problem solving/reasoning/creativity

Reasoning, problem-solving and ideation

World Economic Forum published top 10 job skills for tomorrow. Our blog series has already covered the first nine, and now it is time for the tenth one; reasoning, problem-solving and ideation. As all the other ten job skills, also they are soft skills.  

Of the job skills already presented in our blog series, reasoning, problem-solving and ideation is linked to analytical thinking and innovation , c omplex problem-solving , and creativity, originality and initiative .  

Reasoning means the ability to proceed from hypothesis to conclusion in a logical and sensible way. The skills needed in problem-solving in turn help solve problems quickly and effectively. Problem-solving requires both an ability to correctly define a problem and finding a solution to it. Ideation in turn is often regarded as creativity, the ability to come up with new ideas and ways of doing, of testing the ideas and thus solving problems.  

Service design in problem-solving and ideation   

Problem-solving and ideation automatically brings into my mind the service design process’s double diamond; a process model for service design developed by the British Design Council (link: https://www.designcouncil.org.uk/news-opinion/what-framework-innovation-design-councils-evolved-double-diamond ).   

Picture: Double diamond  

The first diamond in the double diamond helps define the problem that is to be solved (discover – define), the second diamond helps to develop a solution to the problem (develop – define). In both parts of the double diamond first as much information is gathered as is possible (divergent thinking,) and after that the information is analyzed and crystallized into a solution (convergent thinking). The double diamond is widely used and in addition service design companies have further developed their own versions of it.  

There is a wide range of methods and tools available for service design, and you can pick the ones that are best suited to your own work. The tools available for problem-solving and ideation support creativity and finding new solutions. If you want to learn more about the methods and tools for service design, I warmly recommend This is Service Design Doing by Marc Stickdorn et al (link: https://www.adlibris.com/fi/kirja/this-is-service-design-doing-9781491927182?gclid=Cj0KCQiAtJeNBhCVARIsANJUJ2ERp6R_g54Bx0tbQJL4pvv9qo6xrrb08B5MNm8cBnM6ZSHRisTGLogaAgMoEALw_wcB ).  

Solutions to the correct problems  

In service design the problem to be solved is first defined. This helps to ensure that when ideation begins, we are indeed solving the correct problem. Although problem-solving is important, it is even more important that we are solving the correct problems. For this the discover – define phase of the double diamond are useful. Especially in the discover phase an open-minded, curious and empathetic approach is important. In the discover -define phase we are not yet finding a solution to the problem, we are concentrating on defining the problem that we will solve in the next phase.   

  – Anna Sahinoja  

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Article • 10 min read

Creative Problem Solving

Finding innovative solutions to challenges.

By the Mind Tools Content Team

Imagine that you're vacuuming your house in a hurry because you've got friends coming over. Frustratingly, you're working hard but you're not getting very far. You kneel down, open up the vacuum cleaner, and pull out the bag. In a cloud of dust, you realize that it's full... again. Coughing, you empty it and wonder why vacuum cleaners with bags still exist!

James Dyson, inventor and founder of Dyson® vacuum cleaners, had exactly the same problem, and he used creative problem solving to find the answer. While many companies focused on developing a better vacuum cleaner filter, he realized that he had to think differently and find a more creative solution. So, he devised a revolutionary way to separate the dirt from the air, and invented the world's first bagless vacuum cleaner. [1]

Creative problem solving (CPS) is a way of solving problems or identifying opportunities when conventional thinking has failed. It encourages you to find fresh perspectives and come up with innovative solutions, so that you can formulate a plan to overcome obstacles and reach your goals.

In this article, we'll explore what CPS is, and we'll look at its key principles. We'll also provide a model that you can use to generate creative solutions.

About Creative Problem Solving

Alex Osborn, founder of the Creative Education Foundation, first developed creative problem solving in the 1940s, along with the term "brainstorming." And, together with Sid Parnes, he developed the Osborn-Parnes Creative Problem Solving Process. Despite its age, this model remains a valuable approach to problem solving. [2]

The early Osborn-Parnes model inspired a number of other tools. One of these is the 2011 CPS Learner's Model, also from the Creative Education Foundation, developed by Dr Gerard J. Puccio, Marie Mance, and co-workers. In this article, we'll use this modern four-step model to explore how you can use CPS to generate innovative, effective solutions.

Why Use Creative Problem Solving?

Dealing with obstacles and challenges is a regular part of working life, and overcoming them isn't always easy. To improve your products, services, communications, and interpersonal skills, and for you and your organization to excel, you need to encourage creative thinking and find innovative solutions that work.

CPS asks you to separate your "divergent" and "convergent" thinking as a way to do this. Divergent thinking is the process of generating lots of potential solutions and possibilities, otherwise known as brainstorming. And convergent thinking involves evaluating those options and choosing the most promising one. Often, we use a combination of the two to develop new ideas or solutions. However, using them simultaneously can result in unbalanced or biased decisions, and can stifle idea generation.

For more on divergent and convergent thinking, and for a useful diagram, see the book "Facilitator's Guide to Participatory Decision-Making." [3]

Core Principles of Creative Problem Solving

CPS has four core principles. Let's explore each one in more detail:

• Divergent and convergent thinking must be balanced. The key to creativity is learning how to identify and balance divergent and convergent thinking (done separately), and knowing when to practice each one.
• Ask problems as questions. When you rephrase problems and challenges as open-ended questions with multiple possibilities, it's easier to come up with solutions. Asking these types of questions generates lots of rich information, while asking closed questions tends to elicit short answers, such as confirmations or disagreements. Problem statements tend to generate limited responses, or none at all.
• Defer or suspend judgment. As Alex Osborn learned from his work on brainstorming, judging solutions early on tends to shut down idea generation. Instead, there's an appropriate and necessary time to judge ideas during the convergence stage.
• Focus on "Yes, and," rather than "No, but." Language matters when you're generating information and ideas. "Yes, and" encourages people to expand their thoughts, which is necessary during certain stages of CPS. Using the word "but" – preceded by "yes" or "no" – ends conversation, and often negates what's come before it.

How to Use the Tool

Let's explore how you can use each of the four steps of the CPS Learner's Model (shown in figure 1, below) to generate innovative ideas and solutions.

Figure 1 – CPS Learner's Model

Explore the Vision

Identify your goal, desire or challenge. This is a crucial first step because it's easy to assume, incorrectly, that you know what the problem is. However, you may have missed something or have failed to understand the issue fully, and defining your objective can provide clarity. Read our article, 5 Whys , for more on getting to the root of a problem quickly.

Gather Data

Once you've identified and understood the problem, you can collect information about it and develop a clear understanding of it. Make a note of details such as who and what is involved, all the relevant facts, and everyone's feelings and opinions.

Formulate Questions

When you've increased your awareness of the challenge or problem you've identified, ask questions that will generate solutions. Think about the obstacles you might face and the opportunities they could present.

Explore Ideas

Generate ideas that answer the challenge questions you identified in step 1. It can be tempting to consider solutions that you've tried before, as our minds tend to return to habitual thinking patterns that stop us from producing new ideas. However, this is a chance to use your creativity .

Brainstorming and Mind Maps are great ways to explore ideas during this divergent stage of CPS. And our articles, Encouraging Team Creativity , Problem Solving , Rolestorming , Hurson's Productive Thinking Model , and The Four-Step Innovation Process , can also help boost your creativity.

See our Brainstorming resources within our Creativity section for more on this.

Formulate Solutions

This is the convergent stage of CPS, where you begin to focus on evaluating all of your possible options and come up with solutions. Analyze whether potential solutions meet your needs and criteria, and decide whether you can implement them successfully. Next, consider how you can strengthen them and determine which ones are the best "fit." Our articles, Critical Thinking and ORAPAPA , are useful here.

4. Implement

Formulate a plan.

Once you've chosen the best solution, it's time to develop a plan of action. Start by identifying resources and actions that will allow you to implement your chosen solution. Next, communicate your plan and make sure that everyone involved understands and accepts it.

There have been many adaptations of CPS since its inception, because nobody owns the idea.

For example, Scott Isaksen and Donald Treffinger formed The Creative Problem Solving Group Inc . and the Center for Creative Learning , and their model has evolved over many versions. Blair Miller, Jonathan Vehar and Roger L. Firestien also created their own version, and Dr Gerard J. Puccio, Mary C. Murdock, and Marie Mance developed CPS: The Thinking Skills Model. [4] Tim Hurson created The Productive Thinking Model , and Paul Reali developed CPS: Competencies Model. [5]

Sid Parnes continued to adapt the CPS model by adding concepts such as imagery and visualization , and he founded the Creative Studies Project to teach CPS. For more information on the evolution and development of the CPS process, see Creative Problem Solving Version 6.1 by Donald J. Treffinger, Scott G. Isaksen, and K. Brian Dorval. [6]

Creative Problem Solving (CPS) Infographic

See our infographic on Creative Problem Solving .

Creative problem solving (CPS) is a way of using your creativity to develop new ideas and solutions to problems. The process is based on separating divergent and convergent thinking styles, so that you can focus your mind on creating at the first stage, and then evaluating at the second stage.

There have been many adaptations of the original Osborn-Parnes model, but they all involve a clear structure of identifying the problem, generating new ideas, evaluating the options, and then formulating a plan for successful implementation.

[1] Entrepreneur (2012). James Dyson on Using Failure to Drive Success [online]. Available here . [Accessed May 27, 2022.]

[2] Creative Education Foundation (2015). The CPS Process [online]. Available here . [Accessed May 26, 2022.]

[3] Kaner, S. et al. (2014). 'Facilitator′s Guide to Participatory Decision–Making,' San Francisco: Jossey-Bass.

[4] Puccio, G., Mance, M., and Murdock, M. (2011). 'Creative Leadership: Skils That Drive Change' (2nd Ed.), Thousand Oaks, CA: Sage.

[5] OmniSkills (2013). Creative Problem Solving [online]. Available here . [Accessed May 26, 2022].

[6] Treffinger, G., Isaksen, S., and Dorval, B. (2010). Creative Problem Solving (CPS Version 6.1). Center for Creative Learning, Inc. & Creative Problem Solving Group, Inc. Available here .

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Problem-Solving, Decision-Making, and Intelligence

Introduction, problem-solving and creativity, decision-making and reasoning, human and artificial intelligence.

The topics of problem-solving, creativity, decision-making, reasoning and intelligence are closely related, to the point of overlapping. For instance, scholars still face difficulties in unanimously defining the constructs of creativity and intelligence (Jaarsveld & Lachmann, 2017). Smith et al. (2009) defined creativity as “anything novel with a potential of value or utility” (as cited in Goldstein, 2019, p. 377). However, such a utilitarian approach to creativity does not describe a creative aspect of art or music since these spheres do not provide an objective, measurable value. Intelligence has once been perceived as an ability to solve well-defined problems through algorithms, such as proving a theorem (Jaarsveld et al., 2012). This approach has been challenged by Kaufmann (2013), who argued that distinguishing intelligence from creativity disservices both (as cited in Silvia, 2015). Given these circumstances, it is necessary to provide a theoretical clarification, so employers and team leaders could select staff members suitable for achieving specific professional goals.

Problem-solving strategies largely depend on the type of problem and thinking necessary for solving it. Well-defined problems, which have clear initial and end states, are usually solved through convergent thinking. Cropley (2006) defined convergent thinking as applying logical and conventional search to produce an already known answer (as cited in Jaarsveld & Lachmann, 2017). On the contrary, divergent thinking produces new approaches and brings novel, unusual, or surprising answers to unknown, ill-defined problems (Jaarsveld & Lachmann, 2017). Whereas convergent thinking relies on pure intelligence, divergent thinking utilizes creativity.

Creative problem-solving has a theoretical explanation that breaks it down into distinctive stages. According to Basadur et al. (2020), the process of creative problem-solving consists of four stages linked by brain networks (as cited in Goldstein, 2019, p. 378). First, the brain generates a problem or recognizes its existence; next, it formulates the problem and develops ideas. At the third stage, the brain evaluates all generated ideas and selects the most appropriate solution. Finally, the selected solution becomes an actual product; this stage may take multiple cycles of trials and errors (Goldstein, 2019). Therefore, creative problem-solving should be perceived as a lengthy process that involves practice and goes far beyond idea generation. One cannot become competent and solve problems creatively without practicing and making mistakes.

Therefore, nurturing creativity in problem-solving requires meeting two critical conditions. Most importantly, creative problem-solving demands a right for trying and making errors in the process. Goldstein (2019) provided a case of Jorge Odon, a car mechanic who invented a childbirth assistance device based on the idea of getting a cork out of the bottle without breaking it. Odon’s work took years of development in order to turn an idea into a functional product. Secondly, inhibition, a process of limiting cognition to goal-relevant information, is counter-productive to creativity (Radel et al., 2015). For example, soccer is a simple game where the team which scores more goals wins. However, elite young soccer players also showed better results in creativity tests (Vestberg, 2017). In this regard, inhibiting talents with strict playing discipline would eliminate their advantage in creativity over the average players. Overall, one should realize that creativity is a valuable ability, but at the same time, not mandatory for solving well-defined problems.

It is difficult to deny that humans are not always rational in their decisions. However, even the wrong, irrational human decisions are still based on certain rationale. Goldstein (2019) defined two conclusion- and decision-making methods: inductive and deductive reasoning. Inductive reasoning derives judgments from observations, whereas deductive reasoning utilizes syllogisms and general logic (Goldstein, 2019). Both methods explain how a particular decision was made; however, they are both vulnerable to cognitive distortions, which may lead to errors in decision-making.

Decisions based on deductive reasoning are subjected to heuristics, rules of thumb, which the human brain applies to save time and energy. While heuristics often result in correct decisions, they may sometimes lead to undesirable consequences. For example, groupthink bias stemming from the confirmation heuristic led to the Challenger space shuttle crash in 1986 (Murata et al., 2015). More specifically, the group interested in Challenger’s launch disregarded potential risks associated with cold weather and applied pressure on the O-ring (rubber seal) manufacturer. The group created an illusion of unanimity and invulnerability that ultimately led to O-ring rupture and Challenger’s crash during the launch procedure.

Deductive reasoning based on drawing conclusions from the premises can also be flawed. In particular, belief bias deprives the arguments based on deductive reasoning from impartiality. Goldstein (2019) provided an example of a statement, which can be shaped into the following syllogism:

• All Congress members from New York oppose the new tax law;
• Some of the Congress members who oppose the new tax law take bribes;
• Some Congress members from New York take bribes.

This example shows how belief bias can lead to false and negative conclusions about a whole group. In this regard, one should try to eliminate biases in reasoning, especially if working in a diverse population setting. The American Psychological Association (APA, 2017) Code of conduct explicitly states that psychologists must be aware of and respect differences based on age, gender, national origin, and other group identity factors. Therefore, heuristics- and belief-based flaws in reasoning may lead to significant ethical conflicts and reputational risks.

Human intelligence is a multifaceted construct, which can be defined in several ways. For the sake of clarity, this section will use the traditional definition of intelligence as a problem-solving ability (Simon and Newel, 1971, as cited in Jaarsveld et al., 2012). Therefore, human intelligence is an individual’s ability to solve problems by applying convergent and divergent thinking. The latter part is important, as Jaarsveld et al. (2012) and multiple other scholars perceived creativity, associated with divergent thinking, as an element of intelligence. As such, it is possible to claim that human intelligence, at least its creative aspect, can be enhanced through practical experience. Furthermore, Carroll (1982) discussed the development of the IQ measure and argued that intelligence consists of at least seven independent factors (ac cited in Jaarsveld & Lachmann, 2017). In this regard, intelligence should not be limited to the hereditary dimension.

The subject of artificial intelligence (AI) plays an important part in discussions about the future of humanity. In 1956, Herb Simon and Alan Newel created a real “thinking machine” that used humanlike reasoning to solve problems (Goldstein, 2019, p. 15). In this regard, AI development may have significant socioeconomic and ethical implications, as AI will continue to improve in logical and creative reasoning. At the same time, AI already possesses a colossal advantage in computing power compared to the human brain. Therefore, many human employees will likely lose their jobs since human competitive advantage over machines will become non-existent.

Problem-solving, creativity, decision-making, and intelligence are overlapping constructs, often indistinguishable even for scholars. For instance, intelligence was traditionally defined as the problem-solving ability, and only recently have scholars started attributing creativity to it. Intelligence substantially affects decision-making, especially reasoning behind decisions. Regardless of theoretical aspects, it is important to understand that intelligence, especially its creative aspect, can be enhanced through practical experience. Finally, it is necessary to avoid bias in reasoning in order to prevent the emergence of severe ethical issues.

American Psychological Association. (2017). Ethical principles of psychologists and Code of conduct . Web.

Goldstein, B. E. (2019). Cognitive psychology: Connecting mind, research, and everyday experience (5th ed.). Cengage.

Jaarsveld, S., Lachmann, T., & van Leeuwen, C. (2012). Creative reasoning across developmental levels: Convergence and divergence in problem creation. Intelligence , 40 (2), 172-188. Web.

Jaarsveld, S., & Lachmann, T. (2017). Intelligence and creativity in problem solving: The importance of test features in cognition research. Frontiers in Psychology , 8 , 134. Web.

Murata, A., Nakamura, T., & Karwowski, W. (2015). Influence of cognitive biases in distorting decision making and leading to critical unfavorable incidents. Safety , 1 (1), 44-58. Web.

Radel, R., Davranche, K., Fournier, M., & Dietrich, A. (2015). The role of (dis) inhibition in creativity: Decreased inhibition improves idea generation. Cognition , 134 , 110-120. Web.

Silvia, P. J. (2015). Intelligence and creativity are pretty similar after all. Educational Psychology Review , 27 (4), 599-606. Web.

Vestberg, T., Reinebo, G., Maurex, L., Ingvar, M., & Petrovic, P. (2017). Core executive functions are associated with success in young elite soccer players. PloS One , 12 (2), e0170845. Web.

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Module 5: Thinking and Analysis

Solving problems creatively, learning outcomes.

• Describe the role of creative thinking skills in problem-solving

Problem-Solving with Creative Thinking

Creative problem-solving is a type of problem-solving. It involves searching for new and novel solutions to problems. Unlike critical thinking, which scrutinizes assumptions and uses reasoning, creative thinking is about generating alternative ideas—practices and solutions that are unique and effective. It’s about facing sometimes muddy and unclear problems and seeing how things can be done differently—how new solutions can be imagined. [1]

You have to remain open-minded, focus on your organizational skills, and learn to communicate your ideas well when you are using creative thinking to solve problems. If an employee at a café you own suggests serving breakfast in addition to the already-served lunch and dinner, keeping an open mind means thinking through the benefits of this new plan (e.g., potential new customers and increased profits) instead of merely focusing on the possible drawbacks (e.g., possible scheduling problems, added start-up costs, loss of lunch business). Implementing this plan would mean a new structure for buying, workers’ schedules and pay, and advertising, so you would have to organize all these component areas. And finally, you would need to communicate your ideas on how to make this new plan work not only to the staff who will work the new shift, but also to the public who frequent your café and the others you want to encourage to try your new hours.

We need creative solutions throughout the workplace—whether board room, emergency room, or classroom. It was no fluke that the 2001 revised Bloom’s cognitive taxonomy, originally developed in 1948, placed a new word at the apex— creating . That  creating is the highest level of thinking skills.

Bloom’s Taxonomy is an important learning theory used by psychologists, cognitive scientists, and educators to demonstrate levels of thinking. Many assessments and lessons you’ve seen during your schooling have likely been arranged with Bloom’s in mind. Researchers recently revised it to place creativity—invention—as the highest level

“Because we’ve always done it that way” is not a valid reason to not try a new approach. It may very well be that the old process is a very good way to do things, but it also may just be that the old, comfortable routine is not as effective and efficient as a new process could be.

The good news is that we can always improve upon our problem-solving and creative-thinking skills—even if we don’t consider ourselves to be artists or creative. The following information may surprise and encourage you!

• Creative thinking (a companion to critical thinking) is an invaluable skill for college students. It’s important because it helps you look at problems and situations from a fresh perspective. Creative thinking is a way to develop novel or unorthodox solutions that do not depend wholly on past or current solutions. It’s a way of employing strategies to clear your mind so that your thoughts and ideas can transcend what appear to be the limitations of a problem. Creative thinking is a way of moving beyond barriers. [2]
• As a creative thinker, you are curious, optimistic, and imaginative. You see problems as interesting opportunities, and you challenge assumptions and suspend judgment. You don’t give up easily. You work hard. [3]

Is this you? Even if you don’t yet see yourself as a competent creative thinker or problem-solver, you can learn solid skills and techniques to help you become one.

Creative Problem-Solving: Fiction and Facts

As you continue to develop your creative thinking skills, be alert to perceptions about creative thinking that could slow down progress. Remember that creative thinking and problem-solving are ways to transcend the limitations of a problem and see past barriers. It’s a way to think outside the box.

creative problem-solving: a practice that seeks new and novel solutions to problems, often by using imagination rather than linear reason

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• "Critical and Creative Thinking, MA." University of Massachusetts Boston . 2016. Web. 16 Feb 2016. ↵
• Mumaw, Stefan. "Born This Way: Is Creativity Innate or Learned?" Peachpit. Pearson, 27 Dec 2012. Web. 16 Feb 2016. ↵
• Harris, Robert. "Introduction to Creative Thinking." Virtual Salt. 2 Apr 2012. Web. 16 Feb 2016. ↵
• Ibid. ↵
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• Text adaptation. Authored by : Claire. Provided by : Ivy Tech. Located at : http://ivytech.edu/ . License : CC BY: Attribution

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Reasoning, Problem Solving, Creativity and Decision Making

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Reasoning plays a significant role in adjusting to one’s environment. It not only controls one’s cognitive activities, but also the total behavior and personality is affected by the proper or improper development of one’s reasoning ability.

Reasoning is step-wise thinking with a purpose or goal in mind. Reasoning is the term applied to highly purposeful controlled selective thinking. Reasoning is the word used to describe the mental recognition of cause-and –effect relationships.

According to Munn , “Reasoning is combining past experiences in order to solve a problem which cannot be solved by mere reproduction of earlier solution”.

Types Of Reasoning

1. inductive reasoning : (particular to general).

In this type of reasoning, we usually follow the process of induction. Induction is a way of providing a statement or generalizing or rules by specific cases to general conclusion.

2. Deductive reasoning : (General to particular)

Deductive reasoning is just opposite to inductive reasoning. Here, one’s start completely agreeing with some already discovered, or per-establish generalized fact or principle tries to apply it to particular cases.

Problem Solving

Problem solving as a deliberate in serious act, involve the use of scientific method, higher thinking and systematic step for the realization of the set goals.

According to Skinner (1968) “ Problem solving is a process of overcoming difficulties that appear to interface with the specific goals”.

Scientific Method of Problem Solving

Problem identification :(The first step in the problem solving behavior of an individual concerned is identification or awareness of the difficulty or problems that need a solution such as psychological, economic, political, and socio-cultural problems)

Problem understanding :(The problem felt by the individual should be properly identified by a careful analysis: Concept, theory, and previous experienced)

Collection of the relevant information :(In this step, the individual is required to collect all the relevant information about the problem through all possible sources: cause and effect relationship, variable or facts)

Formulation of hypothesis (In the light of the collected relevant information and nature of this problem, one may in some serious cognitive activities to think of the various possibilities for the solution of this problems.)

Selection of a proper solution : (In this important step, all the possible solution, through out of in the previous step are closely analyzed and evaluate: follow systematic and scientific method)

Verification of concluded solution or testing hypothesis : (The solution arrived at conclusion draw most be further verified by utilizing it in the solution of the various problems: test hypothesis)

Obstacle in Problem Solving Methods

• Mental set and habit
• Functional fixedness
• Information bias
• Irrelevant information
• Over confidence
• Representation bias.

Creativity is the ability to produce ideas. It is a phenomenon whereby something new and somehow valuable is formed. Creativity is the ability and disposition to produce novelty. Creativity as ability( Knowledge, idea, perception, past experience etc).

Relation to intelligence( Education, concept and idea). Creativity as process ( Lifelong and continuous process)

Decision Making

In psychology, decision making is a cognitive ability related to specific issues. It is regarded as the cognitive process resulting in the selection of a belief or a course of action among several alternative possibilities. Every decision making process produce a final choice. Decision making is the process of choosing between two or more alternatives. Individual make choices based on their personal performance, values and goals. Every decision has a outcome and involve risk.

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• Identify your goals
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• Consider the consequences
• Make your decision
• Evaluate your decision

Creativity in problem solving: integrating two different views of insight

• Original Paper
• Open access
• Published: 02 September 2021
• volume  54 ,  pages 83–96 ( 2022 )

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• Per Øystein Haavold   ORCID: orcid.org/0000-0002-6786-9400 1 &
• Bharath Sriraman 2  

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Even after many decades of productive research, problem solving instruction is still considered ineffective. In this study we address some limitations of extant problem solving models related to the phenomenon of insight during problem solving. Currently, there are two main views on the source of insight during problem solving. Proponents of the first view argue that insight is the consequence of analytic thinking and a sequence of conscious and stepwise steps. The second view suggests that insight is the result of unconscious processes that come about only after an impasse has occurred. Extant models of problem solving within mathematics education tend to highlight the first view of insight, while Gestalt inspired creativity research tends to emphasize the second view of insight. In this study, we explore how the two views of insight—and the corresponding set of models—can describe and explain different aspects of the problem solving process. Our aim is to integrate the two different views on insight, and demonstrate how they complement each other, each highlighting different, but important, aspects of the problem solving process. We pursue this aim by studying how expert and novice mathematics students worked on two ill-defined mathematical problems. We apply both a problem solving model and a creativity model in analyzing students’ work on the two problems, in order to compare and contrast aspects of insight during the students’ work. The results of this study indicate that sudden and unconscious insight seems to be crucial to the problem solving process, and the occurrence of such insight cannot be fully explained by problem solving models and analytic views of insight. We therefore propose that extant problem solving models should adopt aspects of the Gestalt inspired views of insight.

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1 Introduction

Most mathematics educators would probably agree that the development of students’ problem solving abilities is an important objective of instruction. Thus there has been a considerable amount of research on problem solving in the last several decades (Lester, 2013 ). In general, researchers into problem solving have usually defined the term problem as tasks or questions that an individual or group of individuals do not immediately know how to answer (Lester, 2013 ). However, this definition says very little about how to teach individuals to become better problem solvers (Lester, 2013 ). Several models of problem solving have therefore been developed to describe and explain factors and processes involved in problem solving—most of which have drawn heavily on Pólya’s ( 1949 ) famous four-stage model of problem solving. Nevertheless, problem solving instruction is still considered ineffective. There are many reasons for this perception, but one key issue is the lack of concern for the complexity and the many factors involved in problem solving processes (Lester, 2013 ).

The focus of this paper is one of the more subtle yet essential factors involved in problem solving. Ever since the Gestaltists first began studying problem solving nearly 100 years ago, insight in problem solving has been of interest to psychologists (Hadamard, 1945 ; Ohlsson, 2011 ; Poincaré, 1948 ; Weisberg, 2015 ). Here, it is important to note that insight ( Einsicht ) within the Gestalt approach, and much of the literature on insight and problem solving, have a broader meaning than the standard definition in English. According to the Gestaltists, an individual’s comprehension of a problem cannot be reduced to a collection of individual perceptual features. Instead, the individual perceives a particular Gestalt of the problem, which can be interpreted as the totality of the relations between its parts. Insight, to the Gestaltists, was therefore considered a mental restructuring of the problem into a more productive Gestalt (Ohlsson, 2011 ; Wertheimer, 1959 ). In this study, we draw on the Gestalt view and consider insight as a perceptual and conceptual restructuring of a problem in a more productive manner. This view of insight has also been described as mentally crossing a ‘logical gap’, and it has often been referred to as a sudden and unexpected feeling of comprehension during an attempt at solving a problem (Ohlsson, 2011 ; Sternberg & Davidson, 1995 ).

Currently there are two main views on the source of insight during problem solving. Proponents of the first view argue that insight is the consequence of analytic thinking in which the problem is matched with information in memory. The solution typically unfolds in a sequence of conscious steps, and the individual has a feeling of steady incremental progress. Insight is gained gradually and consciously. The Gestaltists called this reproductive thinking (Weisberg, 2015 ). The second view, termed productive thinking by the Gestaltists, suggests that insight is the result of a particular set of processes distinct from conscious analytical thinking. Here, insight is the result of unconscious processes that come about only after an impasse has occurred. Furthermore, insight is gained quickly, often spontaneously, and as a result of mental restructuring of the problem (Weisberg, 2015 ). Extant models of problem solving within mathematics education tend to highlight the first view of insight. Lester and Kehle ( 2003 ), for example, characterizes successful problem solving as “coordinating previous experiences, knowledge, familiar representations and patterns of inference, and intuition…” (p. 510). Although unconscious processes such as intuition are sometimes mentioned, they are usually not explained or elaborated in problem solving models, and the emphasis is on analytic and conscious cognitive processes. On the other hand, within the field of creativity research and theoretical models of creativity—in particular Gestaltist inspired research—analytic thinking is considered unable to produce novelty. Highly inspired by the Gestaltists, the focus has therefore often been on more spontaneous processes that can result in a new interpretation of the problem (Weisberg, 2015 ).

In this study, we investigated how the two views of insight—a and corresponding set of models—can describe and explain different aspects of the problem solving process. The aim of our study was to integrate the two different views on insight, and demonstrate how they complement each other, each highlighting different, but important aspects of the problem solving process. We pursued this aim by studying how expert and novice mathematics students at a large research university in Norway approached and worked on two ambiguous and ill-defined mathematical problems. We then applied both a problem solving model and a creativity model in our analysis of students’ work on the two problems, in order to compare and contrast aspects of insight during the students’ work. More specifically, we set out to answer the following research question:

How do expert and novice students approach and attempt to gain insight into ill-defined mathematical problems?

To work towards our aim, we made use of a novice-expert comparison, which has proven to be useful within cognitive research (National Research Council, 2000 ). Expertise has commonly been described as 10 years of intense preparation in some field (Ericsson & Lehmann, 1996 ), or “proficiency taken to its highest level” (Glaser, 1987 ). However, expertise has also been defined in terms of cognitive development and knowledge structures (Hoffman, 1998 ), and described as a continuum or multiple stages rather than a dichotomy between experts and novices (e.g. Dreyfus & Dreyfus, 2005 ). In this study, we therefore differentiated between expert and novice students according to educational background and mathematical attainment. The main rationale for this choice was to contrast mathematical performance with two different theoretical perspectives of insight during problem solving.

We also made use of ill-defined problems , which are those problems for which there are conflicting assumptions, evidence, and opinions that may lead to different solution (e.g., Kitchener, 1983 ; Krutetskii, 1976 ). They force the problem solver to deal with uncertainty, and facilitate multiple possible approaches by looking at the problem in new and productive ways. Ill-defined problems are therefore particularly useful for facilitating perceptual restructuring and insight during the problem solving process (Webb et al., 2016 ).

1.1 Problem solving models

Problem solving has long been of interest to mathematics education researchers. At the root of this research, and most problem-solving frameworks, lies the work of the eminent mathematician George Polya (Schoenfeld, 1985a ). In his work How to Solve It , Pólya ( 1949 ) presented a four-step model of problem solving which consisted of the four steps understanding , planning , implementing , and looking back . The model outlines problem solving as a systematic and gradual process that facilitates insight primarily by building on prior knowledge and conscious evaluation. Because of the structured and pedagogical approach to problem solving and the explicit focus on prior knowledge, Polya’s four step model has become the most popular approach to teaching and learning problem solving (Liljedahl et al., 2016 ).

One of the shortcomings of Polya’s model is that research generated under its umbrella focused almost entirely on heuristics, or rules of thumb for making progress on difficult problems, while ignoring “managerial skills necessary to regulate one’s activity (metacognitive skills)” (Lester, 1985 , p. 62). Lester ( 1985 ) and Schoenfeld ( 1985a ) suggested that metacognitive activity (knowledge of one’s thought processes or self-regulation) underlies the application of heuristics and algorithms. As a result, Polya’s model was modified (Lester, 1985 ; Schoenfeld, 1985a ) to include a cognitive component and a metacognitive component. In the cognitive component, the four phases of understanding, planning, implementing, and looking back are labeled as orientation , organization , execution , and verification respectively. The metacognitive component consists of three classes of variables attributed to Flavell and Wellman ( 1977 ). This model purports to describe the four cognitive categories in terms of ‘points’ where metacognitive actions occur during problem-solving (see Fig.  1 ).

The cognitive-metacognitive model (Lester, 1985 )

The cognitive component Orientation refers to strategic behavior to assess and understand a problem. It includes comprehension strategies, analysis of information, initial and subsequent representation, and assessment of familiarity and chance of success. Organization refers to identification of goals, global planning, and local planning. The category of execution refers to regulation of behavior to conform to plans. It includes performance of local actions, monitoring progress and consistency of local plans, and trade-off decisions (speed vs. accuracy). Finally, verification consists of evaluating decisions made and evaluating the outcomes of the executed plans. It includes evaluation of actions carried out in the orientation, organization, and execution categories. The metacognitive component of the model is comprised of three classes of variables, namely person variables, task variables, and strategy variables. Person variables refer to an individual’s belief system and affective characteristics that may influence performance. Task variables refer to features of a task, such as the content, context, structure, syntax and process. An individual’s awareness of features of a task influences performance. Finally, strategy variables are those that refer to an individual’s awareness of strategies that help in comprehension, organizing, executing plans, and checking and evaluation.

The main purpose of this model is to show that metacognitive actions can influence cognitive behavior at all phases of problem solving (Lester, 1985 ; Schoenfeld, 1985a ). The introduction of metacognitive actions is an important modification of Polya’s model (Liljedahl et al., 2016 ). In contrast to Polya’s non-specific heuristics, the introduction of metacognitive components is an acknowledgment that problem solving is an emergent process that depends on the individual’s prior knowledge and internal dialogue. Unlike Pólya ( 1949 ), who prescribed heuristics applicable to all problems and problem solvers, Schoenfeld ( 1985a ) and Lester ( 1985 ) portray problem solving heuristics as personal objects that are limited to the individual’s existing knowledge and understanding of the problem (Liljedahl et al., 2016 ).

Nevertheless, both the original model by Pólya ( 1949 ) and the revised model (Lester, 1985 ; Schoenfeld, 1985a ) lay out problem solving as a conscious and incremental process in which the problem solver gains insight primarily through past experience and conscious evaluation. Generally, the first step in the process, after gaining an initial understanding of the problem, would entail attempts at matching the problem with prior knowledge and evaluating whether a solution method could be transferred to the new problem. If this attempt is unsuccessful, the problem solver would then move on to applying heuristic methods. Through the use of heuristics, the problem solver attempts to modify the present state of the problem so that he/she can advance towards the final goal (Weisberg, 2015 ). Of course, the process is not nearly this simple or linear, but it provides a general overview of the analytic approach to problem solving. Insight, or restructuring of the problem in a new and more productive manner, is gradually gained through a stepwise and conscious process.

However, within most of creativity research, which leans heavily on the Gestalt view of insight, this view of gradually gaining insight is rejected (Weisberg, 2015 ). Problem solving models, and similar reproductive approaches to insight in problem solving, do not explain how existing knowledge and analytic thinking can produce novel ideas, which are usually necessary for solving problems that require some form of insight. The argument is essentially that a logical system can only produce information that is already present, at least implicitly, in the premises, and that is therefore not novel (Weisberg, 2015 ). Therefore, insight has to be the result of some kind of special cognitive process different from the conscious and evaluative approach that characterizes analytic thinking (Ohlsson, 2011 ).

1.2 Creativity models

From the perspective of creativity research then, when one tries to solve a problem the individual will first try solutions based on similarities with other problems and consciously evaluate the progress. However, those attempts will often fail as problems that require some form of novelty will not be solved by transferring methods from similar problems. The problem solver will eventually reach an impasse. It is at this point that the person may suddenly and unconsciously gain insight through a mental restructuring of the problem and come up with a solution. This notion of insight as a result of sudden and unconscious illumination is usually attributed to the Gestalt psychologists, and it is currently the dominant view of creative thinking (Ohlsson, 2011 ).

According to the Gestaltists, creative thinking and insight follow a sequence of four stages, namely, preparation - incubation - illumination , and verification (Wallas, 1926 ; Hadamard, 1945 ; Poincaré, 1948 ). The first stage consists of working hard to understand the problem at hand. Poincaré calls this the preliminary period of conscious work. The second stage occurs when the problem is put aside for a period of time and the mind is occupied with other things. The third stage is where the solution suddenly appears while the individual is perhaps engaged in other unrelated activities. "This appearance of sudden illumination is a manifest sign of long, unconscious prior work." (Poincaré, 1948 , p. 16). However, the creative process does not end here. There is a fourth and final stage, namely verification, which includes expressing the results by language or writing. At this stage one verifies the result, makes it precise, and looks for possible extensions through utilization of the result.

More recently, Ohlsson ( 2011 ) reformulated the four step Gestalt model of creativity as the insight sequence in an effort to draw a clear distinction between problem solving through analytic thinking and problem solving through sudden insight. Furthermore, while the Gestaltists were concerned with insight and creative thought on timescales of months and years, proponents of more recent Gestalt inspired research that uses the insight sequence, consider aspects of insight and creativity also on much shorter timescales (Beghetto & Karwowski, 2019 ; Ohlsson, 2011 ). The insight sequence describes successful problem solving as a chain consisting of the following events: attempted solution \(\to\) consistent failure \(\to\) impasse \(\to\) restructuring \(\to\) insight \(\to\) Solution. Unlike problem solving models that describe insight as something gained gradually through analytic and conscious thinking, the insight sequence emphasizes impasse and sudden (and unconscious) cognitive restructuring as the basis for insight (Weisberg, 2015 ). Presently, this restructuring is thought to occur by an impasse that causes an altered balance in a lower layer of cognitive processing systems, which leads to a new, and possibly more productive, representation in a higher and more conscious layer (Ohlsson, 2011 ).

An important idea in the setting of perceptual restructuring is cognitive flexibility . Cognitive flexibility refers to our ability to switch between different mental sets, tasks and strategies in light of uncertainty and impasse (Ionescu, 2012 ). According to Nijstad et al. ( 2010 ), cognitive flexibility is a key element for achieving creative insights, problem solutions, or ideas through the use of flexible switching among categories, approaches, and sets, and through the use of remote (rather than close) associations. Cognitive fixation , on the other hand, is the counterpart to flexibility. The notion of people struggling to come up with creative solutions because they fixate, or fail to abandon non-productive strategies, has its roots a long way back in psychological literature and features particularly in the writings of the Gestalt school (Haylock, 1987 ).

Although cognitive flexibility seems to relate to the intuitive concept, we still lack a clear definition and comprehension of the phenomenon (Ionescu, 2012 ). For example, in a review of the literature, Ionescu ( 2012 ) identified several behaviors that are considered flexible, as follows: switching between tasks or multitasking; changing behavior in light of a new rule; finding a new solution to a problem; and creating new knowledge or tools. In this paper, we consider flexibility as the ability to break away from inappropriate approaches, i.e., particular methods and strategies, within a single problem (Haylock, 1987 ). Regarding cognitive fixation, Haylock ( 1987 ) concluded that there are two particularly important types of fixation in mathematical problem solving: algorithmic fixation and content universe fixation . Algorithmic fixation is closely related to the Einstellung effect , and it refers to individuals continuing to use an initially successful algorithm or method learnt beforehand or developed through the sequence of tasks themselves. The other type of fixation, content universe fixation, refers to situations where students’ thinking about mathematical problems is restricted unnecessarily to an insufficient range of elements that may be used or related to the problem (Haylock, 1987 ). The overcoming of these kinds of fixations, and thus allowing the mind to range over a wider set of possibilities than might first come to one’s conscious awareness, is an important aspect of successful problem solving.

1.3 Expert and novice problem solvers

Besides the use of metacognition to describe phases of problem-solving performance, another widespread approach within the problem solving research paradigm has been to describe in detail solutions used by ‘expert’ problem solvers and compare this to solutions of ‘novices’ (Simon & Simon, 1978 ). The rationale behind this genre of research was to identify strategies used by experts, and develop prescriptive models to teach students how to problem solve like experts. The main findings of studies in the ‘expert-novice’ genre were that experts and novices differed in their problem solving strategies because of the following:

Knowledge for understanding and representing problems (Orientation).

Strategic knowledge (Organization).

Repertoires of known procedures and familiar patterns (Execution and Verification).

Experts are adept at creating a representation of the problem, and understanding it in terms of fundamental principles. While experts tend to focus on structural properties of problems, novices place a greater emphasis on surface properties. Furthermore, novices are often not able to construct problem representations that are helpful in achieving solutions. This description fits into the orientation category of Lester’s ( 1985 ) cognitive-metacognitive model. Experts also solve problems by using a process of successful refinements. Global planning and qualitative analysis characterize their strategies, before generating specific equations to solve the problems. Novices, on the other hand, tend to go directly from the problem text in search of equations that could be used. This behavior fits into the organization category of Lester’s model. Finally, experts have developed a repertoire of problem types and solution methods besides having an extensive knowledge of basic principles. Novices are lacking much of this knowledge and experience. This observation fits into the execution and verification categories of Lester’s model.

Expert and novice differences have also been studied within creativity research. In general, it is believed that the more knowledge we have in a domain, the more flexible problem solvers we are in that domain (Ionescu, 2012 ). The most common explanation for this aspect is that experts have acquired, over many years of practice, a vast knowledge base of techniques, methods, strategies, etc., when solving problems. This large knowledge base enables the expert often to solve novel problems by small modifications to what they already know, which in turn requires relatively minor cognitive effort (Ohlsson, 2011 ). However, it has also been argued that expertise could lead to less flexibility and more cognitive fixations. Expertise is generally considered to be domain specific, as skills tend to go from higher levels of generality to greater specificity as a result of practice (Ohlsson, 2011 ). As a result, it is conceivable that expertise can lead to less flexibility and a greater fixation on a narrow pattern of previous experiences. Others have found non-linear relationships between expertise and flexibility. In a series of clever studies on the relationship between expertise and flexibility among chess experts, Bilalic et al. ( 2008 ) found a clear difference between ordinary (3 SDs above average performance) and super experts (5 SDs above average performance). While ordinary chess experts demonstrated cognitive fixation, possibly caused by knowledge specificity, the super experts demonstrated cognitive flexibility and not fixations induced by previous mental sets. Somewhat similarly, Elgrably and Leikin ( 2021 ) recently investigated the relationship between different types of mathematical expertise and creativity. Two groups of students—expert problem solvers in mathematics and mathematics majors in university—were given a problem-posing-through investigation-task. The results showed that the expert problem solvers posed three times as many problems, with more flexible and original properties, than the mathematics majors. These findings are in line with much of the literature that indicates a clear, yet somewhat nuanced relationship between mathematical knowledge and flexibility (e.g., Haavold et al., 2020 ).

2.1 Data collection and materials

To answer our research question and work towards the aim of the study, we investigated how expert and novice mathematics students approached and attempted to gain insight into two ill-defined mathematical problems. We report here on data from task-based interviews with small (3–4) groups of students. Each session lasted for about 60 min, in which the students worked on two ill-defined mathematical problems. During the interview, the interviewer answered clarification questions, but deflected more task specific and content related questions back to the students. We opted to make use of group based protocols as they are particularly appropriate for observing decision-making and students’ real social cognitive behavior (Schoenfeld, 1985b ).

The participants in the study consisted of two different groups of students aged 22–24 years, both of which are in their fifth and final year of their study programmes. All participants volunteered and were recruited by the first author of this paper via postings on the university’s learning management system (Canvas). The first group (novice group) consisted of 12 students, divided into four groups of three, enrolled in a 5 year pre-service teacher education programme specifically aimed at teaching in primary school and lower secondary school. The students in the novice group were not mathematics specialists, and had studied only 1 year of mathematics after upper secondary school. The mathematical content in their previous mathematics studies was focused on elementary mathematical topics such as geometry, algebra, and numeracy—with a particular didactical emphasis. The expert group consisted of four master’s students who excelled at graduate level mathematics. We classified this group as experts as they all were, at the time, working on their master’s degree in mathematics and had demonstrated proficiency (i.e., high grades—85th percentile) in advanced mathematics courses in calculus, number theory, algebra, and statistics.

Two ill-defined problems were given to the students. Each of them provided different types of misdirection and extensions of the problem space for the problem solvers.

Problem 1: the Roman inheritance problem The first problem comes from The Moscow Puzzles and is usually referred to as the Roman problem:

A dying Roman knowing his wife was pregnant, left a will saying that if she had a son, he would inherit two-thirds of the estate and the widow one-third, but if she had a daughter, the daughter would get one-third and the widow two-thirds. Soon after his death, his widow had twins- a boy and a girl, a possibility the will had not foreseen. What division of the estate keeps as closely as possible to the terms of the will?

There isn’t a single right answer to this problem as the constraints are not fully exhaustive. This presents the students with a problem that can be repeatedly restructured and facilitate many approaches, and insight is predicated on recognizing this ambiguity. The Roman jurist, Salvian Julian, proposed for instance that the father’s intent is that the daughter should receive half as much as her mother, and the son twice as much. The inheritance should be divided into seven parts, and the mother should get two parts, the son four parts, and the daughter one part. However, an opposing view is that the father wished the mother to inherit at least 1/3 of the estate, but Salvian Julian would give her only 2/7. Therefore, give instead the mother 1/3 and divide the rest between son and daughter according to the intended ratio of four to one. The solution of the problem depends on which of the constraints the line of reasoning is based on.

Problem 2: wrong arithmetic, but correct result The second problem was based on the idea of mathematical pathologies, which refer to examples that are specifically designed to violate properties that are perceived as valid (Sriraman & Dickman, 2017 ):

Sometimes the wrong method gives us the right answer. When does this method work?

This example is ‘cooked up’ knowingly to violate common properties of fraction multiplication. To gain insight into this problem, the students need to accept the counterintuitive properties as a premise and break away from established mental sets related to arithmetic. So when does this method work? One possible approach is to use algebra to identify the constraints of each digit:

which boils down to

and finally

As ten is on the left side, there are now four cases that can be investigated further: \(b-a=5, b-a=-5, c=5, and d=5.\) For each of these cases, new constraints can be imposed and the situation further investigated.

2.2 Data analysis

Ill-defined problems contain conflicting or incomplete constraints, and they necessitate restructuring of the problem in a new and more productive manner—which is how we define insight in this paper. To identify how the students attempted to gain insight into the two ill-defined problems, we carried out a three-step analysis (e.g., Simon, 2019 ) in which the interviews and students’ written work were analyzed retrospectively using approaches from qualitative content analysis (Mayring, 2015 ).

In the first step, we investigated the students’ work on each problem through an inductive analysis. The goal was to isolate and identify each individual solution that the students attempted. We refer to this step as approaches as it includes students’ solution attempts at solving the particular task, the type of strategies and reasoning employed by the students, and explicit assumptions made by the students. As we mentioned earlier, insight is predicated on some form of mental restructuring that allows the problem solver to view the problem a new and more productive manner. Although we cannot observe the cognitive processes directly, we can observe and identify the individuals’ approaches, in the form of actions and utterances, which indicate how they conceive the problem’s starting and goal state, constraints and operators. In other words, each approach indicates a particular mental structuring or restructuring of the problem (Weisberg, 2015 ).

In the second step, we made use of a mixed content analysis (Mayring, 2015 ) and looked more closely at the students’ approaches from both creativity and problem solving perspectives. More specifically, from a problem solving perspective, we first imposed the four stages of orientation, organization, execution, and verification (Lester, 1985 ) on to the previously identified approaches, and examined how the students moved between approaches. This step was accomplished by further categorizing all the observed behavior, i.e., utterances and actions, for each of the identified approaches. All behavior related to assessing or understanding the problem was coded as orientation. We then coded all behavior related to organizing and execution as a common category, as it can be very difficult to distinguish planning and execution of plans (Schoenfeld, 1985a ). The last category, verification, referred to all behavior related to evaluation of decisions made and the outcome of the executed plans. After the deductive coding, we made use of inductive coding with two goals in mind, as follows: (1) identify common characteristics of each phase across both problems for both groups of students respectively, and (2) identify how the groups of students moved between problem solving phases during the problem solving process.

To investigate the students’ work from a creativity perspective, we made use of a creativity model based on the Gestalt view of insight in the second step of our analysis. As mentioned earlier, the Gestaltists viewed insight as dependent on sudden and cognitive restructuring (Weisberg, 2015 ). Although cognitive flexibility can refer to various categories and sets, in this study we considered the identified approaches as a particular mental structuring, or restructuring, of the problem. Cognitive flexibility then, in this context, becomes the ability to switch between different approaches to the ill-defined problems. Furthermore, and as Nijstad et al. ( 2010 ) point out, the use of remote associations is a particular characteristic of cognitive flexibility. Thus, we looked more closely at (1) how many different approaches the students’ in each group made use of, (2) to what extent and in what way each approach differed from previous approaches in terms of strategies used and assumptions made, and 3) to what extent and in what way impasses during the problem solving process occurred—indicating the occurrence of fixations. Here, it is important to point out that we did not consider the success of each approach. It is often necessary to produce several attempts at solving an ill-defined problem in the absence of a priori knowledge of a valid solution, before finally solving it. Failed attempts are therefore often crucial to the creative process, as creative products are generated in the course of a dynamic process of exploration and assessment across both failed and successful attempts (Corazza, 2016 ).

In the third and final step, we attempted to develop explanatory inferences and work towards the aim of the paper. Here we compare and contrast how the two models—and corresponding views of insight—can describe and explain different aspects of the problem solving process. More specifically, we attempted to identify how and to what extent each of the two different models can describe and explain how the two groups of students gained, or failed to gain, insight into the ill-defined problems.

3.1 Problem 1: the Roman inheritance problem

Expert students The expert group approached the problem in two ways. At the start of the first approach, the students read the problem several times, first individually and then aloud, and discussed what they were “supposed to actually find out” as one student said. Simultaneously, they wrote down some of the constraints that they had identified in the problem: the wife should get more than the daughter, but less than the son. They then quickly reasoned what the wife’s proportion of the will would be if the total sum were halved. As one student said, “the wife should get exactly half of one third plus two third”. They concluded the wife should get half, and the rest be split between the daughter and the son. However, they quickly concluded that this was incorrect as this would either leave the son with less than the wife, or an inheritance exceeding the upper limit.

After rejecting the first approach, the expert students made a second attempt at solving the problem. They went back to talking about the information and conditions of the problem. They then decided to set up an equation, as this would “impose the all the necessary conditions on to the problem and we can solve it” as one student said. The right side of the equation had to be 1, as this represented the entire inheritance. The mother’s share was set as x, the son as y and the daughter z. They then substituted the variables and solved the equation (see Fig.  2 ).

Experts’ equation solution to the Roman problem

The students concluded that this was the right result. One of the students said: “The wife gets \(\frac{2}{7}\) , the daughter gets \(\frac{1}{7}\) , and the son gets \(\frac{4}{7}\) . This is the right result I guess”. However, this solution takes into account only the ratio between the wife, son and daughter, and not the share of the inheritance each person was promised. The students in the expert group mentioned this inconsistency a few times, but as one of the students said: “this is a bit weird, but I guess this is how you solve the problem”.

Novice students We identified three approaches for the novice groups.

As did the experts, all four novice groups first read the problem several times. However, unlike the experts, none of the novice groups discussed the information or constraints in the problem. Instead, they immediately started proposing possible solution strategies. The first approach all four novice groups attempted was some form of fraction expansion, followed by an empirically test to see if a more fine grained partition could make the inheritance division correct. The students would first set up a preliminary model, for instance imposing the constraints that the son would get more than the wife, and the wife would get more than the daughter. Then, they would adjust the model according to the results using bar charts, matrices or other heuristic approaches, and compare them to the conditions of the task. All four groups of students came up with at least three different partitions, before concluding that they were not able to build a model that satisfied all conditions of the task (see Fig. 3 ).

Example of novices’ model solution for the Roman problem

After concluding that the first approach did not satisfy all the conditions of the problem, all four novice groups immediately moved on to what we identified as a second approach. In the second approach, the novice students would use one of the son, the wife or daughter as a starting point based on the information in the task, and then quantify what share of the inheritance the others would get. For instance, if the son would receive \(\frac{2}{3}\) of the inheritance, then the wife would get \(\frac{2}{9}\) and the daughter would get \(\frac{1}{9}\) . The students would then use the daughter or the wife as the starting point, respectively, and quantify how much the others would get. However, after trying different starting points, all four novice groups concluded that this approach would not provide a correct solution.

The third approach we observed for all four novice groups was similar to the expert group’s second approach. The students wrote down and identified the ratios between the wife, the son and the daughter as the key constraints of the task. This approach was observed immediately after the novice students concluded their second approach was inappropriate, and it was also clear that this approach was inspired by the second approach. As one student said: “We have to take into account all constraints. At the same time. Not one by one. The son should get twice as much as the wife, and the wife should get twice as much as the daughter.” However, unlike the expert students, the novice students did not explicitly formulate equations that represented the conditions of the problem. Instead, they reasoned more informally. As one student said: “the wife should get twice as much as the daughter, and the son should get twice as much as the wife. The daughter then gets one part, the wife two parts, and the son four parts. That gives us seven parts in total”. All three novice groups concluded that this was the solution closest to the intentions of the will, but still not a satisfactory solution. After the third approach, three of the novice groups discussed the overall intentions of the will and which of their approaches was most in line with the wishes of the dying Roman. All three novice groups concluded that it was impossible to find a solution that was in full accordance with the will. However, all three groups also concluded that the main intention of the will was that the son should get more than the wife, and the wife should get more than the daughter.

3.2 Problem 2: wrong arithmetic, but correct result

Expert students The expert students approached the problem in two ways. First, the expert students read the problem, first individually and then out aloud. The experts then spent a few minutes talking about how “weird the expression was”, while verifying that both sides of the equation were equal, and the proposed method was correct. The students quickly agreed on both the meaning and goal of the problem. As one student said: “oh, they’ve just placed the digits together, and we need to find out when fraction multiplication gives this kind of product.” After verifying that the expression was indeed correct, the students proposed a hypothesis for which type of numbers this method was correct based on the example given. The students quickly mentioned that the sums of the digits in both the numerators and denominators were nine, and that nine was also a common factor of both 18 and 45. However, this hypothesis was not pursued further. Instead, the students quickly rejected the first approach and decided to represent the problem algebraically, which we have identified as their second approach.

After setting up the algebraic expression seen in Fig.  4 , the students repeatedly stated that this expression wasn’t appropriate. As one student said: “you can’t use correct algebra on something that is incorrect. The left side is ok, but the right side is completely wrong”. One of the students mentioned that they could have further identified constraints on each of the four “unknowns”, but he quickly decided that such a pursuit was pointless as it was “not correct mathematics”. The students then concluded that they couldn’t find any other solutions, as it couldn’t be solved algebraically and it was difficult to generalize any sort of pattern from just one case.

Experts’ algebraic solution for the Wrong arithmetic, but right result problem

Novice students Each of the four novice groups approached the problem in two ways. As with the Roman inheritance problem, all four novice groups first read the problem both individually and out loud. However, unlike the experts, the novice students did not explicitly discuss and agree on the meaning and goal of the problem. Instead, they seemed to spend a few minutes on their own trying to understand the problem. This period of apparent uncertainty was then interrupted by one of the students in the group proposing a particular solution strategy. For all four novice groups this involved a proposed hypothesis regarding the relationship between the numbers, which they refined empirically without considering the mathematical structure of the problem. For instance, the students explored commutativity and tried \(\frac{8}{5}\times \frac{1}{4}=\frac{81}{54}\) , they added the same numbers to denominators and numerators, and attempted to work with more or less randomly chosen fractions that, according to one student, were “in the same ballpark” as the fractions in the task. Common to all these hypotheses were that they were inferred from the specific numerical example in the problem text, and they were not based on any systematic investigation of the structural properties of the expression. One student, for example, evaluated the hypothesis according to “how close they came to giving an equal left and right side”. The students switched back and forth between several different hypotheses, but did not explicitly consider how the right side of the expression was constructed mathematically. Eventually, all four novice groups concluded that this approach was not “fruitful”, as one student said.

Eventually, all four novice groups rejected the first approach. Although there were some variations between the four groups, it seemed the second approach was an informal line of reasoning similar in structure to the novice students’ third approach on the Roman inheritance problem. Furthermore, the second approach seemed to evolve out of the seemingly superficial hypotheses proposed in the first approach. As one student said, “We need to make things easier… we’re just looking for connections between the numbers here, but there can so many of them.” In the second approach, the novice students seemed to look for specific examples that would satisfy the conditions of the problem and thus identify possible structural relationships. For instance, three of the novice groups realized eventually that they could just “turn the fractions upside down and maintain the same ratio between them” as one student said. Two of the groups also listed several trivial solutions that satisfied the criterion 1 × 1 = 1. The main difference between the novices’ first and second approaches, was that the first approach seemed to focus on identifying properties in the numbers given in the task, while the second approach seemed to focus on finding other examples that also satisfied the proposed method (see Fig. 5 ).

Example of novices’ empirical model solution to the Wrong arithmetic, but right result problem

3.3 Problem solving model

During the orientation phase of both tasks, both the experts and novices first read the task instructions individually and aloud. Both groups of students seemed to prefer to read the problem first and gain an initial understanding of it before talking about it to the other students. However, after reading the problem carefully, either quietly or aloud, the rest of the orientation phase was different for the experts and novices. While the experts wrote down and discussed the goals and conditions of the problems, seemingly to make sure everyone had the same understanding of the problem and its goal, the novices immediately began working on a solution strategy proposed by one of the students. Furthermore, after rejecting their first more informal approach, the experts went back to the orientation phase to make sure they all understood the problem correctly and had identified all the relevant conditions of the problem. There were also similarities and differences between the experts and novices in the organization and execution phases. For both problems, the experts first quickly proposed and rejected a hypothesis that seemed to be based on surface properties and incomplete constraints of the problems. For example, regarding problem two, there seemed to be no deeper analysis of the problem behind the first approach other than trying to identify common properties of the numbers on both sides of the equation sign. Similarly, the novices also first proposed hypotheses that seemed to be based on surface properties and incomplete constraints of the two problems. However, after rejecting the first approach, the experts then quickly sought a generalized and formalized solution, by representing and applying algebraic expressions and equations. The novices, on the other hand, continued to formulate hypotheses that they tested empirically, or they looked for numerical examples that satisfied given constraints of the problems. Finally, during the verification phase, there were also some noticeable differences between the two groups of students. The expert students quickly concluded, without any form of justification, that their first approach, for both problems, was incorrect. The students then similarly concluded quickly that their second approach was either correct or that the problem couldn’t be solved, for problem 1 and problem 2 respectively. Unlike the expert students, who evaluated each approach quickly and conclusively after the organization and execution phase, the novices seemed to evaluate the approach continuously and gradually come to a conclusion regarding its correctness.

These observations are in line with much of the existing literature on expert vs. novice problem solvers (Lester & Kehle, 2003 ; Schoenfeld, 1985a ). The experts placed a greater focus on understanding the problem, global planning, and creating representations that captured the structural properties of the problems. The novices, on the other hand, tended to go directly from the problem text in search of solution strategies that could be productive. Furthermore, the novices tended to create representations of the problems that were either incomplete or focused on surface properties. We also noticed that the experts quickly determined whether or not a particular approach was correct, while the novices seemed to explore each approach to a much greater extent before assessing its validity. This could be a result of a more extensive knowledge base. How the two groups of students moved between the different problem solving phases is also similar to results in the literature regarding expert and novice problem solving. Schoenfeld ( 1985a ) found, for example, that novices tend to spend much time on what he called the explore phase, which can be said to be an unstructured exploration of the problem analogous to orientation and organization. Experts, on the other hand, tend to display greater control and monitoring as they cycle more purposefully between the different problem solving phases. In this study, the experts’ problem solving behavior seemed to consist of repeating cycles of orientation → organizing/execution → verification. The novices, on the other hand, seemed to stick to cycling back and forth between the organizing/execution phase and the verification phase, after a single and initial orientation phase.

3.4 Creativity model

For the experts, we identified two approaches for each of the two problems. For the novices, we identified three approaches for the first problem and two approaches for the second problem. Immediately, a purely quantitative analysis would seem to indicate that the novices displayed greater cognitive flexibility during the problem solving process. However, a more detailed analysis reveals a more nuanced picture. For both problems, the experts’ first approach seemed to be unstructured exploration based on either surface or an incomplete set of properties of the problem. The second approach, on the other hand, for both problems, was a more general and structured approach, where all the relational properties of the problem were represented using algebraic equations. For example, the experts’ first approach to the Roman inheritance problem seemed to conclude that the wife’s part of the inheritance would simply be the midpoint of the two different situations described in the will. The second approach, on the other hand, was an equation that seemingly covered all the relational properties described in the problem. The experts’ work on both problems indicates a prominent mental shift between the first and the second approaches. It seems they were able to quickly break away from an inappropriate approach and instead pursue a more appropriate approach. Furthermore, the second approach is vastly different from the first approach in terms of both assumptions and strategies. As Nijstad et al. ( 2010 ) pointed out, sudden switching between remote mental sets—such as assumptions and strategies within a particular approach—is a key feature of cognitive flexibility. The novices, on the other hand, seemed to switch between approaches that were related to each other. For example, the novices’ two approaches on the Wrong arithmetic, but correct result problem were both based on unstructured exploration around arithmetic properties. This pattern indicates that although the novices were able to break away from unproductive approaches, the closely related approaches indicate less cognitive flexibility than that shown by the experts. This interpretation is in line with much of the relevant literature which concludes that extensive knowledge is positively associated with flexible problem solving (Ionescu, 2012 ).

Turning to the issue of cognitive fixation, we observed several incidents of ostensible impasses from which the experts and novices were unable to break. For both problems, the novices stuck to empirical investigations of hypotheses and informal reasoning. Although the novices shifted fluidly between different assumptions and strategies for both problems, the fact that they stuck to a particular set of approaches, indicates to some extent the presence of algorithmic fixation (Haylock, 1987 ). Although algorithmic fixation primarily refers to the inappropriate continued use of a particular algorithm, this kind of fixation also includes a more general predisposition to solve a problem in a specific manner even though better or more appropriate methods of solving the problem exist. Creating, for example, algebraic representations for both problems, in particular the second problem, would have helped the novices determine the relevant structural properties. The experts also experienced incidents of prolonged impasse that could indicate cognitive fixations. However, unlike the novices who displayed tendencies of algorithmic fixation, the experts seemed to primarily display tendencies of content universe fixation (Haylock, 1987 ). Working on the first problem, the experts concluded quickly that their second approach was “the correct solution”, as one student said, even though the constraints of the problem were not fully exhaustive and the ill-defined nature of the problem allowed multiple interpretations. For the second problem, the experts repeatedly stated that the algebraic expression (see Fig.  4 ) they had created was not appropriate, as they believe you could not apply “correct algebra on something that is incorrect”, as one student said. However, within the context of the problem, creating an equation that captures all the relevant structural properties is perfectly appropriate. In fact, analyzing the algebraic expression would have help the students’ identify the constraints of each digit. Overall though, the findings in the context of creativity is also in line with much of the literature. Both the experts and the novices displayed both flexibility and fixation during the problem solving process—although somewhat differently.

3.5 How students gained insight

Immediately, it would appear that the findings in this study are in line with much of the literature on expert and novice problem solving. Furthermore, both the experts’ and the novices’ work seemed to progress largely in a stepwise manner, as described and explained both by the problem solving model utilized in this study (Lester, 1985 ) and the analytic view of insight (Weisberg, 2015 ). One instance of this aspect can be seen in the novices’ work on the first problem. While their second approach was premised only on a single constraint of the problem, their third approach took into account all the relational properties between the wife, the daughter and the son simultaneously. In this instance, the novice students’ clearly modified their approach in a gradual and stepwise manner and further insight was gained as a result. A second important instance can be found in the experts’ work. For both problems, the experts returned to the orientation phase after their first approach, and then produced a new and more effective approach. This chain of events indicates that the experts’ first ineffective approach and return to the orientation phase somehow led to a productive mental restructuring of the problem—or greater insight in other words—which in turn resulted in a more effective approach.

However, a more finely-grained scrutiny of the students’ work reveals several limitations of the problem solving model. One such discrepancy is the emphasis on past experiences during problem solving (Liljedahl et al., 2016 ). Problem solving models (Lester, 1985 ; Pólya, 1949 ; Schoenfeld, 1985a ), and the analytic view of insight (Weisberg, 2015 ), highlight the importance of past experiences during problem solving and argue that insight is a consequence of matching the problem with information in memory. In this study, we did not observe a single incident in which either group explicitly referenced past experiences or compared the problem to other problems. It could be argued that the ill-defined structure of the problems themselves was unfamiliar, but it is still noticeable that neither group of students performed any sort overt assessment of familiarity with the task (Lester, 1985 ).

Another ostensible discrepancy can be found in the novices’ many approaches to the problems. Although the novices did not move between the different problem solving phases to the same extent as the experts, they did not stick to one particular approach “come hell or high water”—as Schoenfeld ( 1985a ) observed to be common among novice problem solvers. Instead, the novices moved seemingly effortlessly between different approaches, constantly adapting to the ambiguity of the ill-defined problems. This behavior is a clear indication of cognitive flexibility (Ionescu, 2012 ). Furthermore, each of these apparent mental restructurings of the problems seemed to follow small impasses in the problem solving process—as predicted by the Gestaltists (Weisberg, 2015 ).

Insight as a consequence of impasses and sudden mental restructuring, as opposed to a stepwise and conscious process, was even more prominent in the experts’ work. The experts’ work on both problems indicates a significant mental shift between the first and the second approach. After trying and concluding that their first and more informal approach was inappropriate, the experts quickly decided to pursue a completely different and more structured approach. Although this behavior can be projected on to the four phases of the problem solving model (Lester, 1985 ), as seen earlier, the model itself cannot qualitatively explain the drastic shift in terms of assumptions and strategies. The experts’ second approach was in no way a further refinement of their first approach, and they did not explicitly reference past experiences. Instead, it seemed the second approach appeared suddenly, unconsciously and as a response to the failure of the first approach. This chain of events is similar to what Ohlsson ( 2011 ) refers to as the insight sequence , which describes insight as something gained after an attempted solution fails and a sudden and meaningful mental restructuring is required. After an impasse has occurred, insight is gained after dealing with the problem from a completely novel perspective.

Finally, our analyses also indicate occurrences in which both groups of students failed to gain insight. For example, while the novices applied mostly empirical and informal reasoning, the experts sought generalized and formalized solutions. Although much of the literature explains this as a consequence of the experts’ more extensive knowledge base (Lester & Kehle, 2003 ; Schoenfeld, 1985a ), neither problem used in this study required advanced mathematics. The algebraic representations that the experts made use of were fairly simple and seemingly within the grasp of individuals who have taken at least upper secondary algebra. An alternative explanation can therefore be cognitive fixation (Haylock, 1987 ), in which individuals fail to abandon ineffective approaches and move beyond impasses. This was perhaps seen most clearly in the experts’ work on the second problem. After creating an algebraic representation of the structural properties of the problem, the experts quickly rejected, in unison, the approach as inappropriate. We propose that this is a clear example of an unnecessary restriction to an insufficient range of elements (Haylock, 1987 ). In other words, the experts imposed an unnecessary set of restrictions on to the problem solving process based on their conceptions of the situation, rather than the properties of the problem itself. Now, it can be argued that this fixation can be linked to the experts’ past experiences. However, the problem solving model, and the analytic view of insight, do not explain or describe how the problem solver can break away from established mental sets. In fact, the problem solving model, and the analytic view of insight, emphasize the use of prior knowledge and reliance on past experiences when first attacking a problem (Liljedahl, 2016). When facing a new problem, in particular an ill-defined problem such as those made use of in this study, the focus on past experiences could actually be a hindrance to making progress (Weisberg, 2015 ).

4 Final thoughts

In this study, we aimed to integrate two different views on insight during problem solving, and explore how they each highlight different aspects of the problem solving process. Looking back, applying both problem solving and creativity models on to the experts’ and novices’ work reveals and explains different aspects of the students’ problem solving processes. While the problem solving model helps us analyze and understand parts of the problem solving process, there are crucial aspects of the students’ work that it does not explain. In this study, we observed what we claim to be the occurrence of cognitive flexibility, cognitive fixation, and more importantly, sudden, and seemingly unconscious, insight during the problem solving process—for both experts and novices. The results of this study therefore dovetail with what the Gestaltists said all along: Sudden and unconscious insight seems to be crucial to the problem solving process, and the occurrence of such insight cannot be fully explained by standardized problem solving models and an analytic view of insight. Current researchers inspired by the Gestaltists have dubbed this understanding of insight as the special process view of insight (Ohlsson, 2011 ; Weisberg, 2015 ), as it asserts that the thought processes underlying insight are distinctly different from those thought processes underlying analytic thinking.

We suggest, based on the results of this study and the review of the relevant literature, that research into problem solving within mathematics education would benefit from adopting aspects of Gestalt inspired views of insight. Although we do not go as far as some who claim that adherence to any sort of heuristics can be a hindrance to the problem solving process, we do agree that there are no prescriptive heuristics for some of the more unconscious, yet highly important, cognitive aspects of problem solving (Liljedahl et al., 2016 ). So, what happens during the moment of insight or subconscious work? What is the source of creative thought? Although we do not fully understand mental restructuring and creative thought, Ohlsson ( 2011 ) has proposed redistribution theory as a Gestalt-inspired response. Here, the problem solver first creates an initial inappropriate representation of the problem. This particular interpretation activates one or more incorrect solutions, which the problem solver then works through. At some point, after working through the incorrect solutions, the problem solver reaches an impasse. It is at this point that the initial, and inappropriate, representation of the problem could be inhibited. This inhibition of the original representation of the problem might then result in a new representation of the problem, which causes the problem solver to realize that the problem can be thought of in a different way—in other words, a mental restructuring has occurred. Somewhat ironically, the Gestalt inspired method of problem solving can therefore also be said to rely heavily on past experience. What is entailed is not to match the problem with past experiences to find an appropriate solution, but rather to relax unnecessary constraints and inhibit knowledge that is not necessary. We propose that this line of reasoning can add to extant problem solving models in at least two ways, as follows: 1) Most problem solving models highlight the importance of assessing the familiarity of the problem (Lester, 1985 ; Liljedahl et al., 2016 ; Pólya, 1949 ; Schoenfeld, 1985a ). However, the heuristic emphasis seems to be on identifying similarities between the problem at hand and past experiences. We suggest that identifying divergences between the problem at hand and past experiences is also important, as it may help the problem solver recognize unnecessary constraints. 2) Working through numerous incorrect approaches and solutions can be helpful to the overall problem solving process, as it may lead to an impasse and a subsequent more appropriate restructuring of the problem. We suggest that problem solving models should also emphasize the value of working hard on problems for an extended period of time, and even failed attempts.

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Haavold, P.Ø., Sriraman, B. Creativity in problem solving: integrating two different views of insight. ZDM Mathematics Education 54 , 83–96 (2022). https://doi.org/10.1007/s11858-021-01304-8

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How to be creative - solve problems with new ideas

Key message: Creativity builds on knowledge. Solving a problem creatively means closing a knowledge gap with a new idea.

What is creativity?

If we want to solve a problem, we first need an idea and then a method that fits the idea: An idea is a concise mental picture of something, an idea that describes the essence (principle) of something. A method is the way to a desired goal (a precise description of this way). To find an optimal solution to the problem, we must have several alternative ideas. Then we can choose the best one and implement it.

"A common misconception about the CPS process is that it is a free-running, unstructured, almost mystical process [CPS = creative problem solving] . This belief is encouraged by anecdotes of creative breakthroughs occurring during periods of unconscious sleep or via an unconscious thought process." (Titus, P. A. (2000). Marketing und der kreative Problemlösungsprozess. Zeitschrift für Marketing-Ausbildung, 22, p. 226)

Ideas arise on the basis of knowledge. But why do many people think that new ideas arise unconsciously, that they "fall from the sky"?

" Our brain is actually not designed to learn and retain knowledge, especially rarely used detailed knowledge. Every human brain is evolutionarily more adapted to skill and the accumulation and generalization of experience that enables survival in its environment." (Rost, F. (2018) Lern- und Arbeitstechniken für das Studium. Springer VS, Wiesbaden, p. 34)

In dangerous situations, we do not have time to think for a long time, which is why our brain tries to intuitively (unconsciously) do the right thing, i.e. what corresponds to our previous experience. Our brain thinks unconsciously in "tracks", i.e. in a predetermined direction. Therefore, if we want to develop new ideas, we have to leave these "tracks".

When we think about a difficult problem for a long time, we intuitively think in such "tracks". It's not until we take a break and restart the thought process that we can leave the old "tracks" and come up with a new idea. Then it seems as if the idea "fell from the sky". Example: With a difficult problem, it makes no sense to brainstorm with people who know nothing about the problem.

"... hard problems are those for which a solver (human or computer) has insufficient knowledge and resources ..." (Wang, P. (2007). The logic of intelligence. In Artificial General Intelligence eds. B. Goertzel and C. Pennachin. New York, NY: Springer, p. 39)

With sufficient knowledge, we would easily find or have ideas how to solve our problem or we would realize that it is unsolvable. How can we come to the necessary knowledge? There are two possibilities:

1. We find ideas that others have used to solve a similar problem

Nowadays we can find almost everything on the internet. In order to search with the right keywords, it is always good to ask a basic question or to look at the definition of the term we are interested in. I did a search with the sentence "How to find ideas someone else has used to solve our problem " and I found this article:

" Before any project, we need to stop and ask ourselves, has somebody already solved this problem. Then we need to genuinely ask ourselves, is there a good reason why our solution needs to be different." ( www.producttalk.org/2013/08/find-someone-else-who-has-solved-your-problem-before , 03.10.2020)

If find an idea that someone else has already used to solve our problem, we should still try to improve this idea.

2. We need to be creative and have new ideas to solve our problem

"I define creativity more specifically as the process of having original ideas that have value. ... Creativity is about producing something new. ... It does have to be new to the maker at least and not just a copy or a repetition." ( www.interaliamag.org/interviews/ken-robinson , 21.09.20)

"Too often ... ' creativity ' means having great, original ideas. ... the ideas are often judged more by their novelty than by their potential usefulness ..." ( https://hbr.org/2002/08/creativity-is-not-enough , 26.09.20)

Creativity is when someone produces "something new" that is useful/has value. "Something new" is useful when it improves our lives and can be implemented to solve a problem.

Creativity is the ability to have new ideas that can be used to achieve a goal. Is this the definition of creativity?

Scientists also create new knowledge that is useful. So are all scientists creative? If a scientist researches a question for so long that in the end he can't help but come up with a new idea, that can't be called creative.

Example: Two scientists are looking for the solution to a problem. Neither of them knows what the other is doing. One scientist needs 10 years to solve the problem, the other only 2 years. Which of the two is more creative?

If you want to evaluate a person's creativity, you have to evaluate not only the quality of his idea, but also the time it took him to come up with the idea. So a person is particularly creative if he or she can develop new ideas that are very useful in a short time on the basis of very little information. However, it is also true that difficult problems can rarely be solved in a short time.

Definition: Creativity is the ability to have new and useful ideas based on incomplete information.

Why do we need knowledge to have new ideas?

"... in Leo Tolstoy’s novel ' War and Peace ' , Prince Andrei Bolkonsky explains the concept of war ʹ ... In war, you do not know the position of your enemy; some things you might be able to observe, some things you have to divine (but that depends on your ability to do so!) and many things cannot even be guessed at. ... If you decide to attack, you cannot know whether the necessary conditions are met for you to succeed. ʹ In essence, war is characterized by a high degree of uncertainty. A good commander ... can add to that what he or she sees, tentatively fill in the blanks ... A bad commander extrapolates from what he sees and thus arrives at improper conclusions." (Dörner, D., & Funke, J. (2017). Complex problem solving: What it is and what it is not. Frontiers in Psychology, 8,1153, p. 7)

A good commander has a lot of knowledge and experience. With his existing knowledge, he can carry out an analysis of the problem and thus identify where there are gaps in knowledge.

The method "trial and error" is the simplest method to fill a knowledge gap with an idea and thus to find a solution for a problem but nobody wants to rely on his luck and choose the ideas for the trials randomly.

"If trials are chosen randomly, the probability of arriving at a valuable solution to a complex, high-interaction problem is very low. More efficient orderings utilize knowledge to direct this search process." (Nickerson, J. a. and Zenger, T.R. (2004). A Knowledge-Based Theory of the Firm—The Problem-Solving Perspective. Organization Science, Vol. 15 No. 6, p. 620)

"The mobilisation of prior knowledge is not sufficient to solve novel problems in many everyday situations. Gaps in knowledge must be filled by observation and exploration of the problem situation. This often involves interaction with a new system to discover rules that in turn must be applied to solve the problem. Instead of a straightforward application of previously mastered knowledge, existing knowledge needs to be reorganised and combined with new knowledge using a range of reasoning skills. " (Pisa 2012 Field Trial Problem Solving Framework, www.oecd.org/pisa/pisaproducts/46962005.pdf , 01.11.20, p. 15)

When an expert examines a problem, he imagines what knowledge he is missing and what possibilities (what facts and rules) there are to fill his knowledge gaps. He uses his sense of possibility and becomes creative (see below at "abstract thinking and logical reasoning").

"This sense of possibility entails ... being able to construe an unknown whole that could accommodate a known part. The whole has to align ... with the laws of physics and chemistry. Otherwise, the entire venture is ill-founded. A sense of possibility does not aim for the moon but imagines something that might be possible but has not been considered possible or even potentially possible so far." (Dörner, D., & Funke, J. (2017). Complex problem solving: What it is and what it is not. Frontiers in Psychology, 8,1153, p. 7)

(How to recognize the whole of a situation/system, see on Learn-Study-Work "How to analyze situations/systems".)

Rules are conditional statements (if premise then consequence). When I want to calculate the area of a rectangle, I can use the formula A = L x W (rule: If I multiply the length by the width, then I get the area) . But that is only possible if I know the values for L and W (facts).

"In the broadest sense, a rule could be any statement which says that a certain conclusion must be valid whenever a certain premise is satisfied, i.e. any statement that could be read as a sentence of the form ' if ... then ... ' ... it should be noted that there are a number of rather different interpretations of the term ' rule ' outside of first-order logic." (Hitzler P., Krötzsch M., Rudolph, S. (2009). Foundations of Semantic Web Technologies. Chapman & Hall/CRC, p. 213 - 216)

Example: If I want to fly I should study all things that go up: hot air, some gases, birds, leaves in the wind, etc. When I understand the rules why these things go up, then maybe I can use these rules to fly myself. Birds can fly. It is possible that they can fly because their wings have a special shape. So I should examine the shape of their wings.

"... for all flying animals in nature [apply] the same physical laws [rules] for propulsion and lift ... : Propulsion by the flapping of the wing, lift by the negative pressure on the curved upper surface of the wing (Bernoulli's principle)." ( www.planet-schule.de/wissenspool/tierische-flugpioniere/inhalt/hintergrund.html#kap2 , 23.04.22)

In the years 1486 - 1513, Leonardo da Vinci was engaged in flying, e.g. he designed a swinging aircraft. However, he lacked the necessary knowledge to put the design into practice.

Otto Lilienthal was the "first to systematically investigate and describe aerodynamic principles. His groundbreaking book ʹDer Vogelflug als Grundlage der Fliegekunstʹ ... was used by the Wright brothers, for example, for their aircraft developments: ʹThe most important finding (...) was the discovery that curved wings provided greater lift than flat onesʹ, they noted." ( https://aeroreport.de/de/good-to-know/wie-ein-vogel , 05.01.23)

He writes in his book:

"If we now look back at what has been presented in this work, a number of theorems [rules] derived from experiments stand out in it ... The insight of the correctness of these theorems requires only an understanding of the simplest concepts of mechanics ... " ( www.luftfahrt-bibliothek.de/datenarchiv/otto-lilienthal-der-vogelflug-als-grundlage-der-fliegekunst.pdf , 25.04.22, S. 182)

In the field of science, the possibilities that are supposed to fill gaps in knowledge are called hypotheses.

"Much scientific research is based on investigating known unknowns. In other words, scientists develop a hypothesis to be tested, and then in an ideal situation experiments are best designed to test the null hypothesis. ... it is common for the researcher to believe that the result that will be obtained will be within a range of known possibilities. Occasionally, however, the result is completely unexpected—it was an unknown unknown." ( Logan, D. C. (2009). Known knowns, known unknowns, unknown unknowns and the propagation of scientific enquiry. Journal of Experimental Botany, 60 (3), p. 712)

The difficulty lies in imagining the "unknown unknows" as well.

What kind of knowledge is required to have new ideas to solve a problem?

1. Domain specific knowledge

"... experts do not possess better general cognitive skills than novices – such as better memory capacity – but rather experts have better domain knowledge based on their experience ..." (Pisa 2012 Field Trial Problem Solving Framework, www.oecd.org/pisa/pisaproducts/46962005.pdf , 01.11.20, p. 40/41)

If a person has a good expertise, he knows many rules and facts in the relevant domain and he/she also knows where and how they can be applied to solve problems.

"... expertise in a domain helps people develop a sensitivity to patterns of meaningful information that are not available to novices. ...

Differences in how physics experts and novices approach problems can also be seen when they are asked to sort problems, written on index cards, according to the approach that could be used to solve them (Chi et al., 1981). Experts’ problem piles are arranged on the basis of the principles [rules] that can be applied to solve the problems; novices’ piles are arranged on the basis of the problems’ surface attributes. ...

Experts have not only acquired knowledge, but are also good at retrieving the knowledge that is relevant to a particular task." (National Research Council. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press, p. 33, 38, 43, www.nap.edu/read/9853/chapter/5 , 22.07.23)

So it is important to understand the rules behind the surface properties.

2. General knowledge

We need general knowledge to be able to find analogies that inspire us to have new ideas.

"The ability to perceive similarities and analogies is one of the most fundamental aspects of human cognition. It is crucial for recognition, classification, and learning and it plays an important role in scientific discovery and creativity. ... analogical reasoning involves the identification and transfer of structural information from a known system (the source) to a new and relatively unknown system (the target)." ( Vosniadou, S. and Ortony, A. ( 1989). Similarity and analogical reasoning, Cambridge : Cambridge University Press , p. 1)

"An analogy is said to exist when the pattern of relations amongst one set of elements ... is shared with that of another set." (Barr, N. (2014). Reasoned connections: Complex creativity and dual-process theories of cognition. UWSpace, p. 12)

If two objects (systems) have analogies, it may be possible to apply the rules of the known system to the unknown system. People with a great general knowledge know the most important rules in many fields and can transfer them from one field to another.

3. Problem-solving knowledge

A lot of expertise and general knowledge is good , but we also need to know how to apply this knowledge and how to control ourselves . That is why we need problem-solving skills .

"... ' skills ' means the ability to apply knowledge and use know-how to complete tasks and solve problems." ( https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CELEX:32017H0615(01)&from=DE#d1e32-20-1 , 14.11.20)

"Knowledge and skill, then, are necessary elements of understanding, but not sufficient in themselves. Understanding requires more: the ability to thoughtfully and actively ' do ' the work with discernment, as well as the ability to self-assess, justify, and critique such ' doings ' ." (Wiggins, G., & McTighe, J. (1998). Understanding by design. Alexandria, VA: ASCD, p. 6)

Problem-solving knowledge is very helpful for someone who wants to solve problems optimally.

"The reason the executive so often rejects new ideas is that he is a busy man whose chief day-in, day-out task is to handle an ongoing stream of problems. He receives an unending flow of questions on which decisions must be made. Constantly he is forced to deal with problems to which solutions are more or less urgent and the answers to which are far from clear-cut. It may seem splendid to a subordinate to supply his boss with a lot of brilliant new ideas to help him in his job. But advocates of creativity must once and for all understand the pressing facts of the executive’s life: Every time an idea is submitted to him, it creates more problems for him—and he already has enough." ( https://hbr.org/2002/08/creativity-is-not-enough , 26.09.20)

Many people think problem-solving knowledge and creativity is only for big problems. They think they don't have time to optimise the solutions of their daily live problems. They see that they have to invest time, but they do not understand that they can also save a lot of time by solving problems optimally. Learning starts with small steps. First solve some small problems optimally, then you can try to solve a larger problem optimally.

"Harrington (1995) described that, for any organization to keep pace with the fast-changing environment, it needs to take full advantage of both CI [continuous improvement] and breakthrough improvement. Organizations which are just starting their improvement process activities should first direct their efforts to CI, establishing a working base. Then they should expand their improvement effort to include breakthrough improvement." (Singh, J., and Singh, H. (2015). Continuous improvement philosophy–literature review and directions. Benchmarking: An International Journal, Vol. 22 No. 1, p. 95)

Creativity techniques

Creativity techniques can also help to have new ideas. Unfortunately, I don't know much about these techniques. However, if you read my texts here on this web site, you will find that they contain many new ideas.

Why are abstract thinking and logical reasoning important for creativity?

"One of the challenges facing artificial intelligence research today is designing systems capable of utilizing systematic reasoning to generalize to new tasks. The Abstraction and Reasoning Corpus (ARC) measures such a capability through a set of visual reasoning tasks." ( https://link.springer.com/chapter/10.1007/978-3-030-93409-5_54 , 02.06.23)

A problem solution has four main elements: the initial situation, the goal, the obstacle that prevents the achievement of the goal and the method that should lead to the goal. If there are knowledge gaps in one or more of these elements, we cannot solve the problem. For each knowledge gap, we must use our sense of possibility (see above). That means we have to list all the possibilities that could fill that gap. First, we do this at a high abstract level. Then we concretize each of these abstract possibilities. We need thinking to understand the problem and to imagine possible solutions and we need reasoning to decide which option leads to our goal.

"Reasoning denotes acute and exact mentalizing involving logical deductions. ... Thinking, however, is ... the construction of an initially unknown reality. ... Once our sense of possibility has helped grasping a situation, problem solvers need to call on their reasoning skills. Not every situation requires the same action, and we may want to act this way or another to reach this or that goal. This appears logical, but it is a logic based on constantly shifting grounds: We cannot know whether necessary conditions are met, sometimes the assumptions we have made later turn out to be incorrect, and sometimes we have to revise our assumptions or make completely new ones. It is necessary to constantly switch between our sense of possibility and our sense of reality, that is, to switch between thinking and reasoning." (Dörner, D., & Funke, J. (2017). Complex problem solving: What it is and what it is not. Frontiers in Psychology, 8,1153, p. 7)

"Despite the onslaught of creative machines like generative AI, they won’t conquer human creativity and ingenuity. Just as the advent of photography didn’t kill painting, creating a new art form instead, generative AI enables new avenues of art and creativity." ( https://nextconf.eu/2023/05/lets-get-physical/?ct=t(nl_07_23_next23 , 08.07.23)

Artificial intelligence agents (computer programs) can quickly calculate many trials, but they are less creative when asked to fill knowledge gaps.

"Ideally, we would like Artificial intelligence (AI) agents to interact with an arbitrary environment where only partial information is available with uncertainty. ... AI agents are much better at quantitative computation than humans but are less capable of perceiving concepts qualitatively. This is the reason why AI agents can beat human experts in deterministic games such as chess and go but are struggling to complete an everyday task ..." ( https://openresearch-repository.anu.edu.au/handle/1885/154259 , 11.10.20)

Continue with the next step of the problem-solving process:

How to evaluate the ideas and turn them into methods

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