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26 Whole ClassMath Games: Adaptable For All Students

Samantha Dock

Whole class math games are becoming progressively more popular among educators and students as a strategy to increase learner engagement and comprehension. This approach transforms traditional math lessons into interactive and immersive learning experiences. 

In this article, we list 26 of the best math teacher-approved whole class math games for you to try with your students today.

What are whole class math games? 

Whole class math games are a way to leverage gamification, such as challenges, levels, and points, to motivate and engage all learners in math in a fun way. Whole class math games should be student-centered learning and accessible to all students and be flexible to include all learner abilities and additional needs. 

Through fun math games, teachers can motivate and inspire students to work collaboratively, to solidify their understanding of key math concepts and to have fun during math class.

26 Whole Class Math Games

Play these 26 fun and engaging math games with your pre-kindergarten to 8th grade students.

Benefits of whole class math games

By using whole class math games strategically, they can be an effective learning strategy . The benefits of whole class math games include:

• Intrinsic motivation: Games can provide intrinsic motivation by offering immediate feedback, rewards, and a sense of accomplishment.
• Reinforce math content: Games naturally encourage repeated practice and exploration. Learners are more likely to continue practicing and retaining what they have learned over time by embedding math concepts within engaging game contexts. 
• Personalize learning and feedback: Gamification facilitates a tailored learning experience where students can progress at their own pace and receive individualized feedback, addressing their unique learning needs and preferences.
• Promote perseverance: Students are motivated to persevere with challenging mathematical problems.
• Promote problem-solving , strategic thinking and critical reasoning
• Foster collaboration: Games encourage students to work together, discuss strategies and learn from one another.
• Math can be fun: Games can help to reduce math anxiety by presenting math in an accessible way in a low-stakes environment. 

26 whole class math games to engage all students

Here, we’ve listed 26 whole class math games including mental math games , multiplication games and more:

6 No prep whole class math games

1. clap and count.

This is a great no prep and quick whole class math game to get students engaged and moving! 

Grade level: PreK-3

How to play: The math teacher or selected student picks a number and says it aloud or writes it on the whiteboard. The class is then expected to clap and count up to that number. This game helps students practice their number sense.

Ideas to adapt: You can adapt this to include exercising, as well! (Example: the number 5 is selected, so students have to do 5 push ups).

This game can be used to count in varying increments e.g. 2s, 5s, 10s. 

Another quick, no-prep whole class math game that gets students to collaborate and strategize! This is a great game to play if you have a few extra minutes at the end of class or the students need a brain break.

Grade level: Grades 3-8

How to play: In this game, have students count to the number 21. If two or more students say the same number, start over from 1. The same student cannot say two numbers in a row. 

Ideas to adapt: You can change the number higher or lower depending on the number of students in your class or have students close their eyes to make it more challenging!

3. Mystery Number

This activity reinforces math fluency and vocabulary. This can be a whole class math game or have students buddy up.

Grade level: Grades 1-4

How to play: One person in the group thinks of a number and gives the other person hints about what the number is. For example, a hint can be, “The number is bigger than 3, but less than 17.”

Ideas to adapt: You can have students write down their number on a mini dry erase board or their notebook so they do not forget it.

War is a simple yet effective card game to get students to compare two quantities.

Grade level: Grades 1-8

How to play: Typically, this game is played with a deck of cards that is split into two even piles. The cards are face down and students pick the top card to compare with their partner. The person with the higher card value gets to keep both cards. If the two cards have the same value, the students place 3 cards face down and flip over the fourth one. The student with the higher value card gets to keep all of the cards. In order to win the game, students must collect all of the cards in the pile.

Ideas to adapt: This game is easily adaptable to each grade level. You can make and print out cards with the concept that you are learning. For example, you can create 52 cards with integers on them, or fractions and decimals.

Another no-prep card game to quickly engage students and get them practicing number bonds and math fluency. 24 lays the groundwork for computational thinking.

Grade level: Grades 1-9

How to play: In 24, students are in groups of 2-4 and are asked to make the number 24 using all four numbers on the card and any operation. The student who can make the number 24 first wins the cards. The student with the most cards at the end wins.

6. Dominoes

There are so many variations of dominoes that you can use with your students! 

How to play: One way to play is to have all the dominoes flipped over and each student picks a domino, writes down each side as an addend and adds them together. You can also use the domino to create addition fact families to help with fluency in numbers and operations, categorizing them into even and odd numbers, or sorting them by the sum.

Another way to adapt dominoes is to lay them down so that you have a row that adds up to 10. To begin with, the dominoes are face down. Each player takes turns picking up a tile and making a new row or putting it at the end of a row to make 10. In this game, the student with the most rows of 10 at the end wins. This helps to support students’ conceptual and computational growth. 

Ideas to adapt: Creating magic squares is another way to use dominoes to engage students in learning and enhance their math fact fluency.

9 Whole class math games for any grade

7. jeopardy.

Grade level: Grades K-8

Jeopardy is a classic game that teachers typically use as a unit review.

How to play: Students are split up into 4 teams and must answer the questions on the board. The questions are separated into 5 categories and given a dollar amount. The higher the dollar amount, the more difficult the question is. The group with the highest dollar amount at the end of the game wins.

This is a great game that can be played in person or virtually.

How to play: Give students a blank BINGO board and have all the potential answers on the front board. Ask students to write down the answers on their board wherever they want. Make sure they know to write down each answer only once. 

Once all BINGO boards have been created, pick a question for the students to answer. You can create a PowerPoint with all the questions on separate slides and ask students to pick a number. When you click on that number the question will be presented to the students to answer. Students will mark off the answer on their BINGO cards and the first student to get 5 in a row wins!

9. Trashketball

Trashketball is another engaging review game.

How to play: This game works best if there are about 6 single-sided pages printed in a packet for each student to complete. Students work on the first page, check their answers with the teacher, and if they get all their answers correct, they can crumble their piece of paper into a ball and stand at the 2 or 3 point line to attempt a basket. You can use the recycling can on top of a desk as the basket and put tape on the ground to mark off where the 2 and 3 point lines are. The student with the most points at the end of class wins.

Ideas to adapt: If a student has an incorrect answer, you can tell them which question was wrong or tell them, “2 problems on this page are incorrect.” This allows them to conduct error analysis on their own work.

10. Scavenger hunt

Many premade scavenger hunt worksheets can be found online. You can also create your own and adapt it to be for whatever your class is learning at that time!

How to play: For this, students will need paper or a recording sheet to show their work, a pencil, and a clipboard. You will need to hang up the question/answer pages around the room. On these pages, the top half should have the answer to the previous problem and the bottom half should have the next question to be answered. The goal is to have students practice as many questions as possible to review the material.

This whole class math game is similar to the scavenger hunt.

How to play: Question pages should be posted around the room and students will need a recording sheet, pencil, and clipboard to lean on. Students are given a set amount of time to work on the page in front of them (for example, students have 30 seconds to simplify the algebraic expression). After 30 seconds, the math teacher will say “scoot” and students will shift to the next question page, have 30 seconds to complete the problem before moving on to the next one. The goal is to have students practice fast math facts and fluency.

12. Whodunit

A “Whodunit” activity is a mystery-solving game where participants work together to uncover clues, solve puzzles, and ultimately identify the culprit behind a fictional crime. This activity does involve a lot of set up but is a great way to get students to buy into the material and actively participate in their learning experience.

How to play: Small groups will work together to solve math problems and be rewarded with clues to identify the culprit. You can find many pre-made downloadable options online to reduce planning time.

13. Relay race

Grade level: Grades K-8 

Playing a relay race in math can be a fun and engaging way to review concepts or practice mathematical skills. 

How to play: Divide the class into equal size teams and set up stations. Students will take turns sending one team member at a time to the task station which could include solving equations, completing math puzzles, answering word problems, or performing mental math calculations. 

After the race, gather the students to review the tasks and discuss any challenges or interesting strategies used during the relay. You can also review the correct answers to the math problems to ensure understanding.

14. Quiz, quiz, trade

“Quiz, Quiz, Trade” is a cooperative learning strategy that promotes formative assessment , active engagement and peer-to-peer teaching. It’s particularly effective for reviewing math concepts in a fun and interactive way. 

How to play: Prepare a set of question cards related to the math concepts you want to review. Each student should have a card and each card should have a math problem or a question on one side and the answer on the other. Arrange students into pairs or small groups around the room and distribute one question card to each student, ensuring each student has a different question.

Then have students hold their cards up with the question side facing out. Each student quizzes their partner with the question on their card. They can read the question aloud or show it to their partner. The partner tries to answer the question without looking at the back of the card. If they answer correctly, they receive praise and encouragement from their partner. 

After both partners have quizzed each other, they trade cards, find a new partner, and repeat the process. “Quiz, Quiz, Trade” effectively reviews math concepts while promoting active engagement, collaboration, and peer teaching among students. It’s adaptable to various grade levels and can be customized to focus on specific math topics or skills.

15. Escape room

Creating an educational escape room in math is a fantastic way to engage students in problem-solving, critical thinking, and collaboration while reinforcing mathematical concepts. 

How to play: There are many escape rooms online for teachers to use. Breakoutedu is a great resource to build physical escape rooms to review mathematical concepts or assign virtual escape rooms for students to complete as a class, with a small group or partner, or individually.

5 Number sense and operations whole class math games

16. human number line.

Grade level: Grades K-7

A human number line activity is an interactive and kinesthetic way to teach or reinforce concepts related to numbers, number sense, and mathematical operations. 

How to play: In this activity, students physically represent numbers along a designated line or axis, allowing them to visualize numerical relationships and engage in hands-on learning.

17. Guess who

Playing “Guess Who?” in math creatively reinforces mathematical concepts such as properties, characteristics, or attributes of numbers, shapes, or other mathematical objects. 

How to play: Create or print out game boards featuring various mathematical objects or concepts. For example, you could have boards with numbers, geometric shapes, mathematical operations, or math-related images. 

Players engage in critical thinking as they analyze mathematical properties and make educated guesses based on the information they gather. Also, players practice using mathematical vocabulary and describing mathematical properties in a clear and concise manner. The game promotes active engagement and participation as players interact with each other and work towards solving the mystery.

18. 4 in a row

Ideal for intervention groups or students developing fluency in adding 9, 10, or 11 and place value. Suitable for math workshops or stations to enhance fluency. 

How to play: In this game, pairs of students share one game board. Laminate the board or use a photocopy with counters or markers. The first player rolls a die and adds 10 (or 11 or 9!) to cover a spot. The next student rolls the die and adds 10 (or 11 or 9). The goal is to achieve 4 in a row strategically, considering available numbers.

Ideas to adapt: You can use base 10 blocks or other “hands on” manipulatives to show students how adding 10 impacts the tens place and adding 11 impacts both the tens and the ones. Adding 9 is tricky for some students, so you can show them by adding 10 and then taking away one cube.

Grade level: Grades 5-8

Nerdle is a daily math puzzle inspired by The New York Times’ word puzzle, Wordle. Nerdle challenges players to guess a randomly selected calculation within six attempts. 

How to play: After each guess, players receive feedback on the tiles: green for correct tiles in the correct position, yellow for correct tiles in the wrong position, and gray for incorrect tiles. 

Players refine their guesses using this feedback, aiming to correctly guess the calculation or exhaust their attempts within the allotted six tries. Nerdle offers a fun and intellectually stimulating way to exercise math and deduction skills while enjoying a guessing challenge.

20. Equation Scrabble

Equation Scrabble is a versatile math-centered game for 1-4 players to sharpen math facts and number sense, or delve into specific skills like fractions, decimals, large numbers, negatives, variables, and exponents. 

How to play: Similar to Scrabble but with numbers and variables, students form addition, subtraction, multiplication, and division equations, earning points based on the complexity of their equations. You can find printable versions online, or make your own.

Ideas to adapt: Adjust the game by removing pieces to tailor it to focus on a single operation or skill, offering a flexible and engaging math activity for diverse learning needs.

2 Mental math and problem-solving whole class math games

21. multiplication baseball.

Grade level: Grades 3-4

Multiplication baseball infuses the excitement of baseball with multiplication practice. 

How to play: Players form two teams, one batting and the other fielding. The batting team’s players take turns answering multiplication flashcards to advance around the bases, scoring runs for correct answers and accumulating outs for incorrect ones. 

This game not only sharpens multiplication fluency and mental math skills but also fosters strategic thinking and teamwork as players strategize to score runs and defend against the opposing team. It’s a dynamic way to reinforce multiplication skills while enjoying the spirit of competition on the “field.”

22. Hamburger dice game

Grade level: Grades 4-9

In this engaging activity, students utilize problem-solving strategies and mental math skills as they collaborate to construct a hamburger. 

How to play: Working in small groups or pairs, students are equipped with three dice and a burger building sheet. Taking turns, they roll the dice and creatively manipulate the numbers rolled to match the desired ingredients for their burgers. 

However, there’s a twist—the burger assembly must commence with the bottom bun equivalent to 12 and conclude with the top bun equivalent to 10, allowing flexibility in arranging the toppings between. The objective is clear: the first student to successfully assemble their burger according to the given criteria emerges as the winner, blending mathematical thinking with culinary creativity in a fun and competitive manner.

4 Geometry and measurement whole class math games

The ideas below include different fun math activities suited to geometry and measurement. These are not games in themselves but when presented with an element of competition, the activities below can leverage gamification to engage and inspire students.

23. Pattern blocks

Grade level: Grades K-5

Pattern blocks are versatile mathematical manipulatives that students can use to explore various mathematical concepts and develop important skills. These colorful blocks, typically available in shapes such as triangles, squares, rhombuses, trapezoids, and hexagons, allow students to engage in hands-on learning experiences. 

Students can use pattern blocks to develop spatial reasoning by exploring geometric relationships, identifying shapes, and creating patterns and designs. They also support the development of mathematical concepts such as symmetry, congruence, fractions, and area. 

How to play: Through activities involving sorting, classifying, composing, decomposing, and transforming shapes, students enhance their understanding of geometry, spatial visualization, problem-solving, and critical thinking skills. Pattern blocks provide a tangible and interactive way for students to deepen their mathematical understanding and foster a love for learning in mathematics.

24. Stained glass window activity

Grade level: Grades 7-9

This activity encourages students to tap into their artistic side while practicing their understanding of graphing linear equations. 

How to play: Students will create a stained glass window by graphing linear equations.

Teachers will need to distribute coordinate planes or graph paper, provide pupils with the linear equations, have students graph them and add color and outlines to their designs with colored pencils and Sharpies. 

Ideas to adapt: You can provide equations already in slope-intercept form and or where students need to solve for y before graphing. 

25. Geoboards

Geoboards are hands-on mathematical tools comprising a board with pegs arranged in a grid pattern, allowing students to stretch rubber bands to create shapes and patterns. 

How to play: With geoboards, students delve into various mathematical concepts, from geometry to fractions and measurement. They explore geometric shapes, angles, symmetry, and spatial relationships while enhancing spatial reasoning skills. 

Additionally, geoboards facilitate understanding of fractions by partitioning shapes and practicing measurement through area and perimeter calculations. Moreover, students engage in problem-solving activities and unleash their creativity as they design geometric patterns and solve puzzles, making geoboards versatile tools for interactive and exploratory learning in mathematics.

26. Use LEGO bricks

Grade level: Grades 3-5

Students can use LEGO bricks to enhance their understanding of math and use them similarly to base ten blocks .

How to play: By building structures with LEGO bricks, students can explore concepts such as halves, thirds, fourths, and more by partitioning bricks into equal parts. They can create models where different colored bricks represent different fractions, allowing them to see and compare fractional relationships. 

Additionally, students can use LEGO bricks to perform operations with fractions, such as adding, subtracting, multiplying, and dividing, by combining or separating brick groups. This hands-on approach with LEGO bricks provides a concrete and tangible way for students to grasp abstract fraction concepts, fostering deeper comprehension and retention.

Teaching tips for effective whole class math games

Implementing whole classroom math games presents various challenges, necessitating careful lesson plans and delivery. Below are 7 teaching tips for effective whole class math games:

• Give clear directions: this is crucial to ensure students understand the game rules and objectives.
• Use grouping effectively: this will promote collaboration and engagement. 
• Incorporate grade-level appropriate math problems : this maintains relevance and fosters learning. 
• Diversify game formats, including board games, card games , and digital games: this will accommodate different learning preferences and enhance student interest. 
• Ensure equal participation: teachers will need to monitor carefully to involve all students actively.
• Assess and monitor progress: this enables teachers to tailor instruction and provide timely feedback, reinforcing learning. 
• Consider cognitive load theory : Reduce the complexity of tasks and optimize students’ cognitive resources for effective learning. If the game is too complicated, students may be unable to effectively solidify math concepts.

Successful implementation of whole class math games hinges on addressing these challenges through strategic planning, differentiated instruction, and ongoing support for student engagement and learning.

The array of whole class math games presented in this article reflects the growing trend of gamification in education, offering educators valuable tools to enhance student engagement and comprehension. 

By infusing traditional lessons with interactive elements such as competition, rewards, and collaboration, whole class math games motivate students while reinforcing mathematical concepts. We hope you found inspiration for your classroom, regardless of the grade level or math ability of your students.

Looking for more math games for your classroom?

• Math games for 2nd grade  
• Math games for 3rd grade
• Math games for 4th grade  
• Math games for 5th grade
• Math games for 6th grade
• Math games for 7th grade 
• Math games for 8th grade

Whole class math games FAQ

Some examples of whole classroom math games include Jeopardy, BINGO, relay races, trashketball, and a scavenger hunt.

Three quick whole classroom math games could be clap and count, 21, or mystery number.

Almost any whole class math game can be adapted for each grade level. Some popular games among middle schoolers are Jeopardy, trashketball, scavenger hunt, and escape room.

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Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

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This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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Team Work makes the Math Work

I don't need to tell you that the pandemic has had some significate affects in our students learning, especially in math. Analysis shows that on average students are 5 months behind in math learning and that number increases for historically disadvantaged students to as 7 months of learning loss.  Now that students are back in the classroom one step we can take to increase math learning is to make math a group or team activity.  Group work helps student improve their critical thinking and problem-solving skills; it helps them express their understanding in a safe atmosphere that is interactive supportive and efficient.  Groupwork helps students use mathematical vocabulary and incorporate life experiences into their understanding of the math problem. 

A few things to think about when doing group math work.

1.    Grouping students: A group is a social unit that is comprised of two or more people who have a common goal. Here are three different way to group students and the pros and cons to each.

• Student Choice –

                                                   i.     Pros: positive attitudes towards each other – higher outcomes – less planning

                                                 ii.     Cons: less diversity – some students not selected – off task talking

• Random Grouping –

                                                   i.     Pros: less planning – group diversity – builds communication skills

                                                 ii.     Cons: negative attitude about selection – lower group cohesion

• Educator Selected Academics –

                                                   i.     Pros: organized based on needs, improves dept of learning

                                                 ii.     Cons: widens achievement gaps, planning, labeling

• Group roles: Students should know that they will not be able so succeed on their own unless the others will.  Students should have roles to help in the solving of the problem, these roles should change as students form new groups.
• Leader – keeps students on task, delegates
• Recorder – writes out responses
• Challenger – questions
• Thinker – produces ideas
• Supporter – eases tension and promotes people ideas
• Math problems
• Multi-step to encourage deeper thinking
• Relatable so students  understand the necessity of learning (if children are in the problem have      the students change the names to their own)
• Provide information on problem solving strategies so they can select the best way to find a solution.

        Problem solving strategies

1.    Guess and Check

2.    Draw a picture

3.    Act it out

4.    Create a grid

5.    Find a pattern

6.    Use logic

7.    Work backwards

8.    Simplify problem  

These steps will help student improve their math understanding quicker than any worksheet or web-game. Check out MANGO Math for fun hands-on math games that engage and educated students while they enjoy math.

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Center for Teaching

Group work: using cooperative learning groups effectively.

Many instructors from disciplines across the university use group work to enhance their students’ learning. Whether the goal is to increase student understanding of content, to build particular transferable skills, or some combination of the two, instructors often turn to small group work to capitalize on the benefits of peer-to-peer instruction. This type of group work is formally termed cooperative learning, and is defined as the instructional use of small groups to promote students working together to maximize their own and each other’s learning (Johnson, et al., 2008).

Cooperative learning is characterized by positive interdependence, where students perceive that better performance by individuals produces better performance by the entire group (Johnson, et al., 2014). It can be formal or informal, but often involves specific instructor intervention to maximize student interaction and learning. It is infinitely adaptable, working in small and large classes and across disciplines, and can be one of the most effective teaching approaches available to college instructors.

What can it look like?

What’s the theoretical underpinning, is there evidence that it works.

• What are approaches that can help make it effective?

Informal cooperative learning groups In informal cooperative learning, small, temporary, ad-hoc groups of two to four students work together for brief periods in a class, typically up to one class period, to answer questions or respond to prompts posed by the instructor.

Additional examples of ways to structure informal group work

Think-pair-share

The instructor asks a discussion question. Students are instructed to think or write about an answer to the question before turning to a peer to discuss their responses. Groups then share their responses with the class.

Peer Instruction

This modification of the think-pair-share involves personal responses devices (e.g. clickers). The question posted is typically a conceptually based multiple-choice question. Students think about their answer and vote on a response before turning to a neighbor to discuss. Students can change their answers after discussion, and “sharing” is accomplished by the instructor revealing the graph of student response and using this as a stimulus for large class discussion. This approach is particularly well-adapted for large classes.

In this approach, groups of students work in a team of four to become experts on one segment of new material, while other “expert teams” in the class work on other segments of new material. The class then rearranges, forming new groups that have one member from each expert team. The members of the new team then take turns teaching each other the material on which they are experts.

Formal cooperative learning groups

In formal cooperative learning students work together for one or more class periods to complete a joint task or assignment (Johnson et al., 2014). There are several features that can help these groups work well:

• The instructor defines the learning objectives for the activity and assigns students to groups.
• The groups are typically heterogeneous, with particular attention to the skills that are needed for success in the task.
• Within the groups, students may be assigned specific roles, with the instructor communicating the criteria for success and the types of social skills that will be needed.
• Importantly, the instructor continues to play an active role during the groups’ work, monitoring the work and evaluating group and individual performance.
• Instructors also encourage groups to reflect on their interactions to identify potential improvements for future group work.

This video shows an example of formal cooperative learning groups in David Matthes’ class at the University of Minnesota:

There are many more specific types of group work that fall under the general descriptions given here, including team-based learning , problem-based learning , and process-oriented guided inquiry learning .

The use of cooperative learning groups in instruction is based on the principle of constructivism, with particular attention to the contribution that social interaction can make. In essence, constructivism rests on the idea that individuals learn through building their own knowledge, connecting new ideas and experiences to existing knowledge and experiences to form new or enhanced understanding (Bransford, et al., 1999). The consideration of the role that groups can play in this process is based in social interdependence theory, which grew out of Kurt Koffka’s and Kurt Lewin’s identification of groups as dynamic entities that could exhibit varied interdependence among members, with group members motivated to achieve common goals. Morton Deutsch conceptualized varied types of interdependence, with positive correlation among group members’ goal achievements promoting cooperation.

Lev Vygotsky extended this work by examining the relationship between cognitive processes and social activities, developing the sociocultural theory of development. The sociocultural theory of development suggests that learning takes place when students solve problems beyond their current developmental level with the support of their instructor or their peers. Thus both the idea of a zone of proximal development, supported by positive group interdependence, is the basis of cooperative learning (Davidson and Major, 2014; Johnson, et al., 2014).

Cooperative learning follows this idea as groups work together to learn or solve a problem, with each individual responsible for understanding all aspects. The small groups are essential to this process because students are able to both be heard and to hear their peers, while in a traditional classroom setting students may spend more time listening to what the instructor says.

Cooperative learning uses both goal interdependence and resource interdependence to ensure interaction and communication among group members. Changing the role of the instructor from lecturing to facilitating the groups helps foster this social environment for students to learn through interaction.

David Johnson, Roger Johnson, and Karl Smith performed a meta-analysis of 168 studies comparing cooperative learning to competitive learning and individualistic learning in college students (Johnson et al., 2006). They found that cooperative learning produced greater academic achievement than both competitive learning and individualistic learning across the studies, exhibiting a mean weighted effect size of 0.54 when comparing cooperation and competition and 0.51 when comparing cooperation and individualistic learning. In essence, these results indicate that cooperative learning increases student academic performance by approximately one-half of a standard deviation when compared to non-cooperative learning models, an effect that is considered moderate. Importantly, the academic achievement measures were defined in each study, and ranged from lower-level cognitive tasks (e.g., knowledge acquisition and retention) to higher level cognitive activity (e.g., creative problem solving), and from verbal tasks to mathematical tasks to procedural tasks. The meta-analysis also showed substantial effects on other metrics, including self-esteem and positive attitudes about learning. George Kuh and colleagues also conclude that cooperative group learning promotes student engagement and academic performance (Kuh et al., 2007).

Springer, Stanne, and Donovan (1999) confirmed these results in their meta-analysis of 39 studies in university STEM classrooms. They found that students who participated in various types of small-group learning, ranging from extended formal interactions to brief informal interactions, had greater academic achievement, exhibited more favorable attitudes towards learning, and had increased persistence through STEM courses than students who did not participate in STEM small-group learning.

The box below summarizes three individual studies examining the effects of cooperative learning groups.

What are approaches that can help make group work effective?

Preparation

Articulate your goals for the group work, including both the academic objectives you want the students to achieve and the social skills you want them to develop.

Determine the group conformation that will help meet your goals.

• In informal group learning, groups often form ad hoc from near neighbors in a class.
• In formal group learning, it is helpful for the instructor to form groups that are heterogeneous with regard to particular skills or abilities relevant to group tasks. For example, groups may be heterogeneous with regard to academic skill in the discipline or with regard to other skills related to the group task (e.g., design capabilities, programming skills, writing skills, organizational skills) (Johnson et al, 2006).
• Groups from 2-6 are generally recommended, with groups that consist of three members exhibiting the best performance in some problem-solving tasks (Johnson et al., 2006; Heller and Hollabaugh, 1992).
• To avoid common problems in group work, such as dominance by a single student or conflict avoidance, it can be useful to assign roles to group members (e.g., manager, skeptic, educator, conciliator) and to rotate them on a regular basis (Heller and Hollabaugh, 1992). Assigning these roles is not necessary in well-functioning groups, but can be useful for students who are unfamiliar with or unskilled at group work.

Choose an assessment method that will promote positive group interdependence as well as individual accountability.

• In team-based learning, two approaches promote positive interdependence and individual accountability. First, students take an individual readiness assessment test, and then immediately take the same test again as a group. Their grade is a composite of the two scores. Second, students complete a group project together, and receive a group score on the project. They also, however, distribute points among their group partners, allowing student assessment of members’ contributions to contribute to the final score.
• Heller and Hollabaugh (1992) describe an approach in which they incorporated group problem-solving into a class. Students regularly solved problems in small groups, turning in a single solution. In addition, tests were structured such that 25% of the points derived from a group problem, where only those individuals who attended the group problem-solving sessions could participate in the group test problem.  This approach can help prevent the “free rider” problem that can plague group work.
• The University of New South Wales describes a variety of ways to assess group work , ranging from shared group grades, to grades that are averages of individual grades, to strictly individual grades, to a combination of these. They also suggest ways to assess not only the product of the group work but also the process.  Again, having a portion of a grade that derives from individual contribution helps combat the free rider problem.

Helping groups get started

Explain the group’s task, including your goals for their academic achievement and social interaction.

Explain how the task involves both positive interdependence and individual accountability, and how you will be assessing each.

Assign group roles or give groups prompts to help them articulate effective ways for interaction. The University of New South Wales provides a valuable set of tools to help groups establish good practices when first meeting. The site also provides some exercises for building group dynamics; these may be particularly valuable for groups that will be working on larger projects.

Monitoring group work

Regularly observe group interactions and progress , either by circulating during group work, collecting in-process documents, or both. When you observe problems, intervene to help students move forward on the task and work together effectively. The University of New South Wales provides handouts that instructors can use to promote effective group interactions, such as a handout to help students listen reflectively or give constructive feedback , or to help groups identify particular problems that they may be encountering.

Assessing and reflecting

In addition to providing feedback on group and individual performance (link to preparation section above), it is also useful to provide a structure for groups to reflect on what worked well in their group and what could be improved. Graham Gibbs (1994) suggests using the checklists shown below.

The University of New South Wales provides other reflective activities that may help students identify effective group practices and avoid ineffective practices in future cooperative learning experiences.

Bransford, J.D., Brown, A.L., and Cocking, R.R. (Eds.) (1999). How people learn: Brain, mind, experience, and school . Washington, D.C.: National Academy Press.

Bruffee, K. A. (1993). Collaborative learning: Higher education, interdependence, and the authority of knowledge. Baltimore, MD: Johns Hopkins University Press.

Cabrera, A. F., Crissman, J. L., Bernal, E. M., Nora, A., Terenzini, P. T., & Pascarella, E. T. (2002). Collaborative learning: Its impact on college students’ development and diversity. Journal of College Student Development, 43 (1), 20-34.

Davidson, N., & Major, C. H. (2014). Boundary crossing: Cooperative learning, collaborative learning, and problem-based learning. Journal on Excellence in College Teaching, 25 (3&4), 7-55.

Dees, R. L. (1991). The role of cooperative leaning in increasing problem-solving ability in a college remedial course. Journal for Research in Mathematics Education, 22 (5), 409-21.

Gokhale, A. A. (1995). Collaborative Learning enhances critical thinking. Journal of Technology Education, 7 (1).

Heller, P., and Hollabaugh, M. (1992) Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups. American Journal of Physics 60, 637-644.

Johnson, D.W., Johnson, R.T., and Smith, K.A. (2006). Active learning: Cooperation in the university classroom (3 rd edition). Edina, MN: Interaction.

Johnson, D.W., Johnson, R.T., and Holubec, E.J. (2008). Cooperation in the classroom (8 th edition). Edina, MN: Interaction.

Johnson, D.W., Johnson, R.T., and Smith, K.A. (2014). Cooperative learning: Improving university instruction by basing practice on validated theory. Journl on Excellence in College Teaching 25, 85-118.

Jones, D. J., & Brickner, D. (1996). Implementation of cooperative learning in a large-enrollment basic mechanics course. American Society for Engineering Education Annual Conference Proceedings.

Kuh, G.D., Kinzie, J., Buckley, J., Bridges, B., and Hayek, J.C. (2007). Piecing together the student success puzzle: Research, propositions, and recommendations (ASHE Higher Education Report, No. 32). San Francisco, CA: Jossey-Bass.

Love, A. G., Dietrich, A., Fitzgerald, J., & Gordon, D. (2014). Integrating collaborative learning inside and outside the classroom. Journal on Excellence in College Teaching, 25 (3&4), 177-196.

Smith, M. E., Hinckley, C. C., & Volk, G. L. (1991). Cooperative learning in the undergraduate laboratory. Journal of Chemical Education 68 (5), 413-415.

Springer, L., Stanne, M. E., & Donovan, S. S. (1999). Effects of small-group learning on undergraduates in science, mathematics, engineering, and technology: A meta-analysis. Review of Educational Research, 96 (1), 21-51.

Uribe, D., Klein, J. D., & Sullivan, H. (2003). The effect of computer-mediated collaborative learning on solving ill-defined problems. Educational Technology Research and Development, 51 (1), 5-19.

Vygotsky, L. S. (1962). Thought and Language. Cambridge, MA: MIT Press.

Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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30 Thought-Provoking Math Puzzles for Middle Schoolers

Critical thinking, trial and error, and pure logic abound.

Tired of your tried-and-true math routine? Chances are if you’re feeling the itch to incorporate new activities into your math time, your students are as well. Mixing it up in math class can bring fresh perspectives to stale concepts or standards, and your students will enjoy stretching their brains in different ways with these middle school math puzzles. Critical thinking, trial and error, and pure logic abound in these 30 though-provoking puzzles. Get ready to reignite your middle schoolers’ excitement for math!

(Just a heads up, WeAreTeachers may collect a share of sales from the links on this page. We only recommend items our team loves!)

Sudoku is way more than just an activity to pass the time on long-haul flights. This math puzzle is actually a fantastic problem-solving activity for middle schoolers. Kick-starting your typical math class with a Sudoku puzzle will have your students thinking critically, practicing trial and error, and looking at math in a totally different way. Plus, you can differentiate by providing Easy, Medium, and Difficult puzzles.

Learn more: Sodoku Puzzles To Print

2. 5 Pirates Puzzle

Ahoy and shiver me timbers! This logic puzzle is perfect for a small-group activity to get your middle schoolers working together to solve the conundrum of how pirates plan to share treasure among themselves. Multiple scenarios will play out in this puzzle, so scaffolding with problem-solving strategies is a must.

Learn more:  5 Pirates Puzzles/Math Is Fun

3. Fives Challenge Puzzle

This puzzle is perfect for reviewing addition, multiplication, division, and subtraction and would be a great activity to do when gearing up to teach order of operations. Students could work in pairs or small groups to riddle out each target number.

Learn more:  Fives Challenge Puzzle/Math = Love

4. Beehive Puzzle

Perfect for a station during math rotation or for a rainy-day recess activity, this logic puzzle involves creating a beehive shape without having any squares of the same color touching each other. Students can practice trial and error as well as problem-solving.

Learn more:  Beehive Puzzle/Math = Love

5. Guess My Number

Guess My Number is just as much a riddle as it is a math puzzle. Students use their number sense to determine the number in question. As an extension activity, students can come up with their own clues and trade them with a classmate to solve.

Learn more:  Guess My Number/Education.com

6. Math Riddles

Perfect for a morning warmup, these middle school math puzzles activate all kinds of math knowledge. You can poll the class and have them show their work before clicking to reveal the correct answer. This site even has more challenging puzzles if your middle schoolers fly through the easier ones.

Learn more:  Math Riddles/Get Riddles

My seventh graders loved playing this puzzle as an early-finisher activity. Though the idea is simple (move the tiles until two of the same numbers touch), it’s actually great for recognizing exponents and also for thinking strategically.

Learn more:  2048/Prodigy

8. Magic Squares

Magic Squares have been around for thousands of years, and they come in all shapes and sizes. The 3×3 grid is a great size to introduce to your students and then work up to larger and more complex grids. You can even bring this puzzle off the paper and have your students write the grid out in sidewalk chalk, or write the numbers on water bottle caps to make a fun tactile activity.

Learn more:  Magic Squares/Prodigy

9. Impossible Domino Bridge

Using dominoes to build a seemingly impossible bridge is a perfect activity for the first day or week of a new school year. Your students can work together in small groups and get to know one another as they attempt to construct the bridge that looks like it could turn into a game of Jenga at any moment.

Learn more:  Impossible Domino Bridge/Math = Love

10. Math Picture Puzzles

Your students communicate through emojis anyway, so why not get math involved? This self-checking site allows them to work independently (on the honor system) and also choose between three levels of difficulty. Students can take this idea to the next level, create their own emojis, and arrange them in number sentences for their classmates to solve.

Learn more: Picture Puzzles/MathEasily.com

11. What Is the Weight?

Sometimes you just need a quick resource to get your students working on solving a math puzzle. This puzzle comes from an app, so you can have it downloaded on your students’ iPads or tablets. Middle schoolers will focus on determining the weights of different animals, which is good practice for estimating and working with customary/metric units of measurement.

Learn more: Brain Teasers/Mental Up

12. Colorku

Math doesn’t always have to be just about numbers. This board game uses colors and patterns to focus on analyzing sequences, and would be great to have on hand for those rainy-day recesses as well as for inclusion in a math station. Further, Colorku can be used as a calm-down tool or even a fidget tool.

Buy it: Colorku at Amazon

13. Rubik’s Cube

Rubik’s Cubes made a major comeback in popularity when I taught fifth grade. My students would happily sit together at recess to race each other to see who could solve the cube faster. Though entertaining, Rubik’s Cubes are also suited to teach students about growth mindset, spacial awareness, and 3D space.

Buy it: Rubik’s Cube at Amazon

14. SafeCracker

Though this puzzle looks like something out of an Indiana Jones quest, it’s actually a tactilely engaging tool that will delight even your most resistant math learners. The goal is to align the wheel into columns where the sum adds up to 40. You might need to get more than one of these middle school math puzzles for your classroom.

Buy it: SafeCracker at Amazon

15. “T” Brain Teaser Puzzle

In addition to sparking structural design creativity, this boxed wooden puzzle challenges middle schoolers to engage in trial and error as they work at fitting 50+ pieces into a cube. Much of math is learning how to persevere through tricky problems or procedures, and this puzzle definitely fosters that.

Buy it: T Brain Teaser at Amazon

16. Multistep Equation Puzzle

Solve-and-sort puzzles add flair to repeatedly solving different variations of a math problem for practice. In this free puzzle, students will need to not only solve the equations with variables on both sides, they will also need to sort the problem based on if their solution is positive or negative in order to uncover the secret word.

Get it: Solve-and-Sort Puzzle/Teachers Pay Teachers

In this variation of a classic Sudoku puzzle, students practice critical thinking and exercise their knowledge of how the four math operations work. The best thing about these types of puzzles is that the differentiation potential is endless. Students can solve smaller puzzles with addition, or use only prime numbers in a more complex multiplication problem.

Learn more: Yohaku

18. Jigmaze

One of the Standards for Mathematical Practices is perseverance, and all teachers know that this is a tough one to instill in students, even more so if students are struggling in foundational skills. This type of puzzle can be used to strengthen perseverance as students physically arrange and rearrange pieces of a broken maze.

Learn more: Jigmaze/Math = Love

19. Flexagons

Flexagons, octaflexagons, and dodecaflexagons (say that one 10 times fast!) are a mathematical take on traditional origami. Through constructing these paper creations, your students will get exposure to geometrical terms such as faces ,  equilateral triangles , and all manner of types of 3D shapes.

Get it: Flexagons/Medium

20. Möbius Strip

Though the high-level mathematical equation may be well above your students’ heads (and mine too, if I’m being honest), the STEAM-centered concept of a Möbius strip can be a fun one to explore and create (no need to go into cosines and conversational belts). Middle school math puzzles for the win!

Get it for free: Make a Möbius/STEAMsational

In this complex-looking puzzle, the goal is for the sum of each vertical or horizontal line to match the number given at the beginning of the row or column. This site comes with a great explanation on exactly what that means and how to achieve it. A Kakuro puzzle would be a great “learn as you go” activity for students where they really must pay close attention to the instructions to be able to understand the goal.

Learn more: Kakuro/Braingle

22. Number Searches

This school district’s site has tons of grade-specific number puzzles that would be perfect for when you need to be out of the classroom and have a substitute teacher. They are ready to be printed and contain easy explanations for your students. Check out the number searches, patterns, and 3D riddles.

Learn more: Number Searches/Cranbury School District

23. Two Truths and One Lie

The tried-and-true icebreaker used at many a staff meeting and the first week of school, Two Truths and One Lie can also be used to review and practice tons of mathematical concepts. These middle school math puzzles cover concepts such as negative numbers, fractions, and a ton more.

Buy it: Two Truths & One Lie Math Edition at Amazon

24. Adding Integers Puzzle

The objective of this cuttable resource is for students to solve the integer problem and match up expressions that end up having the same sum. The multiple size options are great for differentiation or to make this independent activity into a small-group collaborative activity.

Buy it: Adding Integers at Teachers Pay Teachers

25. Perfect Square Roots

For upper middle school students, this square-roots puzzle helps with the recognition of perfect square roots. Rather than simply memorizing the perfect square roots, students work to identify and spell out the specific square root and ensure that it fits within the crossword. In this way, the puzzle is self-checking as well.

Buy it: Square Roots Crossword at Teachers Pay Teachers

26. Factor Tree Challenge

Factor trees are an effective way to visually show students the factors of numbers. Trees allow a chain of multiple factors, so you can start with a large number and end up with “branches” that show all of the factors. Once your middle schoolers are familiar with this concept, have them explore this self-checking challenge (and many others as well) that will test their knowledge of abstract factors.

Learn more: Prime Challenges/Transum

27. Ludicross

Another take on Sudoku, Ludicross is interactive in that students can drag and drop the number into position with the goal of making the sum of the numbers in both diagonals the same. Like several of the other puzzles mentioned in this list, students can take this number puzzle to the next level by creating their own and swapping with a classmate to solve.

Learn more: Ludicross/Transum

28. Interactive Mobiles

These colorfully shaped mobiles are a unique way for students to make pattern associations. Because these puzzles are self-paced, students can begin with a simple puzzle and work their way up to complex mobiles with three or more shapes.

Try it: Mobiles/SolveMe Puzzles

29. Deleting Sheep

This logic puzzle is a doozy! The objective is to remove only two numbers in each row with the result being that each horizontal and vertical line equals 30. Trial and error and problem-solving skills abound in this puzzle, and it will keep your middle schoolers engaged for quite some time.

Get it: Deleting Sheep/Dover Publications

30. Pips Puzzle

Have any spare decks of cards lying around your classroom? This inexpensive item provides a different take on a Magic Square. Students can work in small groups, and maybe you can ignite a little class competition to see which groups can complete the challenge the fastest.

Buy it: Pips Puzzle/Math = Love

Looking for more engaging math resources? Try these Magical Math Puzzles and Number Tricks To Wow Your Students .

Plus, get all the latest teaching tips and tricks when you sign up for our free newsletters .

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Group Work That Works

Educators weigh in on solutions to the common pitfalls of group work.

Mention group work and you’re confronted with pointed questions and criticisms. The big problems, according to our audience: One or two students do all the work; it can be hard on introverts; and grading the group isn’t fair to the individuals.

But the research suggests that a certain amount of group work is beneficial.

“The most effective creative process alternates between time in groups, collaboration, interaction, and conversation... [and] times of solitude, where something different happens cognitively in your brain,” says Dr. Keith Sawyer, a researcher on creativity and collaboration, and author of Group Genius: The Creative Power of Collaboration .

So we looked through our archives and reached out to educators on Facebook to find out what solutions they’ve come up with for these common problems.

Making Sure Everyone Participates

“How many times have we put students in groups only to watch them interact with their laptops instead of each other? Or complain about a lazy teammate?” asks Mary Burns, a former French, Latin, and English middle and high school teacher who now offers professional development in technology integration.

Unequal participation is perhaps the most common complaint about group work. Still, a review of Edutopia’s archives—and the tens of thousands of insights we receive in comments and reactions to our articles—revealed a handful of practices that educators use to promote equal participation. These involve setting out clear expectations for group work, increasing accountability among participants, and nurturing a productive group work dynamic.

Norms: At Aptos Middle School in San Francisco, the first step for group work is establishing group norms. Taji Allen-Sanchez, a sixth- and seventh-grade science teacher, lists expectations on the whiteboard, such as “everyone contributes” and “help others do things for themselves.”

For ambitious projects, Mikel Grady Jones, a high school math teacher in Houston, takes it a step further, asking her students to sign a group contract in which they agree on how they’ll divide the tasks and on expectations like “we all promise to do our work on time.” Heather Wolpert-Gawron, an English middle school teacher in Los Angeles, suggests creating a classroom contract with your students at the start of the year, so that agreed-upon norms can be referenced each time a new group activity begins.

Group size: It’s a simple fix, but the size of the groups can help establish the right dynamics. Generally, smaller groups are better because students can’t get away with hiding while the work is completed by others.

“When there is less room to hide, nonparticipation is more difficult,” says Burns. She recommends groups of four to five students, while Brande Tucker Arthur, a 10th-grade biology teacher in Lynchburg, Virginia, recommends even smaller groups of two or three students.

Meaningful roles: Roles can play an important part in keeping students accountable, but not all roles are helpful. A role like materials manager, for example, won’t actively engage a student in contributing to a group problem; the roles must be both meaningful and interdependent.

At University Park Campus School , a grade 7–12 school in Worcester, Massachusetts, students take on highly interdependent roles like summarizer, questioner, and clarifier. In an ongoing project, the questioner asks probing questions about the problem and suggests a few ideas on how to solve it, while the clarifier attempts to clear up any confusion, restates the problem, and selects a possible strategy the group will use as they move forward.

At Design 39, a K–8 school in San Diego, groups and roles are assigned randomly using Random Team Generator , but ClassDojo , Team Shake , and drawing students’ names from a container can also do the trick. In a practice called vertical learning, Design 39 students conduct group work publicly, writing out their thought processes on whiteboards to facilitate group feedback. The combination of randomizing teams and public sharing exposes students to a range of problem-solving approaches, gets them more comfortable with making mistakes, promotes teamwork, and allows kids to utilize different skill sets during each project.

Rich tasks: Making sure that a project is challenging and compelling is critical. A rich task is a problem that has multiple pathways to the solution and that one person would have difficulty solving on their own.

In an eighth-grade math class at Design 39, one recent rich task explored the concept of how monetary investments grow: Groups were tasked with solving exponential growth problems using simple and compound interest rates.

Rich tasks are not just for math class. When Dan St. Louis, the principal of University Park, was a teacher, he asked his English students to come up with a group definition of the word Orwellian . They did this through the jigsaw method, a type of grouping strategy that John Hattie’s study Visible Learning ranked as highly effective.

“Five groups of five students might each read a different news article about the modern world,” says St. Louis. “Then each student would join a new group of five where they need to explain their previous group’s article to each other and make connections to each. Using these connections, the group must then construct a definition of the word Orwellian .” For another example of the jigsaw approach, see this video from Cult of Pedagogy.

Supporting Introverts

Teachers worry about the impact of group work on introverts. Some of our educators suggest that giving introverts choice in who they’re grouped with can help them feel more comfortable.

“Even the quietest students are usually comfortable and confident when they are with peers with whom they connect,” says Shelly Kunkle, a veteran teacher at Wasawee Middle School in North Webster, Indiana. Wolpert-Gawron asks her students to list four peers they want to work with and then makes sure to pair them with one from their list.

Having defined roles within groups—like clarifier or questioner—also provides structure for students who may be less comfortable within complex social dynamics, and ensures that introverts don’t get overshadowed by their more extroverted peers.

Finally, be mindful that introverted students often simply need time to recharge. “Many introverts do not mind and even enjoy interacting in groups as long as they get some quiet time and solitude to recharge. It’s not about being shy or feeling unsafe in a large group,” says Barb Larochelle, a recently retired high school English teacher in Edmonton, Alberta, who taught for 29 years.

“I planned classes with some time to work quietly alone, some time to interact in smaller groups or as a whole class, and some time to get up and move around a little. A whole class of any one of those is going to be hard on one group, but a balance works well.”

Assessing Group Work

Grading group work is problematic. Often, you don’t have a clear understanding of what each student knows, and a single student’s lack of effort can torpedo the group grade. To some degree, strategies that assign meaningful roles or that require public presentations from groups provide a window in to each student’s knowledge and contributions.

But not all classwork needs to be graded. Suzanna Kruger, a high school science teacher in Seaside, Oregon, doesn’t grade group work—there are plenty of individual assignments that receive grades, and plenty of other opportunities for formative assessment.

John McCarthy, a former high school English and social studies teacher and current education consultant and adjunct professor at Madonna University for the graduate department for education, suggests using group presentations or group products as a non-graded review for a test. But if you want to grade group work, he recommends making all academic assessments within group work individual assessments. For example, instead of grading a group presentation, McCarthy grades each student on an essay, which the students then use to create their group presentation.

Laura Moffit, a fifth-grade teacher in Wilmington, North Carolina, uses self and peer evaluations to shed light on how each student is contributing to group work—starting with a lesson on how to do an objective evaluation. “Just have students circle :), :|, or :( on three to five statements about each partner, anonymously,” Moffit commented on Facebook. “Then give the evaluations back to each group member. Finding out what people really think of your performance is a wake-up call.”

And Ted Malefyt, a middle school science teacher in Hamilton, Michigan, carries a clipboard with the class list formatted in a spreadsheet and walks around checking in on students while they do group work.

“Using this spreadsheet, you have your own record of which student is meeting your expectations and who needs extra help,” explains Malefyt. “As formative assessment takes place, quickly document with simple checkmarks.”

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Teachers’ noticing to promote students’ mathematical dialogue in group work

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• Published: 07 June 2022
• Volume 26 , pages 509–531, ( 2023 )

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• Marie Sjöblom   ORCID: orcid.org/0000-0002-8533-5058 1 ,
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How can teachers refine their strategies for purposefully engaging students in mathematical discussions when students are working in groups and the teacher enters an ongoing group conversation? In three educational design research cycles, four teachers collaborated with a researcher for one year to analyse, design and evaluate strategies for engaging students in small-group mathematical discussions. The idea of noticing (Mason in Researching your own practice: the discipline of noticing, RoutledgeFalmer, London, 2002; Sherin et al. in Mathematics teacher noticing: seeing through teachers’ eyes, Taylor & Francis, New York, 2011) was used to organize the findings—by paying attention to aspects in the mathematical discussions and interpreting the interactions, teachers could together refine their own actions/responses to better support students’ work. The Inquiry Co-operation Model of Alrø and Skovsmose (Dialogue and learning in mathematics education: intention, reflection, critique, Kluwer Academic Publishers, Dordrecht, 2004) was used as a theoretical base for understanding qualities in mathematical discussions. Ehrenfeld and Horn’s (Educ Stud Math 103(7):251–272, 2020) model of initiation-entry-focus-exit and participation was for interpreting and organizing the findings on teachers’ actions. The results show that teachers became more aware of the importance of explicit instructions and their own role as facilitators of mathematical questions to students, by directing specific mathematical questions to all students within the groups. In this article, by going back and forth between what happened in the teachers’ professional development group and in the classrooms, it was possible to simultaneously follow the teachers’ development processes and what changed in students’ mathematical discussions.

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Group work for mathematics learning

In recent decades, participating in mathematical discussions while working together with other students has been a preferred alternative for students to learn mathematics (Brandt & Schütte, 2010 ; Cobb et al., 2001 ; Sfard, 2015 ; Sfard & Kieran, 2001 ). The adoption of socio-cultural theories of learning (Radford, 2013 ; Sfard, 2015 ; Vygotsky, 1978 ) as well as the related emphasis in different curricula on the development of competences such as mathematical reasoning or communication have challenged silent individual work. Organized interactions between students are seen as central for learning. Arguments that teachers can use small-group work to provide the context for social and cognitive engagement with mathematics (Walshaw & Anthony, 2008a ) have also contributed to the use of group work as an important pedagogical strategy.

However, research has shown that the extent to which group work indeed leads to better mathematics learning is questionable. Deen and Zuidema ( 2008 ) found that although students have more opportunities to talk compared to whole-class discussions, “group work proves to be not a sufficient condition for learning” (p. 171). One problem is that teachers only get a glimpse of students’ learning during small-group work when they enter an ongoing conversation. For instance, Esmonde and Langer-Osuna ( 2013 ), discussed the difference between group work when a teacher is present compared to when it is student-led. Teachers get little insight into what happens when they are not present, and it is hard to gain an oversight of all groups in a classroom simultaneously. Furthermore, from the students’ point of view, group work is not always seen positively. Fuentes ( 2013 ) found that student interaction is affected by lack of communication between all students in a group, poor communication patterns, and other norms that impact students’ learning negatively. Horn ( 2017 ) concluded that the learning students can gain from group work should be weighed against, for instance, the social risks that students might be exposed to. One problem could be that certain students are left out of conversations, or do not participate actively, and hence have fewer opportunities to learn (Barnes, 2005 ; Cohen, 1994 ).

The role of the teacher is important for mathematical group work to promote meaningful conversations and support students’ learning through monitoring student group work (Ehrenfeld & Horn, 2020 ; Stein et al., 2008 ). So, what can teachers do to support group work? Effective group work has many aspects to take into consideration, for instance:

teachers need to select appropriate tasks that allow all students access to the mathematics; use instructional strategies that prompt participation by all students; and support high quality mathematics conversations within groups. Not accomplishing any one of these can exclude some students from participating in interactions necessary to support their learning. (Staples, 2008 , p. 351)

Hintz and Tyson ( 2015 ) also found that teachers can model behaviour and encourage students by being curious about their mathematical thinking and asking questions to amplify their ideas; and Esmonde and Langer-Osuna ( 2013 ) stressed the importance of teachers carefully listening to student groups to learn more about their interactional styles as well as their mathematical thinking. When students explain their thinking, they get the opportunity to “hear and respond to one another’s ideas about the mathematics” (Hintz & Tyson, 2015 , p. 305).

In spite of their documented importance, teachers’ routines for monitoring student small-group work are understudied (Ehrenfeld & Horn, 2020 ). More generally, research on small-group mathematical conversations in upper secondary school is limited (Walshaw and Antony 2008b ). Staples ( 2008 ) concluded that more research is needed on effective use of group work, especially at the high school level.

Although group work is both considered important and is practiced in many classrooms, the question of what makes learning together in small groups powerful and durable still remains open (Sfard, 2015 ), and it is not yet researched enough if and how students can benefit from dialogic teaching (Resnick et al., 2019 ). In particular, it is an open question how teachers can promote discussions in mathematical small-group work, and what can be done to support teachers in doing so.

Aim and research question

This article aims to contribute to the research on how teachers can promote discussions in mathematical small-group work. This is done by focussing on the process of change in teachers’ awareness as the teachers move back and forth between their classrooms and students’ group discussions, and a professional development group in which their teaching was discussed. Frequently research concentrates its gaze in one main site of practice: the classroom to study students, teachers or their interactions; or spaces of teachers’ professional development and their relations to colleagues and researchers (Lefstein et al., 2020 ). In this article, the moving between these settings allows us to capture how teachers’ changed awareness takes place through the process of noticing (Mason, 2002 ; Sherin et al., 2011 ). Noticing is often defined by three parts – attending to what is happening, interpreting it, and deciding how to act/respond (Jacobs et al., 2011 ; Kazemi et al., 2011 ). By noticing what happens in the classroom, teachers can create a “movement or shift of attention” (Mason, 2011 , p. 45). However, as much research using noticing is done for pre-service teachers, focussing cognitive processes and not how in-service teachers can respond to situations or make decisions within the classroom (Santagata et al., 2021 ), there is a need to better understand how this movement of teachers’ attention can be implemented.

In this article, the process of noticing is used in two settings that complement each other. In the classroom setting, teachers pay attention to certain pre-decided aspects of mathematical discussions, interpret them and decide how to act/respond. However, noticing is also used outside the classroom, in a professional development group setting. Here attention is directed towards aspects of the mathematical discussions. The latter are reviewed and analysed through video recordings, and the spontaneous interpretations from the classroom are complemented by interpretations connected to theories and reflections within the teacher group. Decisions on how to promote student interactions are discussed in a cyclic process in cooperation between teachers and researcher.

By going back and forth between these two settings, it is possible to follow how teachers’ changes in awareness affect how students’ interactions are promoted and vice versa. The teachers’ analysis of students’ interactions can help teachers understand more about their roles as teachers. Following teachers’ encounters with students’ mathematical conversations in small groups can contribute to changes in teachers’ roles and deliberate actions in students’ group work (Goldsmith & Seago, 2011 ).

The research question guiding the study is: how can teachers refine their strategies for purposefully engaging students in mathematical discussions when students are working in groups and the teachers enter an ongoing group conversation?

Supporting and understanding mathematical interactions in group work

In order to understand how teachers can promote mathematical discussions, two analytical frameworks were connected to the noticing processes and used in this study: the Inquiry Co-operation model (IC-model) of Alrø and Skovsmose ( 2004 ), and the initiation-entry-focus-exit and participation framework, in this article called the Teacher Moves model (TM-model) of Ehrenfeld and Horn ( 2020 ). The notion of teachers’ noticing is central in articulating the way teachers were moving between settings. Even though noticing rarely is used in educational design research or connected to other theoretical frameworks from mathematics education (Santagata et al., 2021 ), here it will be used to analyse the changes in teachers’ awareness with support of the two analytical frameworks. Before going into details about the study, we briefly present the frameworks.

The IC-model: dialogic acts to understand interactions

One way of understanding how conversations in the classroom produce opportunities to learn mathematics is proposed by Alrø and Skovsmose ( 2004 ) with their Inquiry Co-operation Model, which captures the characteristics of mathematical dialogue. The model contains eight dialogic acts (written in italics in this article) which support mathematics learning when used actively as part of a conversation: getting-in-contact (preparing for interaction), locating (understanding the problem), identifying (finding the mathematics in the problem), advocating (examining ideas), thinking aloud (making perspectives and thoughts visible), reformulating (clarifying and rephrasing), challenging (questioning ideas) and evaluating (looking back at the problem).

In previous studies (e.g., Alrø & Skovsmose, 2004 ; Mueller et al., 2020 ; Sjöblom, 2015 ), the IC-model has been used to analyse mathematical dialogue by looking at which IC-acts are present and how they affect the conversation. The IC-acts could be connected to students developing mathematical abilities and through the conversations learning mathematics. Malasari et al. ( 2020 ) used the model in relation to mathematical literacy proficiency, claiming that medium- or high-achieving students who receive teaching based on the IC-model learn more compared to conventional teaching. This was explained by the fact that these students maximised the advocating stage in exchanging ideas, asking questions and supporting each other. However, Malasari et al. ( 2020 ) found that this was not the case for low-achieving students, who were unable to maximize the advocating stage. Also, Weng and Jankvist ( 2017 ) found the IC-model especially useful for teachers when talking about mathematics to mathematically gifted students.

Likwambe ( 2018 ) used the IC-model to explore the nature of dialogue in a calculus lecture room from the lecturer’s viewpoint, where the different dialogic acts resulted in questions from the lecturer that made students think mathematically. Likwambe did not use the model from a student viewpoint.

Sjöblom ( 2018 ) used the IC-model to study how student-to-student interaction shifted focus from finding correct answers in mathematics to also trying to understand each other’s mathematical reasoning and pose questions to each other. However, when students talk to each other, it is not likely that all IC-acts are occurring (Alrø & Skovsmose, 2004 ; Sjöblom, 2015 ). Hence, teachers are left with the challenge of facilitating the emergence of a variety of dialogical acts to promote richer mathematical conversations for all students, particularly in situations where students work in groups and their conversations are expected to result in learning. Depending on many complex factors, such as the classroom context, students’ cooperation and previous knowledge, what acts to promote differ from situation to situation, classroom to classroom.

The TM-model: teachers’ moves to promote student interaction

Ehrenfeld and Horn ( 2020 ) developed a framework on initiation-entry-focus-exit and participation (in this article called the TM-model) for illuminating common moves teachers work with when they engage with groups of students solving mathematical problems. In the model, “moves” are related to actions that teachers take when interacting with students working in small groups. In the framework, there are five key categories of moves (written in bold in this article), namely:

Initiation of conversation with students

Entry into student conversations

Focus of the interactions

Exit from student conversation

Participation patterns

Ehrenfeld and Horn ( 2020 ) claimed that these moves can be used to understand the complexity of group work and inform teachers’ and researchers’ understandings of how to support students’ collaborative mathematical sense-making. However, as all classrooms are different, it is not possible to use the framework to find a model for what is the best practice for teachers. One part of the complexity also relates to the fact that when student interaction is the focus and is made an important activity in the mathematics classroom, teachers also give away a bit of their control of the lesson, as the students get to decide, for instance, what questions to ask, when to involve the teacher and how to work together in their groups.

The different moves are useful for studying what teachers do when engaging student groups. The initiation move is about how teachers approach groups and initiate conversations. The entry move is about what teachers first say when they enter the student conversation. The focus move is about what teachers focus on in the interaction with the group—is it about participation or mathematics or something else? The exit move is about how teachers exit the conversation and if they leave students with open-ended or close-ended questions. Finally, the participation move is about students’ opportunities to participate actively in conversations.

Using the frameworks together

The two frameworks have several connections. For instance, the way teachers initiate and enter conversations can be connected to the IC-act getting-in-contact . When analysing the focus of the conversations, Ehrenfeld and Horn ( 2020 ) concluded that this was not always about mathematical content, but also related to participation, instructions on tasks, or answering technical questions. When it was more about mathematics it could, for instance, be about listening or asking for a summary of the results (opening up for reformulating, challenging or evaluating ), or mathematically asking students questions/answering student questions to locate , identify , advocate or think-aloud . As regards participation , this could again be connected to getting-in-contact and involving all students in mathematical conversations.

In this article, when working with teacher noticing (Mason, 2002 ; Sherin et al., 2011 ), the two frameworks contribute in different ways to understand the interaction. The IC-model can help both to decide what to attend to in the interaction and to interpret what is going on in the mathematical discussions. The TM-model can help both interpreting and designing teachers’ actions/responses. According to Miller “attention is always limited” ( 2011 , p. 53), and teachers need to make a selection of what parts of students’ interactions to attend to, and in this article, attention is limited to the IC-acts and the TM-moves.

In the analysis, the IC-model helped characterizing and interpreting what happened in the mathematical conversations by identifying dialogic acts used by students and teachers. It was also used to help teachers identify what kind of dialogic acts they needed to attend to and promote, for instance getting-in-contact is important for participation . By working with IC-acts so that all students were encouraged to ask questions and include everyone in the discussions, the IC-model was considered a way of avoiding students being seen as outsiders (Barnes, 2005 ).

In the analysis, the TM-model was a tool to organize data and interpret the actions/responses of teachers when they engaged with student groups, for instance how they initiated conversations or how they worked with participation patterns. As it is a framework that illuminates teachers’ common moves, it was used to understand and organize the story of what happened in the classroom. Thereafter, by reflecting in the teacher professional development meetings about teachers’ and students’ interactions and connecting this to the IC-acts, conclusions could be drawn about how teachers’ actions/responses affected the conversations and what needed to be changed.

Methodology

Research design.

In this study, the methodology was chosen to help teachers notice and become more aware of how their own actions in the context of their classrooms affected students’ opportunities for mathematical discussions. Building on educational design research, EDR, (Cobb et al., 2003 ; McKenney & Reeves, 2012 ; van Den Akker et al., 2006 ), the study was conducted together with a group of four teachers. The teachers, who worked together in a public upper secondary school in a city in Sweden, volunteered to participate in the project during one school year since they wanted to learn more about interaction in mathematics. The study was led by the first author of this article, who had the double role of leading the teachers’ professional development processes and being researcher in the EDR-project.

In EDR there are three phases in each cycle: the analysis/exploration phase, the design/construction phase, and the evaluation/reflection phase (McKenney & Reeves, 2012 ). The results of EDR-projects are both theoretical as well as practical, in that they are to lead to both theoretical understandings and a maturing intervention (McKenney & Reeves, 2012 ). The group conducted three EDR-cycles during the school year, and all three cycles were closely connected to the context of teachers’ classrooms as well as their professional development group. The work was organized with clear goals and followed the structure of EDR-methodology. The overarching goal was to promote interaction with a focus on encouraging students to actively listen to each other, ask questions and present their mathematical thinking while working together in small groups. The 16–19-year-old students were attending university preparatory programmes. During the year, teachers met regularly with the first author almost every week at scheduled times during working hours and conducted tasks in their classrooms in-between.

In the teacher group, the focus was both on students’ and teachers’ actions, since being able to promote student interaction requires teachers to be aware of their own role in and impact on students’ mathematical discussions. The teachers were involved in all three phases of the EDR-cycles—actively exploring the needs of their students in group work, deciding and designing what to try out in their classrooms and taking part in analysing the results and reflecting on what to try in the next cycle. Hence, EDR was used as a way of organizing and supporting the teachers’ noticing processes. In the analysis/exploration phase, teachers attended to certain aspects of students’ interactions in the classroom, and interpreted the results both in the classroom but also together in the professional development meetings. Thereafter, in the design/construction phase they planned and implemented lesson activities as an action/response to the students’ needs. The design phase included plans for mathematical tasks, how teachers structured their lessons and gave students instructions, and how teachers talked with students in small-group conversations. Finally, in the evaluation/reflection phase, the teachers again interpreted and decided how to act/respond in the coming cycle.

Data collection

The data collection included data from both the classroom setting and the teacher professional development setting. It consisted of audio recordings of 30 teacher meetings, on average 45 min each, in which the designs were planned and analysed within the group of teachers. Two of the teachers were also video recorded when the designed lessons were conducted. Footnote 1 The teachers taught mathematics in either the Natural Science (NS) programme, year 10, or the Social Science (SS) programme, year 12. Six groups of students, usually three groups in each classroom and with every group having three or four students, were included in the recordings, with separate cameras for each group. The groups were randomly chosen amongst the students that had consented to be part in the research project. The video recordings were used in the analyses within the teacher group as a way of promoting rich discussions about productive learning environments (Schoenfeld, 2017 ). Table 1 provides a summary of the collected data.

Methods of analysis: a two-step procedure

The material was analysed in a two-step process. In the first step, parts of the classroom data were discussed during the three cycles within the teacher group. Sherin and Dyer ( 2017 ) claim that watching and discussing videos provides opportunities for teacher learning and it is also a common method in teacher noticing studies (Santagata et al., 2021 ). Hence, it was a conscious choice to watch videos together, as this supported the work of noticing student interaction. The analysis together with the teachers was done differently in each cycle, depending on the teachers’ choices about what to focus and notice. In the first cycle, the first author of this article selected and showed clips of student interactions, focussing on how they did or did not listen to each other and how the IC-act of getting-in-contact played out, since this was what the teachers wanted to attend to initially. In the second cycle, the teachers selected and watched one film each of a group work situation (approximately 1 h per film) and summarized to each other in the coming meeting how the interaction worked out. Here, the area of attention was questions that could be related to several different IC-acts. In the third cycle, short clips of what happened before, during and after the teacher entered the groups were discussed. By analysing the teachers’ objectives as they designed the activities, what took place in the classroom, and then reflecting upon what happened in the classroom within the teacher group, it was possible to understand more about what happened when a teacher entered a conversation. Again, the first author of this article selected and showed video clips of teachers with focus/attention on how teachers engaged students in group work in their classrooms as this was what teachers focused in the third cycle.

The second step of the analysis was conducted by the three authors of this article once the three cycles were completed. By identifying and selecting instances of teachers’ work with engagements in students’ conversations in the teacher meeting transcripts, a general picture of what happened when teachers visited groups emerged. All instances in the teacher meeting transcripts that contained discussions about going in and out of groups were highlighted. These instances were then connected to transcripts of video recordings in the classroom showing how teachers’ actions while engaging with groups developed during the EDR-project. Not all video recordings from the classrooms were used, instead a selection was made with a starting point of situations discussed in detail in the teacher meetings. The TM-model was used to organize the data material, and the IC-model to analyse the mathematical discussions. By using the TM-moves, teachers’ actions were organized and described. Some of the teacher meetings contained questions from the first author of this article to the teachers about their views on mathematical dialogue. These questions made it possible over time to find out what teachers claimed had changed in their awareness and teaching in relation to promoting interaction in students’ small-group work.

Analysis and findings in the EDR-cycles: teachers visiting student groups

Across the three cycles, teachers tried, with various foci, to promote student interaction. The cyclic EDR-process gave the teachers a chance to rethink their actions, and to progress in how they acted when encouraging students to engage in mathematical conversations. In the following sections, we summarize teachers’ noticing processes by going back and forth between the classroom setting and the professional development setting to follow the process of how teachers reflected together and became more aware and refined their strategies for purposefully engaging students in mathematical discussions. The transcripts used in the cycles have been selected from both these two settings to illustrate the processes.

Cycle 1: Focus on listening

The first cycle did not initially focus on teachers’ actions in small-group work, but rather on listening. The teachers wanted to attend to and interpret not only how students talked, but also how students’ listening affected the group work. Since the teachers were not aware of what challenges to be solved in the beginning of the project, the objective in the first cycle was an open attempt to find out what affected student-to-student interactions. The teachers did not deliberately plan how to act in the student groups, since it was first towards the end of the cycle that they became aware that interesting things happened within the groups as they passed by or briefly talked to them.

There were two recurring scenarios when teachers interacted with student groups, that were selected for the video analysis at the teacher meetings. The first scenario was teachers passing by without saying anything, often being ignored by the students (not getting-in-contact ). In the first cycle, no predetermined actions about how teachers initiated or entered student conversations were planned for in the design process. Even if a group was stuck, they often did not ask the teacher for help. On one occasion, a group had discussed for more than five minutes whether a square root is always positive or if it can be both positive and negative ( identifying, advocating, challenging ). When the teacher asked them how they were doing, one of the students who had not been particularly active in the group work, answered, “good” and the teacher went on to another group, not understanding that there were unresolved problems in the group (C1, 181,024, NS2, 20:27). However, it was not only the students who were avoiding interaction. In the analysis, teachers explained that they did not want to disturb students while they were active and that when there were discussions going on, they often choose just to listen and not to interfere to give the students a chance to finish their process (Teacher meeting, C1, 181,128, 10:35). They also said that there were other groups needing help more (Teacher meeting, C1, 181,114, 25:40), thus prioritising which groups to interact with.

The second common scenario, connected to the TM-moves about focus and participation , was a teacher entering a group with one of the students, usually the one who had talked most during group work, interacting with the teacher while the others were quiet. The group focus changed from discussion to “presentation mode”, where the same student explained to the teacher either what the group had found out, or his/her own thoughts. One example was when four students were discussing how to solve, assess and grade the solution given by an anonymous student to a problem in a previous mathematics test in the class. Before the teacher arrived, all four students were active, although two of them, Student 1 and Student 2, were talking more. The following conversation took place (C1, 181,109, SS3, 27:16) when discussing how to understand and grade the student solution:

Transcript 1

Have you managed to grade the task?

Yes, no, we have more… this [student] has not reached a complete solution. The student has only simplified. I do not know what to say.

Ehhh, I could not follow what this person had done.

No. Can you find something that is mathematically wrong even though you cannot follow [the solution]?

Yes… I mean… it is… no… mathematically wrong? I didn’t focus on that. Have you seen, said something about that? [Student 1 asks the group]

But we are supposed to write E-points and C-points, Footnote 2 right?

Yes, exactly.

On every task, if you had been the teacher…

[interrupting]… okay, I would not have given this [solution] any points at all.

In this transcript, the teacher initiated and entered the discussion with an open question to better understand what the students had been doing. Initially, Student 1 was the only one talking, trying to explain and present what they had done. Student 1 often used the pronoun “I”, indicating that he was presenting only his own thoughts. The teacher consequently directed questions to the entire group, using the Swedish plural form of “you”. In the middle of the conversation, Student 1, realizing that he did not know the answer, reached out to the group for help, but when he understood what to do, he tried to take over the conversation again. The teacher then exited the discussion and left the students to continue their work. Students 3 and 4 did not participate actively in the conversation with the teacher. During the teacher’s engagement, little mathematics was discussed and few IC-acts coded; there was one attempt at getting-in contact when Student 1 reached out to the group and another when the teacher posed questions to get students to work with locating what to do with the scoring. It was not until the teacher left the group that the other students became active again.

In this example, it was hard for the teacher to know what had happened before he entered and after he left the conversation. When analysing the video clip, the teacher remembered that he had thought that it was mainly Student 1 who had talked during the group work (Teacher meeting, C1, 181,128, 41:30).

Analysing the focus of the interactions revealed that several of the student discussions, both in this group and in others, were about finding the correct answer, and depending on whether the students succeeded with this or not, they expressed either feeling good or stupid. This affected teachers’ options for initiating or entering conversations. Sometimes students tried to push teachers away to disguise that they did not understand.

Analysing the participation patterns in the videos, teachers reflected that one important technique for getting quiet students to participate actively in the conversations, could be to pose a question that included the quiet student. This was clear both from instances when students asked each other questions and from when teachers entered the conversations and asked questions. On several occasions, the teachers found that questions made discussions continue and deepened the mathematical reasoning. Questions are also important parts of all IC-acts (Alrø & Skovsmose, 2004 ). One conclusion, in relation to the research question and teachers’ actions/responses in the classroom, was that the teachers needed to think more about how questioning could more purposefully engage students in mathematical discussions, leading them to focus on the use of questions in the second cycle.

Cycle 2: Focus on the use of questions

In the second cycle, teachers’ objective was to attend to and make students more aware of the importance of mathematical questions, and in that way increase the use of IC-acts in the conversations. The intention was to get both students and teachers to use questioning as a way of changing focus and participation patterns within the groups. One reason for focusing on this was to support active participation of all students. Bishop ( 2012 ) wrote that students “do not always know how to, or choose to, interact equitably, productively, and positively” (p. 70), and that it therefore is important to have discussions about how small group work is done.

The teachers designed, as an action/response to the results in the first cycle, a meta-discussion in their classrooms—students first thought by themselves, then discussed in their small groups, and then everyone in the class discussed together. Focus was to ask students meta-questions (Pimm & Keynes, 1994 ) to make them more aware of participation and questions in mathematical dialogue. The discussions were about when students choose to actively participate in group discussions, whose responsibility it is that everyone is active, and what the difference is between how -questions and why -questions. When designing the meta-discussion, the IC-model served as inspiration, as different kinds of questions can be connected to different IC-acts depending both on the mathematical content and on the purpose of the questioning. Asking, “how can we find the information we need?” could be about identifying or locating , while asking, “why are you using that strategy?” could be about advocating , challenging or evaluating. A conclusion the teachers came to from the meta-discussion was that students often choose to be quiet when they do not understand. The teachers tried to argue that it is precisely when one does not understand that one should ask questions and not be quiet. Many students still claimed that it was better to just listen to those who actually had an idea about a solution. One of the teachers reflected afterwards that teachers raising thoughts about asking more why-questions was useful for the students in the long run (Teacher meeting, C2, 190121b, 3:50), because if it could help normalize the asking of questions and make it okay to admit that one does not know, then such a strategy could help prevent students from feeling stupid when they cannot find the answer to a question. Such a discussion could help when initiating conversations with the students. If students were not afraid of making mistakes, they might choose to involve the teachers more frequently when there was a need for help.

The teachers also designed an activity in which the mathematical tasks were more open than usual, so that students had more ways of solving the problems. One task was about building mathematical models in different ways to predict the future; for example, to determine how many cars there might be at different points in time. The openness was supposed to change the focus from finding the correct answer to seeing several possible ways of thinking. The teachers decided in the design phase that when they engaged with the groups, they would be aware of and attend to their own use of why -questions and try to use them to initiate the discussions and hinder students from going into “presentation mode”. They wanted to challenge students to think and justify their answers and so decided to interfere a little more with the students compared to the first cycle. A strategy that one of the teachers used spontaneously (it was not planned in the teacher group), was to ask quiet students in the groups, “did you understand why X said that?” or, “I see that X is not convinced about your reasoning, can you talk more to her and explain why you reason the way you do?” This encouraged the quiet students to take part in the discussions and the teachers could promote participation patterns that supported more equal participation.

When designing the lesson about mathematical models, the group of teachers came up with asking “why did you chose that model?” which they considered to be a good why -question for the students. One of the teachers reflected:

Maybe if they get the why question from us, then they will not have to come up with any of their own. Maybe there is a danger in it, but maybe we should pose some why-questions. Just to steer them a little in that direction. (Teacher meeting, C2, 190128, 32:11)

In the following lessons, the teachers related to questioning in different ways. One of the teachers was still mostly not actively participating in the small-group mathematical discussions and students were not involving this teacher when they became stuck on a problem. The TM-moves about initiating and entering were not really visible. The teacher tried to ask questions, but mostly without mathematical content, such as, “how are you doing?” or “are you finding something out?” Several of the groups did not understand the concept ‘mathematical model’ or how to work with models on their calculators/computers, but despite their uncertainties, they still did not ask the teacher, and so it was hard for the teacher to find out what they had become stuck on and thus to determine what mathematical questions to ask.

Transcript 2 is from a group with three students, discussing different models for working out how many cars there might be in year 2010 and 2018, respectively. Student 3 is trying a linear model on her calculator, while the others, although following her work, are mostly inactive. Then they get stuck on what other models they could use but do not tell the teacher about their problem, so the teacher’s attempts at initiation are not successful (C2, 190,212 NS1, 19:05):

Transcript 2

Okay, do we have another line or mathematical model we can work with?

We can use those [points towards models written on the white board].

Yes, because we got one.

y equals c e to the power of x [ \({ce}^{x}\) ].

Can we work with that one?

[Teacher is passing by; all students look busy and Student 6 starts reading from the handout with the task]

What limitations does the model you chose have?

Have you tried different [models]?

We have worked with one.

We are working with that now. [Teacher walks on to another group]

In this transcript, Student 5’s last claim can be interpreted in two different ways—either that the students are about to start working with more models, or that they are doing it. For the next fifteen minutes the students struggled to come up with another mathematical model and they did not ask the teacher any questions even though she walked by several times. The teacher, seeing that they were working, did not interfere. Fifteen minutes later (C2, 190212 NS1, 35:03) the following conversation occurred:

Transcript 3

Have you found something out?

We have only figured out how to work with one model.

Yes, have you tried another model?

No, we do not know how to work with it.

You use your calculator [shows them].

Okay [laughing].

Makes sense, makes sense.

[Teacher walks on to another group]

Wait, what did she say?

You are doing it on your calculator. No. We are so slow.

Then the group started working. Since the group did not ask the teacher any questions on the first visit, it was hard for her to understand what the group had become stuck on. Also, in other groups in the classroom students’ problem was to work out how to use their calculators correctly, something the teacher assumed they already knew how to do, since they had worked with similar tasks in Chemistry and Physics.

At the end of Transcript 3, Student 5 remarked that they were “so slow”. One question here is about whether asking the teacher questions is admitting that you do not know how to do the mathematics and whether there is a social risk in asking questions. The same students said in the meta-discussion that they did not consider asking the teacher to be an option. The students in this class focussed often on finding the right answers, not on engaging in conversations as a learning opportunity. During the activity, there were episodes in which students worked individually, trying to find an answer, without interacting at all.

In the analysis afterwards, the teachers interpreted what happened in the interaction. They came to realize that their use of questions while initiating/entering group conversations could directly affect what happened, since the questioning helped teachers understand what students were struggling with and hence made it easier to support students’ mathematical discussions. Questioning could also have an indirect effect, since the way teachers asked questions and discussed the use of questions with students could make students more aware of their own questions. Hence, teachers wondered whether they could be role models for how to ask mathematical questions. Reflecting on the results in the second cycle during the video analysis phase, teachers concluded that there were two important factors about their actions that affected what happened in the group work after their visits. Firstly, how long they stayed in the groups had an effect on how much they discovered and understood of the students’ discussions. Secondly, the types of questions they asked the students affected what direction the mathematical discussion took both during and after their visit. Asking more mathematical questions revealed more information to the teachers and made it easier for them to support the students to deeper mathematical discussions. Questions could therefore affect all IC-acts and TM-moves positively or negatively, depending on how they were used. The conclusion related to the research questions in the second cycle was, that the way teachers used questions within the groups to purposefully engage students in mathematical discussions was crucial. This needed to be further investigated in relation to what happened when they visited student groups in the third cycle and acted/responded to students’ interactions.

Cycle 3: Focus on teachers’ moves

In the third cycle, the objective and focus of teachers’ designs were mainly about noticing what teachers should do when talking to students in small groups. In cycle two, the teachers had tried to make students more aware of the benefits of asking questions, but it was harder to change students’ use of questions than their own. Therefore, in the third cycle, the teachers chose to focus on their own role and how to engage student groups in mathematical conversations in a more structured way. The teachers discussed how to enter and initiate conversations with groups, how to get students to focus on mathematical issues and how to get everyone to participate . There was no focus on the TM-move about how to exit conversations in this study.

The design for this cycle included teachers predicting what mathematical questions it would be possible to ask in relation to the difficulties they anticipated students would have within particular mathematical content areas. These questions would then be used when teachers entered the groups. For instance, in a task on inscribed angles, the teachers knew that students often find it hard to understand what arc the inscribed angle originates from and so they prepared a mathematical control question on this to initiate a mathematical discussion. They thought this would help the locating and identifying processes, as well as challenge students’ thinking if they were on the wrong track. The control question could then be used as a first question within each group that would help the teachers to see whether the students had understood the problem and to find an appropriate way into the students’ mathematical conversations.

Before the activities, groups were rearranged to consist of only three students instead of four. This was to influence participation patterns, as it was assumed that it would force students to sit closer together and avoid talking in pairs (as a result of the first two cycles). Within the groups, teachers were to attend to and actively engage all three students in discussions while they were present within the group. A quiet student would get a direct question. The teachers’ focus was on getting-in-contact and getting all three students engaged in advocating and challenging , by asking them questions that promoted these dialogical acts. By reformulating what students had said and posing a question, this was to help them to think aloud. They decided, as a general rule, to talk to the students as a group and not individually (Teacher meeting, C3, 190,401, 12:30).

In the third cycle, teachers followed the same type of strategy when planning three mathematics lessons in succession. The result was that students experienced the same kind of arrangement concerning tasks and ways of working together three times in a row. The assumption behind this was that it is not enough to try an arrangement only once, and that students need to grow accustomed to a particular way of working. Teachers constructed group activities that consisted of an individual task to prepare everyone for interaction, followed by a group task (involving mathematical why -questions) in which students discussed together, and a summary in which the group had to think about what they had learnt and what questions they had used. Teachers also tried to limit the time students were given for the group work, so that periods of inactivity would be minimized. In the first round of filming in the third cycle, there were several examples of what happened when the teachers did not achieve the intention of talking to all three students. In one instance, when a teacher was talking to two of the students in a group, one student appeared to be listening to the conversation and nodded and acted as if she understood, but as soon as the teacher left the group this proved not to be the case. In Transcript 4, this group of students was discussing how to find coordinates that fulfil three mathematical inequalities simultaneously (C3, 190,503, SS2, 17:24):

Transcript 4

Now we must find three different coordinates that are part of the system of inequalities.

What does that mean?

Is it just three random coordinates? … [inaudible]

[Teacher arrives]

In which area? Is it in this area? [to the teacher]

Yes. If you fill in the handout exactly as you did in your individual papers.

How is it now? You filled in down here [to Student 7; filling in one area]. I filled in here [filling in another area]. And you [looking at the paper from Student 9] filled in here [filling in the last area].

Then it is that triangle, right?

Yes, exactly. It is that one? [points to triangle]

And now we see why we fill in the area that we are not interested in. We see easily what area is left.

God, this is good.

Okay. Three coordinates here.

They are points. Just mark three points somewhere so we can see where they are.

But wait. Now I do not understand what we were doing.

In this transcript, the conversation was not initiated by the teacher. Instead, students posed a question to the teacher as he passed by. The students wanted help to locate what they should be doing. Student 8 led the discussion and the teacher offered information and supported the process. The teacher talked to the entire group, but only Student 7 and Student 8 were taking part in the discussion, while Student 9 followed what Student 8 was doing on the paper in front of her. Student 9 was quiet during the conversation and the teacher left without listening to her or asking her questions, not knowing if she had understood or not.

When the group of teachers realized this during the video analysis, it encouraged them to be even more aware of the participation patterns, and especially how quiet students acted. If students were quiet, teachers interpreted the situation and considered that this might be because they did not understand or because they might have a question. They decided to act/respond to this by making sure to involve everyone in the conversations while visiting the groups in the following lessons. The analysis showed how teachers were more aware of the importance of talking to all students and subsequently did this purposefully when visiting the groups. This gave them the opportunity to spend a little longer time in the groups, and make sure all students were on track.

When reflecting on the results in the third cycle in relation to the research question, it was not only how long teachers spent in the groups, but also what they attended, interpreted and focussed on when talking to the students that affected their opportunities to engage students in mathematical discussions. They did not only ask “how are you doing?”, but instead planned purposeful specific mathematical questions that would get students to actively discuss mathematics. Teachers often used students’ names as a response to group work that did not include all students. For instance, when one student was in ‘presentation mode’, the teacher calmly asked another student in the group, “X, what do you think about this?”, with the intention of getting everyone engaged in the discussion.

Summarizing what was noticed

In the three cycles, results from going back and forth between the classroom setting and the professional development setting were used to answer how teachers can refine their strategies for purposefully engaging students in mathematical discussions. These results emanate from the analysis of the EDR-cycles and are both theoretical and practical as they should be following EDR-methodology (McKenney & Reeves, 2012 ). In Table 2 , both the professional development setting and the classroom setting inform the conclusions as part of teachers’ noticing processes across all three EDR-cycles. The column “attending” summarizes what the teacher group focused on in the three cycles, “interpreting” what was discussed in teacher professional meetings, and “acting/responding” what they decided to do or think more about in the future. The results in the latter column would not have been achieved if the previous two had not been scrutinised.

In the next sections, we discuss what theoretical and practical results came out from the teachers’ noticing processes across the three EDR-cycles.

Theoretical results

In the study, the more theoretical results relate to the use and combination of the two theoretical frameworks (Alrø & Skovsmose, 2004 ; Ehrenfeld & Horn, 2020 ) to clarify what was happening both in the classroom setting as well as in the teacher professional development setting. The changes in teachers’ awareness were a result of the noticing processes in both settings. In some ways, the analysis became more complex, since it included larger amounts of data, but since interaction in mathematics classrooms is multifaceted, the combination of being in different settings with the two theoretical models could be one way to deepen the understanding of this complexity.

In the study, the IC-model helped to create a common understanding of what is quality in mathematical discussions and identify what parts of students’ interactions teachers wanted to encourage. Across the three cycles, paying attention to the teachers’ as well as to students’ engagement in IC-acts was a way of refining teachers’ awareness of different options for promoting small-group discussions. Across the cycles, much attention was paid to the IC-act of getting-in-contact . When everyone was involved in the conversations, teachers considered students’ activity to be increasing their opportunities for learning mathematics. It also improved teachers’ possibilities to attending and interpreting students’ mathematical thinking and understanding what was mathematically challenging for them. By purposefully asking how- and why-questions, students were engaged in locating, identifying, advocating, challenging or evaluating depending on what mathematical content and purpose the questions had. When teachers asked students questions, they often reformulated what students had said, and asked them to think aloud. Hence, the IC-model provided insights for the different kinds of questions teachers ask students, which could help them to refine their strategies for promoting mathematical discussions.

The TM-model was helpful for interpreting and understanding the findings about what teachers did to promote student interaction. Both students and teachers initiated conversations. The video analysis revealed that students were sometimes reluctant to talk to the teachers, as they did not want to admit that they did not know what to do. Another result, however not visible in the codes of the TM-model, was that a good way to prepare for an entry was to start a conversation with a specific, pre-planned mathematical question, connected to some part of the problem-solving process that the teachers anticipated might be difficult for the students. There were many similarities in this study to Ehrenfeld and Horn’s ( 2020 ) study with respect to focus , with findings about how teachers asked both mathematics and non-mathematics-related questions. For instance, the mathematics probing in the model could be related to how teachers in this study worked with how- and why-questions. As in Ehrenfeld and Horn’s ( 2020 ) study, much of the students’ focus was on results and finding the right answers. This was often because teachers frequently asked for results when they engaged with the groups, but students also did this in their conversations without the teacher, revealing that they put a lot of effort into finding the right answers. When the participation patterns did not include all students, teachers concluded that this could lead to some students not understanding what was going on. While Ehrenfeld and Horn ( 2020 ) did not always include everyone in the conversation through the TM-moves, the teacher group in this study felt that in their context it was important before exiting the group to talk to all three students to make sure that the understanding was there.

Both the IC-model and the TM-model are conceptual frameworks that can make visible the complexity of teachers’ engagement in students’ mathematical group work. By analysing what IC-acts appeared in the mathematical dialogue and what TM-moves teachers made when entering an ongoing student discussion, it was possible to follow and draw conclusions about how teachers’ actions affected and promoted student interaction, and hence what teachers needed to do to refine their teaching. Both models informed the noticing-process, and hence, another result of this study is the combination of theoretical frameworks from mathematics education with teacher noticing as a way of understanding mathematical dialogue.

Practical results

The more practical results in this study relate to the actions/responses that the teachers worked with as a result of the structured noticing process. These are results connected to the specific EDR-cycles and the specific focus areas chosen by the participating teachers, and hence, the results must be seen in relation to the context of their classrooms and their professional development group. As the summary in Table 2 shows, the practical results of how teachers worked with refining their teaching, consisted in teachers becoming more aware of the importance of clear instructions, trying to get an overview of all groups and their own role as facilitators of mathematical questions, and to listen not only to the students talking but also to the students being quiet and involve them in conversations. Also, one conclusion was that a group size of three students in a group promoted cooperation between all students within the group and also made it possible for teachers to talk to all students. The teachers realized the importance of purposefully asking mathematical questions to all students, as well as planning their instructions and actions when entering a group discussion.

The noticing process in itself can also be seen as a practical way of changing teachers’ awareness when it comes to promoting students’ mathematical group work. Comparing the cycles shows that in the first cycle the focus of attention was unclear, changing to become more specific for each cycle. Considering that there always are limitations in what to attend to (Miller, 2011 ), it seemed important to choose the focus areas by interpreting what happened and step-by-step trying to improve the actions/responses and specificity in coming cycles.

In the interpretation stages of noticing, teachers became aware of a variety of factors affecting the small group work, both when they were present in the groups and when students worked alone. The video analysis process revealed certain aspects affecting students’ interactions that were hard for teachers to recognize when visiting groups for a few minutes, which is in line with Schoenfeld’s ( 2017 ) claim that video enables teachers to capture and discuss classroom phenomena that they otherwise might miss. Hence, for teachers to be part of the analysis process, it was important for them to understand the challenges related to mathematical conversations that occurred in the classroom. As in Fuentes’ ( 2013 ) study, there were problems with lack of communication between all students and sometimes poor communication patterns, for instance, with students not participating actively or going into ‘presentation mode’ when teachers were visiting the groups. Another problem was the norms that impeded students’ interactions, such as students only focussing on finding correct answers or feeling stupid when they could not solve a task. These problems were clarified and interpreted at the teacher meetings when analysing video recordings. Thereafter, this resulted in the practical action/responses.

Concluding remarks

Previous research presents both benefits and problems with small-group work in mathematics. Although group work can provide an engaging context for mathematics learning (Alrø & Skovsmose, 2004 ; Walshaw & Anthony, 2008a ), there is uncertainty about what happens when the teacher is not present and how teachers can best promote student interaction (Ehrenfeld & Horn, 2020 ; Sfard, 2015 ; Staples 2008 ; Walshaw and Antony 2008b ). This article contributes to the field by analysing how a group of teachers working with noticing student interaction (Mason, 2002 ; Sherin et al., 2011 ) and using EDR-methodology, can refine the way they purposefully engage students in mathematical discussions. This is connected to attending certain aspects of the interaction, interpreting what is happening in the interaction, and deciding how to act/respond differently as a teacher.

In addition to many other studies that either focus the classroom setting or the professional development setting, this study moves back and forth between both settings. In contrast to for instance Ehrenfeld and Horn’s ( 2020 ) study, this study focuses not only what is happening in the classroom, but also, by following discussions in teacher meetings, why it is happening. This gives insights to how teachers’ noticing processes can be connected to changing teachers’ awareness and promoting students’ interactions in mathematics. It also contributes to the identified gap in research (Santagata et al., 2021 ) regarding the third step in noticing (responding/acting), as the setup with the two settings creates a possible way of making informed decisions and try new ways of teaching across the EDR-cycles.

However, changing the way teachers act in classrooms is a long-term process that is not easily achieved. By working with professional development in a structured way, the teachers were able to reflect upon their own role in relation to the students, which previous research claims is important to succeed with school improvement work (Harris, 2014 ; Timperley, 2011 ). Thus, this project should not be seen as a finished quest, but rather as the beginning of a process.

Availability of data and material

Data not available due to research secrecy, according to the Swedish Research Council ethical guidelines.

Code availability

Data and coding not available due to research secrecy, according to the Swedish Research Council ethical guidelines.

The teacher group did not have time to watch videos from all four classrooms and hence the data collection was limited to two classrooms. The two teachers most positive to being video recorded volunteered to have cameras in their classrooms.

In Sweden grades are on a scale from A to E, and in mathematics tests it is usual to give points on A, C and E-levels.

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Sjöblom, M., Valero, P. & Olander, C. Teachers’ noticing to promote students’ mathematical dialogue in group work. J Math Teacher Educ 26 , 509–531 (2023). https://doi.org/10.1007/s10857-022-09540-9

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#MathMonday: Rubik’s Cube

Fifty years ago, Hungarian sculptor and architect Erno Rubik invented a three-dimensional geometric model with 27 colorful wooden blocks and rubber bands that allowed the blocks to move while retaining the cube shape. It took Rubik a month to solve his prototype cube, turning the cube’s six faces until each one was a single color.

In 1980, the Rubik’s Cube made its international debut and has since become a pop culture icon.

At the beginning, Professor Rubik was simply designing a tool for his architecture students to appreciate space, “ with its incredibly rich possibilities, space alteration by architectural objects, objects’ transformation in space – sculpture and design – and movement in space and in time….”

But solving this puzzle to make all nine squares on each of the six sides the same color is also a  mathematical matter . Rubik’s Cube has proven to be a useful tool for teaching algebraic group theory, specifically the two basic ideas of “commutators” and “conjugates” that facilitate solving the cube.

A  commutator  is a sequence of moves designed to isolate and swap or rotate specific pieces on the Rubik’s Cube while leaving the rest of the cube unchanged. Cube solutions are mainly based on commutators. 

A  conjugate  involves using a move sequence to bring pieces into a position where a known algorithm can be applied effectively, and then returning the pieces to their original location.

In  solutions guides , the Rubik’s Cube has its own standard notations—Up, Down, Right, Left, Face and Back (U, D, R, L, F, B)—that refer to the different layers of the cube. These notations are relative and determined based on the cube’s orientation. “Face” is always the layer that faces you when holding the cube. An apostrophe indicates the layer should rotate counterclockwise 90 degrees; otherwise moves are clockwise. Commutator and conjugate algorithms using these notations are directions used to solve the Cube.

There are over 43  quintillion —43,252,003,274,489,856,000—Cube combinations. For comparison, there are only 9.2 quintillion ways to fill out a  March Madness bracket  and only 7.5 quintillion  grains of sand on all the Earth’s beaches and deserts . 

Surprisingly, shortly after the toy’s debut in 1981, it was estimated that the fewest moves needed to solve Rubik’s Cube was 52.

Today, it is confirmed that the fewest moves needed is just 20! This was proven in 2010 when Google donated 36 CPU-years (central processing unit) of idle computer time to run a program that solved every one of the more than 43 quintillion starting positions of the Rubik’s Cube in fewer than 21 moves.

The official Rubik’s Cube website estimates that only 5.8% of the world’s population can solve the Rubik’s Cube .

If, like me, you are not among that elite groups of cube solvers, the solution guide of algorithms on the toy’s website makes solving it possible. You do not need an intimate understanding of group theory to be successful, but understanding math vocabulary and concepts is definitely necessary. These concepts include face, surface, front, back, rotate, clockwise, edge, corner, ¼ (fractions!), turn and inverse. Happy puzzle-solving!

Looking to turn student math achievement trends upward? ExcelinEd takes the puzzle out of implementing a  comprehensive K-8 math model policy  based on  fundamental principles  from the National Mathematics Advisory Panel.

DID YOU KNOW?

The Rubik’s Cube was inducted into the National Toy Hall of Fame in 2014. More than 500 million Rubik’s Cubes have been sold, and more than 50 books, hundreds of papers and countless websites have been dedicated to describing how to solve the puzzle of Rubik’s Cube. 

If you are a great Cuber, consider joining a World Cube Association  competition . Try your skill to join the ranking board. Perhaps you can knock off leader Max Park, who solved the cube in 3.13 seconds, or second-ranked Luke Garrett, who solved it in 3.44 seconds. It’s unlikely either of them will ever beat the bot designed by Mitsubishi Electric engineers that  recently solved the puzzle in 0.305 seconds , so fast that the cube itself had trouble keeping up with the machine.

All of these times are amazingly faster than the month it took Rubik to solve his own prototype cube!

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