluate information (7.1)
Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

Take a few minutes to write down everything that you know about dogs.
Do you believe that:
Psychic ability exists?
Hypnosis is an altered state of consciousness?
Magnet therapy is effective for relieving pain?
Aerobic exercise is an effective treatment for depression?
UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words think and knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge is our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept : a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition : knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called Challenging Your Preconceptions, highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017). Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

Personal values and beliefs. Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
Racism, sexism, ageism and other forms of prejudice and bigotry. These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
Self-interest. When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase Caveat Emptor (let the buyer beware) .

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism : a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

What percentage of kidnappings are committed by strangers?
Which area of the house is riskiest: kitchen, bathroom, or stairs?
What is the most common cancer in the US?
What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning : a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument : a set of statements in which the beginning statements lead to a conclusion

deductively valid argument : an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing inductive reasoning on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is inductively strong if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

Statement #1: The forecaster on Channel 2 said it is going to rain today.
Statement #2: The forecaster on Channel 5 said it is going to rain today.
Statement #3: It is very cloudy and humid.
Statement #4: You just heard thunder.
Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

A particular class will be interesting/useful/difficult
You will be able to finish writing a paper by next week if you go out tonight
Your laptop’s battery will last through the next trip to the library
You will not miss anything important if you skip class tomorrow
Your instructor will not notice if you skip class tomorrow
You will be able to find a book that you will need for a paper
There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A heuristic is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1. They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the availability heuristic (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning : a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument : an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic : a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic : judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic: judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

Please take a few minutes to list a number of problems that you are facing right now.
Now write about a problem that you recently solved.
What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a problem as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem : a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation : noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation : when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness : a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set : a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight : a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the confirmation bias (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm : a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic : a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias : people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

Identify the existence of a problem. In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
Define the problem. Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
Formulate strategy. Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
Represent and organize information. Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
Allocate resources. This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
Monitor and evaluate solutions. Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as an effective problem-solving sequence, not the effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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7.3 Problem-Solving

Learning objectives.

By the end of this section, you will be able to:

Describe problem solving strategies
Define algorithm and heuristic
Explain some common roadblocks to effective problem solving

People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

The study of human and animal problem solving processes has provided much insight toward the understanding of our conscious experience and led to advancements in computer science and artificial intelligence. Essentially much of cognitive science today represents studies of how we consciously and unconsciously make decisions and solve problems. For instance, when encountered with a large amount of information, how do we go about making decisions about the most efficient way of sorting and analyzing all the information in order to find what you are looking for as in visual search paradigms in cognitive psychology. Or in a situation where a piece of machinery is not working properly, how do we go about organizing how to address the issue and understand what the cause of the problem might be. How do we sort the procedures that will be needed and focus attention on what is important in order to solve problems efficiently. Within this section we will discuss some of these issues and examine processes related to human, animal and computer problem solving.

PROBLEM-SOLVING STRATEGIES

When people are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

Problems themselves can be classified into two different categories known as ill-defined and well-defined problems (Schacter, 2009). Ill-defined problems represent issues that do not have clear goals, solution paths, or expected solutions whereas well-defined problems have specific goals, clearly defined solutions, and clear expected solutions. Problem solving often incorporates pragmatics (logical reasoning) and semantics (interpretation of meanings behind the problem), and also in many cases require abstract thinking and creativity in order to find novel solutions. Within psychology, problem solving refers to a motivational drive for reading a definite “goal” from a present situation or condition that is either not moving toward that goal, is distant from it, or requires more complex logical analysis for finding a missing description of conditions or steps toward that goal. Processes relating to problem solving include problem finding also known as problem analysis, problem shaping where the organization of the problem occurs, generating alternative strategies, implementation of attempted solutions, and verification of the selected solution. Various methods of studying problem solving exist within the field of psychology including introspection, behavior analysis and behaviorism, simulation, computer modeling, and experimentation.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them (table below). For example, a well-known strategy is trial and error. The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

When one is faced with too much information
When the time to make a decision is limited
When the decision to be made is unimportant
When there is access to very little information to use in making the decision
When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Further problem solving strategies have been identified (listed below) that incorporate flexible and creative thinking in order to reach solutions efficiently.

Additional Problem Solving Strategies :

Abstraction – refers to solving the problem within a model of the situation before applying it to reality.
Analogy – is using a solution that solves a similar problem.
Brainstorming – refers to collecting an analyzing a large amount of solutions, especially within a group of people, to combine the solutions and developing them until an optimal solution is reached.
Divide and conquer – breaking down large complex problems into smaller more manageable problems.
Hypothesis testing – method used in experimentation where an assumption about what would happen in response to manipulating an independent variable is made, and analysis of the affects of the manipulation are made and compared to the original hypothesis.
Lateral thinking – approaching problems indirectly and creatively by viewing the problem in a new and unusual light.
Means-ends analysis – choosing and analyzing an action at a series of smaller steps to move closer to the goal.
Method of focal objects – putting seemingly non-matching characteristics of different procedures together to make something new that will get you closer to the goal.
Morphological analysis – analyzing the outputs of and interactions of many pieces that together make up a whole system.
Proof – trying to prove that a problem cannot be solved. Where the proof fails becomes the starting point or solving the problem.
Reduction – adapting the problem to be as similar problems where a solution exists.
Research – using existing knowledge or solutions to similar problems to solve the problem.
Root cause analysis – trying to identify the cause of the problem.

The strategies listed above outline a short summary of methods we use in working toward solutions and also demonstrate how the mind works when being faced with barriers preventing goals to be reached.

One example of means-end analysis can be found by using the Tower of Hanoi paradigm . This paradigm can be modeled as a word problems as demonstrated by the Missionary-Cannibal Problem :

Missionary-Cannibal Problem

Three missionaries and three cannibals are on one side of a river and need to cross to the other side. The only means of crossing is a boat, and the boat can only hold two people at a time. Your goal is to devise a set of moves that will transport all six of the people across the river, being in mind the following constraint: The number of cannibals can never exceed the number of missionaries in any location. Remember that someone will have to also row that boat back across each time.

Hint : At one point in your solution, you will have to send more people back to the original side than you just sent to the destination.

The actual Tower of Hanoi problem consists of three rods sitting vertically on a base with a number of disks of different sizes that can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top making a conical shape. The objective of the puzzle is to move the entire stack to another rod obeying the following rules:

1. Only one disk can be moved at a time.
2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.
3. No disc may be placed on top of a smaller disk.

Figure 7.02. Steps for solving the Tower of Hanoi in the minimum number of moves when there are 3 disks.

Figure 7.03. Graphical representation of nodes (circles) and moves (lines) of Tower of Hanoi.

The Tower of Hanoi is a frequently used psychological technique to study problem solving and procedure analysis. A variation of the Tower of Hanoi known as the Tower of London has been developed which has been an important tool in the neuropsychological diagnosis of executive function disorders and their treatment.

GESTALT PSYCHOLOGY AND PROBLEM SOLVING

As you may recall from the sensation and perception chapter, Gestalt psychology describes whole patterns, forms and configurations of perception and cognition such as closure, good continuation, and figure-ground. In addition to patterns of perception, Wolfgang Kohler, a German Gestalt psychologist traveled to the Spanish island of Tenerife in order to study animals behavior and problem solving in the anthropoid ape.

As an interesting side note to Kohler’s studies of chimp problem solving, Dr. Ronald Ley, professor of psychology at State University of New York provides evidence in his book A Whisper of Espionage (1990) suggesting that while collecting data for what would later be his book The Mentality of Apes (1925) on Tenerife in the Canary Islands between 1914 and 1920, Kohler was additionally an active spy for the German government alerting Germany to ships that were sailing around the Canary Islands. Ley suggests his investigations in England, Germany and elsewhere in Europe confirm that Kohler had served in the German military by building, maintaining and operating a concealed radio that contributed to Germany’s war effort acting as a strategic outpost in the Canary Islands that could monitor naval military activity approaching the north African coast.

While trapped on the island over the course of World War 1, Kohler applied Gestalt principles to animal perception in order to understand how they solve problems. He recognized that the apes on the islands also perceive relations between stimuli and the environment in Gestalt patterns and understand these patterns as wholes as opposed to pieces that make up a whole. Kohler based his theories of animal intelligence on the ability to understand relations between stimuli, and spent much of his time while trapped on the island investigation what he described as insight , the sudden perception of useful or proper relations. In order to study insight in animals, Kohler would present problems to chimpanzee’s by hanging some banana’s or some kind of food so it was suspended higher than the apes could reach. Within the room, Kohler would arrange a variety of boxes, sticks or other tools the chimpanzees could use by combining in patterns or organizing in a way that would allow them to obtain the food (Kohler & Winter, 1925).

While viewing the chimpanzee’s, Kohler noticed one chimp that was more efficient at solving problems than some of the others. The chimp, named Sultan, was able to use long poles to reach through bars and organize objects in specific patterns to obtain food or other desirables that were originally out of reach. In order to study insight within these chimps, Kohler would remove objects from the room to systematically make the food more difficult to obtain. As the story goes, after removing many of the objects Sultan was used to using to obtain the food, he sat down ad sulked for a while, and then suddenly got up going over to two poles lying on the ground. Without hesitation Sultan put one pole inside the end of the other creating a longer pole that he could use to obtain the food demonstrating an ideal example of what Kohler described as insight. In another situation, Sultan discovered how to stand on a box to reach a banana that was suspended from the rafters illustrating Sultan’s perception of relations and the importance of insight in problem solving.

Grande (another chimp in the group studied by Kohler) builds a three-box structure to reach the bananas, while Sultan watches from the ground. Insight , sometimes referred to as an “Ah-ha” experience, was the term Kohler used for the sudden perception of useful relations among objects during problem solving (Kohler, 1927; Radvansky & Ashcraft, 2013).

Solving puzzles.

Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below (see figure) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

How long did it take you to solve this sudoku puzzle? (You can see the answer at the end of this section.)

Here is another popular type of puzzle (figure below) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Did you figure it out? (The answer is at the end of this section.) Once you understand how to crack this puzzle, you won’t forget.

Take a look at the “Puzzling Scales” logic puzzle below (figure below). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

A puzzle involving a scale is shown. At the top of the figure it reads: “Sam Loyds Puzzling Scales.” The first row of the puzzle shows a balanced scale with 3 blocks and a top on the left and 12 marbles on the right. Below this row it reads: “Since the scales now balance.” The next row of the puzzle shows a balanced scale with just the top on the left, and 1 block and 8 marbles on the right. Below this row it reads: “And balance when arranged this way.” The third row shows an unbalanced scale with the top on the left side, which is much lower than the right side. The right side is empty. Below this row it reads: “Then how many marbles will it require to balance with that top?”

What steps did you take to solve this puzzle? You can read the solution at the end of this section.

Pitfalls to problem solving.

Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but she just needs to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in the table below.

Were you able to determine how many marbles are needed to balance the scales in the figure below? You need nine. Were you able to solve the problems in the figures above? Here are the answers.

The first puzzle is a Sudoku grid of 16 squares (4 rows of 4 squares) is shown. Half of the numbers were supplied to start the puzzle and are colored blue, and half have been filled in as the puzzle’s solution and are colored red. The numbers in each row of the grid, left to right, are as follows. Row 1: blue 3, red 1, red 4, blue 2. Row 2: red 2, blue 4, blue 1, red 3. Row 3: red 1, blue 3, blue 2, red 4. Row 4: blue 4, red 2, red 3, blue 1.The second puzzle consists of 9 dots arranged in 3 rows of 3 inside of a square. The solution, four straight lines made without lifting the pencil, is shown in a red line with arrows indicating the direction of movement. In order to solve the puzzle, the lines must extend beyond the borders of the box. The four connecting lines are drawn as follows. Line 1 begins at the top left dot, proceeds through the middle and right dots of the top row, and extends to the right beyond the border of the square. Line 2 extends from the end of line 1, through the right dot of the horizontally centered row, through the middle dot of the bottom row, and beyond the square’s border ending in the space beneath the left dot of the bottom row. Line 3 extends from the end of line 2 upwards through the left dots of the bottom, middle, and top rows. Line 4 extends from the end of line 3 through the middle dot in the middle row and ends at the right dot of the bottom row.

Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. Roadblocks to problem solving include a mental set, functional fixedness, and various biases that can cloud decision making skills.

References:

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Review Questions:

1. A specific formula for solving a problem is called ________.

a. an algorithm

b. a heuristic

c. a mental set

d. trial and error

2. Solving the Tower of Hanoi problem tends to utilize a ________ strategy of problem solving.

a. divide and conquer

b. means-end analysis

d. experiment

3. A mental shortcut in the form of a general problem-solving framework is called ________.

4. Which type of bias involves becoming fixated on a single trait of a problem?

a. anchoring bias

b. confirmation bias

c. representative bias

d. availability bias

5. Which type of bias involves relying on a false stereotype to make a decision?

6. Wolfgang Kohler analyzed behavior of chimpanzees by applying Gestalt principles to describe ________.

a. social adjustment

b. student load payment options

c. emotional learning

d. insight learning

7. ________ is a type of mental set where you cannot perceive an object being used for something other than what it was designed for.

a. functional fixedness

c. working memory

Critical Thinking Questions:

1. What is functional fixedness and how can overcoming it help you solve problems?

2. How does an algorithm save you time and energy when solving a problem?

Personal Application Question:

1. Which type of bias do you recognize in your own decision making processes? How has this bias affected how you’ve made decisions in the past and how can you use your awareness of it to improve your decisions making skills in the future?

anchoring bias

availability heuristic

confirmation bias

functional fixedness

hindsight bias

problem-solving strategy

representative bias

trial and error

working backwards

Answers to Exercises

algorithm: problem-solving strategy characterized by a specific set of instructions

anchoring bias: faulty heuristic in which you fixate on a single aspect of a problem to find a solution

availability heuristic: faulty heuristic in which you make a decision based on information readily available to you

confirmation bias: faulty heuristic in which you focus on information that confirms your beliefs

functional fixedness: inability to see an object as useful for any other use other than the one for which it was intended

heuristic: mental shortcut that saves time when solving a problem

hindsight bias: belief that the event just experienced was predictable, even though it really wasn’t

mental set: continually using an old solution to a problem without results

problem-solving strategy: method for solving problems

representative bias: faulty heuristic in which you stereotype someone or something without a valid basis for your judgment

trial and error: problem-solving strategy in which multiple solutions are attempted until the correct one is found

working backwards: heuristic in which you begin to solve a problem by focusing on the end result

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Learn Creative Problem Solving Techniques to Stimulate Innovation in Your Organization

By Kate Eby | October 20, 2017 (updated August 27, 2021)

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In today’s competitive business landscape, organizations need processes in place to make strong, well-informed, and innovative decisions. Problem solving - in particular creative problem solving (CPS) - is a key skill in learning how to accurately identify problems and their causes, generate potential solutions, and evaluate all the possibilities to arrive at a strong corrective course of action. Every team in any organization, regardless of department or industry, needs to be effective, creative, and quick when solving problems.

In this article, we’ll discuss traditional and creative problem solving, and define the steps, best practices, and common barriers associated. After that, we’ll provide helpful methods and tools to identify the cause(s) of problematic situations, so you can get to the root of the issue and start to generate solutions. Then, we offer nearly 20 creative problem solving techniques to implement at your organization, or even in your personal life. Along the way, experts weigh in on the importance of problem solving, and offer tips and tricks.

What Is Problem Solving and Decision Making?

Problem solving is the process of working through every aspect of an issue or challenge to reach a solution. Decision making is choosing one of multiple proposed solutions — therefore, this process also includes defining and evaluating all potential options. Decision making is often one step of the problem solving process, but the two concepts are distinct.

Collective problem solving is problem solving that includes many different parties and bridges the knowledge of different groups. Collective problem solving is common in business problem solving because workplace decisions typically affect more than one person.

Problem solving, especially in business, is a complicated science. Not only are business conflicts multifaceted, but they often involve different personalities, levels of authority, and group dynamics. In recent years, however, there has been a rise in psychology-driven problem solving techniques, especially for the workplace. In fact, the psychology of how people solve problems is now studied formally in academic disciplines such as psychology and cognitive science.

Joe Carella is the Assistant Dean for Executive Education at the University of Arizona . Joe has over 20 years of experience in helping executives and corporations in managing change and developing successful business strategies. His doctoral research and executive education engagements have seen him focus on corporate strategy, decision making and business performance with a variety of corporate clients including Hershey’s, Chevron, Fender Musical Instruments Corporation, Intel, DP World, Essilor, BBVA Compass Bank.

He explains some of the basic psychology behind problem solving: “When our brain is engaged in the process of solving problems, it is engaged in a series of steps where it processes and organizes the information it receives while developing new knowledge it uses in future steps. Creativity is embedded in this process by incorporating diverse inputs and/or new ways of organizing the information received.”

Laura MacLeod is a Professor of Social Group Work at City University of New York, and the creator of From The Inside Out Project® , a program that coaches managers in team leadership for a variety of workplaces. She has a background in social work and over two decades of experience as a union worker, and currently leads talks on conflict resolution, problem solving, and listening skills at conferences across the country.

MacLeod thinks of problem solving as an integral practice of successful organizations. “Problem solving is a collaborative process — all voices are heard and connected, and resolution is reached by the group,” she says. “Problems and conflicts occur in all groups and teams in the workplace, but if leaders involve everyone in working through, they will foster cohesion, engagement, and buy in. Everybody wins.”

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What Is the First Step in Solving a Problem?

Although problem solving techniques vary procedurally, experts agree that the first step in solving a problem is defining the problem. Without a clear articulation of the problem at stake, it is impossible to analyze all the key factors and actors, generate possible solutions, and then evaluate them to pick the best option.

Dr. Elliott Jaffa is a behavioral and management psychologist with over 25 years of problem solving training and management experience. “Start with defining the problem you want to solve,” he says, “And then define where you want to be, what you want to come away with.” He emphasizes these are the first steps in creating an actionable, clear solution.

Bryan Mattimore is Co-Founder of Growth Engine, an 18-year old innovation agency based in Norwalk, CT. Bryan has facilitated over 1,000 ideation sessions and managed over 200 successful innovation projects leading to over $3 billion in new sales. His newest book is 21 Days to a Big Idea . When asked about the first critical component to successful problem solving, Mattimore says, “Defining the challenge correctly, or ‘solving the right problem’ … The three creative techniques we use to help our clients ‘identify the right problem to be solved’ are questioning assumptions, 20 questions, and problem redefinition. A good example of this was a new product challenge from a client to help them ‘invent a new iron. We got them to redefine the challenge as first: a) inventing new anti-wrinkle devices, and then b) inventing new garment care devices.”

What Are Problem Solving Skills?

To understand the necessary skills in problem solving, you should first understand the types of thinking often associated with strong decision making. Most problem solving techniques look for a balance between the following binaries:

Convergent vs. Divergent Thinking: Convergent thinking is bringing together disparate information or ideas to determine a single best answer or solution. This thinking style values logic, speed, and accuracy, and leaves no chance for ambiguity. Divergent thinking is focused on generating new ideas to identify and evaluate multiple possible solutions, often uniting ideas in unexpected combinations. Divergent thinking is characterized by creativity, complexity, curiosity, flexibility, originality, and risk-taking.
Pragmatics vs. Semantics: Pragmatics refer to the logic of the problem at hand, and semantics is how you interpret the problem to solve it. Both are important to yield the best possible solution.
Mathematical vs. Personal Problem Solving: Mathematical problem solving involves logic (usually leading to a single correct answer), and is useful for problems that involve numbers or require an objective, clear-cut solution. However, many workplace problems also require personal problem solving, which includes interpersonal, collaborative, and emotional intuition and skills.

The following basic methods are fundamental problem solving concepts. Implement them to help balance the above thinking models.

Reproductive Thinking: Reproductive thinking uses past experience to solve a problem. However, be careful not to rely too heavily on past solutions, and to evaluate current problems individually, with their own factors and parameters.
Idea Generation: The process of generating many possible courses of action to identify a solution. This is most commonly a team exercise because putting everyone’s ideas on the table will yield the greatest number of potential solutions.

However, many of the most critical problem solving skills are “soft” skills: personal and interpersonal understanding, intuitiveness, and strong listening.

Mattimore expands on this idea: “The seven key skills to be an effective creative problem solver that I detail in my book Idea Stormers: How to Lead and Inspire Creative Breakthroughs are: 1) curiosity 2) openness 3) a willingness to embrace ambiguity 4) the ability to identify and transfer principles across categories and disciplines 5) the desire to search for integrity in ideas, 6) the ability to trust and exercise “knowingness” and 7) the ability to envision new worlds (think Dr. Seuss, Star Wars, Hunger Games, Harry Potter, etc.).”

“As an individual contributor to problem solving it is important to exercise our curiosity, questioning, and visioning abilities,” advises Carella. “As a facilitator it is essential to allow for diverse ideas to emerge, be able to synthesize and ‘translate’ other people’s thinking, and build an extensive network of available resources.”

MacLeod says the following interpersonal skills are necessary to effectively facilitate group problem solving: “The abilities to invite participation (hear all voices, encourage silent members), not take sides, manage dynamics between the monopolizer, the scapegoat, and the bully, and deal with conflict (not avoiding it or shutting down).”

Furthermore, Jaffa explains that the skills of a strong problem solver aren’t measurable. The best way to become a creative problem solver, he says, is to do regular creative exercises that keep you sharp and force you to think outside the box. Carella echoes this sentiment: “Neuroscience tells us that creativity comes from creating novel neural paths. Allow a few minutes each day to exercise your brain with novel techniques and brain ‘tricks’ – read something new, drive to work via a different route, count backwards, smell a new fragrance, etc.”

What Is Creative Problem Solving? History, Evolution, and Core Principles

Creative problem solving (CPS) is a method of problem solving in which you approach a problem or challenge in an imaginative, innovative way. The goal of CPS is to come up with innovative solutions, make a decision, and take action quickly. Sidney Parnes and Alex Osborn are credited with developing the creative problem solving process in the 1950s. The concept was further studied and developed at SUNY Buffalo State and the Creative Education Foundation.

The core principles of CPS include the following:

Balance divergent and convergent thinking
Ask problems as questions
Defer or suspend judgement
Focus on “Yes, and…” rather than “No, but…”

According to Carella, “Creative problem solving is the mental process used for generating innovative and imaginative ideas as a solution to a problem or a challenge. Creative problem solving techniques can be pursued by individuals or groups.”

When asked to define CPS, Jaffa explains that it is, by nature, difficult to create boundaries for. “Creative problem solving is not cut and dry,” he says, “If you ask 100 different people the definition of creative problem solving, you’ll get 100 different responses - it’s a non-entity.”

Business presents a unique need for creative problem solving. Especially in today’s competitive landscape, organizations need to iterate quickly, innovate with intention, and constantly be at the cutting-edge of creativity and new ideas to succeed. Developing CPS skills among your workforce not only enables you to make faster, stronger in-the-moment decisions, but also inspires a culture of collaborative work and knowledge sharing. When people work together to generate multiple novel ideas and evaluate solutions, they are also more likely to arrive at an effective decision, which will improve business processes and reduce waste over time. In fact, CPS is so important that some companies now list creative problem solving skills as a job criteria.

MacLeod reiterates the vitality of creative problem solving in the workplace. “Problem solving is crucial for all groups and teams,” she says. “Leaders need to know how to guide the process, hear all voices and involve all members - it’s not easy.”

“This mental process [of CPS] is especially helpful in work environments where individuals and teams continuously struggle with new problems and challenges posed by their continuously changing environment,” adds Carella.

Problem Solving Best Practices

By nature, creative problem solving does not have a clear-cut set of do’s and don’ts. Rather, creating a culture of strong creative problem solvers requires flexibility, adaptation, and interpersonal skills. However, there are a several best practices that you should incorporate:

Use a Systematic Approach: Regardless of the technique you use, choose a systematic method that satisfies your workplace conditions and constraints (time, resources, budget, etc.). Although you want to preserve creativity and openness to new ideas, maintaining a structured approach to the process will help you stay organized and focused.
View Problems as Opportunities: Rather than focusing on the negatives or giving up when you encounter barriers, treat problems as opportunities to enact positive change on the situation. In fact, some experts even recommend defining problems as opportunities, to remain proactive and positive.
Change Perspective: Remember that there are multiple ways to solve any problem. If you feel stuck, changing perspective can help generate fresh ideas. A perspective change might entail seeking advice of a mentor or expert, understanding the context of a situation, or taking a break and returning to the problem later. “A sterile or familiar environment can stifle new thinking and new perspectives,” says Carella. “Make sure you get out to draw inspiration from spaces and people out of your usual reach.”
Break Down Silos: To invite the greatest possible number of perspectives to any problem, encourage teams to work cross-departmentally. This not only combines diverse expertise, but also creates a more trusting and collaborative environment, which is essential to effective CPS. According to Carella, “Big challenges are always best tackled by a group of people rather than left to a single individual. Make sure you create a space where the team can concentrate and convene.”
Employ Strong Leadership or a Facilitator: Some companies choose to hire an external facilitator that teaches problem solving techniques, best practices, and practicums to stimulate creative problem solving. But, internal managers and staff can also oversee these activities. Regardless of whether the facilitator is internal or external, choose a strong leader who will value others’ ideas and make space for creative solutions. Mattimore has specific advice regarding the role of a facilitator: “When facilitating, get the group to name a promising idea (it will crystalize the idea and make it more memorable), and facilitate deeper rather than broader. Push for not only ideas, but how an idea might specifically work, some of its possible benefits, who and when would be interested in an idea, etc. This fleshing-out process with a group will generate fewer ideas, but at the end of the day will yield more useful concepts that might be profitably pursued.” Additionally, Carella says that “Executives and managers don’t necessarily have to be creative problem solvers, but need to make sure that their teams are equipped with the right tools and resources to make this happen. Also they need to be able to foster an environment where failing fast is accepted and celebrated.”
Evaluate Your Current Processes: This practice can help you unlock bottlenecks, and also identify gaps in your data and information management, both of which are common roots of business problems.

MacLeod offers the following additional advice, “Always get the facts. Don’t jump too quickly to a solution – working through [problems] takes time and patience.”

Mattimore also stresses that how you introduce creative problem solving is important. “Do not start by introducing a new company-wide innovation process,” he says. “Instead, encourage smaller teams to pursue specific creative projects, and then build a process from the ground up by emulating these smaller teams’ successful approaches. We say: ‘You don’t innovate by changing the culture, you change the culture by innovating.’”

Barriers to Effective Problem Solving

Learning how to effectively solve problems is difficult and takes time and continual adaptation. There are several common barriers to successful CPS, including:

Confirmation Bias: The tendency to only search for or interpret information that confirms a person’s existing ideas. People misinterpret or disregard data that doesn’t align with their beliefs.
Mental Set: People’s inclination to solve problems using the same tactics they have used to solve problems in the past. While this can sometimes be a useful strategy (see Analogical Thinking in a later section), it often limits inventiveness and creativity.
Functional Fixedness: This is another form of narrow thinking, where people become “stuck” thinking in a certain way and are unable to be flexible or change perspective.
Unnecessary Constraints: When people are overwhelmed with a problem, they can invent and impose additional limits on solution avenues. To avoid doing this, maintain a structured, level-headed approach to evaluating causes, effects, and potential solutions.
Groupthink: Be wary of the tendency for group members to agree with each other — this might be out of conflict avoidance, path of least resistance, or fear of speaking up. While this agreeableness might make meetings run smoothly, it can actually stunt creativity and idea generation, therefore limiting the success of your chosen solution.
Irrelevant Information: The tendency to pile on multiple problems and factors that may not even be related to the challenge at hand. This can cloud the team’s ability to find direct, targeted solutions.
Paradigm Blindness: This is found in people who are unwilling to adapt or change their worldview, outlook on a particular problem, or typical way of processing information. This can erode the effectiveness of problem solving techniques because they are not aware of the narrowness of their thinking, and therefore cannot think or act outside of their comfort zone.

According to Jaffa, the primary barrier of effective problem solving is rigidity. “The most common things people say are, ‘We’ve never done it before,’ or ‘We’ve always done it this way.’” While these feelings are natural, Jaffa explains that this rigid thinking actually precludes teams from identifying creative, inventive solutions that result in the greatest benefit.

“The biggest barrier to creative problem solving is a lack of awareness – and commitment to – training employees in state-of-the-art creative problem-solving techniques,” Mattimore explains. “We teach our clients how to use ideation techniques (as many as two-dozen different creative thinking techniques) to help them generate more and better ideas. Ideation techniques use specific and customized stimuli, or ‘thought triggers’ to inspire new thinking and new ideas.”

MacLeod adds that ineffective or rushed leadership is another common culprit. “We're always in a rush to fix quickly,” she says. “Sometimes leaders just solve problems themselves, making unilateral decisions to save time. But the investment is well worth it — leaders will have less on their plates if they can teach and eventually trust the team to resolve. Teams feel empowered and engagement and investment increases.”

Strategies for Problem Cause Identification

As discussed, most experts agree that the first and most crucial step in problem solving is defining the problem. Once you’ve done this, however, it may not be appropriate to move straight to the solution phase. Rather, it is often helpful to identify the cause(s) of the problem: This will better inform your solution planning and execution, and help ensure that you don’t fall victim to the same challenges in the future.

Below are some of the most common strategies for identifying the cause of a problem:

Root Cause Analysis: This method helps identify the most critical cause of a problem. A factor is considered a root cause if removing it prevents the problem from recurring. Performing a root cause analysis is a 12 step process that includes: define the problem, gather data on the factors contributing to the problem, group the factors based on shared characteristics, and create a cause-and-effect timeline to determine the root cause. After that, you identify and evaluate corrective actions to eliminate the root cause.

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Problem Solving Techniques and Strategies

In this section, we’ll explain several traditional and creative problem solving methods that you can use to identify challenges, create actionable goals, and resolve problems as they arise. Although there is often procedural and objective crossover among techniques, they are grouped by theme so you can identify which method works best for your organization.

Divergent Creative Problem Solving Techniques

Brainstorming: One of the most common methods of divergent thinking, brainstorming works best in an open group setting where everyone is encouraged to share their creative ideas. The goal is to generate as many ideas as possible – you analyze, critique, and evaluate the ideas only after the brainstorming session is complete. To learn more specific brainstorming techniques, read this article .

Mind Mapping: This is a visual thinking tool where you graphically depict concepts and their relation to one another. You can use mind mapping to structure the information you have, analyze and synthesize it, and generate solutions and new ideas from there. The goal of a mind map is to simplify complicated problems so you can more clearly identify solutions.

Appreciative Inquiry (AI): The basic assumption of AI is that “an organization is a mystery to be embraced.” Using this principle, AI takes a positive, inquisitive approach to identifying the problem, analyzing the causes, and presenting possible solutions. The five principles of AI emphasize dialogue, deliberate language and outlook, and social bonding.

Lateral Thinking: This is an indirect problem solving approach centered on the momentum of idea generation. As opposed to critical thinking, where people value ideas based on their truth and the absence of errors, lateral thinking values the “movement value” of new ideas: This means that you reward team members for producing a large volume of new ideas rapidly. With this approach, you’ll generate many new ideas before approving or rejecting any.

Problem Solving Techniques to Change Perspective

Constructive Controversy: This is a structured approach to group decision making to preserve critical thinking and disagreement while maintaining order. After defining the problem and presenting multiple courses of action, the group divides into small advocacy teams who research, analyze, and refute a particular option. Once each advocacy team has presented its best-case scenario, the group has a discussion (advocacy teams still defend their presented idea). Arguing and playing devil’s advocate is encouraged to reach an understanding of the pros and cons of each option. Next, advocacy teams abandon their cause and evaluate the options openly until they reach a consensus. All team members formally commit to the decision, regardless of whether they advocated for it at the beginning. You can learn more about the goals and steps in constructive controversy here .

Carella is a fan of this approach. “Create constructive controversy by having two teams argue the pros and cons of a certain idea,” he says. “It forces unconscious biases to surface and gives space for new ideas to formulate.”

Abstraction: In this method, you apply the problem to a fictional model of the current situation. Mapping an issue to an abstract situation can shed extraneous or irrelevant factors, and reveal places where you are overlooking obvious solutions or becoming bogged down by circumstances.

Analogical Thinking: Also called analogical reasoning , this method relies on an analogy: using information from one problem to solve another problem (these separate problems are called domains). It can be difficult for teams to create analogies among unrelated problems, but it is a strong technique to help you identify repeated issues, zoom out and change perspective, and prevent the problems from occurring in the future. .

CATWOE: This framework ensures that you evaluate the perspectives of those whom your decision will impact. The factors and questions to consider include (which combine to make the acronym CATWOE):

Customers: Who is on the receiving end of your decisions? What problem do they currently have, and how will they react to your proposed solution?
Actors: Who is acting to bring your solution to fruition? How will they respond and be affected by your decision?
Transformation Process: What processes will you employ to transform your current situation and meet your goals? What are the inputs and outputs?
World View: What is the larger context of your proposed solution? What is the larger, big-picture problem you are addressing?
Owner: Who actually owns the process? How might they influence your proposed solution (positively or negatively), and how can you influence them to help you?
Environmental Constraints: What are the limits (environmental, resource- and budget-wise, ethical, legal, etc.) on your ideas? How will you revise or work around these constraints?

Complex Problem Solving

Soft Systems Methodology (SSM): For extremely complex problems, SSM can help you identify how factors interact, and determine the best course of action. SSM was borne out of organizational process modeling and general systems theory, which hold that everything is part of a greater, interconnected system: This idea works well for “hard” problems (where logic and a single correct answer are prioritized), and less so for “soft” problems (i.e., human problems where factors such as personality, emotions, and hierarchy come into play). Therefore, SSM defines a seven step process for problem solving:

Begin with the problem or problematic situation
Express the problem or situation and build a rich picture of the themes of the problem
Identify the root causes of the problem (most commonly with CATWOE)
Build conceptual models of human activity surrounding the problem or situation
Compare models with real-world happenings
Identify changes to the situation that are both feasible and desirable
Take action to implement changes and improve the problematic situation

SSM can be used for any complex soft problem, and is also a useful tool in change management .

Failure Mode and Effects Analysis (FMEA): This method helps teams anticipate potential problems and take steps to mitigate them. Use FMEA when you are designing (redesigning) a complex function, process, product, or service. First, identify the failure modes, which are the possible ways that a project could fail. Then, perform an effects analysis to understand the consequences of each of the potential downfalls. This exercise is useful for internalizing the severity of each potential failure and its effects so you can make adjustments or safeties in your plan.

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Problem Solving Based on Data or Logic (Heuristic Methods)

TRIZ: A Russian-developed problem solving technique that values logic, analysis, and forecasting over intuition or soft reasoning. TRIZ (translated to “theory of inventive problem solving” or TIPS in English) is a systematic approach to defining and identifying an inventive solution to difficult problems. The method offers several strategies for arriving at an inventive solution, including a contradictions matrix to assess trade-offs among solutions, a Su-Field analysis which uses formulas to describe a system by its structure, and ARIZ (algorithm of inventive problem solving) which uses algorithms to find inventive solutions.

Inductive Reasoning: A logical method that uses evidence to conclude that a certain answer is probable (this is opposed to deductive reasoning, where the answer is assumed to be true). Inductive reasoning uses a limited number of observations to make useful, logical conclusions (for example, the Scientific Method is an extreme example of inductive reasoning). However, this method doesn’t always map well to human problems in the workplace — in these instances, managers should employ intuitive inductive reasoning , which allows for more automatic, implicit conclusions so that work can progress. This, of course, retains the principle that these intuitive conclusions are not necessarily the one and only correct answer.

Process-Oriented Problem Solving Methods

Plan Do Check Act (PDCA): This is an iterative management technique used to ensure continual improvement of products or processes. First, teams plan (establish objectives to meet desired end results), then do (implement the plan, new processes, or produce the output), then check (compare expected with actual results), and finally act (define how the organization will act in the future, based on the performance and knowledge gained in the previous three steps).

Means-End Analysis (MEA): The MEA strategy is to reduce the difference between the current (problematic) state and the goal state. To do so, teams compile information on the multiple factors that contribute to the disparity between the current and goal states. Then they try to change or eliminate the factors one by one, beginning with the factor responsible for the greatest difference in current and goal state. By systematically tackling the multiple factors that cause disparity between the problem and desired outcome, teams can better focus energy and control each step of the process.

Hurson’s Productive Thinking Model: This technique was developed by Tim Hurson, and is detailed in his 2007 book Think Better: An Innovator’s Guide to Productive Thinking . The model outlines six steps that are meant to give structure while maintaining creativity and critical thinking: 1) Ask “What is going on?” 2) Ask “What is success?” 3) Ask “What is the question?” 4) Generate answers 5) Forge the solution 6) Align resources.

Control Influence Accept (CIA): The basic premise of CIA is that how you respond to problems determines how successful you will be in overcoming them. Therefore, this model is both a problem solving technique and stress-management tool that ensures you aren’t responding to problems in a reactive and unproductive way. The steps in CIA include:

Control: Identify the aspects of the problem that are within your control.
Influence: Identify the aspects of the problem that you cannot control, but that you can influence.
Accept: Identify the aspects of the problem that you can neither control nor influence, and react based on this composite information.

GROW Model: This is a straightforward problem solving method for goal setting that clearly defines your goals and current situation, and then asks you to define the potential solutions and be realistic about your chosen course of action. The steps break down as follows:

Goal: What do you want?
Reality: Where are you now?
Options: What could you do?
Will: What will you do?

OODA Loop: This acronym stands for observe, orient, decide, and act. This approach is a decision-making cycle that values agility and flexibility over raw human force. It is framed as a loop because of the understanding that any team will continually encounter problems or opponents to success and have to overcome them.

There are also many un-named creative problem solving techniques that follow a sequenced series of steps. While the exact steps vary slightly, they all follow a similar trajectory and aim to accomplish similar goals of problem, cause, and goal identification, idea generation, and active solution implementation.

MacLeod offers her own problem solving procedure, which echoes the above steps:

“1. Recognize the Problem: State what you see. Sometimes the problem is covert. 2. Identify: Get the facts — What exactly happened? What is the issue? 3. and 4. Explore and Connect: Dig deeper and encourage group members to relate their similar experiences. Now you're getting more into the feelings and background [of the situation], not just the facts. 5. Possible Solutions: Consider and brainstorm ideas for resolution. 6. Implement: Choose a solution and try it out — this could be role play and/or a discussion of how the solution would be put in place. 7. Evaluate: Revisit to see if the solution was successful or not.”

Many of these problem solving techniques can be used in concert with one another, or multiple can be appropriate for any given problem. It’s less about facilitating a perfect CPS session, and more about encouraging team members to continually think outside the box and push beyond personal boundaries that inhibit their innovative thinking. So, try out several methods, find those that resonate best with your team, and continue adopting new techniques and adapting your processes along the way.

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Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

JGI / Jamie Grill / Getty Images

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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

Discovery of the problem
Deciding to tackle the issue
Seeking to understand the problem more fully
Researching available options or solutions
Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

Perceptually recognizing the problem
Representing the problem in memory
Considering relevant information that applies to the problem
Identifying different aspects of the problem
Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

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Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality . Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

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By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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The Oxford Handbook of Thinking and Reasoning

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21 Problem Solving

Miriam Bassok, Department of Psychology, University of Washington, Seattle, WA

Laura R. Novick, Department of Psychology and Human Development, Vanderbilt University, Nashville, TN

Published: 21 November 2012
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This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1 ) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2 ) research on search in a problem space (the legacy of Newell and Simon) that examined how people generate the problem's solution. It then describes some developments in the field that fueled the integration of these two lines of research: work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. Next, it presents examples of recent work on problem solving in science and mathematics that highlight the impact of visual perception and background knowledge on how people represent problems and search for problem solutions. The final section considers possible directions for future research.

People are confronted with problems on a daily basis, be it trying to extract a broken light bulb from a socket, finding a detour when the regular route is blocked, fixing dinner for unexpected guests, dealing with a medical emergency, or deciding what house to buy. Obviously, the problems people encounter differ in many ways, and their solutions require different types of knowledge and skills. Yet we have a sense that all the situations we classify as problems share a common core. Karl Duncker defined this core as follows: “A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action [i.e., by the performance of obvious operations], then there has to be recourse to thinking” (Duncker, 1945 , p. 1). Consider the broken light bulb. The obvious operation—holding the glass part of the bulb with one's fingers while unscrewing the base from the socket—is prevented by the fact that the glass is broken. Thus, there must be “recourse to thinking” about possible ways to solve the problem. For example, one might try mounting half a potato on the broken bulb (we do not know the source of this creative solution, which is described on many “how to” Web sites).

The above definition and examples make it clear that what constitutes a problem for one person may not be a problem for another person, or for that same person at another point in time. For example, the second time one has to remove a broken light bulb from a socket, the solution likely can be retrieved from memory; there is no problem. Similarly, tying shoes may be considered a problem for 5-year-olds but not for readers of this chapter. And, of course, people may change their goal and either no longer have a problem (e.g., take the guests to a restaurant instead of fixing dinner) or attempt to solve a different problem (e.g., decide what restaurant to go to). Given the highly subjective nature of what constitutes a problem, researchers who study problem solving have often presented people with novel problems that they should be capable of solving and attempted to find regularities in the resulting problem-solving behavior. Despite the variety of possible problem situations, researchers have identified important regularities in the thinking processes by which people (a) represent , or understand, problem situations and (b) search for possible ways to get to their goal.

A problem representation is a model constructed by the solver that summarizes his or her understanding of the problem components—the initial state (e.g., a broken light bulb in a socket), the goal state (the light bulb extracted), and the set of possible operators one may apply to get from the initial state to the goal state (e.g., use pliers). According to Reitman ( 1965 ), problem components differ in the extent to which they are well defined . Some components leave little room for interpretation (e.g., the initial state in the broken light bulb example is relatively well defined), whereas other components may be ill defined and have to be defined by the solver (e.g., the possible actions one may take to extract the broken bulb). The solver's representation of the problem guides the search for a possible solution (e.g., possible attempts at extracting the light bulb). This search may, in turn, change the representation of the problem (e.g., finding that the goal cannot be achieved using pliers) and lead to a new search. Such a recursive process of representation and search continues until the problem is solved or until the solver decides to abort the goal.

Duncker ( 1945 , pp. 28–37) documented the interplay between representation and search based on his careful analysis of one person's solution to the “Radiation Problem” (later to be used extensively in research analogy, see Holyoak, Chapter 13 ). This problem requires using some rays to destroy a patient's stomach tumor without harming the patient. At sufficiently high intensity, the rays will destroy the tumor. However, at that intensity, they will also destroy the healthy tissue surrounding the tumor. At lower intensity, the rays will not harm the healthy tissue, but they also will not destroy the tumor. Duncker's analysis revealed that the solver's solution attempts were guided by three distinct problem representations. He depicted these solution attempts as an inverted search tree in which the three main branches correspond to the three general problem representations (Duncker, 1945 , p. 32). We reproduce this diagram in Figure 21.1 . The desired solution appears on the rightmost branch of the tree, within the general problem representation in which the solver aims to “lower the intensity of the rays on their way through healthy tissue.” The actual solution is to project multiple low-intensity rays at the tumor from several points around the patient “by use of lens.” The low-intensity rays will converge on the tumor, where their individual intensities will sum to a level sufficient to destroy the tumor.

A search-tree representation of one subject's solution to the radiation problem, reproduced from Duncker ( 1945 , p. 32).

Although there are inherent interactions between representation and search, some researchers focus their efforts on understanding the factors that affect how solvers represent problems, whereas others look for regularities in how they search for a solution within a particular representation. Based on their main focus of interest, researchers devise or select problems with solutions that mainly require either constructing a particular representation or finding the appropriate sequence of steps leading from the initial state to the goal state. In most cases, researchers who are interested in problem representation select problems in which one or more of the components are ill defined, whereas those who are interested in search select problems in which the components are well defined. The following examples illustrate, respectively, these two problem types.

The Bird-and-Trains problem (Posner, 1973 , pp. 150–151) is a mathematical word problem that tends to elicit two distinct problem representations (see Fig. 21.2a and b ):

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When the bird reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet? Fig. 21.2 Open in new tab Download slide Alternative representations of Posner's ( 1973 ) trains-and-bird problem. Adapted from Novick and Hmelo ( 1994 ).

Some solvers focus on the back-and-forth path of the bird (Fig. 21.2a ). This representation yields a problem that would be difficult for most people to solve (e.g., a series of differential equations). Other solvers focus on the paths of the trains (Fig. 21.2b ), a representation that yields a relatively easy distance-rate-time problem.

The Tower of Hanoi problem falls on the other end of the representation-search continuum. It leaves little room for differences in problem representations, and the primary work is to discover a solution path (or the best solution path) from the initial state to the goal state .

There are three pegs mounted on a base. On the leftmost peg, there are three disks of differing sizes. The disks are arranged in order of size with the largest disk on the bottom and the smallest disk on the top. The disks may be moved one at a time, but only the top disk on a peg may be moved, and at no time may a larger disk be placed on a smaller disk. The goal is to move the three-disk tower from the leftmost peg to the rightmost peg.

Figure 21.3 shows all the possible legal arrangements of disks on pegs. The arrows indicate transitions between states that result from moving a single disk, with the thicker gray arrows indicating the shortest path that connects the initial state to the goal state.

The division of labor between research on representation versus search has distinct historical antecedents and research traditions. In the next two sections, we review the main findings from these two historical traditions. Then, we describe some developments in the field that fueled the integration of these lines of research—work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. In the fifth section, we present some examples of recent work on problem solving in science and mathematics. This work highlights the role of visual perception and background knowledge in the way people represent problems and search for problem solutions. In the final section, we consider possible directions for future research.

Our review is by no means an exhaustive one. It follows the historical development of the field and highlights findings that pertain to a wide variety of problems. Research pertaining to specific types of problems (e.g., medical problems), specific processes that are involved in problem solving (e.g., analogical inferences), and developmental changes in problem solving due to learning and maturation may be found elsewhere in this volume (e.g., Holyoak, Chapter 13 ; Smith & Ward, Chapter 23 ; van Steenburgh et al., Chapter 24 ; Simonton, Chapter 25 ; Opfer & Siegler, Chapter 30 ; Hegarty & Stull, Chapter 31 ; Dunbar & Klahr, Chapter 35 ; Patel et al., Chapter 37 ; Lowenstein, Chapter 38 ; Koedinger & Roll, Chapter 40 ).

All possible problem states for the three-disk Tower of Hanoi problem. The thicker gray arrows show the optimum solution path connecting the initial state (State #1) to the goal state (State #27).

Problem Representation: The Gestalt Legacy

Research on problem representation has its origins in Gestalt psychology, an influential approach in European psychology during the first half of the 20th century. (Behaviorism was the dominant perspective in American psychology at this time.) Karl Duncker published a book on the topic in his native German in 1935, which was translated into English and published 10 years later as the monograph On Problem-Solving (Duncker, 1945 ). Max Wertheimer also published a book on the topic in 1945, titled Productive Thinking . An enlarged edition published posthumously includes previously unpublished material (Wertheimer, 1959 ). Interestingly, 1945 seems to have been a watershed year for problem solving, as mathematician George Polya's book, How to Solve It , also appeared then (a second edition was published 12 years later; Polya, 1957 ).

The Gestalt psychologists extended the organizational principles of visual perception to the domain of problem solving. They showed that various visual aspects of the problem, as well the solver's prior knowledge, affect how people understand problems and, therefore, generate problem solutions. The principles of visual perception (e.g., proximity, closure, grouping, good continuation) are directly relevant to problem solving when the physical layout of the problem, or a diagram that accompanies the problem description, elicits inferences that solvers include in their problem representations. Such effects are nicely illustrated by Maier's ( 1930 ) nine-dot problem: Nine dots are arrayed in a 3x3 grid, and the task is to connect all the dots by drawing four straight lines without lifting one's pencil from the paper. People have difficulty solving this problem because their initial representations generally include a constraint, inferred from the configuration of the dots, that the lines should not go outside the boundary of the imaginary square formed by the outer dots. With this constraint, the problem cannot be solved (but see Adams, 1979 ). Without this constraint, the problem may be solved as shown in Figure 21.4 (though the problem is still difficult for many people; see Weisberg & Alba, 1981 ).

The nine-dot problem is a classic insight problem (see van Steenburgh et al., Chapter 24 ). According to the Gestalt view (e.g., Duncker, 1945 ; Kohler, 1925 ; Maier, 1931 ; see Ohlsson, 1984 , for a review), the solution to an insight problem appears suddenly, accompanied by an “aha!” sensation, immediately following the sudden “restructuring” of one's understanding of the problem (i.e., a change in the problem representation): “The decisive points in thought-processes, the moments of sudden comprehension, of the ‘Aha!,’ of the new, are always at the same time moments in which such a sudden restructuring of the thought-material takes place” (Duncker, 1945 , p. 29). For the nine-dot problem, one view of the required restructuring is that the solver relaxes the constraint implied by the perceptual form of the problem and realizes that the lines may, in fact, extend past the boundary of the imaginary square. Later in the chapter, we present more recent accounts of insight.

The entities that appear in a problem also tend to evoke various inferences that people incorporate into their problem representations. A classic demonstration of this is the phenomenon of functional fixedness , introduced by Duncker ( 1945 ): If an object is habitually used for a certain purpose (e.g., a box serves as a container), it is difficult to see

A solution to the nine-dot problem.

that object as having properties that would enable it to be used for a dissimilar purpose. Duncker's basic experimental paradigm involved two conditions that varied in terms of whether the object that was crucial for solution was initially used for a function other than that required for solution.

Consider the candles problem—the best known of the five “practical problems” Duncker ( 1945 ) investigated. Three candles are to be mounted at eye height on a door. On the table, for use in completing this task, are some tacks and three boxes. The solution is to tack the three boxes to the door to serve as platforms for the candles. In the control condition, the three boxes were presented to subjects empty. In the functional-fixedness condition, they were filled with candles, tacks, and matches. Thus, in the latter condition, the boxes initially served the function of container, whereas the solution requires that they serve the function of platform. The results showed that 100% of the subjects who received empty boxes solved the candles problem, compared with only 43% of subjects who received filled boxes. Every one of the five problems in this study showed a difference favoring the control condition over the functional-fixedness condition, with average solution rates across the five problems of 97% and 58%, respectively.

The function of the objects in a problem can be also “fixed” by their most recent use. For example, Birch and Rabinowitz ( 1951 ) had subjects perform two consecutive tasks. In the first task, people had to use either a switch or a relay to form an electric circuit. After completing this task, both groups of subjects were asked to solve Maier's ( 1931 ) two-ropes problem. The solution to this problem requires tying an object to one of the ropes and making the rope swing as a pendulum. Subjects could create the pendulum using either the object from the electric-circuit task or the other object. Birch and Rabinowitz found that subjects avoided using the same object for two unrelated functions. That is, those who used the switch in the first task made the pendulum using the relay, and vice versa. The explanations subjects subsequently gave for their object choices revealed that they were unaware of the functional-fixedness constraint they imposed on themselves.

In addition to investigating people's solutions to such practical problems as irradiating a tumor, mounting candles on the wall, or tying ropes, the Gestalt psychologists examined how people understand and solve mathematical problems that require domain-specific knowledge. For example, Wertheimer ( 1959 ) observed individual differences in students' learning and subsequent application of the formula for finding the area of a parallelogram (see Fig. 21.5a ). Some students understood the logic underlying the learned formula (i.e., the fact that a parallelogram can be transformed into a rectangle by cutting off a triangle from one side and pasting it onto the other side) and exhibited “productive thinking”—using the same logic to find the area of the quadrilateral in Figure 21.5b and the irregularly shaped geometric figure in Figure 21.5c . Other students memorized the formula and exhibited “reproductive thinking”—reproducing the learned solution only to novel parallelograms that were highly similar to the original one.

The psychological study of human problem solving faded into the background after the demise of the Gestalt tradition (during World War II), and problem solving was investigated only sporadically until Allen Newell and Herbert Simon's ( 1972 ) landmark book Human Problem Solving sparked a flurry of research on this topic. Newell and Simon adopted and refined Duncker's ( 1945 ) methodology of collecting and analyzing the think-aloud protocols that accompany problem solutions and extended Duncker's conceptualization of a problem solution as a search tree. However, their initial work did not aim to extend the Gestalt findings

Finding the area of ( a ) a parallelogram, ( b ) a quadrilateral, and ( c ) an irregularly shaped geometric figure. The solid lines indicate the geometric figures whose areas are desired. The dashed lines show how to convert the given figures into rectangles (i.e., they show solutions with understanding).

pertaining to problem representation. Instead, as we explain in the next section, their objective was to identify the general-purpose strategies people use in searching for a problem solution.

Search in a Problem Space: The Legacy of Newell and Simon

Newell and Simon ( 1972 ) wrote a magnum opus detailing their theory of problem solving and the supporting research they conducted with various collaborators. This theory was grounded in the information-processing approach to cognitive psychology and guided by an analogy between human and artificial intelligence (i.e., both people and computers being “Physical Symbol Systems,” Newell & Simon, 1976 ; see Doumas & Hummel, Chapter 5 ). They conceptualized problem solving as a process of search through a problem space for a path that connects the initial state to the goal state—a metaphor that alludes to the visual or spatial nature of problem solving (Simon, 1990 ). The term problem space refers to the solver's representation of the task as presented (Simon, 1978 ). It consists of ( 1 ) a set of knowledge states (the initial state, the goal state, and all possible intermediate states), ( 2 ) a set of operators that allow movement from one knowledge state to another, ( 3 ) a set of constraints, and ( 4 ) local information about the path one is taking through the space (e.g., the current knowledge state and how one got there).

We illustrate the components of a problem space for the three-disk Tower of Hanoi problem, as depicted in Figure 21.3 . The initial state appears at the top (State #1) and the goal state at the bottom right (State #27). The remaining knowledge states in the figure are possible intermediate states. The current knowledge state is the one at which the solver is located at any given point in the solution process. For example, the current state for a solver who has made three moves along the optimum solution path would be State #9. The solver presumably would know that he or she arrived at this state from State #5. This knowledge allows the solver to recognize a move that involves backtracking. The three operators in this problem are moving each of the three disks from one peg to another. These operators are subject to the constraint that a larger disk may not be placed on a smaller disk.

Newell and Simon ( 1972 ), as well as other contemporaneous researchers (e.g., Atwood & Polson, 1976 ; Greeno, 1974 ; Thomas, 1974 ), examined how people traverse the spaces of various well-defined problems (e.g., the Tower of Hanoi, Hobbits and Orcs). They discovered that solvers' search is guided by a number of shortcut strategies, or heuristics , which are likely to get the solver to the goal state without an extensive amount of search. Heuristics are often contrasted with algorithms —methods that are guaranteed to yield the correct solution. For example, one could try every possible move in the three-disk Tower of Hanoi problem and, eventually, find the correct solution. Although such an exhaustive search is a valid algorithm for this problem, for many problems its application is very time consuming and impractical (e.g., consider the game of chess).

In their attempts to identify people's search heuristics, Newell and Simon ( 1972 ) relied on two primary methodologies: think-aloud protocols and computer simulations. Their use of think-aloud protocols brought a high degree of scientific rigor to the methodology used by Duncker ( 1945 ; see Ericsson & Simon, 1980 ). Solvers were required to say out loud everything they were thinking as they solved the problem, that is, everything that went through their verbal working memory. Subjects' verbalizations—their think-aloud protocols—were tape-recorded and then transcribed verbatim for analysis. This method is extremely time consuming (e.g., a transcript of one person's solution to the cryptarithmetic problem DONALD + GERALD = ROBERT, with D = 5, generated a 17-page transcript), but it provides a detailed record of the solver's ongoing solution process.

An important caveat to keep in mind while interpreting a subject's verbalizations is that “a protocol is relatively reliable only for what it positively contains, but not for that which it omits” (Duncker, 1945 , p. 11). Ericsson and Simon ( 1980 ) provided an in-depth discussion of the conditions under which this method is valid (but see Russo, Johnson, & Stephens, 1989 , for an alternative perspective). To test their interpretation of a subject's problem solution, inferred from the subject's verbal protocol, Newell and Simon ( 1972 ) created a computer simulation program and examined whether it solved the problem the same way the subject did. To the extent that the computer simulation provided a close approximation of the solver's step-by-step solution process, it lent credence to the researcher's interpretation of the verbal protocol.

Newell and Simon's ( 1972 ) most famous simulation was the General Problem Solver or GPS (Ernst & Newell, 1969 ). GPS successfully modeled human solutions to problems as different as the Tower of Hanoi and the construction of logic proofs using a single general-purpose heuristic: means-ends analysis . This heuristic captures people's tendency to devise a solution plan by setting subgoals that could help them achieve their final goal. It consists of the following steps: ( 1 ) Identify a difference between the current state and the goal (or subgoal ) state; ( 2 ) Find an operator that will remove (or reduce) the difference; (3a) If the operator can be directly applied, do so, or (3b) If the operator cannot be directly applied, set a subgoal to remove the obstacle that is preventing execution of the desired operator; ( 4 ) Repeat steps 1–3 until the problem is solved. Next, we illustrate the implementation of this heuristic for the Tower of Hanoi problem, using the problem space in Figure 21.3 .

As can be seen in Figure 21.3 , a key difference between the initial state and the goal state is that the large disk is on the wrong peg (step 1). To remove this difference (step 2), one needs to apply the operator “move-large-disk.” However, this operator cannot be applied because of the presence of the medium and small disks on top of the large disk. Therefore, the solver may set a subgoal to move that two-disk tower to the middle peg (step 3b), leaving the rightmost peg free for the large disk. A key difference between the initial state and this new subgoal state is that the medium disk is on the wrong peg. Because application of the move-medium-disk operator is blocked, the solver sets another subgoal to move the small disk to the right peg. This subgoal can be satisfied immediately by applying the move-small-disk operator (step 3a), generating State #3. The solver then returns to the previous subgoal—moving the tower consisting of the small and medium disks to the middle peg. The differences between the current state (#3) and the subgoal state (#9) can be removed by first applying the move-medium-disk operator (yielding State #5) and then the move-small-disk operator (yielding State #9). Finally, the move-large-disk operator is no longer blocked. Hence, the solver moves the large disk to the right peg, yielding State #11.

Notice that the subgoals are stacked up in the order in which they are generated, so that they pop up in the order of last in first out. Given the first subgoal in our example, repeated application of the means-ends analysis heuristic will yield the shortest-path solution, indicated by the large gray arrows. In general, subgoals provide direction to the search and allow solvers to plan several moves ahead. By assessing progress toward a required subgoal rather than the final goal, solvers may be able to make moves that otherwise seem unwise. To take a concrete example, consider the transition from State #1 to State #3 in Figure 21.3 . Comparing the initial state to the goal state, this move seems unwise because it places the small disk on the bottom of the right peg, whereas it ultimately needs to be at the top of the tower on that peg. But comparing the initial state to the solver-generated subgoal state of having the medium disk on the middle peg, this is exactly where the small disk needs to go.

Means-ends analysis and various other heuristics (e.g., the hill-climbing heuristic that exploits the similarity, or distance, between the state generated by the next operator and the goal state; working backward from the goal state to the initial state) are flexible strategies that people often use to successfully solve a large variety of problems. However, the generality of these heuristics comes at a cost: They are relatively weak and fallible (e.g., in the means-ends solution to the problem of fixing a hole in a bucket, “Dear Liza” leads “Dear Henry” in a loop that ends back at the initial state; the lyrics of this famous song can be readily found on the Web). Hence, although people use general-purpose heuristics when they encounter novel problems, they replace them as soon as they acquire experience with and sufficient knowledge about the particular problem space (e.g., Anzai & Simon, 1979 ).

Despite the fruitfulness of this research agenda, it soon became evident that a fundamental weakness was that it minimized the importance of people's background knowledge. Of course, Newell and Simon ( 1972 ) were aware that problem solutions require relevant knowledge (e.g., the rules of logical proofs, or rules for stacking disks). Hence, in programming GPS, they supplemented every problem they modeled with the necessary background knowledge. This practice highlighted the generality and flexibility of means-ends analysis but failed to capture how people's background knowledge affects their solutions. As we discussed in the previous section, domain knowledge is likely to affect how people represent problems and, therefore, how they generate problem solutions. Moreover, as people gain experience solving problems in a particular knowledge domain (e.g., math, physics), they change their representations of these problems (e.g., Chi, Feltovich, & Glaser, 1981 ; Haverty, Koedinger, Klahr, & Alibali, 2000 ; Schoenfeld & Herrmann, 1982 ) and learn domain-specific heuristics (e.g., Polya, 1957 ; Schoenfeld, 1979 ) that trump the general-purpose strategies.

It is perhaps inevitable that the two traditions in problem-solving research—one emphasizing representation and the other emphasizing search strategies—would eventually come together. In the next section we review developments that led to this integration.

The Two Legacies Converge

Because Newell and Simon ( 1972 ) aimed to discover the strategies people use in searching for a solution, they investigated problems that minimized the impact of factors that tend to evoke differences in problem representations, of the sort documented by the Gestalt psychologists. In subsequent work, however, Simon and his collaborators showed that such factors are highly relevant to people's solutions of well-defined problems, and Simon ( 1986 ) incorporated these findings into the theoretical framework that views problem solving as search in a problem space.

In this section, we first describe illustrative examples of this work. We then describe research on insight solutions that incorporates ideas from the two legacies described in the previous sections.

Relevance of the Gestalt Ideas to the Solution of Search Problems

In this subsection we describe two lines of research by Simon and his colleagues, and by other researchers, that document the importance of perception and of background knowledge to the way people search for a problem solution. The first line of research used variants of relatively well-defined riddle problems that had the same structure (i.e., “problem isomorphs”) and, therefore, supposedly the same problem space. It documented that people's search depended on various perceptual and conceptual inferences they tended to draw from a specific instantiation of the problem's structure. The second line of research documented that people's search strategies crucially depend on their domain knowledge and on their prior experience with related problems.

Problem Isomorphs

Hayes and Simon ( 1977 ) used two variants of the Tower of Hanoi problem that, instead of disks and pegs, involved monsters and globes that differed in size (small, medium, and large). In both variants, the initial state had the small monster holding the large globe, the medium-sized monster holding the small globe, and the large monster holding the medium-sized globe. Moreover, in both variants the goal was for each monster to hold a globe proportionate to its own size. The only difference between the problems concerned the description of the operators. In one variant (“transfer”), subjects were told that the monsters could transfer the globes from one to another as long as they followed a set of rules, adapted from the rules in the original Tower of Hanoi problem (e.g., only one globe may be transferred at a time). In the other variant (“change”), subjects were told that the monsters could shrink and expand themselves according to a set of rules, which corresponded to the rules in the transfer version of the problem (e.g., only one monster may change its size at a time). Despite the isomorphism of the two variants, subjects conducted their search in two qualitatively different problem spaces, which led to solution times for the change variant being almost twice as long as those for the transfer variant. This difference arose because subjects could more readily envision and track an object that was changing its location with every move than one that was changing its size.

Recent work by Patsenko and Altmann ( 2010 ) found that, even in the standard Tower of Hanoi problem, people's solutions involve object-bound routines that depend on perception and selective attention. The subjects in their study solved various Tower of Hanoi problems on a computer. During the solution of a particular “critical” problem, the computer screen changed at various points without subjects' awareness (e.g., a disk was added, such that a subject who started with a five-disc tower ended with a six-disc tower). Patsenko and Altmann found that subjects' moves were guided by the configurations of the objects on the screen rather than by solution plans they had stored in memory (e.g., the next subgoal).

The Gestalt psychologists highlighted the role of perceptual factors in the formation of problem representations (e.g., Maier's, 1930 , nine-dot problem) but were generally silent about the corresponding implications for how the problem was solved (although they did note effects on solution accuracy). An important contribution of the work on people's solutions of the Tower of Hanoi problem and its variants was to show the relevance of perceptual factors to the application of various operators during search for a problem solution—that is, to the how of problem solving. In the next section, we describe recent work that documents the involvement of perceptual factors in how people understand and use equations and diagrams in the context of solving math and science problems.

Kotovsky, Hayes, and Simon ( 1985 ) further investigated factors that affect people's representation and search in isomorphs of the Tower of Hanoi problem. In one of their isomorphs, three disks were stacked on top of each other to form an inverted pyramid, with the smallest disc on the bottom and the largest on top. Subjects' solutions of the inverted pyramid version were similar to their solutions of the standard version that has the largest disc on the bottom and the smallest on top. However, the two versions were solved very differently when subjects were told that the discs represent acrobats. Subjects readily solved the version in which they had to place a small acrobat on the shoulders of a large one, but they refrained from letting a large acrobat stand on the shoulders of a small one. In other words, object-based inferences that draw on people's semantic knowledge affected the solution of search problems, much as they affect the solution of the ill-defined problems investigated by the Gestalt psychologists (e.g., Duncker's, 1945 , candles problem). In the next section, we describe more recent work that shows similar effects in people's solutions to mathematical word problems.

The work on differences in the representation and solution of problem isomorphs is highly relevant to research on analogical problem solving (or analogical transfer), which examines when and how people realize that two problems that differ in their cover stories have a similar structure (or a similar problem space) and, therefore, can be solved in a similar way. This research shows that minor differences between example problems, such as the use of X-rays versus ultrasound waves to fuse a broken filament of a light bulb, can elicit different problem representations that significantly affect the likelihood of subsequent transfer to novel problem analogs (Holyoak & Koh, 1987 ). Analogical transfer has played a central role in research on human problem solving, in part because it can shed light on people's understanding of a given problem and its solution and in part because it is believed to provide a window onto understanding and investigating creativity (see Smith & Ward, Chapter 23 ). We briefly mention some findings from the analogy literature in the next subsection on expertise, but we do not discuss analogical transfer in detail because this topic is covered elsewhere in this volume (Holyoak, Chapter 13 ).

Expertise and Its Development

In another line of research, Simon and his colleagues examined how people solve ecologically valid problems from various rule-governed and knowledge-rich domains. They found that people's level of expertise in such domains, be it in chess (Chase & Simon, 1973 ; Gobet & Simon, 1996 ), mathematics (Hinsley, Hayes, & Simon, 1977 ; Paige & Simon, 1966 ), or physics (Larkin, McDermott, Simon, & Simon, 1980 ; Simon & Simon, 1978 ), plays a crucial role in how they represent problems and search for solutions. This work, and the work of numerous other researchers, led to the discovery (and rediscovery, see Duncker, 1945 ) of important differences between experts and novices, and between “good” and “poor” students.

One difference between experts and novices pertains to pattern recognition. Experts' attention is quickly captured by familiar configurations within a problem situation (e.g., a familiar configuration of pieces in a chess game). In contrast, novices' attention is focused on isolated components of the problem (e.g., individual chess pieces). This difference, which has been found in numerous domains, indicates that experts have stored in memory many meaningful groups (chunks) of information: for example, chess (Chase & Simon, 1973 ), circuit diagrams (Egan & Schwartz, 1979 ), computer programs (McKeithen, Reitman, Rueter, & Hirtle, 1981 ), medicine (Coughlin & Patel, 1987 ; Myles-Worsley, Johnston, & Simons, 1988 ), basketball and field hockey (Allard & Starkes, 1991 ), and figure skating (Deakin & Allard, 1991 ).

The perceptual configurations that domain experts readily recognize are associated with stored solution plans and/or compiled procedures (Anderson, 1982 ). As a result, experts' solutions are much faster than, and often qualitatively different from, the piecemeal solutions that novice solvers tend to construct (e.g., Larkin et al., 1980 ). In effect, experts often see the solutions that novices have yet to compute (e.g., Chase & Simon, 1973 ; Novick & Sherman, 2003 , 2008 ). These findings have led to the design of various successful instructional interventions (e.g., Catrambone, 1998 ; Kellman et al., 2008 ). For example, Catrambone ( 1998 ) perceptually isolated the subgoals of a statistics problem. This perceptual chunking of meaningful components of the problem prompted novice students to self-explain the meaning of the chunks, leading to a conceptual understanding of the learned solution. In the next section, we describe some recent work that shows the beneficial effects of perceptual pattern recognition on the solution of familiar mathematics problems, as well as the potentially detrimental effects of familiar perceptual chunks to understanding and reasoning with diagrams depicting evolutionary relationships among taxa.

Another difference between experts and novices pertains to their understanding of the solution-relevant problem structure. Experts' knowledge is highly organized around domain principles, and their problem representations tend to reflect this principled understanding. In particular, they can extract the solution-relevant structure of the problems they encounter (e.g., meaningful causal relations among the objects in the problem; see Cheng & Buehner, Chapter 12 ). In contrast, novices' representations tend to be bound to surface features of the problems that may be irrelevant to solution (e.g., the particular objects in a problem). For example, Chi, Feltovich, and Glaser ( 1981 ) examined how students with different levels of physics expertise group mechanics word problems. They found that advanced graduate students grouped the problems based on the physics principles relevant to the problems' solutions (e.g., conservation of energy, Newton's second law). In contrast, undergraduates who had successfully completed an introductory course in mechanics grouped the problems based on the specific objects involved (e.g., pulley problems, inclined plane problems). Other researchers have found similar results in the domains of biology, chemistry, computer programming, and math (Adelson, 1981 ; Kindfield, 1993 / 1994 ; Kozma & Russell, 1997 ; McKeithen et al., 1981 ; Silver, 1979 , 1981 ; Weiser & Shertz, 1983 ).

The level of domain expertise and the corresponding representational differences are, of course, a matter of degree. With increasing expertise, there is a gradual change in people's focus of attention from aspects that are not relevant to solution to those that are (e.g., Deakin & Allard, 1991 ; Hardiman, Dufresne, & Mestre, 1989 ; McKeithen et al., 1981 ; Myles-Worsley et al., 1988 ; Schoenfeld & Herrmann, 1982 ; Silver, 1981 ). Interestingly, Chi, Bassok, Lewis, Reimann, and Glaser ( 1989 ) found similar differences in focus on structural versus surface features among a group of novices who studied worked-out examples of mechanics problems. These differences, which echo Wertheimer's ( 1959 ) observations of individual differences in students' learning about the area of parallelograms, suggest that individual differences in people's interests and natural abilities may affect whether, or how quickly, they acquire domain expertise.

An important benefit of experts' ability to focus their attention on solution-relevant aspects of problems is that they are more likely than novices to recognize analogous problems that involve different objects and cover stories (e.g., Chi et al., 1989 ; Novick, 1988 ; Novick & Holyoak, 1991 ; Wertheimer, 1959 ) or that come from other knowledge domains (e.g., Bassok & Holyoak, 1989 ; Dunbar, 2001 ; Goldstone & Sakamoto, 2003 ). For example, Bassok and Holyoak ( 1989 ) found that, after learning to solve arithmetic-progression problems in algebra, subjects spontaneously applied these algebraic solutions to analogous physics problems that dealt with constantly accelerated motion. Note, however, that experts and good students do not simply ignore the surface features of problems. Rather, as was the case in the problem isomorphs we described earlier (Kotovsky et al., 1985 ), they tend to use such features to infer what the problem's structure could be (e.g., Alibali, Bassok, Solomon, Syc, & Goldin-Meadow, 1999 ; Blessing & Ross, 1996 ). For example, Hinsley et al. ( 1977 ) found that, after reading no more than the first few words of an algebra word problem, expert solvers classified the problem into a likely problem category (e.g., a work problem, a distance problem) and could predict what questions they might be asked and the equations they likely would need to use.

Surface-based problem categorization has a heuristic value (Medin & Ross, 1989 ): It does not ensure a correct categorization (Blessing & Ross, 1996 ), but it does allow solvers to retrieve potentially appropriate solutions from memory and to use them, possibly with some adaptation, to solve a variety of novel problems. Indeed, although experts exploit surface-structure correlations to save cognitive effort, they have the capability to realize that a particular surface cue is misleading (Hegarty, Mayer, & Green, 1992 ; Lewis & Mayer, 1987 ; Martin & Bassok, 2005 ; Novick 1988 , 1995 ; Novick & Holyoak, 1991 ). It is not surprising, therefore, that experts may revert to novice-like heuristic methods when solving problems under pressure (e.g., Beilock, 2008 ) or in subdomains in which they have general but not specific expertise (e.g., Patel, Groen, & Arocha, 1990 ).

Relevance of Search to Insight Solutions

We introduced the notion of insight in our discussion of the nine-dot problem in the section on the Gestalt tradition. The Gestalt view (e.g., Duncker, 1945 ; Maier, 1931 ; see Ohlsson, 1984 , for a review) was that insight problem solving is characterized by an initial work period during which no progress toward solution is made (i.e., an impasse), a sudden restructuring of one's problem representation to a more suitable form, followed immediately by the sudden appearance of the solution. Thus, solving problems by insight was believed to be all about representation, with essentially no role for a step-by-step solution process (i.e., search). Subsequent and contemporary researchers have generally concurred with the Gestalt view that getting the right representation is crucial. However, research has shown that insight solutions do not necessarily arise suddenly or full blown after restructuring (e.g., Weisberg & Alba, 1981 ); and even when they do, the underlying solution process (in this case outside of awareness) may reflect incremental progress toward the goal (Bowden & Jung-Beeman, 2003 ; Durso, Rea, & Dayton, 1994 ; Novick & Sherman, 2003 ).

“Demystifying insight,” to borrow a phrase from Bowden, Jung-Beeman, Fleck, and Kounios ( 2005 ), requires explaining ( 1 ) why solvers initially reach an impasse in solving a problem for which they have the necessary knowledge to generate the solution, ( 2 ) how the restructuring occurred, and ( 3 ) how it led to the solution. A detailed discussion of these topics appears elsewhere in this volume (van Steenburgh et al., Chapter 24 ). Here, we describe briefly three recent theories that have attempted to account for various aspects of these phenomena: Knoblich, Ohlsson, Haider, and Rhenius's ( 1999 ) representational change theory, MacGregor, Ormerod, and Chronicle's ( 2001 ) progress monitoring theory, and Bowden et al.'s ( 2005 ) neurological model. We then propose the need for an integrated approach to demystifying insight that considers both representation and search.

According to Knoblich et al.'s ( 1999 ) representational change theory, problems that are solved with insight are highly likely to evoke initial representations in which solvers place inappropriate constraints on their solution attempts, leading to an impasse. An impasse can be resolved by revising one's representation of the problem. Knoblich and his colleagues tested this theory using Roman numeral matchstick arithmetic problems in which solvers must move one stick to a new location to change a false numerical statement (e.g., I = II + II ) into a statement that is true. According to representational change theory, re-representation may occur through either constraint relaxation or chunk decomposition. (The solution to the example problem is to change II + to III – , which requires both methods of re-representation, yielding I = III – II ). Good support for this theory has been found based on measures of solution rate, solution time, and eye fixation (Knoblich et al., 1999 ; Knoblich, Ohlsson, & Raney, 2001 ; Öllinger, Jones, & Knoblich, 2008 ).

Progress monitoring theory (MacGregor et al., 2001 ) was proposed to account for subjects' difficulty in solving the nine-dot problem, which has traditionally been classified as an insight problem. According to this theory, solvers use the hill-climbing search heuristic to solve this problem, just as they do for traditional search problems (e.g., Hobbits and Orcs). In particular, solvers are hypothesized to monitor their progress toward solution using a criterion generated from the problem's current state. If solvers reach criterion failure, they seek alternative solutions by trying to relax one or more problem constraints. MacGregor et al. found support for this theory using several variants of the nine-dot problem (also see Ormerod, MacGregor, & Chronicle, 2002 ). Jones ( 2003 ) suggested that progress monitoring theory provides an account of the solution process up to the point an impasse is reached and representational change is sought, at which point representational change theory picks up and explains how insight may be achieved. Hence, it appears that a complete account of insight may require an integration of concepts from the Gestalt (representation) and Newell and Simon's (search) legacies.

Bowden et al.'s ( 2005 ) neurological model emphasizes the overlap between problem solving and language comprehension, and it hinges on differential processing in the right and left hemispheres. They proposed that an impasse is reached because initial processing of the problem produces strong activation of information irrelevant to solution in the left hemisphere. At the same time, weak semantic activation of alternative semantic interpretations, critical for solution, occurs in the right hemisphere. Insight arises when the weakly activated concepts reinforce each other, eventually rising above the threshold required for conscious awareness. Several studies of problem solving using compound remote associates problems, involving both behavioral and neuroimaging data, have found support for this model (Bowden & Jung-Beeman, 1998 , 2003 ; Jung-Beeman & Bowden, 2000 ; Jung-Beeman et al., 2004 ; also see Moss, Kotovsky, & Cagan, 2011 ).

Note that these three views of insight have received support using three quite distinct types of problems (Roman numeral matchstick arithmetic problems, the nine-dot problem, and compound remote associates problems, respectively). It remains to be established, therefore, whether these accounts can be generalized across problems. Kershaw and Ohlsson ( 2004 ) argued that insight problems are difficult because the key behavior required for solution may be hindered by perceptual factors (the Gestalt view), background knowledge (so expertise may be important; e.g., see Novick & Sherman, 2003 , 2008 ), and/or process factors (e.g., those affecting search). From this perspective, solving visual problems (e.g., the nine-dot problem) with insight may call upon more general visual processes, whereas solving verbal problems (e.g., anagrams, compound remote associates) with insight may call upon general verbal/semantic processes.

The work we reviewed in this section shows the relevance of problem representation (the Gestalt legacy) to the way people search the problem space (the legacy of Newell and Simon), and the relevance of search to the solution of insight problems that require a representational change. In addition to this inevitable integration of the two legacies, the work we described here underscores the fact that problem solving crucially depends on perceptual factors and on the solvers' background knowledge. In the next section, we describe some recent work that shows the involvement of these factors in the solution of problems in math and science.

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

Although the use of puzzle problems continues in research on problem solving, especially in investigations of insight, many contemporary researchers tackle problem solving in knowledge-rich domains, often in academic disciplines (e.g., mathematics, biology, physics, chemistry, meteorology). In this section, we provide a sampling of this research that highlights the importance of visual perception and background knowledge for successful problem solving.

The Role of Visual Perception

We stated at the outset that a problem representation (e.g., the problem space) is a model of the problem constructed by solvers to summarize their understanding of the problem's essential nature. This informal definition refers to the internal representations people construct and hold in working memory. Of course, people may also construct various external representations (Markman, 1999 ) and even manipulate those representations to aid in solution (see Hegarty & Stull, Chapter 31 ). For example, solvers often use paper and pencil to write notes or draw diagrams, especially when solving problems from formal domains (e.g., Cox, 1999 ; Kindfield, 1993 / 1994 ; S. Schwartz, 1971 ). In problems that provide solvers with external representation, such as the Tower of Hanoi problem, people's planning and memory of the current state is guided by the actual configurations of disks on pegs (Garber & Goldin-Meadow, 2002 ) or by the displays they see on a computer screen (Chen & Holyoak, 2010 ; Patsenko & Altmann, 2010 ).

In STEM (science, technology, engineering, and mathematics) disciplines, it is common for problems to be accompanied by diagrams or other external representations (e.g., equations) to be used in determining the solution. Larkin and Simon ( 1987 ) examined whether isomorphic sentential and diagrammatic representations are interchangeable in terms of facilitating solution. They argued that although the two formats may be equivalent in the sense that all of the information in each format can be inferred from the other format (informational equivalence), the ease or speed of making inferences from the two formats might differ (lack of computational equivalence). Based on their analysis of several problems in physics and math, Larkin and Simon further argued for the general superiority of diagrammatic representations (but see Mayer & Gallini, 1990 , for constraints on this general conclusion).

Novick and Hurley ( 2001 , p. 221) succinctly summarized the reasons for the general superiority of diagrams (especially abstract or schematic diagrams) over verbal representations: They “(a) simplify complex situations by discarding unnecessary details (e.g., Lynch, 1990 ; Winn, 1989 ), (b) make abstract concepts more concrete by mapping them onto spatial layouts with familiar interpretational conventions (e.g., Winn, 1989 ), and (c) substitute easier perceptual inferences for more computationally intensive search processes and sentential deductive inferences (Barwise & Etchemendy, 1991 ; Larkin & Simon, 1987 ).” Despite these benefits of diagrammatic representations, there is an important caveat, noted by Larkin and Simon ( 1987 , p. 99) at the very end of their paper: “Although every diagram supports some easy perceptual inferences, nothing ensures that these inferences must be useful in the problem-solving process.” We will see evidence of this in several of the studies reviewed in this section.

Next we describe recent work on perceptual factors that are involved in people's use of two types of external representations that are provided as part of the problem in two STEM disciplines: equations in algebra and diagrams in evolutionary biology. Although we focus here on effects of perceptual factors per se, it is important to note that such factors only influence performance when subjects have background knowledge that supports differential interpretation of the alternative diagrammatic depictions presented (Hegarty, Canham, & Fabricant, 2010 ).

In the previous section, we described the work of Patsenko and Altmann ( 2010 ) that shows direct involvement of visual attention and perception in the sequential application of move operators during the solution of the Tower of Hanoi problem. A related body of work documents similar effects in tasks that require the interpretation and use of mathematical equations (Goldstone, Landy, & Son, 2010 ; Landy & Goldstone, 2007a , b). For example, Landy and Goldstone ( 2007b ) varied the spatial proximity of arguments to the addition (+) and multiplication (*) operators in algebraic equations, such that the spatial layout of the equation was either consistent or inconsistent with the order-of-operations rule that multiplication precedes addition. In consistent equations , the space was narrower around multiplication than around addition (e.g., g*m + r*w = m*g + w*r ), whereas in inconsistent equations this relative spacing was reversed (e.g., s * n+e * c = n * s+c * e ). Subjects' judgments of the validity of such equations (i.e., whether the expressions on the two sides of the equal sign are equivalent) were significantly faster and more accurate for consistent than inconsistent equations.

In discussing these findings and related work with other external representations, Goldstone et al. ( 2010 ) proposed that experience with solving domain-specific problems leads people to “rig up” their perceptual system such that it allows them to look at the problem in a way that is consistent with the correct rules. Similar logic guides the Perceptual Learning Modules developed by Kellman and his collaborators to help students interpret and use algebraic equations and graphs (Kellman et al., 2008 ; Kellman, Massey, & Son, 2009 ). These authors argued and showed that, consistent with the previously reviewed work on expertise, perceptual training with particular external representations supports the development of perceptual fluency. This fluency, in turn, supports students' subsequent use of these external representations for problem solving.

This research suggests that extensive experience with particular equations or graphs may lead to perceptual fluency that could replace the more mindful application of domain-specific rules. Fisher, Borchert, and Bassok ( 2011 ) reported results from algebraic-modeling tasks that are consistent with this hypothesis. For example, college students were asked to represent verbal statements with algebraic equations, a task that typically elicits systematic errors (e.g., Clement, Lochhead, & Monk, 1981 ). Fisher et al. found that such errors were very common when subjects were asked to construct “standard form” equations ( y = ax ), which support fluent left-to-right translation of words to equations, but were relatively rare when subjects were asked to construct nonstandard division-format equations (x = y/a) that do not afford such translation fluency.

In part because of the left-to-right order in which people process equations, which mirrors the linear order in which they process text, equations have traditionally been viewed as sentential representations. However, Landy and Goldstone ( 2007a ) have proposed that equations also share some properties with diagrammatic displays and that, in fact, in some ways they are processed like diagrams. That is, spatial information is used to represent and to support inferences about syntactic structure. This hypothesis received support from Landy and Goldstone's ( 2007b ) results, described earlier, in which subjects' judgments of the validity of equations were affected by the Gestalt principle of grouping: Subjects did better when the grouping was consistent rather than inconsistent with the underlying structure of the problem (order of operations). Moreover, Landy and Goldstone ( 2007a ) found that when subjects wrote their own equations they grouped numbers and operators (+, *, =) in a way that reflected the hierarchical structure imposed by the order-of-operations rule.

In a recent line of research, Novick and Catley ( 2007 ; Novick, Catley, & Funk, 2010 ; Novick, Shade, & Catley, 2011 ) have examined effects of the spatial layout of diagrams depicting the evolutionary history of a set of taxa on people's ability to reason about patterns of relationship among those taxa. We consider here their work that investigates the role of another Gestalt perceptual principle—good continuation—in guiding students' reasoning. According to this principle, a continuous line is perceived as a single entity (Kellman, 2000 ). Consider the diagrams shown in Figure 21.6 . Each is a cladogram, a diagram that depicts nested sets of taxa that are related in terms of levels of most recent common ancestry. For example, chimpanzees and starfish are more closely related to each other than either is to spiders. The supporting evidence for their close relationship is their most recent common ancestor, which evolved the novel character of having radial cleavage. Spiders do not share this ancestor and thus do not have this character.

Cladograms are typically drawn in two isomorphic formats, which Novick and Catley ( 2007 ) referred to as trees and ladders. Although these formats are informationally equivalent (Larkin & Simon, 1987 ), Novick and Catley's ( 2007 ) research shows that they are not computationally equivalent (Larkin & Simon, 1987 ). Imagine that you are given evolutionary relationships in the ladder format, such as in Figure 21.6a (but without the four characters—hydrostatic skeleton, bilateral symmetry, radial cleavage, and trocophore larvae—and associated short lines indicating their locations on the cladogram), and your task is to translate that diagram to the tree format. A correct translation is shown in Figure 21.6b . Novick and Catley ( 2007 ) found that college students were much more likely to get such problems correct when the presented cladogram was in the nested circles (e.g., Figure 21.6d ) rather than the ladder format. Because the Gestalt principle of good continuation makes the long slanted line at the base of the ladder appear to represent a single hierarchical level, a common translation error for the ladder to tree problems was to draw a diagram such as that shown in Figure 21.6c .

The difficulty that good continuation presents for interpreting relationships depicted in the ladder format extends to answering reasoning questions as well. Novick and Catley (unpublished data) asked comparable questions about relationships depicted in the ladder and tree formats. For example, using the cladograms depicted in Figures 21.6a and 21.6b , consider the following questions: (a) Which taxon—jellyfish or earthworm—is the closest evolutionary relation to starfish, and what evidence supports your answer? (b) Do the bracketed taxa comprise a clade (a set of taxa consisting of the most recent common ancestor and all of its descendants), and what evidence supports your answer? For both such questions, students had higher accuracy and evidence quality composite scores when the relationships were depicted in the tree than the ladder format.

Four cladograms depicting evolutionary relationships among six animal taxa. Cladogram ( a ) is in the ladder format, cladograms ( b ) and ( c ) are in the tree format, and cladogram ( d ) is in the nested circles format. Cladograms ( a ), ( b ), and ( d ) are isomorphic.

If the difficulty in extracting the hierarchical structure of the ladder format is due to good continuation (which leads problem solvers to interpret continuous lines that depict multiple hierarchical levels as depicting only a single level), then a manipulation that breaks good continuation at the points where a new hierarchical level occurs should improve understanding. Novick et al. ( 2010 ) tested this hypothesis using a translation task by manipulating whether characters that are the markers for the most recent common ancestor of each nested set of taxa were included on the ladders. Figure 21.6a shows a ladder with such characters. As predicted, translation accuracy increased dramatically simply by adding these characters to the ladders, despite the additional information subjects had to account for in their translations.

The Role of Background Knowledge

As we mentioned earlier, the specific entities in the problems people encounter evoke inferences that affect how people represent these problems (e.g., the candle problem; Duncker, 1945 ) and how they apply the operators in searching for the solution (e.g., the disks vs. acrobats versions of the Tower of Hanoi problem; Kotovsky et al., 1985 ). Such object-based inferences draw on people's knowledge about the properties of the objects (e.g., a box is a container, an acrobat is a person who can be hurt). Here, we describe the work of Bassok and her colleagues, who found that similar inferences affect how people select mathematical procedures to solve problems in various formal domains. This work shows that the objects in the texts of mathematical word problems affect how people represent the problem situation (i.e., the situation model they construct; Kintsch & Greeno, 1985 ) and, in turn, lead them to select mathematical models that have a corresponding structure. To illustrate, a word problem that describes constant change in the rate at which ice is melting off a glacier evokes a model of continuous change, whereas a word problem that describes constant change in the rate at which ice is delivered to a restaurant evokes a model of discrete change. These distinct situation models lead subjects to select corresponding visual representations (e.g., Bassok & Olseth, 1995 ) and solutions methods, such as calculating the average change over time versus adding the consecutive changes (e.g., Alibali et al., 1999 ).

In a similar manner, people draw on their general knowledge to infer how the objects in a given problem are related to each other and construct mathematical solutions that correspond to these inferred object relations. For example, a word problem that involves doctors from two hospitals elicits a situation model in which the two sets of doctors play symmetric roles (e.g., work with each other), whereas a mathematically isomorphic problem that involves mechanics and cars elicits a situation model in which the sets play asymmetric roles (e.g., mechanics fix cars). The mathematical solutions people construct to such problems reflect this difference in symmetry (Bassok, Wu, & Olseth, 1995 ). In general, people tend to add objects that belong to the same taxonomic category (e.g., doctors + doctors) but divide functionally related objects (e.g., cars ÷ mechanics). People establish this correspondence by a process of analogical alignment between semantic and arithmetic relations, which Bassok and her colleagues refer to as “semantic alignment” (Bassok, Chase, & Martin, 1998 ; Doumas, Bassok, Guthormsen, & Hummel, 2006 ; Fisher, Bassok, & Osterhout, 2010 ).

Semantic alignment occurs very early in the solution process and can prime arithmetic facts that are potentially relevant to the problem solution (Bassok, Pedigo, & Oskarsson, 2008 ). Although such alignments can lead to erroneous solutions, they have a high heuristic value because, in most textbook problems, object relations indeed correspond to analogous mathematical relations (Bassok et al., 1998 ). Interestingly, unlike in the case of reliance on specific surface-structure correlations (e.g., the keyword “more” typically appears in word problems that require addition; Lewis & Mayer, 1987 ), people are more likely to exploit semantic alignment when they have more, rather than less modeling experience. For example, Martin and Bassok ( 2005 ) found very strong semantic-alignment effects when subjects solved simple division word problems, but not when they constructed algebraic equations to represent the relational statements that appeared in the problems. Of course, these subjects had significantly more experience with solving numerical word problems than with constructing algebraic models of relational statements. In a subsequent study, Fisher and Bassok ( 2009 ) found semantic-alignment effects for subjects who constructed correct algebraic models, but not for those who committed modeling errors.

Conclusions and Future Directions

In this chapter, we examined two broad components of the problem-solving process: representation (the Gestalt legacy) and search (the legacy of Newell and Simon). Although many researchers choose to focus their investigation on one or the other of these components, both Duncker ( 1945 ) and Simon ( 1986 ) underscored the necessity to investigate their interaction, as the representation one constructs for a problem determines (or at least constrains) how one goes about trying to generate a solution, and searching the problem space may lead to a change in problem representation. Indeed, Duncker's ( 1945 ) initial account of one subject's solution to the radiation problem was followed up by extensive and experimentally sophisticated work by Simon and his colleagues and by other researchers, documenting the involvement of visual perception and background knowledge in how people represent problems and search for problem solutions.

The relevance of perception and background knowledge to problem solving illustrates the fact that, when people attempt to find or devise ways to reach their goals, they draw on a variety of cognitive resources and engage in a host of cognitive activities. According to Duncker ( 1945 ), such goal-directed activities may include (a) placing objects into categories and making inferences based on category membership, (b) making inductive inferences from multiple instances, (c) reasoning by analogy, (d) identifying the causes of events, (e) deducing logical implications of given information, (f) making legal judgments, and (g) diagnosing medical conditions from historical and laboratory data. As this list suggests, many of the chapters in the present volume describe research that is highly relevant to the understanding of problem-solving behavior. We believe that important advancements in problem-solving research would emerge by integrating it with research in other areas of thinking and reasoning, and that research in these other areas could be similarly advanced by incorporating the insights gained from research on what has more traditionally been identified as problem solving.

As we have described in this chapter, many of the important findings in the field have been established by a careful investigation of various riddle problems. Although there are good methodological reasons for using such problems, many researchers choose to investigate problem solving using ecologically valid educational materials. This choice, which is increasingly common in contemporary research, provides researchers with the opportunity to apply their basic understanding of problem solving to benefit the design of instruction and, at the same time, allows them to gain a better understanding of the processes by which domain knowledge and educational conventions affect the solution process. We believe that the trend of conducting educationally relevant research is likely to continue, and we expect a significant expansion of research on people's understanding and use of dynamic and technologically rich external representations (e.g., Kellman et al., 2008 ; Mayer, Griffith, Jurkowitz, & Rothman, 2008 ; Richland & McDonough, 2010 ; Son & Goldstone, 2009 ). Such investigations are likely to yield both practical and theoretical payoffs.

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Article • 7 min read

What Is Problem Solving?

Find a solution to any problem you face..

By the Mind Tools Content Team

We all spend a lot of our time solving problems, both at work and in our personal lives.

Some problems are small, and we can quickly sort them out ourselves. But others are complex challenges that take collaboration, creativity, and a considerable amount of effort to solve.

At work, the types of problems we face depend largely on the organizations we're in and the jobs we do. A manager in a cleaning company, for example, might spend their day untangling staffing issues, resolving client complaints, and sorting out problems with equipment and supplies. An aircraft designer, on the other hand, might be grappling with a problem about aerodynamics, or trying to work out why a new safety feature isn't working. Meanwhile, a politician might be exploring solutions to racial injustice or climate change.

But whatever issues we face, there are some common ways to tackle them effectively. And we can all boost our confidence and ability to succeed by building a strong set of problem-solving skills.

Mind Tools offers a large collection of resources to help you do just that!

How Well Do You Solve Problems?

Start by taking an honest look at your existing skills. What's your current approach to solving problems, and how well is it working? Our quiz, How Good Is Your Problem Solving? lets you analyze your abilities, and signposts ways to address any areas of weakness.

Define Every Problem

The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem – not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the real issue might be a general lack of training, or an unreasonable workload across the team.

Tools like 5 Whys , Appreciation and Root Cause Analysis get you asking the right questions, and help you to work through the layers of a problem to uncover what's really going on.

However, defining a problem doesn't mean deciding how to solve it straightaway. It's important to look at the issue from a variety of perspectives. If you commit yourself too early, you can end up with a short-sighted solution. The CATWOE checklist provides a powerful reminder to look at many elements that may contribute to the problem, keeping you open to a variety of possible solutions.

Understanding Complexity

As you define your problem, you'll often discover just how complicated it is. There are likely several interrelated issues involved. That's why it's important to have ways to visualize, simplify and make sense of this tangled mess!

Affinity Diagrams are great for organizing many different pieces of information into common themes, and for understanding the relationships between them.

Another popular tool is the Cause-and-Effect Diagram . To generate viable solutions, you need a solid understanding of what's causing the problem.

When your problem occurs within a business process, creating a Flow Chart , Swim Lane Diagram or a Systems Diagram will help you to see how various activities and inputs fit together. This may well highlight a missing element or bottleneck that's causing your problem.

Quite often, what seems to be a single problem turns out to be a whole series of problems. The Drill Down technique prompts you to split your problem into smaller, more manageable parts.

General Problem-Solving Tools

When you understand the problem in front of you, you’re ready to start solving it. With your definition to guide you, you can generate several possible solutions, choose the best one, then put it into action. That's the four-step approach at the heart of good problem solving.

There are various problem-solving styles to use. For example:

Constructive Controversy is a way of widening perspectives and energizing discussions.
Inductive Reasoning makes the most of people’s experiences and know-how, and can speed up solution finding.
Means-End Analysis can bring extra clarity to your thinking, and kick-start the process of implementing solutions.

Specific Problem-Solving Systems

Some particularly complicated or important problems call for a more comprehensive process. Again, Mind Tools has a range of approaches to try, including:

Simplex , which involves an eight-stage process: problem finding, fact finding, defining the problem, idea finding, selecting and evaluating, planning, selling the idea, and acting. These steps build upon the basic, four-step process described above, and they create a cycle of problem finding and solving that will continually improve your organization.
Appreciative Inquiry , which is a uniquely positive way of solving problems by examining what's working well in the areas surrounding them.
Soft Systems Methodology , which takes you through four stages to uncover more details about what's creating your problem, and then define actions that will improve the situation.

Further Problem-Solving Strategies

Good problem solving requires a number of other skills – all of which are covered by Mind Tools.

For example, we have a large section of resources to improve your Creativity , so that you come up with a range of possible solutions.

By strengthening your Decision Making , you'll be better at evaluating the options, selecting the best ones, then choosing how to implement them.

And our Project Management collection has valuable advice for strengthening the whole problem-solving process. The resources there will help you to make effective changes – and then keep them working long term.

Problems are an inescapable part of life, both in and out of work. So we can all benefit from having strong problem-solving skills.

It's important to understand your current approach to problem solving, and to know where and how to improve.

Define every problem you encounter – and understand its complexity, rather than trying to solve it too soon.

There's a range of general problem-solving approaches, helping you to generate possible answers, choose the best ones, and then implement your solution.

Some complicated or serious problems require more specific problem-solving systems, especially when they relate to business processes.

By boosting your creativity, decision-making and project-management skills, you’ll become even better at solving all the problems you face.

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What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

Identify the issue : Recognize the problem that needs to be solved.
Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

Brainstorming with others
Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
Analyzing cause and effect
Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

Creating a list of potential ideas to solve the problem
Grouping and categorizing similar solutions
Prioritizing potential solutions based on feasibility, cost, and resources required
Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
Decision-making matrices
Pros and cons lists
Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

Communicating the solution to relevant parties
Setting timelines and milestones
Assigning tasks and responsibilities
Monitoring the solution and making adjustments as necessary
Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

Generate a diverse range of solutions
Encourage all team members to participate
Foster creative thinking

When brainstorming, remember to:

Reserve judgment until the session is over
Encourage wild ideas
Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

5 Whys : Ask “why” five times to get to the underlying cause.
Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

List your problem’s strengths, such as relevant resources or strong partnerships.
Identify its weaknesses, such as knowledge gaps or limited resources.
Explore opportunities, like trends or new technologies, that could help solve the problem.
Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

Write the problem in the center of a blank page.
Draw branches from the central problem to related sub-problems or contributing factors.
Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
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20 Effective Math Strategies To Approach Problem-Solving

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations.

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies:

Draw a model
Use different approaches
Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better.

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem.

Here are 20 strategies to help students develop their problem-solving skills.

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand:

The context
What the key information is
How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information.

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed. For example, if the word problem asks how many are left, the problem likely requires subtraction. Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary. Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer. Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it. The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer. Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process. It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

math problem that needs a problem solving strategy

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives . When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts. The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution. This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71. Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved. It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve. Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense.

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions.

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work.

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable. Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten. For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10. When the estimate is clear the two numbers are close. This means your answer is reasonable.

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables. To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly. Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills. If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems.

Polya’s 4 steps include:

Understand the problem
Devise a plan
Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall.

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom.

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving

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Explore the range of problem solving resources for 2nd to 8th grade students.

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Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice.

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra.

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

Ultimate Guide to Metacognition [FREE]

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Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

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STEM Problem Solving: Inquiry, Concepts, and Reasoning

Published: 29 January 2022
Volume 32 , pages 381–397, ( 2023 )

Cite this article

Aik-Ling Tan ORCID: orcid.org/0000-0002-4627-4977 1 ,
Yann Shiou Ong ORCID: orcid.org/0000-0002-6092-2803 1 ,
Yong Sim Ng ORCID: orcid.org/0000-0002-8400-2040 1 &
Jared Hong Jie Tan 1

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Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey’s idea of practical and science inquiry and Bereiter’s idea of referent-centred and problem-centred knowledge, we examine how integrated STEM problem solving offers opportunities for learners to shuttle between practical and science inquiry and the kinds of knowledge that result from each form of inquiry. We hypothesize that connecting science inquiry with practical inquiry narrows the gap between science and everyday experiences to overcome isolation and fragmentation of science learning. In this study, we examine classroom talk as students engage in problem solving to increase crop yield. Qualitative content analysis of the utterances of six classes of 113 eighth graders and their teachers were conducted for 3 hours of video recordings. Analysis showed an almost equal amount of science and practical inquiry talk. Teachers and students applied their everyday experiences to generate solutions. Science talk was at the basic level of facts and was used to explain reasons for specific design considerations. There was little evidence of higher-level scientific conceptual knowledge being applied. Our observations suggest opportunities for more intentional connections of science to practical problem solving, if we intend to apply higher-order scientific knowledge in problem solving. Deliberate application and reference to scientific knowledge could improve the quality of solutions generated.

Science Camps for Introducing Nature of Scientific Inquiry Through Student Inquiries in Nature: Two Applications with Retention Study

Ways of thinking in STEM-based problem solving

Framing and Assessing Scientific Inquiry Practices

Avoid common mistakes on your manuscript.

1 Introduction

As we enter to second quarter of the twenty-first century, it is timely to take stock of both the changes and demands that continue to weigh on our education system. A recent report by World Economic Forum highlighted the need to continuously re-position and re-invent education to meet the challenges presented by the disruptions brought upon by the fourth industrial revolution (World Economic Forum, 2020 ). There is increasing pressure for education to equip children with the necessary, relevant, and meaningful knowledge, skills, and attitudes to create a “more inclusive, cohesive and productive world” (World Economic Forum, 2020 , p. 4). Further, the shift in emphasis towards twenty-first century competencies over mere acquisition of disciplinary content knowledge is more urgent since we are preparing students for “jobs that do not yet exist, technology that has not yet been invented, and problems that has yet exist” (OECD, 2018 , p. 2). Tan ( 2020 ) concurred with the urgent need to extend the focus of education, particularly in science education, such that learners can learn to think differently about possibilities in this world. Amidst this rhetoric for change, the questions that remained to be answered include how can science education transform itself to be more relevant; what is the role that science education play in integrated STEM learning; how can scientific knowledge, skills and epistemic practices of science be infused in integrated STEM learning; what kinds of STEM problems should we expose students to for them to learn disciplinary knowledge and skills; and what is the relationship between learning disciplinary content knowledge and problem solving skills?

In seeking to understand the extent of science learning that took place within integrated STEM learning, we dissected the STEM problems that were presented to students and examined in detail the sense making processes that students utilized when they worked on the problems. We adopted Dewey’s ( 1938 ) theoretical idea of scientific and practical/common-sense inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge building process to interpret teacher-students’ interactions during problem solving. There are two primary reasons for choosing these two theoretical frameworks. Firstly, Dewey’s ideas about the relationship between science inquiry and every day practical problem-solving is important in helping us understand the role of science subject matter knowledge and science inquiry in solving practical real-world problems that are commonly used in STEM learning. Secondly, Bereiter’s ideas of referent-centred and problem-centred knowledge augment our understanding of the types of knowledge that students can learn when they engage in solving practical real-world problems.

Taken together, Dewey’s and Bereiter’s ideas enable us to better understand the types of problems used in STEM learning and their corresponding knowledge that is privileged during the problem-solving process. As such, the two theoretical lenses offered an alternative and convincing way to understand the actual types of knowledge that are used within the context of integrated STEM and help to move our understanding of STEM learning beyond current focus on examining how engineering can be used as an integrative mechanism (Bryan et al., 2016 ) or applying the argument of the strengths of trans-, multi-, or inter-disciplinary activities (Bybee, 2013 ; Park et al., 2020 ) or mapping problems by the content and context as pure STEM problems, STEM-related problems or non-STEM problems (Pleasants, 2020 ). Further, existing research (for example, Gale et al., 2000 ) around STEM education focussed largely on description of students’ learning experiences with insufficient attention given to the connections between disciplinary conceptual knowledge and inquiry processes that students use to arrive at solutions to problems. Clarity in the role of disciplinary knowledge and the related inquiry will allow for more intentional design of STEM problems for students to learn higher-order knowledge. Applying Dewey’s idea of practical and scientific inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge, we analysed six lessons where students engaged with integrated STEM problem solving to propose answers to the following research questions: What is the extent of practical and scientific inquiry in integrated STEM problem solving? and What conceptual knowledge and problem-solving skills are learnt through practical and science inquiry during integrated STEM problem solving?

2 Inquiry in Problem Solving

Inquiry, according to Dewey ( 1938 ), involves the direct control of unknown situations to change them into a coherent and unified one. Inquiry usually encompasses two interrelated activities—(1) thinking about ideas related to conceptual subject-matter and (2) engaging in activities involving our senses or using specific observational techniques. The National Science Education Standards released by the National Research Council in the US in 1996 defined inquiry as “…a multifaceted activity that involves making observations; posing questions; examining books and other sources of information to see what is already known; planning investigations; reviewing what is already known in light of experimental evidence; using tools to gather, analyze, and interpret data; proposing answers, explanations, and predictions; and communicating the results. Inquiry requires identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations” (p. 23). Planning investigation; collecting empirical evidence; using tools to gather, analyse and interpret data; and reasoning are common processes shared in the field of science and engineering and hence are highly relevant to apply to integrated STEM education.

In STEM education, establishing the connection between general inquiry and its application helps to link disciplinary understanding to epistemic knowledge. For instance, methods of science inquiry are popular in STEM education due to the familiarity that teachers have with scientific methods. Science inquiry, a specific form of inquiry, has appeared in many science curriculum (e.g. NRC, 2000 ) since Dewey proposed in 1910 that learning of science should be perceived as both subject-matter and a method of learning science (Dewey, 1910a , 1910b ). Science inquiry which involved ways of doing science should also encompass the ways in which students learn the scientific knowledge and investigative methods that enable scientific knowledge to be constructed. Asking scientifically orientated questions, collecting empirical evidence, crafting explanations, proposing models and reasoning based on available evidence are affordances of scientific inquiry. As such, science should be pursued as a way of knowing rather than merely acquisition of scientific knowledge.

Building on these affordances of science inquiry, Duschl and Bybee ( 2014 ) advocated the 5D model that focused on the practice of planning and carrying out investigations in science and engineering, representing two of the four disciplines in STEM. The 5D model includes science inquiry aspects such as (1) deciding on what and how to measure, observe and sample; (2) developing and selecting appropriate tools to measure and collect data; (3) recording the results and observations in a systematic manner; (4) creating ways to represent the data and patterns that are observed; and (5) determining the validity and the representativeness of the data collected. The focus on planning and carrying out investigations in the 5D model is used to help teachers bridge the gap between the practices of building and refining models and explanation in science and engineering. Indeed, a common approach to incorporating science inquiry in integrated STEM curriculum involves student planning and carrying out scientific investigations and making sense of the data collected to inform engineering design solution (Cunningham & Lachapelle, 2016 ; Roehrig et al., 2021 ). Duschl and Bybee ( 2014 ) argued that it is needful to design experiences for learners to appreciate that struggles are part of problem solving in science and engineering. They argued that “when the struggles of doing science is eliminated or simplified, learners get the wrong perceptions of what is involved when obtaining scientific knowledge and evidence” (Duschl & Bybee, 2014 , p. 2). While we concur with Duschl and Bybee about the need for struggles, in STEM learning, these struggles must be purposeful and grade appropriate so that students will also be able to experience success amidst failure.

The peculiar nature of science inquiry was scrutinized by Dewey ( 1938 ) when he cross-examined the relationship between science inquiry and other forms of inquiry, particularly common-sense inquiry. He positioned science inquiry along a continuum with general or common-sense inquiry that he termed as “logic”. Dewey argued that common-sense inquiry serves a practical purpose and exhibits features of science inquiry such as asking questions and a reliance on evidence although the focus of common-sense inquiry tends to be different. Common-sense inquiry deals with issues or problems that are in the immediate environment where people live, whereas the objects of science inquiry are more likely to be distant (e.g. spintronics) from familiar experiences in people’s daily lives. While we acknowledge the fundamental differences (such as novel discovery compared with re-discovering science, ‘messy’ science compared with ‘sanitised’ science) between school science and science that is practiced by scientists, the subject of interest in science (understanding the world around us) remains the same.

The unfamiliarity between the functionality and purpose of science inquiry to improve the daily lives of learners does little to motivate learners to learn science (Aikenhead, 2006 ; Lee & Luykx, 2006 ) since learners may not appreciate the connections of science inquiry in their day-to-day needs and wants. Bereiter ( 1992 ) has also distinguished knowledge into two forms—referent-centred and problem-centred. Referent-centred knowledge refers to subject-matter that is organised around topics such as that in textbooks. Problem-centred knowledge is knowledge that is organised around problems, whether they are transient problems, practical problems or problems of explanations. Bereiter argued that referent-centred knowledge that is commonly taught in schools is limited in their applications and meaningfulness to the lives of students. This lack of familiarity and affinity to referent-centred knowledge is likened to the science subject-matter knowledge that was mentioned by Dewey. Rather, it is problem-centred knowledge that would be useful when students encounter problems. Learning problem-centred knowledge will allow learners to readily harness the relevant knowledge base that is useful to understand and solve specific problems. This suggests a need to help learners make the meaningful connections between science and their daily lives.

Further, Dewey opined that while the contexts in which scientific knowledge arise could be different from our daily common-sense world, careful consideration of scientific activities and applying the resultant knowledge to daily situations for use and enjoyment is possible. Similarly, in arguing for problem-centred knowledge, Bereiter ( 1992 ) questioned the value of inert knowledge that plays no role in helping us understand or deal with the world around us. Referent-centred knowledge has a higher tendency to be inert due to the way that the knowledge is organised and the way that the knowledge is encountered by learners. For instance, learning about the equation and conditions for photosynthesis is not going to help learners appreciate how plants are adapted for photosynthesis and how these adaptations can allow plants to survive changes in climate and for farmers to grow plants better by creating the best growing conditions. Rather, students could be exposed to problems of explanations where they are asked to unravel the possible reasons for low crop yield and suggest possible ways to overcome the problem. Hence, we argue here that the value of the referent knowledge is that they form the basis and foundation for the students to be able to discuss or suggest ways to overcome real life problems. Referent-centred knowledge serves as part of the relevant knowledge base that can be harnessed to solve specific problems or as foundational knowledge students need to progress to learn higher-order conceptual knowledge that typically forms the foundations or pillars within a discipline. This notion of referent-centred knowledge serving as foundational knowledge that can be and should be activated for application in problem-solving situation is shown by Delahunty et al. ( 2020 ). They found that students show high reliance on memory when they are conceptualising convergent problem-solving tasks.

While Bereiter argues for problem-centred knowledge, he cautioned that engagement should be with problems of explanation rather than transient or practical problems. He opined that if learners only engage in transient or practical problem alone, they will only learn basic-category types of knowledge and fail to understand higher-order conceptual knowledge. For example, for photosynthesis, basic-level types of knowledge included facts about the conditions required for photosynthesis, listing the products formed from the process of photosynthesis and knowing that green leaves reflect green light. These basic-level knowledges should intentionally help learners learn higher-level conceptual knowledge that include learners being able to draw on the conditions for photosynthesis when they encounter that a plant is not growing well or is exhibiting discoloration of leaves.

Transient problems disappear once a solution becomes available and there is a high likelihood that we will not remember the problem after that. Practical problems, according to Bereiter are “stuck-door” problems that could be solved with or without basic-level knowledge and often have solutions that lacks precise definition. There are usually a handful of practical strategies, such as pulling or pushing the door harder, kicking the door, etc. that will work for the problems. All these solutions lack a well-defined approach related to general scientific principles that are reproducible. Problems of explanations are the most desirable types of problems for learners since these are problems that persist and recur such that they can become organising points for knowledge. Problems of explanations consist of the conceptual representations of (1) a text base that serves to represent the text content and (2) a situation model that shows the portion of the world in which the text is relevant. The idea of text base to represent text content in solving problems of explanations is like the idea of domain knowledge and structural knowledge (refers to knowledge of how concepts within a domain are connected) proposed by Jonassen ( 2000 ). He argued that both types of knowledges are required to solve a range of problems from well-structured problems to ill-structured problems with a simulated context, to simple ill-structured problems and to complex ill-structured problems.

Jonassen indicated that complex ill-structured problems are typically design problems and are likely to be the most useful forms of problems for learners to be engaged in inquiry. Complex ill-structured design problems are the “wicked” problems that Buchanan ( 1992 ) discussed. Buchanan’s idea is that design aims to incorporate knowledge from different fields of specialised inquiry to become whole. Complex or wicked problems are akin to the work of scientists who navigate multiple factors and evidence to offer models that are typically oversimplified, but they apply them to propose possible first approximation explanations or solutions and iteratively relax constraints or assumptions to refine the model. The connections between the subject matter of science and the design process to engineer a solution are delicate. While it is important to ensure that practical concerns and questions are taken into consideration in designing solutions (particularly a material artefact) to a practical problem, the challenge here lies in ensuring that creativity in design is encouraged even if students initially lack or neglect the scientific conceptual understanding to explain/justify their design. In his articulation of wicked problems and the role of design thinking, Buchanan ( 1992 ) highlighted the need to pay attention to category and placement. Categories “have fixed meanings that are accepted within the framework of a theory or a philosophy and serve as the basis for analyzing what already exist” (Buchanan, 1992 , p. 12). Placements, on the other hand, “have boundaries to shape and constrain meaning, but are not rigidly fixed and determinate” (p. 12).

The difference in the ideas presented by Dewey and Bereiter lies in the problem design. For Dewey, scientific knowledge could be learnt from inquiring into practical problems that learners are familiar with. After all, Dewey viewed “modern science as continuous with, and to some degree an outgrowth and refinement of, practical or ‘common-sense’ inquiry” (Brown, 2012 ). For Bereiter, he acknowledged the importance of familiar experiences, but instead of using them as starting points for learning science, he argued that practical problems are limiting in helping learners acquire higher-order knowledge. Instead, he advocated for learners to organize their knowledge around problems that are complex, persistent and extended and requiring explanations to better understand the problems. Learners are to have a sense of the kinds of problems to which the specific concept is relevant before they can be said to have grasp the concept in a functionally useful way.

To connect between problem solving, scientific knowledge and everyday experiences, we need to examine ways to re-negotiate the disciplinary boundaries (such as epistemic understanding, object of inquiry, degree of precision) of science and make relevant connections to common-sense inquiry and to the problem at hand. Integrated STEM appears to be one way in which the disciplinary boundaries of science can be re-negotiated to include practices from the fields of technology, engineering and mathematics. In integrated STEM learning, inquiry is seen more holistically as a fluid process in which the outcomes are not absolute but are tentative. The fluidity of the inquiry process is reflected in the non-deterministic inquiry approach. This means that students can use science inquiry, engineering design, design process or any other inquiry approaches that fit to arrive at the solution. This hybridity of inquiry between science, common-sense and problems allows for some familiar aspects of the science inquiry process to be applied to understand and generate solutions to familiar everyday problems. In attempting to infuse elements of common-sense inquiry with science inquiry in problem-solving, logic plays an important role to help learners make connections. Hypothetically, we argue that with increasing exposure to less familiar ways of thinking such as those associated with science inquiry, students’ familiarity with scientific reasoning increases, and hence such ways of thinking gradually become part of their common-sense, which students could employ to solve future relevant problems. The theoretical ideas related to complexities of problems, the different forms of inquiry afforded by different problems and the arguments for engaging in problem solving motivated us to examine empirically how learners engage with ill-structured problems to generate problem-centred knowledge. Of particular interest to us is how learners and teachers weave between practical and scientific reasoning as they inquire to integrate the components in the original problem into a unified whole.

3.1 Context

The integrated STEM activity in our study was planned using the S-T-E-M quartet instructional framework (Tan et al., 2019 ). The S-T-E-M quartet instructional framework positions complex, persistent and extended problems at its core and focusses on the vertical disciplinary knowledge and understanding of the horizontal connections between the disciplines that could be gained by learners through solving the problem (Tan et al., 2019 ). Figure 1 depicts the disciplinary aspects of the problem that was presented to the students. The activity has science and engineering as the two lead disciplines. It spanned three 1-h lessons and required students to both learn and apply relevant scientific conceptual knowledge to solve a complex, real-world problem through processes that resemble the engineering design process (Wheeler et al., 2019 ).

Connections across disciplines in integrate STEM activity

Frequency of different types of reasoning

In the first session (1 h), students were introduced to the problem and its context. The problem pertains to the issue of limited farmland in a land scarce country that imports 90% of food (Singapore Food Agency [SFA], 2020 ). The students were required to devise a solution by applying knowledge of the conditions required for photosynthesis and plant growth to design and build a vertical farming system to help farmers increase crop yield with limited farmland. This context was motivated by the government’s effort to generate interests and knowledge in farming to achieve the 30 by 30 goal—supplying 30% of country’s nutritional needs by 2030. The scenario was a fictitious one where they were asked to produce 120 tonnes of Kailan (a type of leafy vegetable) with two hectares of land instead of the usual six hectares over a specific period. In addition to the abovementioned constraints, the teacher also discussed relevant success criteria for evaluating the solution with the students. Students then researched about existing urban farming approaches. They were given reading materials pertaining to urban farming to help them understand the affordances and constraints of existing solutions. In the second session (6 h), students engaged in ideation to generate potential solutions. They then designed, built and tested their solution and had opportunities to iteratively refine their solution. Students were given a list of materials (e.g. mounting board, straws, ice-cream stick, glue, etc.) that they could use to design their solutions. In the final session (1 h), students presented their solution and reflected on how well their solution met the success criteria. The prior scientific conceptual knowledge that students require to make sense of the problem include knowledge related to plant nutrition, namely, conditions for photosynthesis, nutritional requirements of Kailin and growth cycle of Kailin. The problem resembles a real-world problem that requires students to engage in some level of explanation of their design solution.

A total of 113 eighth graders (62 boys and 51 girls), 14-year-olds, from six classes and their teachers participated in the study. The students and their teachers were recruited as part of a larger study that examined the learning experiences of students when they work on integrated STEM activities that either begin with a problem, a solution or are focused on the content. Invitations were sent to schools across the country and interested schools opted in for the study. For the study reported here, all students and teachers were from six classes within a school. The teachers had all undergone 3 h of professional development with one of the authors on ways of implementing the integrated STEM activity used in this study. During the professional development session, the teachers learnt about the rationale of the activity, familiarize themselves with the materials and clarified the intentions and goals of the activity. The students were mostly grouped in groups of three, although a handful of students chose to work independently. The group size of students was not critical for the analysis of talk in this study as the analytic focus was on the kinds of knowledge applied rather than collaborative or group think. We assumed that the types of inquiry adopted by teachers and students were largely dependent on the nature of problem. Eighth graders were chosen for this study since lower secondary science offered at this grade level is thematic and integrated across biology, chemistry and physics. Furthermore, the topic of photosynthesis is taught under the theme of Interactions at eighth grade (CPDD, 2021 ). This thematic and integrated nature of science at eighth grade offered an ideal context and platform for integrated STEM activities to be trialled.

The final lessons in a series of three lessons in each of the six classes was analysed and reported in this study. Lessons where students worked on their solutions were not analysed because the recordings had poor audibility due to masking and physical distancing requirements as per COVID-19 regulations. At the start of the first lesson, the instructions given by the teacher were:

You are going to present your models. Remember the scenario that you were given at the beginning that you were tasked to solve using your model. …. In your presentation, you have to present your prototype and its features, what is so good about your prototype, how it addresses the problem and how it saves costs and space. So, this is what you can talk about during your presentation. ….. pay attention to the presentation and write down questions you like to ask the groups after the presentation… you can also critique their model, you can evaluate, critique and ask questions…. Some examples of questions you can ask the groups are? Do you think your prototype can achieve optimal plant growth? You can also ask questions specific to their models.

3.2 Data collection

Parental consent was sought a month before the start of data collection. The informed consent adhered to confidentiality and ethics guidelines as described by the Institutional Review Board. The data collection took place over a period of one month with weekly video recording. Two video cameras, one at the front and one at the back of the science laboratory were set up. The front camera captured the students seated at the front while the back video camera recorded the teacher as well as the groups of students at the back of the laboratory. The video recordings were synchronized so that the events captured from each camera can be interpreted from different angles. After transcription of the raw video files, the identities of students were substituted with pseudonyms.

3.3 Data analysis

The video recordings were analysed using the qualitative content analysis approach. Qualitative content analysis allows for patterns or themes and meanings to emerge from the process of systematic classification (Hsieh & Shannon, 2005 ). Qualitative content analysis is an appropriate analytic method for this study as it allows us to systematically identify episodes of practical inquiry and science inquiry to map them to the purposes and outcomes of these episodes as each lesson unfolds.

In total, six h of video recordings where students presented their ideas while the teachers served as facilitator and mentor were analysed. The video recordings were transcribed, and the transcripts were analysed using the NVivo software. Our unit of analysis is a single turn of talk (one utterance). We have chosen to use utterances as proxy indicators of reasoning practices based on the assumption that an utterance relates to both grammar and context. An utterance is a speech act that reveals both meaning and intentions of the speaker within specific contexts (Li, 2008 ).

Our research analytical lens is also interpretative in nature and the validity of our interpretation is through inter-rater discussion and agreement. Each utterance at the speaker level in transcripts was examined and coded either as relevant to practical reasoning or scientific reasoning based on the content. The utterances could be a comment by the teacher, a question by a student or a response by another student. Deductive coding is deployed with the two codes, practical reasoning and scientific reasoning derived from the theoretical ideas of Dewey and Bereiter as described earlier. Practical reasoning refers to utterances that reflect commonsensical knowledge or application of everyday understanding. Scientific reasoning refers to utterances that consist of scientifically oriented questions, scientific terms, or the use of empirical evidence to explain. Examples of each type of reasoning are highlighted in the following section. Each coded utterance is then reviewed for detailed description of the events that took place that led to that specific utterance. The description of the context leading to the utterance is considered an episode. The episodes and codes were discussed and agreed upon by two of the authors. Two coders simultaneously watched the videos to identify and code the episodes. The coders interpreted the content of each utterance, examine the context where the utterance was made and deduced the purpose of the utterance. Once each coder has established the sense-making aspect of the utterance in relation to the context, a code of either practical reasoning or scientific reasoning is assigned. Once that was completed, the two coders compared their coding for similarities and differences. They discussed the differences until an agreement was reached. Through this process, an agreement of 85% was reached between the coders. Where disagreement persisted, codes of the more experienced coder were adopted.

4 Results and Discussion

The specific STEM lessons analysed were taken from the lessons whereby students presented the model of their solutions to the class for peer evaluation. Every group of students stood in front of the class and placed their model on the bench as they presented. There was also a board where they could sketch or write their explanations should they want to. The instructions given by the teacher to the students were to explain their models and state reasons for their design.

4.1 Prevalence of Reasoning

The 6h of videos consists of 1422 turns of talk. Three hundred four turns of talk (21%) were identified as talk related to reasoning, either practical reasoning or scientific reasoning. Practical reasoning made up 62% of the reasoning turns while 38% were scientific reasoning (Fig. 2 ).

The two types of reasoning differ in the justifications that are used to substantiate the claims or decisions made. Table 1 describes the differences between the two categories of reasoning.

4.2 Applications of Scientific Reasoning

Instances of engagement with scientific reasoning (for instance, using scientific concepts to justify, raising scientifically oriented questions, or providing scientific explanations) revolved around the conditions for photosynthesis and the concept of energy conversion when students were presenting their ideas or when they were questioned by their peers. For example, in explaining the reason for including fish in their plant system, one group of students made connection to cyclical energy transfer: “…so as the roots of the plants submerged in the water, faeces from the fish will be used as fertilizers so that the plant can grow”. The students considered how organic matter that is still trapped within waste materials can be released and taken up by plants to enhance the growth. The application of scientific reasoning made their design one that is innovative and sustainable as evaluated by the teacher. Some students attempted more ecofriendly designs by considering energy efficiencies through incorporating water turbines in their farming systems. They applied the concept of different forms of energy and energy conversion when their peers inquired about their design. The same scientific concepts were explained at different levels of details by different students. At one level, the students explained in a purely descriptive manner of what happens to the different entities in their prototypes, with implied changes to the forms of energy─ “…spins then generates electricity. So right, when the water falls down, then it will spin. The water will fall on the fan blade thing, then it will spin and then it generates electricity. So, it saves electricity, and also saves water”. At another level, students defended their design through an explanation of energy conversion─ “…because when the water flows right, it will convert gravitational potential energy so, when it reaches the bottom, there is not really much gravitational potential energy”. While these instances of applying scientific reasoning indicated that students have knowledge about the scientific phenomena and can apply them to assist in the problem-solving process, we are not able to establish if students understood the science behind how the dynamo works to generate electricity. Students in eighth grade only need to know how a generator works at a descriptive level and the specialized understanding how a dynamo works is beyond the intended learning outcomes at this grade level.

The application of scientific concepts for justification may not always be accurate. For instance, the naïve conception that students have about plants only respiring at night and not in the day surfaced when one group of students tried to justify the growth rates of Kailan─ “…I mean, they cannot be making food 24/7 and growing 24/7. They have nighttime for a reason. They need to respire”. These students do not appreciate that plants respire in the day as well, and hence respiration occurs 24/7. This naïve conception that plants only respire at night is one that is common among learners of biology (e.g. Svandova, 2014 ) since students learn that plant gives off oxygen in the day and takes in oxygen at night. The hasty conclusion to that observation is that plants carry out photosynthesis in the day and respire at night. The relative rates of photosynthesis and respiration were not considered by many students.

Besides naïve conceptions, engagement with scientific ideas to solve a practical problem offers opportunities for unusual and alternative ideas about science to surface. For instance, another group of students explained that they lined up their plants so that “they can take turns to absorb sunlight for photosynthesis”. These students appear to be explaining that the sun will move and depending on the position of the sun, some plants may be under shade, and hence rates of photosynthesis are dependent on the position of the sun. However, this idea could also be interpreted as (1) the students failed to appreciate that sunlight is everywhere, and (2) plants, unlike animals, particularly humans, do not have the concept of turn-taking. These diverse ideas held by students surfaced when students were given opportunities to apply their knowledge of photosynthesis to solve a problem.

4.3 Applications of Practical Reasoning

Teachers and students used more practical reasoning during an integrated STEM activity requiring both science and engineering practices as seen from 62% occurrence of practical reasoning compared with 38% for scientific reasoning. The intention of the activity to integrate students’ scientific knowledge related to plant nutrition to engineering practice of building a model of vertical farming system could be the reason for the prevalence of practical reasoning. The practical reasoning used related to structural design considerations of the farming system such as how water, light and harvesting can be carried out in the most efficient manner. Students defended the strengths of designs using logic based on their everyday experiences. In the excerpt below (transcribed verbatim), we see students applied their everyday experiences when something is “thinner” (likely to mean narrower), logically it would save space. Further, to reach a higher level, you use a machine to climb up.

Excerpt 1. “Thinner, more space” Because it is more thinner, so like in terms of space, it’s very convenient. So right, because there is – because it rotates right, so there is this button where you can stop it. Then I also installed steps, so that – because there are certain places you can’t reach even if you stop the – if you stop the machine, so when you stop it and you climb up, and then you see the condition of the plants, even though it costs a lot of labour, there is a need to have an experienced person who can grow plants. Then also, when like – when water reach the plants, cos the plants I want to use is soil-based, so as the water reach the soil, the soil will xxx, so like the water will be used, and then we got like – and then there’s like this filter that will filter like the dirt.

In the examples of practical reasoning, we were not able to identify instances where students and teachers engaged with discussion around trade-off and optimisation. Understanding constraints, trade-offs and optimisations are important ideas in informed design matrix for engineering as suggested by Crismond and Adams ( 2012 ). For instance, utterances such as “everything will be reused”, “we will be saving space”, “it looks very flimsy” or “so that it can contains [sic] the plants” were used. These utterances were made both by students while justifying their own prototypes and also by peers who challenged the design of others. Longer responses involving practical reasoning were made based on common-sense, everyday logic─ “…the product does not require much manpower, so other than one or two supervisors like I said just now, to harvest the Kailan, hence, not too many people need to be used, need to be hired to help supervise the equipment and to supervise the growth”. We infer that the higher instances of utterances related to practical reasoning could be due to the presence of more concrete artefacts that is shown, and the students and teachers were more focused on questioning the structure at hand. This inference was made as instructions given by the teacher at the start of students’ presentation focus largely on the model rather than the scientific concepts or reasoning behind the model.

4.4 Intersection Between Scientific and Practical Reasoning

Comparing science subject matter knowledge and problem-solving to the idea of categories and placement (Buchanan, 1992 ), subject matter is analogous to categories where meanings are fixed with well-established epistemic practices and norms. The problem-solving process and design of solutions are likened to placements where boundaries are less rigid, hence opening opportunities for students’ personal experiences and ideas to be presented. Placements allow students to apply their knowledge from daily experiences and common-sense logic to justify decisions. Common-sense knowledge and logic are more accessible, and hence we observe higher frequency of usage. Comparatively, while science subject matter (categories) is also used, it is observed less frequently. This could possibly be due either to less familiarity with the subject matter or lack of appropriate opportunity to apply in practical problem solving. The challenge for teachers during implementation of a STEM problem-solving activity, therefore, lies in the balance of the application of scientific and practical reasoning to deepen understanding of disciplinary knowledge in the context of solving a problem in a meaningful manner.

Our observations suggest that engaging students with practical inquiry tasks with some engineering demands such as the design of modern farm systems offers opportunities for them to convert their personal lived experiences into feasible concrete ideas that they can share in a public space for critique. The peer critique following the sharing of their practical ideas allows for both practical and scientific questions to be asked and for students to defend their ideas. For instance, after one group of students presented their prototype that has silvered surfaces, a student asked a question: “what is the function of the silver panels?”, to which his peers replied : “Makes the light bounce. Bounce the sunlight away and then to other parts of the tray.” This question indicated that students applied their knowledge that shiny silvered surfaces reflect light, and they used this knowledge to disperse the light to other trays where the crops were growing. An example of a practical question asked was “what is the purpose of the ladder?”, to which the students replied: “To take the plants – to refill the plants, the workers must climb up”. While the process of presentation and peer critique mimic peer review in the science inquiry process, the conceptual knowledge of science may not always be evident as students paid more attention to the design constraints such as lighting, watering, and space that was set in the activity. Given the context of growing plants, engagement with the science behind nutritional requirements of plants, the process of photosynthesis, and the adaptations of plants could be more deliberately explored.

5 Conclusion

The goal of our work lies in applying the theoretical ideas of Dewey and Bereiter to better understand reasoning practices in integrate STEM problem solving. We argue that this is a worthy pursue to better understand the roles of scientific reasoning in practical problem solving. One of the goals of integrated STEM education in schools is to enculture students into the practices of science, engineering and mathematics that include disciplinary conceptual knowledge, epistemic practices, and social norms (Kelly & Licona, 2018 ). In the integrated form, the boundaries and approaches to STEM learning are more diverse compared with monodisciplinary ways of problem solving. For instance, in integrated STEM problem solving, besides scientific investigations and explanations, students are also required to understand constraints, design optimal solutions within specific parameters and even to construct prototypes. For students to learn the ways of speaking, doing and being as they participate in integrated STEM problem solving in schools in a meaningful manner, students could benefit from these experiences.

With reference to the first research question of What is the extent of practical and scientific reasoning in integrated STEM problem solving, our analysis suggests that there are fewer instances of scientific reasoning compared with practical reasoning. Considering the intention of integrated STEM learning and adopting Bereiter’s idea that students should learn higher-order conceptual knowledge through engagement with problem solving, we argue for a need for scientific reasoning to be featured more strongly in integrated STEM lessons so that students can gain higher order scientific conceptual knowledge. While the lessons observed were strong in design and building, what was missing in generating solutions was the engagement in investigations, where learners collected or are presented with data and make decisions about the data to allow them to assess how viable the solutions are. Integrated STEM problems can be designed so that science inquiry can be infused, such as carrying out investigations to figure out relationships between variables. Duschl and Bybee ( 2014 ) have argued for the need to engage students in problematising science inquiry and making choices about what works and what does not.

With reference to the second research question , What is achieved through practical and scientific reasoning during integrated STEM problem solving? , our analyses suggest that utterance for practical reasoning are typically used to justify the physical design of the prototype. These utterances rely largely on what is observable and are associated with basic-level knowledge and experiences. The higher frequency of utterances related to practical reasoning and the nature of the utterances suggests that engagement with practical reasoning is more accessible since they relate more to students’ lived experiences and common-sense. Bereiter ( 1992 ) has urged educators to engage learners in learning that is beyond basic-level knowledge since accumulation of basic-level knowledge does not lead to higher-level conceptual learning. Students should be encouraged to use scientific knowledge also to justify their prototype design and to apply scientific evidence and logic to support their ideas. Engagement with scientific reasoning is preferred as conceptual knowledge, epistemic practices and social norms of science are more widely recognised compared with practical reasoning that are likely to be more varied since they rely on personal experiences and common-sense. This leads us to assert that both context and content are important in integrated STEM learning. Understanding the context or the solution without understanding the scientific principles that makes it work makes the learning less meaningful since we “…cannot strip learning of its context, nor study it in a ‘neutral’ context. It is always situated, always relayed to some ongoing enterprise”. (Bruner, 2004 , p. 20).

To further this discussion on how integrated STEM learning experiences harness the ideas of practical and scientific reasoning to move learners from basic-level knowledge to higher-order conceptual knowledge, we propose the need for further studies that involve working with teachers to identify and create relevant problems-of-explanations that focuses on feasible, worthy inquiry ideas such as those related to specific aspects of transportation, alternative energy sources and clean water that have impact on the local community. The design of these problems can incorporate opportunities for systematic scientific investigations and scaffolded such that there are opportunities to engage in epistemic practices of the constitute disciplines of STEM. Researchers could then examine the impact of problems-of-explanations on students’ learning of higher order scientific concepts. During the problem-solving process, more attention can be given to elicit students’ initial and unfolding ideas (practical) and use them as a basis to start the science inquiry process. Researchers can examine how to encourage discussions that focus on making meaning of scientific phenomena that are embedded within specific problems. This will help students to appreciate how data can be used as evidence to support scientific explanations as well as justifications for the solutions to problems. With evidence, learners can be guided to work on reasoning the phenomena with explanatory models. These aspects should move engagement in integrated STEM problem solving from being purely practice to one that is explanatory.

6 Limitations

There are four key limitations of our study. Firstly, the degree of generalisation of our observations is limited. This study sets out to illustrate what how Dewey and Bereiter’s ideas can be used as lens to examine knowledge used in problem-solving. As such, the findings that we report here is limited in its ability to generalise across different contexts and problems. Secondly, the lessons that were analysed came from teacher-frontal teaching and group presentation of solution and excluded students’ group discussions. We acknowledge that there could potentially be talk that could involve practical and scientific reasonings within group work. There are two practical consideration for choosing to analyse the first and presentation segments of the suite of lesson. Firstly, these two lessons involved participation from everyone in class and we wanted to survey the use of practical and scientific reasoning by the students as a class. Secondly, methodologically, clarity of utterances is important for accurate analysis and as students were wearing face masks during the data collection, their utterances during group discussions lack the clarity for accurate transcription and analysis. Thirdly, insights from this study were gleaned from a small sample of six classes of students. Further work could involve more classes of students although that could require more resources devoted to analysis of the videos. Finally, the number of students varied across groups and this could potentially affect the reasoning practices during discussions.

Data Availability

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to acknowledge the contributions of the other members of the research team who gave their comment and feedback in the conceptualization stage.

This study is funded by Office of Education Research grant OER 24/19 TAL.

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Tan, AL., Ong, Y.S., Ng, Y.S. et al. STEM Problem Solving: Inquiry, Concepts, and Reasoning. Sci & Educ 32 , 381–397 (2023). https://doi.org/10.1007/s11191-021-00310-2

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Problem Solving, Critical Thinking, and Analytical Reasoning Skills Sought by Employers

In this section:

Problem Solving

Critical Thinking

Analytical Reasoning

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Critical thinking, analytical reasoning, and problem-solving skills are required to perform well on tasks expected by employers. 1 Having good problem-solving and critical thinking skills can make a major difference in a person’s career. 2

Every day, from an entry-level employee to the Chairman of the Board, problems need to be resolved. Whether solving a problem for a client (internal or external), supporting those who are solving problems, or discovering new problems to solve, the challenges faced may be simple/complex or easy/difficult.

A fundamental component of every manager's role is solving problems. So, helping students become a confident problem solver is critical to their success; and confidence comes from possessing an efficient and practiced problem-solving process.

Employers want employees with well-founded skills in these areas, so they ask four questions when assessing a job candidate 3 :

Evaluation of information: How well does the applicant assess the quality and relevance of information?
Analysis and Synthesis of information: How well does the applicant analyze and synthesize data and information?
Drawing conclusions: How well does the applicant form a conclusion from their analysis?
Acknowledging alternative explanations/viewpoints: How well does the applicant consider other options and acknowledge that their answer is not the only perspective?

When an employer says they want employees who are good at solving complex problems, they are saying they want employees possessing the following skills:

Analytical Thinking — A person who can use logic and critical thinking to analyze a situation.
Critical Thinking – A person who makes reasoned judgments that are logical and well thought out.
Initiative — A person who will step up and take action without being asked. A person who looks for opportunities to make a difference.
Creativity — A person who is an original thinker and have the ability to go beyond traditional approaches.
Resourcefulness — A person who will adapt to new/difficult situations and devise ways to overcome obstacles.
Determination — A person who is persistent and does not give up easily.
Results-Oriented — A person whose focus is on getting the problem solved.

Two of the major components of problem-solving skills are critical thinking and analytical reasoning. These two skills are at the top of skills required of applicants by employers.

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Critical Thinking 4

“Mentions of critical thinking in job postings have doubled since 2009, according to an analysis by career-search site Indeed.com.” 5 Making logical and reasoned judgments that are well thought out is at the core of critical thinking. Using critical thinking an individual will not automatically accept information or conclusions drawn from to be factual, valid, true, applicable or correct. “When students are taught how to use critical thinking to tap into their creativity to solve problems, they are more successful than other students when they enter management-training programs in large corporations.” 6

A strong applicant should question and want to make evidence-based decisions. Employers want employees who say things such as: “Is that a fact or just an opinion? Is this conclusion based on data or gut feel?” and “If you had additional data could there be alternative possibilities?” Employers seek employees who possess the skills and abilities to conceptualize, apply, analyze, synthesize, and evaluate information to reach an answer or conclusion.

Employers require critical thinking in employees because it increases the probability of a positive business outcome. Employers want employees whose thinking is intentional, purposeful, reasoned, and goal directed.

Recruiters say they want applicants with problem-solving and critical thinking skills. They “encourage applicants to prepare stories to illustrate their critical-thinking prowess, detailing, for example, the steps a club president took to improve attendance at weekly meetings.” 7

Employers want students to possess analytical reasoning/thinking skills — meaning they want to hire someone who is good at breaking down problems into smaller parts to find solutions. “The adjective, analytical, and the related verb analyze can both be traced back to the Greek verb, analyein — ‘to break up, to loosen.’ If a student is analytical, you are good at taking a problem or task and breaking it down into smaller elements in order to solve the problem or complete the task.” 9

Analytical reasoning connotes a person's general aptitude to arrive at a logical conclusion or solution to given problems. Just as with critical thinking, analytical thinking critically examines the different parts or details of something to fully understand or explain it. Analytical thinking often requires the person to use “cause and effect, similarities and differences, trends, associations between things, inter-relationships between the parts, the sequence of events, ways to solve complex problems, steps within a process, diagraming what is happening.” 10

Analytical reasoning is the ability to look at information and discern patterns within it. “The pattern could be the structure the author of the information uses to structure an argument, or trends in a large data set. By learning methods of recognizing these patterns, individuals can pull more information out of a text or data set than someone who is not using analytical reasoning to identify deeper patterns.” 11

Employers want employees to have the aptitude to apply analytical reasoning to problems faced by the business. For instance, “a quantitative analyst can break down data into patterns to discern information, such as if a decrease in sales is part of a seasonal pattern of ups and downs or part of a greater downward trend that a business should be worried about. By learning to recognize these patterns in both numbers and written arguments, an individual gains insights into the information that someone who simply takes the information at face value will miss.” 12

Managers with excellent analytical reasoning abilities are considered good at, “evaluating problems, analyzing them from more than one angle and finding a solution that works best in the given circumstances”. 13 Businesses want managers who can apply analytical reasoning skills to meet challenges and keep a business functioning smoothly

A person with good analytical reasoning and pattern recognition skills can see trends in a problem much easier than anyone else.

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{{item.title}}, my essentials, ask for help, contact edconnect, directory a to z, how to guides, maths trains brains, five games to have fun with maths in years 7 and 8.

These games will help enrich your child's reasoning and problem-solving skills as they enhance their knowledge of how numbers work, quantifying collections, patterning, algebra and probability. Help your child expand their mathematical skills using these five fun games.

Prime Climb

A colourful board game for ages 10 and up, Prime Climb combines strategy and luck as players battle to be the first to land both pawns in the 101 circle at the centre of the game board.

By rolling dice and choosing which operation you’d like to use, players can bump their opponent’s pawns off the board in this exciting race to 101!

A classic game, chess is steeped in opportunities to deepen mathematical skills and understanding. Players take turns moving one chess piece at a time until one player is able to capture their opponent's king

A great game to develop mathematical reasoning and patient problem solving, chess promotes your child’s understanding of concepts such as position, angles and probability.

Does your child like codebreaking? They may enjoy Mastermind, where they can pit themselves against an opponent and use deduction to unlock a secret code recorded in coloured pegs on a board.

Fun fact: Mind lends itself to strategy, mathematical reasoning and algebra, with mathematicians studying the game since the late 1970s to come up with elegant and efficient solutions.

Monopoly isn’t just a great family board game, it can also be a great experience to enrich an understanding of working with money. Players have to budget, invest and explore the value of acquiring assets - in a fun, risk-free way!

By applying skills and understanding in position, probability and patterning, players can give themselves a competitive edge.

Cribbage is a classic game that can lead to some amazing battles. The game is based on scoring points by collecting particular combinations of cards. Each card is given a numerical value, with the players aiming to keep the sum of the cards at 31 or lower. The first player to reach 121 points wins.

Aside from the opportunity to develop their understanding of operations and quantifying collections, Cribbage can help your teen enhance their problem-solving skills and mathematical reasoning. It also requires relatively little equipment to play - just a deck of cards and a cribbage board for keeping score, which your teen can make.

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Researchers from the University of Notre Dame team and Tencent AI Lab have introduced a new benchmark named MathChat to address this gap. MathChat evaluates LLMs’ performance in multi-turn interactions and open-ended question-answering. This benchmark aims to push the boundaries of what LLMs can achieve in mathematical reasoning by focusing on dialogue-based tasks. MathChat includes tasks inspired by educational methodologies, such as follow-up questioning and error correction, which are critical for developing models that can understand and respond to dynamic mathematical queries.

The MathChat benchmark includes follow-up question-answering, error correction, analysis, and problem generation. These tasks require models to engage in multi-turn dialogues, identify and correct mistakes, analyze errors, and generate new problems based on given solutions. This comprehensive approach ensures that models are tested on various abilities beyond simple problem-solving. By encompassing multiple aspects of mathematical reasoning, MathChat provides a more accurate assessment of a model’s capabilities in handling real-world mathematical interactions.

In their experiments, the researchers found that while current state-of-the-art LLMs perform well on single-turn tasks, they struggle significantly with multi-turn and open-ended tasks. For instance, models fine-tuned on extensive single-turn QA data showed limited ability to handle the more complex demands of MathChat. Introducing a synthetic dialogue-based dataset, MathChatsync, significantly improved model performance, highlighting the importance of training with diverse conversational data. This dataset focuses on improving interaction and instruction-following capabilities, essential for multi-turn reasoning.

The researchers evaluated various LLMs on the MathChat benchmark, observing that while these models excel in single-turn question answering, they underperform in scenarios requiring sustained reasoning and dialogue understanding. For example, MetaMath achieved 77.18% accuracy in the first round of follow-up QA but dropped to 32.16% in the second round and 19.31% in the third. Similarly, WizardMath started with 83.20% accuracy, which fell to 44.81% and 36.86% in subsequent rounds. DeepSeek-Math and InternLM2-Math also exhibited significant performance drops in multi-round interactions, with the latter achieving 83.80% accuracy in single-round tasks but much lower in follow-up rounds. The MathChatsync fine-tuning led to substantial improvements: Mistral-MathChat achieved an overall average score of 0.661, compared to 0.623 for Gemma-MathChat, indicating the effectiveness of diverse, dialogue-based training data.

In conclusion, this research identifies a critical gap in current LLM capabilities and proposes a new benchmark and dataset to address this challenge. The MathChat benchmark and MathChatsync dataset represent significant steps in developing models that can effectively engage in multi-turn mathematical reasoning, paving the way for more advanced and interactive AI applications in mathematics. The study highlights the necessity of diverse training data and comprehensive evaluation to enhance the capabilities of LLMs in real-world mathematical problem-solving scenarios. This work underscores the potential for LLMs to transform mathematical education and research by providing more interactive and effective tools.

Check out the Paper . All credit for this research goes to the researchers of this project. Also, don’t forget to follow us on Twitter . Join our Telegram Channel , Discord Channel , and LinkedIn Gr oup .

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🚀 Excited to share our latest research MathChat! 📊 We explore the new frontiers in interactive math problem-solving. Check it out! 🧵👇 MathChat is a benchmark designed to evaluate LLMs on mathematical multi-turn interaction and open-ended generation. https://t.co/d6UFlPZyC9 — Zhenwen Liang (@LiangZhenwen) May 31, 2024

Nikhil is an intern consultant at Marktechpost. He is pursuing an integrated dual degree in Materials at the Indian Institute of Technology, Kharagpur. Nikhil is an AI/ML enthusiast who is always researching applications in fields like biomaterials and biomedical science. With a strong background in Material Science, he is exploring new advancements and creating opportunities to contribute.

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35 problem-solving techniques and methods for solving complex problems
6. Discovery & Action Dialogue (DAD) One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions. With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so.
7 Module 7: Thinking, Reasoning, and Problem-Solving
Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of ...
The Problem-Solving Process
Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue. The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything ...
Problem-Solving Strategies: Definition and 5 Techniques to Try
In insight problem-solving, the cognitive processes that help you solve a problem happen outside your conscious awareness. 4. Working backward. Working backward is a problem-solving approach often ...
7.3 Problem-Solving
Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below (see figure) is a 4×4 grid.
Definitive Guide to Problem Solving Techniques
Most problem solving techniques look for a balance between the following binaries: ... Inductive Reasoning: A logical method that uses evidence to conclude that a certain answer is probable (this is opposed to deductive reasoning, where the answer is assumed to be true). Inductive reasoning uses a limited number of observations to make useful ...
Problem-Solving Strategies and Obstacles
Problem-solving is a vital skill for coping with various challenges in life. This webpage explains the different strategies and obstacles that can affect how you solve problems, and offers tips on how to improve your problem-solving skills. Learn how to identify, analyze, and overcome problems with Verywell Mind.
Problem Solving
Abstract. Problem solving refers to cognitive processing directed at achieving a goal when the problem solver does not initially know a solution method. A problem exists when someone has a goal but does not know how to achieve it. Problems can be classified as routine or nonroutine, and as well defined or ill defined.
Problem Solving
This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2) research on search ...
Problem solving (video)
The video explains different problem-solving methods, including trial and error, algorithm strategy, and heuristics. It also discusses concepts like means-end analysis, working backwards, fixation, and insight. These techniques help us tackle both well-defined and ill-defined problems effectively.
What Are Critical Thinking Skills and Why Are They Important?
Problem-solving: Problem-solving is perhaps the most important skill that critical thinkers can possess. The ability to solve issues and bounce back from conflict is what helps you succeed, be a leader, and effect change. One way to properly solve problems is to first recognize there's a problem that needs solving.
Critical Thinking and Problem-Solving
Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking ...
What Is Problem Solving?
The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem - not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the ...
Fluency, Reasoning & Problem Solving: What They REALLY Are
This is important for two reasons: 1) It splits up reasoning skills and problem solving into two different entities. 2) It demonstrates that fluency is not something to be rushed through to get to the 'problem solving' stage but is rather the foundation of problem solving.
What is Problem Solving? (Steps, Techniques, Examples)
The problem-solving process typically includes the following steps: Identify the issue: Recognize the problem that needs to be solved. Analyze the situation: Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present. Generate potential solutions: Brainstorm a list of possible ...
What is Problem Solving? Steps, Process & Techniques
1. Define the problem. Diagnose the situation so that your focus is on the problem, not just its symptoms. Helpful problem-solving techniques include using flowcharts to identify the expected steps of a process and cause-and-effect diagrams to define and analyze root causes.. The sections below help explain key problem-solving steps.
14 Effective Problem-Solving Strategies
14 types of problem-solving strategies. Here are some examples of problem-solving strategies you can practice using to see which works best for you in different situations: 1. Define the problem. Taking the time to define a potential challenge can help you identify certain elements to create a plan to resolve them.
Problem Solving Reasoning
Problem Solving Reasoning is a logical reasoning part where candidates will be given various questions and they need to perform various operations such as addition, division, greater than, lesser than, etc are interchanged or substituted to find the correct answer. Almost all the government examinations ask questions on the problem solving reasoning section.
Problem solving
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue ...
20 Effective Math Strategies For Problem Solving
Here are five strategies to help students check their solutions. 1. Use the Inverse Operation. For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7.
Skills and Strategies for Critical Thinking and Quantitative Reasoning
Quantitative reasoning refers to one's ability to understand, interpret, and analyze numerical information to support decision-making and problem solving (Kjelvik & Schultheis, 2019 ). Accordingly, quantitative reasoning skills include numerical comprehension, data analysis, and evidence-based arguments (Mayes, 2019 ).
STEM Problem Solving: Inquiry, Concepts, and Reasoning
Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey's idea of ...
Problem Solving, Critical Thinking, and Analytical Reasoning Skills
Critical thinking, analytical reasoning, and problem-solving skills are required to perform well on tasks expected by employers. 1 Having good problem-solving and critical thinking skills can make a major difference in a person's career. 2. ... Two of the major components of problem-solving skills are critical thinking and analytical reasoning.
Enhance Team Problem-Solving with Logical Reasoning
Here's how you can apply logical reasoning skills in a team setting. Powered by AI and the LinkedIn community. 1. Identify Issues. 2. Gather Data. 3. Analyze Thoroughly. 4.
Boost Civil Engineering Problem-Solving with Logical Reasoning
Here's how you can use logical reasoning skills to enhance problem-solving in civil engineering projects. Powered by AI and the LinkedIn community. 1. Define Issue. Be the first to add your ...
Boost Problem-Solving with Logical Reasoning
Here's how you can enhance problem identification and analysis with logical reasoning skills. Powered by AI and the LinkedIn community. 1. Define Issue. Be the first to add your personal ...
Five games to have fun with maths in Years 7 and 8
Five games to have fun with maths in Years 7 and 8. These games will help enrich your child's reasoning and problem-solving skills as they enhance their knowledge of how numbers work, quantifying collections, patterning, algebra and probability. Help your child expand their mathematical skills using these five fun games.
Boost Problem-Solving with Logical Reasoning
Enhancing problem-solving skills is a vital aspect of life coaching that can lead to significant personal and professional growth. Logical reasoning, a critical component in this process, is the ...
Enhancing Problem-Solving Skills with Inductive Reasoning
Problem Solving Most occupations require good problem-solving skills. For instance, architects and engineers must solve many complicated problems as they design and construct modern buildings that are aesthetically pleasing, functional, and that meet stringent safety requirements. Two goals of this chapter are: 1. to help you become a better problem solver and 2. to demonstrate that problem ...
From Static to Conversational: MathChat and MathChatsync Open New Doors
Mathematical reasoning has long been a critical area of research within computer science. With the advancement of large language models (LLMs), there has been significant progress in automating mathematical problem-solving. This involves the development of models that can interpret, solve, and explain complex mathematical problems, making these technologies increasingly relevant in educational ...

35 problem-solving techniques and methods for solving complex problems

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How do you identify problems?

Complete problem-solving methods

Problem-solving warm-up activities

Tips for more effective problem solving

Clearly define the problem

Don’t jump to conclusions

Try different approaches

Don’t take it personally

Get the right people in the room

Document everything

Bring a facilitator

Develop your problem-solving skills

Design a great agenda

1. Six Thinking Hats

2. Lightning Decision Jam

3. Problem Definition Process

4. The 5 Whys

5. World Cafe

6. Discovery & Action Dialogue (DAD)

7. Design Sprint 2.0

8. Open space technology

Techniques to identify and analyze problems

10. The Creativity Dice

11. Fishbone Analysis

12. Problem Tree

13. SWOT Analysis

14. Agreement-Certainty Matrix

16. Speed Boat

17. The Journalistic Six

18. LEGO Challenge

19. What, So What, Now What?

20. Journalists

Problem-solving techniques for developing solutions

21. Mindspin

22. Improved Solutions

23. Four Step Sketch

24. 15% Solutions

25. How-Now-Wow Matrix

26. Impact and Effort Matrix

27. Dotmocracy

28. Check-in / Check-out

29. Doodling Together

30. Show and Tell

31. Constellations

32. Draw a Tree

How do I conclude a problem-solving process?

33. One Breath Feedback

34. Who What When Matrix

35. Response cards

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7 Module 7: Thinking, Reasoning, and Problem-Solving

READING WITH PURPOSE

Analyze, Evaluate, and Create

7.1. Different kinds of thought and knowledge

Factual and conceptual knowledge

Procedural knowledge

Metacognitive knowledge

The Dunning-Kruger Effect

Critical thinking

A Critical Thinker’s Toolkit

7.2 Reasoning and Judgment

Deductive reasoning

Inductive reasoning and judgment

7.3 Problem Solving

Defining and Mentally Representing Problems in Order to Solve Them

Solving Problems by Trial and Error

Solving Problems with Algorithms and Heuristics

An Effective Problem-Solving Sequence

Share This Book

7.3 Problem-Solving

PROBLEM-SOLVING STRATEGIES

Additional Problem Solving Strategies :

Figure 7.02. Steps for solving the Tower of Hanoi in the minimum number of moves when there are 3 disks.

Figure 7.03. Graphical representation of nodes (circles) and moves (lines) of Tower of Hanoi.