problem solving focus model

Find, Organize, Clarify, Understand, Select (FOCUS – PDCA)

Published: November 7, 2018 by Ken Feldman

problem solving focus model

PDCA (Plan-Do-Check-Act) is a popular iterative methodology to fix a problem or improve a process. 

Developed and promoted by Drs. Deming and Shewhart, it’s used as a cycle of examining a problem, collecting some data, improving the process and then monitoring it to be sure your improvement was successful. If not, you repeat the cycle. 

The preceding FOCUS expands the methodology and combines with PDCA to form a comprehensive approach to problem-solving and process improvement.

Overview: What is FOCUS PDCA? 

The FOCUS PDCA approach was developed for the healthcare industry. It’s an extension of the classic PDCA methodology, where FOCUS is a set of activities that precedes those used in the PDCA cycle. 

The components of FOCUS PDCA are:

The PDCA, used in the context of FOCUS, is a variation of the original PDCA. 

3 benefits of FOCUS PDCA 

Any improvement methodology will have benefits for the organization. Some of the specific ones related to using FOCUS PDCA are discussed below. 

It’s comprehensive 

The FOCUS PDCA approach starts at the beginning by identifying the problem and ends with a control plan in place to ensure your improvements don’t disappear over time.

It’s simple 

Most of the tools used in FOCUS PDCA do not involve complex statistical analysis. Many are intuitive and don’t require deep analytical skills. This means almost everyone can serve on the team without worrying about whether or not they have the necessary skills. A good attitude, an openness to collaborating, and being open to change are the primary skills your team members will need. 

It provides a framework 

This approach provides a simple 9-step framework and guidelines for consistently addressing and resolving process problems.

Why is FOCUS PDCA important to understand? 

As a simple but powerful tool for improvement, your understanding of how to use it will be beneficial both to you as well as your organization. 

It keeps you focused

As the acronym suggests, by focusing on a specific problem and using a focused problem-solving approach, you will get better results and improvements. 

It fosters engagement 

As the CFO of a well known corporation was fond of saying, “The best ideas come from our people.” Understanding and applying FOCUS PDCA will give you the opportunity to engage a wide range of business employees and foster a culture of continuous improvement.

It helps you understand your process 

The use of FOCUS PDCA forces you to gain greater insight and understanding of your process. Knowing what to do — and how to do it better — will make your organization better able to satisfy your customers. 

An industry example of FOCUS PDCA 

A large healthcare organization had a run of problems regarding the wrong administration of meds to patients on the hospital floor. They initially used a FMEA to explore the specific problem areas. After identifying the possible source, they formed a team to develop specific recommendations to eliminate the problem.

The team utilized the FOCUS PDCA approach to identify, define, understand, and eventually improve the process. Through the use of technology, they were able to come up with a number of solutions. One was the delivery of meds to the patient floor via robotic carts with a safety mechanism that prevented disbursing the wrong med. They used a signature verification technology to prevent mistakes caused by handwritten scripts. They also implemented better drug labeling to prevent accidental administration of the wrong drug or wrong dosage.

3 best practices when thinking about FOCUS PDCA 

Like any improvement method, there is the right way to use it and a not-so-right way to use it. Here are some suggestions for the right way to do it.

1. Involve the right people

Your team should be selected to take advantage of the most appropriate people for the problem at hand. Don’t necessarily rely on volunteers, but hand-select those you feel have the right skills and knowledge to make maximum contribution to the team. 

2. Avoid coming in with a solution 

The purpose of FOCUS PDCA is to fully understand the nature and root causes of the problem. The solution will result from that deep understanding. Don’t come in with preconceived notions of root causes and solutions. Trust the process of FOCUS PDCA.

3. Communicate  

Frequent communication will prevent many unexpected surprises. Keep in close communication with the appropriate level of management, important stakeholders, and other people involved in the process. 

Frequently Asked Questions (FAQ) about FOCUS PDCA

1. what does focus pdca stand for .

Find, Organize, Clarify, Understand, Select, and then Plan, Do, Check, and Act. 

2. Can FOCUS PDCA be used in any function? 

While FOCUS PDCA was originally developed for application in healthcare, it’s easily adaptable and flexible for solving any problem in any organization. 

3. Is there a difference between FOCUS PDCA and FOCUS PDSA? 

Not really. In the original development of PDCA, the C stood for Check. Dr. W. Edwards Deming revised the acronym a little by substituting Study for Check.

Summing up FOCUS PDCA 

PDCA is a common tool for solving problems and improving processes. The healthcare field expanded the approach by adding five preceding activities they referred to as FOCUS. 

In total, the FOCUS PDCA approach is a powerful yet simple method for addressing a business problem, and through the active involvement of your people, you can identify and implement improvement solutions at your organization.

About the Author

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Ken Feldman

Focus Model

Five levels of Focus Model explained - toolshero

Focus Model: this article provides a practical explanation of the Focus Model . After reading, you’ll have better insight into the various ways in which you can think or communicate about something.

What is the Focus Model?

The five levels of the Focus Model are a variation of David Rock’s ‘Choose Your Focus’ model in his book “Quiet Leadership” . The Focus Model describes five different ways to think or communicate about something. When you are aware of these five levels and realise what the basis of your thoughts or communications is, you can opt to move to a different, more useful level.

The drama level

The least useful thought and communication level is the drama level. The drama level is a level that includes emotion and venting.

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For instance, the majority of one’s energy is spent on repeating: ‘He said, she said, do you know what happened next?’ This is more about the contents of a situation. Although venting can sometimes be beneficial, it’s rarely useful and can hinder you in moving past the emotion to solve the problem.

The problem level

This level focuses on the problem. Both the drama and the details of a situation can contribute to a better understanding of the problem, but it’s important that a conversation (or your thoughts) go beyond the drama and details to better understand the problem or core problem to be solved.

The detail level

The detail level is aimed at the specific characteristics of a situation and the small actions, events or decisions that led to the current situation. Here, details can be useful, depending on how they are used – they are useful when they contribute to understanding a broader problem.

However, when they cause more drama while the emotion around each detail surfaces, the details are merely interesting.

The strategy / planning level

This level is aimed at solving a problem. It is solution-oriented rather than problem or detail-based. Although some detail is required in planning, the focus is higher and more strategic than just the problem itself.

The vision level

The vision level is like looking at the forest (rather than focusing on the trees – which is drama and detail thinking). A vision is what keeps our broader intentions in their place. Why do you do this? Why is this important? What’s the broader goal you want to achieve? When you are guided by the vision, the problems have more context and become less personal. You can then spend these efforts on achieving the broader vision instead of spending your energy on smaller problems.

Five levels of Focus Model - toolshero

Why the Focus Model?

The ‘Choose your focus model’ helps to focus thought processes. It helps to identify your type of thinking at any given moment and offers the possibility to subsequently choose what you wish to focus your attention on. This tool can be useful for any type of conversation, for instance for team meetings or when approaching a difficult thought task.

This model is so easy that it can be applied to any type of conversation. For instance, the author does this by writing it down on paper or a whiteboard, so the concept is visible. The most frequently occurring impact of this model is that people feel lost in details, aren’t clear on what they are trying to achieve or how.

Additionally, the model helps you to recognise the perspective of the thoughts and subsequently lets you choose a different perspective, or has everyone in the conversation speak from the same perspective.

As David Rock says, “Quiet leaders are very disciplined in their conversations. They are diligently focused on ensuring that each conversation is as productive as possible in each step and if not, they fix this. They know it’s important to organise the process of each conversation before diving into the contents of a dialogue .”

Getting started with the Focus Model

You can use the Choose Your Focus Model to consciously aim your thoughts at vision and planning. You can apply the model in various ways, such as:

First write down the focus areas on a flip chart. Explain these at the start of the meeting. Ask whether the participants would find it useful to focus on clarifying the vision and planning the work. It’s also important to ask for permission to interrupt discussions when you notice these have drifted away from the selected focus areas. If you ask for permission first, people will accept that you interrupt them.

When you see discussions that are headed towards details or problems and that will occur at a certain moment, simply point to the flip chart with the five focus areas. This is usually a clear sign for the group. If you must interrupt, ask the question, “What is the focus area in this discussion?”

The Choose Your Focus Model describes five focus level for your thoughts:

Vision Planning Detail — Problem Drama

The essence of this model is that the three levels above the line (vision – planning – detail) are the productive, solution-oriented levels. In various coaching techniques, you go through these three levels when you are looking to achieve something.

You must have a positive vision that serves as the basis for planning how to get there, the basis for details on how to start and what you should do in practice.

The other two bottom levels (Problem – Drama) are the problem-oriented levels, that are very natural to people but aren’t very productive if you wish to maintain your focus to bring about change.

The Choose Your Focus Model can also be effective in focusing your own thoughts. Have you spent sufficient time on problems today? Perhaps it’s time to consider what you truly want to achieve in the greater whole?

The focus areas are:

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problem solving focus model

David Summerton Business Consulting

problem solving focus model

Solving Problems With The FOCUS Model

Solving Problems With The FOCUS Model

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

Nothing stays the way you want it to for very long and perfection is something that is always short-lived. To stay ahead of your competitors, you need to be able to refine your processes on an ongoing basis, so that your products and services remain efficient and, most importantly, your customers remain with you.

The FOCUS model is valuable in problem solving because it uses a team-based approach to business-process improvement which works well for solving cross-departmental process issues. The approach also encourages staff to rely on objective data rather than on personal opinions which increases the quality of the outcome.

The FOCUS model has five steps:

Step 1: Find the Problem

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.

To get the ball rolling and, very importantly, to generate some energy and confidence, try looking at the following issues to stimulate some debate:

Step 2: Organise a Team to deal with 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.

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.

Working on a very big problem or issue can be an attractive process but, due to the size and scale of the topic, positive results can be hard to record relatively quickly. Far better to break down large problems into smaller pieces so that some quick gains can be realized, which will them help increase motivation and application.

Writing a Problem Statement effectively anchors the activity and gives some direction – written as an objective statement.

Key questions to ask in the process are:

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. It is here 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? It is important at this stage to identify any bottlenecks or failures in the process that could be causing problems.

As understanding grows and develops, potential solutions to the problem may become apparent. Beware of jumping to “obvious” conclusions – these could overlook important parts of the problem, potentially creating a whole new process that fails to solve the problem.

Generate as many possible solutions as you can through normal structured thinking combined with a rule that no ideas are out of scope. The richer the mix of potential solutions the better!

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, making sure that they link back to the Problem Statement in Step 1.

Once identified, carefully consider the possible consequences of moving ahead, always remembering that one option might be to leave things as they are currently and wait for more information/clarification on the issue at hand.

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What is Solution-Focused Therapy: 3 Essential Techniques

What is Solution-Focused Therapy: 3 Essential Techniques

You’re at an important business meeting, and you’re there to discuss some problems your company is having with its production.

At the meeting, you explain what’s causing the problems: The widget-producing machine your company uses is getting old and slowing down. The machine is made up of hundreds of small parts that work in concert, and it would be much more expensive to replace each of these old, worn-down parts than to buy a new widget-producing machine.

You are hoping to convey to the other meeting attendees the impact of the problem, and the importance of buying a new widget-producing machine. You give a comprehensive overview of the problem and how it is impacting production.

One meeting attendee asks, “So which part of the machine, exactly, is getting worn down?” Another says, “Please explain in detail how our widget-producing machine works.” Yet another asks, “How does the new machine improve upon each of the components of the machine?” A fourth attendee asks, “Why is it getting worn down? We should discuss how the machine was made in order to fully understand why it is wearing down now.”

You are probably starting to feel frustrated that your colleagues’ questions don’t address the real issue. You might be thinking, “What does it matter how the machine got worn down when buying a new one would fix the problem?” In this scenario, it is much more important to buy a new widget-producing machine than it is to understand why machinery wears down over time.

When we’re seeking solutions, it’s not always helpful to get bogged down in the details. We want results, not a narrative about how or why things became the way they are.

This is the idea behind solution-focused therapy . For many people, it is often more important to find solutions than it is to analyze the problem in great detail. This article will cover what solution-focused therapy is, how it’s applied, and what its limitations are.

Before you continue, we thought you might like to download our three Positive Psychology Exercises for free . These science-based exercises will explore fundamental aspects of positive psychology including strengths, values, and self-compassion, and will give you the tools to enhance the wellbeing of your clients, students, or employees.

This Article Contains:

What is solution-focused therapy, theory behind the solution-focused approach, solution-focused model, popular techniques and interventions, sfbt treatment plan: an example, technologies to execute an sfbt treatment plan (incl. quenza), limitations of sfbt counseling, what does sfbt have to do with positive psychology, a take-home message.

Solution-focused therapy, also called solution-focused brief therapy (SFBT), is a type of therapy that places far more importance on discussing solutions than problems (Berg, n.d.). Of course, you must discuss the problem to find a solution, but beyond understanding what the problem is and deciding how to address it, solution-focused therapy will not dwell on every detail of the problem you are experiencing.

Solution-focused brief therapy doesn’t require a deep dive into your childhood and the ways in which your past has influenced your present. Instead, it will root your sessions firmly in the present while working toward a future in which your current problems have less of an impact on your life (Iveson, 2002).

This solution-centric form of therapy grew out of the field of family therapy in the 1980s. Creators Steve de Shazer and Insoo Kim Berg noticed that most therapy sessions were spent discussing symptoms, issues, and problems.

De Shazer and Berg saw an opportunity for quicker relief from negative symptoms in a new form of therapy that emphasized quick, specific problem-solving rather than an ongoing discussion of the problem itself.

The word “brief” in solution-focused brief therapy is key. The goal of SFBT is to find and implement a solution to the problem or problems as soon as possible to minimize time spent in therapy and, more importantly, time spent struggling or suffering (Antin, 2018).

SFBT is committed to finding realistic, workable solutions for clients as quickly as possible, and the efficacy of this treatment has influenced its spread around the world and use in multiple contexts.

SFBT has been successfully applied in individual, couples, and family therapy. The problems it can address are wide-ranging, from the normal stressors of life to high-impact life events.

The only realm in which SFBT is generally not recommended is that of the more extreme mental health issues, such as schizophrenia or major depressive disorder (Antin, 2018).

The solution-focused approach of SFBT is founded in de Shazer and Berg’s idea that the solutions to one’s problems are typically found in the “exceptions” to the problem, meaning the times when the problem is not actively affecting the individual (Iveson, 2002).

This approach is a logical one—to find a lasting solution to a problem, it is rational to look first at those times in which the problem lacks its usual potency.

For example, if a client is struggling with excruciating shyness, but typically has no trouble speaking to his or her coworkers, a solution-focused therapist would target the client’s interactions at work as an exception to the client’s usual shyness. Once the client and therapist have discovered an exception, they will work as a team to find out how the exception is different from the client’s usual experiences with the problem.

The therapist will help the client formulate a solution based on what sets the exception scenario apart, and aid the client in setting goals and implementing the solution.

You may have noticed that this type of therapy relies heavily on the therapist and client working together. Indeed, SFBT works on the assumption that every individual has at least some level of motivation to address their problem or problems and to find solutions that improve their quality of life .

This motivation on the part of the client is an essential piece of the model that drives SFBT (Miller & Rollnick, 2013).

Solution-Focused Therapy change

Solution-focused theorists and therapists believe that generally, people develop default problem patterns based on their experiences, as well as default solution patterns.

These patterns dictate an individual’s usual way of experiencing a problem and his or her usual way of coping with problems (Focus on Solutions, 2013).

The solution-focused model holds that focusing only on problems is not an effective way of solving them. Instead, SFBT targets clients’ default solution patterns, evaluates them for efficacy, and modifies or replaces them with problem-solving approaches that work (Focus on Solutions, 2013).

In addition to this foundational belief, the SFBT model is based on the following assumptions:

Based on these assumptions, the model instructs therapists to do the following in their sessions with clients:

SFBT therapists aim to bring out the skills, strengths, and abilities that clients already possess rather than attempting to build new competencies from scratch. This assumption of a client’s competence is one of the reasons this therapy can be administered in a short timeframe—it is much quicker to harness the resources clients already have than to create and nurture new resources.

Beyond these basic activities, there are many techniques and exercises used in SFBT to promote problem-solving and enhance clients’ ability to work through their own problems.

asking questions solution-focused therapy

Working with a therapist is generally recommended when you are facing overwhelming or particularly difficult problems, but not all problems require a licensed professional to solve.

For each technique listed below, it will be noted if it can be used as a standalone technique.

Asking good questions is vital in any form of therapy, but SFBT formalized this practice into a technique that specifies a certain set of questions intended to provoke thinking and discussion about goal-setting and problem-solving.

One such question is the “coping question.” This question is intended to help clients recognize their own resiliency and identify some of the ways in which they already cope with their problems effectively.

There are many ways to phrase this sort of question, but generally, a coping question is worded something like, “How do you manage, in the face of such difficulty, to fulfill your daily obligations?” (Antin, 2018).

Another type of question common in SFBT is the “miracle question.” The miracle question encourages clients to imagine a future in which their problems are no longer affecting their lives. Imagining this desired future will help clients see a path forward, both allowing them to believe in the possibility of this future and helping them to identify concrete steps they can take to make it happen.

This question is generally asked in the following manner: “Imagine that a miracle has occurred. This problem you are struggling with is suddenly absent from your life. What does your life look like without this problem?” (Antin, 2018).

If the miracle question is unlikely to work, or if the client is having trouble imagining this miracle future, the SFBT therapist can use “best hopes” questions instead. The client’s answers to these questions will help establish what the client is hoping to achieve and help him or her set realistic and achievable goals.

The “best hopes” questions can include the following:

To identify the exceptions to the problems plaguing clients, therapists will ask “exception questions.” These are questions that ask about clients’ experiences both with and without their problems. This helps to distinguish between circumstances in which the problems are most active and the circumstances in which the problems either hold no power or have diminished power over clients’ moods or thoughts.

Exception questions can include:

Another question frequently used by SFBT practitioners is the “scaling question.”

It asks clients to rate their experiences (such as how their problems are currently affecting them, how confident they are in their treatment, and how they think the treatment is progressing) on a scale from 0 (lowest) to 10 (highest). This helps the therapist to gauge progress and learn more about clients’ motivation and confidence in finding a solution.

For example, an SFBT therapist may ask, “On a scale from 0 to 10, how would you rate your progress in finding and implementing a solution to your problem?” (Antin, 2018).

Do One Thing Different

This exercise can be completed individually, but the handout may need to be modified for adult or adolescent users.

This exercise is intended to help the client or individual to learn how to break his or her problem patterns and build strategies to simply make things go better.

The handout breaks the exercise into the following steps (Coffen, n.d.):

Following these eight steps and answering the questions thoughtfully will help people recognize their strengths and resources, identify ways in which they can overcome problems, plan and set goals to address problems, and practice useful skills.

While this handout can be extremely effective for SFBT, it can also be used in other therapies or circumstances.

To see this handout and download it for you or your clients, click here .

Presupposing Change

one thing different solution-focused therapy

The “presupposing change” technique has great potential in SFBT, in part because when people are experiencing problems, they have a tendency to focus on the problems and ignore the positive changes in their life.

It can be difficult to recognize the good things happening in your life when you are struggling with a painful or particularly troublesome problem.

This technique is intended to help clients be attentive to the positive things in their lives, no matter how small or seemingly insignificant. Any positive change or tiny step of progress should be noted, so clients can both celebrate their wins and draw from past wins to facilitate future wins.

Presupposing change is a strikingly simple technique to use: Ask questions that assume positive changes. This can include questions like, “What’s different or better since I saw you last time?”

If clients are struggling to come up with evidence of positive change or are convinced that there has been no positive change, the therapist can ask questions that encourage clients to think about their abilities to effectively cope with problems, like, How come things aren’t worse for you? What stopped total disaster from occurring? How did you avoid falling apart? (Australian Institute of Professional Counsellors, 2009).

The most powerful word in the Solution Focused Brief Therapy vocabulary – The Solution Focused Universe

A typical treatment plan in SFBT will include several factors relevant to the treatment, including:

All of these are common and important components of a successful treatment plan. Some of these components (e.g., diagnosis and medications) may be unaddressed or acknowledged only as a formality in SFBT due to its usual focus on less severe mental health issues. Others are vital to treatment progress and potential success in SFBT, including goals, objectives, measurement criteria, and client strengths.

Quenza Problem-Solving Exercise

To this end, therapists are increasingly leveraging the benefits of technology to help develop, execute, and evaluate the outcomes of treatment plans efficiently.

Among these technologies are many digital platforms that therapists can use to carry out some steps in clients’ treatment plans outside of face-to-face sessions.

For example, by adopting a versatile blended care platform such as Quenza , an SFBT practitioner may carry out some of the initial steps in the assessment/diagnosis phase of a treatment plan, such as by inviting the client to complete a digital diagnostic questionnaire.

Likewise, the therapist may use the platform to send digital activities to the client’s smartphone, such as an end-of-day reflection inviting the client to recount their application of the ‘Do One Thing Different’ technique to overcome a problem.

These are just a few ideas for how you might use a customizable blended care tool such as Quenza to help carry out several of the steps in an SFBT treatment plan.

Empathy solution-focused therapy

Some of the potential disadvantages for therapists include (George, 2010):

Some of the potential limitations for clients include (Antin, 2018):

Generally, SFBT can be an excellent treatment for many of the common stressors people experience in their lives, but it may be inappropriate if clients want to concentrate more on their symptoms and how they got to where they are today. As noted earlier, it is also generally not appropriate for clients with major mental health disorders.

First, both SFBT and positive psychology share a focus on the positive—on what people already have going for them and on what actions they can take. While problems are discussed and considered in SFBT, most of the time and energy is spent on discussing, thinking about, and researching what is already good, effective, and successful.

Second, both SFBT and positive psychology consider the individual to be his or her own best advocate, the source of information on his or her problems and potential solutions, and the architect of his or her own treatment and life success. The individual is considered competent, able, and “enough” in both SFBT and positive psychology.

This assumption of the inherent competence of individuals has run both subfields into murky waters and provoked criticism, particularly when systemic and societal factors are considered. While no respectable psychologist would disagree that an individual is generally in control of his or her own actions and, therefore, future, there is considerable debate about what level of influence other factors have on an individual’s life.

While many of these criticisms are valid and bring up important points for discussion, we won’t dive too deep into them in this piece. Suffice it to say that both SFBT and positive psychology have important places in the field of psychology and, like any subfield, may not apply to everyone and to all circumstances.

However, when they do apply, they are both capable of producing positive, lasting, and life-changing results.

Solution-focused therapy puts problem-solving at the forefront of the conversation and can be particularly useful for clients who aren’t suffering from major mental health issues and need help solving a particular problem (or problems). Rather than spending years in therapy, SFBT allows such clients to find solutions and get results quickly.

Have you ever tried Solution-Focused Brief Therapy, as a therapist or as a client? What did you think of the focus on solutions? Do you think SFBT misses anything important by taking the spotlight off the client’s problem(s)? Let us know in the comments section.

We hope you enjoyed reading this article. Don’t forget to download our three Positive Psychology Exercises for free .

Antin, L. (2018). Solution-focused brief therapy (SFBT). Good Therapy. Retrieved from

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Article feedback

What our readers think.


Thank you. I’m about to start an MMFT internship, and SFBT is the model I prefer. You put everything in perspective.


Great insights. I have a client who has become a bit disengaged with our work together. This gives me a really helpful new approach for our upcoming sessions. He’s very focused on the problem and wanting a “quick fix.” This might at least get us on that path. Thank you!


Hi Courtney, great paper! I will like to know more about the limitations to SFT and noticed that you provided an intext citation to Antin 2016. Would you be able to provide the full reference? Thank you!

Nicole Celestine

Thank you for bringing this to our attention. The reference has now been updated in the reference list — this should be Antin (2018):

– Nicole | Community Manager

Randy H.

The only thing tat was revealed to me while reading this article is the client being able to recognize the downfall of what got them into their problem in the first place. I felt that maybe a person should understand the problem to the extent that they may understand how to recognize what led to the problem in the first place. Understanding the process of how something broke down would give one knowledge and wisdom that may be able to be applied in future instances when something may go wrong again. Even if the thing is new (machine or person) having the wisdom and understanding of the cause that led to the effect may help prevent and or overcome an arising problem in the future. Not being able to recognize the process that brought down the machine and or human may be like adhering to ignorance, although they say ignorance is bliss in case of an emergency it would be better to be informed rather then blindly ignorant, as the knowledge of how the problem surfaced in the first place may alleviate unwarranted suffering sooner rather than later. But then again looking at it this way I may work myself out of a job if my clients never came back to see me. However is it about me or them or the greater societal structural good that we can induce through our education, skills, training, experience, and good will good faith effort to instill social justice coupled with lasting change for the betterment of human society and the world as a whole.

Matthew McMahon

Very very helpful, thank you for writing. Just one point “While no respectable psychologist would disagree that an individual is generally in control of his or her own actions and, therefore, future, there is considerable debate about what level of influence other factors have on an individual’s life.” I think any psychologist that has worked in neurological dysfunction would probably acknowledge consciousness and ‘voluntary control’ are not that straight-forward. Generally though, I suppose there’s that whole debate of if we are ever in control of our actions or even our thoughts. It may well boil down to what we mean by ‘we’, as in what are we? A bundle of fibres acting on memories and impulses? A unique body of energy guided by intangible forces? Maybe I am not a respectable psychologist 🙂


This article provided me with insight on how to proceed with a role-play session in my CBT graduate course. Thank you!

Hi Derrick, That’s fantastic that you were able to find some guidance in this post. Best of luck with your grad students! – Nicole | Community Manager

Fisokuhle Thwala

Thank You…Great input and clarity . I now have light…


I was looking everywhere for a simple explanation for my essay and this is it!! thank you so much for this is was very useful and I learned a lot.

Penelope Wauterz

Very well done. Thank you for the multitude of insights.

Will My Marriage Last

Thank you for such a good passage discussed. I really have a great time understanding it.

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FOCUS PDCA is a management method, developed in the healthcare industry, used to improve processes. Created by the Hospital Corporation of America (HCA), it is a systematic process improvement method. Through FOCUS PDCA, a knowledge of how a process is currently performing to meet customer needs and expectations is used to plan and test process changes. FOCUS PDCA is an extension of the Deming or Shewhart Cycle which includes Plan-Do-Check-Act. The HCA FOCUS piece precedes the PDCA with a five-part plan:

Although created for the medical industry , FOCUS PDCA can be used by any type of organization. The FOCUS portion identifies the area needing improvement, brings together a team that is capable of achieving improvement, and selects a solution. The PDCA portion identifies what needs to be done to implement the solution, makes the necessary changes, and verifies that the desired result has been achieved.


The FOCUS PDCA method works well in a number of situations, and in particular when there is a highly technical or complex work environment or task. The advantages include:

Changes and success are based on measurable data and observations.

The Steps Defined by FOCUS PDCA

The FOCUS PDCA acronym describes the basic components of the improvement process. These are:

FIND a Process to Improve

In some cases, the problem may be obvious. A process may not be meeting quality goals, or environmental emissions may be too high. However, in other cases, there may not be a readily apparent need for improvement. For example, a Failure Mode and Effect Analysis may reveal a previously unknown problem. The guiding principle of FOCUS PDCA should be to bring all processes fully into alignment with the strategic goals of the organization.


The team should be composed of people who understand the process, but they do not need to be experts. The team should include those who are closest to the process, such as those who operate and maintain it. It is also worthwhile to include those who provide inputs to the process, and those who receive its outputs — that is, the people “upstream” and “downstream” of the process.

CLARIFY the Current Understanding of the Process

Collect data and information about the process. This may be done through physical measurements, but it should also include talking with those who are involved with the process. Get the answers to the six basic questions: who, what, where, when, why, and how.

UNDERSTAND Variation in the Process

Every process includes variation, and some variation is usually acceptable. When variation interrupts the normal flow of work, it may be a symptom of problems in the process, and often causes other problems as well. In this step, the question “why” is asked to find the cause of variation. This is sometimes called “Five Whys” — the general principle is to ask “why” a sufficient number of times (often five) to identify the real cause of the variation.

SELECT a Strategy for Improvement

This step may involve brainstorming or other methods of developing ideas. Then, based on what was learned in the previous two steps, the best solution is selected. Keep in mind that the solution must be in alignment with the overall organization's strategy, it must add value for the final customer, and it must be both technically and financially feasible.

The final four steps are PDCA, a modified version of the Deming Cycle . It is a continuous process used to improve a process or system.

In a traditional PDCA the “Plan” step is where the problem is defined and a solution developed. In FOCUS PDCA, that has already been accomplished. Instead, the “Plan” step is where the implementation of the solution begins. Based on the solution identified in the final step of FOCUS, a plan for implementing that solution is developed. The plan specifies what will be done, how it will be done, when it will be done, and who is responsible for each task needed to complete the necessary changes. In addition, a means for data collection and measuring success is established.

If it was not done as a part of FOCUS, data must be collected to characterize the condition of the process before changes are made. Then the required changes are made — that is, the plan is implemented.

Did the changes have the desired result? Data is collected, the process is observed, and the changes are evaluated by comparing the actual results with the desired results. If the results are not as desired, then the previous steps are reviewed. This includes going back to the beginning to be sure the problem definition was correct, and the root cause was accurately identified.

The final step is to take the steps necessary to maintain the improvement. For example, the changes should be incorporated into the standards governing the process. In addition, a means of monitoring the process or system should be established so that variations from the new normal can be readily detected.

The purpose of FOCUS PDCA is to provide a structure that guides the process of problem-solving and process improvement. By using this approach, a comprehensive analysis, response, action plan, and feedback loop is established to ensure success.

Keeping people safe during the FOCUS PDCA process, and after changes have been made, is a top priority. In many cases, this means identifying safe areas and using tools such as floor marking to keep people separate from moving machinery and vehicles. You can learn more about effective, durable floor marking with a free Best Practice Guide To Floor Marking .

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Start implementing Kaizen in your facility using the Plan-Do-Check-Act cycle and achieve continual improvement.

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Facts, Optimism, Cope, Understanding, Solve (FOCUS)

Man in hospital

FOCUS was developed to assist individuals who assume caregiver roles when family members acquire a sudden and severe disability (e.g., spinal cord injury, brain injury) in which personal, social and vocational roles are disrupted.

By using the components of the FOCUS acronym, the model is designed to assist caregivers in prioritizing the many problems that typically occur in the wake of acquired disability

FOCUS Approach to Problem Solving

Facts/Problem Definition The objective is to clearly state the problem and break it into manageable parts. Present facts in unambiguous, concrete terms, separate from assumptions, and differentiate relevant from irrelevant information. Seek all available facts to answer who, what, when, where, why, and how of the situation. Conduct thoughtful assessment of caregiver problems and use procedures to assist in prioritizing their importance.

Optimism/Problem Orientation The objective is to develop a sense of optimism regarding problem-solving ability. This includes instilling a belief that one is sufficiently skilled to solve problems, as well as instilling a sense of motivation to engage in problem-solving process while simultaneously regulating emotional experiences to maintain a sense of confidence. Normalize the experience of multiple problems, decisions, and challenges associated with disability. Help caregiver understand that these can be addressed with a systematic problem-solving approach.

Creativity/Generation of Alternative Solutions The objective is to actively brainstorm multiple solutions to the problem of highest priority. Caregivers are instructed to think of multiple solutions and write each down. It is crucial that judgments be withheld at this point in the process.

Understanding/Decision-making The objective is to outline the process needed to make an informed, wise, and appropriate choice that maximizes the probability of a positive outcome. Making a decision about the “best” strategy to try and solve the problem requires a thoughtful consideration of gains and benefits of the best available strategies.

Solve/Implementation & Verification The objective is solving the problem and then systematically reviewing the outcome to determine how the solution worked and the degree to which the actual outcome approximates the expected one. This self-monitoring is crucial for learning what made a solution effective or ineffective and how to implement a similar solution or use a different solution in the future

Target Population

Caregivers of persons who acquire sudden and severe disabilities

Research Outcomes

FOCUS has been applied successfully through face-to-face, telephone and videoconferencing sessions

Outcomes Research References

Berry, J. W., Elliott, T., Grant, J., Edwards, G., & Fine, P. R. (2012). Does problem solving training for family caregivers benefit care recipients with severe disabilities? A latent growth model of the Project CLUES randomized clinical trial. Rehabilitation Psychology , 57, 98-112.

Elliott, T., & Berry, J. W. (2009). Brief problem-solving training for family caregivers of persons with recent-onset spinal cord injury: A randomized controlled trial. Journal of Clinical Psychology , 65, 406-422

Elliott, T., Berry, J. W., & Grant, J. S. (2009). Problem-solving training for family caregivers of women with disabilities: A randomized clinical trial. Behaviour Research and Therapy , 47, 548-558

Elliott, T., Brossart, D., Berry, J. W., & Fine, P. R. (2008). Problem-solving training via videoconferencing for family caregivers of persons with spinal cord injuries: a randomized controlled trial. Behaviour Research and Therapy , 46, 1220–1229.

Pfeiffer, K., Beische, D., Hautzinger, M., Berry, J. W., Wengert, J., Hoffrichter, R., Becker, C., van Schayck, R., & Elliott, T. (2014). Telephone-based problem-solving intervention for family caregivers of stroke survivors: A randomized controlled trial. Journal of Consulting and Clinical Psychology , 82, 628-643.

Rivera, P., Elliott, T., Berry, J., & Grant, J. (2008). Problem-solving training for family caregivers of persons with traumatic brain injuries: a randomized controlled trial. Archives of Physical Medicine and Rehabilitation , 89, 931–941.

Clinical Approach References

Kurylo, M., Elliott, T., & Shewchuk, R. (2001). FOCUS on the family caregiver: a problem-solving training intervention. Journal of Counseling and Development , 79, 275–281.

Rivera, P., Shewchuk, R., & Elliott, T. (2003). Project FOCUS: Using videoconferencing to provide problem solving training to family caregivers of persons with spinal cord injuries. Topics in Spinal Cord Injury Rehabilitation , 9(1), 53-62

Intervention Issues

In the Practice Section

In the Caregiver Briefcase

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I blog on Brain-Based Learning, Metacognition, EdTech, and Social-Emotional Learning. I am the author of the Crush School Series of Books , which help students understand how their brains process information and learn. I also wrote The Power of Three: How to Simplify Your Life to Amplify Your Personal and Professional Success , but be warned that it's meant for adults who want to thrive and are comfortable with four letter words.

The   Focus Method: Problem Solving Approach

The Focus Method Problem Solving Approach is a tool designed to help students improve their problem solving skills. When students experience a problem in class or at school they often do not know the ways they can solve them.  If they have a difficult time understanding a topic or mastering a skill they tend to get stuck on the problem, and not knowing what to do, they do not progress in their learning. This problem solving approach is a straight forward way of guiding students to a solution by empowering them to figure out the steps and actions they can take to be successful. The beauty of this method is that it works in school as well as life and can become a lifelong skill after it has been used just a handful of times.

DOWNLOAD t he Focus Method Problem Solving Approach for FREE below. Enjoy! 

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8D Related Topics

What are the Eight Disciplines (8D)?

Quality Glossary Definition: Eight disciplines (8D) model

The eight disciplines (8D) model is a problem solving approach typically employed by quality engineers or other professionals, and is most commonly used by the automotive industry but has also been successfully applied in healthcare, retail, finance, government, and manufacturing. The purpose of the 8D methodology is to identify, correct, and eliminate recurring problems, making it useful in product and process improvement.

The 8D problem solving model establishes a permanent corrective action based on statistical analysis of the problem and focuses on the origin of the problem by determining its root causes. Although it originally comprised eight stages, or disciplines, the eight disciplines system was later augmented by an initial planning stage.

How to Use the 8D Approach

Eight Disciplines (8D)

List of the Eight Disciplines (8D)

You can also search articles , case studies , and publications  for 8D resources.

Introduction To 8D Problem Solving

A Disciplined Approach ( Quality Progress ) Nothing causes anxiety for a team quite like the release of a corrective action preventive action (CAPA) system and accompanying eight disciplines (8D) model. Follow this step-by-step explanation of 8D to reassure your team and get results.

In the Loop ( Quality Progress ) An 8D report is a quality report suppliers use to inform a customer about the status of complaint-related actions. Use this refresher to help track the status of customer complaints.

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problem solving focus model

The Six Step Problem Solving Model

Problem solving models are used to address the many challenges that arise in the workplace. While many people regularly solve problems, there are a range of different approaches that can be used to find a solution.

Complex challenges for teams, working groups and boards etc., are usually solved more quickly by using a shared, collaborative, and systematic approach to problem solving.

Advantages of Six-Step Problem Solving

The Six-Step method provides a focused procedure for the problem solving (PS) group.

All six steps are followed in order – as a cycle, beginning with “1. Identify the Problem.” Each step must be completed before moving on to the next step.

The steps are repeatable. At any point the group can return to an earlier step, and proceed from there. For example, once the real problem is identified – using “2. Determine the Root Cause(s) of the Problem”, the group may return to the first step to redefine the problem.

The Six Steps

The process is one of continuous improvement. The goal is not to solve but to evolve, adjusting the solution continually as new challenges emerge, through repeating the Six Step Process.

Step One: Define the Problem

Step One is about diagnosing the problem – the context, background and symptoms of the issue. Once the group has a clear grasp of what the problem is, they investigate the wider symptoms to discover the implications of the problem, who it affects, and how urgent/important it is to resolve the symptoms.

At this stage groups will use techniques such as:

As this step continues, the PS group will constantly revise the definition of the problem. As more symptoms are found, it clarifies what the real problem is.

Step Two: Determine the Root Cause(s) of the Problem

Once all the symptoms are found and the problem diagnosed and an initial definition agreed, the PS group begins to explore what has caused the problem. In this step the problem solving team will use tools such as:

These techniques help collate the information in a structured way, and focus in on the underlying causes of the problem. This is called the root cause.

At this stage, the group may return to step one to revise the definition of the problem.

Step Three: Develop Alternative Solutions

Analytical, creative problem solving is about creating a variety of solutions, not just one. Often the most obvious answer is not the most effective solution to the problem. The PS group focuses on:

At this stage it is not about finding one solution, but eliminating the options that will prove less effective at dealing with both the symptoms and the root cause.

Step Four: Select a Solution

In the fourth step, groups evaluate all the selected, potential solutions, and narrow it down to one. This step applies two key questions.

Feasibility is ascertained by deciding if a solution:

Which solution is favoured?

Acceptance by the people who will use and implement the solution is key to success.

This is where the previous steps come into play. To users and implementers, a solution may seem too radical, complex or unrealistic. The previous two steps help justify the choices made by the PS group, and offer a series of different, viable solutions for users and implementers to discuss and select from.

Step Five: Implement the Solution

Once the solution has been chosen, initial project planning begins and establishes:

The group may use tools, such as a Gantt chart, timeline or log frame. Between Steps Five and during Step Six the operational/technical implementation of the chosen solution takes place.

Step Six: Evaluate the Outcome

The project implementation now needs to be monitored by the group to ensure their recommendations are followed. Monitoring includes checking:

Many working groups skip Step Six as they believe that the project itself will cover the issues above, but this often results in the desired outcome not being achieved.

Effective groups designate feedback mechanisms to detect if the project is going off course. They also ensure the project is not introducing new problems. This step relies on:

In Step Six, as the results of the project emerge, evaluation helps the group decide if they need to return to a previous step or continue with the implementation. Once the solution goes live, the PS group should continue to monitor the solutions progress, and be prepared to re-initiate the Six Step process when it is required.

Overall, the Six Step method is a simple and reliable way to solve a problem. Using a creative, analytical approach to problem solving is an intuitive and reliable process.

It helps keep groups on track, and enables a thorough investigation of the problem and solution search. It involves implementers and users, and finds a justifiable, monitorable solution based on data.

You can read more about the Six-Step Problem Solving Model in our free eBook ‘ Top 5 Problem Solving Tools ’. Download it now for your PC, Mac, laptop, tablet, Kindle, eBook reader or Smartphone.

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Solution-Focused Approach to Coaching: Questions, Interventions, Techniques [+ Example Session]

Ayisha Amatullah Coaching Models

The standard approach to problem-solving insists there is a cause and effect between problems and solutions. However, the Solutions-Focused approach to coaching skips over the continuous delving and search for what causes problems and goes straight for the solution. It focuses on solutions, not problems, strengths, not weaknesses, and on what’s going well, rather than what’s gone wrong. 

In This Post

What is the solution-focused approach?

Key concepts of the solution-focused approach, solution-focused questions and interventions, solution-focused questions using the oskar coaching model, solution-focused example coaching session.

Solution-focused is a present and future-forward approach to helping individuals reach a goal or solve a problem without focusing on the problem.  

Solution-Focused is a proven and practical approach to positive change within people, relationships, and organizations. It goes against the standard method of solving problems by focusing solely on the solution. 

Problems are often looked at as challenges. And when we think of challenges, we often believe that they are challenging to overcome. The solution-focused points out the solutions, the skills, strengths, and resources in people, which motivates them to accomplish their goals. 

Solution-focused is an evidence-based approach that stems from solution-focused therapy. Solution-focused therapy was founded in the late 1970s by sociologists Steve de Shazerv and Insoo Kim Berg in collaboration with their colleagues at the Milwaukee Brief Family Therapy Center. 

Solution-Focused Coaching

Solution-focused then moved on to be used as a coaching tool in organizations for workplace problems and to deal with executives, teams, and people. 

In recent years, it has been used in life coaching. Ayisha Amatullah, the founder of Universal Coach Institute, was one of the first to use and teach solution-focused for life coaching and developed a solution-focused life coaching model. 

Solution-focused focuses on the following:

The more the positives are discussed, the easier the situation appears, and the more motivated the individual will be to move forward.

When to use the solution-focused approach in coaching?

Solution-focused can be used in many types of situations. However, the best times to use the solution-focused approach in coaching are when an individual:

Benefits of using the solution-focused approach in coaching

Important concepts the coach and individual should live by when arriving at solutions include: 

In the solution-focused approach to coaching, questions are asked in a way to shift the client’s attention away from the stressful problem towards the solution. Instead of asking questions that emphasize the problems, difficulties, and causes, the coach asks questions that explore the individual’s goals, exceptions that have led to success in the past, questions about times when the problem was less severe, the ideal outcome, and questions about existing resources.

Problem-Focused Question: 

Can you tell me about the problem?

Solution-Focused Questions: 

What do you want to change?

solution focused questions interventions techniques

Counter Finding

In the solution-focused conversation, the coach or the helper looks for anything that counts, called “counters.” Counters are the resources that are already present and are likely helpful in helping the individual find solutions. 

Counters include: 

The goal is to collect as many counters as possible to help search for what works.

Examples of Counter Finding Questions:

Past Success Questions

Coping Questions

Reframing Questions

Often when we want to solve a problem, we indulge in problem-talk. We complain and talk badly about the problem. In solution-focused, we use reframing questions to get the individual to look at the situation in another way and engage them in problem-free talk. 

Future Perfect

The “future perfect” is a technique used to help the individual describe how they would like the situation to be. On the surface, it may appear that the goal of the “future perfect” is to create hope through visualization. However, the future perfect technique has a deeper purpose. A trained professional will listen to the individual’s “future perfect” and be able to find counters, exceptions, possible solutions, and even a possible action plan.  

The most popular solution-focused future perfect intervention is the Miracle Question. The Miracle Question is a method of questioning used to aid an individual in envisioning how the future will be different when the problem is no longer present. 

Miracle Question

“Suppose tonight you go to bed and go to sleep as usual. And during the night, a miracle happens. And the problem vanishes. And the issues that concern you are resolved, but you’re still asleep. Therefore, you don’t know that the miracle has happened. When you wake up tomorrow, what will be the first things that will tell you that the miracle has happened? How will you know that the transformation has occurred?”

Recommended Reading: The Miracle Question with Examples, Worksheets, Exercises, & Demo Video

Scaling invites an individual to measure and track their progress and experience. Scaling in Solution-Focused is used to identify what the client already has working for them in reference to the future perfect. 

Exception Questions

When an individual has a problem, that problem is not present all the time. Most problems are only happening occasionally. There are times when the problem is not happening at all or is happening to a lesser degree. Helping an individual to notice these times can help reduce the feeling of being overwhelmed by the problem and help identify things they or others are already doing to solve the problem.

The goal is for the client to repeat what has worked in the past and help them gain confidence in making improvements for the future. 

Example Exception Questions:

Affirming is about providing positive feedback of what the coach or helping professional heard. The professional will reflect and repeat back the counters, possible solutions, exceptions, strengths, and attributes the individual has revealed during the session to help the individual make a decision on where to start to move towards the solution. 

OSKAR is a well-known solution-focused coaching model. It was created by Mark McKergow and Paul Z. Jackson and published in their book: The Solutions Focus: Making Coaching and Change SIMPLE .

[Full Disclosure: As an affiliate, I receive a commission if you purchase this book on Amazon once you click the link] 

OSKAR stands for Outcome, Scaling, Know-How, Affirm, and Review.

problem solving focus model

Know-How & Resources

Affirm & Action

Review: What’s better?

If you would like to learn how to use the miracle question in coaching check out the Life and Solution-Focused Coach Training program .

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Ayisha Amatullah

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Is Your Team Solving Problems, or Just Identifying Them?

problem solving focus model

Eight ways to encourage your team to think more critically.

Some teams are really good at identifying problems. When colleagues propose new ideas, team members readily ask tough questions and point out risks. But they ought to be providing constructive feedback as well. How can you encourage team members to think more creatively about solving problems? For starters, they need to see you doing it. Be a role model. Say: “We’re going to talk about solutions; I don’t want to hear about obstacles just yet. And I am going to get us started.” Ask others to contribute to the conversation. Be disarming. Make sure they know their ideas need not be perfect. When you encounter skepticism, ask probing questions. What could we do differently? How could risks be mitigated? Simple things like creating a trigger word to remind employees to be solutions-oriented can make a big difference. That way, if the conversation veers off course, colleagues can help get it back on track.

Some teams are really good at spotting potential problems. When colleagues present new ideas or propose new initiatives, team members readily ask tough questions and point out possible risks. But team members ought to provide constructive feedback as well. How can you, the manager, help change the culture on your team from one that’s focused on identifying problems to one that fixes them? How can you set new norms that engender a positive tone? And what’s the best way to reward employees for thinking critically while also making helpful suggestions?

What the Experts Say

Having a team that’s quick to identify problems and voice potential obstacles is not necessarily a bad thing. “Intellectually honest resistance” to a new idea is worth airing, according to Liane Davey, professional speaker and author of the book The Good Fight . But when your team is overly focused on finding problems instead of solving them, it can be detrimental to productivity and morale. “Talent is attracted to possibility, opportunity, and agency,” she says. “You will lose great people if your team is always talking about why it can’t , rather than about how it can .” And yet, says Heidi Grant, social psychologist and author of the book  Reinforcements: How to Get People to Help You , the best teams balance the two. As the manager, your job is to “create an environment that allows for both creativity and analytical thinking” in order to come up with solutions that are informed by reality. Here’s how.

Recognize underlying issues.

For starters, you need to appreciate that your team’s tendencies are not unusual. There are several deep-rooted dynamics at work, according to Grant. When faced with a new challenge or idea, many of us react by “getting into the details and focusing on obstacles,” she says. “We ruminate on the problem and its many facets rather than thinking of ways around it.” This predisposition “gets compounded when we work with other people — there’s a social element” that often exacerbates a group’s inclination to think in negative terms. This social aspect is more or less evident depending on the personalities that compose your team.

Hierarchy also plays a role. “Managers and people in power think about the ‘why’ — the vision,” Grant says. “The less power you have, the more you tend to think about the details.” (Perhaps it’s because those people are often the ones who need to deal with the nitty-gritty in the execution stage.) Understanding these dynamics will help you map out the process of changing your team’s culture.

Reflect on your goal.

You need to be clear about the changes you’re looking for from your team. “You want your team to be more ‘solutions-focused,’ which is a bit like saying you want your team to be more innovative or more agile,” says Grant. Many managers aspire to those things, “but it’s not obvious how to get from here to there.” Consider how your team currently responds to new ideas and proposals. What, or who, are the sources of opposition? Where does your team get stuck? Which details cause the most agita? Then, think about what you’d like your team to do differently. This will help you define the specific behaviors you seek.

Reflect, too, on why you wish to change your team’s culture, says Davey. “As a leader, you need to make sure you’re devoting time and energy to things on the horizon and the bigger picture,” she says. “You can’t spend all of your time on today. You need to keep time and mindshare reserved for tomorrow.”

Talk to your team.

Next, Grant recommends talking with your team about your observations and what you’d like to see them do differently. Explain that you want the team to do a better job of “looking for alternate routes,” rather than dwelling on the details of a problem. Ask team members for their take on what stands in the way of that and then listen carefully to how they respond. You might hear, for instance, that team members believe they’re under a lot of time pressure, or perhaps they feel that new ideas aren’t welcome.

Maybe the team fixates on problems because people feel overwhelmed, says Davey. They might resent you asking them to focus on solutions when they’re already overstretched. “They’re thinking, ‘I can’t cope with the status quo, how am I going to manage tomorrow?’” If that’s the case, you need to think about how to “solve the bandwidth question”; otherwise, “you’re not going to get buy-in.” Ask what you can do to help. What tasks can you remove from their plates? “You need to be constantly pruning the workload ,” she says. “Retire old ways of working so that you have room for new ones.”

Set new norms.

Changing your team’s culture requires getting people on board with new ways of thinking and speaking, according to Grant. To accomplish this, you need to set new norms “that deliberately lift up other ways of working.” Norms are powerful because we’re heavily influenced by other people’s behavior, she says. Simple things like “beginning each meeting with a positive reflection ” or creating “a trigger word to remind people to be solutions-oriented” can make a big difference, she says. That way, if the conversation veers off course, colleagues can help get it back on track.

In that spirit, Grant recommends empowering employees to hold others on the team accountable and speak up if someone is “being too problem-focused.” She acknowledges that encouraging employees to call out colleagues will be hard. “It doesn’t come naturally.” But ultimately, it’s worthwhile because “it will help speed up the shift in how people work together.”

Role model.

In order to inspire your team to think more creatively about solving problems, “others need to see you doing it,” says Grant. “You need to put your ideas out there.” Be direct and straightforward. “Say, ‘We’re going to talk about solutions now; I don’t want to hear about obstacles just yet. And I am going to get us started.’” Be disarming. Make sure team members know that their ideas don’t need to be perfect. “When people are afraid of making a mistake or they’re worried about being evaluated negatively, they get risk averse.” The implicit message ought to be: This is a safe place to propose new ideas. Use “your body language, tone, and words to invite others into the conversation.”

Bring in new information.

Davey recommends “using external information to trigger creative conversations.” For instance, at your next team meeting, you might say, “I read an interesting article about a trend in our industry. How do you think this will affect us? What opportunities does this trend create? If this trend continues, what might we need to pay attention to? What hard choices might we need to make?” Asking questions takes the pressure off team members to have specific answers, says Davey. “There’s no need to be prescriptive.” It spurs “people to think about how they respond to how the world is changing,” she says.

Including outside voices can also be effective. Invite a consultant or someone from the accounting or legal department to attend a team brainstorming session, Davey says. “They have data and credibility to contribute” and might spark new strands of conversation.

Deal with challenges productively.

When you encounter resistance to a new idea, it’s important to listen — but also to make sure that team members’ fault-finding does not monopolize the conversation, says Davey. Say, for instance, your colleague discounts a possible new strategy because “the company tried it once decades ago and it didn’t work.” First, you must “validate their feelings and their perspective.” Say something like, “‘You’re concerned that we tried it before, and it wasn’t successful. That’s a good point.’” If you fail to acknowledge your colleague’s objection, “the other person might feel bruised and not heard.”

Second, you need to figure out a way to address the resistance in a productive way. You could either create a so-called “parking lot” where you place concerns (writing them on a white board that you’ll return to later in the meeting, for example). Or, even better, start a dialog to explore possible solutions. “Ask questions to continue the conversation.” Davey suggests: “‘Hypothetically, if we could do it again, what would it look like? How could risks be mitigated? What would we have to solve for?’” The goal, she says, is to combat “lazy cynicism” by ensuring that there’s “fact-based rigor” behind any concerns.

Reward positive behaviors.

When you observe team members seeking to solve problems productively, you need to “publicly affirm that they’re doing the right thing,” says Grant. “New habits don’t form unless they’re rewarded.” Acknowledge great ideas and creative thinking. Be genuine. “Say the positive thing you’re thinking out loud” in order to “increase the sense that norms are shifting.” Other team members will take notice of the boss’s support and approval. “Social affirmation is powerful for changing group behavior.” Davey agrees. “There’s a certain amount of pride” that employees feel when their manager says, “‘This is what we’re looking for.’”

Principles to Remember

Advice in Practice

Case study #1: stimulate new ways of thinking by role modeling and setting solutions-oriented norms..

Kean Graham, CEO of MonetizeMore, a midsize Canadian ad tech company, says that years ago, his team was overly focused on identifying problems, rather than remedying them.

“Team members would bring up issues without any recommendations for how to fix them,” he recalls. “When I tried to get people to think about solutions, people weren’t willing to engage, or they would just give me a list of reasons why an idea wouldn’t work.”

This mindset dented productivity. “It prevented problems from being solved quickly,” he says. “In fact, many would linger, causing much more damage than necessary.”

Kean knew that he had to make a change. First, he reflected on the challenge. He thought about things he wanted the team to do differently and specific behaviors he wanted to see from employees. He adjusted the company’s culture doc to reflect a renewed “focus on being solutions-oriented.”

Then he talked to his team about it. “I told people that we were going to try a new approach and we, as a team, needed a new mentality,” he says. “I said, ‘From now on, we can’t complain about problems without providing a possible solution.’”

Kean knew that he needed to model this new orientation and be ready with fresh ideas and solutions. “It’s important that I’m the best example of the culture we want to exhibit,” he says. “I am naturally a solutions-oriented person, and I’ve made even more of a point to focus on an actionable solution as quickly as possible.”

He also worked on setting new norms and even created a special term to encourage his employees to think differently. For instance, he calls discussing problems without a fix a “dead-end.” And he encourages team members to hold one another accountable. “During meetings, if we hear someone only mention a problem, we remind them to not give us dead-ends,” he says. “We ask instead for that person to suggest a solution so that they take ownership of the process to make sure it gets solved.”

Kean makes sure to publicly acknowledge and appreciate team members’ creativity. “It’s important to be positive, especially when you’re trying to change group behavior,” he says. “A lot of that is just saying, ‘Great idea,’ or, ‘I like where you’re going with that.’”

Eventually, with practice, most employees have shifted their mindset. “Now it’s second nature,” he says. “Our new culture is solutions-oriented, and employees tend to have a high locus of control. They are now proactive about problem solving and feel more empowered to come up with solutions on their own.”  

Case Study #2: Ask probing questions and encourage team members to take ownership of solutions.

Declan Edwards, founder and CEO of BU Coaching, an Australian consulting startup that focuses on employee emotional well-being, says his team could once be described as a group of “people searching for fires but with no tools to put them out.”

“They were great at identifying issues, but they had never been encouraged to solve anything for themselves,” he says. “As a result, they kept bringing all the problems to me and my co-founder. Before we knew it, we were spending more time fighting fires than actually building the company.”

Declan felt burnt out and resentful. “I remember attending a team meeting where there was a whole range of problems being brought forward, and no one was taking responsibility for solving them.”

He realized he needed to change the culture. He wanted his team both to think more creatively and to take more ownership for solving problems. “When people have a hand in creating the solution, they are instantly more invested in making it work.”

To encourage new ways of thinking, Declan made his expectation clear. “I said to the team, ‘For every problem you bring to the table, you must also bring one proposed solution,’” he recalls.

Declan says that he was wary of putting undue pressure on his team. “So, I highlighted that it didn’t have to be a perfect solution, but it had to be something that would at least get the ball rolling.”

At first, employees needed guidance. But over time, they adapted to a new way of thinking and acting. Today, when an employee presents a problem, Declan encourages the team to have a short discussion about it — but he makes sure the conversation never devolves into a complaining session. “Confirming that the problem is real validates people’s perspectives,” he says.

Next, Declan asks a series of probing questions. What needs to be done? What are our options? What opportunities and risks are there? What is your recommendation? What resources do you need? What are the next steps you’re taking to implement this solution?

To spark new ideas, Declan often relies on outside sources. They offer fresh perspectives and new information, he says. “We have a team of consultants and business advisors supporting us, and we regularly use resources such as podcasts, articles, and more formal training programs to ensure our entire team is at the top of our game,” he says.

Today, employees arrive at meetings with solutions and ideas to share. “Culture change takes time, so there are definitely still fires to be put out; however, now it doesn’t feel as if all of that is set on my shoulders,” he says. “It now feels as though we have a united front that is creative, collaborative, and solves problems together.”

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What Is Creative Problem-Solving & Why Is It Important?

Business team using creative problem-solving

One of the biggest hindrances to innovation is complacency—it can be more comfortable to do what you know than venture into the unknown. Business leaders can overcome this barrier by mobilizing creative team members and providing space to innovate.

There are several tools you can use to encourage creativity in the workplace. Creative problem-solving is one of them, which facilitates the development of innovative solutions to difficult problems.

Here’s an overview of creative problem-solving and why it’s important in business.

What Is Creative Problem-Solving?

Research is necessary when solving a problem. But there are situations where a problem’s specific cause is difficult to pinpoint. This can occur when there’s not enough time to narrow down the problem’s source or there are differing opinions about its root cause.

In such cases, you can use creative problem-solving , which allows you to explore potential solutions regardless of whether a problem has been defined.

Creative problem-solving is less structured than other innovation processes and encourages exploring open-ended solutions. It also focuses on developing new perspectives and fostering creativity in the workplace . Its benefits include:

Design Thinking and Innovation | Uncover creative solutions to your business problems | Learn more

Creative problem-solving is traditionally based on the following key principles :

1. Balance Divergent and Convergent Thinking

Creative problem-solving uses two primary tools to find solutions: divergence and convergence. Divergence generates ideas in response to a problem, while convergence narrows them down to a shortlist. It balances these two practices and turns ideas into concrete solutions.

2. Reframe Problems as Questions

By framing problems as questions, you shift from focusing on obstacles to solutions. This provides the freedom to brainstorm potential ideas.

3. Defer Judgment of Ideas

When brainstorming, it can be natural to reject or accept ideas right away. Yet, immediate judgments interfere with the idea generation process. Even ideas that seem implausible can turn into outstanding innovations upon further exploration and development.

4. Focus on "Yes, And" Instead of "No, But"

Using negative words like "no" discourages creative thinking. Instead, use positive language to build and maintain an environment that fosters the development of creative and innovative ideas.

Creative Problem-Solving and Design Thinking

Whereas creative problem-solving facilitates developing innovative ideas through a less structured workflow, design thinking takes a far more organized approach.

Design thinking is a human-centered, solutions-based process that fosters the ideation and development of solutions. In the online course Design Thinking and Innovation , Harvard Business School Dean Srikant Datar leverages a four-phase framework to explain design thinking.

The four stages are:

The four stages of design thinking: clarify, ideate, develop, and implement

Creative problem-solving primarily operates in the ideate phase of design thinking but can be applied to others. This is because design thinking is an iterative process that moves between the stages as ideas are generated and pursued. This is normal and encouraged, as innovation requires exploring multiple ideas.

Creative Problem-Solving Tools

While there are many useful tools in the creative problem-solving process, here are three you should know:

Creating a Problem Story

One way to innovate is by creating a story about a problem to understand how it affects users and what solutions best fit their needs. Here are the steps you need to take to use this tool properly.

1. Identify a UDP

Create a problem story to identify the undesired phenomena (UDP). For example, consider a company that produces printers that overheat. In this case, the UDP is "our printers overheat."

2. Move Forward in Time

To move forward in time, ask: “Why is this a problem?” For example, minor damage could be one result of the machines overheating. In more extreme cases, printers may catch fire. Don't be afraid to create multiple problem stories if you think of more than one UDP.

3. Move Backward in Time

To move backward in time, ask: “What caused this UDP?” If you can't identify the root problem, think about what typically causes the UDP to occur. For the overheating printers, overuse could be a cause.

Following the three-step framework above helps illustrate a clear problem story:

You can extend the problem story in either direction if you think of additional cause-and-effect relationships.

4. Break the Chains

By this point, you’ll have multiple UDP storylines. Take two that are similar and focus on breaking the chains connecting them. This can be accomplished through inversion or neutralization.

Even if creating a problem story doesn't provide a solution, it can offer useful context to users’ problems and additional ideas to be explored. Given that divergence is one of the fundamental practices of creative problem-solving, it’s a good idea to incorporate it into each tool you use.


Brainstorming is a tool that can be highly effective when guided by the iterative qualities of the design thinking process. It involves openly discussing and debating ideas and topics in a group setting. This facilitates idea generation and exploration as different team members consider the same concept from multiple perspectives.

Hosting brainstorming sessions can result in problems, such as groupthink or social loafing. To combat this, leverage a three-step brainstorming method involving divergence and convergence :

Alternate Worlds

The alternate worlds tool is an empathetic approach to creative problem-solving. It encourages you to consider how someone in another world would approach your situation.

For example, if you’re concerned that the printers you produce overheat and catch fire, consider how a different industry would approach the problem. How would an automotive expert solve it? How would a firefighter?

Be creative as you consider and research alternate worlds. The purpose is not to nail down a solution right away but to continue the ideation process through diverging and exploring ideas.

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Continue Developing Your Skills

Whether you’re an entrepreneur, marketer, or business leader, learning the ropes of design thinking can be an effective way to build your skills and foster creativity and innovation in any setting.

If you're ready to develop your design thinking and creative problem-solving skills, explore Design Thinking and Innovation , one of our online entrepreneurship and innovation courses. If you aren't sure which course is the right fit, download our free course flowchart to determine which best aligns with your goals.

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About the Author

Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.

problem solving focus model

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

problem solving focus model

Frequently Asked Questions

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

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by Editor-in-Chief and therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

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You can become a better problem solving by:

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.

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ZDM – Mathematics Education volume  53 ,  pages 737–752 ( 2021 ) Cite this article

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Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real, not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions: (1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also, successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry environments can be characterised and compared with our model.

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1 Introduction

Problem solving (PS)—in the sense of working on non-routine tasks for which the solver knows no previously learned scheme or algorithm designed to solve them (cf. Schoenfeld, 1985 , 1992b )—is an important aspect of doing mathematics (Halmos, 1980 ) as well as learning and teaching mathematics (Liljedahl et al. 2016 ). As one of several reasons, PS is used as a means to help students learn how to think mathematically (Schoenfeld, 1992b ). Hence, PS is part of mathematics curricula in almost all countries (e.g., KMK, 2004 ; NCTM, 2000 , 2014 ). Accordingly, PS has been a focus of interest of researchers for several decades, Pólya ( 1945 ) being one of the most prominent scholars interested in this activity.

Problem-solving processes (PS processes) can be characterised by their inner or their outer structure (Philipp, 2013 , pp. 39–40). The inner structure refers to (meta)cognitive processes such as heuristics, checks, or beliefs, whereas the outer structure refers to observable actions that can be characterised in phases like ‘understanding the problem’ or ‘devising a plan’, as well as the chronological sequence of such phases in a PS process. Our focus in this paper is on the outer structure, as it is directly accessible to teachers and researchers via observation.

In the research literature, there are various characterisations of PS processes. However, almost all of the existing models are normative , which means they represent idealised processes. They characterise PS processes according to distinct phases, in a predetermined sequence, which is why they are sometimes called ‘prescriptive’ instead of normative. These phases and their sequencing have been formulated as a norm for PS processes. Normative models are generally used as a pedagogical tool to guide students’ PS processes and to help them to become better problem solvers. The normative models in current research have mostly been derived from theoretical considerations. Nevertheless, real PS processes look different; they contain errors, detours, and cycles, and they do not follow a predetermined sequence. Actual processes like these are not considered in normative models. Accordingly, there are almost no models that guide teachers and researchers in observing, understanding, and analysing PS processes in their ‘non-smooth’ occurrences (cf. Fernandez et al. 1994 ; Rott, 2014 ). Our aim in this paper, therefore, is to address this research gap by suggesting a descriptive model.

A descriptive model enables not only the representation of real PS processes, but also reveals additional potential for analyses. Our model allows one systematically to compare several PS processes simultaneously by means of accumulation, which is an approach that to our knowledge has not been proposed before in the mathematics education community. In Sect.  6 , we show how this approach can be used to reveal ‘bumps and bruises’ of real students’ PS processes to illustrate the practical value of our descriptive model (Sect.  5.3 , Fig.  5 ). We show how our model allows one to discriminate problem-solving processes from routine processes when students work on tasks. We illustrate how differences between successful and unsuccessful processes can be identified using our model. We also reveal how students’ PS processes, working in a paper-and-pencil environment compared to working in a digital (dynamic geometry) environment , can be characterised and compared by means of our model.

Our descriptive model is based on intertwining theoretical considerations, in the form of a review of existing models, as well as on a video-study researching the processes shown by mathematics pre-service teachers working on geometrical problems.

2 Theoretical background

In this section, we first describe and compare aspects of existing models of PS processes (which are mostly normative) to characterize their potential and their limitations for analysing students’ PS processes (2.1). We then discuss why looking specifically at students’ PS processes in geometry and in dynamic geometry contexts is of particular value for developing a descriptive model of PS processes (2.2).

2.1 Models of problem-solving processes

Looking at models from mathematics, mathematics education, and psychology that describe the progression of PS processes, we find phase models, evolved by authors observing their own PS processes or those of people with whom the authors are familiar. So, the vast majority of existing PS process models are not based on ‘uninvolved’ empirical data (e.g., videotaped PS processes of students); they were actually not designed for the analysis of empirical data or to describe externally observed processes, which emphasises the need for a descriptive model.

2.1.1 Classic models of problem-solving processes

Two ‘basic types’ of phase models for PS processes have evolved in psychology and mathematics education. Any further models can be assigned to one or the other of these basic types: (1) the intuitive or creative type and (2) the logical type (Neuhaus, 2002 ).

Intuitive or creative models of PS processes originate in Poincaré’s ( 1908 ) introspective reflection on his own PS processes. Building on his thoughts, the mathematician Hadamard ( 1945 ) and the psychologist Wallas ( 1926 ) described PS processes with a particular focus on subconscious activities. Their ideas are most often summarised in a four-phase model: (i) After working on a difficult problem for some time and not finding a solution ( preparation ), (ii) the problem solver does and thinks of different things ( incubation ). (iii) After some more time—hours, or even weeks—suddenly, a genius idea appears ( illumination ), providing a solution or at least a significant step towards a solution of the problem; (iv) this idea has to be checked for correctness ( verification ).

So-called logical models of PS processes were introduced by Dewey ( 1910 ), describing five phases: (i) encountering a problem ( suggestions ), (ii) specifying the nature of the problem ( intellectualization ), (iii) approaching possible solutions ( the guiding idea and hypothesis ), (iv) developing logical consequences of the approach ( reasoning (in the narrower sense) ), and (v) accepting or rejecting the idea by experiments ( testing the hypothesis by action ). Unlike in Wallas’ model, there are no subconscious activities described in Dewey’s model. Pólya’s ( 1945 ) famous four-phase model—(i) understanding the problem , (ii) devising a plan , (iii) carrying out the plan , and (iv) looking back —manifests, according to Neuhaus ( 2002 ), references to Dewey’s work.

Research in mathematics education mainly focuses on logical models for describing PS processes, following Pólya or more recent variants of his model (see below). This is due to the fact that PS processes of the intuitive or creative type might take hours, days, or even weeks to allow for genuine incubation phases, and PS activities in the context of schooling and university teaching are mostly shorter and more contained. Therefore, we focus on logical models. In the following, we compare prominent PS process phase models that emerged in the last decades (see Fig.  1 ).

figure 1

Different phase models of problem-solving processes

2.1.2 Recent models of problem-solving processes

In Fig.  1 , different models are presented (for more details see the appendix). These build on and alter distinct aspects of Pólya’s model, especially envisioned phases and possible transitions between these phases. They mark this distinction by using different terminology for these nuanced differences in the phases. The models by Mason et al. ( 1982 ), Schoenfeld ( 1985 , Chapter 4), and Wilson et al. ( 1993 ; Fernandez et al. 1994 ) are normative; they are mostly used for teaching purposes, that is, to instruct students in becoming better problem solvers. Compared to actual PS processes, these models comprise simplifications; looking at and analysing students’ PS processes requires models which are suited to portray these uneven and cragged processes.

In several studies, actual PS processes are analysed; however, only a few of these studies use any of these normative models that describe the outer structure of PS processes. Even fewer studies present a descriptive model as part of their results. Some of the rare studies that attempt to derive such a model are presented in more detail in the appendix; their essential ideas are presented below (Artzt & Armour-Thomas, 1992 ; Jacinto & Carreira, 2017 ; Yimer & Ellerton, 2010 ).

2.1.3 Comparing models of problem-solving processes

In this section, we compare the previously mentioned as well as additional phase models with foci on (a) the different types of phases and (b) linearity or non-linearity of the portrayed PS processes. Figure  1 illustrates similarities and differences in these models, starting with those of Dewey ( 1910 ) and Pólya ( 1945 ) as these authors were the first to suggest such models. Schoenfeld ( 1985 ) and Mason et al. ( 1982 ) introduced this discussion to the mathematics education community, referring back to ideas of Pólya. Then, we discuss those of Wilson et al. ( 1993 ), and Yimer and Ellerton ( 2010 ), as examples of more recent models in mathematics education.

Different types of phases

The presented models comprise three, four, or more different phases. However, we do not think that this number is important per se; instead, it is interesting to see which activities are encompassed in these phases of the different models and in the extent and manner in which they follow Pólya’s formulation, adopt it, or go beyond his ideas. In Fig.  1 , we indicated Pólya’s phases with differently patterned layers in the background.

Dewey’s ( 1910 ) model starts with a phase (named “suggestions”) in which the problem solvers come into contact with a problem without already analysing or working on it. Such a phase is seldom found in phase models in the context of mathematics education. In mathematics though this phase at the beginning is typical and important, as Dewey already pointed out. In the context of teaching, on the other hand, PS mostly starts with a task handed to the students by their teachers. Analysing and working on the problem is expected right from the beginning; this is part of the nature of the provided task. So, in educational research the phase of “suggestions” is rarely mentioned, as it normally does not occur in students’ PS processes.

“Understanding the problem”, Pólya’s ( 1945 ) first phase, is comparable to the second phase (“intellectualization”) of Dewey’s model. In this phase, problem solvers are meant to make sense of the given problem and its conditions. Such a phase is used in all models, though often labelled slightly differently (see Fig.  1 for a juxtaposition). Artzt and Armour-Thomas, ( 1992 ) facing the empirical data of their study, differentiated this phase of “understanding the problem” into a first step, where students are meant to apprehend the task (“understanding”), and a second step, where students are actually expected to comprehend the problem (“analysing”); a similar differentiation is presented by Jacinto and Carreira ( 2017 ) into “grasping, noticing” and “interpreting” a problem.

The next two phases incorporate the actual work on the problem. Pólya describes these phases as “devising” and “carrying out a plan”. Especially the planning phase encompasses many different activities, such as looking for similar problems or generalizations. These two phases are also integral parts of the models by Wilson et al. ( 1993 ), and Yimer and Ellerton ( 2010 ) (see Fig.  1 ), or Jacinto and Carreira ( 2017 , there called “plan” and “create”). Mason et al. ( 1982 ) chose to combine both phases, calling this combined phase “attack”. According to their educational and research experience, they noted that both phases cannot be distinguished in most cases; therefore, a differentiation would not be helpful for learning PS and describing PS processes. Schoenfeld ( 1985 ), on the contrary, further differentiated those phases by splitting Pólya’s second phase into a structured “planning” (or “design”) phase and an unstructured “exploration” phase. When “planning”, one might adopt a known procedure or try a combination of known procedures in a new problem context. However, when known procedures do not help, working heuristically (e.g., looking at examples, counter-examples, or extreme cases) might be a way to approach the given problem in “exploration” (Schoenfeld, 1985 , p. 106). According to Schoenfeld, exploration is the “heuristic heart” of PS processes.

The last phase in Pólya’s model is “looking back”, the moment when a solution should be checked, other approaches should be explored, and methods used should be reflected upon. This phase is also present in other models (see Fig.  1 ). In their empirical approach, Yimer and Ellerton ( 2010 ), for example, differentiated this phase into two steps, namely, “evaluation” (i.e., checking the results), which refers to looking back on the recently solved problem, and “internalization” (i.e., reflecting the solution and the methods used), which focuses on what has been learnt by solving this problem and looks forward to using this recent experience for solving future problems. Jacinto and Carreira ( 2017 ) used the same “verifying” phase as Pólya, but added a “disseminating” phase for presenting solutions, as their final phase.

Other researchers (see the appendix) came to insights similar to those of these researchers, using slightly different terminologies when describing these phases or combinations of these phases.

Sequence of phases: linear or non-linear problem-solving processes

Other important aspects are transitions from one phase to another, and how such transitions occur. The graphical representations of different models in Fig.  1 not only indicate slightly different phases (and distinct labels for these phases), but also illustrate different understandings of how these phases are related and sequenced.

There are strictly linear models like Pólya’s ( 1945 ), which outline four phases that should be passed through when solving a problem, in the given order. Of course, Pólya as a mathematician knew that PS processes are not always linear; in his normative model, however, he proposed such a stepwise procedure, which has often been criticised (cf. Wilson et al. 1993 ). Mason et al. ( 1982 ) and Schoenfeld ( 1985 ) discarded this strict linearity, including forward and backward steps between analysing, planning, and exploring (or attacking, respectively) a problem. Thereafter, PS processes linearly proceed towards the looking back equivalents of their models. Wilson et al. ( 1993 ) presented a fully “dynamic, cyclic interpretation of Polya’s stages” (p. 60) and included forward and backward steps between all phases, even after “looking back”. The same is true for Yimer and Ellerton ( 2010 ), who included transitions between all phases in their model.

As we illustrate later, transitions from one phase to another reflect also characteristic features of routine and non-routine processes in general, and can be also distinctive for students’ PS processes in traditional paper-and-pencil environments compared to Dynamic Geometry Software (DGS) contexts. Our descriptive model of PS processes, which we present in Sect.  5 , also evolved by comparative analyses of students’ PS processes in both learning contexts. Thus, we comment briefly in 2.2 on what existing research has found in this respect so far.

2.2 Problem solving in geometry and dynamic geometry software

Overall, geometry is especially suited for learning mathematical PS in general and PS strategies or heuristics in particular (see Schoenfeld, 1985 ). Notably, many geometric problems can be illustrated in models, sketches, and drawings, or can be solved looking at special cases or working backwards (ibid.). Additionally, the objects of action (at least in Euclidean geometry) and the permitted actions (e.g., constructions with compasses and ruler) are easy to understand. Therefore, in our empirical study (see Sect.  4 ), we opted for PS processes in geometry contexts, knowing that other contexts could be equally fruitful.

One particular tool to support learning and working in the context of geometry, since the 1980s, is DGS, which is characterised by three features, namely, dragmode, macro-constructions, and locus of points (Sträßer, 2002 ). With these features, DGS can be used not only for verification purposes, but also for guided discoveries as well as working heuristically (e.g., Jacinto & Carreira, 2017 ). However, as Gawlick ( 2002 ) pointed out, to profit from such an environment, students—especially low achievers—need some time to get accustomed to handling the software. Comparing DGS and paper-and-pencil environments, Koyuncu et al. ( 2015 ) observed that in a study with two pre-service teachers, “[b]oth participants had a tendency toward using algebraic solutions in the [paper-and-pencil based] environment, whereas they used geometric solutions in the [DGS based] environment.” (p. 857 f.). These potential differences between PS processes in paper-and-pencil versus DGS environments are interesting for research and practice. Therefore, we compared students’ PS processes in these two environments in our empirical study.

3 Research questions

With regard to research on PS processes, it is striking that there is only a small number of studies, often with a low number of participants, that present and apply a descriptive model of PS processes. Further, the identified models are not suited for comparing PS processes across groups of students, but can only describe cases. Last but not least, in most empirical studies, the selection of phases that are included, and the assumption of (non-)linearity, are not discussed and/or justified. In all these respects, we see a research gap. Contributing to filling this gap was one of the motivations for the study presented here. Based on the existing research literature, we formulated two main research questions:

What elements of the already discussed PS process models can be used for a descriptive model? In particular, what is necessary so that such a descriptive model enables

a recognition of types of phases and an identification of phases in actual PS processes as well as

an identification of the sequence (i.e., the order, linear or non-linear) of phases and transitions between phases?

Can the model be used to describe and discriminate among different types of PS processes, for example

routine and non-routine processes,

successful and not successful processes, or

paper-and-pencil vs. DGS processes?

These questions guided our study and the motivation for developing a descriptive model of PS processes. Next, we present the methodology, before we discuss results of our empirical study and present our model.

4 Methodology

In a previous empirical study, we looked at PS processes of pre-service teacher students in geometry contexts. The data in this study were enormously rich and challenged us in their analyses in many ways. Existing PS models did not allow us to harvest fully this rich data corpus and we realised that with respect to our empirical data, we needed a descriptive model. So we formulated the research questions listed above in order to explore the potential and necessary extensions of the existing normative PS process models. We changed our perspective and focused on the development of an empirically grounded theoretical model. We required an approach that would allow us to mine the data of our empirical study and to provide a conceptualisation that could be helpful for further research on students’ problem-solving processes. The methodological approach we used is described in the following.

4.1 Our empirical study

About 250 pre-service teacher students attended a course on Elementary Geometry , which was conceived and conducted by the third author at a university in Northern Germany. The course lasted for one semester (14 weeks); each week, a two-hour lecture for all students as well as eight 2-h tutorials for up to 30 students each, supervised by tutors (advanced students), took place. Four tutorials (U1, Ulap2, U3, and Ulap4) were involved in this study: in U1 and U3 the students worked in a paper-and-pencil environment, in Ulap2 and Ulap4 the students used laptop computers to work in a DGS environment. (The abbreviations consist of U, the first letter ‘Uebung’, German for tutorial, with an added ‘lap’ for groups which used laptop computers as well as an individual number.) Students worked on weekly exercises, which were discussed in the tutorials. In addition, over the course of the semester, in groups of three or four, the students worked on five geometric problems in the tutorials (approx. 45 min for each problem), accompanied by as little tutor help as possible. In this paper, we focus on these five problems. See the appendix for additional information regarding the organisation of our study.

The five problems were chosen so that students had the opportunity to solve a variety of non-routine tasks, which at the same time did not require too much advanced knowledge that students might not have.

For each of the five problems, two groups from each of the four tutorials were observed. Each problem was therefore worked on by four groups with and four groups without DGS (minus some data loss because of students missing tutorials or technical difficulties). The collected data were videos of the groups working on these problems ( processes ), notes by the students ( products ), as well as observers’ notes. Overall, 33 processes (15 from paper-and-pencil as well as 18 from DGS groups) from all five problems, with a combined duration of 25 h, were analysed. For space reasons, we cannot discuss all five problems in detail here. Instead, we present three of the five problems here; the other two can be found in the appendix.

4.1.1 The problems

Regarding the ‘Shortest Detour’ (Fig.  2 , top), as long as A and B are on different sides of the straight line, a line segment from A to B is the shortest way. When A and B are on the same side of g , an easy (not the only) way to solve this problem is by reflecting one of the points, e.g. A , on g and then constructing the line segment from the reflection of A to B , as reflections preserve lengths.

figure 2

Three of the five problems used in our study

Part a) of the ‘Three Beaches’ problem (Fig.  2 , bottom), finding the incircle of an equilateral triangle, should be a routine-procedure as this topic had been discussed in the lecture. Students working on part b) of this problem needed to realize that in an equilateral triangle, all points have the same sum of distances to the sides (Viviani’s problem). This could be justified by showing that the three perpendiculars of a point to a side in such a triangle add up to the height of this triangle, for example by geometrical addition or by calculating areas.

Like Problem (4), Problem (3) (Fig.  2 , middle) contained an a)-part which is a routine task—finding the circumcircle of a (non-regular) triangle—and a b)-part that constitutes a problem for the students.

These tasks were chosen because they actually represented problems for our students, and expected PS processes appeared neither too long nor too short for a reasonable workload by students and for our analyses. Further, the problems covered the content of the accompanying lecture, and the problems could be solved both with and without DGS.

Differences between working with and without DGS: With DGS many examples can be generated quickly, so that an overview of the situation and the solution can be obtained in a short time. For the justifications, however, with and without DGS, students had to reflect, think, and reason to find appropriate arguments.

4.2 Framework for the analysis of the empirical data

For the analyses of our students’ PS processes, we used the protocol analysis framework by Schoenfeld ( 1985 , Chapter 9) with adaptations and operationalizations by Rott ( 2014 ), following two phases of coding.

Process coding: With his framework, Schoenfeld ( 1985 ) intended to “identify major turning points in a solution. This is done by parsing a [PS process] into macroscopic chunks called episodes” (p. 314). An episode is “a period of time during which an individual or a problem-solving group is engaged in one large task […] or a closely related body of tasks in the service of the same goal […]” (p. 292). Please note, the term “episode” refers to coded process data, whereas “phase” refers to parts of PS models. Schoenfeld (p. 296) continued: “Once a protocol has been parsed into episodes, each episode is characterized” using one of six categories (see also Schoenfeld, 1992a , p. 189):

Reading or rereading the problem.

Analysing the problem (in a coherent and structured way).

Exploring aspects of the problem (in a much less structured way than in Analysis).

Planning all or part of a solution.

Implementing a plan.

Verifying a solution.

According to Schoenfeld ( 1985 ), Planning-Implementation can be coded simultaneously.

The idea of episodes as macroscopic chunks implies a certain length, thus individual statements do not comprise an episode; for example, quickly checking an interim result is not coded as a verification episode. Also, PS processes are coded by watching videos, not by reading transcripts (Schoenfeld, 1992a ).

Schoenfeld’s framework was chosen to answer our first research question, for two reasons. (i) The episode types he proposed cover a lot of the variability of phases also identified by us (see Sect.  2.1.3 ). (ii) Coding episodes and coding episode types in independent steps offers the possibility of adding inductively new types of episodes.

After parsing a PS process into episodes, we coded the episodes with Schoenfeld’s categories (deductive categories), but also generated new episode types to characterize these episodes (inductive categories). While coding, we observed initial difficulties in coding the deductive episodes reliably; especially differentiating between Analysis and Exploration episode types was difficult (as predicted by Schoenfeld, 1992a , p. 194). We noticed that Schoenfeld’s ( 1985 , Chapter 9) empirical framework referred to his theoretical model of PS processes (ibid., Chapter 4) which was based on Pólya’s ( 1945 ) list of questions and guidelines. Recognizing an analogy between Schoenfeld’s framework and Pólya’s work (see Fig.  1 ), we were able to operationalize their descriptions in a coding manual (see Rott, 2014 ).

When the deductive episode types did not fit our observations, we inductively added a new episode type. This happened three times. Especially in the DGS environment, where students showed behaviour that was not directly related to solving the task, new types of activities occurred. For example, students talked about the software and how to use it. This kind of behaviour was coded by us as Organization . When it took students more than 30 s to write down their findings (without developing any new results or ideas), this episode was coded as Writing . Discussions about things which were not related to mathematics, but for example daily life, were coded as Digression . These codings were used only when activities did not align with numbers 2–6 of Schoenfeld’s list.

This coding of the videotapes was done independently by different research assistants and the first author. We then applied the “percentage of agreement” ( P A ) approach to compute the interrater-agreement as described in the TIMSS 1999 video study (Jacobs et al. 2003 , pp. 99–105), gaining more than P A  = 0.7 for parsing PS processes into episodes and more than P A  = 0.85 for characterizing the episode types. More importantly, every process was coded by at least two raters. Whenever those codes did not coincide, we attained agreement by recoding together (as in Schoenfeld’s study, 1992a , p. 194).

Product coding : To be able to compare successful and unsuccessful PS processes, students’ products produced in the 45-min sessions were rated. Because the focus was on processes, product rating finally was reduced to a dichotomous right/wrong coding without going into detail regarding students’ argumentations (these will be analysed and the results reported in forthcoming papers). Rating was done independently by a research assistant and the first author with an interrater-agreement of Cohen’s kappa > 0.9. Differing cases were discussed and recoded consensually.

5 Results of our empirical study and implications for our descriptive model

In this section, we briefly illustrate results of our data analyses, which underline the need to go beyond existing models. We summarize key findings of our empirical study and illustrate how these have contributed to the development of our descriptive model of PS processes. After this, we highlight how answering our research questions based on our theoretical and empirical analyses contributes to the development of our descriptive model. Finally, we present and describe our descriptive model.

5.1 Sample problem-solving processes and codings to illustrate the procedure of analysis

To illustrate our analyses and codings of students’ PS processes, we present three sample processes, the first two in detail and the third one only briefly. The first two were paper-and-pencil processes and stem from the same group of students, belonging to parts a) and b) of the ‘Three Beaches’ problem. The third process shows a group of students working on the ‘Shortest Detour’ problem with DGS. Our codings of the different episodes are highlighted in italics .

5.1.1 Group U1-C, Three Beaches (part a))

After reading the Three Beaches problem (00:25–01:30), the three students of group C from tutorial U1 try to understand it. They remember the Airport problem in which they had to find a point with the same distance to all three vertices of a triangle and they try to identify the differences between both problems. The students wonder whether they should again use the perpendicular bisectors of the sides of the triangle or the bisectors of the angles of the triangle ( Analysis , 01:30–05:05). They agree to use the bisectors of the angles and construct their solutions with compasses and ruler. One of the students claims that in the case of an equilateral triangle, perpendicular and angle bisectors would be identical and convinces the others by constructing a triangle and both bisectors with compasses and ruler ( Planning-Implementation , 05:05–06:05). Finally, the students verify their solution by discussing the meaning of the distance from a point to the sides of a triangle, as they initially were not sure how to measure this distance (06:05–07:40). Even though the Analysis episode was quite long (see Fig.  3 ), this part of the task was actually not a problem for the students as they remembered a way to solve it.

figure 3

Process codings of the group U1-C, working on the ‘Three Beaches’ problem

5.1.2 Group U1-C, three beaches (part b))

After reading part b) of the problem (07:45–07:55), the students discuss whether the requested point is the same as in a) ( Analysis , 07:55–10:25). They agree to try out and construct a triangle each, place points in it, draw perpendiculars to the sides, and measure the distances. One student asks whether it is allowed to place the point on a vertex and thus have two distances become zero ( Exploration , 10:25–15:30). After this, the students discuss the meaning of distance, particularly the meaning of a distance related to a side of a triangle. They agree that any point on a side, even the vertex, would satisfy the condition of the problem, thus being a suitable site for the ‘house’ ( Analysis , 15:30–16:40). The students wonder why the distance from one vertex to its opposing side (the height of the triangle) is as large as the sum of the distances from the centre of the incircle (from part a)). They remember that the angle bisectors intersect each other in a ratio of 1/3 to 2/3. Thereafter, they continue to place points in their triangles (not on sides) and measure their distances. They finally agree on the [wrong] hypothesis that any point on the angle bisectors is a point with a minimal sum of the distances to the sides; other points in the triangle would have a slightly larger sum [because of inaccuracies in their drawings]. They realize, however, that they cannot give any reasons for their solution ( Exploration , 16:40–32:30). The codings are represented in Fig.  3 (right).

5.1.3 Group Ulap2-TV, shortest detour

Ulap2-TV working on the shortest detour problem (Fig.  4 ) is an example of a process with more transitions. The students solve the first case of the problem ( A and B on different sides of g ) within 5 min ( Planning-Implementation , Verification ) and then explore the second case ( A and B on the same side of g ) for more than 17 min before solving the problem.

figure 4

Process coding of the group Ulap2-TV, working on the ‘Shortest Detour’ problem

We selected these three PS processes from our study, as they are examples of our empirical data in several aspects: They illustrate both learning environments (paper-and-pencil and DGS), they incorporate all types of episodes (except for Digression ) and, therefore, all types of phases discussed in the PS research literature, and they include linear and cyclic progressions (see below). The routine process (Three Beaches, part a)) is rather atypical as the students take a lot of time analysing the task, before implementing routine techniques ( Planning-Implementation ). The two PS processes (Three Beaches, part b) and Shortest Detour) are typical for our students, spending a lot of time in Exploration episodes. In the DGS environment, we see that the students take some time to handle the software ( Organization ). Compared to free-hand drawings in paper-and-pencil environments, the students in the DGS environment need to think about constructions ( Planning ) before exploring the situation.

5.2 From theoretical models and empirical results to a descriptive model of problem-solving processes

In the following, the coded episodes from all 33 PS processes of our empirical study are used to answer the first research question. What parts or phases of the established models are suited to describe the analysed processes? Which transitions between phases can be observed? The systematic comparison of PS models from the literature (Sect.  2.1.3 ) is the theoretical underpinning of answering these questions. This process aims at generating a descriptive process model suitable for representing students’ actual PS processes.

5.2.1 Different types of episodes that are suited to describing empirical processes

Within the observed processes, all of Schoenfeld’s episode types could be identified with high interrater agreement. Thus, based on our data, we saw no need to merge phases like Understanding and Planning , even though some models suggest doing so.

More specifically, structured approaches of Planning could be differentiated from unstructured approaches which we call Explorations as suggested by Schoenfeld ( 1985 , Chapters 4 & 9) (in 6 out of 33 non-routine processes, both Exploration and Planning were coded).

Furthermore, in some processes, Planning and Implementation episodes can be differentiated from each other (as suggested by Pólya, 1945 ); there are, however, processes in which those two episode types cannot be distinguished as the problem solvers often do not announce their plans (as predicted by Mason et al. 1982 ). In those PS processes, these two episode types are merged to Planning-Implementation (as done by Schoenfeld as well).

Verification episodes are rare, but can be found in our data. As our students do not show signs of trying to reflect on their use of PS strategies, we decided not to distinguish this episode type into ‘checking’ and ‘reflection’.

Incubation and illumination could not be observed in our sample. This was expected as the students did not have the time to incubate.

Altogether, the following theoretically recorded phases could be identified in our empirical data and are part of our model: understanding (analysis), exploration, planning, implementation (sometimes as planning-implementation), and verification.

5.2.2 Transitions between phases: linearity and non-linearity of the processes

Apart from the phases that occur, the transitions between these phases are of interest. Transitions have been coded between nearly all possible ordered pairs of episode types. If the phases proceed according to Pólya’s or Schoenfeld’s model ( Analysis → Exploration → Planning → Implementation → Verification ), we consider this as a linear process. If phases are omitted within a process but this order is still intact we regard this process still as ‘linear’. In contrast, a process is considered by us as non-linear or cyclic, if this order is violated (e.g., Planning → Exploration ). We also checked whether non-linear processes are cyclic in the sense of Wilson et al. (backward steps are possible after all types of episodes), or whether they are cyclic in the sense of Schoenfeld and Mason et al. (backward steps only before Implementation ).

The first sample process (Three Beaches, part a) illustrates a strictly linear approach as in Pólya’s model, represented in the descending order of the time bars (Fig.  3 , left). The second example (Three Beaches, part b) shows a cyclic process as after the first Exploration , an Analysis was coded (Fig.  3 , right). The third example (Shortest Detour) starts in a linear way; then, after a first Verification , the students go back to Planning-Implementation and Exploration episodes. Thus, overall, their process is cyclic (and not in a way that would fit Schoenfeld’s model as the linear order is broken after a Verification ).

We checked all our process codings for their order of episodes (see Table 1 ). In our sample, a third of the processes are non-linear; thus, a strictly linear model is not suited to describing our students’ PS processes.

5.3 Deriving a model for describing problem-solving processes

Using the results of our empirical study as described in Sects.  5.1 and 5.2 , our findings result in a descriptive model of PS processes. We consider this model as an answer to our first research question. We identified phases from (mostly normative) models in our data, then empirically refined these phases, and took the relevance of their sequencing into account as illustrated in Fig.  5 .

figure 5

Descriptive model of problem-solving processes

In our descriptive model (see Fig.  5 ), we distinguish between structured ( Planning ) and unstructured ( Exploration ) approaches in accordance with the model of Schoenfeld ( 1985 ). It is also possible to differentiate between explicit planning ( Planning and Implementation coded separately) as well as implicit planning, which means (further) developing a plan while executing it ( Planning and Implementation coded jointly), as suggested by Mason et al. ( 1982 ). Our descriptive model combines ideas from different models in the literature. Furthermore, linear processes can be displayed (using only arrows that point downwards in the direction of the solution) as can non-linear processes (using at least one arrow that points upwards). Therefore, with this model, linear and non-linear PS processes can explicitly be distinguished from each other. Please note that we use ‘(verified) solution’ with a restriction in brackets, as not all processes lead to a verified or even correct solution. Our model is a model of the outer structure as it describes the observable sequence of the different phases.

In the following, we illustrate how far our descriptive model can also respond to our second research question. We use it to describe, as well as to distinguish different types of PS processes.

6 Using our descriptive model to analyse problem-solving processes

Below, we illustrate how our descriptive model (Fig.  5 ) can be used to analyse and compare students’ PS processes. We first reconstruct different processes of student groups and then propose a new way to represent typical transitions in students’ PS processes.

6.1 Representing students’ problem-solving processes

In contrast to the process coding by Schoenfeld, which contains specific information about the duration of episodes, our analyses are more abstract. We focus on the empirically found types of episodes and transitions between these episodes. This is done following Schoenfeld ( 1985 ), who emphasised: “The juncture between episodes is, in most cases, where managerial decisions (or their absence) will make or break a solution” (p. 300). Focusing on the transitions between episodes is one important characteristic that distinguishes different types of PS processes. Using our descriptive model allows one to do this.

For each process, the transitions between episodes can be displayed with our model (Fig.  5 ). In the following, we consider only the five content-related episode types, but not Reading , Organization , Writing , and Digression, as activities of the latter types of episodes do not contribute to the solution and they are not ordered as in Pólya’s or Schoenfeld’s phases.

For example, the routine process of group U1-C (Three Beaches, part a), see Sect.  5.1 ), starts with an Analysis , followed by a merged Planning-Implementation and a Verification or, in short: [A,P-I,V]; thereafter, this process ends. This means, there are four different transitions in this process indicated by arrows: Start → A, A → P-I, P-I → V, and V → End. Thus, in Fig.  6 (left), these transitions are illustrated with arrows. In this case, these transitions each occur only once, which is indicated by a circled number 1.

figure 6

Translation from Schoenfeld codings to a representation using the descriptive model; the circled numbers indicate the number of times a transition occurs

The second example (U1-C, Three Beaches, part b)) consists of the following episodes: Analysis–Exploration–Analysis–Exploration [A,E,A,E]. This means that there are five transitions in this process: Start → A, A → E, E → A, A → E, and E → End (see Fig.  6 , middle). Please notice that the transition A → E is observed twice.

The final example shows group Ulap2-TV (Shortest Detour), which starts with a Planning-Implementation and proceeds through [P-I,V,P,E,P-I,V] with a total of seven transitions, two of which are P-I → V (ignoring Organization and Writing , Fig.  6 , right).

This reduction to transitions, neglecting the exact order and the duration of episodes, enables one to do a specific comparison of processes and an accumulation of several PS processes (e.g., from all DGS processes, see Sect.  6.2 ). The focus is now on transitions and how often they happen, which indicates different types of PS processes as shown below. This ‘translation’ from the Schoenfeld coding to the representation in our descriptive model has been done for all 33 processes. The directions of the arrows indicate from which phase to which the transitions are occurring, e.g., from analysis to planning; the numbers on the arrows show how often these transitions were coded (they do not indicate an order).

The three selected processes already show clearly different paths, for example, linear vs. cyclic (see Sect. 2.2.4).

6.2 Characterizing types of problem-solving processes by accumulation

Students’ PS processes can be successful or non-successful or conducted in paper-and-pencil or DGS contexts. Looking at different groups of students simultaneously can be fruitful, as such accumulations allow one to look at patterns in existing transitions. Our descriptive model allows one to consider several processes at once, via accumulation.

Representations of single processes, as presented in Fig.  6 and in the boxes in Fig.  7 , can be combined by adding up all coded transitions (which would be impossible with time bars used by Schoenfeld). For such an accumulation, we count all transitions between types of episodes and display them in numbers next to the arrows representing the number of those transitions. For example, six of the processes in the outer boxes start with a transition from the given problem to Planning , while one process begins with an Analysis . This is shown in the centre box by the numbers 6 and 1 in the arrows from the given problem to Planning and Analysis , respectively (see Fig.  7 for the combination of all processes regarding task 3a). Arrows were drawn only where transitions actually occurred in this task. Looking at the arrows that start at the ‘given problem’ or that lead to the ‘(verified) solution’, one can see how many processes were accumulated. All episode types (small boxes) must have the same number of transitions towards as well as from this episode type.

figure 7

Centre rectangle: Accumulation of seven different group processes regarding task 3a)

To show the usability of our model, we distinguish between working on routine tasks and on problems in Sect.  6.2.1 ; thereafter, the routine processes are not further considered.

6.2.1 Routine vs. non-routine processes

In our study, two sub-tasks (3a) and 4a)) were routine tasks in which the students were asked to find special points in triangles. If we look at the accumulations of those processes in our model, clear patterns emerge: There are no Exploration episodes at all, either in the seven processes of task 3a) (Fig.  8 , left) nor in the eight processes of tasks 4a) (Fig.  8 , middle). Instead, there are Planning and/or Implementation episodes in all 15 processes. In some of those processes, Planning and Implementation can clearly be coded as two separate episodes. In other processes, it is not possible to discriminate between these episode types as two distinct episodes in the empirical data (see Fig.  8 ).

figure 8

Accumulation of seven processes for the routine task 3a) (left) and eight processes for task 4a) (middle), 15 processes in total (right)

Most processes (12 out of 15) show no need for analysing the task but start directly with Planning and/or Implementation . Even though there are five Verification episodes, these verifications are often only short checking activities with no reflection in the sense of Pólya; however, the length and quality of an episode cannot be seen in the model. Additionally, all of these 15 processes are linear (as can be seen by the arrows, which point only downwards).

In contrast to these routine tasks, non-routine processes are often non-linear and contain at least one Exploration episode. In Fig.  9 , in direct comparison to Fig.  8 , the seven PS processes of problem 3a) (left), the eight PS processes of problem 3b) (middle), and an accumulation of the 15 PS processes (right) are shown. Overall, in these 15 processes, 17 Exploration episodes were coded, which can be seen in Fig.  9 (right): 4 processes start with an Exploration ; 12 times there is an Exploration after an Analysis episode; and once after Planning-Implementation .

figure 9

Accumulation of seven processes for problem 3b) (left) and eight processes for problem 4b) (middle), 15 problem-solving processes in total (right)

In Fig.  10 (right), an accumulation is given of all 33 PS processes of all five problems. The differences of the routine and the PS processes (e.g., the latter containing Exploration episodes and being cyclic) can be seen by comparing Figs.  8 and 9 .

figure 10

Accumulation of transitions in problem-solving processes, paper-and-pencil (left) vs. DGS (middle); all problem-solving processes (right)

6.2.2 Successful and unsuccessful problem-solving processes

One of Schoenfeld's ( 1985 ) major results was the importance of self-regulatory activities in PS processes. Schoenfeld was not able to characterize successful PS processes; however, he identified characteristics of processes that did not end in a verified solution. The unsuccessful problem solvers were most often those who missed out on self-regulatory activities (i.e., controlling interim results or planning next steps); they engaged in a behaviour that Schoenfeld called “wild goose chase” and that he described this way:

Approximately 60% of the protocols were of the type [...], where the students read the problem, picked a solution direction (often with little analysis or rationalization), and then pursued that approach until they ran out of time. In contrast, successful solution attempts came in a variety of shapes and sizes—but they consistently contained a significant amount of self-regulatory activity, which could clearly be seen as contributing to the problem solvers’ success. (Schoenfeld, 1992a , p. 195)

We made similar observations looking at the processes of our students; several of them, who did not show any signs of structured actions or process evaluations, were not able to solve the tasks. Thus, to test if this observation was statistically significant, we had to operationalize the PS type “wild goose chase”, as Schoenfeld had provided no operational definition for this phenomenon. A process is considered by us to be a “wild goose chase”, if it consists of only Exploration or Analysis & Exploration episodes, whereas processes that are not of this type contain planning and/or verifying activities (only considering content-related episode types). In our descriptive model, by definition, wild goose chase processes look like the process manifested by U1-C (Three Beaches, part b) (Fig.  6 , middle).

To check if the kind of behaviour in these processes is interrelated with success or failure of the related products (see Sect.  4.2 ), a chi-square test was used (because of the nominal character of the process categories, no Pearson or Spearman correlation could be calculated). The null hypothesis was ‘there is no correlation between the PS type wild goose chase and (no) success in the product’.

The entries in Table 2 consist of the observed numbers of process–product combinations; the expected numbers assuming statistical independence (calculated by the marginal totals) are added in parentheses. The entries in the main diagonal are apparently higher than the expected values. The test shows a significant correlation ( p  < 0.01) between the problem solvers’ behaviour and their success. Therefore, the null hypothesis can be rejected, there is a correlation between showing wild goose chase-behaviour in PS processes and not being successful in solving the problem.

6.2.3 Paper-and-pencil vs. DGS environment processes

Looking at the processes of the non-routine tasks indicates that the tasks were ‘problems’ for the students, as these processes showed no signs of routine behaviour (see Sect.  6.2.1 ). Instead, we see many transitions between different episodes and the typical cyclic structure of PS processes. Comparing accumulations of all 15 paper-and-pencil with all 18 DGS PS processes, we see some interesting differences, which our model helps to reveal (see Fig.  10 ). The time the students worked on the problem was set in the tutorials and, therefore, identical in both environments and in all processes. At the end of this paper, we discuss three aspects that our comparisons revealed; more detailed analyses are planned for forthcoming papers.

We coded more transitions in DGS than in paper-and-pencil processes (73 transitions in 18 DGS processes, in short: 73/18 or on average 4 transitions per DGS process compared to 52/15 or 3.5 transitions per paper-and-pencil process). If transitions are a sign of self-regulation (Schoenfeld, 1985 ; Wilson et al. 1993 ), our students in the DGS environment seem to better regulate their processes (please note that Organization episodes are not counted here; including them would further add transitions to DGS processes). However, there might be more transitions (and thus episodes) in DGS processes because of having more time for exploring situations and generating examples, which does not take as much time as in paper-and-pencil processes.

We see more Planning (and Implementation ) episodes in DGS than in paper-and-pencil processes (9/18 or Planning in 50% of the DGS processes compared to 2/15 or 13% in paper-and-pencil processes). Using Schoenfeld’s conceptualization of Planning and Exploration episodes, the DGS processes seem to be more structured—especially since there are less Exploration episodes in DGS than in paper-and-pencil processes (17/18 compared to 21/15), even though there are more episodes in the DGS environment (see above). There seems to be a need for students in the DGS environment to plan their actions, especially when it comes to complex constructions that cannot be sketched freehandedly as in the paper-and-pencil environment. Considering the success of the students (6 solutions in the DGS environment compared to 3 in the paper-and-pencil environment), this hypothesis is supported. As already existing research indicates, better regulated PS processes should be more successful. Please note that successful solutions cannot be obtained by stating only correct hypotheses, which would favour the DGS environment; solutions coded as ‘correct’ had to be argued for.

We double checked our codings to make sure that this result was not an artefact of the coding, that the students actually planned their actions, not only using the DGS (which was coded in Organization episodes). This result could be due to our setting, as our student peer groups had only one computer and thus needed to talk about their actions. In future studies, it should be investigated if this phenomenon can be replicated in environments in which each student has his or her own computer.

We also observed more Verification episodes in DGS compared to paper-and-pencil processes (7/18 or 39% compared to 2/15 or 13%). There could be different reasons for this observation, e.g., students not trusting the technology, or just the simplicity of using the dragmode to check results compared to making drawings in the paper-and-pencil environment.

The results of using our descriptive model for comparisons of PS processes appear to be insightful. The purpose of this section was to illustrate these insights and the use of our empirical model of PS processes. Accumulating PS processes of several groups is a key to enabling comparisons such as the ones presented.

7 Discussion

The goal of this paper was to present a descriptive model of PS processes, that is, a model suited to the description and analyses of empirically observed PS processes. So far, existing research has mainly discussed and applied normative models for PS processes, which are generally used to instruct people, particularly students, about ideal ways of approaching problems. There exist a few, well accepted, models of PS processes in mathematics education (Fig.  1 ); however, these models only partly allow represention of and emphasis on the non-linearity of real and empirical PS processes, and they do not have the potential to compare processes across groups of students. For the generation of our descriptive model of PS processes, following our first research question, (1) the existing models were compared. It turned out that similarities and fine differences exist between the current normative models, especially regarding the phases of PS processes and their sequencing. We identified which elements of the existing models could be useful for the generation of a descriptive model, linking theoretical considerations from research literature with regard to our empirical data. Analysing PS processes of students working on geometric problems, we observed that distinctive episodes (esp. the distinction between Planning and Exploration ) and transitions between episodes, were essential. Classifying the episodes was mostly possible with the existing models, but characterising their transitions and sequencing required extension of the existing models, which resulted in a juxtaposition of components for our descriptive phase model (e.g., allowing us to code, separately or in combination, Planning-Implementation or to regard the (non-)linearity of processes).

Our generated descriptive model turned out not only to provide valuable insights into problems solving processes of students, but also with respect to our second research question, (2), to compare, contrast, and characterise the idiosyncratic characteristics of students’ PS processes (using Explorations or not, linear or cyclic processes, including Verification and Planning or not). Our developed descriptive model can be used to analyse processes of students ‘at once’, in accumulation, which allowed us to group and characterise comparisons of students’ processes, which was not possible with the existing models. As demonstrated in Sect.  6.2 , our model further allows one to distinguish students’ PS processes while working on routine versus problem tasks. Applying our descriptive model to routine tasks, we detected linear processes, whereas in problem tasks cyclic processes were characteristic. Furthermore, in routine tasks, no Exploration episodes could be coded. Most of the students expressed no need for analysing the task but started directly with Planning and/or Implementation.

Our descriptive model also allows one to recognize a type of PS behaviour already described by Schoenfeld ( 1992a ) as “wild goose chases”. Our data illustrated that wild goose chase processes are statistically correlated with unsuccessful attempts at solving the given problems.

In addition, our descriptive model indicated differences between paper and pencil and DGS processes. In the latter context, students showed more transitions, more Planning (and Implementation ), and more Verification episodes. This result revealed significantly different approaches that students embarked on when working on problems in paper and pencil or DGS environments. These findings might indicate that in the DGS environment in our study, students better regulated their processes (cf. Schoenfeld, 1985 , 1992b ; Wilson et al. 1993 )—a hypothesis yet to be confirmed.

A limitation of our study might be the difficulty of the problems given to our students; only 9 of 33 processes ended with a correct solution. In future studies, problems should be used that better differentiate between successful and unsuccessful problem solvers. Also, our descriptive model has so far been grounded only in university students’ geometric PS processes. Even though geometry is particularly suited for learning mathematical PS in general and heuristics in specific (see Schoenfeld, 1985 ), other contexts and fields of mathematics might highlight other challenges students face. Further empirical evidence is needed to see how far our model is also useful and suitable to describe other contexts with respect to specifics of their mathematical fields. Following some of our ideas and insights, Rott ( 2014 ) has already conducted such a study: fifth graders working on problems from geometry, number theory, combinatorics, and arithmetic. Similar results as in the study presented here, were seen and indicate the value of our descriptive model. More research in this regard is a desideratum.

Regarding teaching, using our model can be helpful to discuss with students on a meta-level these documented distinct phases of PS processes, transitions between them, and the possibility of going back to each phase during a PS process. This might help students to be aware of their processes, of different ways to gain a solution and justification, and to be more flexible during PS processes. More reflection on this aspect is also a desideratum for future research.

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