essay on problem solving skills in everyday life

10 Everyday uses for Problem Solving Skills

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Many employers are recognizing the value and placing significant investments in developing the problem solving skills of their employees.  While we often think about these skills in the work context, problem solving isn’t just helpful in the workplace.  Here are 10 everyday uses for problem solving skills that can you may not have thought about

1. Stuck in traffic and late for work, again

With busy schedules and competing demands for your time, getting where you need to be on time can be a real challenge.  When traffic backs up, problem solving skills can help you figure out alternatives to avoid congestion, resolve the immediate situation and develop a solution to avoid encountering the situation in the future.

2. What is that stain on the living room carpet?

Parents, pet owners and spouses face this situation all the time.  The living room carpet was clean yesterday but somehow a mysterious stain has appeared and nobody is claiming it.  In order to clean it effectively, first you need to figure out what it is.  Problem solving can help you track down the culprit, diagnose the cause of the stain and develop an action plan to get your home clean and fresh again.

3. What is that smell coming from my garden shed?

Drawing from past experiences, the seasoned problem solver in you suspects that the source of the peculiar odor likely lurks somewhere within the depths of the shed. Your challenge now lies in uncovering the origin of this scent, managing its effects, and formulating a practical plan to prevent such occurrences in the future.

4. I don’t think the car is supposed to make that thumping noise

As with many problems in the workplace, this may be a situation to bring in problem solving experts in the form of your trusted mechanic.  If that isn’t an option, problem solving skills can be helpful to diagnose and assess the impact of the situation to ensure you can get where you need to be.

5. Creating a budget

Tap into your problem-solving prowess as you embark on the journey of budgeting. Begin by determining what expenses to include in your budget, and strategize how to account for unexpected financial surprises. The challenge lies in crafting a comprehensive budget that not only covers your known expenses but also prepares you for the uncertainties that may arise.

6. My daughter has a science project – due tomorrow

Sometimes the challenge isn’t impact, its urgency.  Problem solving skills can help you quickly assess the situation and develop an action plan to get that science project done and turned in on time.

7. What should I get my spouse for his/her birthday?

As with many problems, this one may not have a “right answer” or apparent solution.  Its time to apply those problem solving skills to evaluate the effects of past decisions combined with current environmental signals and available resources to select the perfect gift to put a smile on your significant other’s face.

8. The office printer suddenly stopped working, and there are important documents that need to be printed urgently.

Uh oh, time to think quickly.  There is an urgent situation that must be addressed to get things back to normal, a cause to be identified (what’s causing the printer issue), and an action plan to resolve it.  Problem solving skills can help you avoid stress and ensure that your documents are printed on time.

9. I’m torn between two cars! Which one should I choose?

In a world brimming with countless choices, employ decision analysis as your trusty tool to navigate the sea of options. Whether you’re selecting a car (or any other product), the challenge is to methodically identify and evaluate the best choices that align with your unique needs and preferences.

10. What’s for dinner?

Whether you are planning to eat alone, with family or entertaining friends and colleagues, meal planning can be a cause of daily stress.  Applying problem solving skills can put the dinner dilemma into perspective and help get the food on the table and keep everyone happy.

Problem Solving skills aren’t just for the workplace – they can be applied in your everyday life.  Kepner-Tregoe can help you and your team develop your problem solving skills through a combination of training and consulting with our problem solving experts.

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Everyday problem solving across the adult life span: solution diversity and efficacy

Everyday problem solving involves examining the solutions that individuals generate when faced with problems that take place in their everyday experiences. Problems can range from medication adherence and meal preparation to disagreeing with a physician over a recommended medical procedure or compromising with extended family members over where to host Thanksgiving dinner. Across the life span, research has demonstrated divergent patterns of change in performance based on the type of everyday problems used as well as based on the way that problem-solving efficacy is operationally defined. Advancing age is associated with worsening performance when tasks involve single-solution or fluency-based definitions of effectiveness. However, when efficacy is defined in terms of the diversity of strategies used, as well as by the social and emotional impact of solution choice on the individual, performance is remarkably stable and sometimes even improves in the latter half of life. This article discusses how both of these approaches to everyday problem solving inform research on the influence that aging has on everyday functioning.

Introduction

As mentioned in many chapters in this review volume, a common theme of the cognitive aging literature is a steady decline in functioning. As we get older, we experience changes in processing speed, 1 – 3 memory, 4 reasoning, 5 attention, 6 , 7 and executive functioning. 8 – 10 Underlying such decline is a series of structural changes in the brain 11 , 12 as well as shifts in what motivates us to think and act as we grow older. 13 – 15 Despite these declines, older adults are often autonomous, independent, and well adjusted. They live full lives and occupy as many roles in society as younger individuals, if not more roles, and they are relied upon as authority figures—leaders, advisors, employers, parents, and grandparents. This divergence between declines in lab-assessed cognitive functioning and maintained interpersonal influence is what fuels research on everyday problem solving across the adult half of the life span. 16 , 17 The need to assess everyday functioning independently of traditional measures of primary mental abilities led to the creation of everyday problem solving batteries that displayed higher levels of ecological validity, more closely resembling challenges that are part of our day-to-day existence. Research in this field focuses on defining the conditions under which older adults may have difficulties with these problems (e.g., physical limitations and comprehension of sophisticated instructions) so that interventions can be established to ameliorate such difficulties and promote a higher quality of life. In addition, research in this field also focuses on examining how those processes that we use to solve problems change across the adult life span and lead us to implement different types of strategies based upon the goals that we set at each stage of life. 18 , 19 Here, research is reviewed that discusses the challenges faced by older individuals when managing everyday problems as well as the differences that have been found in how young and older adults approach the process of solving everyday problems.

Everyday problems are the circumstances that we find ourselves in on a daily basis that involve using the skills, accumulated knowledge, and resources (e.g., time, money, and friends) that we have available to us to reach our goals and to side step obstacles to these goals. 17 Everyday problems vary in terms of their problem space, or the possible solutions that an individual can reach given the contextual features and demands of the situation. 16 Everyday problems sometimes have a clear outcome, or goal state, that all individuals will work toward. For instance, if you return to your automobile at the airport to find that you have a flat tire, the steps that are required to effectively resolve the problem so that you can be on your way are quite clear. Success depends upon your ability to implement these steps (e.g., use of physical strength to remove lug nuts). On the other hand, everyday problems sometimes create obstacles that cannot be directly removed and require a careful balance of knowing not just what to do but when to do it. For example, should you find yourself in a disagreement with your partner on what to give your child as a gift on a birthday, you can each give the child your respective preferred gifts. However, how will you resolve the negativity that emerged as a result of conflicting preferences? What if you have a limited number of resources to devote to a gift and a compromise is necessary to resolve the conflict? If you have to involve others in the problem-solving process, it is challenging to ensure that all parties have the same goals in mind. Moreover, you cannot proceed forward to acquire the gift until you have buy-in from others. When problems are ill-defined, the timing of your actions is important because you may have to refrain from acting until a negotiation can take place. In such a situation, it is important to recognize how to regulate your own emotions and how to influence the emotions and thought processes of others. 18

In this review, how everyday problem solving changes across the adult half of the life span will be discussed. Included is (a) a description of the methods used to assess everyday problem-solving performance and the diversity in findings that emerges when age's impact on everyday problem solving is gauged using well-defined versus ill-defined problems as well as different operational definitions of efficacy, (b) a description of the contextual factors that lead to age differences in everyday problem solving, and (c) a brief assessment of the future directions of the field.

Assessing age differences in everyday problem-solving performance

As mentioned earlier, decades of research on cognitive aging demonstrate gradual decline in functioning over time. This decline, however, runs counter to the notion that with age comes wisdom, or at least an accumulation of experiences that can help us determine how to continue to function when faced with problems. Baltes referred to these divergent trends as multidirectionality, and he believed that two distinct systems of cognition existed to capture these trends: pragmatic knowledge and structural mechanics. 20 Accumulated experience or pragmatic knowledge (sometimes called tacit knowledge or crystallized intelligence) continues to grow throughout the life span given the novel circumstances and ever-expanding culture to which we are exposed year by year. Conversely, the hard-wired, biologically determined mechanisms that support cognition (sometimes called fluid intelligence) slowly degrade as our cells and tissues wear out over time. Key to successful aging is how the change that takes place in these two systems balance against one another. 21 – 23 Ultimately, each person has a limited pool of resources to devote to all aspects of their life at any given moment. 20 Consequently, throughout the life span, we set goals for ourselves that shape our behavior by prioritizing some pursuits over others. This selection process limits the number of goals that we consider at any given stage of our life so as to make it possible to optimize the investment of our resources to maintain the greatest level of successful functioning as is possible. 20 With respect to everyday problem solving, this poses some interesting questions: (a) To what extent does decline in cognitive functioning (especially rooted in one's neural mechanics) affect the resources that are available to individuals in the latter half of life when faced with complex everyday problems that are vital to autonomous functioning, and (b) How does one's changing goals and accumulated experience across the life span influence the process by which we solve problems? These two questions have dominated research on everyday problem solving and aging over the past 25 years.

Consistent with the need to map cognitive decline onto everyday problem solving, many everyday problem-solving tasks consist of a pool of well-defined problems from multiple domains (e.g., nutrition, health, and finance) that reflect the activities that an autonomous individual will engage in when caring for themselves. 24 A well-defined problem is a problem that has a constrained problem space (i.e., number of possible responses limited by the features of the question asked) with a single correct solution. For instance, the Everyday Cognition Battery (ECB) includes items that ask participants to compare the nutritional value of two brands of chili. 25 Participants are asked to use nutrition labels to determine which brand has less fat and to compare the statistics provided on each label for each of the other listed categories of dietary information. They might also be asked specific questions about what the categories listed on the labels refer to as a way to assess the participants' general knowledge about food labels. Additional items in the battery focus on medication use and understanding financial information. Overall, the problems included in the ECB capture functioning that contributes to many of the domains found in the assessment of Instrumental Activities of Daily Living. 26 Studies using this battery or similar tasks with similar well-defined problems demonstrate decline in everyday functioning with advancing age. 27 Also, performance on the different components of the battery (e.g., everyday inductive reasoning or everyday knowledge) are significantly correlated with performance on corresponding psychometric tests of intelligence, 28 like lab-based assessments of inductive reasoning and verbal knowledge commonly used to track intelligence. 29 However, performance on the practical problems included in the ECB better predicts actual everyday functioning than does performance on the standard psychometric tests of intelligence. Overall, well-defined problems are used to trace how age-related cognitive decline affects the specific abilities that are vital to maintaining an individual's autonomy over and above those abilities measured by tests of primary mental abilities. 24 , 28 , 30 Often, researchers who use tasks that include well-defined everyday problems are trying to replicate specific activities from everyday life within the lab to systematically examine where functional deficits may occur. These tasks provide useful information as to which skills might be most affected in an individual, 31 opening up the possibility for future skills training geared toward forestalling further losses. 32 , 33

In addition to assessing individuals' ability to generate the single correct solution found in each well-defined everyday problem, other researchers assess everyday problem-solving performance by posing ill-defined hypothetical problems and counting the number of safe and effective solutions that can be generated by each participant. In an ill-defined problem, participants are asked to identify all of the ways that they might overcome an obstacle to a goal to reach an acceptable outcome. 34 , 35 The participants' solutions are then examined by coders to ensure that they are in fact safe and effective ways to resolve the problem before being tallied as an indicator of everyday problem-solving ability. A sample problem from such a test might ask the participants to consider ways in which a man with heart problems might still be able to complete summer maintenance and landscaping duties around his house even though his doctor has told him to refrain from strenuous physical activity and even though he does not have enough money to hire someone to do the work for him. 36 Researchers use everyday problem-solving assessments that consist of ill-defined problems in order to allow participant-specific experiences to inform the solutions that are generated. For instance, suppose that one is asked to balance a checkbook in a well-defined everyday problem-solving task, this activity requires that the individual demonstrate addition and subtraction skills. If an ill-defined problem involved balancing one's checkbook or managing finances, then recommending the addition of deposits and subtraction of expenditures would be an effective solution. However, recommending that one seek assistance from someone who has experience balancing a checkbook would also be an effective solution. Ultimately, tasks using ill-defined problems have the potential to capture other solutions that a person may find that go beyond the most common strategy for resolving the issue at hand.

From young adulthood until middle age, the fluency of solution generation increases, possibly reflecting the appropriate balance between gains stemming from pragmatic life experience and only minimal structural or neurological decline. However, performance declines slightly in one's 50s and throughout the remainder of one's years. 35 Similarly, when social problem solving was examined via ill-defined problems, again an inverted U-shaped function characterized performance, with solution fluency peaking in one's 40s and 50s, and declining thereafter. 37 This finding is important because perceived quality of life is closely linked to one's ability to function independently 38 – 40 and solve everyday problems that might emerge on occasion 41 , 42 that are both linked to mortality. 43 – 45 Despite there being numerous studies that link advancing age to declines in everyday problem-solving ability, 46 , 47 other studies have identified areas in which we improve in everyday problem solving with age. Studies that do not demonstrate the similar levels of age-related decline in problem-solving performance often rely on an operational definition of problem solving efficacy that differs from a focus on solution fluency (i.e., the number of safe and effective solutions generated). Table 1 includes some of the ways that researchers have defined everyday problem-solving effectiveness.

Operationalizing effective everyday problem solving

Everyday problem-solving performance dependent on manner of assessment

What defines success when solving everyday problems? In the aforementioned studies, when faced with a well-defined problem, success was based on whether the participant provided the single best solution. When faced with an ill-defined problem, success was based on the overall number of safe and effective solutions that the individual offered as potential ways of managing the problem. In general, with these definitions of success, advancing age is associated with a decline in everyday problem solving performance. 46 Although these definitions of effectiveness provide a useful metric for problem solving success, they are not without their limitations. The one-solution definition of success assumes that there may only be one way to solve a well-defined problem and that such problems are generally solved in isolation instead of with the assistance of others or with supplemental information. Additionally, the solution-fluency definition of success assumes that the solutions that are generated reflect the maximum number of solutions accessible to the participants when in actuality they may reflect those solutions that the participants believed to be most relevant to or efficacious for a given problem.

When you examine the actual strategies that young and older individuals use (or recommend) to solve problems, older adults may fare better than expected because the previously mentioned techniques for assessing everyday problem-solving performance underestimate the value of the behaviors evinced by older adults when they are coping with an everyday problem. 17 , 18 Specifically, the conventional ways of operationalizing everyday problem solving success fail to account for the quality of individual solutions that are generated. They also do not account for the evolving nature of the everyday problem solving process, including the temporal and environmental limitations on direct action that might be imposed on the problem solver by the problem space. Finally, they ignore the impact that nominated solutions have on the participants' well-being and on their ability to meet the goals that they have set for resolving the problem. Given these limitations, additional definitions of everyday problem-solving success have emerged in order to broaden the scope with which age differences in the everyday problem-solving process are examined.

For instance, Cornelius and Caspi defined everyday problem solving success in terms of the degree to which participants' recommended solutions matched those of an expert panel consisting of developmental psychologists as well as young, middle-aged, and older adult lay people. They asked participants ranging in age from 20 to 78 years to consider 48 hypothetical, ill-defined problems from six domains of everyday functioning (i.e., family, friend, work, home, consumer, and information gathering) included in their Everyday Problem Solving Inventory (EPSI). 48 Participants indicated the extent to which they might use each of four specific strategies, tailored to each problem, in an attempt to reach a resolution: purposeful action (self-initiated action to directly resolve the problem), cognitive analysis (planning action and thinking about the situation to better understand it), passive-dependent behavior (doing nothing to change the situation or relying on others to step in), and avoidant thinking and denial (distracting one's attention away from the problem, avoiding responsibility for the problem, or denying one's emotions). Other studies conducted at this point in time were also relying on similar coping-based techniques for operationalizing the diversity of problem-solving strategies that may be employed to manage stressors like those in the EPSI. 49 , 50 After the participants responded, their recommended strategies were compared to those selected by an expert panel as ideal, and an effectiveness correlation was calculated and examined by age group. Overall, older individuals chose strategies that more closely matched those recommended by the expert panel than did younger age groups. This finding runs counter to previous findings that have been discussed demonstrating that problem-solving ability peaks in mid-life and then declines.

This age-related enhancement in everyday problem solving was later replicated using the same set of problems from the EPSI, parsimoniously redistributing them into achievement-oriented and interpersonal domains. 51 Again, older adults were more effective than young adults in their overall choice of strategies for solving everyday problems. Moreover, older adults were more effective than young adults in choosing strategies to resolve hypothetical social conflicts (e.g., how to react when your peers gossip about one of your closest friends). The major reason for this outcome is that older adults were more likely to implement combinations of strategies that included both problem-focused solutions (e.g., purposeful action) and emotion-focused solutions (avoidance and passive dependence). As had been noted by Blanchard-Fields and her colleagues in prior research, older adults approach everyday problems involving interpersonal conflict in ways that are fundamentally different from young adults and in ways that possibly reflect age-appropriate differences in social motivation and experience that guide older adults to be more mindful of the emotions evoked by problems. 19 , 52

Although older adults are less accurate than young adults when solving well-defined instrumental everyday problems and less fluent when generating solutions for ill-defined everyday problems in tasks that do not recognize the value of emotion recognition in the problem-solving process, 46 older adults display a consistent advantage over young adults when problem solving success is defined in terms of one's ability to implement a diverse repertoire of strategies that meet the practical and emotional challenges created by problems. 18 This divergence in outcomes emerges because of the differences that exist in the two dominant approaches to research on everyday problem solving and aging. The method of investigation used, including the operational definition of problem-solving success, influences the conclusions that are drawn about how everyday problem solving performance changes across the adult half of the life span. This can make it quite challenging to compare outcomes across tasks. 53 Ultimately, though, each technique seeks to characterize the diversity of solutions offered by the population to manage everyday problems. Errors that individuals make while completing well-defined problems can help inform the research and development conducted by those who design consumer products, financial forms, and even home environments by specifying which parameters need to be changed to promote a more user friendly experience for people of all ages. Additionally, the breadth of emotion-focused coping strategies offered by individuals facing challenging interpersonal conflicts can be used to develop age-specific norms that can inform mental health professionals of those strategies that would be most relevant to patients at different points in their life. In their own ways, each technique strives to add more information to the existing literature on ways that people of all ages can elevate their sense of well-being while continuing to maintain their autonomy and social functioning.

Contextual factors that contribute to age differences in everyday problem solving

Over the past two decades, researchers have recognized that everyday problem solving can be influenced by many person-specific (e.g., sensory abilities and level of cognitive functioning) and age-typical (e.g., communion-oriented goals or time constraints imposed by thoughts about the end of life) contextual factors. Consequently, the literature is replete with examples of studies that attempt to measure the correlational impact of contextual factors on everyday functioning or that directly manipulate context to track how such interventions affect solution quality and strategy preference. These studies are valuable to the field because they inform practitioners (e.g., medical doctors, nurses, rehabilitation therapists, mental health professionals, and financial advisors) about the roles that cognitive ability and personal motivation play in driving adult decision making. Earlier, it was already noted that cognitive functioning predicts everyday problem-solving performance. 25 , 28 , 33 In fact, recent research suggests that individual differences in cognitive functioning mediate the relationship between poor health status and poor everyday problem-solving performance. 47 Some possible factors that underlie this mediation effect include wide variation (or inconsistency) in response time and age-associated decline in sensory abilities. 54 – 56 These findings tie back to Baltes' hypothesis that we become most susceptible to functional deficits in old age when our neurological architecture degrades to the point where we have difficulty implementing the knowledge that we have gained from our past experiences as we cope with current obstacles to our goals. 20 Consistent with the idea that our own personal goals and our appraisals of problems matter and shape our choices, other researchers have proceeded forward knowing that, although cognitive ability can factor into everyday problem-solving performance, personal relevance and social context also influence how we solve everyday problems. Figure 1 illustrates the mediating role that social context can have on everyday problem-solving performance.

Contextual factors that influence solution implementation in everyday problem-solving tasks.

When faced with a challenge that is not personally meaningful to us, it is reasonable to expect that our feelings of self-efficacy toward our solutions might be less than they would have otherwise been if we were faced with a problem that was more relevant to our own personal history. This prediction is supported by the work of Artistico et al ., who identified that age differences in everyday problem-solving performance map on to the divergent feelings of self-efficacy held by young and older adults when solving problems that either were age relevant or were not relevant to their own age group. 57 , 58 Problems that are more relevant to our current stage of life might be easier to resolve because (a) the problem's context may be more familiar to use, (b) solutions to past similar problems are still accessible, and (c) our peers may also be familiar with these problems and could offer instrumental and emotional support. Intuitively, how much personal experience we have with a given problem should predict how successful we are at solving it. However, Berg et al . have found that experience with the problem itself matters less in producing age differences in everyday problem solving than does the heuristic-oriented (or experiential-based) reasoning implemented by older adults and not younger adults when completing problem solving tasks. 59 Specifically, Berg et al . demonstrated that older adults may be less motivated than young adults to produce as exhaustive of a list of potential solutions to problems or to consider as much information when generating solutions. This behavioral tendency of older adults has appeared in decision making research over the past 20 years and is discussed throughout this review volume. Ultimately, future research will continue to examine whether this practice is being driven by cognitive decline or by a fundamental shift in the reward structure that motivates decisions in the latter half of life. 60 – 62

The impact that personal relevance has on everyday problem solving may stem from how it facilitates several other appraisal processes that force us to examine the contextual features of problems in more details. Specifically, we have to assess what may be the source of a problem (e.g., domain and cause), our goals for coping with the problem, and the types of solutions that may lead to the best outcomes for the problem. Blanchard-Fields and colleagues have identified that younger and older adults choose similar forms of purposeful action- and planning-oriented strategies (also known as problem-focused strategies) when faced with instrumental problems, or problems that merely present some obstacle to the individual who is trying to achieve some personal goal that does not involve other people or relationships, neither directly nor indirectly. 18 , 63 This finding is not surprising given that it is most adaptive to combat the source of the problem directly in these types of situations (e.g., a flat tire on a car or a broken iPod). 64 Interestingly, younger adults are more likely than older adults to use emotion-focused strategies when faced with instrumental problems. 63 This may occur because young people do not have the same personal freedom (e.g., time constraints) and monetary resources at their disposal as older individuals do to invest in clearing obstacles to achievement-oriented problems. If you cannot do something to resolve the problem yourself, you may have to depend on others for assistance. Consequently, passive dependence or attempts to avoid or deny the existence of the obstacle can be an effective way to reduce the disappointment or frustration that one experiences when a goal is thwarted. 18 Developmentally, we would expect young and older adults to offer different solutions to instrumental problems.

Likewise, when faced with interpersonal problems, we might predict that how one responds may depend upon where they currently fall along the developmental spectrum. Early in life, individuals are focused on gathering information, seeking novel experiences, and meeting new people. In the latter half of life, however, the focus shifts toward investing resources in our current relationships to maintain strong socioemotional bonds. 15 , 64 This means that young adults have more social capital to spare and can afford to engage in more argumentative or confrontational strategies for resolving interpersonal problems than can older adults. 65 Whereas young adults focus on balancing short-term negativity with long-term happiness, older adults are focused on being happy today. Simply put, older adults are more likely than young adults to focus on strategies that squelch those negative emotions that are toxic or threaten relationships because they do not have as much time left in life to enjoy these relationships. 15 , 18 For example, in a seminal paper in everyday problem solving and aging, Blanchard-Fields and colleagues found that older adults were more likely thanyoung adults to engage inavoidant-denial strategies when faced with interpersonal problems that were emotionally evocative. 66 Older adults appear to be more keenly aware of when it is important to step away from a conflict to cool off and when it is valuable to delay reacting so as to avoid from fanning the flames. 52 , 67 Consistent with this interpretation, older adults have been found to experience less anger during interpersonal conflicts. 68 , 69 More-over, from middle adulthood through old age, there is a greater emphasis placed on secondary control striving, or the need to internally regulate our reactivity to an environment that might fall outside of our control. 70 With respect to interpersonal everyday problems, the latter half of life is when we realize that attempting to change the behaviors of others might be counterproductive because doing so might exacerbate conflict. A substantial component of this is recognizing that interpersonal harmony requires working within the boundaries of relationships and considering the mutual goals that we have with our relationship partners.

Future directions

The research reviewed up to this point has highlighted how the methods used to assess everyday problem solving contribute to age differences in everyday problem solving performance. It has also presented some of the factors that are responsible for eliciting age differences in strategy selection when researchers focus on the dynamics of how young and older adults react to everyday problems. With the emergence of socioemotional selectivity theory in cognitive aging, there has been a renewed focus on the role that age differences in goals play in motivating everyday decisions. 15 Specifically, an increased emphasis has been placed on trying to gain a deeper understanding of the ways that emotion regulatory and relational communion goals drive older adults' behavioral tendencies when faced with stress or everyday problems. Strough, Berg, and Sansone were among the first to provide evidence that young and older adults approached interpersonal interactions with different goal sets in mind. 71 Their research suggested that older adults were more supportive of social others than were young adults when pursuing the resolution to everyday problems. They interpreted this to suggest that older adults were more focused on generativity, whereas young adults were focused on independence. What is most remarkable about this finding is that, in old age, when individuals face the greatest potential for cognitive and physical decline as well as thwarted instrumental goals, they realign their priorities in order to provide support to their friends and family. Future research in this field will examine how older adults capitalize on their interpersonal focus to live happy and healthy lives. 72 It will also characterize the relational contexts under which older adults are most at risk for stress or which predispose older adults to health problems.

Emerging from the discussion on how we should define successful everyday problem solving was the recognition that individuals would experience the greatest sense of well-being when they selected problem solving strategies that matched their personal goals for the situation in which they found themselves. 18 , 71 , 73 In other words, if your chosen course of action allowed you to meet your goal for the problem, then you will be successful at resolving that problem. Although shockingly simple in theory, in practice, this perspective creates some challenging methodological and statistical hurdles for the researcher to negotiate. For instance, if you want to assess the match between goals and strategies in real time, you have to collect information on the participant's current goals, carefully distinguishing between short-term interests and longer-term life philosophies. Next, you have to wait for a problem to arise and then track how the participant resolves it. If the problem involves someone else, then you have to determine if there is a match between the goals of both parties involved and then examine the strategies of each individual to see how they contribute to individual and collective goals. Time-sampling studies, which ask people to report their goals, the obstacles that they experience to their goals, their emotional reactions to such obstacles, as well as the strategies that they are using to manage these obstacles are currently in progress. 74 Using archival data from a study in which participants ranging in age from 15 to 84 years were asked to discuss a problem from their own lives, Hoppmann, Coats, and Blanchard-Fields found that younger individuals were most likely to match autonomy goals (e.g., independence of action) with self-focused strategies, whereas older adults matched generativity goals with other-focused strategies. 75 These findings highlight the importance of considering goals when trying to account for why young and older adults may arrive at different resolutions to their problems.

Although numerous studies have examined the types of problem- and emotion-focused coping strategies that young and older adults endorse when faced with everyday problems, few studies have examined the interpersonal interactions that occur between individuals as they collaboratively solve everyday problems. As mentioned earlier, older adults display passive dependent strategies at times during a conflict when action may worsen the negativity experienced by both interaction partners, whereas young adults are willing to be confrontational. For instance, when working with a friend to generate as many solutions as possible to hypothetical interpersonal problems, older adults were more likely than young adults to recommend help seeking and careful planning, whereas young adults were more likely to recommend verbally aggressive self-assertion. 76 In other words, when collaborating with a friend, older adults are more likely than young adults to agree that interpersonally destructive strategies are not the best way to resolve conflict. This may reflect a shared recognition in the importance of reducing the potential for making the problem worse. Aside from looking at the strategies nominated by collaborators, Berg et al . have also examined the ways that collaborators treat one another while working to solve problems. In a study examining how partners in older couples collaboratively coped with prostate cancer, husbands and wives both benefited emotionally from working with one another if they were satisfied with their marriage. 77 In a second study, middle-aged and older married couples were asked to discuss an ongoing conflict and to also collaborate with one another to complete an instrumental planning task. 78 Older couples experienced less negative affect during the conflict if they were satisfied with their marriage. However, contrary to the prediction that older couples may behave more passively toward one another during conflict, older couples did express negativity toward one another (especially wives) during the discussion of their conflict. Additionally, when working on the instrumental task, members of older adult couples were warm when attempting to exert control over their partner during the task. When taken together, these findings suggest that jointly reported marital satisfaction can be important for fostering collaborative efforts between partners when coping with health problems, resolving an interpersonal spat, and even when dealing with the daily chores and errands of everyday life. 79 Future research should continue to examine the dynamic role of partner involvement in everyday problem solving performance to identify which relationship factors are most valuable for predicting long-term health and well-being.

Conclusions

One of the central themes of research examining everyday problem solving across the life span has been to identify the trajectory of change in performance throughout the years as we gain experience and knowledge while simultaneously displaying cognitive and physical declines. The impact of cognitive decline on everyday problem solving is most evident when examining the outcomes of studies that use tasks consisting of well-defined problems. Studies using tasks consisting of ill-defined problems produce mixed evidence of both decline and maintenance, depending on the manner with which problem-solving success is operationally defined. Based on more recent findings, however, it is clear that those individuals in their latter half of life are motivated by interpersonal factors that are important to young people but just are not prioritized to the same degree. Future research needs to further clarify the role that interpersonal interaction plays in promoting successful everyday problem solving. Outside of the lab environment, older adults continue to make autonomous choices while also working interdependently with members of their social network. Although cognitive and physical decline are inevitable to some degree for all of us, it seems that a shared decision space between close partners may go a long way to promote sustained well-being, physical health, and everyday cognition.

Conflicts of interest The author declares no conflicts of interest.

Personal Problem Solving Reflective Essay

Recognize and define problems, gather relevant evidence, interpret data, make accurate conclusions, merge new competencies with my problem solving skill set.

Problem solving is a powerful practice because it addresses different challenges in life. My problem solving skill set can make it easier for me to deal with various obstacles and difficulties. However, the skill step has a major gap that requires immediate improvements.

The gap that required some improvements is “Defining and Gathering Evidence”. This paper develops an actionable plan that can be useful towards improving the above gap in my problem solving skill set.

The first step towards a successful problem solving practice is being able to understand the targeted issue. My first objective is learning how to recognize various problems. This strategy makes it easier for individuals to find the most appropriate means for addressing every challenge.

This step also analyzes the nature of the targeted problem. The person solving the problem must prioritize the issues surrounding the problem. It becomes easier to solve a clearly understood problem. This strategy will become a critical aspect of my future problem solving skill set.

The next action plan is gathering the correct evidence based on the identified problem. Individuals must analyze every underlying assumption. Professionals should use appropriate languages in order to improve their levels of communication.

This step will focus on the facts and issues associated with the problem. Individuals should gather the required information using both qualitative and quantitative methods. Such methods will present the required data and facts. The practice will become a critical aspect of my future problem solving strategy. The targeted evidence and information will determine the success of every problem solving process.

The other useful step is learning how to interpret data accurately. This practice determines the success of different problem solving processes. Proper data interpretation makes it easier for individual to appraise different evidences. They also evaluate various arguments before finding the best solution to a specific problem. These new skills will become a critical part of my problem solving strategy.

The next step is making accurate decisions and conclusions based on the collected data. The gathered evidence must be relevant, accurate, informative, and meaningful. These new skills will make it easier for me to understand the nature of different problems.

I will use such skills to make accurate assumptions and conclusions throughout every problem solving process. People must render precise verdicts about various things in life. This strategy can be useful whenever dealing with various problems.

The above competencies will improve the above gap in my problem solving skill set. This action plan will encourage me to undertake more problem solving exercises. The strategy will equip me with new skills and competencies. I will use these competencies to define and analyze several problems.

The next step is to define and gather relevant evidence for every problem. Another good practice is gathering the required evidence using various systematic methods. Professionals can collate the gathered information using the PICOC (Population, Intervention, Comparison, Outcome, and Context) Method. This strategy will be critical towards supporting my career objectives.

Effective communication, negotiation, persistence, logical reasoning, and persuasion can address various problems. Practice makes it easier for individuals to achieve their goals and potentials. My goal is to undertake more problem solving exercises in order to put my new competencies into practice. The above action plan will make my problem solving skill set complete.

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IvyPanda. (2019, December 24). Personal Problem Solving. https://ivypanda.com/essays/personal-problem-solving/

"Personal Problem Solving." IvyPanda , 24 Dec. 2019, ivypanda.com/essays/personal-problem-solving/.

IvyPanda . (2019) 'Personal Problem Solving'. 24 December.

IvyPanda . 2019. "Personal Problem Solving." December 24, 2019. https://ivypanda.com/essays/personal-problem-solving/.

1. IvyPanda . "Personal Problem Solving." December 24, 2019. https://ivypanda.com/essays/personal-problem-solving/.

Bibliography

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Problem-Solving Strategies and Obstacles

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

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

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

What Is Problem-Solving?

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

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

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

The problem-solving process involves:

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

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

Problem-Solving Mental Processes

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

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

Problem-Solving Strategies

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

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

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

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

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

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

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

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

Trial and Error

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

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

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

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

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

How to Apply Problem-Solving Strategies in Real Life

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

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

Obstacles to Problem-Solving

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

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

How to Improve Your Problem-Solving Skills

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

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

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

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

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Global Cognition

Critical thinking in everyday life.

by Winston Sieck updated September 19, 2021

Have you ever been listening to one of your teacher’s lessons and thought that it had no relevance to your own life?

You’re not alone. Just about every student has felt the same way.

Sure, you use critical thinking skills in the classroom to solve word problems in math, write essays in English, and create hypotheses in science.

But how will you use critical thinking in everyday life?

First, keep in mind that critical thinking is simply a “deliberate thought process.”

Basically, it means that you are using reason and logic to come to a conclusion about an issue or decision you are tangling with.

And clear, sound reasoning is something that will help you every day.

To help you make the leap from classroom to real world, here are 3 concrete examples of critical thinking in everyday life.

Fake News vs. Real News

Take a moment to reflect on your media skills. Do you think you have what it takes to sort out a real news source from a piece of clever advertising?

According to a recent study from Stanford University, a whopping 82% of the teens surveyed could not distinguish between an ad labeled “sponsored content” and a legitimate news story.

Part of the problem may come from schools cutting back on formal instruction of critical thinking skills and an assumption that today’s “digital native” teens can automatically tell the difference without practice or instruction.

You are good at lots of things. But, you know, you’ve practiced those things you’re good at. So, how can you practice telling fact from fiction?

One way (outside of school) is to chat with your family and friends about media sources. Find out how they stay informed, and why they choose those outlets. Ask each other routine questions for evaluating sources .

Do your Friends Know Everything?

It’s tempting to believe that the world begins and ends with your friends. Don’t get me wrong. Friends are definitely important. However, it pays to reflect a little on how a group influences our lives.

To practice critical thinking in everyday life, take a close look at your group of friends. Are there things that are “forbidden” in your social circle? Are you expected to act a certain way, dress a certain way?

Think a certain way?

It’s natural that when a group defines something as “cool”, all the people in the group work to fit into that definition. Regardless of what they individually believe.

The problem is that virtually every situation can be defined in multiple ways. What is “dumb” to one person may be “cool” to another.

Develop your ability to redefine the way you see the world around you. On your own terms.

Find a time when your friend group sees the negative in a situation. Is there a positive way to view it instead? Or at least a way that makes it seem not quite so bad?

You may not be ready to speak up with your independent view. And that’s ok. Just practice thinking differently from the group to strengthen your mind.

Critical Thinking in the Driver’s Seat

One of the core critical thinking skills you need every day is the ability to examine the implications and consequences of a belief or action. In its deepest form, this ability can help you form your own set of beliefs in everything from climate change to religion.

But this skill can also save your life (and your car insurance rate) behind the wheel.

Imagine you are cruising down the freeway when your phone alerts you to an incoming text message. The ability to examine your potential actions and their accompanying consequences will help you make the best choice for how to handle the situation.

Do you look at the text and risk getting into an accident? Do you wait and risk not responding to an urgent matter? Or do you pull over to look at the text and risk being late for your appointment?

The same skill can be applied when you are looking for a place to park, when to pull onto a busy street, or whether to run the yellow light.

Better yet, the more practiced you are at looking at the implications of your driving habits, the faster you can make split second decisions behind the wheel.

Why Critical Thinking in Everyday Life Matters

Literally everyone can benefit from critical thinking because the need for it is all around us.

In a philosophical paper , Peter Facione makes a strong case that critical thinking skills are needed by everyone, in all societies who value safety, justice, and a host of other positive values:

“Considered as a form of thoughtful judgment or reflective decision-making, in a very real sense critical thinking is pervasive. There is hardly a time or a place where it would not seem to be of potential value. As long as people have purposes in mind and wish to judge how to accomplish them, as long as people wonder what is true and what is not, what to believe and what to reject, strong critical thinking is going to be necessary.”

So, in other words, as long as you remain curious, purposeful, and ambitious, no matter what your interests, you’re going to need critical thinking to really own your life.

About Winston Sieck

Dr. Winston Sieck is a cognitive psychologist working to advance the development of thinking skills. He is founder and president of Global Cognition, and director of Thinker Academy .

Reader Interactions

July 27, 2019 at 7:20 am

Wonderful article.. Useful in daily life… I have never imagined the way critical thinking is useful to make judgments

December 9, 2020 at 9:38 pm

My name is Anthony Lambert I am student at miller Motte. Critical Thinking is one my classes. I thank you for giving me the skills of critical thinking.

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From Dilemmas to Solutions: Problem-Solving Examples to Learn From

• Daria Burnett
• May 21, 2023

Introduction to Problem-Solving

Life is full of challenges and dilemmas, both big and small.

But if there’s one skill that can help you navigate these, it’s problem-solving .

So, what exactly is problem-solving? And why is it such a crucial skill in daily life?

Understanding the Concept of Problem-Solving

Problem-solving is a mental process that involves identifying, analyzing, and resolving challenges or difficulties.

It’s like a journey that starts with a problem and ends with a solution.

It’s a skill that’s not just used in the field of psychology but in all aspects of life.

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Whether you’re trying to decide on the best route to work, dealing with a disagreement with a friend, or figuring out how to fix a leaky faucet, you’re using your problem-solving skills.

When you’re faced with a problem, your brain goes through a series of steps to find a solution.

This process can be conscious or unconscious and can involve logical thinking, creativity, and prior knowledge.

Effective problem-solving can lead to better decisions and outcomes, making it a valuable tool in your personal and professional life.

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Importance of Problem-Solving in Daily Life

Why is problem-solving so important in daily life? Well, it’s simple.

Problems are a part of life.

They arise in different shapes and sizes, and in different areas of life, including work, relationships, health, and personal growth.

Having strong problem-solving skills can help you navigate these challenges effectively and efficiently.

In your personal life, problem-solving can help you manage stress and conflict, make better decisions, and achieve your goals.

In the workplace, it can help you navigate complex projects, improve processes, and foster innovation.

Problem-solving is also a key skill in many professions and industries, from engineering and science to healthcare and customer service.

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Moreover, problem-solving can contribute to your overall mental well-being.

It can give you a sense of control and agency, reduce feelings of stress and anxiety, and foster a positive attitude.

It’s also a key component of resilience, the ability to bounce back from adversity.

In conclusion, problem-solving is a fundamental skill in life.

It’s a tool you can use to tackle challenges, make informed decisions, and drive change.

By understanding the concept of problem-solving and recognizing its importance in daily life, you’re taking the first step toward becoming a more effective problem solver.

As we delve deeper into this topic, you’ll discover practical problem-solving examples, learn about different problem-solving techniques, and gain insights on how to improve your own problem-solving skills.

So, stay tuned and continue your exploration of introduction to psychology with us.

Stages of Problem-Solving

The process of problem-solving can be broken down into three key stages: identifying the problem , developing possible solutions , and implementing the best solution .

Each stage requires a different set of skills and strategies.

By understanding these stages, you can enhance your problem-solving abilities and tackle various challenges more effectively.

Identifying the Problem

The first step in problem-solving is recognizing that a problem exists.

This involves defining the issue clearly and understanding its root cause.

You might need to gather information, ask questions, and analyze the situation from multiple perspectives.

It can be helpful to write down the problem and think about how it impacts you or others involved.

For instance, if you’re struggling with time management, the problem might be that you have too many obligations and not enough time.

Or perhaps your methods of organizing your tasks aren’t effective.

It’s important to be as specific as possible when identifying the problem, as this will guide the rest of the problem-solving process.

Developing Possible Solutions

Once you’ve identified the problem, the next step is to brainstorm possible solutions.

This is where creativity comes into play.

Don’t limit yourself; even ideas that seem unrealistic or out of the box can lead to effective solutions.

Consider different strategies and approaches.

You could try using techniques like mind mapping, listing pros and cons, or consulting with others for fresh perspectives.

Remember, the goal is to generate a variety of options, not to choose a solution at this stage.

Implementing the Best Solution

The final stage of problem-solving is to select the best solution and put it into action.

Review the options you’ve developed, evaluate their potential effectiveness, and make a decision.

Keep in mind that the “best” solution isn’t necessarily the perfect one (as there might not be a perfect solution), but rather the one that seems most likely to achieve your desired outcome given the circumstances.

Once you’ve chosen a solution, plan out the steps needed to implement it and then take action.

Monitor the results and adjust your approach as necessary.

If the problem persists, don’t be discouraged; return to the previous stages, reassess the problem and your potential solutions, and try again.

Remember, problem-solving is a dynamic process that often involves trial and error.

It’s an essential skill in many areas of life, from everyday challenges to workplace dilemmas.

To learn more about the psychology behind problem-solving and decision-making, check out our introduction to psychology article.

Problem-Solving Examples

Understanding the concept of problem-solving is one thing, but seeing it in action is another.

To help you grasp the practical application of problem-solving strategies, let’s explore three different problem-solving examples from daily life, the workplace, and relationships.

Daily Life Problem-Solving Example

Imagine you’re trying to lose weight but struggle with late-night snacking.

The issue isn’t uncommon, but it’s hindering your progress towards your weight loss goal.

• Identifying the Problem : Late-night snacking is causing you to consume extra calories, preventing weight loss.
• Developing Possible Solutions : You could consider eating an earlier dinner, having a healthier snack option, or practicing mindful eating.
• Implementing the Best Solution : After trying out different solutions, you find that preparing a healthy snack in advance minimizes your calorie intake and satisfies your late-night cravings, helping you stay on track with your weight loss goal.

Workplace Problem-Solving Example

Let’s consider a scenario where a team at work is failing to meet project deadlines consistently.

• Identifying the Problem : The team is not completing projects on time, causing delays in the overall project timeline.
• Developing Possible Solutions : The team could consider improving their time management skills, using project management tools, or redistributing tasks among team members.
• Implementing the Best Solution : After trying out different strategies, the team finds that using a project management tool helps them stay organized, delegate tasks effectively, and complete projects within the given timeframe.

For more insights on effective management styles that can help in problem-solving at the workplace, check out our articles on autocratic leadership , democratic leadership style , and laissez faire leadership .

Relationship Problem-Solving Example

In a romantic relationship, conflicts can occasionally arise.

Let’s imagine a common issue where one partner feels the other isn’t spending enough quality time with them.

• Identifying the Problem : One partner feels neglected due to a lack of quality time spent together.
• Developing Possible Solutions : The couple could consider scheduling regular date nights, engaging in shared hobbies, or setting aside a specific time each day for undisturbed conversation.
• Implementing the Best Solution : The couple decides to implement a daily “unplugged” hour where they focus solely on each other without distractions. This results in improved relationship satisfaction.

For more on navigating relationship challenges, check out our articles on anxious avoidant attachment and emotional awareness .

These problem-solving examples illustrate how the process of identifying a problem, developing possible solutions, and implementing the best solution can be applied to various situations.

By understanding and applying these strategies, you can improve your problem-solving skills and navigate challenges more effectively.

Techniques for Effective Problem-Solving

As you navigate the world of problem-solving, you’ll find that there are multiple techniques you can use to arrive at a solution.

Each technique offers a unique approach to identifying issues, generating potential solutions, and choosing the best course of action.

In this section, we’ll explore three common techniques: Brainstorming , Root Cause Analysis , and SWOT Analysis .

Brainstorming

Brainstorming is a free-thinking method used to generate a large number of ideas related to a specific problem.

You do this by suspending criticism and allowing your creativity to flow.

The aim is to produce as many ideas as possible, even if they seem far-fetched.

You then evaluate these ideas to identify the most beneficial solutions.

By using brainstorming, you can encourage out-of-the-box thinking and possibly discover innovative solutions to challenging problems.

Root Cause Analysis

Root Cause Analysis (RCA) is a method used to identify the underlying causes of a problem.

The goal is to address these root causes rather than the symptoms of the problem.

This technique helps to prevent the same issue from recurring in the future.

There are several RCA methods, such as the “5 Whys” technique, where you ask “why” multiple times until you uncover the root cause of the problem.

By identifying and addressing the root cause, you tackle the problem at its source, which can lead to more effective and long-lasting solutions.

SWOT Analysis

SWOT Analysis is a strategic planning technique that helps you identify your Strengths, Weaknesses, Opportunities, and Threats related to a problem.

This approach encourages you to examine the problem from different angles, helping you understand the resources you have at your disposal (Strengths), the areas where you could improve (Weaknesses), the external factors that could benefit you (Opportunities), and the external factors that could cause problems (Threats).

With this comprehensive understanding, you can develop a well-informed strategy to solve the problem.

Each of these problem-solving techniques provides a distinct approach to identifying and resolving issues.

By understanding and utilizing these methods, you can enhance your problem-solving skills and increase your effectiveness in dealing with challenges.

For more insights into effective problem-solving and other psychological topics, explore our introduction to psychology .

Improving Your Problem-Solving Skills

Learning to solve problems effectively is a skill that can be honed with time and practice.

The following are some ways to enhance your problem-solving capabilities.

Practice Makes Perfect

The saying “practice makes perfect” holds true when it comes to problem-solving.

The more problems you tackle, the better you’ll become at devising and implementing effective solutions.

Seek out opportunities to practice your problem-solving skills both in everyday life and in more complex situations.

This could involve resolving a dispute at work, figuring out a puzzle, or even strategizing in a board game.

Each problem you encounter is a new opportunity to apply and refine your skills.

Learning from Others’ Experiences

There’s much to be gained from observing how others approach problem-solving.

Whether it’s reading about problem solving examples from renowned psychologists or discussing strategies with colleagues, you can learn valuable techniques and perspectives from the experiences of others.

Consider participating in group activities that require problem-solving, such as escape rooms or team projects.

Observe how team members identify problems, brainstorm solutions, and decide on the best course of action.

Embracing a Growth Mindset

A key component of effective problem-solving is adopting a growth mindset.

This mindset, coined by psychologist Carol Dweck, is the belief that abilities and intelligence can be developed through dedication and hard work.

When you embrace a growth mindset, you view challenges as opportunities to learn and grow rather than as insurmountable obstacles.

Believing in your ability to develop and enhance your problem-solving skills over time can make the process less daunting and more rewarding.

So, when you encounter a problem, instead of thinking, “I can’t do this,” try thinking, “I can’t do this yet, but with effort and practice, I can learn.”

For more on the growth mindset, you might want to check out our article on what is intrinsic motivation which includes how a growth mindset can fuel your motivation to improve.

By practicing regularly, learning from others, and embracing a growth mindset, you can continually improve your problem-solving skills and become more adept at overcoming challenges you encounter.

Introduction to Problem Solving Skills

What is problem solving and why is it important.

The ability to solve problems is a basic life skill and is essential to our day-to-day lives, at home, at school, and at work. We solve problems every day without really thinking about how we solve them. For example: it’s raining and you need to go to the store. What do you do? There are lots of possible solutions. Take your umbrella and walk. If you don't want to get wet, you can drive, or take the bus. You might decide to call a friend for a ride, or you might decide to go to the store another day. There is no right way to solve this problem and different people will solve it differently.

Problem solving is the process of identifying a problem, developing possible solution paths, and taking the appropriate course of action.

Why is problem solving important? Good problem solving skills empower you not only in your personal life but are critical in your professional life. In the current fast-changing global economy, employers often identify everyday problem solving as crucial to the success of their organizations. For employees, problem solving can be used to develop practical and creative solutions, and to show independence and initiative to employers.

Throughout this case study you will be asked to jot down your thoughts in idea logs. These idea logs are used for reflection on concepts and for answering short questions. When you click on the "Next" button, your responses will be saved for that page. If you happen to close the webpage, you will lose your work on the page you were on, but previous pages will be saved. At the end of the case study, click on the "Finish and Export to PDF" button to acknowledge completion of the case study and receive a PDF document of your idea logs.

What Does Problem Solving Look Like?

The ability to solve problems is a skill, and just like any other skill, the more you practice, the better you get. So how exactly do you practice problem solving? Learning about different problem solving strategies and when to use them will give you a good start. Problem solving is a process. Most strategies provide steps that help you identify the problem and choose the best solution. There are two basic types of strategies: algorithmic and heuristic.

Algorithmic strategies are traditional step-by-step guides to solving problems. They are great for solving math problems (in algebra: multiply and divide, then add or subtract) or for helping us remember the correct order of things (a mnemonic such as “Spring Forward, Fall Back” to remember which way the clock changes for daylight saving time, or “Righty Tighty, Lefty Loosey” to remember what direction to turn bolts and screws). Algorithms are best when there is a single path to the correct solution.

But what do you do when there is no single solution for your problem? Heuristic methods are general guides used to identify possible solutions. A popular one that is easy to remember is IDEAL [ Bransford & Stein, 1993 ] :

• I dentify the problem
• D efine the context of the problem
• E xplore possible strategies
• A ct on best solution

IDEAL is just one problem solving strategy. Building a toolbox of problem solving strategies will improve your problem solving skills. With practice, you will be able to recognize and use multiple strategies to solve complex problems.

Watch the video

What is the best way to get a peanut out of a tube that cannot be moved? Watch a chimpanzee solve this problem in the video below [ Geert Stienissen, 2010 ].

[PDF transcript]

Describe the series of steps you think the chimpanzee used to solve this problem.

• [Page 2: What does Problem Solving Look Like?] Describe the series of steps you think the chimpanzee used to solve this problem.

Think of an everyday problem you've encountered recently and describe your steps for solving it.

• [Page 2: What does Problem Solving Look Like?] Think of an everyday problem you've encountered recently and describe your steps for solving it.

Developing Problem Solving Processes

Problem solving is a process that uses steps to solve problems. But what does that really mean? Let's break it down and start building our toolbox of problem solving strategies.

What is the first step of solving any problem? The first step is to recognize that there is a problem and identify the right cause of the problem. This may sound obvious, but similar problems can arise from different events, and the real issue may not always be apparent. To really solve the problem, it's important to find out what started it all. This is called identifying the root cause .

Example: You and your classmates have been working long hours on a project in the school's workshop. The next afternoon, you try to use your student ID card to access the workshop, but discover that your magnetic strip has been demagnetized. Since the card was a couple of years old, you chalk it up to wear and tear and get a new ID card. Later that same week you learn that several of your classmates had the same problem! After a little investigation, you discover that a strong magnet was stored underneath a workbench in the workshop. The magnet was the root cause of the demagnetized student ID cards.

The best way to identify the root cause of the problem is to ask questions and gather information. If you have a vague problem, investigating facts is more productive than guessing a solution. Ask yourself questions about the problem. What do you know about the problem? What do you not know? When was the last time it worked correctly? What has changed since then? Can you diagram the process into separate steps? Where in the process is the problem occurring? Be curious, ask questions, gather facts, and make logical deductions rather than assumptions.

Watch Adam Savage from Mythbusters, describe his problem solving process [ ForaTv, 2010 ]. As you watch this section of the video, try to identify the questions he asks and the different strategies he uses.

Adam Savage shared many of his problem solving processes. List the ones you think are the five most important. Your list may be different from other people in your class—that's ok!

• [Page 3: Developing Problem Solving Processes] Adam Savage shared many of his problem solving processes. List the ones you think are the five most important.

“The ability to ask the right question is more than half the battle of finding the answer.” — Thomas J. Watson , founder of IBM

Voices From the Field: Solving Problems

In manufacturing facilities and machine shops, everyone on the floor is expected to know how to identify problems and find solutions. Today's employers look for the following skills in new employees: to analyze a problem logically, formulate a solution, and effectively communicate with others.

In this video, industry professionals share their own problem solving processes, the problem solving expectations of their employees, and an example of how a problem was solved.

Meet the Partners:

• Taconic High School in Pittsfield, Massachusetts, is a comprehensive, fully accredited high school with special programs in Health Technology, Manufacturing Technology, and Work-Based Learning.
• Berkshire Community College in Pittsfield, Massachusetts, prepares its students with applied manufacturing technical skills, providing hands-on experience at industrial laboratories and manufacturing facilities, and instructing them in current technologies.
• H.C. Starck in Newton, Massachusetts, specializes in processing and manufacturing technology metals, such as tungsten, niobium, and tantalum. In almost 100 years of experience, they hold over 900 patents, and continue to innovate and develop new products.
• Nypro Healthcare in Devens, Massachusetts, specializes in precision injection-molded healthcare products. They are committed to good manufacturing processes including lean manufacturing and process validation.

Making Decisions

Now that you have a couple problem solving strategies in your toolbox, let's practice. In this exercise, you are given a scenario and you will be asked to decide what steps you would take to identify and solve the problem.

Scenario: You are a new employee and have just finished your training. As your first project, you have been assigned the milling of several additional components for a regular customer. Together, you and your trainer, Bill, set up for the first run. Checking your paperwork, you gather the tools and materials on the list. As you are mounting the materials on the table, you notice that you didn't grab everything and hurriedly grab a few more items from one of the bins. Once the material is secured on the CNC table, you load tools into the tool carousel in the order listed on the tool list and set the fixture offsets.

Bill tells you that since this is a rerun of a job several weeks ago, the CAD/CAM model has already been converted to CNC G-code. Bill helps you download the code to the CNC machine. He gives you the go-ahead and leaves to check on another employee. You decide to start your first run.

What problems did you observe in the video?

• [Page 5: Making Decisions] What problems did you observe in the video?
• What do you do next?
• Try to fix it yourself.
• Ask your trainer for help.

As you are cleaning up, you think about what happened and wonder why it happened. You try to create a mental picture of what happened. You are not exactly sure what the end mill hit, but it looked like it might have hit the dowel pin. You wonder if you grabbed the correct dowel pins from the bins earlier.

You can think of two possible next steps. You can recheck the dowel pin length to make sure it is the correct length, or do a dry run using the CNC single step or single block function with the spindle empty to determine what actually happened.

• Check the dowel pins.
• Use the single step/single block function to determine what happened.

You notice that your trainer, Bill, is still on the floor and decide to ask him for help. You describe the problem to him. Bill asks if you know what the end mill ran into. You explain that you are not sure but you think it was the dowel pin. Bill reminds you that it is important to understand what happened so you can fix the correct problem. He suggests that you start all over again and begin with a dry run using the single step/single block function, with the spindle empty, to determine what it hit. Or, since it happened at the end, he mentions that you can also check the G-code to make sure the Z-axis is raised before returning to the home position.

• Run the single step/single block function.
• Edit the G-code to raise the Z-axis.

You finish cleaning up and check the CNC for any damage. Luckily, everything looks good. You check your paperwork and gather the components and materials again. You look at the dowel pins you used earlier, and discover that they are not the right length. As you go to grab the correct dowel pins, you have to search though several bins. For the first time, you are aware of the mess - it looks like the dowel pins and other items have not been put into the correctly labeled bins. You spend 30 minutes straightening up the bins and looking for the correct dowel pins.

Finally finding them, you finish setting up. You load tools into the tool carousel in the order listed on the tool list and set the fixture offsets. Just to make sure, you use the CNC single step/single block function, to do a dry run of the part. Everything looks good! You are ready to create your first part. The first component is done, and, as you admire your success, you notice that the part feels hotter than it should.

You wonder why? You go over the steps of the process to mentally figure out what could be causing the residual heat. You wonder if there is a problem with the CNC's coolant system or if the problem is in the G-code.

• Look at the G-code.

After thinking about the problem, you decide that maybe there's something wrong with the setup. First, you clean up the damaged materials and remove the broken tool. You check the CNC machine carefully for any damage. Luckily, everything looks good. It is time to start over again from the beginning.

You again check your paperwork and gather the tools and materials on the setup sheet. After securing the new materials, you use the CNC single step/single block function with the spindle empty, to do a dry run of the part. You watch carefully to see if you can figure out what happened. It looks to you like the spindle barely misses hitting the dowel pin. You determine that the end mill was broken when it hit the dowel pin while returning to the start position.

After conducting a dry run using the single step/single block function, you determine that the end mill was damaged when it hit the dowel pin on its return to the home position. You discuss your options with Bill. Together, you decide the best thing to do would be to edit the G-code and raise the Z-axis before returning to home. You open the CNC control program and edit the G-code. Just to make sure, you use the CNC single step/single block function, to do another dry run of the part. You are ready to create your first part. It works. You first part is completed. Only four more to go.

As you are cleaning up, you notice that the components are hotter than you expect and the end mill looks more worn than it should be. It dawns on you that while you were milling the component, the coolant didn't turn on. You wonder if it is a software problem in the G-code or hardware problem with the CNC machine.

It's the end of the day and you decide to finish the rest of the components in the morning.

• You decide to look at the G-code in the morning.
• You leave a note on the machine, just in case.

You decide that the best thing to do would be to edit the G-code and raise the Z-axis of the spindle before it returns to home. You open the CNC control program and edit the G-code.

While editing the G-code to raise the Z-axis, you notice that the coolant is turned off at the beginning of the code and at the end of the code. The coolant command error caught your attention because your coworker, Mark, mentioned having a similar issue during lunch. You change the coolant command to turn the mist on.

• You decide to talk with your supervisor.
• You discuss what happened with a coworker over lunch.

As you reflect on the residual heat problem, you think about the machining process and the factors that could have caused the issue. You try to think of anything and everything that could be causing the issue. Are you using the correct tool for the specified material? Are you using the specified material? Is it running at the correct speed? Is there enough coolant? Are there chips getting in the way?

Wait, was the coolant turned on? As you replay what happened in your mind, you wonder why the coolant wasn't turned on. You decide to look at the G-code to find out what is going on.

From the milling machine computer, you open the CNC G-code. You notice that there are no coolant commands. You add them in and on the next run, the coolant mist turns on and the residual heat issues is gone. Now, its on to creating the rest of the parts.

Have you ever used brainstorming to solve a problem? Chances are, you've probably have, even if you didn't realize it.

You notice that your trainer, Bill, is on the floor and decide to ask him for help. You describe the problem with the end mill breaking, and how you discovered that items are not being returned to the correctly labeled bins. You think this caused you to grab the incorrect length dowel pins on your first run. You have sorted the bins and hope that the mess problem is fixed. You then go on to tell Bill about the residual heat issue with the completed part.

Together, you go to the milling machine. Bill shows you how to check the oil and coolant levels. Everything looks good at the machine level. Next, on the CNC computer, you open the CNC G-code. While looking at the code, Bill points out that there are no coolant commands. Bill adds them in and when you rerun the program, it works.

Bill is glad you mentioned the problem to him. You are the third worker to mention G-code issues over the last week. You noticed the coolant problems in your G-code, John noticed a Z-axis issue in his G-code, and Sam had issues with both the Z-axis and the coolant. Chances are, there is a bigger problem and Bill will need to investigate the root cause .

Talking with Bill, you discuss the best way to fix the problem. Bill suggests editing the G-code to raise the Z-axis of the spindle before it returns to its home position. You open the CNC control program and edit the G-code. Following the setup sheet, you re-setup the job and use the CNC single step/single block function, to do another dry run of the part. Everything looks good, so you run the job again and create the first part. It works. Since you need four of each component, you move on to creating the rest of them before cleaning up and leaving for the day.

It's a new day and you have new components to create. As you are setting up, you go in search of some short dowel pins. You discover that the bins are a mess and components have not been put away in the correctly labeled bins. You wonder if this was the cause of yesterday's problem. As you reorganize the bins and straighten up the mess, you decide to mention the mess issue to Bill in your afternoon meeting.

You describe the bin mess and using the incorrect length dowels to Bill. He is glad you mentioned the problem to him. You are not the first person to mention similar issues with tools and parts not being put away correctly. Chances are there is a bigger safety issue here that needs to be addressed in the next staff meeting.

In any workplace, following proper safety and cleanup procedures is always important. This is especially crucial in manufacturing where people are constantly working with heavy, costly and sometimes dangerous equipment. When issues and problems arise, it is important that they are addressed in an efficient and timely manner. Effective communication is an important tool because it can prevent problems from recurring, avoid injury to personnel, reduce rework and scrap, and ultimately, reduce cost, and save money.

You now know that the end mill was damaged when it hit the dowel pin. It seems to you that the easiest thing to do would be to edit the G-code and raise the Z-axis position of the spindle before it returns to the home position. You open the CNC control program and edit the G-code, raising the Z-axis. Starting over, you follow the setup sheet and re-setup the job. This time, you use the CNC single step/single block function, to do another dry run of the part. Everything looks good, so you run the job again and create the first part.

At the end of the day, you are reviewing your progress with your trainer, Bill. After you describe the day's events, he reminds you to always think about safety and the importance of following work procedures. He decides to bring the issue up in the next morning meeting as a reminder to everyone.

In any workplace, following proper procedures (especially those that involve safety) is always important. This is especially crucial in manufacturing where people are constantly working with heavy, costly, and sometimes dangerous equipment. When issues and problems arise, it is important that they are addressed in an efficient and timely manner. Effective communication is an important tool because it can prevent problems from recurring, avoid injury to personnel, reduce rework and scrap, and ultimately, reduce cost, and save money. One tool to improve communication is the morning meeting or huddle.

The next morning, you check the G-code to determine what is wrong with the coolant. You notice that the coolant is turned off at the beginning of the code and also at the end of the code. This is strange. You change the G-code to turn the coolant on at the beginning of the run and off at the end. This works and you create the rest of the parts.

Throughout the day, you keep wondering what caused the G-code error. At lunch, you mention the G-code error to your coworker, John. John is not surprised. He said that he encountered a similar problem earlier this week. You decide to talk with your supervisor the next time you see him.

You are in luck. You see your supervisor by the door getting ready to leave. You hurry over to talk with him. You start off by telling him about how you asked Bill for help. Then you tell him there was a problem and the end mill was damaged. You describe the coolant problem in the G-code. Oh, and by the way, John has seen a similar problem before.

Your supervisor doesn't seem overly concerned, errors happen. He tells you "Good job, I am glad you were able to fix the issue." You are not sure whether your supervisor understood your explanation of what happened or that it had happened before.

The challenge of communicating in the workplace is learning how to share your ideas and concerns. If you need to tell your supervisor that something is not going well, it is important to remember that timing, preparation, and attitude are extremely important.

It is the end of your shift, but you want to let the next shift know that the coolant didn't turn on. You do not see your trainer or supervisor around. You decide to leave a note for the next shift so they are aware of the possible coolant problem. You write a sticky note and leave it on the monitor of the CNC control system.

How effective do you think this solution was? Did it address the problem?

In this scenario, you discovered several problems with the G-code that need to be addressed. When issues and problems arise, it is important that they are addressed in an efficient and timely manner. Effective communication is an important tool because it can prevent problems from recurring and avoid injury to personnel. The challenge of communicating in the workplace is learning how and when to share your ideas and concerns. If you need to tell your co-workers or supervisor that there is a problem, it is important to remember that timing and the method of communication are extremely important.

You are able to fix the coolant problem in the G-code. While you are glad that the problem is fixed, you are worried about why it happened in the first place. It is important to remember that if a problem keeps reappearing, you may not be fixing the right problem. You may only be addressing the symptoms.

You decide to talk to your trainer. Bill is glad you mentioned the problem to him. You are the third worker to mention G-code issues over the last week. You noticed the coolant problems in your G-code, John noticed a Z-axis issue in his G-code, and Sam had issues with both the Z-axis and the coolant. Chances are, there is a bigger problem and Bill will need to investigate the root cause .

Over lunch, you ask your coworkers about the G-code problem and what may be causing the error. Several people mention having similar problems but do not know the cause.

You have now talked to three coworkers who have all experienced similar coolant G-code problems. You make a list of who had the problem, when they had the problem, and what each person told you.

When you see your supervisor later that afternoon, you are ready to talk with him. You describe the problem you had with your component and the damaged bit. You then go on to tell him about talking with Bill and discovering the G-code issue. You show him your notes on your coworkers' coolant issues, and explain that you think there might be a bigger problem.

You supervisor thanks you for your initiative in identifying this problem. It sounds like there is a bigger problem and he will need to investigate the root cause. He decides to call a team huddle to discuss the issue, gather more information, and talk with the team about the importance of communication.

Root Cause Analysis

Root cause analysis ( RCA ) is a method of problem solving that identifies the underlying causes of an issue. Root cause analysis helps people answer the question of why the problem occurred in the first place. RCA uses clear cut steps in its associated tools, like the "5 Whys Analysis" and the "Cause and Effect Diagram," to identify the origin of the problem, so that you can:

• Determine what happened.
• Determine why it happened.
• Fix the problem so it won’t happen again.

RCA works under the idea that systems and events are connected. An action in one area triggers an action in another, and another, and so on. By tracing back these actions, you can discover where the problem started and how it developed into the problem you're now facing. Root cause analysis can prevent problems from recurring, reduce injury to personnel, reduce rework and scrap, and ultimately, reduce cost and save money. There are many different RCA techniques available to determine the root cause of a problem. These are just a few:

• Root Cause Analysis Tools
• 5 Whys Analysis
• Fishbone or Cause and Effect Diagram
• Pareto Analysis

How Huddles Work

Communication is a vital part of any setting where people work together. Effective communication helps employees and managers form efficient teams. It builds trusts between employees and management, and reduces unnecessary competition because each employee knows how their part fits in the larger goal.

One tool that management can use to promote communication in the workplace is the huddle . Just like football players on the field, a huddle is a short meeting where everyone is standing in a circle. A daily team huddle ensures that team members are aware of changes to the schedule, reiterated problems and safety issues, and how their work impacts one another. When done right, huddles create collaboration, communication, and accountability to results. Impromptu huddles can be used to gather information on a specific issue and get each team member's input.

The most important thing to remember about huddles is that they are short, lasting no more than 10 minutes, and their purpose is to communicate and identify. In essence, a huddle’s purpose is to identify priorities, communicate essential information, and discover roadblocks to productivity.

Who uses huddles? Many industries and companies use daily huddles. At first thought, most people probably think of hospitals and their daily patient update meetings, but lots of managers use daily meetings to engage their employees. Here are a few examples:

• Brian Scudamore, CEO of 1-800-Got-Junk? , uses the daily huddle as an operational tool to take the pulse of his employees and as a motivational tool. Watch a morning huddle meeting .
• Fusion OEM, an outsourced manufacturing and production company. What do employees take away from the daily huddle meeting .
• Biz-Group, a performance consulting group. Tips for a successful huddle .

Brainstorming

One tool that can be useful in problem solving is brainstorming . Brainstorming is a creativity technique designed to generate a large number of ideas for the solution to a problem. The method was first popularized in 1953 by Alex Faickney Osborn in the book Applied Imagination . The goal is to come up with as many ideas as you can in a fixed amount of time. Although brainstorming is best done in a group, it can be done individually. Like most problem solving techniques, brainstorming is a process.

• Define a clear objective.
• Have an agreed a time limit.
• During the brainstorming session, write down everything that comes to mind, even if the idea sounds crazy.
• If one idea leads to another, write down that idea too.
• Combine and refine ideas into categories of solutions.
• Assess and analyze each idea as a potential solution.

When used during problem solving, brainstorming can offer companies new ways of encouraging staff to think creatively and improve production. Brainstorming relies on team members' diverse experiences, adding to the richness of ideas explored. This means that you often find better solutions to the problems. Team members often welcome the opportunity to contribute ideas and can provide buy-in for the solution chosen—after all, they are more likely to be committed to an approach if they were involved in its development. What's more, because brainstorming is fun, it helps team members bond.

• Watch Peggy Morgan Collins, a marketing executive at Power Curve Communications discuss How to Stimulate Effective Brainstorming .
• Watch Kim Obbink, CEO of Filter Digital, a digital content company, and her team share their top five rules for How to Effectively Generate Ideas .

Importance of Good Communication and Problem Description

Communication is one of the most frequent activities we engage in on a day-to-day basis. At some point, we have all felt that we did not effectively communicate an idea as we would have liked. The key to effective communication is preparation. Rather than attempting to haphazardly improvise something, take a few minutes and think about what you want say and how you will say it. If necessary, write yourself a note with the key points or ideas in the order you want to discuss them. The notes can act as a reminder or guide when you talk to your supervisor.

Tips for clear communication of an issue:

• Provide a clear summary of your problem. Start at the beginning, give relevant facts, timelines, and examples.
• Avoid including your opinion or personal attacks in your explanation.
• Avoid using words like "always" or "never," which can give the impression that you are exaggerating the problem.
• If this is an ongoing problem and you have collected documentation, give it to your supervisor once you have finished describing the problem.
• Remember to listen to what's said in return; communication is a two-way process.

Not all communication is spoken. Body language is nonverbal communication that includes your posture, your hands and whether you make eye contact. These gestures can be subtle or overt, but most importantly they communicate meaning beyond what is said. When having a conversation, pay attention to how you stand. A stiff position with arms crossed over your chest may imply that you are being defensive even if your words state otherwise. Shoving your hands in your pockets when speaking could imply that you have something to hide. Be wary of using too many hand gestures because this could distract listeners from your message.

The challenge of communicating in the workplace is learning how and when to share your ideas or concerns. If you need to tell your supervisor or co-worker about something that is not going well, keep in mind that good timing and good attitude will go a long way toward helping your case.

Like all skills, effective communication needs to be practiced. Toastmasters International is perhaps the best known public speaking organization in the world. Toastmasters is open to anyone who wish to improve their speaking skills and is willing to put in the time and effort to do so. To learn more, visit Toastmasters International .

Methods of Communication

Communication of problems and issues in any workplace is important, particularly when safety is involved. It is therefore crucial in manufacturing where people are constantly working with heavy, costly, and sometimes dangerous equipment. As issues and problems arise, they need to be addressed in an efficient and timely manner. Effective communication is an important skill because it can prevent problems from recurring, avoid injury to personnel, reduce rework and scrap, and ultimately, reduce cost and save money.

There are many different ways to communicate: in person, by phone, via email, or written. There is no single method that fits all communication needs, each one has its time and place.

In person: In the workplace, face-to-face meetings should be utilized whenever possible. Being able to see the person you need to speak to face-to-face gives you instant feedback and helps you gauge their response through their body language. Be careful of getting sidetracked in conversation when you need to communicate a problem.

Email: Email has become the communication standard for most businesses. It can be accessed from almost anywhere and is great for things that don’t require an immediate response. Email is a great way to communicate non-urgent items to large amounts of people or just your team members. One thing to remember is that most people's inboxes are flooded with emails every day and unless they are hyper vigilant about checking everything, important items could be missed. For issues that are urgent, especially those around safety, email is not always be the best solution.

Phone: Phone calls are more personal and direct than email. They allow us to communicate in real time with another person, no matter where they are. Not only can talking prevent miscommunication, it promotes a two-way dialogue. You don’t have to worry about your words being altered or the message arriving on time. However, mobile phone use and the workplace don't always mix. In particular, using mobile phones in a manufacturing setting can lead to a variety of problems, cause distractions, and lead to serious injury.

Written: Written communication is appropriate when detailed instructions are required, when something needs to be documented, or when the person is too far away to easily speak with over the phone or in person.

There is no "right" way to communicate, but you should be aware of how and when to use the appropriate form of communication for your situation. When deciding the best way to communicate with a co-worker or manager, put yourself in their shoes, and think about how you would want to learn about the issue. Also, consider what information you would need to know to better understand the issue. Use your good judgment of the situation and be considerate of your listener's viewpoint.

Did you notice any other potential problems in the previous exercise?

• [Page 6:] Did you notice any other potential problems in the previous exercise?

Summary of Strategies

In this exercise, you were given a scenario in which there was a problem with a component you were creating on a CNC machine. You were then asked how you wanted to proceed. Depending on your path through this exercise, you might have found an easy solution and fixed it yourself, asked for help and worked with your trainer, or discovered an ongoing G-code problem that was bigger than you initially thought.

When issues and problems arise, it is important that they are addressed in an efficient and timely manner. Communication is an important tool because it can prevent problems from recurring, avoid injury to personnel, reduce rework and scrap, and ultimately, reduce cost, and save money. Although, each path in this exercise ended with a description of a problem solving tool for your toolbox, the first step is always to identify the problem and define the context in which it happened.

There are several strategies that can be used to identify the root cause of a problem. Root cause analysis (RCA) is a method of problem solving that helps people answer the question of why the problem occurred. RCA uses a specific set of steps, with associated tools like the “5 Why Analysis" or the “Cause and Effect Diagram,” to identify the origin of the problem, so that you can:

Once the underlying cause is identified and the scope of the issue defined, the next step is to explore possible strategies to fix the problem.

If you are not sure how to fix the problem, it is okay to ask for help. Problem solving is a process and a skill that is learned with practice. It is important to remember that everyone makes mistakes and that no one knows everything. Life is about learning. It is okay to ask for help when you don’t have the answer. When you collaborate to solve problems you improve workplace communication and accelerates finding solutions as similar problems arise.

One tool that can be useful for generating possible solutions is brainstorming . Brainstorming is a technique designed to generate a large number of ideas for the solution to a problem. The method was first popularized in 1953 by Alex Faickney Osborn in the book Applied Imagination. The goal is to come up with as many ideas as you can, in a fixed amount of time. Although brainstorming is best done in a group, it can be done individually.

Depending on your path through the exercise, you may have discovered that a couple of your coworkers had experienced similar problems. This should have been an indicator that there was a larger problem that needed to be addressed.

In any workplace, communication of problems and issues (especially those that involve safety) is always important. This is especially crucial in manufacturing where people are constantly working with heavy, costly, and sometimes dangerous equipment. When issues and problems arise, it is important that they be addressed in an efficient and timely manner. Effective communication is an important tool because it can prevent problems from recurring, avoid injury to personnel, reduce rework and scrap, and ultimately, reduce cost and save money.

One strategy for improving communication is the huddle . Just like football players on the field, a huddle is a short meeting with everyone standing in a circle. A daily team huddle is a great way to ensure that team members are aware of changes to the schedule, any problems or safety issues are identified and that team members are aware of how their work impacts one another. When done right, huddles create collaboration, communication, and accountability to results. Impromptu huddles can be used to gather information on a specific issue and get each team member's input.

To learn more about different problem solving strategies, choose an option below. These strategies accompany the outcomes of different decision paths in the problem solving exercise.

• View Problem Solving Strategies Select a strategy below... Root Cause Analysis How Huddles Work Brainstorming Importance of Good Problem Description Methods of Communication

Communication is one of the most frequent activities we engage in on a day-to-day basis. At some point, we have all felt that we did not effectively communicate an idea as we would have liked. The key to effective communication is preparation. Rather than attempting to haphazardly improvise something, take a few minutes and think about what you want say and how you will say it. If necessary, write yourself a note with the key points or ideas in the order you want to discuss them. The notes can act as a reminder or guide during your meeting.

• Provide a clear summary of the problem. Start at the beginning, give relevant facts, timelines, and examples.

In person: In the workplace, face-to-face meetings should be utilized whenever possible. Being able to see the person you need to speak to face-to-face gives you instant feedback and helps you gauge their response in their body language. Be careful of getting sidetracked in conversation when you need to communicate a problem.

There is no "right" way to communicate, but you should be aware of how and when to use the appropriate form of communication for the situation. When deciding the best way to communicate with a co-worker or manager, put yourself in their shoes, and think about how you would want to learn about the issue. Also, consider what information you would need to know to better understand the issue. Use your good judgment of the situation and be considerate of your listener's viewpoint.

"Never try to solve all the problems at once — make them line up for you one-by-one.” — Richard Sloma

Problem Solving: An Important Job Skill

Problem solving improves efficiency and communication on the shop floor. It increases a company's efficiency and profitability, so it's one of the top skills employers look for when hiring new employees. Recent industry surveys show that employers consider soft skills, such as problem solving, as critical to their business’s success.

The 2011 survey, "Boiling Point? The skills gap in U.S. manufacturing ," polled over a thousand manufacturing executives who reported that the number one skill deficiency among their current employees is problem solving, which makes it difficult for their companies to adapt to the changing needs of the industry.

In this video, industry professionals discuss their expectations and present tips for new employees joining the manufacturing workforce.

Quick Summary

• [Quick Summary: Question1] What are two things you learned in this case study?
• What question(s) do you still have about the case study?
• [Quick Summary: Question2] What question(s) do you still have about the case study?
• Is there anything you would like to learn more about with respect to this case study?
• [Quick Summary: Question3] Is there anything you would like to learn more about with respect to this case study?

Critical Thinking in Everyday Life: 9 Strategies

• Developing as Rational Persons: Viewing Our Development in Stages
• How to Study and Learn (Part One)
• How to Study and Learn (Part Two)
• How to Study and Learn (Part Three)
• How to Study and Learn (Part Four)
• The Art of Close Reading (Part One)
• The Art of Close Reading (Part Two)
• The Art of Close Reading (Part Three)
• Looking To The Future With a Critical Eye: A Message for High School Graduates
• Becoming a Critic Of Your Thinking
• For Young Students (Elementary/K-6)

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In this article, we will explain 9 strategies that any motivated person can use to develop as a thinker. As we explain the strategy, we will describe it as if we were talking directly to such a person. Further details to our descriptions may need to be added for those who know little about critical thinking. Here are the 9:

There is nothing magical about our ideas. No one of them is essential. Nevertheless, each represents a plausible way to begin to do something concrete to improve thinking in a regular way. Though you probably can’t do all of these at the same time, we recommend an approach in which you experiment with all of these over an extended period of time.

First Strategy: Use “Wasted” Time. All humans waste some time; that is, fail to use all of their time productively or even pleasurably. Sometimes we jump from one diversion to another, without enjoying any of them. Sometimes we become irritated about matters beyond our control. Sometimes we fail to plan well causing us negative consequences we could easily have avoided (for example, we spend time unnecessarily trapped in traffic — though we could have left a half hour earlier and avoided the rush). Sometimes we worry unproductively. Sometimes we spend time regretting what is past. Sometimes we just stare off blankly into space.

The key is that the time is “gone” even though, if we had thought about it and considered our options, we would never have deliberately spent our time in the way we did. So why not take advantage of the time you normally waste by practicing your critical thinking during that otherwise wasted time? For example, instead of sitting in front of the TV at the end of the day flicking from channel to channel in a vain search for a program worth watching, spend that time, or at least part of it, thinking back over your day and evaluating your strengths and weaknesses. For example, you might ask yourself questions like these:

When did I do my worst thinking today? When did I do my best? What in fact did I think about today? Did I figure anything out? Did I allow any negative thinking to frustrate me unnecessarily? If I had to repeat today what would I do differently? Why? Did I do anything today to further my long-term goals? Did I act in accordance with my own expressed values? If I spent every day this way for 10 years, would I at the end have accomplished something worthy of that time?

It would be important of course to take a little time with each question. It would also be useful to record your observations so that you are forced to spell out details and be explicit in what you recognize and see. As time passes, you will notice patterns in your thinking. Second Strategy: A Problem A Day. At the beginning of each day (perhaps driving to work or going to school) choose a problem to work on when you have free moments. Figure out the logic of the problem by identifying its elements. In other words, systematically think through the questions: What exactly is the problem? How can I put it into the form of a question. How does it relate to my goals, purposes, and needs?

Third Strategy: Internalize Intellectual Standards. Each week, develop a heightened awareness of one of the universal intellectual standards (clarity, precision, accuracy, relevance, depth, breadth, logicalness, significance). Focus one week on clarity, the next on accuracy, etc. For example, if you are focusing on clarity for the week, try to notice when you are being unclear in communicating with others. Notice when others are unclear in what they are saying.

When you are reading, notice whether you are clear about what you are reading. When you orally express or write out your views (for whatever reason), ask yourself whether you are clear about what you are trying to say. In doing this, of course, focus on four techniques of clarification : 1) Stating what you are saying explicitly and precisely (with careful consideration given to your choice of words), 2) Elaborating on your meaning in other words, 3) Giving examples of what you mean from experiences you have had, and 4) Using analogies , metaphors, pictures, or diagrams to illustrate what you mean. In other words, you will frequently STATE, ELABORATE, ILLUSTRATE, AND EXEMPLIFY your points. You will regularly ask others to do the same.

Fourth Strategy: Keep An Intellectual Journal. Each week, write out a certain number of journal entries. Use the following format (keeping each numbered stage separate):

Strategy Five: Reshape Your Character. Choose one intellectual trait---intellectual perseverance, autonomy, empathy, courage, humility, etc.--- to strive for each month, focusing on how you can develop that trait in yourself. For example, concentrating on intellectual humility, begin to notice when you admit you are wrong. Notice when you refuse to admit you are wrong, even in the face of glaring evidence that you are in fact wrong. Notice when you become defensive when another person tries to point out a deficiency in your work, or your thinking. Notice when your intellectual arrogance keeps you from learning, for example, when you say to yourself “I already know everything I need to know about this subject.” Or, “I know as much as he does. Who does he think he is forcing his opinions on me?” By owning your “ignorance,” you can begin to deal with it.

Strategy Six: Deal with Your Egocentrism. Egocentric thinking is found in the disposition in human nature to think with an automatic subconscious bias in favor of oneself. On a daily basis, you can begin to observe your egocentric thinking in action by contemplating questions like these: Under what circumstances do I think with a bias in favor of myself? Did I ever become irritable over small things? Did I do or say anything “irrational” to get my way? Did I try to impose my will upon others? Did I ever fail to speak my mind when I felt strongly about something, and then later feel resentment? Once you identify egocentric thinking in operation, you can then work to replace it with more rational thought through systematic self-reflection, thinking along the lines of: What would a rational person feel in this or that situation? What would a rational person do? How does that compare with what I want to do? (Hint: If you find that you continually conclude that a rational person would behave just as you behaved you are probably engaging in self-deception.)

Strategy Seven: Redefine the Way You See Things . We live in a world, both personal and social, in which every situation is “defined,” that is, given a meaning. How a situation is defined determines not only how we feel about it, but also how we act in it, and what implications it has for us. However, virtually every situation can be defined in more than one way. This fact carries with it tremendous opportunities. In principle, it lies within your power and mine to make our lives more happy and fulfilling than they are. Many of the negative definitions that we give to situations in our lives could in principle be transformed into positive ones. We can be happy when otherwise we would have been sad.

We can be fulfilled when otherwise we would have been frustrated. In this strategy, we practice redefining the way we see things, turning negatives into positives, dead-ends into new beginnings, mistakes into opportunities to learn. To make this strategy practical, we should create some specific guidelines for ourselves. For example, we might make ourselves a list of five to ten recurrent negative contexts in which we feel frustrated, angry, unhappy, or worried. We could then identify the definition in each case that is at the root of the negative emotion. We would then choose a plausible alternative definition for each and then plan for our new responses as well as new emotions. For example, if you tend to worry about all problems, both the ones you can do something about and those that you can’t; you can review the thinking in this nursery rhyme: “For every problem under the sun, there is a solution or there is none. If there be one, think til you find it. If there be none, then never mind it.”

Let’s look at another example. You do not have to define your initial approach to a member of the opposite sex in terms of the definition “his/her response will determine whether or not I am an attractive person.” Alternatively, you could define it in terms of the definition “let me test to see if this person is initially drawn to me—given the way they perceive me.” With the first definition in mind, you feel personally put down if the person is not “interested” in you; with the second definition you explicitly recognize that people respond not to the way a stranger is, but the way they look to them subjectively. You therefore do not take a failure to show interest in you (on the part of another) as a “defect” in you.

Strategy Eight: Get in touch with your emotions: Whenever you feel some negative emotion, systematically ask yourself: What, exactly, is the thinking leading to this emotion? For example, if you are angry, ask yourself, what is the thinking that is making me angry? What other ways could I think about this situation? For example, can you think about the situation so as to see the humor in it and what is pitiable in it? If you can, concentrate on that thinking and your emotions will (eventually) shift to match it.

Strategy Nine: Analyze group influences on your life: Closely analyze the behavior that is encouraged, and discouraged, in the groups to which you belong. For any given group, what are you "required" to believe? What are you "forbidden" to do? Every group enforces some level of conformity. Most people live much too much within the view of themselves projected by others. Discover what pressure you are bowing to and think explicitly about whether or not to reject that pressure.

Conclusion: The key point to keep in mind when devising strategies is that you are engaged in a personal experiment. You are testing ideas in your everyday life. You are integrating them, and building on them, in the light of your actual experience. For example, suppose you find the strategy “Redefine the Way You See Things” to be intuitive to you. So you use it to begin. Pretty soon you find yourself noticing the social definitions that rule many situations in your life. You recognize how your behavior is shaped and controlled by the definitions in use:

• “I’m giving a party,” (Everyone therefore knows to act in a “partying” way)
• “The funeral is Tuesday,” (There are specific social behaviors expected at a funeral)
• “Jack is an acquaintance, not really a friend.” (We behave very differently in the two cases)

You begin to see how important and pervasive social definitions are. You begin to redefine situations in ways that run contrary to some commonly accepted definitions. You notice then how redefining situations (and relationships) enables you to “Get in Touch With Your Emotions.” You recognize that the way you think (that is, define things) generates the emotions you experience. When you think you are threatened (i.e., define a situation as “threatening”), you feel fear. If you define a situation as a “failure,” you may feel depressed. On the other hand, if you define that same situation as a “lesson or opportunity to learn” you feel empowered to learn. When you recognize this control that you are capable of exercising, the two strategies begin to work together and reinforce each other.

Next consider how you could integrate strategy #9 (“Analyze group influences on your life”) into your practice. One of the main things that groups do is control us by controlling the definitions we are allowed to operate with. When a group defines some things as “cool” and some as “dumb, ” the members of the group try to appear “cool” and not appear “dumb.” When the boss of a business says, “That makes a lot of sense,” his subordinates know they are not to say, “No, it is ridiculous.” And they know this because defining someone as the “boss” gives him/her special privileges to define situations and relationships.

You now have three interwoven strategies: you “Redefine the Way You See Things,” “Get in touch with your emotions,” and “Analyze group influences on your life.” The three strategies are integrated into one. You can now experiment with any of the other strategies, looking for opportunities to integrate them into your thinking and your life. If you follow through on some plan analogous to what we have described, you are developing as a thinker. More precisely, you are becoming a “Practicing” Thinker. Your practice will bring advancement. And with advancement, skilled and insightful thinking may becomes more and more natural to you.

Paul, R. & Elder, L. (2001). Modified from the book by Paul, R. & Elder, L. (2001). Critical Thinking: Tools for Taking Charge of Your Learning and Your Life .

For full copies of this and many other critical thinking articles, books, videos, and more, join us at the Center for Critical Thinking Community Online - the world's leading online community dedicated to critical thinking!   Also featuring interactive learning activities, study groups, and even a social media component, this learning platform will change your conception of intellectual development.

High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

Chapter: part one: connecting mathematics with work and life, part one— connecting mathematics with work and life.

Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics. (NRC, 1989, p. 1)

The above statement remains true today, although it was written almost ten years ago in the Mathematical Sciences Education Board's (MSEB) report Everybody Counts (NRC, 1989). In envisioning a future in which all students will be afforded such opportunities, the MSEB acknowledges the crucial role played by formulae and algorithms, and suggests that algorithmic skills are more flexible, powerful, and enduring when they come from a place of meaning and understanding. This volume takes as a premise that all students can develop mathematical understanding by working with mathematical tasks from workplace and everyday contexts . The essays in this report provide some rationale for this premise and discuss some of the issues and questions that follow. The tasks in this report illuminate some of the possibilities provided by the workplace and everyday life.

Contexts from within mathematics also can be powerful sites for the development of mathematical understanding, as professional and amateur mathematicians will attest. There are many good sources of compelling problems from within mathematics, and a broad mathematics education will include experience with problems from contexts both within and outside mathematics. The inclusion of tasks in this volume is intended to highlight particularly compelling problems whose context lies outside of mathematics, not to suggest a curriculum.

The operative word in the above premise is "can." The understandings that students develop from any encounter with mathematics depend not only on the context, but also on the students' prior experience and skills, their ways of thinking, their engagement with the task, the environment in which they explore the task—including the teacher, the students, and the tools—the kinds of interactions that occur in that environment, and the system of internal and external incentives that might be associated with the activity. Teaching and learning are complex activities that depend upon evolving and rarely articulated interrelationships among teachers, students, materials, and ideas. No prescription for their improvement can be simple.

This volume may be beneficially seen as a rearticulation and elaboration of a principle put forward in Reshaping School Mathematics :

Principle 3: Relevant Applications Should be an Integral Part of the Curriculum.

Students need to experience mathematical ideas in the context in which they naturally arise—from simple counting and measurement to applications in business and science. Calculators and computers make it possible now to introduce realistic applications throughout the curriculum.

The significant criterion for the suitability of an application is whether it has the potential to engage students' interests and stimulate their mathematical thinking. (NRC, 1990, p. 38)

Mathematical problems can serve as a source of motivation for students if the problems engage students' interests and aspirations. Mathematical problems also can serve as sources of meaning and understanding if the problems stimulate students' thinking. Of course, a mathematical task that is meaningful to a student will provide more motivation than a task that does not make sense. The rationale behind the criterion above is that both meaning and motivation are required. The motivational benefits that can be provided by workplace and everyday problems are worth mentioning, for although some students are aware that certain mathematics courses are necessary in order to gain entry into particular career paths, many students are unaware of how particular topics or problem-solving approaches will have relevance in any workplace. The power of using workplace and everyday problems to teach mathematics lies not so much in motivation, however, for no con-

text by itself will motivate all students. The real power is in connecting to students' thinking.

There is growing evidence in the literature that problem-centered approaches—including mathematical contexts, "real world" contexts, or both—can promote learning of both skills and concepts. In one comparative study, for example, with a high school curriculum that included rich applied problem situations, students scored somewhat better than comparison students on algebraic procedures and significantly better on conceptual and problem-solving tasks (Schoen & Ziebarth, 1998). This finding was further verified through task-based interviews. Studies that show superior performance of students in problem-centered classrooms are not limited to high schools. Wood and Sellers (1996), for example, found similar results with second and third graders.

Research with adult learners seems to indicate that "variation of contexts (as well as the whole task approach) tends to encourage the development of general understanding in a way which concentrating on repeated routine applications of algorithms does not and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. 163). This conclusion is consistent with the notion that using a variety of contexts can increase the chance that students can show what they know. By increasing the number of potential links to the diverse knowledge and experience of the students, more students have opportunities to excel, which is to say that the above premise can promote equity in mathematics education.

There is also evidence that learning mathematics through applications can lead to exceptional achievement. For example, with a curriculum that emphasizes modeling and applications, high school students at the North Carolina School of Science and Mathematics have repeatedly submitted winning papers in the annual college competition, Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).

The relationships among teachers, students, curricular materials, and pedagogical approaches are complex. Nonetheless, the literature does supports the premise that workplace and everyday problems can enhance mathematical learning, and suggests that if students engage in mathematical thinking, they will be afforded opportunities for building connections, and therefore meaning and understanding.

In the opening essay, Dale Parnell argues that traditional teaching has been missing opportunities for connections: between subject-matter and context, between academic and vocational education, between school and life, between knowledge and application, and between subject-matter disciplines. He suggests that teaching must change if more students are to learn mathematics. The question, then, is how to exploit opportunities for connections between high school mathematics and the workplace and everyday life.

Rol Fessenden shows by example the importance of mathematics in business, specifically in making marketing decisions. His essay opens with a dialogue among employees of a company that intends to expand its business into

Japan, and then goes on to point out many of the uses of mathematics, data collection, analysis, and non-mathematical judgment that are required in making such business decisions.

In his essay, Thomas Bailey suggests that vocational and academic education both might benefit from integration, and cites several trends to support this suggestion: change and uncertainty in the workplace, an increased need for workers to understand the conceptual foundations of key academic subjects, and a trend in pedagogy toward collaborative, open-ended projects. Further-more, he observes that School-to-Work experiences, first intended for students who were not planning to attend a four-year college, are increasingly being seen as useful in preparing students for such colleges. He discusses several such programs that use work-related applications to teach academic skills and to prepare students for college. Integration of academic and vocational education, he argues, can serve the dual goals of "grounding academic standards in the realistic context of workplace requirements and introducing a broader view of the potential usefulness of academic skills even for entry level workers."

Noting the importance and utility of mathematics for jobs in science, health, and business, Jean Taylor argues for continued emphasis in high school of topics such as algebra, estimation, and trigonometry. She suggests that workplace and everyday problems can be useful ways of teaching these ideas for all students.

There are too many different kinds of workplaces to represent even most of them in the classrooms. Furthermore, solving mathematics problems from some workplace contexts requires more contextual knowledge than is reasonable when the goal is to learn mathematics. (Solving some other workplace problems requires more mathematical knowledge than is reasonable in high school.) Thus, contexts must be chosen carefully for their opportunities for sense making. But for students who have knowledge of a workplace, there are opportunities for mathematical connections as well. In their essay, Daniel Chazan and Sandra Callis Bethell describe an approach that creates such opportunities for students in an algebra course for 10th through 12th graders, many of whom carried with them a "heavy burden of negative experiences" about mathematics. Because the traditional Algebra I curriculum had been extremely unsuccessful with these students, Chazan and Bethell chose to do something different. One goal was to help students see mathematics in the world around them. With the help of community sponsors, Chazen and Bethell asked students to look for mathematics in the workplace and then describe that mathematics and its applications to their classmates.

The tasks in Part One complement the points made in the essays by making direct connections to the workplace and everyday life. Emergency Calls (p. 42) illustrates some possibilities for data analysis and representation by discussing the response times of two ambulance companies. Back-of-the-Envelope Estimates (p. 45) shows how quick, rough estimates and calculations

are useful for making business decisions. Scheduling Elevators (p. 49) shows how a few simplifying assumptions and some careful reasoning can be brought together to understand the difficult problem of optimally scheduling elevators in a large office building. Finally, in the context of a discussion with a client of an energy consulting firm, Heating-Degree-Days (p. 54) illuminates the mathematics behind a common model of energy consumption in home heating.

Cronin, T. P. (1988). High school students win "college" competition. Consortium: The Newsletter of the Consortium for Mathematics and Its Applications , 26 , 3, 12.

Miller, D. E. (1995). North Carolina sweeps MCM '94. SIAM News , 28 (2).

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education . Washington, DC: National Academy Press.

National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum . Washington, DC: National Academy Press.

Schoen, H. L. & Ziebarth, S. W. (1998). Assessment of students' mathematical performance (A Core-Plus Mathematics Project Field Test Progress Report). Iowa City: Core Plus Mathematics Project Evaluation Site, University of Iowa.

Strässer, R., Barr, G. Evans, J. & Wolf, A. (1991). Skills versus understanding. In M. Harris (Ed.), Schools, mathematics, and work (pp. 158-168). London: The Falmer Press.

Wood, T. & Sellers, P. (1996). Assessment of a problem-centered mathematics program: Third grade. Journal for Research in Mathematics Education , 27 (3), 337-353.

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

The study of mathematics stands, in many ways, as a gateway to student success in education. This is becoming particularly true as our society moves inexorably into the technological age. Therefore, it is vital that more students develop higher levels of competency in mathematics. 1

The standards and expectations for students must be high, but that is only half of the equation. The more important half is the development of teaching techniques and methods that will help all students (rather than just some students) reach those higher expectations and standards. This will require some changes in how mathematics is taught.

Effective education must give clear focus to connecting real life context with subject-matter content for the student, and this requires a more ''connected" mathematics program. In many of today's classrooms, especially in secondary school and college, teaching is a matter of putting students in classrooms marked "English," "history," or "mathematics," and then attempting to fill their heads with facts through lectures, textbooks, and the like. Aside from an occasional lab, workbook, or "story problem," the element of contextual teaching and learning is absent, and little attempt is made to connect what students are learning with the world in which they will be expected to work and spend their lives. Often the frag-

mented information offered to students is of little use or application except to pass a test.

What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.

I recently had occasion to interview 75 students representing seven different high schools in the Northwest. In nearly all cases, the students were juniors identified as vocational or general education students. The comment of one student stands out as representative of what most of these students told me in one way or another: "I know it's up to me to get an education, but a lot of times school is just so dull and boring. … You go to this class, go to that class, study a little of this and a little of that, and nothing connects. … I would like to really understand and know the application for what I am learning." Time and again, students were asking, "Why do I have to learn this?" with few sensible answers coming from the teachers.

My own long experience as a community college president confirms the thoughts of these students. In most community colleges today, one-third to one-half of the entering students are enrolled in developmental (remedial) education, trying to make up for what they did not learn in earlier education experiences. A large majority of these students come to the community college with limited mathematical skills and abilities that hardly go beyond adding, subtracting, and multiplying with whole numbers. In addition, the need for remediation is also experienced, in varying degrees, at four-year colleges and universities.

What is the greatest sin committed in the teaching of mathematics today? It is the failure to help students use the magnificent power of the brain to make connections between the following:

• subject-matter content and the context of use;
• academic and vocational education;
• school and other life experiences;
• knowledge and application of knowledge; and
• one subject-matter discipline and another.

Why is such failure so critical? Because understanding the idea of making the connection between subject-matter content and the context of application

is what students, at all levels of education, desperately require to survive and succeed in our high-speed, high-challenge, rapidly changing world.

Educational policy makers and leaders can issue reams of position papers on longer school days and years, site-based management, more achievement tests and better assessment practices, and other "hot" topics of the moment, but such papers alone will not make the crucial difference in what students know and can do. The difference will be made when classroom teachers begin to connect learning with real-life experiences in new, applied ways, and when education reformers begin to focus upon learning for meaning.

A student may memorize formulas for determining surface area and measuring angles and use those formulas correctly on a test, thereby achieving the behavioral objectives set by the teacher. But when confronted with the need to construct a building or repair a car, the same student may well be left at sea because he or she hasn't made the connection between the formulas and their real-life application. When students are asked to consider the Pythagorean Theorem, why not make the lesson active, where students actually lay out the foundation for a small building like a storage shed?

What a difference mathematics instruction could make for students if it were to stress the context of application—as well as the content of knowledge—using the problem-solving model over the freezer model. Teaching conducted upon the connected model would help more students learn with their thinking brain, as well as with their memory brain, developing the competencies and tools they need to survive and succeed in our complex, interconnected society.

One step toward this goal is to develop mathematical tasks that integrate subject-matter content with the context of application and that are aimed at preparing individuals for the world of work as well as for post-secondary education. Since many mathematics teachers have had limited workplace experience, they need many good examples of how knowledge of mathematics can be applied to real life situations. The trick in developing mathematical tasks for use in classrooms will be to keep the tasks connected to real life situations that the student will recognize. The tasks should not be just a contrived exercise but should stay as close to solving common problems as possible.

As an example, why not ask students to compute the cost of 12 years of schooling in a public school? It is a sad irony that after 12 years of schooling most students who attend the public schools have no idea of the cost of their schooling or how their education was financed. No wonder that some public schools have difficulty gaining financial support! The individuals being served by the schools have never been exposed to the real life context of who pays for the schools and why. Somewhere along the line in the teaching of mathematics, this real life learning opportunity has been missed, along with many other similar contextual examples.

The mathematical tasks in High School Mathematics at Work provide students (and teachers) with a plethora of real life mathematics problems and

challenges to be faced in everyday life and work. The challenge for teachers will be to develop these tasks so they relate as close as possible to where students live and work every day.

Parnell, D. (1985). The neglected majority . Washington, DC: Community College Press.

Parnell, D. (1995). Why do I have to learn this ? Waco, TX: CORD Communications.

D ALE P ARNELL is Professor Emeritus of the School of Education at Oregon State University. He has served as a University Professor, College President, and for ten years as the President and Chief Executive Officer of the American Association of Community Colleges. He has served as a consultant to the National Science Foundation and has served on many national commissions, such as the Secretary of Labor's Commission on Achieving Necessary Skills (SCANS). He is the author of the book The Neglected Majority which provided the foundation for the federally-funded Tech Prep Associate Degree Program.

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

"OK, the agenda of the meeting is to review the status of our launch into Japan. You can see the topics and presenters on the list in front of you. Gregg, can you kick it off with a strategy review?"

"Happy to, Bob. We have assessed the possibilities, costs, and return on investment of opening up both store and catalog businesses in other countries. Early research has shown that both Japan and Germany are good candidates. Specifically, data show high preference for good quality merchandise, and a higher-than-average propensity for an active outdoor lifestyle in both countries. Education, age, and income data are quite different from our target market in the U.S., but we do not believe that will be relevant because the cultures are so different. In addition, the Japanese data show that they have a high preference for things American, and, as you know, we are a classic American company. Name recognition for our company is 14%, far higher than any of our American competition in Japan. European competitors are virtually unrecognized, and other Far Eastern competitors are perceived to be of lower quality than us. The data on these issues are quite clear.

"Nevertheless, you must understand that there is a lot of judgment involved in the decision to focus on Japan. The analyses are limited because the cultures are different and we expect different behavioral drivers. Also,

much of the data we need in Japan are simply not available because the Japanese marketplace is less well developed than in the U.S. Drivers' license data, income data, lifestyle data, are all commonplace here and unavailable there. There is little prior penetration in either country by American retailers, so there is no experience we can draw upon. We have all heard how difficult it will be to open up sales operations in Japan, but recent sales trends among computer sellers and auto parts sales hint at an easing of the difficulties.

"The plan is to open three stores a year, 5,000 square feet each. We expect to do $700/square foot, which is more than double the experience of American retailers in the U.S. but 45% less than our stores. In addition, pricing will be 20% higher to offset the cost of land and buildings. Asset costs are approximately twice their rate in the U.S., but labor is slightly less. Benefits are more thoroughly covered by the government. Of course, there is a lot of uncertainty in the sales volumes we are planning. The pricing will cover some of the uncertainty but is still less than comparable quality goods already being offered in Japan.

"Let me shift over to the competition and tell you what we have learned. We have established long-term relationships with 500 to 1000 families in each country. This is comparable to our practice in the U.S. These families do not know they are working specifically with our company, as this would skew their reporting. They keep us appraised of their catalog and shopping experiences, regardless of the company they purchase from. The sample size is large enough to be significant, but, of course, you have to be careful about small differences.

"All the families receive our catalog and catalogs from several of our competitors. They match the lifestyle, income, and education demographic profiles of the people we want to have as customers. They are experienced catalog shoppers, and this will skew their feedback as compared to new catalog shoppers.

"One competitor is sending one 100-page catalog per quarter. The product line is quite narrow—200 products out of a domestic line of 3,000. They have selected items that are not likely to pose fit problems: primarily outerwear and knit shirts, not many pants, mostly men's goods, not women's. Their catalog copy is in Kanji, but the style is a bit stilted we are told, probably because it was written in English and translated, but we need to test this hypothesis. By contrast, we have simply mailed them the same catalog we use in the U.S., even written in English.

"Customer feedback has been quite clear. They prefer our broader assortment by a ratio of 3:1, even though they don't buy most of the products. As the competitors figured, sales are focused on outerwear and knits, but we are getting more sales, apparently because they like looking at the catalog and spend more time with it. Again, we need further testing. Another hypothesis is that our brand name is simply better known.

"Interestingly, they prefer our English-language version because they find it more of an adventure to read the catalog in another language. This is probably

a built-in bias of our sampling technique because we specifically selected people who speak English. We do not expect this trend to hold in a general mailing.

"The English language causes an 8% error rate in orders, but orders are 25% larger, and 4% more frequent. If we can get them to order by phone, we can correct the errors immediately during the call.

"The broader assortment, as I mentioned, is resulting in a significantly higher propensity to order, more units per order, and the same average unit cost. Of course, paper and postage costs increase as a consequence of the larger format catalog. On the other hand, there are production efficiencies from using the same version as the domestic catalog. Net impact, even factoring in the error rate, is a significant sales increase. On the other hand, most of the time, the errors cause us to ship the wrong item which then needs to be mailed back at our expense, creating an impression in the customers that we are not well organized even though the original error was theirs.

"Final point: The larger catalog is being kept by the customer an average of 70 days, while the smaller format is only kept on average for 40 days. Assuming—we need to test this—that the length of time they keep the catalog is proportional to sales volumes, this is good news. We need to assess the overall impact carefully, but it appears that there is a significant population for which an English-language version would be very profitable."

"Thanks, Gregg, good update. Jennifer, what do you have on customer research?"

"Bob, there's far more that we need to know than we have been able to find out. We have learned that Japan is very fad-driven in apparel tastes and fascinated by American goods. We expect sales initially to sky-rocket, then drop like a stone. Later on, demand will level out at a profitable level. The graphs on page 3 [ Figure 2-1 ] show demand by week for 104 weeks, and we have assessed several scenarios. They all show a good underlying business, but the uncertainty is in the initial take-off. The best data are based on the Italian fashion boom which Japan experienced in the late 80s. It is not strictly analogous because it revolved around dress apparel instead of our casual and weekend wear. It is, however, the best information available.

FIGURE 2-1: Sales projections by week, Scenario A

FIGURE 2-2: Size distributions, U.S. vs. Japan

"Our effectiveness in positioning inventory for that initial surge will be critical to our long-term success. There are excellent data—supplied by MITI, I might add—that show that Japanese customers can be intensely loyal to companies that meet their high service expectations. That is why we prepared several scenarios. Of course, if we position inventory for the high scenario, and we experience the low one, we will experience a significant loss due to liquidations. We are still analyzing the long-term impact, however. It may still be worthwhile to take the risk if the 2-year ROI 1 is sufficient.

"We have solid information on their size scales [ Figure 2-2 ]. Seventy percent are small and medium. By comparison, 70% of Americans are large and extra large. This will be a challenge to manage but will save a few bucks on fabric.

"We also know their color preferences, and they are very different than Americans. Our domestic customers are very diverse in their tastes, but 80% of Japanese customers will buy one or two colors out of an offering of 15. We are still researching color choices, but it varies greatly for pants versus shirts, and for men versus women. We are confident we can find patterns, but we also know that it is easy to guess wrong in that market. If we guess wrong, the liquidation costs will be very high.

"Bad news on the order-taking front, however. They don't like to order by phone. …"

In this very brief exchange among decision-makers we observe the use of many critically important skills that were originally learned in public schools. Perhaps the most important is one not often mentioned, and that is the ability to convert an important business question into an appropriate mathematical one, to solve the mathematical problem, and then to explain the implications of the solution for the original business problem. This ability to inhabit simultaneously the business world and the mathematical world, to translate between the two, and, as a consequence, to bring clarity to complex, real-world issues is of extraordinary importance.

In addition, the participants in this conversation understood and interpreted graphs and tables, computed, approximated, estimated, interpolated, extrapolated, used probabilistic concepts to draw conclusions, generalized from

small samples to large populations, identified the limits of their analyses, discovered relationships, recognized and used variables and functions, analyzed and compared data sets, and created and interpreted models. Another very important aspect of their work was that they identified additional questions, and they suggested ways to shed light on those questions through additional analysis.

There were two broad issues in this conversation that required mathematical perspectives. The first was to develop as rigorous and cost effective a data collection and analysis process as was practical. It involved perhaps 10 different analysts who attacked the problem from different viewpoints. The process also required integration of the mathematical learnings of all 10 analysts and translation of the results into business language that could be understood by non-mathematicians.

The second broad issue was to understand from the perspective of the decision-makers who were listening to the presentation which results were most reliable, which were subject to reinterpretation, which were actually judgments not supported by appropriate analysis, and which were hypotheses that truly required more research. In addition, these business people would likely identify synergies in the research that were not contemplated by the analysts. These synergies need to be analyzed to determine if—mathematically—they were real. The most obvious one was where the inventory analysts said that the customers don't like to use the phone to place orders. This is bad news for the sales analysts who are counting on phone data collection to correct errors caused by language problems. Of course, we need more information to know the magnitude—or even the existance—of the problem.

In brief, the analyses that preceded the dialogue might each be considered a mathematical task in the business world:

• A cost analysis of store operations and catalogs was conducted using data from existing American and possibly other operations.
• Customer preferences research was analyzed to determine preferences in quality and life-style. The data collection itself could not be carried out by a high school graduate without guidance, but 80% of the analysis could.
• Cultural differences were recognized as a causes of analytical error. Careful analysis required judgment. In addition, sources of data were identified in the U.S., and comparable sources were found lacking in Japan. A search was conducted for other comparable retail experience, but none was found. On the other hand, sales data from car parts and computers were assessed for relevance.
• Rates of change are important in understanding how Japanese and American stores differ. Sales per square foot, price increases,
• asset costs, labor costs and so forth were compared to American standards to determine whether a store based in Japan would be a viable business.
• "Nielsen" style ratings of 1000 families were used to collect data. Sample size and error estimates were mentioned. Key drivers of behavior (lifestyle, income, education) were mentioned, but this list may not be complete. What needs to be known about these families to predict their buying behavior? What does "lifestyle" include? How would we quantify some of these variables?
• A hypothesis was presented that catalog size and product diversity drive higher sales. What do we need to know to assess the validity of this hypothesis? Another hypothesis was presented about the quality of the translation. What was the evidence for this hypothesis? Is this a mathematical question? Sales may also be proportional to the amount of time a potential customer retains the catalog. How could one ascertain this?
• Despite the abundance of data, much uncertainty remains about what to expect from sales over the first two years. Analysis could be conducted with the data about the possible inventory consequences of choosing the wrong scenario.
• One might wonder about the uncertainty in size scales. What is so difficult about identifying the colors that Japanese people prefer? Can these preferences be predicted? Will this increase the complexity of the inventory management task?
• Can we predict how many people will not use phones? What do they use instead?

As seen through a mathematical lens, the business world can be a rich, complex, and essentially limitless source of fascinating questions.

R OL F ESSENDEN is Vice-President of Inventory Planning and Control at L. L. Bean, Inc. He is also Co-Principal Investigator and Vice-Chair of Maine's State Systemic Initiative and Chair of the Strategic Planning Committee. He has previously served on the Mathematical Science Education Board, and on the National Alliance for State Science and Mathematics Coalitions (NASSMC).

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

In high school education, preparation for work immediately after high school and preparation for post-secondary education have traditionally been viewed as incompatible. Work-bound high-school students end up in vocational education tracks, where courses usually emphasize specific skills with little attention to underlying theoretical and conceptual foundations. 1 College-bound students proceed through traditional academic discipline-based courses, where they learn English, history, science, mathematics, and foreign languages, with only weak and often contrived references to applications of these skills in the workplace or in the community outside the school. To be sure, many vocational teachers do teach underlying concepts, and many academic teachers motivate their lessons with examples and references to the world outside the classroom. But these enrichments are mostly frills, not central to either the content or pedagogy of secondary school education.

Rethinking Vocational and Academic Education

Educational thinking in the United States has traditionally placed priority on college preparation. Thus the distinct track of vocational education has been seen as an option for those students who are deemed not capable of success in the more desirable academic track. As vocational programs acquired a reputation

as a ''dumping ground," a strong background in vocational courses (especially if they reduced credits in the core academic courses) has been viewed as a threat to the college aspirations of secondary school students.

This notion was further reinforced by the very influential 1983 report entitled A Nation at Risk (National Commission on Excellence in Education, 1983), which excoriated the U.S. educational system for moving away from an emphasis on core academic subjects that, according to the report, had been the basis of a previously successful American education system. Vocational courses were seen as diverting high school students from core academic activities. Despite the dubious empirical foundation of the report's conclusions, subsequent reforms in most states increased the number of academic courses required for graduation and reduced opportunities for students to take vocational courses.

The distinction between vocational students and college-bound students has always had a conceptual flaw. The large majority of students who go to four-year colleges are motivated, at least to a significant extent, by vocational objectives. In 1994, almost 247,000 bachelors degrees were conferred in business administration. That was only 30,000 less than the total number (277,500) of 1994 bachelor degree conferred in English, mathematics, philosophy, religion, physical sciences and science technologies, biological and life sciences, social sciences, and history combined . Furthermore, these "academic" fields are also vocational since many students who graduate with these degrees intend to make their living working in those fields.

Several recent economic, technological, and educational trends challenge this sharp distinction between preparation for college and for immediate post-high-school work, or, more specifically, challenge the notion that students planning to work after high school have little need for academic skills while college-bound students are best served by an abstract education with only tenuous contact with the world of work:

• First, many employers and analysts are arguing that, due to changes in the nature of work, traditional approaches to teaching vocational skills may not be effective in the future. Given the increasing pace of change and uncertainty in the workplace, young people will be better prepared, even for entry level positions and certainly for subsequent positions, if they have an underlying understanding of the scientific, mathematical, social, and even cultural aspects of the work that they will do. This has led to a growing emphasis on integrating academic and vocational education. 2
• Views about teaching and pedagogy have increasingly moved toward a more open and collaborative "student-centered" or "constructivist" teaching style that puts a great deal of emphasis on having students work together on complex, open-ended projects. This reform strategy is now widely implemented through the efforts of organizations such as the Coalition of Essential Schools, the National Center for Restructuring Education, Schools, and Teaching at
• Teachers College, and the Center for Education Research at the University of Wisconsin at Madison. Advocates of this approach have not had much interaction with vocational educators and have certainly not advocated any emphasis on directly preparing high school students for work. Nevertheless, the approach fits well with a reformed education that integrates vocational and academic skills through authentic applications. Such applications offer opportunities to explore and combine mathematical, scientific, historical, literary, sociological, economic, and cultural issues.
• In a related trend, the federal School-to-Work Opportunities Act of 1994 defines an educational strategy that combines constructivist pedagogical reforms with guided experiences in the workplace or other non-work settings. At its best, school-to-work could further integrate academic and vocational learning through appropriately designed experiences at work.
• The integration of vocational and academic education and the initiatives funded by the School-to-Work Opportunities Act were originally seen as strategies for preparing students for work after high school or community college. Some educators and policy makers are becoming convinced that these approaches can also be effective for teaching academic skills and preparing students for four-year college. Teaching academic skills in the context of realistic and complex applications from the workplace and community can provide motivational benefits and may impart a deeper understanding of the material by showing students how the academic skills are actually used. Retention may also be enhanced by giving students a chance to apply the knowledge that they often learn only in the abstract. 3
• During the last twenty years, the real wages of high school graduates have fallen and the gap between the wages earned by high school and college graduates has grown significantly. Adults with no education beyond high school have very little chance of earning enough money to support a family with a moderate lifestyle. 4 Given these wage trends, it seems appropriate and just that every high school student at least be prepared for college, even if some choose to work immediately after high school.

Innovative Examples

There are many examples of programs that use work-related applications both to teach academic skills and to prepare students for college. One approach is to organize high school programs around broad industrial or occupational areas, such as health, agriculture, hospitality, manufacturing, transportation, or the arts. These broad areas offer many opportunities for wide-ranging curricula in all academic disciplines. They also offer opportunities for collaborative work among teachers from different disciplines. Specific skills can still be taught in this format but in such a way as to motivate broader academic and theoretical themes. Innovative programs can now be found in many vocational

high schools in large cities, such as Aviation High School in New York City and the High School of Agricultural Science and Technology in Chicago. Other schools have organized schools-within-schools based on broad industry areas.

Agriculturally based activities, such as 4H and Future Farmers of America, have for many years used the farm setting and students' interest in farming to teach a variety of skills. It takes only a little imagination to think of how to use the social, economic, and scientific bases of agriculture to motivate and illustrate skills and knowledge from all of the academic disciplines. Many schools are now using internships and projects based on local business activities as teaching tools. One example among many is the integrated program offered by the Thomas Jefferson High School for Science and Technology in Virginia, linking biology, English, and technology through an environmental issues forum. Students work as partners with resource managers at the Mason Neck National Wildlife Refuge and the Mason Neck State Park to collect data and monitor the daily activities of various species that inhabit the region. They search current literature to establish a hypothesis related to a real world problem, design an experiment to test their hypothesis, run the experiment, collect and analyze data, draw conclusions, and produce a written document that communicates the results of the experiment. The students are even responsible for determining what information and resources are needed and how to access them. Student projects have included making plans for public education programs dealing with environmental matters, finding solutions to problems caused by encroaching land development, and making suggestions for how to handle the overabundance of deer in the region.

These examples suggest the potential that a more integrated education could have for all students. Thus continuing to maintain a sharp distinction between vocational and academic instruction in high school does not serve the interests of many of those students headed for four-year or two-year college or of those who expect to work after high school. Work-bound students will be better prepared for work if they have stronger academic skills, and a high-quality curriculum that integrates school-based learning into work and community applications is an effective way to teach academic skills for many students.

Despite the many examples of innovative initiatives that suggest the potential for an integrated view, the legacy of the duality between vocational and academic education and the low status of work-related studies in high school continue to influence education and education reform. In general, programs that deviate from traditional college-prep organization and format are still viewed with suspicion by parents and teachers focused on four-year college. Indeed, college admissions practices still very much favor the traditional approaches. Interdisciplinary courses, "applied" courses, internships, and other types of work experience that characterize the school-to-work strategy or programs that integrate academic and vocational education often do not fit well into college admissions requirements.

Joining Work and Learning

What implications does this have for the mathematics standards developed by the National Council of Teachers of Mathematics (NCTM)? The general principle should be to try to design standards that challenge rather than reinforce the distinction between vocational and academic instruction. Academic teachers of mathematics and those working to set academic standards need to continue to try to understand the use of mathematics in the workplace and in everyday life. Such understandings would offer insights that could suggest reform of the traditional curriculum, but they would also provide a better foundation for teaching mathematics using realistic applications. The examples in this volume are particularly instructive because they suggest the importance of problem solving, logic, and imagination and show that these are all important parts of mathematical applications in realistic work settings. But these are only a beginning.

In order to develop this approach, it would be helpful if the NCTM standards writers worked closely with groups that are setting industry standards. 5 This would allow both groups to develop a deeper understanding of the mathematics content of work.

The NCTM's Curriculum Standards for Grades 9-12 include both core standards for all students and additional standards for "college-intending" students. The argument presented in this essay suggests that the NCTM should dispense with the distinction between college intending and non-college intending students. Most of the additional standards, those intended only for the "college intending" students, provide background that is necessary or beneficial for the calculus sequence. A re-evaluation of the role of calculus in the high school curriculum may be appropriate, but calculus should not serve as a wedge to separate college-bound from non-college-bound students. Clearly, some high school students will take calculus, although many college-bound students will not take calculus either in high school or in college. Thus in practice, calculus is not a characteristic that distinguishes between those who are or are not headed for college. Perhaps standards for a variety of options beyond the core might be offered. Mathematics standards should be set to encourage stronger skills for all students and to illustrate the power and usefulness of mathematics in many settings. They should not be used to institutionalize dubious distinctions between groups of students.

Bailey, T. & Merritt, D. (1997). School-to-work for the collegebound . Berkeley, CA: National Center for Research in Vocational Education.

Hoachlander, G . (1997) . Organizing mathematics education around work . In L.A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow's America , (pp. 122-136). New York: College Entrance Examination Board.

Levy, F. & Murnane, R. (1992). U.S. earnings levels and earnings inequality: A review of recent trends and proposed explanations. Journal of Economic Literature , 30 , 1333-1381.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform . Washington, DC: Author.

T HOMAS B AILEY is an Associate Professor of Economics Education at Teachers College, Columbia University. He is also Director of the Institute on Education and the Economy and Director of the Community College Research Center, both at Teachers College. He is also on the board of the National Center for Research in Vocational Education.

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

For decades our industrial society has been based on fossil fuels. In today's knowledge-based society, mathematics is the energy that drives the system. In the words of the new WQED television series, Life by the Numbers , to create knowledge we "burn mathematics." Mathematics is more than a fixed tool applied in known ways. New mathematical techniques and analyses and even conceptual frameworks are continually required in economics, in finance, in materials science, in physics, in biology, in medicine.

Just as all scientific and health-service careers are mathematically based, so are many others. Interaction with computers has become a part of more and more jobs, and good analytical skills enhance computer use and troubleshooting. In addition, virtually all levels of management and many support positions in business and industry require some mathematical understanding, including an ability to read graphs and interpret other information presented visually, to use estimation effectively, and to apply mathematical reasoning.

What Should Students Learn for Today's World?

Education in mathematics and the ability to communicate its predictions is more important than ever for moving from low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton , had a section "Focus

on Careers" on October 5, 1997 in which the majority of the ads were for high technology careers (many more than for sales and marketing, for example).

But precisely what mathematics should students learn in school? Mathematicians and mathematics educators have been discussing this question for decades. This essay presents some thoughts about three areas of mathematics—estimation, trigonometry, and algebra—and then some thoughts about teaching and learning.

Estimation is one of the harder skills for students to learn, even if they experience relatively little difficulty with other aspects of mathematics. Many students think of mathematics as a set of precise rules yielding exact answers and are uncomfortable with the idea of imprecise answers, especially when the degree of precision in the estimate depends on the context and is not itself given by a rule. Yet it is very important to be able to get an approximate sense of the size an answer should be, as a way to get a rough check on the accuracy of a calculation (I've personally used it in stores to detect that I've been charged twice for the same item, as well as often in my own mathematical work), a feasibility estimate, or as an estimation for tips.

Trigonometry plays a significant role in the sciences and can help us understand phenomena in everyday life. Often introduced as a study of triangle measurement, trigonometry may be used for surveying and for determining heights of trees, but its utility extends vastly beyond these triangular applications. Students can experience the power of mathematics by using sine and cosine to model periodic phenomena such as going around and around a circle, going in and out with tides, monitoring temperature or smog components changing on a 24-hour cycle, or the cycling of predator-prey populations.

No educator argues the importance of algebra for students aiming for mathematically-based careers because of the foundation it provides for the more specialized education they will need later. Yet, algebra is also important for those students who do not currently aspire to mathematics-based careers, in part because a lack of algebraic skills puts an upper bound on the types of careers to which a student can aspire. Former civil rights leader Robert Moses makes a good case for every student learning algebra, as a means of empowering students and providing goals, skills, and opportunities. The same idea was applied to learning calculus in the movie Stand and Deliver . How, then, can we help all students learn algebra?

For me personally, the impetus to learn algebra was at least in part to learn methods of solution for puzzles. Suppose you have 39 jars on three shelves. There are twice as many jars on the second shelf as the first, and four more jars on the third shelf than on the second shelf. How many jars are there on each shelf? Such problems are not important by themselves, but if they show the students the power of an idea by enabling them to solve puzzles that they'd like to solve, then they have value. We can't expect such problems to interest all students. How then can we reach more students?

Workplace and Everyday Settings as a Way of Making Sense

One of the common tools in business and industry for investigating mathematical issues is the spreadsheet, which is closely related to algebra. Writing a rule to combine the elements of certain cells to produce the quantity that goes into another cell is doing algebra, although the variables names are cell names rather than x or y . Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires.

By exploring mathematics via tasks which come from workplace and everyday settings, and with the aid of common tools like spreadsheets, students are more likely to see the relevance of the mathematics and are more likely to learn it in ways that are personally meaningful than when it is presented abstractly and applied later only if time permits. Thus, this essay argues that workplace and everyday tasks should be used for teaching mathematics and, in particular, for teaching algebra. It would be a mistake, however, to rely exclusively on such tasks, just as it would be a mistake to teach only spreadsheets in place of algebra.

Communicating the results of an analysis is a fundamental part of any use of mathematics on a job. There is a growing emphasis in the workplace on group work and on the skills of communicating ideas to colleagues and clients. But communicating mathematical ideas is also a powerful tool for learning, for it requires the student to sharpen often fuzzy ideas.

Some of the tasks in this volume can provide the kinds of opportunities I am talking about. Another problem, with clear connections to the real world, is the following, taken from the book entitled Consider a Spherical Cow: A Course in Environmental Problem Solving , by John Harte (1988). The question posed is: How does biomagnification of a trace substance occur? For example, how do pesticides accumulate in the food chain, becoming concentrated in predators such as condors? Specifically, identify the critical ecological and chemical parameters determining bioconcentrations in a food chain, and in terms of these parameters, derive a formula for the concentration of a trace substance in each link of a food chain. This task can be undertaken at several different levels. The analysis in Harte's book is at a fairly high level, although it still involves only algebra as a mathematical tool. The task could be undertaken at a more simple level or, on the other hand, it could be elaborated upon as suggested by further exercises given in that book. And the students could then present the results of their analyses to each other as well as the teacher, in oral or written form.

Concepts or Procedures?

When teaching mathematics, it is easy to spend so much time and energy focusing on the procedures that the concepts receive little if any attention. When teaching algebra, students often learn the procedures for using the quadratic formula or for solving simultaneous equations without thinking of intersections of curves and lines and without being able to apply the procedures in unfamiliar settings. Even

when concentrating on word problems, students often learn the procedures for solving "coin problems" and "train problems" but don't see the larger algebraic context. The formulas and procedures are important, but are not enough.

When using workplace and everyday tasks for teaching mathematics, we must avoid falling into the same trap of focusing on the procedures at the expense of the concepts. Avoiding the trap is not easy, however, because just like many tasks in school algebra, mathematically based workplace tasks often have standard procedures that can be used without an understanding of the underlying mathematics. To change a procedure to accommodate a changing business climate, to respond to changes in the tax laws, or to apply or modify a procedure to accommodate a similar situation, however, requires an understanding of the mathematical ideas behind the procedures. In particular, a student should be able to modify the procedures for assessing energy usage for heating (as in Heating-Degree-Days, p. 54) in order to assess energy usage for cooling in the summer.

To prepare our students to make such modifications on their own, it is important to focus on the concepts as well as the procedures. Workplace and everyday tasks can provide opportunities for students to attach meaning to the mathematical calculations and procedures. If a student initially solves a problem without algebra, then the thinking that went into his or her solution can help him or her make sense out of algebraic approaches that are later presented by the teacher or by other students. Such an approach is especially appropriate for teaching algebra, because our teaching of algebra needs to reach more students (too often it is seen by students as meaningless symbol manipulation) and because algebraic thinking is increasingly important in the workplace.

An Example: The Student/Professor Problem

To illustrate the complexity of learning algebra meaningfully, consider the following problem from a study by Clement, Lockhead, & Monk (1981):

Write an equation for the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (p. 288)

The authors found that of 47 nonscience majors taking college algebra, 57% got it wrong. What is more surprising, however, is that of 150 calculus-level students, 37% missed the problem.

A first reaction to the most common wrong answer, 6 S = P , is that the students simply translated the words of the problems into mathematical symbols without thinking more deeply about the situation or the variables. (The authors note that some textbooks instruct students to use such translation.)

By analyzing transcripts of interviews with students, the authors found this approach and another (faulty) approach, as well. These students often drew a diagram showing six students and one professor. (Note that we often instruct students to draw diagrams when solving word problems.) Reasoning

from the diagram, and regarding S and P as units, the student may write 6 S = P , just as we would correctly write 12 in. = 1 ft. Such reasoning is quite sensible, though it misses the fundamental intent in the problem statement that S is to represent the number of students, not a student.

Thus, two common suggestions for students—word-for-word translation and drawing a diagram—can lead to an incorrect answer to this apparently simple problem, if the students do not more deeply contemplate what the variables are intended to represent. The authors found that students who wrote and could explain the correct answer, S = 6 P , drew upon a richer understanding of what the equation and the variables represent.

Clearly, then, we must encourage students to contemplate the meanings of variables. Yet, part of the power and efficiency of algebra is precisely that one can manipulate symbols independently of what they mean and then draw meaning out of the conclusions to which the symbolic manipulations lead. Thus, stable, long-term learning of algebraic thinking requires both mastery of procedures and also deeper analytical thinking.

Paradoxically, the need for sharper analytical thinking occurs alongside a decreased need for routine arithmetic calculation. Calculators and computers make routine calculation easier to do quickly and accurately; cash registers used in fast food restaurants sometimes return change; checkout counters have bar code readers and payment takes place by credit cards or money-access cards.

So it is education in mathematical thinking, in applying mathematical computation, in assessing whether an answer is reasonable, and in communicating the results that is essential. Teaching mathematics via workplace and everyday problems is an approach that can make mathematics more meaningful for all students. It is important, however, to go beyond the specific details of a task in order to teach mathematical ideas. While this approach is particularly crucial for those students intending to pursue careers in the mathematical sciences, it will also lead to deeper mathematical understanding for all students.

Clement, J., Lockhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly , 88 , 286-290.

Harte, J. (1988). Consider a spherical cow: A course in environmental problem solving . York, PA: University Science Books.

J EAN E. T AYLOR is Professor of Mathematics at Rutgers, the State University of New Jersey. She is currently a member of the Board of Directors of the American Association for the Advancement of Science and formerly chaired its Section A Nominating Committee. She has served as Vice President and as a Member-at-Large of the Council of the American Mathematical Society, and served on its Executive Committee and its Nominating Committee. She has also been a member of the Joint Policy Board for Mathematics, and a member of the Board of Advisors to The Geometry Forum (now The Mathematics Forum) and to the WQED television series, Life by the Numbers .

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

Teaching a mathematics class in which few of the students have demonstrated success is a difficult assignment. Many teachers avoid such assignments, when possible. On the one hand, high school mathematics teachers, like Bertrand Russell, might love mathematics and believe something like the following:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. … Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its nature home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world. (Russell, 1910, p. 73)

But, on the other hand, students may not have the luxury, in their circumstances, of appreciating this beauty. Many of them may not see themselves as thinkers because contemplation would take them away from their primary

focus: how to get by in a world that was not created for them. Instead, like Jamaica Kincaid, they may be asking:

What makes the world turn against me and all who look like me? I won nothing, I survey nothing, when I ask this question, the luxury of an answer that will fill volumes does not stretch out before me. When I ask this question, my voice is filled with despair. (Kincaid, 1996, pp. 131-132)

Our Teaching and Issues it Raised

During the 1991-92 and 1992-93 school years, we (a high school teacher and a university teacher educator) team taught a lower track Algebra I class for 10th through 12th grade students. 1 Most of our students had failed mathematics before, and many needed to pass Algebra I in order to complete their high school mathematics requirement for graduation. For our students, mathematics had become a charged subject; it carried a heavy burden of negative experiences. Many of our students were convinced that neither they nor their peers could be successful in mathematics.

Few of our students did well in other academic subjects, and few were headed on to two- or four-year colleges. But the students differed in their affiliation with the high school. Some, called ''preppies" or "jocks" by others, were active participants in the school's activities. Others, "smokers" or "stoners," were rebelling to differing degrees against school and more broadly against society. There were strong tensions between members of these groups. 2

Teaching in this setting gives added importance and urgency to the typical questions of curriculum and motivation common to most algebra classes. In our teaching, we explored questions such as the following:

• What is it that we really want high school students, especially those who are not college-intending, to study in algebra and why?
• What is the role of algebra's manipulative skills in a world with graphing calculators and computers? How do the manipulative skills taught in the traditional curriculum give students a new perspective on, and insight into, our world?
• If our teaching efforts depend on students' investment in learning, on what grounds can we appeal to them, implicitly or explicitly, for energy and effort? In a tracked, compulsory setting, how can we help students, with broad interests and talents and many of whom are not college-intending, see value in a shared exploration of algebra?

An Approach to School Algebra

As a result of thinking about these questions, in our teaching we wanted to avoid being in the position of exhorting students to appreciate the beauty or utility of algebra. Our students were frankly skeptical of arguments based on

utility. They saw few people in their community using algebra. We had also lost faith in the power of extrinsic rewards and punishments, like failing grades. Many of our students were skeptical of the power of the high school diploma to alter fundamentally their life circumstances. We wanted students to find the mathematical objects we were discussing in the world around them and thus learn to value the perspective that this mathematics might give them on their world.

To help us in this task, we found it useful to take what we call a "relationships between quantities" approach to school algebra. In this approach, the fundamental mathematical objects of study in school algebra are functions that can be represented by inputs and outputs listed in tables or sketched or plotted on graphs, as well as calculation procedures that can be written with algebraic symbols. 3 Stimulated, in part, by the following quote from August Comte, we viewed these functions as mathematical representations of theories people have developed for explaining relationships between quantities.

In the light of previous experience, we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. (Quoted in Serres, 1982, p. 85)

The "Sponsor" Project

Using this approach to the concept of function, during the 1992-93 school year, we designed a year-long project for our students. The project asked pairs of students to find the mathematical objects we were studying in the workplace of a community sponsor. Students visited the sponsor's workplace four times during the year—three after-school visits and one day-long excused absence from school. In these visits, the students came to know the workplace and learned about the sponsor's work. We then asked students to write a report describing the sponsor's workplace and answering questions about the nature of the mathematical activity embedded in the workplace. The questions are organized in Table 5-1 .

Using These Questions

In order to determine how the interviews could be structured and to provide students with a model, we chose to interview Sandra's husband, John Bethell, who is a coatings inspector for an engineering firm. When asked about his job, John responded, "I argue for a living." He went on to describe his daily work inspecting contractors painting water towers. Since most municipalities contract with the lowest bidder when a water tower needs to be painted, they will often hire an engineering firm to make sure that the contractor works according to specification. Since the contractor has made a low bid, there are strong

TABLE 5-1: Questions to ask in the workplace

financial incentives for the contractor to compromise on quality in order to make a profit.

In his work John does different kinds of inspections. For example, he has a magnetic instrument to check the thickness of the paint once it has been applied to the tower. When it gives a "thin" reading, contractors often question the technology. To argue for the reading, John uses the surface area of the tank, the number of paint cans used, the volume of paint in the can, and an understanding of the percentage of this volume that evaporates to calculate the average thickness of the dry coating. Other examples from his workplace involve the use of tables and measuring instruments of different kinds.

Some Examples of Students' Work

When school started, students began working on their projects. Although many of the sponsors initially indicated that there were no mathematical dimensions to their work, students often were able to show sponsors places where the mathematics we were studying was to be found. For example, Jackie worked with a crop and soil scientist. She was intrigued by the way in which measurement of weight is used to count seeds. First, her sponsor would weigh a test batch of 100 seeds to generate a benchmark weight. Then, instead of counting a large number of seeds, the scientist would weigh an amount of seeds and compute the number of seeds such a weight would contain.

Rebecca worked with a carpeting contractor who, in estimating costs, read the dimensions of rectangular rooms off an architect's blueprint, multiplied to find the area of the room in square feet (doing conversions where necessary), then multiplied by a cost per square foot (which depended on the type of carpet) to compute the cost of the carpet. The purpose of these estimates was to prepare a bid for the architect where the bid had to be as low as possible without making the job unprofitable. Rebecca used a chart ( Table 5-2 ) to explain this procedure to the class.

Joe and Mick, also working in construction, found out that in laying pipes, there is a "one by one" rule of thumb. When digging a trench for the placement of the pipe, the non-parallel sides of the trapezoidal cross section must have a slope of 1 foot down for every one foot across. This ratio guarantees that the dirt in the hole will not slide down on itself. Thus, if at the bottom of the hole, the trapezoid must have a certain width in order to fit the pipe, then on ground level the hole must be this width plus twice the depth of the hole. Knowing in advance how wide the hole must be avoids lengthy and costly trial and error.

Other students found that functions were often embedded in cultural artifacts found in the workplace. For example, a student who visited a doctor's office brought in an instrument for predicting the due dates of pregnant women, as well as providing information about average fetal weight and length ( Figure 5-1 ).

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

While the complexities of organizing this sort of project should not be minimized—arranging sponsors, securing parental permission, and meeting administrators and parent concerns about the requirement of off-campus, after-school work—we remain intrigued by the potential of such projects for helping students see mathematics in the world around them. The notions of identifying central mathematical objects for a course and then developing ways of identifying those objects in students' experience seems like an important alternative to the use of application-based materials written by developers whose lives and social worlds may be quite different from those of students.

Chazen, D. (1996). Algebra for all students? Journal of Mathematical Behavior , 15 (4), 455-477.

Eckert, P. (1989). Jocks and burnouts: Social categories and identity in the high school . New York: Teachers College Press.

Fey, J. T., Heid, M. K., et al. (1995). Concepts in algebra: A technological approach . Dedham, MA: Janson Publications.

Kieran, C., Boileau, A., & Garancon, M. (1996). Introducing algebra by mean of a technology-supported, functional approach. In N. Bednarz et al. (Eds.), Approaches to algebra , (pp. 257-293). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Kincaid, J. (1996). The autobiography of my mother . New York: Farrar, Straus, Giroux.

Nemirovsky, R. (1996). Mathematical narratives, modeling and algebra. In N. Bednarz et al. (Eds.) Approaches to algebra , (pp. 197-220). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Russell, B. (1910). Philosophical Essays . London: Longmans, Green.

Schwartz, J. & Yerushalmy, M. (1992). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy , (MAA Notes, Vol. 25, pp. 261-289). Washington, DC: Mathematical Association of America.

Serres, M. (1982). Mathematics and philosophy: What Thales saw … In J. Harari & D. Bell (Eds.), Hermes: Literature, science, philosophy , (pp. 84-97). Baltimore, MD: Johns Hopkins.

Thompson, P. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics , 25 , 165-208.

Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions , (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.

D ANIEL C HAZAN is an Associate Professor of Teacher Education at Michigan State University. To assist his research in mathematics teaching and learning, he has taught algebra at the high school level. His interests include teaching mathematics by examining student ideas, using computers to support student exploration, and the potential for the history and philosophy of mathematics to inform teaching.

S ANDRA C ALLIS B ETHELL has taught mathematics and Spanish at Holt High School for 10 years. She has also completed graduate work at Michigan State University and Western Michigan University. She has interest in mathematics reform, particularly in meeting the needs of diverse learners in algebra courses.

Emergency Calls

A city is served by two different ambulance companies. City logs record the date, the time of the call, the ambulance company, and the response time for each 911 call ( Table 1 ). Analyze these data and write a report to the City Council (with supporting charts and graphs) advising it on which ambulance company the 911 operators should choose to dispatch for calls from this region.

TABLE 1: Ambulance dispatch log sheet, May 1–30

This problem confronts the student with a realistic situation and a body of data regarding two ambulance companies' response times to emergency calls. The data the student is provided are typically "messy"—just a log of calls and response times, ordered chronologically. The question is how to make sense of them. Finding patterns in data such as these requires a productive mixture of mathematics common sense, and intellectual detective work. It's the kind of reasoning that students should be able to do—the kind of reasoning that will pay off in the real world.

Mathematical Analysis

In this case, a numerical analysis is not especially informative. On average, the companies are about the same: Arrow has a mean response time of 11.4 minutes compared to 11.6 minutes for Metro. The spread of the data is also not very helpful. The ranges of their distributions are exactly the same: from 6 minutes to 19 minutes. The standard deviation of Arrow's response time is a little longer—4.3 minutes versus 3.4 minutes for Metro—indicating that Arrow's response times fluctuate a bit more.

Graphs of the response times (Figures 1 and 2 ) reveal interesting features. Both companies, especially Arrow, seem to have bimodal distributions, which is to say that there are two clusters of data without much data in between.

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

The distributions for both companies suggest that there are some other factors at work. Might a particular driver be the problem? Might the slow response times for either company be on particular days of the week or at particular times of day? Graphs of the response time versus the time of day (Figures 3 and 4 ) shed some light on these questions.

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

These graphs show that Arrow's response times were fast except between 5:30 AM and 9:00 AM, when they were about 9 minutes slower on average. Similarly, Metro's response times were fast except between about 3:30 PM and 6:30 PM, when they were about 5 minutes slower. Perhaps the locations of the companies make Arrow more susceptible to the morning rush hour and Metro more susceptible to the afternoon rush hour. On the other hand, the employees on Arrow's morning shift or Metro's afternoon shift may not be efficient. To avoid slow responses, one could recommend to the City Council that Metro be called during the morning and that Arrow be called during the afternoon. A little detective work into the sources of the differences between the companies may yield a better recommendation.

Comparisons may be drawn between two samples in various contexts—response times for various services (taxis, computer-help desks, 24-hour hot lines at automobile manufacturers) being one class among many. Depending upon the circumstances, the data may tell very different stories. Even in the situation above, if the second pair of graphs hadn't offered such clear explanations, one might have argued that although the response times for Arrow were better on average the spread was larger, thus making their "extremes" more risky. The fundamental idea is using various analysis and representation techniques to make sense of data when the important factors are not necessarily known ahead of time.

Back-of-the-Envelope Estimates

Practice "back-of-the-envelope" estimates based on rough approximations that can be derived from common sense or everyday observations. Examples:

• Consider a public high school mathematics teacher who feels that students should work five nights a week, averaging about 35 minutes a night, doing focused on-task work and who intends to grade all homework with comments and corrections. What is a reasonable number of hours per week that such a teacher should allocate for grading homework?
• How much paper does The New York Times use in a week? A paper company that wishes to make a bid to become their sole supplier needs to know whether they have enough current capacity. If the company were to store a two-week supply of newspaper, will their empty 14,000 square foot warehouse be big enough?

Some 50 years ago, physicist Enrico Fermi asked his students at the University of Chicago, "How many piano tuners are there in Chicago?" By asking such questions, Fermi wanted his students to make estimates that involved rough approximations so that their goal would be not precision but the order of magnitude of their result. Thus, many people today call these kinds of questions "Fermi questions." These generally rough calculations often require little more than common sense, everyday observations, and a scrap of paper, such as the back of a used envelope.

Scientists and mathematicians use the idea of order of magnitude , usually expressed as the closest power of ten, to give a rough sense of the size of a quantity. In everyday conversation, people use a similar idea when they talk about "being in the right ballpark." For example, a full-time job at minimum wage yields an annual income on the order of magnitude of $10,000 or 10 4 dollars. Some corporate executives and professional athletes make annual salaries on the order of magnitude of $10,000,000 or 10 7 dollars. To say that these salaries differ by a factor of 1000 or 10 3 , one can say that they differ by three orders of magnitude. Such a lack of precision might seem unscientific or unmathematical, but such approximations are quite useful in determining whether a more precise measurement is feasible or necessary, what sort of action might be required, or whether the result of a calculation is "in the right ballpark." In choosing a strategy to protect an endangered species, for example, scientists plan differently if there are 500 animals remaining than if there are 5,000. On the other hand, determining whether 5,200 or 6,300 is a better estimate is not necessary, as the strategies will probably be the same.

Careful reasoning with everyday observations can usually produce Fermi estimates that are within an order of magnitude of the exact answer (if there is one). Fermi estimates encourage students to reason creatively with approximate quantities and uncertain information. Experiences with such a process can help an individual function in daily life to determine the reasonableness of numerical calculations, of situations or ideas in the workplace, or of a proposed tax cut. A quick estimate of some revenue- or profit-enhancing scheme may show that the idea is comparable to suggesting that General Motors enter the summer sidewalk lemonade market in your neighborhood. A quick estimate could encourage further investigation or provide the rationale to dismiss the idea.

Almost any numerical claim may be treated as a Fermi question when the problem solver does not have access to all necessary background information. In such a situation, one may make rough guesses about relevant numbers, do a few calculations, and then produce estimates.

The examples are solved separately below.

Grading Homework

Although many component factors vary greatly from teacher to teacher or even from week to week, rough calculations are not hard to make. Some important factors to consider for the teacher are: how many classes he or she teaches, how many students are in each of the classes, how much experience has the teacher had in general and has the teacher previously taught the classes, and certainly, as part of teaching style, the kind of homework the teacher assigns, not to mention the teacher's efficiency in grading.

Suppose the teacher has 5 classes averaging 25 students per class. Because the teacher plans to write corrections and comments, assume that the students' papers contain more than a list of answers—they show some student work and, perhaps, explain some of the solutions. Grading such papers might take as long as 10 minutes each, or perhaps even longer. Assuming that the teacher can grade them as quickly as 3 minutes each, on average, the teacher's grading time is:

This is an impressively large number, especially for a teacher who already spends almost 25 hours/week in class, some additional time in preparation, and some time meeting with individual students. Is it reasonable to expect teachers to put in that kind of time? What compromises or other changes might the teacher make to reduce the amount of time? The calculation above offers four possibilities: Reduce the time spent on each homework paper, reduce the number of students per class, reduce the number of classes taught each day, or reduce the number of days per week that homework will be collected. If the teacher decides to spend at most 2 hours grading each night, what is the total number of students for which the teacher should have responsibility? This calculation is a partial reverse of the one above:

If the teacher still has 5 classes, that would mean 8 students per class!

The New York Times

Answering this question requires two preliminary estimates: the circulation of The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times is a national newspaper, the number of subscribers outside the New York metropolitan area is probably small compared to the number inside. The population of the New York metropolitan area is roughly ten million people. Since most families buy at most one copy, and not all families buy The New York Times , the circulation might be about 1 million newspapers each day. (A circulation of 500,000 seems too small and 2 million seems too big.) The Sunday and weekday editions probably have different

circulations, but assume that they are the same since they probably differ by less than a factor of two—much less than an order of magnitude. When folded, a weekday edition of the paper measures about 1/2 inch thick, a little more than 1 foot long, and about 1 foot wide. A Sunday edition of the paper is the same width and length, but perhaps 2 inches thick. For a week, then, the papers would stack 6 × 1/2 + 2 = 5 inches thick, for a total volume of about 1 ft × 1 ft × 5/12 ft = 0.5 ft 3 .

The whole circulation, then, would require about 1/2 million cubic feet of paper per week, or about 1 million cubic feet for a two-week supply.

Is the company's warehouse big enough? The paper will come on rolls, but to make the estimates easy, assume it is stacked. If it were stacked 10 feet high, the supply would require 100,000 square feet of floor space. The company's 14,000 square foot storage facility will probably not be big enough as its size differs by almost an order of magnitude from the estimate. The circulation estimate and the size of the newspaper estimate should each be within a factor of 2, implying that the 100,000 square foot estimate is off by at most a factor of 4—less than an order of magnitude.

How big a warehouse is needed? An acre is 43,560 square feet so about two acres of land is needed. Alternatively, a warehouse measuring 300 ft × 300 ft (the length of a football field in both directions) would contain 90,000 square feet of floor space, giving a rough idea of the size.

After gaining some experience with these types of problems, students can be encouraged to pay close attention to the units and to be ready to make and support claims about the accuracy of their estimates. Paying attention to units and including units as algebraic quantities in calculations is a common technique in engineering and the sciences. Reasoning about a formula by paying attention only to the units is called dimensional analysis.

Sometimes, rather than a single estimate, it is helpful to make estimates of upper and lower bounds. Such an approach reinforces the idea that an exact answer is not the goal. In many situations, students could first estimate upper and lower bounds, and then collect some real data to determine whether the answer lies between those bounds. In the traditional game of guessing the number of jelly beans in a jar, for example, all students should be able to estimate within an order of magnitude, or perhaps within a factor of two. Making the closest guess, however, involves some chance.

Fermi questions are useful outside the workplace. Some Fermi questions have political ramifications:

• How many miles of streets are in your city or town? The police chief is considering increasing police presence so that every street is patrolled by car at least once every 4 hours.
• When will your town fill up its landfill? Is this a very urgent matter for the town's waste management personnel to assess in depth?
• In his 1997 State of the Union address, President Clinton renewed his call for a tax deduction of up to $10,000 for the cost of college tuition. He estimates that 16.5 million students stand to benefit. Is this a reasonable estimate of the number who might take advantage of the tax deduction? How much will the deduction cost in lost federal revenue?

Creating Fermi problems is easy. Simply ask quantitative questions for which there is no practical way to determine exact values. Students could be encouraged to make up their own. Examples are: ''How many oak trees are there in Illinois?" or "How many people in the U.S. ate chicken for dinner last night?" "If all the people in the world were to jump in the ocean, how much would it raise the water level?" Give students the opportunity to develop their own Fermi problems and to share them with each other. It can stimulate some real mathematical thinking.

Scheduling Elevators

In some buildings, all of the elevators can travel to all of the floors, while in others the elevators are restricted to stopping only on certain floors. What is the advantage of having elevators that travel only to certain floors? When is this worth instituting?

Scheduling elevators is a common example of an optimization problem that has applications in all aspects of business and industry. Optimal scheduling in general not only can save time and money, but it can contribute to safety (e.g., in the airline industry). The elevator problem further illustrates an important feature of many economic and political arguments—the dilemma of trying simultaneously to optimize several different needs.

Politicians often promise policies that will be the least expensive, save the most lives, and be best for the environment. Think of flood control or occupational safety rules, for example. When we are lucky, we can perhaps find a strategy of least cost, a strategy that saves the most lives, or a strategy that damages the environment least. But these might not be the same strategies: generally one cannot simultaneously satisfy two or more independent optimization conditions. This is an important message for students to learn, in order to become better educated and more critical consumers and citizens.

In the elevator problem, customer satisfaction can be emphasized by minimizing the average elevator time (waiting plus riding) for employees in an office building. Minimizing wait-time during rush hours means delivering many people quickly, which might be accomplished by filling the elevators and making few stops. During off-peak hours, however, minimizing wait-time means maximizing the availability of the elevators. There is no reason to believe that these two goals will yield the same strategy. Finding the best strategy for each is a mathematical problem; choosing one of the two strategies or a compromise strategy is a management decision, not a mathematical deduction.

This example serves to introduce a complex topic whose analysis is well within the range of high school students. Though the calculations require little more than arithmetic, the task puts a premium on the creation of reasonable alternative strategies. Students should recognize that some configurations (e.g., all but one elevator going to the top floor and the one going to all the others) do not merit consideration, while others are plausible. A systematic evaluation of all possible configurations is usually required to find the optimal solution. Such a systematic search of the possible solution space is important in many modeling situations where a formal optimal strategy is not known. Creating and evaluating reasonable strategies for the elevators is quite appropriate for high school student mathematics and lends itself well to thoughtful group effort. How do you invent new strategies? How do you know that you have considered all plausible strategies? These are mathematical questions, and they are especially amenable to group discussion.

Students should be able to use the techniques first developed in solving a simple case with only a few stories and a few elevators to address more realistic situations (e.g., 50 stories, five elevators). Using the results of a similar but simpler problem to model a more complicated problem is an important way to reason in mathematics. Students

need to determine what data and variables are relevant. Start by establishing the kind of building—a hotel, an office building, an apartment building? How many people are on the different floors? What are their normal destinations (e.g., primarily the ground floor or, perhaps, a roof-top restaurant). What happens during rush hours?

To be successful at the elevator task, students must first develop a mathematical model of the problem. The model might be a graphical representation for each elevator, with time on the horizontal axis and the floors represented on the vertical axis, or a tabular representation indicating the time spent on each floor. Students must identify the pertinent variables and make simplifying assumptions about which of the possible floors an elevator will visit.

This section works through some of the details in a particularly simple case. Consider an office building with six occupied floors, employing 240 people, and a ground floor that is not used for business. Suppose there are three elevators, each of which can hold 10 people. Further suppose that each elevator takes approximately 25 seconds to fill on the ground floor, then takes 5 seconds to move between floors and 15 seconds to open and close at each floor on which it stops.

Scenario One

What happens in the morning when everyone arrives for work? Assume that everyone arrives at approximately the same time and enters the elevators on the ground floor. If all elevators go to all floors and if the 240 people are evenly divided among all three elevators, each elevator will have to make 8 trips of 10 people each.

When considering a single trip of one elevator, assume for simplicity that 10 people get on the elevator at the ground floor and that it stops at each floor on the way up, because there may be an occupant heading to each floor. Adding 5 seconds to move to each floor and 15 seconds to stop yields 20 seconds for each of the six floors. On the way down, since no one is being picked up or let off, the elevator does not stop, taking 5 seconds for each of six floors for a total of 30 seconds. This round-trip is represented in Table 1 .

TABLE 1: Elevator round-trip time, Scenario one

Since each elevator makes 8 trips, the total time will be 1,400 seconds or 23 minutes, 20 seconds.

Scenario Two

Now suppose that one elevator serves floors 1–3 and, because of the longer trip, two elevators are assigned to floors 4–6. The elevators serving the top

TABLE 2: Elevator round-trip times, Scenario two

floors will save 15 seconds for each of floors 1–3 by not stopping. The elevator serving the bottom floors will save 20 seconds for each of the top floors and will save time on the return trip as well. The times for these trips are shown in Table 2 .

Assuming the employees are evenly distributed among the floors (40 people per floor), elevator A will transport 120 people, requiring 12 trips, and elevators B and C will transport 120 people, requiring 6 trips each. These trips will take 1200 seconds (20 minutes) for elevator A and 780 seconds (13 minutes) for elevators B and C, resulting in a small time savings (about 3 minutes) over the first scenario. Because elevators B and C are finished so much sooner than elevator A, there is likely a more efficient solution.

Scenario Three

The two round-trip times in Table 2 do not differ by much because the elevators move quickly between floors but stop at floors relatively slowly. This observation suggests that a more efficient arrangement might be to assign each elevator to a pair of floors. The times for such a scenario are listed in Table 3 .

Again assuming 40 employees per floor, each elevator will deliver 80 people, requiring 8 trips, taking at most a total of 920 seconds. Thus this assignment of elevators results in a time savings of almost 35% when compared with the 1400 seconds it would take to deliver all employees via unassigned elevators.

TABLE 3: Elevator round-trip times, Scenario three

Perhaps this is the optimal solution. If so, then the above analysis of this simple case suggests two hypotheses:

• The optimal solution assigns each floor to a single elevator.
• If the time for stopping is sufficiently larger than the time for moving between floors, each elevator should serve the same number of floors.

Mathematically, one could try to show that this solution is optimal by trying all possible elevator assignments or by carefully reasoning, perhaps by showing that the above hypotheses are correct. Practically, however, it doesn't matter because this solution considers only the morning rush hour and ignores periods of low use.

The assignment is clearly not optimal during periods of low use, and much of the inefficiency is related to the first hypothesis for rush hour optimization: that each floor is served by a single elevator. With this condition, if an employee on floor 6 arrives at the ground floor just after elevator C has departed, for example, she or he will have to wait nearly two minutes for elevator C to return, even if elevators A and B are idle. There are other inefficiencies that are not considered by focusing on the rush hour. Because each floor is served by a single elevator, an employee who wishes to travel from floor 3 to floor 6, for example, must go via the ground floor and switch elevators. Most employees would prefer more flexibility than a single elevator serving each floor.

At times when the elevators are not all busy, unassigned elevators will provide the quickest response and the greatest flexibility.

Because this optimal solution conflicts with the optimal rush hour solution, some compromise is necessary. In this simple case, perhaps elevator A could serve all floors, elevator B could serve floors 1-3, and elevator C could serve floors 4-6.

The second hypothesis, above, deserves some further thought. The efficiency of the rush hour solution Table 3 is due in part to the even division of employees among the floors. If employees were unevenly distributed with, say, 120 of the 240 people working on the top two floors, then elevator C would need to make 12 trips, taking a total of 1380 seconds, resulting in almost no benefit over unassigned elevators. Thus, an efficient solution in an actual building must take into account the distribution of the employees among the floors.

Because the stopping time on each floor is three times as large as the traveling time between floors (15 seconds versus 5 seconds), this solution effectively ignores the traveling time by assigning the same number of employees to each elevator. For taller buildings, the traveling time will become more significant. In those cases fewer employees should be assigned to the elevators that serve the upper floors than are assigned to the elevators that serve the lower floors.

The problem can be made more challenging by altering the number of elevators, the number of floors, and the number of individuals working on each floor. The rate of movement of elevators can be determined by observing buildings in the local area. Some elevators move more quickly than others. Entrance and exit times could also be measured by students collecting

data on local elevators. In a similar manner, the number of workers, elevators, and floors could be taken from local contexts.

A related question is, where should the elevators go when not in use? Is it best for them to return to the ground floor? Should they remain where they were last sent? Should they distribute themselves evenly among the floors? Or should they go to floors of anticipated heavy traffic? The answers will depend on the nature of the building and the time of day. Without analysis, it will not be at all clear which strategy is best under specific conditions. In some buildings, the elevators are controlled by computer programs that "learn" and then anticipate the traffic patterns in the building.

A different example that students can easily explore in detail is the problem of situating a fire station or an emergency room in a city. Here the key issue concerns travel times to the region being served, with conflicting optimization goals: average time vs. maximum time. A location that minimizes the maximum time of response may not produce the least average time of response. Commuters often face similar choices in selecting routes to work. They may want to minimize the average time, the maximum time, or perhaps the variance, so that their departure and arrival times are more predictable.

Most of the optimization conditions discussed so far have been expressed in units of time. Sometimes, however, two optimization conditions yield strategies whose outcomes are expressed in different (and sometimes incompatible) units of measurement. In many public policy issues (e.g., health insurance) the units are lives and money. For environmental issues, sometimes the units themselves are difficult to identify (e.g., quality of life).

When one of the units is money, it is easy to find expensive strategies but impossible to find ones that have virtually no cost. In some situations, such as airline safety, which balances lives versus dollars, there is no strategy that minimize lives lost (since additional dollars always produce slight increases in safety), and the strategy that minimizes dollars will be at $0. Clearly some compromise is necessary. Working with models of different solutions can help students understand the consequences of some of the compromises.

Heating-Degree-Days

An energy consulting firm that recommends and installs insulation and similar energy saving devices has received a complaint from a customer. Last summer she paid $540 to insulate her attic on the prediction that it would save 10% on her natural gas bills. Her gas bills have been higher than the previous winter, however, and now she wants a refund on the cost of the insulation. She admits that this winter has been colder than the last, but she had expected still to see some savings.

The facts: This winter the customer has used 1,102 therms, whereas last winter she used only 1,054 therms. This winter has been colder: 5,101 heating-degree-days this winter compared to 4,201 heating-degree-days last winter. (See explanation below.) How does a representative of the energy consulting firm explain to this customer that the accumulated heating-degree-days measure how much colder this winter has been, and then explain how to calculate her anticipated versus her actual savings.

Explaining the mathematics behind a situation can be challenging and requires a real knowledge of the context, the procedures, and the underlying mathematical concepts. Such communication of mathematical ideas is a powerful learning device for students of mathematics as well as an important skill for the workplace. Though the procedure for this problem involves only proportions, a thorough explanation of the mathematics behind the procedure requires understanding of linear modeling and related algebraic reasoning, accumulation and other precursors of calculus, as well as an understanding of energy usage in home heating.

The customer seems to understand that a straight comparison of gas usage does not take into account the added costs of colder weather, which can be significant. But before calculating any anticipated or actual savings, the customer needs some understanding of heating-degree-days. For many years, weather services and oil and gas companies have been using heating-degree-days to explain and predict energy usage and to measure energy savings of insulation and other devices. Similar degree-day units are also used in studying insect populations and crop growth. The concept provides a simple measure of the accumulated amount of cold or warm weather over time. In the discussion that follows, all temperatures are given in degrees Fahrenheit, although the process is equally workable using degrees Celsius.

Suppose, for example, that the minimum temperature in a city on a given day is 52 degrees and the maximum temperature is 64 degrees. The average temperature for the day is then taken to be 58 degrees. Subtracting that result from 65 degrees (the cutoff point for heating), yields 7 heating-degree-days for the day. By recording high and low temperatures and computing their average each day, heating-degree-days can be accumulated over the course of a month, a winter, or any period of time as a measure of the coldness of that period.

Over five consecutive days, for example, if the average temperatures were 58, 50, 60, 67, and 56 degrees Fahrenheit, the calculation yields 7, 15, 5, 0, and 9 heating-degree-days respectively, for a total accumulation of 36 heating-degree-days for the five days. Note that the fourth day contributes 0 heating-degree-days to the total because the temperature was above 65 degrees.

The relationship between average temperatures and heating-degree-days is represented graphically in Figure 1 . The average temperatures are shown along the solid line graph. The area of each shaded rectangle represents the number of heating-degree-days for that day, because the width of each rectangle is one day and the height of each rectangle is the number of degrees below 65 degrees. Over time, the sum of the areas of the rectangles represents the number of heating-degree-days accumulated during the period. (Teachers of calculus will recognize connections between these ideas and integral calculus.)

The statement that accumulated heating-degree-days should be proportional to gas or heating oil usage is based primarily on two assumptions: first, on a day for which the average temperature is above 65 degrees, no heating should be required, and therefore there should be no gas or heating oil usage; second, a day for which the average temperature is 25 degrees (40 heating-degree-days) should require twice as much heating as a day for which the average temperature is 45

FIGURE 1: Daily heating-degree-days

degrees (20 heating-degree-days) because there is twice the temperature difference from the 65 degree cutoff.

The first assumption is reasonable because most people would not turn on their heat if the temperature outside is above 65 degrees. The second assumption is consistent with Newton's law of cooling, which states that the rate at which an object cools is proportional to the difference in temperature between the object and its environment. That is, a house which is 40 degrees warmer than its environment will cool at twice the rate (and therefore consume energy at twice the rate to keep warm) of a house which is 20 degrees warmer than its environment.

The customer who accepts the heating-degree-day model as a measure of energy usage can compare this winter's usage with that of last winter. Because 5,101/4,201 = 1.21, this winter has been 21% colder than last winter, and therefore each house should require 21% more heat than last winter. If this customer hadn't installed the insulation, she would have required 21% more heat than last year, or about 1,275 therms. Instead, she has required only 5% more heat (1,102/1,054 = 1.05), yielding a savings of 14% off what would have been required (1,102/1,275 = .86).

Another approach to this would be to note that last year the customer used 1,054 therms/4,201 heating-degree-days = .251 therms/heating-degree-day, whereas this year she has used 1,102 therms/5,101 heating-degree-days = .216 therms/heating-degree-day, a savings of 14%, as before.

How good is the heating-degree-day model in predicting energy usage? In a home that has a thermometer and a gas meter or a gauge on a tank, students could record daily data for gas usage and high and low temperature to test the accuracy of the model. Data collection would require only a few minutes per day for students using an electronic indoor/outdoor thermometer that tracks high and low temperatures. Of course, gas used for cooking and heating water needs to be taken into account. For homes in which the gas tank has no gauge or doesn't provide accurate enough data, a similar experiment could be performed relating accumulated heating-degree-days to gas or oil usage between fill-ups.

It turns out that in well-sealed modern houses, the cutoff temperature for heating can be lower than 65 degrees (sometimes as low as 55 degrees) because of heat generated by light bulbs, appliances, cooking, people, and pets. At temperatures sufficiently below the cutoff, linearity turns out to be a good assumption. Linear regression on the daily usage data (collected as suggested above) ought to find an equation something like U = -.251( T - 65), where T is the average temperature and U is the gas usage. Note that the slope, -.251, is the gas usage per heating-degree-day, and 65 is the cutoff. Note also that the accumulation of heating-degree-days takes a linear equation and turns it into a proportion. There are some important data analysis issues that could be addressed by such an investigation. It is sometimes dangerous, for example, to assume linearity with only a few data points, yet this widely used model essentially assumes linearity from only one data point, the other point having coordinates of 65 degrees, 0 gas usage.

Over what range of temperatures, if any, is this a reasonable assumption? Is the standard method of computing average temperature a good method? If, for example, a day is mostly near 20 degrees but warms up to 50 degrees for a short time in the afternoon, is 35 heating-degree-days a good measure of the heating required that day? Computing averages of functions over time is a standard problem that can be solved with integral calculus. With knowledge of typical and extreme rates of temperature change, this could become a calculus problem or a problem for approximate solution by graphical methods without calculus, providing background experience for some of the important ideas in calculus.

Students could also investigate actual savings after insulating a home in their school district. A customer might typically see 8-10% savings for insulating roofs, although if the house is framed so that the walls act like chimneys, ducting air from the house and the basement into the attic, there might be very little savings. Eliminating significant leaks, on the other hand, can yield savings of as much as 25%.

Some U.S. Department of Energy studies discuss the relationship between heating-degree-days and performance and find the cutoff temperature to be lower in some modern houses. State energy offices also have useful documents.

What is the relationship between heating-degree-days computed using degrees Fahrenheit, as above, and heating-degree-days computed using degrees Celsius? Showing that the proper conversion is a direct proportion and not the standard Fahrenheit-Celsius conversion formula requires some careful and sophisticated mathematical thinking.

Traditionally, vocational mathematics and precollege mathematics have been separate in schools. But the technological world in which today's students will work and live calls for increasing connection between mathematics and its applications. Workplace-based mathematics may be good mathematics for everyone.

High School Mathematics at Work illuminates the interplay between technical and academic mathematics. This collection of thought-provoking essays—by mathematicians, educators, and other experts—is enhanced with illustrative tasks from workplace and everyday contexts that suggest ways to strengthen high school mathematical education.

This important book addresses how to make mathematical education of all students meaningful—how to meet the practical needs of students entering the work force after high school as well as the needs of students going on to postsecondary education.

The short readable essays frame basic issues, provide background, and suggest alternatives to the traditional separation between technical and academic mathematics. They are accompanied by intriguing multipart problems that illustrate how deep mathematics functions in everyday settings—from analysis of ambulance response times to energy utilization, from buying a used car to "rounding off" to simplify problems.

The book addresses the role of standards in mathematics education, discussing issues such as finding common ground between science and mathematics education standards, improving the articulation from school to work, and comparing SAT results across settings.

Experts discuss how to develop curricula so that students learn to solve problems they are likely to encounter in life—while also providing them with approaches to unfamiliar problems. The book also addresses how teachers can help prepare students for postsecondary education.

For teacher education the book explores the changing nature of pedagogy and new approaches to teacher development. What kind of teaching will allow mathematics to be a guide rather than a gatekeeper to many career paths? Essays discuss pedagogical implication in problem-centered teaching, the role of complex mathematical tasks in teacher education, and the idea of making open-ended tasks—and the student work they elicit—central to professional discourse.

High School Mathematics at Work presents thoughtful views from experts. It identifies rich possibilities for teaching mathematics and preparing students for the technological challenges of the future. This book will inform and inspire teachers, teacher educators, curriculum developers, and others involved in improving mathematics education and the capabilities of tomorrow's work force.

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Home » Blog » General » Teaching Problem-Solving Skills in Everyday Life: A Guide for Educators

Teaching Problem-Solving Skills in Everyday Life: A Guide for Educators

Introduction

Problems are an inevitable part of life, both at home and at school. They can arise from thoughtless actions or purely by accident. How we handle and react to these problems is crucial for our personal growth and well-being. As educators, it is essential to teach students the art of problem-solving to help them navigate everyday challenges. This blog post will introduce you to an easy no-prep activity, discussion questions, and related skills that can be incorporated into your lessons to help students become better problem-solvers.

No-Prep Activity: The Problem-Solving Steps Game

This activity will help students practice the Problem-Solving Steps in a fun and interactive way. The game requires no preparation or materials, making it an ideal choice for busy educators.

• Divide the class into small groups.
• Provide each group with a common everyday problem (e.g., forgetting a homework assignment, accidentally breaking a toy, or dealing with a conflict at recess).
• Ask the groups to discuss and apply the Problem-Solving Steps to their assigned problem:
• Identify the problem
• Think about the size of the problem
• Come up with a few solutions
• Pick the best solution
• Test the solution
• After a set time, have each group present their problem and chosen solution to the class.
• As a class, discuss the different solutions and how well they addressed the problem.

This activity encourages students to think critically about the problems they face and apply the Problem-Solving Steps in a collaborative setting. It can be easily adapted to suit various age groups and learning environments.

Discussion Questions

After completing the Problem-Solving Steps Game, use these discussion questions to further explore the topic and stimulate deeper conversations:

• How did working in a group help you come up with better solutions to the problem?
• Can you think of a situation where you successfully used the Problem-Solving Steps in your own life? How did it help you?
• Why is it important to consider the size of a problem before trying to solve it?
• How can practicing problem-solving skills improve our relationships with others?
• What challenges might you face when trying to apply the Problem-Solving Steps in real-life situations?

Related Skills

Teaching problem-solving skills goes hand-in-hand with other essential social-emotional learning concepts. Here are some related skills that can be integrated into your lessons:

• Emotional regulation: Help students recognize and manage their emotions, so they can approach problems with a calm and focused mindset.
• Empathy: Teach students to consider how their actions and decisions might affect others, fostering a sense of understanding and compassion.
• Communication: Encourage effective verbal and nonverbal communication to express thoughts, feelings, and ideas clearly when solving problems.
• Decision-making: Support students in developing the ability to evaluate options and make informed choices when faced with challenging situations.
• Resilience: Cultivate a growth mindset and the ability to bounce back from setbacks, helping students to persevere through difficult problems.

Now that you have an understanding of how to teach problem-solving skills through engaging activities and discussions, it’s time to put these concepts into practice. To help you get started, we invite you to sign up for free sample materials at Everyday Speech. These resources include video lessons, printable materials, and interactive games that can be easily incorporated into your lessons to support students’ social-emotional learning journey.

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Importance of Mathematics in Everyday Life

Introduction.

At Abakus Center, we firmly believe in the significance of mathematics in everyday life. In this comprehensive article, we will delve into the various reasons why mathematics plays a crucial role in our daily activities. From practical applications to cognitive development, we will explore how embracing mathematics can enhance our problem-solving skills, critical thinking abilities, and overall comprehension of the world around us.

Enhancing Problem-Solving Skills

Mathematics serves as the foundation for developing strong problem-solving skills. By engaging in mathematical exercises, individuals cultivate their ability to analyze complex situations, identify patterns, and formulate logical solutions. This skill set extends beyond mathematical problems, allowing individuals to tackle real-life challenges more effectively.

Promoting Logical Thinking

The study of mathematics nurtures logical thinking, enabling individuals to approach problems systematically and make sound decisions. Mathematics trains our minds to evaluate information objectively, consider multiple perspectives, and draw accurate conclusions based on evidence. These logical thinking skills are invaluable in various domains, such as science, technology, engineering, and finance.

Practical Applications of Mathematics

Mathematics permeates numerous aspects of our daily lives, often without us realizing it. Here are some key areas where mathematics plays a vital role:

1. Finance and Money Management

Understanding mathematics is essential for managing personal finances effectively. Concepts such as budgeting, interest rates, investments, and calculating expenses heavily rely on mathematical principles. Proficiency in mathematics empowers individuals to make informed financial decisions and plan for the future wisely.

Read more "Angles Unveiled: Exploring Their Practical Applications in Real Life"

2. Measurements and Conversions

From cooking in the kitchen to constructing buildings, accurate measurements are fundamental. Mathematics provides us with the necessary tools to measure and convert quantities, ensuring precision and consistency in various fields such as architecture, engineering, and culinary arts.

3. Data Analysis and Statistics

In today's data-driven world, the ability to analyze and interpret data is critical. Mathematical concepts such as statistical analysis, probability, and data modeling enable us to extract meaningful insights from vast amounts of information. From medical research to business analytics, mathematics equips individuals with the skills to make informed decisions based on data.

Cognitive Development and Mental Agility

Engaging with mathematics from an early age has a profound impact on cognitive development and mental agility. The intricate nature of mathematical problem-solving exercises stimulates critical thinking, creativity, and analytical reasoning. Furthermore, mathematics challenges the brain, enhancing memory, concentration, and overall cognitive abilities.

Mathematics and Career Opportunities

A solid foundation in mathematics opens doors to a wide range of career opportunities across various industries. Professions such as engineering, computer science, finance, architecture, and research heavily rely on mathematical expertise. By honing mathematical skills, individuals can pursue fulfilling careers that not only offer financial stability but also provide opportunities for personal growth and innovation.

In conclusion, mathematics is undeniably crucial in our everyday lives. From fostering problem-solving skills and logical thinking to enabling practical applications and boosting cognitive development, the impact of mathematics extends far beyond the confines of the classroom. Embracing mathematics empowers individuals to navigate the complexities of the modern world, equipping them with the tools necessary for success in both personal and professional endeavors.

Remember, mathematics is not merely a subject; it is a powerful tool that enriches our lives and shapes the world we live in.

FAQ: How does mathematics enhance problem-solving skills in everyday life?

Answer: Mathematics trains the mind to analyze complex situations, identify patterns, and formulate logical solutions. By engaging in mathematical exercises, individuals develop critical problem-solving skills that extend beyond mathematical problems, allowing them to tackle real-life challenges more effectively.

FAQ: What are some practical applications of mathematics in daily life?

Answer: Mathematics has numerous practical applications in various domains. It plays a vital role in finance and money management, helping individuals understand budgeting, interest rates, investments, and calculating expenses. Additionally, mathematics is essential for measurements and conversions in fields such as architecture, engineering, and culinary arts. Furthermore, data analysis and statistics heavily rely on mathematical concepts, providing insights in areas like medical research and business analytics.

FAQ: How does mathematics contribute to cognitive development?

Answer: Engaging with mathematics from an early age has a profound impact on cognitive development. Mathematical problem-solving exercises stimulate critical thinking, creativity, and analytical reasoning. By challenging the brain, mathematics enhances memory, concentration, and overall cognitive abilities, promoting well-rounded mental agility.

FAQ: What career opportunities are available for individuals with a strong foundation in mathematics?

Answer: A solid foundation in mathematics opens doors to diverse career opportunities across industries. Professions such as engineering, computer science, finance, architecture, and research heavily rely on mathematical expertise. By honing their mathematical skills, individuals can pursue fulfilling careers that offer financial stability and personal growth while fostering innovation and problem-solving.

FAQ: Is mathematics relevant only in academic settings?

Answer: No, mathematics is not limited to academic settings. It permeates various aspects of everyday life. Understanding mathematics is crucial for managing personal finances, making informed financial decisions, and planning for the future. Additionally, mathematics is essential for accurate measurements in fields like construction and culinary arts. Its practical applications extend to data analysis, enabling individuals to interpret and draw insights from vast amounts of information, benefiting multiple industries.

Caroline Milne - plays a crucial role in our blog as an articles writer, forming a strong and valuable connection with our Company. Hailing from the United Kingdom, her unique perspective and expertise greatly contribute to the content she creates for our website.

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• For Parents

Math in Action: Problem Solving Skills for Everyday Life

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Author: BYJU’S Math Companion Tutor

How is math problem-solving commonly used in everyday life?

• Budgeting: Teach children how to manage allowances, savings, and expenses. Discussing budgeting strategies helps them make wise financial decisions.
• Cooking: Cooking involves precise measurements and conversions. Baking a cake, for instance, is a delicious way to apply math skills.
• Shopping: Explain how discounts, percentages, and sales tax work. Involve children in calculating discounts to make shopping both educational and fun.
• Travel planning: Planning a road trip requires understanding distance, time, and fuel consumption. Map reading and calculating travel expenses provide real-life math lessons.
• Time management: Teach children to use schedules and timetables. Managing their time effectively prepares them for future responsibilities.
• Problem-solving games: Encourage board games like chess, Sudoku, or logic puzzles. These games sharpen analytical thinking and math skills.

Math problem-solving skills that are essential for real-life challenges for children

• Critical thinking and analysis: Encourage children to dissect complex problems into smaller, manageable parts. This skill enables them to analyze situations, identify key variables, and approach challenges with clarity.
• Logical reasoning: Logical thinking helps your little one evaluate the relationships between different components of a problem. It guides them in determining cause-and-effect patterns and making informed decisions.
• Pattern recognition: Patterns are everywhere in our world, from nature’s symmetries to data trends. Teaching children to recognize and use patterns equips them with a powerful tool for problem-solving.
• Creative problem solving: Foster creativity by encouraging children to explore various approaches to a problem. This allows them to think outside the box and devise innovative solutions.
• Numerical fluency: Strong numerical skills are fundamental. Proficiency in addition, subtraction, multiplication, and division forms the basis for solving a wide range of everyday problems.
• Measurement and estimation: Understanding measurement units and making reasonable estimations are essential for tasks like cooking, DIY projects, and understanding sizes and quantities.
• Spatial awareness: Geometry plays a significant role in real-life situations, from arranging furniture to reading maps. Developing spatial skills enhances a child’s ability to navigate physical spaces efficiently.
• Time management: The skill of managing time effectively is essential for scheduling daily activities, setting goals, and adhering to deadlines.
• Probability and risk assessment: Understanding probability helps children assess risks and make decisions in uncertain situations, such as games of chance or investments.
• Measurement conversions: Being able to convert units, such as ounces to grams or miles to kilometers, is valuable in daily life.
• Algebraic thinking: Basic algebraic concepts can be applied to real-life situations, such as solving for an unknown variable in a recipe or a budget.

5 ways children use math in everyday life

• Money matters: Understanding and managing allowances, budgeting for spending, and calculating change while shopping.
• Time management: Reading clocks and calendars, scheduling activities, and tracking time spent on tasks.
• Homework and tests: Solving math problems for assignments and taking math tests and quizzes at school.
• Playing games : Board games, card games, and video games often involve counting, calculating scores, and making strategic moves.
• Sports and activities: Keeping score in sports, tracking statistics, and understanding game strategies that involve math.

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