Talk details.

## 10 Modeling Problem Solving

We’ve discussed in previous chapters how part of a tutor’s task is to model good learning habits. When tutors are organized, use good time management, and leverage resources, we demonstrate the skills that students can use to be successful learners.

Problem-solving is an additional skill that tutors model for students. An organized and- intentional problem-solving approach helps us to efficiently work through challenges, and many of us effectively problem solve without much thought given to our approach. 1 However, it makes sense to take a step back and do our best to model problem-solving best-practices. Remember that repeated demonstration of a tutor’s problem-solving strategies can help students learn from our example.

We know the tutor’s role is not to solve a student’s problem for them. How do we model good problem-solving, without actually solving the problem ourselves? It’s tricky, but not impossible. We can empower students to work their way through any problem by asking good questions and walking them through the steps of the process.

## The Rational Problem-Solving Process

Problem-solving is something many of us have taught ourselves through practice. However, there are many scholars and professionals who have examined and broken down effective problem-solving strategies into a series of logical steps. 2 We can check our own process by reflecting on what has been written about best-practices in problem-solving, and maybe make changes to be more consistent and effective. This can better prepare tutors to guide a student through the process when we apply it in a tutoring session.

## Step 1: Define the Problem

It may seem obvious to state that the first step in solving a problem is to notice that we have a problem. Unless we take time to understand precisely what is wrong, however, we may find ourselves creating a solution that doesn’t actually fix anything. It’s very common to dive straight into devising a solution only to find that we’ve solved the wrong problem. Alternatively, we might develop a solution only to discover that the real problem is bigger than we thought.

A good practice for starting out is to try to define the problem in words. By writing or stating a problem definition, we’re challenged to identify the root cause, and this information can guide us in developing effective solutions.

In a tutoring session, sometimes the problem can take a variety of forms. The problem could be:

• the literal problem given in a student’s homework assignment (a word problem in math, or a case study in biology, for example.)
• a lack of clarity in assignment instructions.
• the student not having a strategy for planning a project or starting a paper.
• the student lacking confidence to tackle their homework or study independently

Keep in mind that the form the “problem” takes will change based on the student’s needs and goals. If the problem is that the student doesn’t understand something, the first step is to identify precisely what they don’t understand. If the problem is that something is missing, then understanding exactly what necessary parts are missing is the first step.

In a tutoring session this may mean asking the student to start the process, or begin describing the concept from the beginning, until they reach the point where things become unclear. Together, you can determine where the gaps are, and begin to develop a problem definition.

## Step 2: Pull from Existing Knowledge

After we’ve identified and defined the problem, the next step is to ask ourselves what we already know about the situation. Take an inventory. What information do we already have? What can we learn from the context? What resources have we been given?

When working with a student, pulling from existing knowledge might involve reviewing the concepts already covered and the student’s existing knowledge of the course material. It may also mean reaching into material and experiences outside of the student’s course.

Some helpful questions to guide this step include:

• What does the student know about topics related to the course material?
• What experience might the student have from prior courses?
• In what context might the student have heard these ideas discussed in their everyday lives or in popular culture?

When we encourage students to step back and really take account of everything they already know about the problem and its context, they can be surprised at how much knowledge they actually bring.

## Step 3: Refer to support materials

Once we’ve pulled from the knowledge we already have, we can expand our search for supporting knowledge to outside resources. Are there reference materials we can access? Are there experts we can consult?

The first thing we can encourage students to do is to refer to their course texts, notes, study guides, and materials provided by the class instructor. These are often the best places to start because they’re most likely to provide relevant information. Once these resources have been referenced, we can also encourage students to look for information and guidance from other academic resources.

Students often forget that they can reference what others have written about their problem. Outside textbooks and supporting texts may offer similar ideas presented in a different way, and this could help the student approach the problem with new understanding or perspective. Online research and reference materials are good places to look for clarification of rules, theories, laws, formulas, processes, and examples. While these sources may not be quite as specific to a student’s class assignment, they can sometimes provide confirmation or clarity in areas where a student might need it.

Students should be made to feel free to leverage other academic supports as well. They are already leveraging one aspect of this support when they come to see a tutor. Other supports may include making use of the library or computer center, visiting their instructor’s office hours to ask questions, or even reaching out to other classmates. It’s always helpful for tutors to remind the student that these other supports are available and to encourage them to use these resources.

If a student is unsure or intimidated by contacting an instructor or a classmate, or is uncomfortable learning how to use other support resources, encouragement from a tutor can often be the nudge a student needs. Remind them of these supports and offer to help them access them where appropriate.

## Step 4: Brainstorm Solutions

There’s usually more than one way to solve a problem, and it’s helpful to brainstorm multiple solutions to find the one that works best.

It’s important that tutors allow students to take an active role in developing their own solutions to the problem. This is where our Socratic questioning skills become really crucial and can help the students to apply what they know to the problem they’ve identified. The tutor’s role here is to facilitate the solution-generating process, contributing where appropriate, and helping to guide the student in a productive direction.

It is possible that the student will suggest a solution that we know will not solve the problem. Depending on the nature and scale of the problem, it may not always be appropriate for us to tell the student that we think it won’t work. Guiding the student through the problem-solving process is about helping students to engage with the process itself. That way, they can feel confident applying it on their own, even when a helpful tutor isn’t around to give hints. It’s up to each tutor in each situation to decide when it is appropriate to expedite the process by providing insights into solutions, and when it is best to allow students to test their solutions to determine their effectiveness.

## Step 5: Test a Solution

Choose a solution and try it out. Maybe it will work! Maybe it doesn’t. Having a variety of solutions to try is why we brainstorm more than one. Though trial and error can sometimes feel frustrating, it is in the testing of our solutions that we often learn the most. We’re able to better understand the parts that work, the parts that don’t, and hopefully learn the reasons why. This can result in solutions that are more efficient and better suited to our needs.

Solution-testing is an opportunity for students to learn from mistakes in a safe, low-risk way. Often mistakes in class result in deducted points, a bad grade, or maybe an embarrassing moment in from t of classmates. As a guide through the problem-solving process, tutors can help students to see mistakes as necessary and helpful steps on the way to a solution that works, rather than as failures. It’s important that the tutor help the student see mistakes as progress, especially when a student becomes discouraged. This helps the student maintain a growth mindset while identifying ways to improve.

## Step 6: Revising the Solution

When a solution doesn’t work, it may not mean the whole idea was bad. Maybe it needs some revisions and refining, but doesn’t always need to be discarded. We can use what we learned from solution-testing to make effective revisions.

This may mean we guide a student back to previous steps in the problem-solving process. Students may once more need to pull from existing knowledge, revisit those support materials, or look at some of the alternative solutions that the student developed.

## Step 7: Revisit the Problem

We’ve got a solution that works! Did it fix our problem? If yes, then great!

Sometimes, however, a solution may “work,” without fixing our problem.

When this happens, we need to revisit the problem definition. Do we really understand it? Is there a detail we didn’t consider when developing our solutions? Did we misinterpret what the problem actually is when we crafted our problem definition?

At this point, perhaps we need to revise the solution once more. Sometimes in our process of researching and brainstorming, we can get off course, and taking time to refer to the initial problem can help us recalibrate our efforts and get us back on track.

Other times we may need re-define our problem. Perhaps after developing and testing several solutions, it becomes clear that the real problem is different than what we initially thought it was. Or perhaps our solutions address parts of the problem, but don’t get to the deeper root of the issue.

When a student has worked through the problem-solving process and still feels stuck, tutors can guide them to revisit the problem and clarify the initial goal. Returning to previous steps of the process as needed is normal and often necessary. Ensuring students that they’re still correctly applying the process, even when they need to jump back and forth between these steps, can help keep them from getting discouraged.

Quickwrite Exercise

Think back to a time you solved a problem in the past. It could be an obstacle you encountered in an academic setting (completing an assignment, researching for a paper, troubleshooting a technical problem) or in your personal life.

Take a moment to reflect:

• Did you use pieces of the rational problem solving process, without knowing?
• If you could go back and approach the problem again, how would you implement this problem solving approach? What would it look like? How would it have been different?

## Facilitating the Problem-Solving Process

The rational problem-solving process is an excellent tool to help tutors guide students through problems big and small. This organized way of approaching the task can help us make sure we’re heading in a productive direction, from solving a math problem to developing a strategy to finish a research paper. How do we ensure we’re empowering students to use this process on their own?

It can be helpful to both tutors and students to use the process as a checklist during a problem-solving session. We can name each step as we move through, and make it clear to the student the purpose of each activity. This doesn’t mean we turn a session of math tutoring into a lesson on the problem-solving process, but explicitly stating the names of each step can make it clear to the student the purpose of each activity, and help them to become familiar with the process. If we “narrate” our process as we go, students can experience a guided problem-solving process during their tutoring session and be encouraged to apply it independently.

Once we’ve guided a student through the process, we can then provide opportunities for the student to take charge. We can prompt the student to move from step to step, supporting them in their problem-solving efforts along the way. This guided practice can help students to become well-versed in the process itself, and to feel more comfortable applying it independently. 3

Something to Try

In your next session, when a student comes to you with a problem, use your Socratic questioning skills to walk the student through the problem solving process. (This may be something you’re already implementing naturally!)

Be deliberate about each step. Assist the student in defining the problem, guide the student to collect their existing knowledge, help the student pull from reference materials available, etc.

How does it work for you?

## Practicing the Problem-Solving Process

Don’t forget, that while this process is an excellent tool for helping students to solve problems during a session, it can also help tutors to problem-solve during a session!

Perhaps you encounter a student faced with a problem you yourself don’t know how to solve. No worries! The problem-solving process works just the same.

We can apply it to challenges with assignments, and we can also apply it to other issues we encounter during a tutoring session. Every student is unique, and it may take some problem-solving to learn how to best work with each student. Identifying the “problem,” pulling from our knowledge, consulting our supports, brainstorming, and testing solutions are all ways tutors can determine how best to assist students.

• Dane, E., Baer, M., Pratt, M. G., and Oldham, G. R. (2011). Rational versus intuitive problem solving: How thinking “off the beaten path” can stimulate creativity. Psychology of Aesthetics, Creativity, and the Arts.  5 (1), 3–12.  https://doi.org/10.1037/a0017698.
• Uzonwanne F.C. (2016). Rational Model of Decision Making. In: Farazmand A. (eds) Global Encyclopedia of Public Administration, Public Policy, and Governance. Springer, Cham. https://doi.org/10.1007/978-3-319-31816-5_2474-1.
• Klegeris, A., Bahniwal, M., and Hurren, H. (2017). Improvement in Generic Problem-Solving Abilities of Students by Use of Tutor-less Problem-Based Learning in a Large Classroom Setting. Life Sciences Education. 12(1), 1-116. https://doi.org/10.1187/cbe.12-06-0081.

## Additional Resources:

McNamera, C. (2020). Problem Solving and Decision Making (Solving Problems and Making Decisions). Free Management Library. Authenticity Consulting LLC. https://managementhelp.org/personalproductivity/problem-solving.htm . Accessed 26 Apr. 2021.

Nezu C., Palmatier, A., and Nezu, A. (2004). Social Problem-Solving Training for Caregivers. In Chang, D’Zurilla, & Sanna (Eds.) Social Problem Solving: Theory, Research, and Training. (223-238). American Psychological Association. https://doi.org/10.1037/10805-013 .

Nezu, A., Nezu, C., and D’Zurilla, T. (2007). Solving Life’s problems: a 5 Step Guide to Enhanced Well-Being. Springer Publishing Company LLC. https://www.springerpub.com/solving-life-s-problems-9780826114891.html .

Scott, G. M., Lonergan, D. C., and Mumford, M.D. (2010).  Conceptual Combination: Alternative Knowledge Structures, Alternative Heuristics. Creativity Research Journal. 17(1), 79-98. https://www.tandfonline.com/doi/abs/10.1207/s15326934crj1701_7 .

Tutor Handbook Copyright © 2021 by Penny Feltner and gapinski is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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## Problem solving through values: A challenge for thinking and capability development

• • This paper introduces the 4W framework of consistent problem solving through values.
• • The 4W suggests when, how and why the explication of values helps to solve a problem.
• • The 4W is significant to teach students to cope with problems having crucial consequences.
• • The paper considers challenges using such framework of thinking in different fields of education.

The paper aims to introduce the conceptual framework of problem solving through values. The framework consists of problem analysis, selection of value(s) as a background for the solution, the search for alternative ways of the solution, and the rationale for the solution. This framework reveals when, how, and why is important to think about values when solving problems. A consistent process fosters cohesive and creative value-based thinking during problem solving rather than teaching specific values. Therefore, the framework discloses the possibility for enabling the development of value-grounded problem solving capability.The application of this framework highlights the importance of responsibility for the chosen values that are the basis for the alternatives which determine actions. The 4W framework is meaningful for the people’s lives and their professional work. It is particularly important in the process of future professionals’ education. Critical issues concerning the development of problem solving through values are discussed when considering and examining options for the implementation of the 4W framework in educational institutions.

## 1. Introduction

The core competencies necessary for future professionals include problem solving based on complexity and collaborative approaches ( OECD, 2018 ). Currently, the emphasis is put on the development of technical, technological skills as well as system thinking and other cognitive abilities (e.g., Barber, 2018 ; Blanco, Schirmbeck, & Costa, 2018 ). Hence, education prepares learners with high qualifications yet lacking in moral values ( Nadda, 2017 ). Educational researchers (e.g., Barnett, 2007 ; Harland & Pickering, 2010 ) stress that such skills and abilities ( the how? ), as well as knowledge ( the what? ), are insufficient to educate a person for society and the world. The philosophy of education underlines both the epistemological and ontological dimensions of learning. Barnett (2007) points out that the ontological dimension has to be above the epistemological one. The ontological dimension encompasses the issues related to values that education should foster ( Harland & Pickering, 2010 ). In addition, values are closely related to the enablement of learners in educational environments ( Jucevičienė et al., 2010 ). For these reasons, ‘ the why ?’ based on values is required in the learning process. The question arises as to what values and how it makes sense to educate them. Value-based education seeks to address these issues and concentrates on values transfer due to their integration into the curriculum. Yazdani and Akbarilakeh (2017) discussed that value-based education could only convey factual knowledge of values and ethics. However, such education does not guarantee the internalization of values. Nevertheless, value-based education indicates problem solving as one of the possibilities to develop values.

Values guide and affect personal behavior encompassing the ethical aspects of solutions ( Roccas, Sagiv, & Navon, 2017 ; Schwartz, 1992 , 2012 ; Verplanken & Holland, 2002 ). Therefore, they represent the essential foundation for solving a problem. Growing evidence indicates the creative potential of values ( Dollinger, Burke, & Gump, 2007 ; Kasof, Chen, Himsel, & Greenberger, 2007 ; Lebedeva et al., 2019) and emphasizes their significance for problem solving. Meanwhile, research in problem solving pays little attention to values. Most of the problem solving models (e.g., Newell & Simon, 1972 ; Jonassen, 1997 ) utilize a rational economic approach. Principally, the research on the mechanisms of problem solving have been conducted under laboratory conditions performing simple tasks ( Csapó & Funke, 2017 ). Moreover, some of the decision-making models share the same steps as problem solving (c.f., Donovan, Guss, & Naslund, 2015 ). This explains why these terms are sometimes used interchangeably ( Huitt, 1992 ). Indeed, decision-making is a part of problem solving, which emerges while choosing between alternatives. Yet, values, moral, and ethical issues are more common in decision-making research (e.g., Keeney, 1994 ; Verplanken & Holland, 2002 ; Hall & Davis, 2007 ; Sheehan & Schmidt, 2015 ). Though, research by Shepherd, Patzelt, and Baron (2013) , Baron, Zhao, and Miao (2015) has affirmed that contemporary business decision makers rather often leave aside ethical issues and moral values. Thus, ‘ethical disengagement fallacy’ ( Sternberg, 2017, p.7 ) occurs as people think that ethics is more relevant to others. In the face of such disengagement, ethical issues lose their prominence.

The analysis of the literature revealed a wide field of problem solving research presenting a range of more theoretical insights rather empirical evidence. Despite this, to date, a comprehensive model that reveals how to solve problems emphasizing thinking about values is lacking. This underlines the relevance of the chosen topic, i.e. a challenge for thinking and for the development of capabilities addressing problems through values. To address this gap, the following issues need to be investigated: When, how, and why a problem solver should take into account values during problem solving? What challenges may occur for using such framework of thinking in different fields of education? Aiming this, the authors of the paper substantiated the conceptual framework of problem solving grounded in consistent thinking about values. The substantiation consists of several parts. First, different approaches to solving problems were examined. Second, searching to reveal the possibilities of values integration into problem solving, value-based approaches significant for problem solving were critically analyzed. Third, drawing on the effect of values when solving a problem and their creative potential, the authors of this paper claim that the identification of values and their choice for a solution need to be specified in the process of problem solving. As a synthesis of conclusions coming from the literature review and conceptual extensions regarding values, the authors of the paper created the coherent framework of problem solving through values (so called 4W).

The novelty of the 4W framework is exposed by several contributions. First, the clear design of overall problem solving process with attention on integrated thinking about values is used. Unlike in most models of problem solving, the first stage encompass the identification of a problem, an analysis of a context and the perspectives that influence the whole process, i.e. ‘What?’. The stage ‘What is the basis for a solution?’ focus on values identification and their choice. The stage ‘Ways how?’ encourages to create alternatives considering values. The stage ‘Why?’ represent justification of a chosen alternative according particular issues. Above-mentioned stages including specific steps are not found in any other model of problem solving. Second, even two key stages nurture thinking about values. The specificity of the 4W framework allows expecting its successful practical application. It may help to solve a problem more informed revealing when and how the explication of values helps to reach the desired value-based solution. The particular significance is that the 4W framework can be used to develop capabilities to solve problems through values. The challenges to use the 4W framework in education are discussed.

## 2. Methodology

To create the 4W framework, the integrative literature review was chosen. According to Snyder (2019) , this review is ‘useful when the purpose of the review is not to cover all articles ever published on the topic but rather to combine perspectives to create new theoretical models’ (p.334). The scope of this review focused on research disclosing problem solving process that paid attention on values. The following databases were used for relevant information search: EBSCO/Hostdatabases (ERIC, Education Source), Emerald, Google Scholar. The first step of this search was conducted using integrated keywords problem solving model , problem solving process, problem solving steps . These keywords were combined with the Boolean operator AND with the second keywords values approach, value-based . The inclusion criteria were used to identify research that: presents theoretical backgrounds and/or empirical evidences; performed within the last 5 years; within an educational context; availability of full text. The sources appropriate for this review was very limited in scope (N = 2).

We implemented the second search only with the same set of the integrated keywords. The inclusion criteria were the same except the date; this criterion was extended up to 10 years. This search presented 85 different sources. After reading the summaries, introductions and conclusions of the sources found, the sources that do not explicitly provide the process/models/steps of problem solving for teaching/learning purposes and eliminates values were excluded. Aiming to see a more accurate picture of the chosen topic, we selected secondary sources from these initial sources.

Several important issues were determined as well. First, most researchers ground their studies on existing problem solving models, however, not based on values. Second, some of them conducted empirical research in order to identify the process of studies participants’ problem solving. Therefore, we included sources without date restrictions trying to identify the principal sources that reveal the process/models/steps of problem solving. Third, decision-making is a part of problem solving process. Accordingly, we performed a search with the additional keywords decision-making AND values approach, value-based decision-making . We used such inclusion criteria: presents theoretical background and/or empirical evidence; no date restriction; within an educational context; availability of full text. These all searches resulted in a total of 16 (9 theoretical and 7 empirical) sources for inclusion. They were the main sources that contributed most fruitfully for the background. We used other sources for the justification the wholeness of the 4W framework. We present the principal results of the conducted literature review in the part ‘The background of the conceptual framework’.

## 3. The background of the conceptual framework

3.1. different approaches of how to solve a problem.

Researchers from different fields focus on problem solving. As a result, there still seems to be a lack of a conventional definition of problem solving. Regardless of some differences, there is an agreement that problem solving is a cognitive process and one of the meaningful and significant ways of learning ( Funke, 2014 ; Jonassen, 1997 ; Mayer & Wittrock, 2006 ). Differing in approaches to solving a problem, researchers ( Collins, Sibthorp, & Gookin, 2016 ; Jonassen, 1997 ; Litzinger et al., 2010 ; Mayer & Wittrock, 2006 ; O’Loughlin & McFadzean, 1999 ; ect.) present a variety of models that differ in the number of distinct steps. What is similar in these models is that they stress the procedural process of problem solving with the focus on the development of specific skills and competences.

For the sake of this paper, we have focused on those models of problem solving that clarify the process and draw attention to values, specifically, on Huitt (1992) , Basadur, Ellspermann, and Evans (1994) , and Morton (1997) . Integrating the creative approach to problem solving, Newell and Simon (1972) presents six phases: phase 1 - identifying the problem, phase 2 - understanding the problem, phase 3 - posing solutions, phase 4 - choosing solutions, phase 5 - implementing solutions, and phase 6 - final analysis. The weakness of this model is that these phases do not necessarily follow one another, and several can coincide. However, coping with simultaneously occurring phases could be a challenge, especially if these are, for instance, phases five and six. Certainly, it may be necessary to return to the previous phases for further analysis. According to Basadur et al. (1994) , problem solving consists of problem generation, problem formulation, problem solving, and solution implementation stages. Huitt (1992) distinguishes four stages in problem solving: input, processing, output, and review. Both Huitt (1992) and Basadur et al. (1994) four-stage models emphasize a sequential process of problem solving. Thus, problem solving includes four stages that are used in education. For example, problem-based learning employs such stages as introduction of the problem, problem analysis and learning issues, discovery and reporting, solution presentation and evaluation ( Chua, Tan, & Liu, 2016 ). Even PISA 2012 framework for problem solving composes four stages: exploring and understanding, representing and formulating, planning and executing, monitoring and reflecting ( OECD, 2013 ).

Drawing on various approaches to problem solving, it is possible to notice that although each stage is named differently, it is possible to reveal some general steps. These steps reflect the essential idea of problem solving: a search for the solution from the initial state to the desirable state. The identification of a problem and its contextual elements, the generation of alternatives to a problem solution, the evaluation of these alternatives according to specific criteria, the choice of an alternative for a solution, the implementation, and monitoring of the solution are the main proceeding steps in problem solving.

## 3.2. Value-based approaches relevant for problem solving

Huitt (1992) suggests that important values are among the criteria for the evaluation of alternatives and the effectiveness of a chosen solution. Basadur et al. (1994) point out to visible values in the problem formulation. Morton (1997) underlines that interests, investigation, prevention, and values of all types, which may influence the process, inspire every phase of problem solving. However, the aforementioned authors do not go deeper and do not seek to disclose the significance of values for problem solving.

Decision-making research shows more possibilities for problem solving and values integration. Sheehan and Schmidt (2015) model of ethical decision-making includes moral sensitivity, moral judgment, moral motivation, and moral action where values are presented in the component of moral motivation. Another useful approach concerned with values comes from decision-making in management. It is the concept of Value-Focused Thinking (VFT) proposed by Keeney (1994) . The author argues that the goals often are merely means of achieving results in traditional models of problem solving. Such models frequently do not help to identify logical links between the problem solving goals, values, and alternatives. Thus, according to Keeney (1994) , the decision-making starts with values as they are stated in the goals and objectives of decision-makers. VFT emphasizes the core values of decision-makers that are in a specific context as well as how to find a way to achieve them by using means-ends analysis. The weakness of VFT is its restriction to this means-ends analysis. According to Shin, Jonassen, and McGee (2003) , in searching for a solution, such analysis is weak as the problem solver focuses simply on removing inadequacies between the current state and the goal state. The strengths of this approach underline that values are included in the decision before alternatives are created. Besides, values help to find creative and meaningful alternatives and to assess them. Further, they include the forthcoming consequences of the decision. As VFT emphasizes the significant function of values and clarifies the possibilities of their integration into problem solving, we adapt this approach in the current paper.

## 3.3. The effect of values when solving a problem

In a broader sense, values provide a direction to a person’s life. Whereas the importance of values is relatively stable over time and across situations, Roccas et al. (2017) argue that values differ in their importance to a person. Verplanken and Holland (2002) investigated the relationship between values and choices or behavior. The research revealed that the activation of a value and the centrality of a value to the self, are the essential elements for value-guided behavior. The activation of values could happen in such cases: when values are the primary focus of attention; if the situation or the information a person is confronted with implies values; when the self is activated. The centrality of a particular value is ‘the degree to which an individual has incorporated this value as part of the self’ ( Verplanken & Holland, 2002, p.436 ). Thus, the perceived importance of values and attention to them determine value-guided behavior.

According to Argandoña (2003) , values can change due to external (changing values in the people around, in society, changes in situations, etc.) and internal (internalization by learning) factors affecting the person. The research by Hall and Davis (2007) indicates that the decision-makers’ applied value profile temporarily changed as they analyzed the issue from multiple perspectives and revealed the existence of a broader set of values. The study by Kirkman (2017) reveal that participants noticed the relevance of moral values to situations they encountered in various contexts.

Values are tightly related to personal integrity and identity and guide an individual’s perception, judgment, and behavior ( Halstead, 1996 ; Schwartz, 1992 ). Sheehan and Schmidt (2015) found that values influenced ethical decision-making of accounting study programme students when they uncovered their own values and grounded in them their individual codes of conduct for future jobs. Hence, the effect of values discloses by observing the problem solver’s decision-making. The latter observations could explain the abundance of ethics-laden research in decision-making rather than in problem solving.

Contemporary researchers emphasize the creative potential of values. Dollinger et al. (2007) , Kasof et al. (2007) , Lebedeva, Schwartz, Plucker, & Van De Vijver, 2019 present to some extent similar findings as they all used Schwartz Value Survey (respectively: Schwartz, 1992 ; ( Schwartz, 1994 ), Schwartz, 2012 ). These studies disclosed that such values as self-direction, stimulation and universalism foster creativity. Kasof et al. (2007) focused their research on identified motivation. Stressing that identified motivation is the only fully autonomous type of external motivation, authors define it as ‘the desire to commence an activity as a means to some end that one greatly values’ (p.106). While identified motivation toward specific values (italic in original) fosters the search for outcomes that express those specific values, this research demonstrated that it could also inhibit creative behavior. Thus, inhibition is necessary, especially in the case where reckless creativity could have painful consequences, for example, when an architect creates a beautiful staircase without a handrail. Consequently, creativity needs to be balanced.

Ultimately, values affect human beings’ lives as they express the motivational goals ( Schwartz, 1992 ). These motivational goals are the comprehensive criteria for a person’s choices when solving problems. Whereas some problem solving models only mention values as possible evaluation criteria, but they do not give any significant suggestions when and how the problem solver could think about the values coming to the understanding that his/her values direct the decision how to solve the problem. The authors of this paper claim that the identification of personal values and their choice for a solution need to be specified in the process of problem solving. This position is clearly reflected in humanistic philosophy and psychology ( Maslow, 2011 ; Rogers, 1995 ) that emphasize personal responsibility for discovering personal values through critical questioning, honest self-esteem, self-discovery, and open-mindedness in the constant pursuit of the truth in the path of individual life. However, fundamental (of humankind) and societal values should be taken into account. McLaughlin (1997) argues that a clear boundary between societal and personal values is difficult to set as they are intertwined due to their existence in complex cultural, social, and political contexts at a particular time. A person is related to time and context when choosing values. As a result, a person assumes existing values as implicit knowledge without as much as a consideration. This is particularly evident in the current consumer society.

Moreover, McLaughlin (1997) stresses that if a particular action should be tolerated and legitimated by society, it does not mean that this action is ultimately morally acceptable in all respects. Education has possibilities to reveal this. One such possibility is to turn to the capability approach ( Sen, 1990 ), which emphasizes what people are effectively able to do and to be. Capability, according to Sen (1990) , reflects a person’s freedom to choose between various ways of living, i.e., the focus is on the development of a person’s capability to choose the life he/she has a reason to value. According to Webster (2017) , ‘in order for people to value certain aspects of life, they need to appreciate the reasons and purposes – the whys – for certain valuing’ (italic in original; p.75). As values reflect and foster these whys, education should supplement the development of capability with attention to values ( Saito, 2003 ). In order to attain this possibility, a person has to be aware of and be able to understand two facets of values. Argandoña (2003) defines them as rationality and virtuality . Rationality refers to values as the ideal of conduct and involves the development of a person’s understanding of what values and why he/she should choose them when solving a problem. Virtuality approaches values as virtues and includes learning to enable a person to live according to his/her values. However, according to McLaughlin (1997) , some people may have specific values that are deep or self-evidently essential. These values are based on fundamental beliefs about the nature and purpose of the human being. Other values can be more or less superficial as they are based on giving priority to one or the other. Thus, virtuality highlights the depth of life harmonized to fundamentally rather than superficially laden values. These approaches inform the rationale for the framework of problem solving through values.

## 4. The 4W framework of problem solving through values

Similar to the above-presented stages of the problem solving processes, the introduced framework by the authors of this paper revisits them (see Fig. 1 ). The framework is titled 4W as its four stages respond to such questions: Analyzing the Problem: W hat ? → Choice of the value(s): W hat is the background for the solution? → Search for the alternative w ays of the solution: How ? → The rationale for problem solution: W hy is this alternative significant ? The stages of this framework cover seven steps that reveal the logical sequence of problem solving through values.

The 4 W framework: problem solving through values.

Though systematic problem solving models are criticized for being linear and inflexible (e.g., Treffinger & Isaksen, 2005 ), the authors of this paper assume a structural view of the problem solving process due to several reasons. First, the framework enables problem solvers to understand the thorough process of problem solving through values. Second, this framework reveals the depth of each stage and step. Third, problem solving through values encourages tackling problems that have crucial consequences. Only by understanding and mastering the coherence of how problems those require a value-based approach need to be addressed, a problem solver will be able to cope with them in the future. Finally, this framework aims at helping to recognize, to underline personal values, to solve problems through thinking about values, and to take responsibility for choices, even value-based. The feedback supports a direct interrelation between stages. It shapes a dynamic process of problem solving through values.

The first stage of problem solving through values - ‘ The analysis of the problem: What? ’- consists of three steps (see Fig. 1 ). The first step is ‘ Recognizing the problematic situation and naming the problem ’. This step is performed in the following sequence. First, the problem solver should perceive the problematic situation he/she faces in order to understand it. Dostál (2015) argues that the problematic situation has the potential to become the problem necessary to be addressed. Although each problem is limited by its context, not every problematic situation turns into a problem. This is related to the problem solver’s capability and the perception of reality: a person may not ‘see’ the problem if his/her capability to perceive it is not developed ( Dorst, 2006 ; Dostál, 2015 ). Second, after the problem solver recognizes the existence of the problematic situation, the problem solver has to identify the presence or absence of the problem itself, i.e. to name the problem. This is especially important in the case of the ill-structured problems since they cannot be directly visible to the problem solver ( Jonassen, 1997 ). Consequently, this step allows to determine whether the problem solver developed or has acquired the capability to perceive the problematic situation and the problem (naming the problem).

The second step is ‘ Analysing the context of the problem as a reason for its rise ’. At this step, the problem solver aims to analyse the context of the problem. The latter is one of the external issues, and it determines the solution ( Jonassen, 2011 ). However, if more attention is paid to the solution of the problem, it diverts attention from the context ( Fields, 2006 ). The problem solver has to take into account both the conveyed and implied contextual elements in the problematic situation ( Dostál, 2015 ). In other words, the problem solver has to examine it through his/her ‘contextual lenses’ ( Hester & MacG, 2017 , p.208). Thus, during this step the problem solver needs to identify the elements that shape the problem - reasons and circumstances that cause the problem, the factors that can be changed, and stakeholders that are involved in the problematic situation. Whereas the elements of the context mentioned above are within the problematic situation, the problem solver can control many of them. Such control can provide unique ways for a solution.

Although the problem solver tries to predict the undesirable results, some criteria remain underestimated. For that reason, it is necessary to highlight values underlying the various possible goals during the analysis ( Fields, 2006 ). According to Hester and MacG (2017) , values express one of the main features of the context and direct the attention of the problem solver to a given problematic situation. Hence, the problem solver should explore the value-based positions that emerge in the context of the problem.

The analysis of these contextual elements focus not only on a specific problematic situation but also on the problem that has emerged. This requires setting boundaries of attention for an in-depth understanding ( Fields, 2006 ; Hester & MacG, 2017 ). Such understanding influences several actions: (a) the recognition of inappropriate aspects of the problematic situation; (b) the emergence of paths in which identified aspects are expected to change. These actions ensure consistency and safeguard against distractions. Thus, the problem solver can now recognize and identify the factors that influence the problem although they are outside of the problematic situation. However, the problem solver possesses no control over them. With the help of such context analysis, the problem solver constructs a thorough understanding of the problem. Moreover, the problem solver becomes ready to look at the problem from different perspectives.

The third step is ‘ Perspectives emerging in the problem ’. Ims and Zsolnai (2009) argue that problem solving usually contains a ‘problematic search’. Such a search is a pragmatic activity as the problem itself induces it. Thus, the problem solver searches for a superficial solution. As a result, the focus is on control over the problem rather than a deeper understanding of the problem itself. The analysis of the problem, especially including value-based approaches, reveals the necessity to consider the problem from a variety of perspectives. Mitroff (2000) builds on Linstone (1989) ideas and claims that a sound foundation of both naming and solving any problem lays in such perspectives: the technical/scientific, the interpersonal/social, the existential, and the systemic (see Table 1 ).

The main characteristics of four perspectives for problem solving

Whereas all problems have significant aspects of each perspective, disregarding one or another may lead to the wrong way of solving the problem. While analysing all four perspectives is essential, this does not mean that they all are equally important. Therefore, it is necessary to justify why one or another perspective is more relevant and significant in a particular case. Such analysis, according to Linstone (1989) , ‘forces us to distinguish how we are looking from what we are looking at’ (p.312; italic in original). Hence, the problem solver broadens the understanding of various perspectives and develops the capability to see the bigger picture ( Hall & Davis, 2007 ).

The problem solver aims to identify and describe four perspectives that have emerged in the problem during this step. In order to identify perspectives, the problem solver search answers to the following questions. First, regarding the technical/scientific perspective: What technical/scientific reasons are brought out in the problem? How and to what extent do they influence a problem and its context? Second, regarding the interpersonal/social perspective: What is the impact of the problem on stakeholders? How does it influence their attitudes, living conditions, interests, needs? Third, regarding the existential perspective: How does the problem affect human feelings, experiences, perception, and/or discovery of meaning? Fourth, regarding the systemic perspective: What is the effect of the problem on the person → community → society → the world? Based on the analysis of this step, the problem solver obtains a comprehensive picture of the problem. The next stage is to choose the value(s) that will address the problem.

The second stage - ‘ The choice of value(s): What is the background for the solution?’ - includes the fourth and the fifth steps. The fourth step is ‘ The identification of value(s) as a base for the solution ’. During this step, the problem solver should activate his/her value(s) making it (them) explicit. In order to do this, the problem solver proceeds several sub-steps. First, the problem solver reflects taking into account the analysis done in previous steps. He/she raises up questions revealing values that lay in the background of this analysis: What values does this analyzed context allow me to notice? What values do different perspectives of the problem ‘offer’? Such questioning is important as values are deeply hidden ( Verplanken & Holland, 2002 ) and they form a bias, which restricts the development of the capability to see from various points of view ( Hall & Paradice, 2007 ). In the 4W framework, this bias is relatively eliminated due to the analysis of the context and exploration of the perspectives of a problem. As a result, the problem solver discovers distinct value-based positions and gets an opportunity to identify the ‘value uncaptured’ ( Yang, Evans, Vladimirova, & Rana, 2017, p.1796 ) within the problem analyzed. The problem solver observes that some values exist in the context (the second step) and the disclosed perspectives (the third step). Some of the identified values do not affect the current situation as they are not required, or their potential is not exploited. Thus, looking through various value-based lenses, the problem solver can identify and discover a congruence between the opportunities offered by the values in the problem’s context, disclosed perspectives and his/her value(s). Consequently, the problem solver decides what values he/she chooses as a basis for the desired solution. Since problems usually call for a list of values, it is important to find out their order of priority. Thus, the last sub-step requires the problem solver to choose between fundamentally and superficially laden values.

In some cases, the problem solver identifies that a set of values (more than one value) can lead to the desired solution. If a person chooses this multiple value-based position, two options emerge. The first option is concerned with the analysis of each value-based position separately (from the fifth to the seventh step). In the second option, a person has to uncover which of his/her chosen values are fundamentally laden and which are superficially chosen, considering the desired outcome in the current situation. Such clarification could act as a strategy where the path for the desired solution is possible going from superficially chosen value(s) to fundamentally laden one. When a basis for the solution is established, the problem solver formulates the goal for the desired solution.

The fifth step is ‘ The formulation of the goal for the solution ’. Problem solving highlights essential points that reveal the structure of a person’s goals; thus, a goal is the core element of problem solving ( Funke, 2014 ). Meantime, values reflect the motivational content of the goals ( Schwartz, 1992 ). The attention on the chosen value not only activates it, but also motivates the problem solver. The motivation directs the formulation of the goal. In such a way, values explicitly become a basis of the goal for the solution. Thus, this step involves the problem solver in formulating the goal for the solution as the desired outcome.

The way how to take into account value(s) when formulating the goal is the integration of value(s) chosen by the problem solver in the formulation of the goal ( Keeney, 1994 ). For this purpose the conjunction of a context for a solution (it is analyzed during the second step) and a direction of preference (the chosen value reveals it) serves for the formulation of the goal (that represents the desired solution). In other words, a value should be directly included into the formulation of the goal. The goal could lose value, if value is not included into the goal formulation and remains only in the context of the goal. Let’s take the actual example concerning COVID-19 situation. Naturally, many countries governments’ preference represents such value as human life (‘it is important of every individual’s life’). Thus, most likely the particular country government’s goal of solving the COVID situation could be to save the lifes of the country people. The named problem is a complex where the goal of its solution is also complex, although it sounds simple. However, if the goal as desired outcome is formulated without the chosen value, this value remains in the context and its meaning becomes tacit. In the case of above presented example - the goal could be formulated ‘to provide hospitals with the necessary equipment and facilities’. Such goal has the value ‘human’s life’ in the context, but eliminates the complexity of the problem that leads to a partial solution of the problem. Thus, this step from the problem solver requires caution when formulating the goal as the desired outcome. For this reason, maintaining value is very important when formulating the goal’s text. To avoid the loss of values and maintain their proposed direction, is necessary to take into account values again when creating alternatives.

The third stage - ‘ Search for the alternative ways for a solution: How? ’ - encompasses the sixth step, which is called ‘ Creation of value-based alternatives ’. Frequently problem solver invokes a traditional view of problem identification, generation of alternatives, and selection of criteria for evaluating findings. Keeney (1994) ; Ims and Zsolnai (2009) criticize this rational approach as it supports a search for a partial solution where an active search for alternatives is neglected. Moreover, a problematic situation, according to Perkins (2009) , can create the illusion of a fully framed problem with some apparent weighting and some variations of choices. In this case, essential and distinct alternatives to the solution frequently become unnoticeable. Therefore, Perkins (2009) suggest to replace the focus on the attempts to comprehend the problem itself. Thinking through the ‘value lenses’ offers such opportunities. The deep understanding of the problem leads to the search for the alternative ways of a solution.

Thus, the aim of this step is for the problem solver to reveal the possible alternative ways for searching a desired solution. Most people think they know how to create alternatives, but often without delving into the situation. First of all, the problem solver based on the reflection of (but not limited to) the analysis of the context and the perspectives of the problem generates a range of alternatives. Some of these alternatives represent anchored thinking as he/she accepts the assumptions implicit in generated alternatives and with too little focus on values.

The chosen value with the formulated goal indicates direction and encourages a broader and more creative search for a solution. Hence, the problem solver should consider some of the initial alternatives that could best support the achievement of the desired solution. Values are the principles for evaluating the desirability of any alternative or outcome ( Keeney, 1994 ). Thus, planned actions should reveal the desirable mode of conduct. After such consideration, he/she should draw up a plan setting out the actions required to implement each of considered alternatives.

Lastly, after a thorough examination of each considered alternative and a plan of its implementation, the problem solver chooses one of them. If the problem solver does not see an appropriate alternative, he/she develops new alternatives. However, the problem solver may notice (and usually does) that more than one alternative can help him/her to achieve the desired solution. In this case, he/she indicates which alternative is the main one and has to be implemented in the first place, and what other alternatives and in what sequence will contribute in searching for the desired solution.

The fourth stage - ‘ The rationale for the solution: Why ’ - leads to the seventh step: ‘ The justification of the chosen alternative ’. Keeney (1994) emphasizes the compatibility of alternatives in question with the values that guide the action. This underlines the importance of justifying the choices a person makes where the focus is on taking responsibility. According to Zsolnai (2008) , responsibility means a choice, i.e., the perceived responsibility essentially determines its choice. Responsible justification allows for discovering optimal balance when choosing between distinct value-based alternatives. It also refers to the alternative solution that best reflects responsibility in a particular value context, choice, and implementation.

At this stage, the problem solver revisits the chosen solution and revises it. The problem solver justifies his/her choice based on the following questions: Why did you choose this? Why is this alternative significant looking from the technical/scientific, the interpersonal/social, the existential, and the systemic perspectives? Could you take full responsibility for the implementation of this alternative? Why? How clearly do envisaged actions reflect the goal of the desired solution? Whatever interests and for what reasons do this alternative satisfies in principle? What else do you see in the chosen alternative?

As mentioned above, each person gives priority to one aspect or another. The problem solver has to provide solid arguments for the justification of the chosen alternative. The quality of arguments, according to Jonassen (2011) , should be judged based on the quality of the evidence supporting the chosen alternative and opposing arguments that can reject solutions. Besides, the pursuit of value-based goals reflects the interests of the individual or collective interests. Therefore, it becomes critical for the problem solver to justify the level of responsibility he/she takes in assessing the chosen alternative. Such a complex evaluation of the chosen alternative ensures the acceptance of an integral rather than unilateral solution, as ‘recognizing that, in the end, people benefit most when they act for the common good’ ( Sternberg, 2012, p.46 ).

## 5. Discussion

The constant emphasis on thinking about values as explicit reasoning in the 4W framework (especially from the choice of the value(s) to the rationale for problem solution) reflects the pursuit of virtues. Virtues form the features of the character that are related to the choice ( Argandoña, 2003 ; McLaughlin, 2005 ). Hence, the problem solver develops value-grounded problem solving capability as the virtuality instead of employing rationality for problem solving.

Argandoña (2003) suggests that, in order to make a sound valuation process of any action, extrinsic, transcendent, and intrinsic types of motives need to be considered. They cover the respective types of values. The 4W framework meets these requirements. An extrinsic motive as ‘attaining the anticipated or expected satisfaction’ ( Argandoña, 2003, p.17 ) is reflected in the formulation of the goal of the solution, the creation of alternatives and especially in the justification of the chosen alternative way when the problem solver revisits the external effect of his/her possible action. Transcendent motive as ‘generating certain effects in others’ ( Argandoña, 2003, p.17 ) is revealed within the analysis of the context, perspectives, and creating alternatives. When the learner considers the creation of alternatives and revisits the chosen alternative, he/she pays more attention to these motives. Two types of motives mentioned so far are closely related to an intrinsic motive that emphasizes learning development within the problem solver. These motives confirm that problem solving is, in fact, lifelong learning. In light of these findings, the 4W framework is concerned with some features of value internalization as it is ‘a psychological outcome of conscious mind reasoning about values’ ( Yazdani & Akbarilakeh, 2017, p.1 ).

The 4W framework is complicated enough in terms of learning. One issue is concerned with the educational environments ( Jucevičienė, 2008 ) required to enable the 4W framework. First, the learning paradigm, rather than direct instruction, lies at the foundation of such environments. Second, such educational environments include the following dimensions: (1) educational goal; (2) learning capacity of the learners; (3) educational content relevant to the educational goal: ways and means of communicating educational content as information presented in advance (they may be real, people among them, as well as virtual); (5) methods and means of developing educational content in the process of learners’ performance; (6) physical environment relevant to the educational goal and conditions of its implementation as well as different items in the environment; (7) individuals involved in the implementation of the educational goal.

Another issue is related to exercising this framework in practice. Despite being aware of the 4W framework, a person may still not want to practice problem solving through values, since most of the solutions are going to be complicated, or may even be painful. One idea worth looking into is to reveal the extent to which problem solving through values can become a habit of mind. Profound focus on personal values, context analysis, and highlighting various perspectives can involve changes in the problem solver’s habit of mind. The constant practice of problem solving through values could first become ‘the epistemic habit of mind’ ( Mezirow, 2009, p.93 ), which means a personal way of knowing things and how to use that knowledge. This echoes Kirkman (2017) findings. The developed capability to notice moral values in situations that students encountered changed some students’ habit of mind as ‘for having “ruined” things by making it impossible not to attend to values in such situations!’ (the feedback from one student; Kirkman, 2017, p.12 ). However, this is not enough, as only those problems that require a value-based approach are addressed. Inevitably, the problem solver eventually encounters the challenges of nurturing ‘the moral-ethical habit of mind’ ( Mezirow, 2009, p.93 ). In pursuance to develop such habits of mind, the curriculum should include the necessity of the practising of the 4W framework.

Thinking based on values when solving problems enables the problem solver to engage in thoughtful reflection in contrast to pragmatic and superficial thinking supported by the consumer society. Reflection begins from the first stage of the 4W framework. As personal values are the basis for the desired solution, the problem solver is also involved in self-reflection. The conscious and continuous reflection on himself/herself and the problematic situation reinforce each step of the 4W framework. Moreover, the fourth stage (‘The rationale for the solution: Why’) involves the problem solver in critical reflection as it concerned with justification of ‘the why , the reasons for and the consequences of what we do’ (italic, bold in original; Mezirow, 1990, p.8 ). Exercising the 4W framework in practice could foster reflective practice. Empirical evidence shows that reflective practice directly impacts knowledge, skills and may lead to changes in personal belief systems and world views ( Slade, Burnham, Catalana, & Waters, 2019 ). Thus, with the help of reflective practice it is possible to identify in more detail how and to what extent the 4W framework has been mastered, what knowledge gained, capabilities developed, how point of views changed, and what influence the change process.

Critical issues related to the development of problem solving through values need to be distinguished when considering and examining options for the implementation of the 4W framework at educational institutions. First, the question to what extent can the 4W framework be incorporated into various subjects needs to be answered. Researchers could focus on applying the 4W framework to specific subjects in the humanities and social sciences. The case is with STEM subjects. Though value issues of sustainable development and ecology are of great importance, in reality STEM teaching is often restricted to the development of knowledge and skills, leaving aside the thinking about values. The special task of the researchers is to help practitioners to apply the 4W framework in STEM subjects. Considering this, researchers could employ the concept of ‘dialogic space’ ( Wegerif, 2011, p.3 ) which places particular importance of dialogue in the process of education emphasizing both the voices of teachers and students, and materials. In addition, the dimensions of educational environments could be useful aligning the 4W framework with STEM subjects. As STEM teaching is more based on solving various special tasks and/or integrating problem-based learning, the 4W framework could be a meaningful tool through which content is mastered, skills are developed, knowledge is acquired by solving pre-prepared specific tasks. In this case, the 4W framework could act as a mean addressing values in STEM teaching.

Second is the question of how to enable the process of problem solving through values. In the current paper, the concept of enabling is understood as an integral component of the empowerment. Juceviciene et al. (2010) specify that at least two perspectives can be employed to explain empowerment : a) through the power of legitimacy (according to Freire, 1996 ); and b) through the perspective of conditions for the acquisition of the required knowledge, capabilities, and competence, i.e., enabling. In this paper the 4W framework does not entail the issue of legitimacy. This issue may occur, for example, when a teacher in economics is expected to provide students with subject knowledge only, rather than adding tasks that involve problem solving through values. Yet, the issue of legitimacy is often implicit. A widespread phenomenon exists that teaching is limited to certain periods that do not have enough time for problem solving through values. The issue of legitimacy as an organizational task that supports/or not the implementation of the 4W framework in any curriculum is a question that calls for further discussion.

Third (if not the first), the issue of an educator’s competence to apply such a framework needs to be addressed. In order for a teacher to be a successful enabler, he/she should have the necessary competence. This is related to the specific pedagogical knowledge and skills, which are highly dependent on the peculiarities of the subject being taught. Nowadays actualities are encouraging to pay attention to STEM subjects and their teacher training. For researchers and teacher training institutions, who will be interested in implementing the 4W framework in STEM subjects, it would be useful to draw attention to ‘a material-dialogic approach to pedagogy’ ( Hetherington & Wegerif, 2018, p.27 ). This approach creates the conditions for a deep learning of STEM subjects revealing additional opportunities for problem solving through values in teaching. Highlighting these opportunities is a task for further research.

In contrast to traditional problem solving models, the 4W framework is more concerned with educational purposes. The prescriptive approach to teaching ( Thorne, 1994 ) is applied to the 4W framework. This approach focuses on providing guidelines that enable students to make sound decisions by making explicit value judgements. The limitation is that the 4W framework is focused on thinking but not executing. It does not include the fifth stage, which would focus on the execution of the decision how to solve the problem. This stage may contain some deviation from the predefined process of the solution of the problem.

## 6. Conclusions

The current paper focuses on revealing the essence of the 4W framework, which is based on enabling the problem solver to draw attention to when, how, and why it is essential to think about values during the problem solving process from the perspective of it’s design. Accordingly, the 4W framework advocates the coherent approach when solving a problem by using a creative potential of values.

The 4W framework allows the problem solver to look through the lens of his/her values twice. The first time, while formulating the problem solving goal as the desired outcome. The second time is when the problem solver looks deeper into his/her values while exploring alternative ways to solve problems. The problem solver is encouraged to reason about, find, accept, reject, compare values, and become responsible for the consequences of the choices grounded on his/her values. Thus, the problem solver could benefit from the 4W framework especially when dealing with issues having crucial consequences.

An educational approach reveals that the 4W framework could enable the development of value-grounded problem solving capability. As problem solving encourages the development of higher-order thinking skills, the consistent inclusion of values enriches them.

The 4W framework requires the educational environments for its enablement. The enablement process of problem solving through values could be based on the perspective of conditions for the acquisition of the required knowledge and capability. Continuous practice of this framework not only encourages reflection, but can also contribute to the creation of the epistemic habit of mind. Applying the 4W framework to specific subjects in the humanities and social sciences might face less challenge than STEM ones. The issue of an educator’s competence to apply such a framework is highly important. The discussed issues present significant challenges for researchers and educators. Caring that the curriculum of different courses should foresee problem solving through values, both practicing and empirical research are necessary.

## Declaration of interests

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Both authors have approved the final article.

• Argandoña A. Fostering values in organizations. Journal of Business Ethics. 2003; 45 (1–2):15–28. https://link.springer.com/content/pdf/10.1023/A:1024164210743.pdf [ Google Scholar ]
• Barber S. A truly “Transformative” MBA: Executive education for the fourth industrial revolution. Journal of Pedagogic Development. 2018; 8 (2):44–55. [ Google Scholar ]
• Barnett R. McGraw-Hill Education; UK): 2007. Will to learn: Being a student in an age of uncertainty. [ Google Scholar ]
• Baron R.A., Zhao H., Miao Q. Personal motives, moral disengagement, and unethical decisions by entrepreneurs: Cognitive mechanisms on the “slippery slope” Journal of Business Ethics. 2015; 128 (1):107–118. doi: 10.1007/s10551-014-2078-y. [ CrossRef ] [ Google Scholar ]
• Basadur M., Ellspermann S.J., Evans G.W. A new methodology for formulating ill-structured problems. Omega. 1994; 22 (6):627–645. doi: 10.1016/0305-0483(94)90053-1. [ CrossRef ] [ Google Scholar ]
• Blanco E., Schirmbeck F., Costa C. International Conference on Remote Engineering and Virtual Instrumentation . Springer; Cham: 2018. Vocational Education for the Industrial Revolution; pp. 649–658. [ Google Scholar ]
• Chua B.L., Tan O.S., Liu W.C. Journey into the problem-solving process: Cognitive functions in a PBL environment. Innovations in Education and Teaching International. 2016; 53 (2):191–202. doi: 10.1080/14703297.2014.961502. [ CrossRef ] [ Google Scholar ]
• Collins R.H., Sibthorp J., Gookin J. Developing ill-structured problem-solving skills through wilderness education. Journal of Experiential Education. 2016; 39 (2):179–195. doi: 10.1177/1053825916639611. [ CrossRef ] [ Google Scholar ]
• Csapó B., Funke J., editors. The nature of problem solving: Using research to inspire 21st century learning. OECD Publishing; 2017. The development and assessment of problem solving in 21st-century schools. (Chapter 1). [ CrossRef ] [ Google Scholar ]
• Dollinger S.J., Burke P.A., Gump N.W. Creativity and values. Creativity Research Journal. 2007; 19 (2-3):91–103. doi: 10.1080/10400410701395028. [ CrossRef ] [ Google Scholar ]
• Donovan S.J., Guss C.D., Naslund D. Improving dynamic decision making through training and self-reflection. Judgment and Decision Making. 2015; 10 (4):284–295. http://digitalcommons.unf.edu/apsy_facpub/2 [ Google Scholar ]
• Dorst K. Design problems and design paradoxes. Design Issues. 2006; 22 (3):4–17. doi: 10.1162/desi.2006.22.3.4. [ CrossRef ] [ Google Scholar ]
• Dostál J. Theory of problem solving. Procedia-Social and Behavioral Sciences. 2015; 174 :2798–2805. doi: 10.1016/j.sbspro.2015.01.970. [ CrossRef ] [ Google Scholar ]
• Fields A.M. Ill-structured problems and the reference consultation: The librarian’s role in developing student expertise. Reference Services Review. 2006; 34 (3):405–420. doi: 10.1108/00907320610701554. [ CrossRef ] [ Google Scholar ]
• Freire P. Continuum; New York: 1996. Pedagogy of the oppressed (revised) [ Google Scholar ]
• Funke J. Problem solving: What are the important questions?. Proceedings of the 36th Annual Conference of the Cognitive Science Society; Austin, TX: Cognitive Science Society; 2014. pp. 493–498. [ Google Scholar ]
• Hall D.J., Davis R.A. Engaging multiple perspectives: A value-based decision-making model. Decision Support Systems. 2007; 43 (4):1588–1604. doi: 10.1016/j.dss.2006.03.004. [ CrossRef ] [ Google Scholar ]
• Hall D.J., Paradice D. Investigating value-based decision bias and mediation: do you do as you think? Communications of the ACM. 2007; 50 (4):81–85. [ Google Scholar ]
• Halstead J.M. Values and values education in schools. In: Halstead J.M., Taylor M.J., editors. Values in education and education in values. The Falmer Press; London: 1996. pp. 3–14. [ Google Scholar ]
• Harland T., Pickering N. Routledge; 2010. Values in higher education teaching. [ Google Scholar ]
• Hester P.T., MacG K. Springer; New York: 2017. Systemic decision making: Fundamentals for addressing problems and messes. [ Google Scholar ]
• Hetherington L., Wegerif R. Developing a material-dialogic approach to pedagogy to guide science teacher education. Journal of Education for Teaching. 2018; 44 (1):27–43. doi: 10.1080/02607476.2018.1422611. [ CrossRef ] [ Google Scholar ]
• Huitt W. Problem solving and decision making: Consideration of individual differences using the Myers-Briggs type indicator. Journal of Psychological Type. 1992; 24 (1):33–44. [ Google Scholar ]
• Ims K.J., Zsolnai L. The future international manager. Palgrave Macmillan; London: 2009. Holistic problem solving; pp. 116–129. [ Google Scholar ]
• Jonassen D. Supporting problem solving in PBL. Interdisciplinary Journal of Problem-based Learning. 2011; 5 (2):95–119. doi: 10.7771/1541-5015.1256. [ CrossRef ] [ Google Scholar ]
• Jonassen D.H. Instructional design models for well-structured and III-structured problem-solving learning outcomes. Educational Technology Research and Development. 1997; 45 (1):65–94. doi: 10.1007/BF02299613. [ CrossRef ] [ Google Scholar ]
• Jucevičienė P. Educational and learning environments as a factor for socioeducational empowering of innovation. Socialiniai mokslai. 2008; 1 :58–70. [ Google Scholar ]
• Jucevičienė P., Gudaitytė D., Karenauskaitė V., Lipinskienė D., Stanikūnienė B., Tautkevičienė G. Technologija; Kaunas: 2010. Universiteto edukacinė galia: Atsakas XXI amžiaus iššūkiams [The educational power of university: the response to the challenges of the 21st century] [ Google Scholar ]
• Kasof J., Chen C., Himsel A., Greenberger E. Values and creativity. Creativity Research Journal. 2007; 19 (2–3):105–122. doi: 10.1080/10400410701397164. [ CrossRef ] [ Google Scholar ]
• Keeney R.L. Creativity in decision making with value-focused thinking. MIT Sloan Management Review. 1994; 35 (4):33–41. [ Google Scholar ]
• Kirkman R. Problem-based learning in engineering ethics courses. Interdisciplinary Journal of Problem-based Learning. 2017; 11 (1) doi: 10.7771/1541-5015.1610. [ CrossRef ] [ Google Scholar ]
• Lebedeva N., Schwartz S., Plucker J., Van De Vijver F. Domains of everyday creativity and personal values. Frontiers in Psychology. 2019; 9 :1–16. doi: 10.3389/fpsyg.2018.02681. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Linstone H.A. Multiple perspectives: Concept, applications, and user guidelines. Systems Practice. 1989; 2 (3):307–331. [ Google Scholar ]
• Litzinger T.A., Meter P.V., Firetto C.M., Passmore L.J., Masters C.B., Turns S.R.…Zappe S.E. A cognitive study of problem solving in statics. Journal of Engineering Education. 2010; 99 (4):337–353. [ Google Scholar ]
• Maslow A.H. Vaga; Vilnius: 2011. Būties psichologija. [Psychology of Being] [ Google Scholar ]
• Mayer R., Wittrock M. Problem solving. In: Alexander P., Winne P., editors. Handbook of educational psychology. Psychology Press; New York, NY: 2006. pp. 287–303. [ Google Scholar ]
• McLaughlin T. The educative importance of ethos. British Journal of Educational Studies. 2005; 53 (3):306–325. doi: 10.1111/j.1467-8527.2005.00297.x. [ CrossRef ] [ Google Scholar ]
• McLaughlin T.H. Technologija; Kaunas: 1997. Šiuolaikinė ugdymo filosofija: demokratiškumas, vertybės, įvairovė [Contemporary philosophy of education: democracy, values, diversity] [ Google Scholar ]
• Mezirow J. Jossey-Bass Publishers; San Francisco: 1990. Fostering critical reflection in adulthood; pp. 1–12. https://my.liberatedleaders.com.au/wp-content/uploads/2017/02/How-Critical-Reflection-triggers-Transformative-Learning-Mezirow.pdf [ Google Scholar ]
• Mezirow J. Contemporary theories of learning. Routledge; 2009. An overview on transformative learning; pp. 90–105. (Chapter 6) [ Google Scholar ]
• Mitroff I. Šviesa; Kaunas: 2000. Kaip neklysti šiais beprotiškais laikais: ar mokame spręsti esmines problemas. [How not to get lost in these crazy times: do we know how to solve essential problems] [ Google Scholar ]
• Morton L. Teaching creative problem solving: A paradigmatic approach. Cal. WL Rev. 1997; 34 :375. [ Google Scholar ]
• Nadda P. Need for value based education. International Education and Research Journal. 2017; 3 (2) http://ierj.in/journal/index.php/ierj/article/view/690/659 [ Google Scholar ]
• Newell A., Simon H.A. Prentice-Hall; Englewood Cliffs, NJ: 1972. Human problem solving. [ Google Scholar ]
• OECD . PISA, OECD Publishing; Paris: 2013. PISA 2012 assessment and analytical framework: Mathematics, reading, science, problem solving and financial literacy . https://www.oecd.org/pisa/pisaproducts/PISA%202012%20framework%20e-book_final.pdf [ Google Scholar ]
• OECD . PISA, OECD Publishing; 2018. PISA 2015 results in focus . https://www.oecd.org/pisa/pisa-2015-results-in-focus.pdf [ Google Scholar ]
• O’Loughlin A., McFadzean E. Toward a holistic theory of strategic problem solving. Team Performance Management: An International Journal. 1999; 5 (3):103–120. [ Google Scholar ]
• Perkins D.N. Decision making and its development. In: Callan E., Grotzer T., Kagan J., Nisbett R.E., Perkins D.N., Shulman L.S., editors. Education and a civil society: Teaching evidence-based decision making. American Academy of Arts and Sciences; Cambridge, MA: 2009. pp. 1–28. (Chapter 1) [ Google Scholar ]
• Roccas S., Sagiv L., Navon M. Values and behavior. Cham: Springer; 2017. Methodological issues in studying personal values; pp. 15–50. [ Google Scholar ]
• Rogers C.R. Houghton Mifflin Harcourt; Boston: 1995. On becoming a person: A therapist’s view of psychotherapy. [ Google Scholar ]
• Saito M. Amartya Sen’s capability approach to education: A critical exploration. Journal of Philosophy of Education. 2003; 37 (1):17–33. doi: 10.1111/1467-9752.3701002. [ CrossRef ] [ Google Scholar ]
• Schwartz S.H. Universals in the content and structure of values: Theoretical advances and empirical tests in 20 countries. In: Zanna M.P., editor. Vol. 25. Academic Press; 1992. pp. 1–65. (Advances in experimental social psychology). [ Google Scholar ]
• Schwartz S.H. Are there universal aspects in the structure and contents of human values? Journal of social issues. 1994; 50 (4):19–45. [ Google Scholar ]
• Schwartz S.H. An overview of the Schwartz theory of basic values. Online Readings in Psychology and Culture. 2012; 2 (1):1–20. doi: 10.9707/2307-0919.1116. [ CrossRef ] [ Google Scholar ]
• Sen A. Development as capability expansion. The community development reader. 1990:41–58. http://www.masterhdfs.org/masterHDFS/wp-content/uploads/2014/05/Sen-development.pdf [ Google Scholar ]
• Sheehan N.T., Schmidt J.A. Preparing accounting students for ethical decision making: Developing individual codes of conduct based on personal values. Journal of Accounting Education. 2015; 33 (3):183–197. doi: 10.1016/j.jaccedu.2015.06.001. [ CrossRef ] [ Google Scholar ]
• Shepherd D.A., Patzelt H., Baron R.A. “I care about nature, but…”: Disengaging values in assessing opportunities that cause harm. The Academy of Management Journal. 2013; 56 (5):1251–1273. doi: 10.5465/amj.2011.0776. [ CrossRef ] [ Google Scholar ]
• Shin N., Jonassen D.H., McGee S. Predictors of well‐structured and ill‐structured problem solving in an astronomy simulation. Journal of Research in Science Teaching. 2003; 40 (1):6–33. doi: 10.1002/tea.10058. [ CrossRef ] [ Google Scholar ]
• Slade M.L., Burnham T.J., Catalana S.M., Waters T. The impact of reflective practice on teacher candidates’ learning. International Journal for the Scholarship of Teaching and Learning. 2019; 13 (2):15. doi: 10.20429/ijsotl.2019.130215. [ CrossRef ] [ Google Scholar ]
• Snyder H. Literature review as a research methodology: An overview and guidelines. Journal of Business Research. 2019; 104 :333–339. doi: 10.1016/j.jbusres.2019.07.039. [ CrossRef ] [ Google Scholar ]
• Sternberg R. Teaching for ethical reasoning. International Journal of Educational Psychology. 2012; 1 (1):35–50. doi: 10.4471/ijep.2012.03. [ CrossRef ] [ Google Scholar ]
• Sternberg R. Speculations on the role of successful intelligence in solving contemporary world problems. Journal of Intelligence. 2017; 6 (1):4. doi: 10.3390/jintelligence6010004. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Thorne D.M. Environmental ethics in international business education: Descriptive and prescriptive dimensions. Journal of Teaching in International Business. 1994; 5 (1–2):109–122. doi: 10.1300/J066v05n01_08. [ CrossRef ] [ Google Scholar ]
• Treffinger D.J., Isaksen S.G. Creative problem solving: The history, development, and implications for gifted education and talent development. The Gifted Child Quarterly. 2005; 49 (4):342–353. doi: 10.1177/001698620504900407. [ CrossRef ] [ Google Scholar ]
• Verplanken B., Holland R.W. Motivated decision making: Effects of activation and self-centrality of values on choices and behavior. Journal of Personality and Social Psychology. 2002; 82 (3):434–447. doi: 10.1037/0022-3514.82.3.434. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Webster R.S. Re-enchanting education and spiritual wellbeing. Routledge; 2017. Being spiritually educated; pp. 73–85. [ Google Scholar ]
• Wegerif R. Towards a dialogic theory of how children learn to think. Thinking Skills and Creativity. 2011; 6 (3):179–190. doi: 10.1016/j.tsc.2011.08.002. [ CrossRef ] [ Google Scholar ]
• Yang M., Evans S., Vladimirova D., Rana P. Value uncaptured perspective for sustainable business model innovation. Journal of Cleaner Production. 2017; 140 :1794–1804. doi: 10.1016/j.jclepro.2016.07.102. [ CrossRef ] [ Google Scholar ]
• Yazdani S., Akbarilakeh M. The model of value-based curriculum for medicine and surgery education in Iran. Journal of Minimally Invasive Surgical Sciences. 2017; 6 (3) doi: 10.5812/minsurgery.14053. [ CrossRef ] [ Google Scholar ]
• Zsolnai L. Transaction Publishers; New Brunswick and London: 2008. Responsible decision making. [ Google Scholar ]

Posing and Solving Mathematical Problems pp 231–254 Cite as

## Conceptual Model-Based Problem Solving

• Yan Ping Xin Ph.D. 6
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Part of the Research in Mathematics Education book series (RME)

While mathematics problem-solving skills are well recognized as critical for virtually all areas of daily life and successful functioning on the job, many students with learning disabilities or difficulties in mathematics (LDM) fail to acquire these skills during their early school studies, thereby subjecting themselves to lifelong challenges with mathematical problem solving. This chapter will introduce a conceptual model-based problem-solving (COMPS) approach that aims to promote elementary students’ generalized word problem-solving skills. With the emphasis on algebraic representation of mathematical relations in cohesive mathematical models, the COMPS program makes connections among mathematical ideas; it offers elementary school teachers a way to bridge the gap between algebraic and arithmetic teaching and learning. The COMPS program may be especially helpful for students with LDM who are likely to experience disadvantages in working memory and information organization. Findings from a series of empirical research studies will be presented, and implications for elementary mathematics education will be discussed pertinent to all students meeting the new Common Core State Standards for Mathematics (CCSSM, 2012).

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Baxter, J., Woodward, J., & Olson, D. (2001). Effects of reform-based mathematics instruction in five third grade classrooms. Elementary School Journal, 101 (5), 529–548.

CrossRef   Google Scholar

Bloom, B. S. (1976). Human characteristics and school learning . New York: McGraw-Hill.

Google Scholar

Blomhøj, M. (2004). Mathematical Modelling – A Theory for Practice . In B. Clarke et al. (Eds.), International Perspectives on Learning and Teaching Mathematics (pp. 145–159). Göteborg: National Center for Mathematics Education.

Blum, W., & Leiss, D. (2005). Modellieren mit der “Tanken”-Aufgaben. Mathematic Lehren, 128 , 18–21.

Boulineau, T., Fore, C., Hagan-Burke, S., & Burke, M. D. (2004). Use of story-mapping to increase the story-grammar text comprehension of elementary students with learning disabilities. Learning Disability Quarterly, 27 , 105–121.

Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. S., et al. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34 (4), 663–689.

Bryant, D. P., Bryant, B. R., Roberts, G., Vaughn, S., Pfannenstiel, K. H., Porterfield, J., et al. (2011). Early numeracy intervention program for first-grade students with mathematics difficulties. Council for Exceptional Children, 78 (1), 7–23.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children’s mathematics: Cognitively guided instruction . Portsmouth, NH: Heinemann.

Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2006). Learning mathematics in elementary and middle schools: A learner-centered approach (4th ed.). Upper Saddle River, NJ: Pearson Merrill Prentice Hall.

Cawley, J., Parmar, R., Foley, T. E., Salmon, S., & Roy, S. (2001). Arithmetic performance of students: Implications for standards and programming. Exceptional Children, 67 , 311–328.

Chen, Z. (1999). Schema induction in children’s analogical problem solving. Journal of Educational Psychology, 91 , 703–715.

Common Core State Standards Initiative. (2012). Introduction: Standards for mathematical practice. Retrieved from http://www.corestandards.org/the-standards/mathematics

Ding, M., & Li, X. (2014). Transition from concrete to abstract representations: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87 (1), 103–121.

Fuchs, D., & Deshler, D. D. (2007). What we need to know about responsiveness to intervention (and shouldn’t be afraid to ask). Learning Disabilities Research and Practice, 22 , 129–136.

Fuchs, L. S., Fuchs, D., & Hollenbeck, K. N. (2007). Expanding responsiveness to intervention to mathematics at first and third grade. Learning Disabilities Research and Practice, 22 , 13–24.

Gick, M., & Holyoak, K. H. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15 , 1–38.

Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York: Macmillan.

Horner, R. D., & Baer, D. M. (1978). Multiple-probe technique: A variation of the multiple baseline. Journal of Applied Behavior Analysis, 11 , 189–196.

Hamson, M. J. (2003). The place of mathematical modeling in mathematics education. In S. J. Lamon, W. A. Parker, & K. Houston (Eds.), Mathematical modeling: A way of life . Chichester, England: Horwood.

Hutchison, L. F. (2012). Addressing the STEM teacher shortage in American schools: Ways to recruit and retain effective STEM teachers. Action in Teacher Education, 34 , 541–550.

Jonassen, D. H. (2003). Designing research-based instruction for story problems. Educational Psychology Review, 15 , 267–296.

Kazdin, A. E. (1982). Single-case research designs: Methods for clinical and applied settings . New York: Oxford.

Lesh, R., Doerr, H. M., Carmona, G., & Hjalmarson, M. (2003). Beyond constructivism. Mathematical Thinking and Learning, 5 (2&3), 211–233.

Lesh, R., Landau, M., & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts & processes (pp. 263–343). New York: Academic Press.

Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81 , 521–531.

Maletsky, E. M., et al. (2004). Harcourt math (Indianath ed.). Chicago: Harcourt.

Mevarech, Z. R., & Kramarski, B. (2008). Mathematical modeling and meta-cognitive instruction . Retrieved November 10, 2008, from http://www.icme-organisers.dk/tsg18/S32MevarechKramarski.pdf

National Assessment of Educational Progress Result. (NAEP, 2013). Are higher and lower performing students making gains? Retrieved November 12, 2013, from http://nationsreportcard.gov/reading_math_2013/#/gains-percentiles

Pressley, M. (1986). The relevance of the good strategy user model to the teaching of mathematics. Educational Psychologist, 21 (1–2), 139–161.

Schmidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems involving multiplication or division. Educational Studies in Mathematics, 28 , 55–72.

Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.

Sherin, B. (2001). How students understand physics equations. Cognition and Instruction, 19 (4), 479–541.

Sowder, L. (1988). Children’s solutions of story problems. Journal of Mathematical Behavior, 7 , 227–238.

Thompson, P. (1989). Artificial intelligence, advanced technology, and learning and teaching algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 135–161). Reston, VA: National Council of Teachers of Mathematics.

Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston: Allyn and Bacon.

Xin, Y. P. (2007). Word-problem-solving tasks presented in textbooks and their relation to student performance: A cross-curriculum comparison case Study. Journal of Educational Research, 100 , 347–359.

Xin, Y. P. (2008). The effect of schema-based instruction in solving word problems: An emphasis on pre-algebraic conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39 , 526–551.

Xin, Y. P. (2012). Conceptual model-based problem-solving (COMPS) approach: Teach students with learning difficulties to solve math problems . Boston: Sense.

Xin, Y. P., Kastberg, S., & Chen, V. (2015). Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Program for Students with LDM . National Science Foundation funded project.

Xin, Y. P., Liu, J., & Zheng, X. (2011). A cross-cultural lesson comparison on teaching the connection between multiplication and division. School Science and Mathematics, 111 (7), 354–367.

Xin, Y. P., Tzur, R., & Si, L. (2008). Nurturing multiplicative reasoning in students with learning disabilities in a computerized conceptual-modeling environment . Grant received from the National Science Foundation.

Xin, Y. P., Wiles, B., & Lin, Y. (2008). Teaching conceptual model-based word-problem story grammar to enhance mathematics problem solving. Journal of Special Education, 42 , 163–178.

Xin, Y. P., & Zhang, D. (2009). Exploring a conceptual model-based approach to teaching situated word problems. Journal of Educational Research, 102 (6), 427–441.

Xin, Y. P., Tzur, R., Si, L., Hord, C., Liu, J., Park, J. Y., Cordova, M., & Ruan, L. Y. (2013, April). A comparison of teacher-delivered instruction and an intelligent tutor-assisted math problem-solving intervention program . Paper presented at the 2013 American Educational Research Association (AERA) Annual Meeting, San Francisco.

Xin, Y. P., Si, L., Hord, C., Zhang, D., Cetintas, S., & Park, J. Y. (2012). The effects of computer-assisted instruction in teaching conceptual model-based problem solving. Learning Disabilities: A Multidisciplinary Journal, 18 (2), 71–85.

Xin, Y. P., Zhang, D., Park, J. Y., Tom, K., Whipple, A., & Si, L. (2011). A comparison of two mathematics problem-solving strategies: Facilitate algebra-readiness. Journal of Educational Research, 104 , 381–395.

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Erkki Pehkonen

University of Georgia, Athens, USA

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Xin, Y.P. (2016). Conceptual Model-Based Problem Solving. In: Felmer, P., Pehkonen, E., Kilpatrick, J. (eds) Posing and Solving Mathematical Problems. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-28023-3_14

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## What Is the Model in Model-Based Planning?

Affiliations.

• 1 Department of Psychology and Center for Brain Science, Harvard University.
• 2 Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology.
• 3 Center for Brains, Minds and Machines, Massachusetts Institute of Technology.
• PMID: 33398907
• DOI: 10.1111/cogs.12928

Flexibility is one of the hallmarks of human problem-solving. In everyday life, people adapt to changes in common tasks with little to no additional training. Much of the existing work on flexibility in human problem-solving has focused on how people adapt to tasks in new domains by drawing on solutions from previously learned domains. In real-world tasks, however, humans must generalize across a wide range of within-domain variation. In this work we argue that representational abstraction plays an important role in such within-domain generalization. We then explore the nature of this representational abstraction in realistically complex tasks like video games by demonstrating how the same model-based planning framework produces distinct generalization behaviors under different classes of task representation. Finally, we compare the behavior of agents with these task representations to humans in a series of novel grid-based video game tasks. Our results provide evidence for the claim that within-domain flexibility in humans derives from task representations composed of propositional rules written in terms of objects and relational categories.

Keywords: Artificial intelligence; Generalization; Human cognition; Reinforcement learning; Representation learning; Transfer learning.

© 2021 Cognitive Science Society, Inc.

## Publication types

• Research Support, Non-U.S. Gov't
• Research Support, U.S. Gov't, Non-P.H.S.
• Concept Formation
• Generalization, Psychological*
• Problem Solving
• Video Games

## Center for Teaching Innovation

Resource library.

• Getting Started with Establishing Ground Rules
• Sample group work rubric
• Problem-Based Learning Clearinghouse of Activities, University of Delaware

## Problem-Based Learning

Problem-based learning  (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning.

## Why Use Problem-Based Learning?

Nilson (2010) lists the following learning outcomes that are associated with PBL. A well-designed PBL project provides students with the opportunity to develop skills related to:

• Working in teams.
• Managing projects and holding leadership roles.
• Oral and written communication.
• Self-awareness and evaluation of group processes.
• Working independently.
• Critical thinking and analysis.
• Explaining concepts.
• Self-directed learning.
• Applying course content to real-world examples.
• Researching and information literacy.
• Problem solving across disciplines.

## Considerations for Using Problem-Based Learning

Rather than teaching relevant material and subsequently having students apply the knowledge to solve problems, the problem is presented first. PBL assignments can be short, or they can be more involved and take a whole semester. PBL is often group-oriented, so it is beneficial to set aside classroom time to prepare students to   work in groups  and to allow them to engage in their PBL project.

Students generally must:

• Examine and define the problem.
• Explore what they already know about underlying issues related to it.
• Determine what they need to learn and where they can acquire the information and tools necessary to solve the problem.
• Evaluate possible ways to solve the problem.
• Solve the problem.
• Report on their findings.

## Getting Started with Problem-Based Learning

• Articulate the learning outcomes of the project. What do you want students to know or be able to do as a result of participating in the assignment?
• Create the problem. Ideally, this will be a real-world situation that resembles something students may encounter in their future careers or lives. Cases are often the basis of PBL activities. Previously developed PBL activities can be found online through the University of Delaware’s PBL Clearinghouse of Activities .
• Establish ground rules at the beginning to prepare students to work effectively in groups.
• Introduce students to group processes and do some warm up exercises to allow them to practice assessing both their own work and that of their peers.
• Consider having students take on different roles or divide up the work up amongst themselves. Alternatively, the project might require students to assume various perspectives, such as those of government officials, local business owners, etc.
• Establish how you will evaluate and assess the assignment. Consider making the self and peer assessments a part of the assignment grade.

Nilson, L. B. (2010).  Teaching at its best: A research-based resource for college instructors  (2nd ed.).  San Francisco, CA: Jossey-Bass.

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## What Is Problem-Solving Therapy?

Arlin Cuncic, MA, is the author of "Therapy in Focus: What to Expect from CBT for Social Anxiety Disorder" and "7 Weeks to Reduce Anxiety." She has a Master's degree in psychology.

Daniel B. Block, MD, is an award-winning, board-certified psychiatrist who operates a private practice in Pennsylvania.

Verywell / Madelyn Goodnight

## Effectiveness

Things to consider, how to get started.

Problem-solving therapy is a form of therapy that provides patients with tools to identify and solve problems that arise from life stressors, both big and small. Its aim is to improve your overall quality of life and reduce the negative impact of psychological and physical illness.

Problem-solving therapy can be used to treat depression , among other conditions. It can be administered by a doctor or mental health professional and may be combined with other treatment approaches.

Problem-solving therapy is based on a model that takes into account the importance of real-life problem-solving. In other words, the key to managing the impact of stressful life events is to know how to address issues as they arise. Problem-solving therapy is very practical in its approach and is only concerned with the present, rather than delving into your past.

This form of therapy can take place one-on-one or in a group format and may be offered in person or online via telehealth . Sessions can be anywhere from 30 minutes to two hours long.

There are two major components that make up the problem-solving therapy framework:

• Applying a positive problem-solving orientation to your life
• Using problem-solving skills

A positive problem-solving orientation means viewing things in an optimistic light, embracing self-efficacy , and accepting the idea that problems are a normal part of life. Problem-solving skills are behaviors that you can rely on to help you navigate conflict, even during times of stress. This includes skills like:

• Knowing how to identify a problem
• Defining the problem in a helpful way
• Trying to understand the problem more deeply
• Setting goals related to the problem
• Generating alternative, creative solutions to the problem
• Choosing the best course of action
• Implementing the choice you have made
• Evaluating the outcome to determine next steps

Problem-solving therapy is all about training you to become adaptive in your life so that you will start to see problems as challenges to be solved instead of insurmountable obstacles. It also means that you will recognize the action that is required to engage in effective problem-solving techniques.

One problem-solving technique, called planful problem-solving, involves following a series of steps to fix issues in a healthy, constructive way:

• Problem definition and formulation : This step involves identifying the real-life problem that needs to be solved and formulating it in a way that allows you to generate potential solutions.
• Generation of alternative solutions : This stage involves coming up with various potential solutions to the problem at hand. The goal in this step is to brainstorm options to creatively address the life stressor in ways that you may not have previously considered.
• Decision-making strategies : This stage involves discussing different strategies for making decisions as well as identifying obstacles that may get in the way of solving the problem at hand.
• Solution implementation and verification : This stage involves implementing a chosen solution and then verifying whether it was effective in addressing the problem.

Other techniques your therapist may go over include:

• Problem-solving multitasking , which helps you learn to think clearly and solve problems effectively even during times of stress
• Stop, slow down, think, and act (SSTA) , which is meant to encourage you to become more emotionally mindful when faced with conflict
• Healthy thinking and imagery , which teaches you how to embrace more positive self-talk while problem-solving

## What Problem-Solving Therapy Can Help With

Problem-solving therapy addresses issues related to life stress and is focused on helping you find solutions to concrete issues. This approach can be applied to problems associated with a variety of psychological and physiological symptoms.

Problem-solving therapy may help address mental health issues, like:

• Chronic stress due to accumulating minor issues
• Complications associated with traumatic brain injury (TBI)
• Emotional distress
• Post-traumatic stress disorder (PTSD)
• Problems associated with a chronic disease like cancer, heart disease, or diabetes
• Self-harm and feelings of hopelessness
• Substance use
• Suicidal ideation

This form of therapy is also helpful for dealing with specific life problems, such as:

• Death of a loved one
• Dissatisfaction at work
• Everyday life stressors
• Family problems
• Financial difficulties
• Relationship conflicts

Your doctor or mental healthcare professional will be able to advise whether problem-solving therapy could be helpful for your particular issue. In general, if you are struggling with specific, concrete problems that you are having trouble finding solutions for, problem-solving therapy could be helpful for you.

## Benefits of Problem-Solving Therapy

The skills learned in problem-solving therapy can be helpful for managing all areas of your life. These can include:

• Being able to identify which stressors trigger your negative emotions (e.g., sadness, anger)
• Confidence that you can handle problems that you face
• Having a systematic approach on how to deal with life's problems
• Having a toolbox of strategies to solve the problems you face
• Increased confidence to find creative solutions
• Knowing how to identify which barriers will impede your progress
• Knowing how to manage emotions when they arise
• Reduced avoidance and increased action-taking
• The ability to accept life problems that can't be solved
• The ability to make effective decisions
• The development of patience (realizing that not all problems have a "quick fix")

This form of therapy was initially developed to help people combat stress through effective problem-solving, and it was later adapted to specifically address clinical depression. Today, much of the research on problem-solving therapy deals with its effectiveness in treating depression.

Problem-solving therapy has been shown to help depression in:

• Older adults
• People coping with serious illnesses like breast cancer

Problem-solving therapy also appears to be effective as a brief treatment for depression, offering benefits in as little as six to eight sessions with a therapist or another healthcare professional. This may make it a good option for someone who is unable to commit to a lengthier treatment for depression.

Problem-solving therapy is not a good fit for everyone. It may not be effective at addressing issues that don't have clear solutions, like seeking meaning or purpose in life. Problem-solving therapy is also intended to treat specific problems, not general habits or thought patterns .

In general, it's also important to remember that problem-solving therapy is not a primary treatment for mental disorders. If you are living with the symptoms of a serious mental illness such as bipolar disorder or schizophrenia , you may need additional treatment with evidence-based approaches for your particular concern.

Problem-solving therapy is best aimed at someone who has a mental or physical issue that is being treated separately, but who also has life issues that go along with that problem that has yet to be addressed.

For example, it could help if you can't clean your house or pay your bills because of your depression, or if a cancer diagnosis is interfering with your quality of life.

Your doctor may be able to recommend therapists in your area who utilize this approach, or they may offer it themselves as part of their practice. You can also search for a problem-solving therapist with help from the American Psychological Association’s (APA) Society of Clinical Psychology .

If receiving problem-solving therapy from a doctor or mental healthcare professional is not an option for you, you could also consider implementing it as a self-help strategy using a workbook designed to help you learn problem-solving skills on your own.

During your first session, your therapist may spend some time explaining their process and approach. They may ask you to identify the problem you’re currently facing, and they’ll likely discuss your goals for therapy.

Problem-solving therapy may be a short-term intervention that's focused on solving a specific issue in your life. If you need further help with something more pervasive, it can also become a longer-term treatment option.

Pierce D. Problem solving therapy - Use and effectiveness in general practice . Aust Fam Physician . 2012;41(9):676-679.

Cuijpers P, Wit L de, Kleiboer A, Karyotaki E, Ebert DD. Problem-solving therapy for adult depression: An updated meta-analysis . Eur Psychiatry . 2018;48(1):27-37. doi:10.1016/j.eurpsy.2017.11.006

Nezu AM, Nezu CM, D'Zurilla TJ. Problem-Solving Therapy: A Treatment Manual . New York; 2013. doi:10.1891/9780826109415.0001

Hatcher S, Sharon C, Parag V, Collins N. Problem-solving therapy for people who present to hospital with self-harm: Zelen randomised controlled trial . Br J Psychiatry . 2011;199(4):310-316. doi:10.1192/bjp.bp.110.090126

Sorsdahl K, Stein DJ, Corrigall J, et al. The efficacy of a blended motivational interviewing and problem solving therapy intervention to reduce substance use among patients presenting for emergency services in South Africa: A randomized controlled trial . Subst Abuse Treat Prev Policy . 2015;10(1):46. doi:doi.org/10.1186/s13011-015-0042-1

Kirkham JG, Choi N, Seitz DP. Meta-analysis of problem solving therapy for the treatment of major depressive disorder in older adults . Int J Geriatr Psychiatry . 2016;31(5):526-535. doi:10.1002/gps.4358

Garand L, Rinaldo DE, Alberth MM, et al. Effects of problem solving therapy on mental health outcomes in family caregivers of persons with a new diagnosis of mild cognitive impairment or early dementia: A randomized controlled trial . Am J Geriatr Psychiatry . 2014;22(8):771-781. doi:10.1016/j.jagp.2013.07.007

Hopko DR, Armento MEA, Robertson SMC, et al. Brief behavioral activation and problem-solving therapy for depressed breast cancer patients: Randomized trial . J Consult Clin Psychol . 2011;79(6):834-849. doi:10.1037/a0025450

Nieuwsma JA, Trivedi RB, McDuffie J, Kronish I, Benjamin D, Williams JW. Brief psychotherapy for depression: A systematic review and meta-analysis . Int J Psychiatry Med . 2012;43(2):129-151. doi:10.2190/PM.43.2.c

By Arlin Cuncic, MA Arlin Cuncic, MA, is the author of "Therapy in Focus: What to Expect from CBT for Social Anxiety Disorder" and "7 Weeks to Reduce Anxiety." She has a Master's degree in psychology.

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Article • 8 min read

## The FOCUS Model

A simple, efficient problem-solving approach.

By the Mind Tools Content Team

Are your business processes perfect, or could you improve them?

In an ever-changing world, nothing stays perfect for long. To stay ahead of your competitors, you need to be able to refine your processes on an ongoing basis, so that your services remain efficient and your customers stay happy.

This article looks the FOCUS Model – a simple quality-improvement tool that helps you do this.

## About the Model

The FOCUS Model, which was created by the Hospital Corporation of America (HCA), is a structured approach to Total Quality Management (TQM) , and it is widely used in the health care industry.

The model is helpful because it uses a team-based approach to problem solving and to business-process improvement, and this makes it particularly useful for solving cross-departmental process issues. Also, it encourages people to rely on objective data rather than on personal opinions, and this improves the quality of the outcome.

It has five steps:

• F ind the problem.
• O rganize a team.
• C larify the problem.
• U nderstand the problem.
• S elect a solution.

## Applying the FOCUS Model

Follow the steps below to apply the FOCUS Model in your organization.

## Step 1: Find the Problem

The first step is to identify a process that needs to be improved. Process improvements often follow the Pareto Principle , where 80 percent of issues come from 20 percent of problems. This is why identifying and solving one real problem can significantly improve your business, if you find the right problem to solve.

According to a popular analogy, identifying problems is like harvesting apples. At first, this is easy – you can pick apples up from the ground and from the lower branches of the tree. But the more fruit you collect, the harder it becomes. Eventually, the remaining fruit is all out of reach, and you need to use a ladder to reach the topmost branches.

Start with a simple problem to get the team up to speed with the FOCUS method. Then, when confidence is high, turn your attention to more complex processes.

If the problem isn't obvious, use these questions to identify possible issues:

• What would our customers want us to improve?
• How can we improve quality ?
• What processes don't work as efficiently as they could?
• Where do we experience bottlenecks in our processes?
• What do our competitors or comparators do that we could do?
• What frustrates and irritates our team?
• What might happen in the future that could become a problem for us?

If you have several problems that need attention, list them all and use Pareto Analysis , Decision Matrix Analysis , or Paired Comparison Analysis to decide which problem to address first. (If you try to address too much in one go, you'll overload team members and cause unnecessary stress.)

## Step 2: Organize a Team

Your next step is to assemble a team to address the problem.

Where possible, bring together team members from a range of disciplines – this will give you a broad range of skills, perspectives, and experience to draw on.

Select team members who are familiar with the issue or process in hand, and who have a stake in its resolution. Enthusiasm for the project will be greatest if people volunteer for it, so emphasize how individuals will benefit from being involved.

If your first choice of team member isn't available, try to appoint someone close to them, or have another team member use tools like Perceptual Positioning and Rolestorming to see the issue from their point of view.

Keep in mind that a diverse team is more likely to find a creative solution than a group of people with the same outlook.

## Step 3: Clarify the Problem

Before the team can begin to solve the problem, you need to define it clearly and concisely.

According to " Total Quality Management for Hospital Nutrition Services ," a key text on the FOCUS Model, an enthusiastic team may be keen to attack an "elephant-sized" problem, but the key to success is to break it down into "sushi-sized" pieces that can be analyzed and solved more easily.

Use the Drill Down technique to break big problems down into their component parts. You can also use the 5 Whys Technique , Cause and Effect Analysis , and Root Cause Analysis to get to the bottom of a problem.

Record the details in a problem statement, which will then serve as the focal point for the rest of the exercise ( CATWOE can help you do this effectively.) Focus on factual events and measurable conditions such as:

• Who does the problem affect?
• What has happened?
• Where is it occurring?
• When does it happen?

The problem statement must be objective, so avoid relying on personal opinions, gut feelings, and emotions. Also, be on guard against "factoids" – statements that appear to be facts, but that are really opinions that have come to be accepted as fact.

## Step 4: Understand the Problem

Once the problem statement has been completed, members of the team gather data about the problem to understand it more fully.

Dedicate plenty of time to this stage, as this is where you will identify the fundamental steps in the process that, when changed, will bring about the biggest improvement.

Consider what you know about the problem. Has anyone else tried to fix a similar problem before? If so, what happened, and what can you learn from this?

Use a Flow Chart or Swim Lane Diagram to organize and visualize each step; this can help you discover the stage at which the problem is happening. And try to identify any bottlenecks or failures in the process that could be causing problems.

As you develop your understanding, potential solutions to the problem may become apparent. Beware of jumping to "obvious" conclusions – these could overlook important parts of the problem, and could create a whole new process that fails to solve the problem.

Generate as many possible solutions as you can through normal structured thinking, brainstorming , reverse brainstorming , and Provocation . Don't criticize ideas initially – just come up with lots of possible ideas to explore.

## Step 5: Select a Solution

The final stage in the process is to select a solution.

Use appropriate decision-making techniques to select the most viable option. Decision Trees , Paired Comparison Analysis , and Decision Matrix Analysis are all useful tools for evaluating your options.

Once you've selected an idea, use tools such as Risk Analysis , "What If" Analysis , and the Futures Wheel to think about the possible consequences of moving ahead, and make a well-considered go/no-go decision to decide whether or not you should run the project.

People commonly use the FOCUS Model in conjunction with the Plan-Do-Check-Act cycle. Use this approach to implement your solutions in a controlled way.

The FOCUS Model is a simple quality-improvement tool commonly used in the health care industry. You can use it to improve any process, but it is particularly useful for processes that span different departments.

The five steps in FOCUS are as follows:

People often use the FOCUS Model in conjunction with the Plan-Do-Check-Act cycle, which allows teams to implement their solution in a controlled way.

Bataldan, P. (1992). 'Building Knowledge for Improvement: an Introductory Guide to the Use of FOCUS-PDCA,' Nashville: TN Quality Resource Group, Hospital Corporation of America.

Schiller, M., Miller-Kovach, M., and Miller-Kovach, K. (1994). 'Total Quality Management for Hospital Nutrition Services,' Aspen Publishers Inc. Available here .

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Have you ever been confronted with a challenging problem and had no idea how to even begin working on it? For instance, let's say you have two upcoming exams on the same day, and you are unsure how to prepare for them. Or, let's say you are solving a complex math problem, but you are stuck and don't know how to proceed. In these moments, problem-solving strategies and models can help us tackle difficult problems by guiding us with well-known approaches or plans to follow.

In this article, we explore problem-solving strategies and models that can be applied to solve problems. Then, we practice applying these models in some example exercises.

## Problem-solving strategies and model descriptions

Oftentimes in mathematics, there is more than one way to solve a problem. Using problem-solving strategies can help you approach problems in a structured and logical manner to improve your efficiency.

Problem-solving strategies are models based on previous experience that provide a recommended approach for solving problems or analyzing potential solutions.

Problem-solving strategies involve steps like understanding, planning, and organizing, for example. While problem-solving strategies cannot guarantee an easier solution to a problem, they do provide techniques and tools that act as a guide for success.

## Types of problem-solving models and strategies

Many models and strategies are developed based on the nature of the problem at hand. In this article, we discuss two well-known models that are designed to address various types of problems, including:

Polya's f our-step problem-solving model

• IDEAL problem-solving model

Let's look at these two models in detail.

A mathematician named George Polya developed a model called the Polya f our-step problem-solving model to approach and solve various kinds of problems. This method has the following steps:

## Understand the problem

Devise a plan, carry out the plan.

John Bransford and Barry Stein also proposed a five-step model named IDEAL to resolve a problem with a sound and methodical approach. The IDEAL model is based on the following steps:

• Identify The Problem
• Define An Outcome
• Explore Possible Strategies
• Anticipate Outcomes & Act
• Look And Learn

Using either of these two models to help you identify and approach problems methodically can help make it easier to solve them.

• Polya's four-step problem-solving model

Polya's f our-step problem-solving model can be used to solve day-to-day problems as well as mathematical and other academic problems. As seen briefly, the steps of this problem-solving model include: understanding the problem, creating and carrying out a plan, and looking back. Let's look at these steps in more detail to understand how they are used.

This is a critical initial step. Simply put, if you don't fully understand the problem, you won't be able to identify a solution. You can understand a problem better by reviewing all of the inputs and available information, including its conditions and circumstances. Reading and understanding the problem helps you to organize the information as well as assign the relevant variables.

The following techniques can be applied during this problem-solving step:

Read the problem out loud to process it better.

List or summarize the important information to find out what is given and what is still missing.

Sketch a detailed diagram as a visual aid, depending on the problem.

Visualize a scenario about the problem to put it into context.

Use keyword analysis to identify the necessary operations (i.e., pay attention to important words and phrases such as "how many," "times," or "total").

Now that you have taken the time to properly understand the problem, you can devise a plan on how to proceed further to solve it. During this second step, you identify what strategy to follow to arrive at a solution. When considering a strategy to use, it's important to consider exactly what it is that you want to know.

Some problem-solving strategies include:

Identify the pattern from the given information and use it.

Use the guess-and-check method.

Work backward by using potential answers.

Apply a specific formula for the problem.

Eliminate the possibilities that don't work out.

Solve a simpler version of the problem first.

Form an equation and solve it.

During this third step, you solve the problem by applying your chosen strategy. For example, if you planned to solve the problem by drawing a graph, then during this step, you draw the graph using the information gathered in the previous steps. Here, you test your problem-solving skills and find if the solution works or not.

Below are some points to keep in mind when solving the problem:

Be systematic in your approach when implementing a strategy.

Check the work and see whether the solution works in all relevant cases.

Be flexible and change the strategy if necessary.

Keep solving and don't give up.

At this fourth step, you check your solution. This can be done by solving the problem in another way or simply by confirming that your solution makes sense. This step helps you decide if any improvements are needed for your solution. You may choose to check after solving an individual problem or after solving an entire set. Checking the problem carefully also helps you to reflect on the process and improve your methods for future problem solving.

The IDEAL problem-solving model was developed by Bransford and Stein as a guide for understanding and solving problems. This method is used in both education and industry. The IDEAL problem-solving model consists of five steps: identifying the problem, describing the outcome, exploring the possible strategies, anticipating the outcome, and looking back to learn. Let us explore these steps in detail by considering them one by one.

I dentify the problem - In this first step, you identify and understand the problem. To do this, you evaluate which information is provided and available, and you identify the unknown variables and missing information.

D escribe the outcome - In this second step, you define the result you are seeking. This matters because a problem might have multiple potential results, so you need to clarify which outcomes in particular you are aiming for. Defining an outcome clarifies the path that must be taken to solving the problem.

E xplore possible strategies - Now that you have considered the desired outcome, you are ready to brainstorm and explore different strategies and techniques to solve your particular problem.

A nticipate outcomes and act - From the previous step, you already have explored different strategies and techniques. During this step, you review and evaluate them in order to choose the best one to act on. Your selection should consider the benefits and drawbacks of the strategy and whether it can ultimately lead to the desired outcome. After making your selection, you act on it and apply the technique to the given problem.

L ook and learn - The final step to solving problems with this method is to consider whether the applied technique worked and if the needed results were obtained. Also, an additional step is learning from the current problem and its methods to make problem solving more efficient in the future.

## Examples of problem-solving models and strategies

Here are some solved examples of the problem-solving models and strategies discussed above.

Find the number when two times the sum of $$3$$ and that number is thrice that number plus $$4$$. Solve this problem with Polya's f our-step problem-solving model .

Solution: We will follow the steps of Polya's f our-step problem-solving model as mentioned above to find the number.

Step 1 : Understand the problem.

By reading and understanding the question, we denote the unknown number as $$x$$.

Step 2 : Devise a plan.

We see that two times $$x$$ is added to $$3$$ to make it equal to thrice the $$x$$ plus $$4$$. So, we can determine that forming an equation to solve the mathematical problem is a reasonable plan. Therefore, we form an equation by going step by step:

First we add $$x$$ with $$3$$ and multiply it with $$2$$.

$$\tag{1}\Rightarrow 2(x+3)$$

Then, we form the second part of the equation for thrice the $$x$$ plus $$4$$.

$$\tag{2}\Rightarrow 3x+4$$

Hence, equating both sides $$(1)$$ and $$(2)$$ we get,

$2(x+3)=3x+4$

Step 3 : Carry out the plan.

Now, we algebraically solve the equation above.

\begin{align}2(x+3) &=3x+4 \\2x+6 &= 3x+4 \\3x-2x &= 6-4 \\x &=2\end{align}

Step 4 : Look back.

By inputting the value of 2 in our equation, we see that two times $$2+3$$ is $$10$$ and three times $$2$$ plus $$4$$ is also 10. Hence, the left side and right side are equal. So, our solution is satisfied.

Hence, the number is $$2$$.

A string is $$48 cm$$ long. It is cut into two pieces such that one piece is three times that of the other piece. What is the length of each piece?

Solution : Let us work on this problem using the IDEAL problem-solving method.

Step 1 : Identify the problem.

We are given a length of a string, and we know that it is cut into two parts, whereby one part is three times longer than the other. As the length of the longer piece of string is dependent on the shorter string, we assume only one variable, say $$x$$.

Step 2 : Describe the outcome.

From the problem, we understand that we need to find the length of each piece of string. And we need the results such that the total length of both the pieces should be $$48 cm$$.

Step 3 : Explore possible strategies.

There are multiple ways to solve this problem. One way to solve it is by using the trial-and-error method. Also, as one length is dependent on another, the other way is to form an equation to solve for the unknown variable algebraically.

Step 4 : Anticipate outcomes and act.

From the above step, we have two methods by which we can solve the given problem. Let's find out which method is more efficient and solve the problem by applying it.

For the trial-and-error method, we need to assume value(s) one at a time for the variable and then solve for it individually until we get the total of 48.

That is, suppose we consider $$x=1$$.

Then, by the condition, the second piece is three times the first piece.

$\Rightarrow 3x=3(1)=3$

Then the length of both pieces should be:

$\Rightarrow 1+3=4\neq 48$

Hence, our assumption is wrong. So, we need to consider another value. For this method, we continue this process until we find the total of $$48$$. We can see that proceeding this way is time-consuming. So, let us apply the other method instead.

In this method, we form an equation and solve it to obtain the unknown variable's value. We know that one piece is three times the other piece. Therefore, let the length of one piece be $$x$$. Then the length of the other piece is $$3x$$.

Now, as the string is $$48 cm$$ long, it should be considered as a sum of both of its pieces.

\begin{align}&\Rightarrow x+3x=48 \\&\Rightarrow 4x=48 \\&\Rightarrow x=\frac{48}{4} \\&\Rightarrow x=12 \\\end{align}

So, the length of one piece is $$12cm$$. The length of the other piece is $$3x=3(12)=36cm$$.

Step 5: Look and learn

Let's take a look to see if our answers are correct. The unknown variable value we obtained is $$12$$. Using it to find the other piece we get a value of $$36$$. Now, adding both of them, we get:

$\Rightarrow 12+36=48$.

Here, we got the correct total length. Hence, our calculations and applied method are right.

## Problem-solving strategies and models - Key takeaways

• Problem-solving strategies are models developed based on previous experience that provide a recommended approach for analyzing potential solutions for problems.
• Two common models include Polya's Four-Step Problem-Solving Model and the IDEAL problem-solving model.
• Polya's Four-Step Problem-Solving Model has the following steps: 1) Understand the problem, 2) Devise a plan, 3) Carry out the plan, and 4) Looking back.
• The IDEAL model is based on the following steps: 1) Identify The Problem, 2) Define An Outcome, 3) Explore Possible Strategies, 4) Anticipate Outcomes and Act, 5) Look And Learn.

## Frequently Asked Questions about Problem-solving Models and Strategies

--> what are problem-solving models.

Problem-solving models are models developed based on previous experience that provide a recommended approach for solving problems or analyzing potential solutions.

## --> What are types of problem-solving?

The most basic types of problem-solving are Polya's four-step problem-solving model and the IDEAL problem-solving model.

## --> What are the strategies to problem-solve efficiently?

The strategies to solve a problem efficiently are to understand it, determine the correct method, solve it and verify and learn from it.

## --> What are the lists of problem-solving models in algebra?

In algebra, any problem can be solved using Polya's four-step problem-solving model and IDEAL problem-solving model.

## --> What are the 5 problem-solving strategies?

The 5 problem-solving strategies are 1. Identify The Problem, 2. Define An Outcome, 3. Explore Possible Strategies, 4. Anticipate Outcomes & Act, 5. Look And Learn.

## Final Problem-solving Models and Strategies Quiz

Problem-solving models and strategies quiz - teste dein wissen.

What are the steps to solve a problem efficiently?

Show answer

1. Understand the problem

Show question

Name the two problem-solving models.

State two problem-solving strategies when devising a plan.

Apply the specific formula for the problem.

Step 01: What do you know?

• Mrs. Grave gives 1 penny on Day 1, 2 pennies on Day 2, and 4 pennies on Day 3.
• Each day the amount of money will double.
• Paul does his tasks for 5 days.

Step 02: What do you want to know?

You curious to figure out how much money will Paul have in total after 5 days of doing his tasks. We want to solve the problem by formulating a simpler one.

Step 01: What does David know?

• The number of players starting the tournament:8
• Only winners can advance to the next round

Step 02: What does David want to know?

David wants to compare the number of players in the second round to the number that starts the tournament. To solve the problem, David can use a diagram.

Step 01: What do you know?

• Slices of tomatoes and cucumber were used.
• The total number of slices used is 60.
• The ratio of cucumbers to tomatoes is 4:6
• The ratio simplifies to 2:3

Step 02: What do you want to know?

We need to know the number of both cucumber and tomato slices. We want to solve the problem by doing a table.

24 cucumber slices and 36 tomato slices is one solution

Trial and error is a problem solving strategy.

The IDEAL method is one type of problem-solving model:

Anticipate outcome is one step of problem-solving for Polya's Four-Step problem-solving model?

Problem solving strategies help us to solve a problem efficiently

We can recheck our provided solution by doing the following:

This can be done by solving the problem in another way or double-check if your solution makes sense.

To understand the problem we need to answer the following question:

What do you know?

To do a plan we need to answer the following question:

What do you want to know?

Simplifying the problem to reach a solution is one problem solving strategy:

## Test your knowledge with multiple choice flashcards

Step 01: What do you know? Mrs. Grave gives 1 penny on Day 1, 2 pennies on Day 2, and 4 pennies on Day 3.Each day the amount of money will double. Paul does his tasks for 5 days.Step 02: What do you want to know?You curious to figure out how much money will Paul have in total after 5 days of doing his tasks. We want to solve the problem by formulating a simpler one.

Step 01: What does David know? The number of players starting the tournament:8 Only winners can advance to the next roundStep 02: What does David want to know?David wants to compare the number of players in the second round to the number that starts the tournament. To solve the problem, David can use a diagram.

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## Principal Leadership Magazine, Vol. 5, Number 4, December 2004

Counseling 101 column, a problem-solving model for improving student achievement.

Problem solving is an alternative to assessments and diagnostic categories as a means to identify students who need special services.

By Andrea Canter

Andrea Canter recently retired from Minneapolis Public Schools where she served as lead psychologist and helped implement a district-wide problem solving model. She currently is a consultant to the National Association of School Psychologists (NASP) and editor of its newspaper, Communiquè . “Counseling 101” is provided by NASP ( www.nasponline.org ).

The implementation of the No Child Left Behind Act (NCLB) has prompted renewed efforts to hold schools and students accountable for meeting high academic standards. At the same time, Congress has been debating the reauthorization of the Individuals With Disabilities Education Act (IDEA), which has heightened concerns that NCLB will indeed “leave behind” many students who have disabilities or other barriers to learning. This convergence of efforts to address the needs of at-risk students while simultaneously implementing high academic standards has focused attention on a number of proposals and pilot projects that are generally referred to as problem-solving models. A more specific approach to addressing academic difficulties, response to intervention (RTI), has often been proposed as a component of problem solving.

## What Is Problem Solving?

A problem-solving model is a systematic approach that reviews student strengths and weaknesses, identifies evidence-based instructional interventions, frequently collects data to monitor student progress, and evaluates the effectiveness of interventions implemented with the student. Problem solving is a model that first solves student difficulties within general education classrooms. If problem-solving interventions are not successful in general education classrooms, the cycle of selecting intervention strategies and collecting data is repeated with the help of a building-level or grade-level intervention assistance or problem-solving team. Rather than relying primarily on test scores (e.g., from an IQ or math test), the student’s response to general education interventions becomes the primary determinant of his or her need for special education evaluation and services (Marston, 2002; Reschly & Tilly, 1999).

## Why Is a New Approach Needed?

Although much of the early implementation of problem-solving models has involved elementary schools, problem solving also has significant potential to improve outcomes for secondary school students. Therefore, it is important for secondary school administrators to understand the basic concepts of problem solving and consider how components of this model could mesh with the needs of their schools and students. Because Congress will likely include RTI options in its reauthorization of special education law and regulations regarding learning disabilities, it is also important for school personnel to be familiar with the pros and cons of the problem-solving model.

Student outcomes. Regardless of state or federal mandates, schools need to change the way they address academic problems. More than 25 years of special education legislation and funding have failed to demonstrate either the cost effectiveness or the validity of aligning instruction to diagnostic classifications (Fletcher et al., 2002; Reschly & Tilly, 1999; Ysseldyke & Marston, 1999). Placement in special education programs has not guaranteed significant academic gains or better life outcomes for students with disabilities. Time-consuming assessments that are intended to differentiate students with disabilities from those with low achievement have not resulted in better instruction for struggling students.

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Dilemma of learning disabilities. The learning disabilities (LD) classification has proven especially problematic. Researchers and policymakers representing diverse philosophies regarding disability are generally in agreement that the current process needs revision (Fletcher et al., 2002). Traditionally, if a student with LD is to be served in special education, an evaluation using individual intelligence tests and norm-referenced achievement tests is required to document an ability/achievement discrepancy. This model has been criticized for the following reasons:

• A reliance on intelligence tests in general and with students from ethnic and linguistic minority populations in particular
• A focus on within-child deficiencies that often ignore quality of instruction and environmental factors
• The limited applicability of norm-referenced information to actual classroom teaching
• The burgeoning identification of students as disabled
• The resulting allocation of personnel to responsibilities (classification) that are significantly removed from direct service to students (Ysseldyke & Marston, 1999).

Wait to fail. A major flaw in the current system of identifying student needs is what has been dubbed the wait to fail approach in which students are not considered eligible for support until their skills are widely discrepant from expectations. This runs counter to years of research demonstrating the importance of early intervention (President’s Commission on Excellence in Special Education, 2002). Thus, a number of students fail to receive any remedial services until they reach the intermediate grades or middle school, by which time they often exhibit motivational problems and behavioral problems as well as academic deficits.

For other students, although problems are noted when they are in the early grades, referral is delayed until they fail graduation or high school standards tests, increasing the probability that they will drop out. Their school records often indicate that teachers and parents expressed concern for these students in the early grades, which sometimes resulted in referral for assessments, but did not result in qualification for special education or other services.

Call for evidence-based programs. One of the major tenets of NCLB is the implementation of scientifically based interventions to improve student performance. The traditional models used by most schools today lack such scientifically based evidence. There are, however, many programs and instructional strategies that have demonstrated positive outcomes for diverse student populations and needs (National Reading Panel, 2000). It is clear that schools need systemic approaches to identify and resolve student achievement problems and access proven instructional strategies.

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## How It Works

Although problem-solving steps can be described in several stages, the steps essentially reflect the scientific method of defining and describing a problem (e.g., Ted does not comprehend grade-level reading material); generating potential solutions (e.g., Ted might respond well to direct instruction in comprehension strategies); and implementing, monitoring, and evaluating the effectiveness of the selected intervention.

Problem-solving models have been implemented in many versions at local and state levels to reflect the unique features and needs of individual schools. However, all problem-solving models share the following components:

• Screening and assessment that is focused on student skills rather than classification
• Measuring response to instruction rather than relying on norm-referenced comparisons
• Using evidence-based strategies within general education classrooms
• Developing a collaborative partnership among general and special educators for consultation and team decision making.

Three-tiered model. One common problem-solving model is the three-tiered model. In this model, tier one includes problem-solving strategies directed by the teacher within the general education classrooms. Tier two includes problem-solving efforts at a team level in which grade-level staff members or a team of various school personnel collaborate to develop an intervention plan that is still within the general education curriculum. Tier three involves referral to a special education team for additional problem solving and, potentially, a special education assessment (Office of Special Education Programs, 2002).

Response to intervention. A growing body of research and public policy discussion has focused on problem-solving models that include evaluating a student’s RTI as an alternative to the IQ-achievement discrepancy approach to identifying learning disabilities (Gresham, 2002). RTI refers to specific procedures that align with the steps of problem solving:

• Implementing evidence-based interventions
• Frequently measuring a student’s progress to determine whether the intervention is effective
• Evaluating the quality of the instructional strategy
• Evaluating the fidelity of its implementation. (For example, did the intervention work? Was it scientifically based? Was it implemented as planned?)

Although there is considerable debate about replacing traditional eligibility procedures with RTI approaches (Vaughn & Fuchs, 2003), there is promising evidence that RTI can systematically improve the effectiveness of instruction for struggling students and provide school teams with evidence-based procedures that measures a student’s progress and his or her need for special services.

New roles for personnel. An important component of problem-solving models is the allocation (or realignment) of personnel who are knowledgeable about the applications of research to classroom practice. Whereas traditional models often limit the availability of certain personnel-for example, school psychologists-to prevention and early intervention activities (e.g., classroom consultation), problem-solving models generally enhance the roles of these service providers through a systemic process that is built upon general education consultation. Problem solving shifts the emphasis from identifying disabilities to implementing earlier interventions that have the potential to reduce referral and placement in special education.

## Outcomes of Problem Solving and RTI

Anticipated benefits of problem-solving models, particularly those using RTI procedures, include emphasizing scientifically proven instructional methods, the early identification and remediation of achievement difficulties, more functional and frequent measurement of student progress, a reduction in inappropriate and disproportionate special education placements of students from diverse cultural and linguistic backgrounds, and a reallocation of instructional and behavior support personnel to better meet the needs of all students (Gresham, 2002; Ysseldyke & Marston, 1999). By using problem solving, some districts have reduced overall special education placements, increased individual and group performance on standards tests, and increased collaboration among special and general educators.

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The enhanced collaboration between general education teachers and support personnel is particularly important at the secondary level because staff members often have limited interaction with school personnel who are outside of their specialty area. Problem solving provides a vehicle to facilitate communication across disciplines to resolve student difficulties in the classroom. Secondary schools, however, face additional barriers to collaboration because each student may have five or more teachers. Special education is often even more separated from general education in secondary school settings. Secondary school teachers also have a greater tendency to see themselves as content specialists and may be less invested in addressing general learning problems, particularly when they teach five or six class periods (and 150 or more students) each day. The sheer size of the student body and the staff can create both funding and logistical difficulties for scheduling training and team meetings.

## Is Problem Solving Worth the Effort?

Data from district-wide and state-level projects in rural, suburban, and urban communities around the country support the need to thoughtfully implement problem-solving models at all grade levels. There are several federally funded demonstration centers that systematically collect information about these approaches. Although national demonstration models may be a few years away, it seems likely that state and federal regulations under IDEA will include problem solving and RTI as accepted experimental options. Problem solving continues to offer much promise to secondary school administrators who are seeking to improve student performance through ongoing assessment and evidence-based instruction. PL

• Fletcher, J., Lyon, R., Barnes, M., Stuebing, K., Francis, D., Olson, R., Shaywitz, S., & Shaywitz, B. (2002). Classification of learning disabilities: An evidence-based evaluation. In R. Bradley, L. Donaldson, & D. Hallahan (Eds.), Identification of learning disabilities (pp. 185-250). Mahwah, NJ: Erlbaum.
• Gresham, F. (2002). Responsiveness to intervention: An alternative approach to the identification of learning disabilities. In R. Bradley, L. Donaldson, & D. Hallahan (Eds.), Identification of learning disabilities (pp. 467-519). Mahwah, NJ: Erlbaum.
• Marston, D. (2002). A functional and intervention-based assessment approach to establishing discrepancy for students with learning disabilities. In R. Bradley, L. Donaldson, & D. Hallahan (Eds.), Identification of learning disabilities (pp. 437-447). Mahwah, NJ: Erlbaum.
• National Reading Panel. (2000). Teaching children to read: An evidence-based assessment of the scientific literature on reading and its implications for reading instruction-Reports of the subgroups. Washington, DC: Author.
• Office of Special Education Programs, U.S. Department of Education. (2002). Specific learning disabilities: Finding common ground (Report of the Learning Disabilities Round Table). Washington, DC: Author.
• President’s Commission on Excellence in Special Education. (2002). A new era: Revitalizing special education for children and their families. Washington, DC: U.S. Department of Education.
• Reschly, D., & Tilly, W. D. III (1999). Reform trends and system design alternatives. In D. Reschly, W. D. Tilly III, & J. Grimes (Eds.), Special education in transition: Functional assessment and noncategorical programming (pp. 19-48). Longmont, CO: Sopris West.
• Vaughn, S., & Fuchs, L. (Eds.) (2003). Special issue: Response to intervention. Learning Disabilities Research & Practice, 18(3).
• Ysseldyke, J., & Marston, D. (1999). Origins of categorical special education services in schools and a rationale for changing them. In D. Reschly, W. D. Tilly III, & J. Grimes (Eds.), Special education in transition: Functional assessment and noncategorical programming (pp. 1-18). Longmont, CO: Sopris West.

## Case Study: Optimizing Success Through Problem Solving

By Marcia Staum and Lourdes Ocampo

Milwaukee Public Schools, the largest school district in Wisconsin, is educating students with Optimizing Success Through Problem Solving (OSPS), a problem-solving initiative that uses a four-step, data-based, decision-making process to enhance school reform efforts. OSPS is patterned after best practices in the prevention literature and focuses on prevention, early intervention, and focused intervention levels.  Problem-solving facilitators provide staff members with the training, modeling, support, and tools they need to effectively use data to drive their instructional decision-making. The OSPS initiative began in the fall of 2000 with seven participating schools. Initially, elementary and middle level schools began to use OSPS, with an emphasis on problem solving for individual student issues. As the initiative matured, increased focus was placed on prevention and early intervention support in the schools. Today, 78 schools participate in the OSPS initiative and are serviced by a team of 18 problem-solving facilitators.

## OSPS in Action: Juneau High School

The administration of Juneau High School, a Milwaukee public charter school with 900 students, invited OSPS to become involved at Juneau for the 2003-2004 school year. Because at the time OSPS had limited involvement with high schools, two problem-solving facilitators were assigned to Juneau for one half-day each week. The problem-solving facilitators immediately joined the Juneau’s learning team, which is a small group of staff members and administrators who make educational decisions aimed at increasing student achievement.

When the problem-solving facilitators became involved with Juneau, the learning team was working to improve student participation on the Wisconsin Knowledge and Concepts Exam (WKCE). The previous year, Juneau’s 10th-grade participation on the exam had been very low. The learning team used OSPS’s four-step problem-solving process to develop and implement a plan that resulted in a 99% student participation rate on the WKCE. After this initial success, the problem-solving model was also used at Juneau to increase parent participation in parent-teacher conferences. According to Myron Cain, Juneau’s principal, “Problem solving has helped the learning team at Juneau go from dialogue into action. In addition, problem solving has supported the school within the Collaborative Support Team process and with teambuilding, which resulted in a better school climate.”

By starting at the prevention level, Juneau found that there was increased commitment from staff members. OSPS is now in the initial stages of working with Juneau to explore alternatives to suspension.  The goal is to create a working plan that will lead to creative ways of decreasing the number of suspensions at Juneau.

Marcia Staum is a school psychologist, and Lourdes Ocampo is a school social worker for Optimizing Success Through Problem Solving.

## What Is Response to Intervention?

Many researchers have recommended that a student’s response to intervention or response to instruction (RTI) should be considered as an alternative or replacement to the traditional IQ-achievement discrepancy approach to identifying learning disabilities (Gresham, 2002; President’s Commission on Excellence in Special Education, 2002). Although there is considerable debate about replacing traditional eligibility procedures with RTI approaches (Vaughn & Fuchs, 2003), there is promising evidence that RTI can systematically improve the effectiveness of instruction for struggling students and provide school teams with evidence-based procedures to measure student progress and need for special services. In fact, Congress has proposed the use of research-based RTI methods (as part of a comprehensive evaluation process to reauthorize IDEA) as an allowable alternative to the use of an IQ-achievement discrepancy procedure in identifying learning disabilities.

RTI refers to specific procedures that align with the steps of problem solving. These steps include the implementation of evidence-based instructional strategies in the general education classroom and the frequent measurement of a student’s progress to determine if the intervention is effective. In settings where RTI is also a criteria for identification of disability, a student’s progress in response to intervention is an important determinant of the need and eligibility for special education services.

It is important for administrators to recognize that RTI can be implemented in various ways depending on a school’s overall service delivery model and state and federal mandates. An RTI approach benefits from the involvement of specially trained personnel, such as school psychologists and curriculum specialists, who have expertise in instructional consultation and evaluation.

• National Center on Student Progress Monitoring, www.studentprogress.org
• National Research Center on Learning Disabilities, www.nrcld.org

This article was adapted from a handout published in Helping Children at Home and School II: Handouts for Families and Educators (NASP, 2004). “Counseling 101” articles and related HCHS II handouts can be downloaded from www.naspcenter.org/principals .

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## The Ultimate Problem Solving Model Guide For Crafting Perfect Solutions

Anthony Metivier | November 11, 2022 | Thinking

Ideally, that model should be easy to remember and quick to implement.

The problem is this:

Not all problems are the same. There’s no such thing as a problem solving chart or diagram that is going to apply to all situations.

So instead of forcing every issue you’re facing into some bogus Six Steps of Problem Solving Formula, let’s get real.

Let’s look at a number of multi-step problem solving formulas. Once you have this list, you can pick the models most likely to work and enjoy much better results thanks to variety.

## The 9 Best Problem Solving Models And Formulas

As you go through this list, consider taking notes. As you do, jot down different times when these different approaches might help you.

Remember: not all problems are created the same, so the exact problem solving steps you follow need to be suited to the task at hand. Flexibility based on knowledge of what’s available is a key part of objective reasoning .

This point is important because there are some steps to follow before we even look at any models.

• Recognize that a problem exists and give it a name
• Represent the problem in the best possible medium (writing, graphics, video)
• List your goals for solving the problem states
• Generate and evaluate possible solutions
• Select the best possible solution
• Execution the best possible solution
• Analyze and determine if the solution you’ve chosen solves the problem

This process is sometimes confused with the standard problem solving model. But in actuality it is the meta-level understanding you need before using any particular model.

## One: The Standard Problem Solving Model

As mentioned, many people consider the steps I just listed the standard model for solving problems. They may even simplify it into a problem solving chart like this:

I would suggest making your initial approach much more robust.

## Two: The Dynamic Problem Solving Model

The Dynamic Problem solving model breaks the process into much more distinct phases.

In phase one, we want to spend as much time as possible going deep into the problem. Look at it in as many ways as possible and from as many perspectives and contexts as you can.

• Name and describe the problem, ideally in multiple media
• Explore its contexts
• Study similar examples
• Research opportunities for interviewing people who have solved the problem before

Next, you want to bring in as many idea generation steps as you can.

• Brainstorming
• Mind mapping
• Gamification
• Rating the solutions

Once you have visualized and gathered your ideas by brainstorming them onto paper or using a mind map, there are many ways you can gamify.

For example, you can:

• Ask “what if” questions
• Ask what Isaksen et al . call “wouldn’t it be nice if” or “wouldn’t it be terrible if” questions
• Hold a contest for the best solution (internally and externally to your organization)
• Go for a walk and try not to think about the problem and solutions and then write about your experience in withholding

Rating can be performed in various ways. You can divide the solutions you can up with into grades such as A+, A, B, etc. Or you can give the solutions you’ve gathered ratings from 1-10 and have as many people participate in the process as possible.

For the final stage:

• First choose and accept the path and ensure all team members are on the same page
• Design the actions you’re going to take
• Schedule the time for implementation and review design
• Schedule the time for review

This 3-stage problem solving model is far more robust than the standard solution.

Three: The Brief Problem Solving Model

Isaac Newton reportedly said that in order to solve a problem, you just need to think about it constantly. But sometimes you don’t have all the time in the world.

Famous scientist Richard Feynman reflected on the scarcity of time when he described the following model:

1 Write down the problem.

2 Think really hard.

3 Write down the answer.

Sometimes finding the best possible solution really is just this simple.

To expand a little on how this model might work in practice, you can:

• Describe the problem broadly and without granular details
• Briefly describe the best possible outcome and ideas for achieving it that come to mind
• List the benefits of having it solved to create inspiration and momentum

If you’re in a hurry, this problem solving example will often work very well.

Four: The W.R.A.P. Problem Solving Model

Of all the faster problem solving model examples I’ve seen, the W.R.A.P. formula presented by Chip and Dan Heath in their book Decisive is my favorite. Although not immediate, it’s quite fast.

W.R.A.P. stands for:

• Widen your options
• Reality test
• Attain distance
• Prepare to fail

The final point is especially important because we often don’t take time to consider what we’ll do if the solutions we choose do not perform to expectation.

Five: Analyze For Advantage

Sometimes you just want to find the most advantageous outcome.

To do so, follow this model:

• List the advantages you want as a result of solving the problem
• List the existing assets, resources and advantages you have right now
• List your current limitations, including any fears
• List how you can refine your existing assets to combat those fears
• Plan for maximum advantage based on your newly optimized assets

## Six: Examine Pros and Cons

Although simplistic, a great model to follow in a hurry is to simply list the pros and cons.

All you need to do is write pros and cons at the top of a piece of paper and draw a line down the center.

As you list the pros and cons, your mind will probably start branching out and coming up with solutions so that the cons cannot take over.

## Seven: Find the Forces

Often when we try to solve problems, we’re not looking deeply enough to find the one root cause. Even if we are, we can fail to find solutions because we’re so focused on finding just one source of the problem.

Often, there are multiple forces or factors at work in causing a problem. To get started finding them, you can follow this model:

• List all the people involved
• List all the technologies involved
• List all the situations involved
• Describe the ways in which these different “forces” act upon creating the problem
• Write out various scenarios in which changes are made to the different elements
• Try to predict and previsualize various outcomes based on changes you could make

## Eight: Peer into the Unconscious

Synectics appeared in the 1950s and assumed that many people struggle to solve problems because the solutions remain outside their conscious awareness. However, their unconscious mind might know the solution and be “hiding” it from the mind for various reasons.

Robert Langs proposed a similar thesis, and wondered if the unconscious mind wasn’t something like an antivirus system of the mind.

This idea is not so far-fetched, even though it can be strange to think that the mind would hide the perfect solution from you if it truly knows it.

In Mindshift , Barbara Oakley discusses research showing how the insular cortex can cause a pain response when a person is faced with certain tasks.

This suggests literally what her book title proposes: a shift of mind.

There are many ways you can do this, and books by her, Langs, and the people behind Synectics are a great place to start for examples of various problem solving models that deal with this level of your mind.

Nine: The Problem Solving Situations Model

Discover Projects offers a great way to ensure that you find the right model for solving your problems. It involves identifying the problem correctly in the first place.

They suggest there are at least 6 types of problems:

• Type I problems: Known by the person with the problem, but only one solution is known.
• Type II problems: A problem that is known by the person presenting the problem and the person hired to solve it, but the method of solution and solution are known only by the presenter.
• Type III problems: the problem is known by the presenter and the solver; more than one method may be used to arrive at the solution, which the presenter knows.
• Type IV problems: the problem is known by the presenter and the solver; the problem may be solved in more than one way; the presenter knows the range of solutions.
• Type V problems: these problems are clearly defined and the problem is known by the presenter and the solver; the method and solution are unknown by the presenter and the solver.
• Type VI problems: these problems are not clearly defined or are undefined, have little if any structure, and are complex; the problem is unknown by both the presenter and the solver; the method and solution are unknown by both the presenter and the solver.

When you do this kind of problem identification analysis (where relevant), many more solutions will arise than you would otherwise perceive.

The greatest aspect of this model is that it helps you find out who might have the solution.

Another way of thinking about this approach is basically what Dan Sullivan is getting at in his Who Not How problem solving model. If you’re able to figure out who can solve the problem, chances are that person also knows how to solve it.

## The Best Problem Solving Model Of Them All

Please don’t feel that what I’m about to say is a trick.

It isn’t. It’s the ultimate solution.

The best problem solving model of them all is the one you practice.

And practice means committing the model to memory, using it consistently and optimizing your approach along the way.

For an easy and fun way to commit any model to mind quickly, I invite you to get my FREE Memory Improvement Kit:

It’s a model of a different kind that helps you remove the issue of forgetting from your life.

Once you’ve done that, you can follow multiple paths to solving the vexations of everyday life quickly.

## Related Posts

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## MODEL DAN METODE PEMBELAJARAN BERBASIS PEMECAHAN MASALAH (PROBLEM SOLVING)

• Di beberapa tempat perbuatan mencoral-coret dinding tembok dengan menggunakan kata-kata yang tidak sopan sering dijumpai. Hal tersebut merusak pemandangan kampung dan menjadikan wilayah tersebut terkesan kumuh. Bagaimanakah menyelesaikan masalah tersebut?
• Perilaku membuang sampah di saluran air atau di sungai seolah-olah menjadi perilaku yang biasa saja. Padahal di Indonesia memiliki undang-undang tentang lingkungan hidup. Bagaimana penyelesaian masalah perilaku membuang sampah sembarangan tersebut ditinjau dari undang-undang lingkungan hidup atau peraturan perundang-undangan yang lain?
• Wilayah terluar, terdepan, dan tertinggal dari NKRI berbatasan dengan negara-negara tetangga. Pembangunan di wilayah tersbut belum memadai dan warga yang tinggal di wilayah tersebut merasa tidak diperhatikan oleh Pemerintah RI. Bagaimana sebaiknya wilayah tersebut dikembangankan dan dibangun?

## = Baca Juga =

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1. 5. DIRECTIONS MODEL 4 WITH EXAMPLES SHADOW BASED PROBLEM

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5. Payne Media and Tech Inquiry Based Problem Solving

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#### COMMENTS

1. Problem-Solving Models: What They Are and How To Use Them

Problem-solving models are step-by-step processes that provide a framework for addressing challenges in the workplace. These models allow you and your team members to make data-based decisions while working toward a logical, structured solution to benefit the entire team or company.

2. PDF Model-based Problem Solving

Model-based Problem Solving Peter Struss 10.1 Introduction The development of the concept of model-based systems was an answer to the limita-tions of rule-based "expert systems", which base problem solving (e.g., diagnosis) on a representation of experiential knowledge in a domain. These limitations are not due

3. CC Liong: Model-Based Problem Solving

Liong's topic is "Model-Based Problem Solving." He uses an example of an American who drives a Pontiac car and buys ice-cream as an example to demonstrate the importance of defining the right problem. After buying vanilla ice cream, the car would not start. When he parked at the curb and bought other flavors of ice cream, he could drive away ...

4. PDF THIRTEEN PROBLEM-SOLVING MODELS

1. Clarify Explore the Vision Identify your goal, desire or challenge. This is a crucial first step because it's easy to assume, incorrectly, that you know what the problem is. However, you may have missed something or have failed to understand the issue fully and defining your objective can provide clarity.

5. What is Problem Solving? Steps, Process & Techniques

Problem solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing, and selecting alternatives for a solution; and implementing a solution. The problem-solving process Problem solving resources Problem Solving Chart The Problem-Solving Process

6. Chapter 10 Model-based Problem Solving

The development of the concept of model-based systems was an answer to the limitations of rule-based expert systems, which base problem solving (e.g., diagnosis) on a representation of experiential knowledge in a domain.

7. How to master the seven-step problem-solving process

Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

8. Modeling Problem Solving

Problem-solving is an additional skill that tutors model for students. An organized and- intentional problem-solving approach helps us to efficiently work through challenges, and many of us effectively problem solve without much thought given to our approach. 1 However, it makes sense to take a step back and do our best to model problem-solving ...

9. Problem solving through values: A challenge for thinking and capability

Abstract. The paper aims to introduce the conceptual framework of problem solving through values. The framework consists of problem analysis, selection of value (s) as a background for the solution, the search for alternative ways of the solution, and the rationale for the solution. This framework reveals when, how, and why is important to ...

10. Chapter 10 Model-based Problem Solving

The development of the concept of model-based systems was an answer to the limitations of rule-based expert systems, which base problem solving (e.g., diagnosis) on a representation of experiential knowledge in a domain. These limitations are not due to the syntactic form of representing knowledge (rules), but they result from the nature of the ...

11. The McKinsey guide to problem solving

The McKinsey guide to problem solving. Become a better problem solver with insights and advice from leaders around the world on topics including developing a problem-solving mindset, solving problems in uncertain times, problem solving with AI, and much more. Become a better problem solver with insights and advice from leaders around the world ...

12. (PDF) Conceptual Model-Based Problem Solving

Algebraic conceptualization of mathematical relations and model-based problem solving can be. students' creation and articulation of their own story problems with a particular problem. grades ...

13. Conceptual Model-Based Problem Solving

The representation that models the underlying mathematical relations in the problem, that is, the conceptual model, facilitates solution planning and accurate problem solving. The conceptual model drives the development of a solution plan that involves selecting and applying appropriate arithmetic operations.

14. What Is the Model in Model-Based Planning?

What Is the Model in Model-Based Planning? 2021 Jan;45 (1):e12928. doi: 10.1111/cogs.12928. Thomas Pouncy , Pedro Tsividis , Samuel J Gershman 33398907 10.1111/cogs.12928 Flexibility is one of the hallmarks of human problem-solving. In everyday life, people adapt to changes in common tasks with little to no additional training.

15. Problem-Based Learning

Problem-based learning (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning. Why Use Problem-Based Learning? Nilson (2010) lists the following learning outcomes that are associated with PBL.

16. The Problem-Solving Process

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue. The best strategy for solving a problem depends largely on the unique situation.

17. How to utilize problem-solving models in education

The MTSS problem-solving model is a data-driven decision-making process that helps educators utilize and analyze interventions based on students' needs on a continual basis. Traditionally, the MTSS problem-solving model only involves four steps: Identifying the student's strengths and needs, based on data.

18. Problem-Solving Therapy: Definition, Techniques, and Efficacy

Problem-solving therapy is a form of therapy that provides patients with tools to identify and solve problems that arise from life stressors, both big and small. Its aim is to improve your overall quality of life and reduce the negative impact of psychological and physical illness.

19. The FOCUS Model

The model is helpful because it uses a team-based approach to problem solving and to business-process improvement, and this makes it particularly useful for solving cross-departmental process issues. Also, it encourages people to rely on objective data rather than on personal opinions, and this improves the quality of the outcome. It has five ...

20. Problem-solving Models and Strategies

Two common models include Polya's Four-Step Problem-Solving Model and the IDEAL problem-solving model. Polya's Four-Step Problem-Solving Model has the following steps: 1) Understand the problem, 2) Devise a plan, 3) Carry out the plan, and 4) Looking back. The IDEAL model is based on the following steps: 1) Identify The Problem, 2) Define An ...

21. Problem-Solving Model for Improving Student Achievement

A problem-solving model is a systematic approach that reviews student strengths and weaknesses, identifies evidence-based instructional interventions, frequently collects data to monitor student progress, and evaluates the effectiveness of interventions implemented with the student.

22. Examining the effects of an intervention on mathematical modeling in

Student problem-solving modeling performance was measured before, right after, and two months after the culmination of the intervention. The person estimates emerging from a Rasch analysis of these data were analyzed using inferential statistics and a multi-level piecewise linear growth model.

23. The Ultimate Problem Solving Model Guide For Crafting Solutions

First choose and accept the path and ensure all team members are on the same page. Design the actions you're going to take. Schedule the time for implementation and review design. Schedule the time for review. This 3-stage problem solving model is far more robust than the standard solution.

24. Model Dan Metode Pembelajaran Berbasis Pemecahan Masalah (Problem Solving)

Model dan Metode pemecahan masalah (Problem Solving) digunakan dalam pembelajaran yang membutuhkan jawaban atau pemecahan masalah.Sebagai metode pembelajaran, metode pemecahan masalah sangat baik bagi pembinaan sikap ilmiah pada siswa.Dengan metode ini, para siswa belajar memecahkan suatu masalah menurut prosedur kerja ilmiah. 1. Pengertian Metode Pemecahan Masalah