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  • Problem Solving in STEM

Solving problems is a key component of many science, math, and engineering classes.  If a goal of a class is for students to emerge with the ability to solve new kinds of problems or to use new problem-solving techniques, then students need numerous opportunities to develop the skills necessary to approach and answer different types of problems.  Problem solving during section or class allows students to develop their confidence in these skills under your guidance, better preparing them to succeed on their homework and exams. This page offers advice about strategies for facilitating problem solving during class.

How do I decide which problems to cover in section or class?

In-class problem solving should reinforce the major concepts from the class and provide the opportunity for theoretical concepts to become more concrete. If students have a problem set for homework, then in-class problem solving should prepare students for the types of problems that they will see on their homework. You may wish to include some simpler problems both in the interest of time and to help students gain confidence, but it is ideal if the complexity of at least some of the in-class problems mirrors the level of difficulty of the homework. You may also want to ask your students ahead of time which skills or concepts they find confusing, and include some problems that are directly targeted to their concerns.

You have given your students a problem to solve in class. What are some strategies to work through it?

  • Try to give your students a chance to grapple with the problems as much as possible.  Offering them the chance to do the problem themselves allows them to learn from their mistakes in the presence of your expertise as their teacher. (If time is limited, they may not be able to get all the way through multi-step problems, in which case it can help to prioritize giving them a chance to tackle the most challenging steps.)
  • When you do want to teach by solving the problem yourself at the board, talk through the logic of how you choose to apply certain approaches to solve certain problems.  This way you can externalize the type of thinking you hope your students internalize when they solve similar problems themselves.
  • Start by setting up the problem on the board (e.g you might write down key variables and equations; draw a figure illustrating the question).  Ask students to start solving the problem, either independently or in small groups.  As they are working on the problem, walk around to hear what they are saying and see what they are writing down. If several students seem stuck, it might be a good to collect the whole class again to clarify any confusion.  After students have made progress, bring the everyone back together and have students guide you as to what to write on the board.
  • It can help to first ask students to work on the problem by themselves for a minute, and then get into small groups to work on the problem collaboratively.
  • If you have ample board space, have students work in small groups at the board while solving the problem.  That way you can monitor their progress by standing back and watching what they put up on the board.
  • If you have several problems you would like to have the students practice, but not enough time for everyone to do all of them, you can assign different groups of students to work on different – but related - problems.

When do you want students to work in groups to solve problems?

  • Don’t ask students to work in groups for straightforward problems that most students could solve independently in a short amount of time.
  • Do have students work in groups for thought-provoking problems, where students will benefit from meaningful collaboration.
  • Even in cases where you plan to have students work in groups, it can be useful to give students some time to work on their own before collaborating with others.  This ensures that every student engages with the problem and is ready to contribute to a discussion.

What are some benefits of having students work in groups?

  • Students bring different strengths, different knowledge, and different ideas for how to solve a problem; collaboration can help students work through problems that are more challenging than they might be able to tackle on their own.
  • In working in a group, students might consider multiple ways to approach a problem, thus enriching their repertoire of strategies.
  • Students who think they understand the material will gain a deeper understanding by explaining concepts to their peers.

What are some strategies for helping students to form groups?  

  • Instruct students to work with the person (or people) sitting next to them.
  • Count off.  (e.g. 1, 2, 3, 4; all the 1’s find each other and form a group, etc)
  • Hand out playing cards; students need to find the person with the same number card. (There are many variants to this.  For example, you can print pictures of images that go together [rain and umbrella]; each person gets a card and needs to find their partner[s].)
  • Based on what you know about the students, assign groups in advance. List the groups on the board.
  • Note: Always have students take the time to introduce themselves to each other in a new group.

What should you do while your students are working on problems?

  • Walk around and talk to students. Observing their work gives you a sense of what people understand and what they are struggling with. Answer students’ questions, and ask them questions that lead in a productive direction if they are stuck.
  • If you discover that many people have the same question—or that someone has a misunderstanding that others might have—you might stop everyone and discuss a key idea with the entire class.

After students work on a problem during class, what are strategies to have them share their answers and their thinking?

  • Ask for volunteers to share answers. Depending on the nature of the problem, student might provide answers verbally or by writing on the board. As a variant, for questions where a variety of answers are relevant, ask for at least three volunteers before anyone shares their ideas.
  • Use online polling software for students to respond to a multiple-choice question anonymously.
  • If students are working in groups, assign reporters ahead of time. For example, the person with the next birthday could be responsible for sharing their group’s work with the class.
  • Cold call. To reduce student anxiety about cold calling, it can help to identify students who seem to have the correct answer as you were walking around the class and checking in on their progress solving the assigned problem. You may even want to warn the student ahead of time: "This is a great answer! Do you mind if I call on you when we come back together as a class?"
  • Have students write an answer on a notecard that they turn in to you.  If your goal is to understand whether students in general solved a problem correctly, the notecards could be submitted anonymously; if you wish to assess individual students’ work, you would want to ask students to put their names on their notecard.  
  • Use a jigsaw strategy, where you rearrange groups such that each new group is comprised of people who came from different initial groups and had solved different problems.  Students now are responsible for teaching the other students in their new group how to solve their problem.
  • Have a representative from each group explain their problem to the class.
  • Have a representative from each group draw or write the answer on the board.

What happens if a student gives a wrong answer?

  • Ask for their reasoning so that you can understand where they went wrong.
  • Ask if anyone else has other ideas. You can also ask this sometimes when an answer is right.
  • Cultivate an environment where it’s okay to be wrong. Emphasize that you are all learning together, and that you learn through making mistakes.
  • Do make sure that you clarify what the correct answer is before moving on.
  • Once the correct answer is given, go through some answer-checking techniques that can distinguish between correct and incorrect answers. This can help prepare students to verify their future work.

How can you make your classroom inclusive?

  • The goal is that everyone is thinking, talking, and sharing their ideas, and that everyone feels valued and respected. Use a variety of teaching strategies (independent work and group work; allow students to talk to each other before they talk to the class). Create an environment where it is normal to struggle and make mistakes.
  • See Kimberly Tanner’s article on strategies to promoste student engagement and cultivate classroom equity. 

A few final notes…

  • Make sure that you have worked all of the problems and also thought about alternative approaches to solving them.
  • Board work matters. You should have a plan beforehand of what you will write on the board, where, when, what needs to be added, and what can be erased when. If students are going to write their answers on the board, you need to also have a plan for making sure that everyone gets to the correct answer. Students will copy what is on the board and use it as their notes for later study, so correct and logical information must be written there.

For more information...

Tipsheet: Problem Solving in STEM Sections

Tanner, K. D. (2013). Structure matters: twenty-one teaching strategies to promote student engagement and cultivate classroom equity . CBE-Life Sciences Education, 12(3), 322-331.

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  • v.8(3); Fall 2009

Teaching Creativity and Inventive Problem Solving in Science

Robert l. dehaan.

Division of Educational Studies, Emory University, Atlanta, GA 30322

Engaging learners in the excitement of science, helping them discover the value of evidence-based reasoning and higher-order cognitive skills, and teaching them to become creative problem solvers have long been goals of science education reformers. But the means to achieve these goals, especially methods to promote creative thinking in scientific problem solving, have not become widely known or used. In this essay, I review the evidence that creativity is not a single hard-to-measure property. The creative process can be explained by reference to increasingly well-understood cognitive skills such as cognitive flexibility and inhibitory control that are widely distributed in the population. I explore the relationship between creativity and the higher-order cognitive skills, review assessment methods, and describe several instructional strategies for enhancing creative problem solving in the college classroom. Evidence suggests that instruction to support the development of creativity requires inquiry-based teaching that includes explicit strategies to promote cognitive flexibility. Students need to be repeatedly reminded and shown how to be creative, to integrate material across subject areas, to question their own assumptions, and to imagine other viewpoints and possibilities. Further research is required to determine whether college students' learning will be enhanced by these measures.

INTRODUCTION

Dr. Dunne paces in front of his section of first-year college students, today not as their Bio 110 teacher but in the role of facilitator in their monthly “invention session.” For this meeting, the topic is stem cell therapy in heart disease. Members of each team of four students have primed themselves on the topic by reading selected articles from accessible sources such as Science, Nature, and Scientific American, and searching the World Wide Web, triangulating for up-to-date, accurate, background information. Each team knows that their first goal is to define a set of problems or limitations to overcome within the topic and to begin to think of possible solutions. Dr. Dunne starts the conversation by reminding the group of the few ground rules: one speaker at a time, listen carefully and have respect for others' ideas, question your own and others' assumptions, focus on alternative paths or solutions, maintain an atmosphere of collaboration and mutual support. He then sparks the discussion by asking one of the teams to describe a problem in need of solution.

Science in the United States is widely credited as a major source of discovery and economic development. According to the 2005 TAP Report produced by a prominent group of corporate leaders, “To maintain our country's competitiveness in the twenty-first century, we must cultivate the skilled scientists and engineers needed to create tomorrow's innovations.” ( www.tap2015.org/about/TAP_report2.pdf ). A panel of scientists, engineers, educators, and policy makers convened by the National Research Council (NRC) concurred with this view, reporting that the vitality of the nation “is derived in large part from the productivity of well-trained people and the steady stream of scientific and technical innovations they produce” ( NRC, 2007 ).

For many decades, science education reformers have promoted the idea that learners should be engaged in the excitement of science; they should be helped to discover the value of evidence-based reasoning and higher-order cognitive skills, and be taught to become innovative problem solvers (for reviews, see DeHaan, 2005 ; Hake, 2005 ; Nelson, 2008 ; Perkins and Wieman, 2008 ). But the means to achieve these goals, especially methods to promote creative thinking in scientific problem solving, are not widely known or used. An invention session such as that led by the fictional Dr. Dunne, described above, may seem fanciful as a means of teaching students to think about science as something more than a body of facts and terms to memorize. In recent years, however, models for promoting creative problem solving were developed for classroom use, as detailed by Treffinger and Isaksen (2005) , and such techniques are often used in the real world of high technology. To promote imaginative thinking, the advertising executive Alex F. Osborn invented brainstorming ( Osborn, 1948 , 1979 ), a technique that has since been successful in stimulating inventiveness among engineers and scientists. Could such strategies be transferred to a class for college students? Could they serve as a supplement to a high-quality, scientific teaching curriculum that helps students learn the facts and conceptual frameworks of science and make progress along the novice–expert continuum? Could brainstorming or other instructional strategies that are specifically designed to promote creativity teach students to be more adaptive in their growing expertise, more innovative in their problem-solving abilities? To begin to answer those questions, we first need to understand what is meant by “creativity.”

What Is Creativity? Big-C versus Mini-C Creativity

How to define creativity is an age-old question. Justice Potter Stewart's famous dictum regarding obscenity “I know it when I see it” has also long been an accepted test of creativity. But this is not an adequate criterion for developing an instructional approach. A scientist colleague of mine recently noted that “Many of us [in the scientific community] rarely give the creative process a second thought, imagining one either ‘has it’ or doesn't.” We often think of inventiveness or creativity in scientific fields as the kind of gift associated with a Michelangelo or Einstein. This is what Kaufman and Beghetto (2008) call big-C creativity, borrowing the term that earlier workers applied to the talents of experts in various fields who were identified as particularly creative by their expert colleagues ( MacKinnon, 1978 ). In this sense, creativity is seen as the ability of individuals to generate new ideas that contribute substantially to an intellectual domain. Howard Gardner defined such a creative person as one who “regularly solves problems, fashions products, or defines new questions in a domain in a way that is initially considered novel but that ultimately comes to be accepted in a particular cultural setting” ( Gardner, 1993 , p. 35).

But there is another level of inventiveness termed by various authors as “little-c” ( Craft, 2000 ) or “mini-c” ( Kaufman and Beghetto, 2008 ) creativity that is widespread among all populations. This would be consistent with the workplace definition of creativity offered by Amabile and her coworkers: “coming up with fresh ideas for changing products, services and processes so as to better achieve the organization's goals” ( Amabile et al. , 2005 ). Mini-c creativity is based on what Craft calls “possibility thinking” ( Craft, 2000 , pp. 3–4), as experienced when a worker suddenly has the insight to visualize a new, improved way to accomplish a task; it is represented by the “aha” moment when a student first sees two previously disparate concepts or facts in a new relationship, an example of what Arthur Koestler identified as bisociation: “perceiving a situation or event in two habitually incompatible associative contexts” ( Koestler, 1964 , p. 95).

In this essay, I maintain that mini-c creativity is not a mysterious, innate endowment of rare individuals. Instead, I argue that creative thinking is a multicomponent process, mediated through social interactions, that can be explained by reference to increasingly well-understood mental abilities such as cognitive flexibility and cognitive control that are widely distributed in the population. Moreover, I explore some of the recent research evidence (though with no effort at a comprehensive literature review) showing that these mental abilities are teachable; like other higher-order cognitive skills (HOCS), they can be enhanced by explicit instruction.

Creativity Is a Multicomponent Process

Efforts to define creativity in psychological terms go back to J. P. Guilford ( Guilford, 1950 ) and E. P. Torrance ( Torrance, 1974 ), both of whom recognized that underlying the construct were other cognitive variables such as ideational fluency, originality of ideas, and sensitivity to missing elements. Many authors since then have extended the argument that a creative act is not a singular event but a process, an interplay among several interactive cognitive and affective elements. In this view, the creative act has two phases, a generative and an exploratory or evaluative phase ( Finke et al. , 1996 ). During the generative process, the creative mind pictures a set of novel mental models as potential solutions to a problem. In the exploratory phase, we evaluate the multiple options and select the best one. Early scholars of creativity, such as J. P. Guilford, characterized the two phases as divergent thinking and convergent thinking ( Guilford, 1950 ). Guilford defined divergent thinking as the ability to produce a broad range of associations to a given stimulus or to arrive at many solutions to a problem (for overviews of the field from different perspectives, see Amabile, 1996 ; Banaji et al. , 2006 ; Sawyer, 2006 ). In neurocognitive terms, divergent thinking is referred to as associative richness ( Gabora, 2002 ; Simonton, 2004 ), which is often measured experimentally by comparing the number of words that an individual generates from memory in response to stimulus words on a word association test. In contrast, convergent thinking refers to the capacity to quickly focus on the one best solution to a problem.

The idea that there are two stages to the creative process is consistent with results from cognition research indicating that there are two distinct modes of thought, associative and analytical ( Neisser, 1963 ; Sloman, 1996 ). In the associative mode, thinking is defocused, suggestive, and intuitive, revealing remote or subtle connections between items that may be correlated, or may not, and are usually not causally related ( Burton, 2008 ). In the analytical mode, thought is focused and evaluative, more conducive to analyzing relationships of cause and effect (for a review of other cognitive aspects of creativity, see Runco, 2004 ). Science educators associate the analytical mode with the upper levels (analysis, synthesis, and evaluation) of Bloom's taxonomy (e.g., Crowe et al. , 2008 ), or with “critical thinking,” the process that underlies the “purposeful, self-regulatory judgment that drives problem-solving and decision-making” ( Quitadamo et al. , 2008 , p. 328). These modes of thinking are under cognitive control through the executive functions of the brain. The core executive functions, which are thought to underlie all planning, problem solving, and reasoning, are defined ( Blair and Razza, 2007 ) as working memory control (mentally holding and retrieving information), cognitive flexibility (considering multiple ideas and seeing different perspectives), and inhibitory control (resisting several thoughts or actions to focus on one). Readers wishing to delve further into the neuroscience of the creative process can refer to the cerebrocerebellar theory of creativity ( Vandervert et al. , 2007 ) in which these mental activities are described neurophysiologically as arising through interactions among different parts of the brain.

The main point from all of these works is that creativity is not some single hard-to-measure property or act. There is ample evidence that the creative process requires both divergent and convergent thinking and that it can be explained by reference to increasingly well-understood underlying mental abilities ( Haring-Smith, 2006 ; Kim, 2006 ; Sawyer, 2006 ; Kaufman and Sternberg, 2007 ) and cognitive processes ( Simonton, 2004 ; Diamond et al. , 2007 ; Vandervert et al. , 2007 ).

Creativity Is Widely Distributed and Occurs in a Social Context

Although it is understandable to speak of an aha moment as a creative act by the person who experiences it, authorities in the field have long recognized (e.g., Simonton, 1975 ) that creative thinking is not so much an individual trait but rather a social phenomenon involving interactions among people within their specific group or cultural settings. “Creativity isn't just a property of individuals, it is also a property of social groups” ( Sawyer, 2006 , p. 305). Indeed, Osborn introduced his brainstorming method because he was convinced that group creativity is always superior to individual creativity. He drew evidence for this conclusion from activities that demand collaborative output, for example, the improvisations of a jazz ensemble. Although each musician is individually creative during a performance, the novelty and inventiveness of each performer's playing is clearly influenced, and often enhanced, by “social and interactional processes” among the musicians ( Sawyer, 2006 , p. 120). Recently, Brophy (2006) offered evidence that for problem solving, the situation may be more nuanced. He confirmed that groups of interacting individuals were better at solving complex, multipart problems than single individuals. However, when dealing with certain kinds of single-issue problems, individual problem solvers produced a greater number of solutions than interacting groups, and those solutions were judged to be more original and useful.

Consistent with the findings of Brophy (2006) , many scholars acknowledge that creative discoveries in the real world such as solving the problems of cutting-edge science—which are usually complex and multipart—are influenced or even stimulated by social interaction among experts. The common image of the lone scientist in the laboratory experiencing a flash of creative inspiration is probably a myth from earlier days. As a case in point, the science historian Mara Beller analyzed the social processes that underlay some of the major discoveries of early twentieth-century quantum physics. Close examination of successive drafts of publications by members of the Copenhagen group revealed a remarkable degree of influence and collaboration among 10 or more colleagues, although many of these papers were published under the name of a single author ( Beller, 1999 ). Sociologists Bruno Latour and Steve Woolgar's study ( Latour and Woolgar, 1986 ) of a neuroendocrinology laboratory at the Salk Institute for Biological Studies make the related point that social interactions among the participating scientists determined to a remarkable degree what discoveries were made and how they were interpreted. In the laboratory, researchers studied the chemical structure of substances released by the brain. By analysis of the Salk scientists' verbalizations of concepts, theories, formulas, and results of their investigations, Latour and Woolgar showed that the structures and interpretations that were agreed upon, that is, the discoveries announced by the laboratory, were mediated by social interactions and power relationships among members of the laboratory group. By studying the discovery process in other fields of the natural sciences, sociologists and anthropologists have provided more cases that further illustrate how social and cultural dimensions affect scientific insights (for a thoughtful review, see Knorr Cetina, 1995 ).

In sum, when an individual experiences an aha moment that feels like a singular creative act, it may rather have resulted from a multicomponent process, under the influence of group interactions and social context. The process that led up to what may be sensed as a sudden insight will probably have included at least three diverse, but testable elements: 1) divergent thinking, including ideational fluency or cognitive flexibility, which is the cognitive executive function that underlies the ability to visualize and accept many ideas related to a problem; 2) convergent thinking or the application of inhibitory control to focus and mentally evaluate ideas; and 3) analogical thinking, the ability to understand a novel idea in terms of one that is already familiar.

LITERATURE REVIEW

What do we know about how to teach creativity.

The possibility of teaching for creative problem solving gained credence in the 1960s with the studies of Jerome Bruner, who argued that children should be encouraged to “treat a task as a problem for which one invents an answer, rather than finding one out there in a book or on the blackboard” ( Bruner, 1965 , pp. 1013–1014). Since that time, educators and psychologists have devised programs of instruction designed to promote creativity and inventiveness in virtually every student population: pre–K, elementary, high school, and college, as well as in disadvantaged students, athletes, and students in a variety of specific disciplines (for review, see Scott et al. , 2004 ). Smith (1998) identified 172 instructional approaches that have been applied at one time or another to develop divergent thinking skills.

Some of the most convincing evidence that elements of creativity can be enhanced by instruction comes from work with young children. Bodrova and Leong (2001) developed the Tools of the Mind (Tools) curriculum to improve all of the three core mental executive functions involved in creative problem solving: cognitive flexibility, working memory, and inhibitory control. In a year-long randomized study of 5-yr-olds from low-income families in 21 preschool classrooms, half of the teachers applied the districts' balanced literacy curriculum (literacy), whereas the experimenters trained the other half to teach the same academic content by using the Tools curriculum ( Diamond et al. , 2007 ). At the end of the year, when the children were tested with a battery of neurocognitive tests including a test for cognitive flexibility ( Durston et al. , 2003 ; Davidson et al. , 2006 ), those exposed to the Tools curriculum outperformed the literacy children by as much as 25% ( Diamond et al. , 2007 ). Although the Tools curriculum and literacy program were similar in academic content and in many other ways, they differed primarily in that Tools teachers spent 80% of their time explicitly reminding the children to think of alternative ways to solve a problem and building their executive function skills.

Teaching older students to be innovative also demands instruction that explicitly promotes creativity but is rigorously content-rich as well. A large body of research on the differences between novice and expert cognition indicates that creative thinking requires at least a minimal level of expertise and fluency within a knowledge domain ( Bransford et al. , 2000 ; Crawford and Brophy, 2006 ). What distinguishes experts from novices, in addition to their deeper knowledge of the subject, is their recognition of patterns in information, their ability to see relationships among disparate facts and concepts, and their capacity for organizing content into conceptual frameworks or schemata ( Bransford et al. , 2000 ; Sawyer, 2005 ).

Such expertise is often lacking in the traditional classroom. For students attempting to grapple with new subject matter, many kinds of problems that are presented in high school or college courses or that arise in the real world can be solved merely by applying newly learned algorithms or procedural knowledge. With practice, problem solving of this kind can become routine and is often considered to represent mastery of a subject, producing what Sternberg refers to as “pseudoexperts” ( Sternberg, 2003 ). But beyond such routine use of content knowledge the instructor's goal must be to produce students who have gained the HOCS needed to apply, analyze, synthesize, and evaluate knowledge ( Crowe et al. , 2008 ). The aim is to produce students who know enough about a field to grasp meaningful patterns of information, who can readily retrieve relevant knowledge from memory, and who can apply such knowledge effectively to novel problems. This condition is referred to as adaptive expertise ( Hatano and Ouro, 2003 ; Schwartz et al. , 2005 ). Instead of applying already mastered procedures, adaptive experts are able to draw on their knowledge to invent or adapt strategies for solving unique or novel problems within a knowledge domain. They are also able, ideally, to transfer conceptual frameworks and schemata from one domain to another (e.g., Schwartz et al. , 2005 ). Such flexible, innovative application of knowledge is what results in inventive or creative solutions to problems ( Crawford and Brophy, 2006 ; Crawford, 2007 ).

Promoting Creative Problem Solving in the College Classroom

In most college courses, instructors teach science primarily through lectures and textbooks that are dominated by facts and algorithmic processing rather than by concepts, principles, and evidence-based ways of thinking. This is despite ample evidence that many students gain little new knowledge from traditional lectures ( Hrepic et al. , 2007 ). Moreover, it is well documented that these methods engender passive learning rather than active engagement, boredom instead of intellectual excitement, and linear thinking rather than cognitive flexibility (e.g., Halpern and Hakel, 2003 ; Nelson, 2008 ; Perkins and Wieman, 2008 ). Cognitive flexibility, as noted, is one of the three core mental executive functions involved in creative problem solving ( Ausubel, 1963 , 2000 ). The capacity to apply ideas creatively in new contexts, referred to as the ability to “transfer” knowledge (see Mestre, 2005 ), requires that learners have opportunities to actively develop their own representations of information to convert it to a usable form. Especially when a knowledge domain is complex and fraught with ill-structured information, as in a typical introductory college biology course, instruction that emphasizes active-learning strategies is demonstrably more effective than traditional linear teaching in reducing failure rates and in promoting learning and transfer (e.g., Freeman et al. , 2007 ). Furthermore, there is already some evidence that inclusion of creativity training as part of a college curriculum can have positive effects. Hunsaker (2005) has reviewed a number of such studies. He cites work by McGregor (2001) , for example, showing that various creativity training programs including brainstorming and creative problem solving increase student scores on tests of creative-thinking abilities.

What explicit instructional strategies are available to promote creative problem solving? In addition to brainstorming, McFadzean (2002) discusses several “paradigm-stretching” techniques that can encourage creative ideas. One method, known as heuristic ideation, encourages participants to force together two unrelated concepts to discover novel relationships, a modern version of Koestler's bisociation ( Koestler, 1964 ). On the website of the Center for Development and Learning, Robert Sternberg and Wendy M. Williams offer 24 “tips” for teachers wishing to promote creativity in their students ( Sternberg and Williams, 1998 ). Among them, the following techniques might apply to a science classroom:

  • Model creativity—students develop creativity when instructors model creative thinking and inventiveness.
  • Repeatedly encourage idea generation—students need to be reminded to generate their own ideas and solutions in an environment free of criticism.
  • Cross-fertilize ideas—where possible, avoid teaching in subject-area boxes: a math box, a social studies box, etc; students' creative ideas and insights often result from learning to integrate material across subject areas.
  • Build self-efficacy—all students have the capacity to create and to experience the joy of having new ideas, but they must be helped to believe in their own capacity to be creative.
  • Constantly question assumptions—make questioning a part of the daily classroom exchange; it is more important for students to learn what questions to ask and how to ask them than to learn the answers.
  • Imagine other viewpoints—students broaden their perspectives by learning to reflect upon ideas and concepts from different points of view.

Although these strategies are all consistent with the knowledge about creativity that I have reviewed above, evidence from well-designed investigations to warrant the claim that they can enhance measurable indicators of creativity in college students is only recently beginning to materialize. If creativity most often occurs in “a mental state where attention is defocused, thought is associative, and a large number of mental representations are simultaneously activated” ( Martindale, 1999 , p. 149), the question arises whether instructional strategies designed to enhance the HOCS also foster such a mental state? Do valid tests exist to show that creative problem solving can be enhanced by such instruction?

How Is Creativity Related to Critical Thinking and the Higher-Order Cognitive Skills?

It is not uncommon to associate creativity and ingenuity with scientific reasoning ( Sawyer, 2005 ; 2006 ). When instructors apply scientific teaching strategies ( Handelsman et al. , 2004 ; DeHaan, 2005 ; Wood, 2009 ) by using instructional methods based on learning research, according to Ebert-May and Hodder ( 2008 ), “we see students actively engaged in the thinking, creativity, rigor, and experimentation we associate with the practice of science—in much the same way we see students learn in the field and in laboratories” (p. 2). Perkins and Wieman (2008) note that “To be successful innovators in science and engineering, students must develop a deep conceptual understanding of the underlying science ideas, an ability to apply these ideas and concepts broadly in different contexts, and a vision to see their relevance and usefulness in real-world applications … An innovator is able to perceive and realize potential connections and opportunities better than others” (pp. 181–182). The results of Scott et al. (2004) suggest that nontraditional courses in science that are based on constructivist principles and that use strategies of scientific teaching to promote the HOCS and enhance content mastery and dexterity in scientific thinking ( Handelsman et al. , 2007 ; Nelson, 2008 ) also should be effective in promoting creativity and cognitive flexibility if students are explicitly guided to learn these skills.

Creativity is an essential element of problem solving ( Mumford et al. , 1991 ; Runco, 2004 ) and of critical thinking ( Abrami et al. , 2008 ). As such, it is common to think of applications of creativity such as inventiveness and ingenuity among the HOCS as defined in Bloom's taxonomy ( Crowe et al. , 2008 ). Thus, it should come as no surprise that creativity, like other elements of the HOCS, can be taught most effectively through inquiry-based instruction, informed by constructivist theory ( Ausubel, 1963 , 2000 ; Duch et al. , 2001 ; Nelson, 2008 ). In a survey of 103 instructors who taught college courses that included creativity instruction, Bull et al. (1995) asked respondents to rate the importance of various course characteristics for enhancing student creativity. Items ranking high on the list were: providing a social climate in which students feels safe, an open classroom environment that promotes tolerance for ambiguity and independence, the use of humor, metaphorical thinking, and problem defining. Many of the responses emphasized the same strategies as those advanced to promote creative problem solving (e.g., Mumford et al. , 1991 ; McFadzean, 2002 ; Treffinger and Isaksen, 2005 ) and critical thinking ( Abrami et al. , 2008 ).

In a careful meta-analysis, Scott et al. (2004) examined 70 instructional interventions designed to enhance and measure creative performance. The results were striking. Courses that stressed techniques such as critical thinking, convergent thinking, and constraint identification produced the largest positive effect sizes. More open techniques that provided less guidance in strategic approaches had less impact on the instructional outcomes. A striking finding was the effectiveness of being explicit; approaches that clearly informed students about the nature of creativity and offered clear strategies for creative thinking were most effective. Approaches such as social modeling, cooperative learning, and case-based (project-based) techniques that required the application of newly acquired knowledge were found to be positively correlated to high effect sizes. The most clear-cut result to emerge from the Scott et al. (2004) study was simply to confirm that creativity instruction can be highly successful in enhancing divergent thinking, problem solving, and imaginative performance. Most importantly, of the various cognitive processes examined, those linked to the generation of new ideas such as problem finding, conceptual combination, and idea generation showed the greatest improvement. The success of creativity instruction, the authors concluded, can be attributed to “developing and providing guidance concerning the application of requisite cognitive capacities … [and] a set of heuristics or strategies for working with already available knowledge” (p. 382).

Many of the scientific teaching practices that have been shown by research to foster content mastery and HOCS, and that are coming more widely into use, also would be consistent with promoting creativity. Wood (2009) has recently reviewed examples of such practices and how to apply them. These include relatively small modifications of the traditional lecture to engender more active learning, such as the use of concept tests and peer instruction ( Mazur, 1996 ), Just-in-Time-Teaching techniques ( Novak et al. , 1999 ), and student response systems known as “clickers” ( Knight and Wood, 2005 ; Crossgrove and Curran, 2008 ), all designed to allow the instructor to frequently and effortlessly elicit and respond to student thinking. Other strategies can transform the lecture hall into a workshop or studio classroom ( Gaffney et al. , 2008 ) where the teaching curriculum may emphasize problem-based (also known as project-based or case-based) learning strategies ( Duch et al. , 2001 ; Ebert-May and Hodder, 2008 ) or “community-based inquiry” in which students engage in research that enhances their critical-thinking skills ( Quitadamo et al. , 2008 ).

Another important approach that could readily subserve explicit creativity instruction is the use of computer-based interactive simulations, or “sims” ( Perkins and Wieman, 2008 ) to facilitate inquiry learning and effective, easy self-assessment. An example in the biological sciences would be Neurons in Action ( http://neuronsinaction.com/home/main ). In such educational environments, students gain conceptual understanding of scientific ideas through interactive engagement with materials (real or virtual), with each other, and with instructors. Following the tenets of scientific teaching, students are encouraged to pose and answer their own questions, to make sense of the materials, and to construct their own understanding. The question I pose here is whether an additional focus—guiding students to meet these challenges in a context that explicitly promotes creativity—would enhance learning and advance students' progress toward adaptive expertise?

Assessment of Creativity

To teach creativity, there must be measurable indicators to judge how much students have gained from instruction. Educational programs intended to teach creativity became popular after the Torrance Tests of Creative Thinking (TTCT) was introduced in the 1960s ( Torrance, 1974 ). But it soon became apparent that there were major problems in devising tests for creativity, both because of the difficulty of defining the construct and because of the number and complexity of elements that underlie it. Tests of intelligence and other personality characteristics on creative individuals revealed a host of related traits such as verbal fluency, metaphorical thinking, flexible decision making, tolerance of ambiguity, willingness to take risks, autonomy, divergent thinking, self-confidence, problem finding, ideational fluency, and belief in oneself as being “creative” ( Barron and Harrington, 1981 ; Tardif and Sternberg, 1988 ; Runco and Nemiro, 1994 ; Snyder et al. , 2004 ). Many of these traits have been the focus of extensive research of recent decades, but, as noted above, creativity is not defined by any one trait; there is now reason to believe that it is the interplay among the cognitive and affective processes that underlie inventiveness and the ability to find novel solutions to a problem.

Although the early creativity researchers recognized that assessing divergent thinking as a measure of creativity required tests for other underlying capacities ( Guilford, 1950 ; Torrance, 1974 ), these workers and their colleagues nonetheless believed that a high score for divergent thinking alone would correlate with real creative output. Unfortunately, no such correlation was shown ( Barron and Harrington, 1981 ). Results produced by many of the instruments initially designed to measure various aspects of creative thinking proved to be highly dependent on the test itself. A review of several hundred early studies showed that an individual's creativity score could be affected by simple test variables, for example, how the verbal pretest instructions were worded ( Barron and Harrington, 1981 , pp. 442–443). Most scholars now agree that divergent thinking, as originally defined, was not an adequate measure of creativity. The process of creative thinking requires a complex combination of elements that include cognitive flexibility, memory control, inhibitory control, and analogical thinking, enabling the mind to free-range and analogize, as well as to focus and test.

More recently, numerous psychometric measures have been developed and empirically tested (see Plucker and Renzulli, 1999 ) that allow more reliable and valid assessment of specific aspects of creativity. For example, the creativity quotient devised by Snyder et al. (2004) tests the ability of individuals to link different ideas and different categories of ideas into a novel synthesis. The Wallach–Kogan creativity test ( Wallach and Kogan, 1965 ) explores the uniqueness of ideas associated with a stimulus. For a more complete list and discussion, see the Creativity Tests website ( www.indiana.edu/∼bobweb/Handout/cretv_6.html ).

The most widely used measure of creativity is the TTCT, which has been modified four times since its original version in 1966 to take into account subsequent research. The TTCT-Verbal and the TTCT-Figural are two versions ( Torrance, 1998 ; see http://ststesting.com/2005giftttct.html ). The TTCT-Verbal consists of five tasks; the “stimulus” for each task is a picture to which the test-taker responds briefly in writing. A sample task that can be viewed from the TTCT Demonstrator website asks, “Suppose that people could transport themselves from place to place with just a wink of the eye or a twitch of the nose. What might be some things that would happen as a result? You have 3 min.” ( www.indiana.edu/∼bobweb/Handout/d3.ttct.htm ).

In the TTCT-Figural, participants are asked to construct a picture from a stimulus in the form of a partial line drawing given on the test sheet (see example below; Figure 1 ). Specific instructions are to “Add lines to the incomplete figures below to make pictures out of them. Try to tell complete stories with your pictures. Give your pictures titles. You have 3 min.” In the introductory materials, test-takers are urged to “… think of a picture or object that no one else will think of. Try to make it tell as complete and as interesting a story as you can …” ( Torrance et al. , 2008 , p. 2).

An external file that holds a picture, illustration, etc.
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Sample figural test item from the TTCT Demonstrator website ( www.indiana.edu/∼bobweb/Handout/d3.ttct.htm ).

How would an instructor in a biology course judge the creativity of students' responses to such an item? To assist in this task, the TTCT has scoring and norming guides ( Torrance, 1998 ; Torrance et al. , 2008 ) with numerous samples and responses representing different levels of creativity. The guides show sample evaluations based upon specific indicators such as fluency, originality, elaboration (or complexity), unusual visualization, extending or breaking boundaries, humor, and imagery. These examples are easy to use and provide a high degree of validity and generalizability to the tests. The TTCT has been more intensively researched and analyzed than any other creativity instrument, and the norming samples have longitudinal validations and high predictive validity over a wide age range. In addition to global creativity scores, the TTCT is designed to provide outcome measures in various domains and thematic areas to allow for more insightful analysis ( Kaufman and Baer, 2006 ). Kim (2006) has examined the characteristics of the TTCT, including norms, reliability, and validity, and concludes that the test is an accurate measure of creativity. When properly used, it has been shown to be fair in terms of gender, race, community status, and language background. According to Kim (2006) and other authorities in the field ( McIntyre et al. , 2003 ; Scott et al. , 2004 ), Torrance's research and the development of the TTCT have provided groundwork for the idea that creative levels can be measured and then increased through instruction and practice.

SCIENTIFIC TEACHING TO PROMOTE CREATIVITY

How could creativity instruction be integrated into scientific teaching.

Guidelines for designing specific course units that emphasize HOCS by using strategies of scientific teaching are now available from the current literature. As an example, Karen Cloud-Hansen and colleagues ( Cloud-Hansen et al. , 2008 ) describe a course titled, “Ciprofloxacin Resistance in Neisseria gonorrhoeae .” They developed this undergraduate seminar to introduce college freshmen to important concepts in biology within a real-world context and to increase their content knowledge and critical-thinking skills. The centerpiece of the unit is a case study in which teams of students are challenged to take the role of a director of a local public health clinic. One of the county commissioners overseeing the clinic is an epidemiologist who wants to know “how you plan to address the emergence of ciprofloxacin resistance in Neisseria gonorrhoeae ” (p. 304). State budget cuts limit availability of expensive antibiotics and some laboratory tests to patients. Student teams are challenged to 1) develop a plan to address the medical, economic, and political questions such a clinic director would face in dealing with ciprofloxacin-resistant N. gonorrhoeae ; 2) provide scientific data to support their conclusions; and 3) describe their clinic plan in a one- to two-page referenced written report.

Throughout the 3-wk unit, in accordance with the principles of problem-based instruction ( Duch et al. , 2001 ), course instructors encourage students to seek, interpret, and synthesize their own information to the extent possible. Students have access to a variety of instructional formats, and active-learning experiences are incorporated throughout the unit. These activities are interspersed among minilectures and give the students opportunities to apply new information to their existing base of knowledge. The active-learning activities emphasize the key concepts of the minilectures and directly confront common misconceptions about antibiotic resistance, gene expression, and evolution. Weekly classes include question/answer/discussion sessions to address student misconceptions and 20-min minilectures on such topics as antibiotic resistance, evolution, and the central dogma of molecular biology. Students gather information about antibiotic resistance in N. gonorrhoeae , epidemiology of gonorrhea, and treatment options for the disease, and each team is expected to formulate a plan to address ciprofloxacin resistance in N. gonorrhoeae .

In this project, the authors assessed student gains in terms of content knowledge regarding topics covered such as the role of evolution in antibiotic resistance, mechanisms of gene expression, and the role of oncogenes in human disease. They also measured HOCS as gains in problem solving, according to a rubric that assessed self-reported abilities to communicate ideas logically, solve difficult problems about microbiology, propose hypotheses, analyze data, and draw conclusions. Comparing the pre- and posttests, students reported significant learning of scientific content. Among the thinking skill categories, students demonstrated measurable gains in their ability to solve problems about microbiology but the unit seemed to have little impact on their more general perceived problem-solving skills ( Cloud-Hansen et al. , 2008 ).

What would such a class look like with the addition of explicit creativity-promoting approaches? Would the gains in problem-solving abilities have been greater if during the minilectures and other activities, students had been introduced explicitly to elements of creative thinking from the Sternberg and Williams (1998) list described above? Would the students have reported greater gains if their instructors had encouraged idea generation with weekly brainstorming sessions; if they had reminded students to cross-fertilize ideas by integrating material across subject areas; built self-efficacy by helping students believe in their own capacity to be creative; helped students question their own assumptions; and encouraged students to imagine other viewpoints and possibilities? Of most relevance, could the authors have been more explicit in assessing the originality of the student plans? In an experiment that required college students to develop plans of a different, but comparable, type, Osborn and Mumford (2006) created an originality rubric ( Figure 2 ) that could apply equally to assist instructors in judging student plans in any course. With such modifications, would student gains in problem-solving abilities or other HOCS have been greater? Would their plans have been measurably more imaginative?

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Originality rubric (adapted from Osburn and Mumford, 2006 , p. 183).

Answers to these questions can only be obtained when a course like that described by Cloud-Hansen et al. (2008) is taught with explicit instruction in creativity of the type I described above. But, such answers could be based upon more than subjective impressions of the course instructors. For example, students could be pretested with items from the TTCT-Verbal or TTCT-Figural like those shown. If, during minilectures and at every contact with instructors, students were repeatedly reminded and shown how to be as creative as possible, to integrate material across subject areas, to question their own assumptions and imagine other viewpoints and possibilities, would their scores on TTCT posttest items improve? Would the plans they formulated to address ciprofloxacin resistance become more imaginative?

Recall that in their meta-analysis, Scott et al. (2004) found that explicitly informing students about the nature of creativity and offering strategies for creative thinking were the most effective components of instruction. From their careful examination of 70 experimental studies, they concluded that approaches such as social modeling, cooperative learning, and case-based (project-based) techniques that required the application of newly acquired knowledge were positively correlated with high effect sizes. The study was clear in confirming that explicit creativity instruction can be successful in enhancing divergent thinking and problem solving. Would the same strategies work for courses in ecology and environmental biology, as detailed by Ebert-May and Hodder (2008) , or for a unit elaborated by Knight and Wood (2005) that applies classroom response clickers?

Finally, I return to my opening question with the fictional Dr. Dunne. Could a weekly brainstorming “invention session” included in a course like those described here serve as the site where students are introduced to concepts and strategies of creative problem solving? As frequently applied in schools of engineering ( Paulus and Nijstad, 2003 ), brainstorming provides an opportunity for the instructor to pose a problem and to ask the students to suggest as many solutions as possible in a brief period, thus enhancing ideational fluency. Here, students can be encouraged explicitly to build on the ideas of others and to think flexibly. Would brainstorming enhance students' divergent thinking or creative abilities as measured by TTCT items or an originality rubric? Many studies have demonstrated that group interactions such as brainstorming, under the right conditions, can indeed enhance creativity ( Paulus and Nijstad, 2003 ; Scott et al. , 2004 ), but there is little information from an undergraduate science classroom setting. Intellectual Ventures, a firm founded by Nathan Myhrvold, the creator of Microsoft's Research Division, has gathered groups of engineers and scientists around a table for day-long sessions to brainstorm about a prearranged topic. Here, the method seems to work. Since it was founded in 2000, Intellectual Ventures has filed hundreds of patent applications in more than 30 technology areas, applying the “invention session” strategy ( Gladwell, 2008 ). Currently, the company ranks among the top 50 worldwide in number of patent applications filed annually. Whether such a technique could be applied successfully in a college science course will only be revealed by future research.

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  • Published: 11 January 2023

The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature

  • Enwei Xu   ORCID: orcid.org/0000-0001-6424-8169 1 ,
  • Wei Wang 1 &
  • Qingxia Wang 1  

Humanities and Social Sciences Communications volume  10 , Article number:  16 ( 2023 ) Cite this article

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Collaborative problem-solving has been widely embraced in the classroom instruction of critical thinking, which is regarded as the core of curriculum reform based on key competencies in the field of education as well as a key competence for learners in the 21st century. However, the effectiveness of collaborative problem-solving in promoting students’ critical thinking remains uncertain. This current research presents the major findings of a meta-analysis of 36 pieces of the literature revealed in worldwide educational periodicals during the 21st century to identify the effectiveness of collaborative problem-solving in promoting students’ critical thinking and to determine, based on evidence, whether and to what extent collaborative problem solving can result in a rise or decrease in critical thinking. The findings show that (1) collaborative problem solving is an effective teaching approach to foster students’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]); (2) in respect to the dimensions of critical thinking, collaborative problem solving can significantly and successfully enhance students’ attitudinal tendencies (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI[0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI[0.58, 0.82]); and (3) the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have an impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. On the basis of these results, recommendations are made for further study and instruction to better support students’ critical thinking in the context of collaborative problem-solving.

Introduction

Although critical thinking has a long history in research, the concept of critical thinking, which is regarded as an essential competence for learners in the 21st century, has recently attracted more attention from researchers and teaching practitioners (National Research Council, 2012 ). Critical thinking should be the core of curriculum reform based on key competencies in the field of education (Peng and Deng, 2017 ) because students with critical thinking can not only understand the meaning of knowledge but also effectively solve practical problems in real life even after knowledge is forgotten (Kek and Huijser, 2011 ). The definition of critical thinking is not universal (Ennis, 1989 ; Castle, 2009 ; Niu et al., 2013 ). In general, the definition of critical thinking is a self-aware and self-regulated thought process (Facione, 1990 ; Niu et al., 2013 ). It refers to the cognitive skills needed to interpret, analyze, synthesize, reason, and evaluate information as well as the attitudinal tendency to apply these abilities (Halpern, 2001 ). The view that critical thinking can be taught and learned through curriculum teaching has been widely supported by many researchers (e.g., Kuncel, 2011 ; Leng and Lu, 2020 ), leading to educators’ efforts to foster it among students. In the field of teaching practice, there are three types of courses for teaching critical thinking (Ennis, 1989 ). The first is an independent curriculum in which critical thinking is taught and cultivated without involving the knowledge of specific disciplines; the second is an integrated curriculum in which critical thinking is integrated into the teaching of other disciplines as a clear teaching goal; and the third is a mixed curriculum in which critical thinking is taught in parallel to the teaching of other disciplines for mixed teaching training. Furthermore, numerous measuring tools have been developed by researchers and educators to measure critical thinking in the context of teaching practice. These include standardized measurement tools, such as WGCTA, CCTST, CCTT, and CCTDI, which have been verified by repeated experiments and are considered effective and reliable by international scholars (Facione and Facione, 1992 ). In short, descriptions of critical thinking, including its two dimensions of attitudinal tendency and cognitive skills, different types of teaching courses, and standardized measurement tools provide a complex normative framework for understanding, teaching, and evaluating critical thinking.

Cultivating critical thinking in curriculum teaching can start with a problem, and one of the most popular critical thinking instructional approaches is problem-based learning (Liu et al., 2020 ). Duch et al. ( 2001 ) noted that problem-based learning in group collaboration is progressive active learning, which can improve students’ critical thinking and problem-solving skills. Collaborative problem-solving is the organic integration of collaborative learning and problem-based learning, which takes learners as the center of the learning process and uses problems with poor structure in real-world situations as the starting point for the learning process (Liang et al., 2017 ). Students learn the knowledge needed to solve problems in a collaborative group, reach a consensus on problems in the field, and form solutions through social cooperation methods, such as dialogue, interpretation, questioning, debate, negotiation, and reflection, thus promoting the development of learners’ domain knowledge and critical thinking (Cindy, 2004 ; Liang et al., 2017 ).

Collaborative problem-solving has been widely used in the teaching practice of critical thinking, and several studies have attempted to conduct a systematic review and meta-analysis of the empirical literature on critical thinking from various perspectives. However, little attention has been paid to the impact of collaborative problem-solving on critical thinking. Therefore, the best approach for developing and enhancing critical thinking throughout collaborative problem-solving is to examine how to implement critical thinking instruction; however, this issue is still unexplored, which means that many teachers are incapable of better instructing critical thinking (Leng and Lu, 2020 ; Niu et al., 2013 ). For example, Huber ( 2016 ) provided the meta-analysis findings of 71 publications on gaining critical thinking over various time frames in college with the aim of determining whether critical thinking was truly teachable. These authors found that learners significantly improve their critical thinking while in college and that critical thinking differs with factors such as teaching strategies, intervention duration, subject area, and teaching type. The usefulness of collaborative problem-solving in fostering students’ critical thinking, however, was not determined by this study, nor did it reveal whether there existed significant variations among the different elements. A meta-analysis of 31 pieces of educational literature was conducted by Liu et al. ( 2020 ) to assess the impact of problem-solving on college students’ critical thinking. These authors found that problem-solving could promote the development of critical thinking among college students and proposed establishing a reasonable group structure for problem-solving in a follow-up study to improve students’ critical thinking. Additionally, previous empirical studies have reached inconclusive and even contradictory conclusions about whether and to what extent collaborative problem-solving increases or decreases critical thinking levels. As an illustration, Yang et al. ( 2008 ) carried out an experiment on the integrated curriculum teaching of college students based on a web bulletin board with the goal of fostering participants’ critical thinking in the context of collaborative problem-solving. These authors’ research revealed that through sharing, debating, examining, and reflecting on various experiences and ideas, collaborative problem-solving can considerably enhance students’ critical thinking in real-life problem situations. In contrast, collaborative problem-solving had a positive impact on learners’ interaction and could improve learning interest and motivation but could not significantly improve students’ critical thinking when compared to traditional classroom teaching, according to research by Naber and Wyatt ( 2014 ) and Sendag and Odabasi ( 2009 ) on undergraduate and high school students, respectively.

The above studies show that there is inconsistency regarding the effectiveness of collaborative problem-solving in promoting students’ critical thinking. Therefore, it is essential to conduct a thorough and trustworthy review to detect and decide whether and to what degree collaborative problem-solving can result in a rise or decrease in critical thinking. Meta-analysis is a quantitative analysis approach that is utilized to examine quantitative data from various separate studies that are all focused on the same research topic. This approach characterizes the effectiveness of its impact by averaging the effect sizes of numerous qualitative studies in an effort to reduce the uncertainty brought on by independent research and produce more conclusive findings (Lipsey and Wilson, 2001 ).

This paper used a meta-analytic approach and carried out a meta-analysis to examine the effectiveness of collaborative problem-solving in promoting students’ critical thinking in order to make a contribution to both research and practice. The following research questions were addressed by this meta-analysis:

What is the overall effect size of collaborative problem-solving in promoting students’ critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills)?

How are the disparities between the study conclusions impacted by various moderating variables if the impacts of various experimental designs in the included studies are heterogeneous?

This research followed the strict procedures (e.g., database searching, identification, screening, eligibility, merging, duplicate removal, and analysis of included studies) of Cooper’s ( 2010 ) proposed meta-analysis approach for examining quantitative data from various separate studies that are all focused on the same research topic. The relevant empirical research that appeared in worldwide educational periodicals within the 21st century was subjected to this meta-analysis using Rev-Man 5.4. The consistency of the data extracted separately by two researchers was tested using Cohen’s kappa coefficient, and a publication bias test and a heterogeneity test were run on the sample data to ascertain the quality of this meta-analysis.

Data sources and search strategies

There were three stages to the data collection process for this meta-analysis, as shown in Fig. 1 , which shows the number of articles included and eliminated during the selection process based on the statement and study eligibility criteria.

figure 1

This flowchart shows the number of records identified, included and excluded in the article.

First, the databases used to systematically search for relevant articles were the journal papers of the Web of Science Core Collection and the Chinese Core source journal, as well as the Chinese Social Science Citation Index (CSSCI) source journal papers included in CNKI. These databases were selected because they are credible platforms that are sources of scholarly and peer-reviewed information with advanced search tools and contain literature relevant to the subject of our topic from reliable researchers and experts. The search string with the Boolean operator used in the Web of Science was “TS = (((“critical thinking” or “ct” and “pretest” or “posttest”) or (“critical thinking” or “ct” and “control group” or “quasi experiment” or “experiment”)) and (“collaboration” or “collaborative learning” or “CSCL”) and (“problem solving” or “problem-based learning” or “PBL”))”. The research area was “Education Educational Research”, and the search period was “January 1, 2000, to December 30, 2021”. A total of 412 papers were obtained. The search string with the Boolean operator used in the CNKI was “SU = (‘critical thinking’*‘collaboration’ + ‘critical thinking’*‘collaborative learning’ + ‘critical thinking’*‘CSCL’ + ‘critical thinking’*‘problem solving’ + ‘critical thinking’*‘problem-based learning’ + ‘critical thinking’*‘PBL’ + ‘critical thinking’*‘problem oriented’) AND FT = (‘experiment’ + ‘quasi experiment’ + ‘pretest’ + ‘posttest’ + ‘empirical study’)” (translated into Chinese when searching). A total of 56 studies were found throughout the search period of “January 2000 to December 2021”. From the databases, all duplicates and retractions were eliminated before exporting the references into Endnote, a program for managing bibliographic references. In all, 466 studies were found.

Second, the studies that matched the inclusion and exclusion criteria for the meta-analysis were chosen by two researchers after they had reviewed the abstracts and titles of the gathered articles, yielding a total of 126 studies.

Third, two researchers thoroughly reviewed each included article’s whole text in accordance with the inclusion and exclusion criteria. Meanwhile, a snowball search was performed using the references and citations of the included articles to ensure complete coverage of the articles. Ultimately, 36 articles were kept.

Two researchers worked together to carry out this entire process, and a consensus rate of almost 94.7% was reached after discussion and negotiation to clarify any emerging differences.

Eligibility criteria

Since not all the retrieved studies matched the criteria for this meta-analysis, eligibility criteria for both inclusion and exclusion were developed as follows:

The publication language of the included studies was limited to English and Chinese, and the full text could be obtained. Articles that did not meet the publication language and articles not published between 2000 and 2021 were excluded.

The research design of the included studies must be empirical and quantitative studies that can assess the effect of collaborative problem-solving on the development of critical thinking. Articles that could not identify the causal mechanisms by which collaborative problem-solving affects critical thinking, such as review articles and theoretical articles, were excluded.

The research method of the included studies must feature a randomized control experiment or a quasi-experiment, or a natural experiment, which have a higher degree of internal validity with strong experimental designs and can all plausibly provide evidence that critical thinking and collaborative problem-solving are causally related. Articles with non-experimental research methods, such as purely correlational or observational studies, were excluded.

The participants of the included studies were only students in school, including K-12 students and college students. Articles in which the participants were non-school students, such as social workers or adult learners, were excluded.

The research results of the included studies must mention definite signs that may be utilized to gauge critical thinking’s impact (e.g., sample size, mean value, or standard deviation). Articles that lacked specific measurement indicators for critical thinking and could not calculate the effect size were excluded.

Data coding design

In order to perform a meta-analysis, it is necessary to collect the most important information from the articles, codify that information’s properties, and convert descriptive data into quantitative data. Therefore, this study designed a data coding template (see Table 1 ). Ultimately, 16 coding fields were retained.

The designed data-coding template consisted of three pieces of information. Basic information about the papers was included in the descriptive information: the publishing year, author, serial number, and title of the paper.

The variable information for the experimental design had three variables: the independent variable (instruction method), the dependent variable (critical thinking), and the moderating variable (learning stage, teaching type, intervention duration, learning scaffold, group size, measuring tool, and subject area). Depending on the topic of this study, the intervention strategy, as the independent variable, was coded into collaborative and non-collaborative problem-solving. The dependent variable, critical thinking, was coded as a cognitive skill and an attitudinal tendency. And seven moderating variables were created by grouping and combining the experimental design variables discovered within the 36 studies (see Table 1 ), where learning stages were encoded as higher education, high school, middle school, and primary school or lower; teaching types were encoded as mixed courses, integrated courses, and independent courses; intervention durations were encoded as 0–1 weeks, 1–4 weeks, 4–12 weeks, and more than 12 weeks; group sizes were encoded as 2–3 persons, 4–6 persons, 7–10 persons, and more than 10 persons; learning scaffolds were encoded as teacher-supported learning scaffold, technique-supported learning scaffold, and resource-supported learning scaffold; measuring tools were encoded as standardized measurement tools (e.g., WGCTA, CCTT, CCTST, and CCTDI) and self-adapting measurement tools (e.g., modified or made by researchers); and subject areas were encoded according to the specific subjects used in the 36 included studies.

The data information contained three metrics for measuring critical thinking: sample size, average value, and standard deviation. It is vital to remember that studies with various experimental designs frequently adopt various formulas to determine the effect size. And this paper used Morris’ proposed standardized mean difference (SMD) calculation formula ( 2008 , p. 369; see Supplementary Table S3 ).

Procedure for extracting and coding data

According to the data coding template (see Table 1 ), the 36 papers’ information was retrieved by two researchers, who then entered them into Excel (see Supplementary Table S1 ). The results of each study were extracted separately in the data extraction procedure if an article contained numerous studies on critical thinking, or if a study assessed different critical thinking dimensions. For instance, Tiwari et al. ( 2010 ) used four time points, which were viewed as numerous different studies, to examine the outcomes of critical thinking, and Chen ( 2013 ) included the two outcome variables of attitudinal tendency and cognitive skills, which were regarded as two studies. After discussion and negotiation during data extraction, the two researchers’ consistency test coefficients were roughly 93.27%. Supplementary Table S2 details the key characteristics of the 36 included articles with 79 effect quantities, including descriptive information (e.g., the publishing year, author, serial number, and title of the paper), variable information (e.g., independent variables, dependent variables, and moderating variables), and data information (e.g., mean values, standard deviations, and sample size). Following that, testing for publication bias and heterogeneity was done on the sample data using the Rev-Man 5.4 software, and then the test results were used to conduct a meta-analysis.

Publication bias test

When the sample of studies included in a meta-analysis does not accurately reflect the general status of research on the relevant subject, publication bias is said to be exhibited in this research. The reliability and accuracy of the meta-analysis may be impacted by publication bias. Due to this, the meta-analysis needs to check the sample data for publication bias (Stewart et al., 2006 ). A popular method to check for publication bias is the funnel plot; and it is unlikely that there will be publishing bias when the data are equally dispersed on either side of the average effect size and targeted within the higher region. The data are equally dispersed within the higher portion of the efficient zone, consistent with the funnel plot connected with this analysis (see Fig. 2 ), indicating that publication bias is unlikely in this situation.

figure 2

This funnel plot shows the result of publication bias of 79 effect quantities across 36 studies.

Heterogeneity test

To select the appropriate effect models for the meta-analysis, one might use the results of a heterogeneity test on the data effect sizes. In a meta-analysis, it is common practice to gauge the degree of data heterogeneity using the I 2 value, and I 2  ≥ 50% is typically understood to denote medium-high heterogeneity, which calls for the adoption of a random effect model; if not, a fixed effect model ought to be applied (Lipsey and Wilson, 2001 ). The findings of the heterogeneity test in this paper (see Table 2 ) revealed that I 2 was 86% and displayed significant heterogeneity ( P  < 0.01). To ensure accuracy and reliability, the overall effect size ought to be calculated utilizing the random effect model.

The analysis of the overall effect size

This meta-analysis utilized a random effect model to examine 79 effect quantities from 36 studies after eliminating heterogeneity. In accordance with Cohen’s criterion (Cohen, 1992 ), it is abundantly clear from the analysis results, which are shown in the forest plot of the overall effect (see Fig. 3 ), that the cumulative impact size of cooperative problem-solving is 0.82, which is statistically significant ( z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]), and can encourage learners to practice critical thinking.

figure 3

This forest plot shows the analysis result of the overall effect size across 36 studies.

In addition, this study examined two distinct dimensions of critical thinking to better understand the precise contributions that collaborative problem-solving makes to the growth of critical thinking. The findings (see Table 3 ) indicate that collaborative problem-solving improves cognitive skills (ES = 0.70) and attitudinal tendency (ES = 1.17), with significant intergroup differences (chi 2  = 7.95, P  < 0.01). Although collaborative problem-solving improves both dimensions of critical thinking, it is essential to point out that the improvements in students’ attitudinal tendency are much more pronounced and have a significant comprehensive effect (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]), whereas gains in learners’ cognitive skill are slightly improved and are just above average. (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

The analysis of moderator effect size

The whole forest plot’s 79 effect quantities underwent a two-tailed test, which revealed significant heterogeneity ( I 2  = 86%, z  = 12.78, P  < 0.01), indicating differences between various effect sizes that may have been influenced by moderating factors other than sampling error. Therefore, exploring possible moderating factors that might produce considerable heterogeneity was done using subgroup analysis, such as the learning stage, learning scaffold, teaching type, group size, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, in order to further explore the key factors that influence critical thinking. The findings (see Table 4 ) indicate that various moderating factors have advantageous effects on critical thinking. In this situation, the subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), learning scaffold (chi 2  = 9.03, P  < 0.01), and teaching type (chi 2  = 7.20, P  < 0.05) are all significant moderators that can be applied to support the cultivation of critical thinking. However, since the learning stage and the measuring tools did not significantly differ among intergroup (chi 2  = 3.15, P  = 0.21 > 0.05, and chi 2  = 0.08, P  = 0.78 > 0.05), we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving. These are the precise outcomes, as follows:

Various learning stages influenced critical thinking positively, without significant intergroup differences (chi 2  = 3.15, P  = 0.21 > 0.05). High school was first on the list of effect sizes (ES = 1.36, P  < 0.01), then higher education (ES = 0.78, P  < 0.01), and middle school (ES = 0.73, P  < 0.01). These results show that, despite the learning stage’s beneficial influence on cultivating learners’ critical thinking, we are unable to explain why it is essential for cultivating critical thinking in the context of collaborative problem-solving.

Different teaching types had varying degrees of positive impact on critical thinking, with significant intergroup differences (chi 2  = 7.20, P  < 0.05). The effect size was ranked as follows: mixed courses (ES = 1.34, P  < 0.01), integrated courses (ES = 0.81, P  < 0.01), and independent courses (ES = 0.27, P  < 0.01). These results indicate that the most effective approach to cultivate critical thinking utilizing collaborative problem solving is through the teaching type of mixed courses.

Various intervention durations significantly improved critical thinking, and there were significant intergroup differences (chi 2  = 12.18, P  < 0.01). The effect sizes related to this variable showed a tendency to increase with longer intervention durations. The improvement in critical thinking reached a significant level (ES = 0.85, P  < 0.01) after more than 12 weeks of training. These findings indicate that the intervention duration and critical thinking’s impact are positively correlated, with a longer intervention duration having a greater effect.

Different learning scaffolds influenced critical thinking positively, with significant intergroup differences (chi 2  = 9.03, P  < 0.01). The resource-supported learning scaffold (ES = 0.69, P  < 0.01) acquired a medium-to-higher level of impact, the technique-supported learning scaffold (ES = 0.63, P  < 0.01) also attained a medium-to-higher level of impact, and the teacher-supported learning scaffold (ES = 0.92, P  < 0.01) displayed a high level of significant impact. These results show that the learning scaffold with teacher support has the greatest impact on cultivating critical thinking.

Various group sizes influenced critical thinking positively, and the intergroup differences were statistically significant (chi 2  = 8.77, P  < 0.05). Critical thinking showed a general declining trend with increasing group size. The overall effect size of 2–3 people in this situation was the biggest (ES = 0.99, P  < 0.01), and when the group size was greater than 7 people, the improvement in critical thinking was at the lower-middle level (ES < 0.5, P  < 0.01). These results show that the impact on critical thinking is positively connected with group size, and as group size grows, so does the overall impact.

Various measuring tools influenced critical thinking positively, with significant intergroup differences (chi 2  = 0.08, P  = 0.78 > 0.05). In this situation, the self-adapting measurement tools obtained an upper-medium level of effect (ES = 0.78), whereas the complete effect size of the standardized measurement tools was the largest, achieving a significant level of effect (ES = 0.84, P  < 0.01). These results show that, despite the beneficial influence of the measuring tool on cultivating critical thinking, we are unable to explain why it is crucial in fostering the growth of critical thinking by utilizing the approach of collaborative problem-solving.

Different subject areas had a greater impact on critical thinking, and the intergroup differences were statistically significant (chi 2  = 13.36, P  < 0.05). Mathematics had the greatest overall impact, achieving a significant level of effect (ES = 1.68, P  < 0.01), followed by science (ES = 1.25, P  < 0.01) and medical science (ES = 0.87, P  < 0.01), both of which also achieved a significant level of effect. Programming technology was the least effective (ES = 0.39, P  < 0.01), only having a medium-low degree of effect compared to education (ES = 0.72, P  < 0.01) and other fields (such as language, art, and social sciences) (ES = 0.58, P  < 0.01). These results suggest that scientific fields (e.g., mathematics, science) may be the most effective subject areas for cultivating critical thinking utilizing the approach of collaborative problem-solving.

The effectiveness of collaborative problem solving with regard to teaching critical thinking

According to this meta-analysis, using collaborative problem-solving as an intervention strategy in critical thinking teaching has a considerable amount of impact on cultivating learners’ critical thinking as a whole and has a favorable promotional effect on the two dimensions of critical thinking. According to certain studies, collaborative problem solving, the most frequently used critical thinking teaching strategy in curriculum instruction can considerably enhance students’ critical thinking (e.g., Liang et al., 2017 ; Liu et al., 2020 ; Cindy, 2004 ). This meta-analysis provides convergent data support for the above research views. Thus, the findings of this meta-analysis not only effectively address the first research query regarding the overall effect of cultivating critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills) utilizing the approach of collaborative problem-solving, but also enhance our confidence in cultivating critical thinking by using collaborative problem-solving intervention approach in the context of classroom teaching.

Furthermore, the associated improvements in attitudinal tendency are much stronger, but the corresponding improvements in cognitive skill are only marginally better. According to certain studies, cognitive skill differs from the attitudinal tendency in classroom instruction; the cultivation and development of the former as a key ability is a process of gradual accumulation, while the latter as an attitude is affected by the context of the teaching situation (e.g., a novel and exciting teaching approach, challenging and rewarding tasks) (Halpern, 2001 ; Wei and Hong, 2022 ). Collaborative problem-solving as a teaching approach is exciting and interesting, as well as rewarding and challenging; because it takes the learners as the focus and examines problems with poor structure in real situations, and it can inspire students to fully realize their potential for problem-solving, which will significantly improve their attitudinal tendency toward solving problems (Liu et al., 2020 ). Similar to how collaborative problem-solving influences attitudinal tendency, attitudinal tendency impacts cognitive skill when attempting to solve a problem (Liu et al., 2020 ; Zhang et al., 2022 ), and stronger attitudinal tendencies are associated with improved learning achievement and cognitive ability in students (Sison, 2008 ; Zhang et al., 2022 ). It can be seen that the two specific dimensions of critical thinking as well as critical thinking as a whole are affected by collaborative problem-solving, and this study illuminates the nuanced links between cognitive skills and attitudinal tendencies with regard to these two dimensions of critical thinking. To fully develop students’ capacity for critical thinking, future empirical research should pay closer attention to cognitive skills.

The moderating effects of collaborative problem solving with regard to teaching critical thinking

In order to further explore the key factors that influence critical thinking, exploring possible moderating effects that might produce considerable heterogeneity was done using subgroup analysis. The findings show that the moderating factors, such as the teaching type, learning stage, group size, learning scaffold, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, could all support the cultivation of collaborative problem-solving in critical thinking. Among them, the effect size differences between the learning stage and measuring tool are not significant, which does not explain why these two factors are crucial in supporting the cultivation of critical thinking utilizing the approach of collaborative problem-solving.

In terms of the learning stage, various learning stages influenced critical thinking positively without significant intergroup differences, indicating that we are unable to explain why it is crucial in fostering the growth of critical thinking.

Although high education accounts for 70.89% of all empirical studies performed by researchers, high school may be the appropriate learning stage to foster students’ critical thinking by utilizing the approach of collaborative problem-solving since it has the largest overall effect size. This phenomenon may be related to student’s cognitive development, which needs to be further studied in follow-up research.

With regard to teaching type, mixed course teaching may be the best teaching method to cultivate students’ critical thinking. Relevant studies have shown that in the actual teaching process if students are trained in thinking methods alone, the methods they learn are isolated and divorced from subject knowledge, which is not conducive to their transfer of thinking methods; therefore, if students’ thinking is trained only in subject teaching without systematic method training, it is challenging to apply to real-world circumstances (Ruggiero, 2012 ; Hu and Liu, 2015 ). Teaching critical thinking as mixed course teaching in parallel to other subject teachings can achieve the best effect on learners’ critical thinking, and explicit critical thinking instruction is more effective than less explicit critical thinking instruction (Bensley and Spero, 2014 ).

In terms of the intervention duration, with longer intervention times, the overall effect size shows an upward tendency. Thus, the intervention duration and critical thinking’s impact are positively correlated. Critical thinking, as a key competency for students in the 21st century, is difficult to get a meaningful improvement in a brief intervention duration. Instead, it could be developed over a lengthy period of time through consistent teaching and the progressive accumulation of knowledge (Halpern, 2001 ; Hu and Liu, 2015 ). Therefore, future empirical studies ought to take these restrictions into account throughout a longer period of critical thinking instruction.

With regard to group size, a group size of 2–3 persons has the highest effect size, and the comprehensive effect size decreases with increasing group size in general. This outcome is in line with some research findings; as an example, a group composed of two to four members is most appropriate for collaborative learning (Schellens and Valcke, 2006 ). However, the meta-analysis results also indicate that once the group size exceeds 7 people, small groups cannot produce better interaction and performance than large groups. This may be because the learning scaffolds of technique support, resource support, and teacher support improve the frequency and effectiveness of interaction among group members, and a collaborative group with more members may increase the diversity of views, which is helpful to cultivate critical thinking utilizing the approach of collaborative problem-solving.

With regard to the learning scaffold, the three different kinds of learning scaffolds can all enhance critical thinking. Among them, the teacher-supported learning scaffold has the largest overall effect size, demonstrating the interdependence of effective learning scaffolds and collaborative problem-solving. This outcome is in line with some research findings; as an example, a successful strategy is to encourage learners to collaborate, come up with solutions, and develop critical thinking skills by using learning scaffolds (Reiser, 2004 ; Xu et al., 2022 ); learning scaffolds can lower task complexity and unpleasant feelings while also enticing students to engage in learning activities (Wood et al., 2006 ); learning scaffolds are designed to assist students in using learning approaches more successfully to adapt the collaborative problem-solving process, and the teacher-supported learning scaffolds have the greatest influence on critical thinking in this process because they are more targeted, informative, and timely (Xu et al., 2022 ).

With respect to the measuring tool, despite the fact that standardized measurement tools (such as the WGCTA, CCTT, and CCTST) have been acknowledged as trustworthy and effective by worldwide experts, only 54.43% of the research included in this meta-analysis adopted them for assessment, and the results indicated no intergroup differences. These results suggest that not all teaching circumstances are appropriate for measuring critical thinking using standardized measurement tools. “The measuring tools for measuring thinking ability have limits in assessing learners in educational situations and should be adapted appropriately to accurately assess the changes in learners’ critical thinking.”, according to Simpson and Courtney ( 2002 , p. 91). As a result, in order to more fully and precisely gauge how learners’ critical thinking has evolved, we must properly modify standardized measuring tools based on collaborative problem-solving learning contexts.

With regard to the subject area, the comprehensive effect size of science departments (e.g., mathematics, science, medical science) is larger than that of language arts and social sciences. Some recent international education reforms have noted that critical thinking is a basic part of scientific literacy. Students with scientific literacy can prove the rationality of their judgment according to accurate evidence and reasonable standards when they face challenges or poorly structured problems (Kyndt et al., 2013 ), which makes critical thinking crucial for developing scientific understanding and applying this understanding to practical problem solving for problems related to science, technology, and society (Yore et al., 2007 ).

Suggestions for critical thinking teaching

Other than those stated in the discussion above, the following suggestions are offered for critical thinking instruction utilizing the approach of collaborative problem-solving.

First, teachers should put a special emphasis on the two core elements, which are collaboration and problem-solving, to design real problems based on collaborative situations. This meta-analysis provides evidence to support the view that collaborative problem-solving has a strong synergistic effect on promoting students’ critical thinking. Asking questions about real situations and allowing learners to take part in critical discussions on real problems during class instruction are key ways to teach critical thinking rather than simply reading speculative articles without practice (Mulnix, 2012 ). Furthermore, the improvement of students’ critical thinking is realized through cognitive conflict with other learners in the problem situation (Yang et al., 2008 ). Consequently, it is essential for teachers to put a special emphasis on the two core elements, which are collaboration and problem-solving, and design real problems and encourage students to discuss, negotiate, and argue based on collaborative problem-solving situations.

Second, teachers should design and implement mixed courses to cultivate learners’ critical thinking, utilizing the approach of collaborative problem-solving. Critical thinking can be taught through curriculum instruction (Kuncel, 2011 ; Leng and Lu, 2020 ), with the goal of cultivating learners’ critical thinking for flexible transfer and application in real problem-solving situations. This meta-analysis shows that mixed course teaching has a highly substantial impact on the cultivation and promotion of learners’ critical thinking. Therefore, teachers should design and implement mixed course teaching with real collaborative problem-solving situations in combination with the knowledge content of specific disciplines in conventional teaching, teach methods and strategies of critical thinking based on poorly structured problems to help students master critical thinking, and provide practical activities in which students can interact with each other to develop knowledge construction and critical thinking utilizing the approach of collaborative problem-solving.

Third, teachers should be more trained in critical thinking, particularly preservice teachers, and they also should be conscious of the ways in which teachers’ support for learning scaffolds can promote critical thinking. The learning scaffold supported by teachers had the greatest impact on learners’ critical thinking, in addition to being more directive, targeted, and timely (Wood et al., 2006 ). Critical thinking can only be effectively taught when teachers recognize the significance of critical thinking for students’ growth and use the proper approaches while designing instructional activities (Forawi, 2016 ). Therefore, with the intention of enabling teachers to create learning scaffolds to cultivate learners’ critical thinking utilizing the approach of collaborative problem solving, it is essential to concentrate on the teacher-supported learning scaffolds and enhance the instruction for teaching critical thinking to teachers, especially preservice teachers.

Implications and limitations

There are certain limitations in this meta-analysis, but future research can correct them. First, the search languages were restricted to English and Chinese, so it is possible that pertinent studies that were written in other languages were overlooked, resulting in an inadequate number of articles for review. Second, these data provided by the included studies are partially missing, such as whether teachers were trained in the theory and practice of critical thinking, the average age and gender of learners, and the differences in critical thinking among learners of various ages and genders. Third, as is typical for review articles, more studies were released while this meta-analysis was being done; therefore, it had a time limit. With the development of relevant research, future studies focusing on these issues are highly relevant and needed.

Conclusions

The subject of the magnitude of collaborative problem-solving’s impact on fostering students’ critical thinking, which received scant attention from other studies, was successfully addressed by this study. The question of the effectiveness of collaborative problem-solving in promoting students’ critical thinking was addressed in this study, which addressed a topic that had gotten little attention in earlier research. The following conclusions can be made:

Regarding the results obtained, collaborative problem solving is an effective teaching approach to foster learners’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]). With respect to the dimensions of critical thinking, collaborative problem-solving can significantly and effectively improve students’ attitudinal tendency, and the comprehensive effect is significant (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

As demonstrated by both the results and the discussion, there are varying degrees of beneficial effects on students’ critical thinking from all seven moderating factors, which were found across 36 studies. In this context, the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have a positive impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. Since the learning stage (chi 2  = 3.15, P  = 0.21 > 0.05) and measuring tools (chi 2  = 0.08, P  = 0.78 > 0.05) did not demonstrate any significant intergroup differences, we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving.

Data availability

All data generated or analyzed during this study are included within the article and its supplementary information files, and the supplementary information files are available in the Dataverse repository: https://doi.org/10.7910/DVN/IPFJO6 .

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Acknowledgements

This research was supported by the graduate scientific research and innovation project of Xinjiang Uygur Autonomous Region named “Research on in-depth learning of high school information technology courses for the cultivation of computing thinking” (No. XJ2022G190) and the independent innovation fund project for doctoral students of the College of Educational Science of Xinjiang Normal University named “Research on project-based teaching of high school information technology courses from the perspective of discipline core literacy” (No. XJNUJKYA2003).

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Xu, E., Wang, W. & Wang, Q. The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature. Humanit Soc Sci Commun 10 , 16 (2023). https://doi.org/10.1057/s41599-023-01508-1

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problem solving in science education

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The acquisition of problem solving competence: evidence from 41 countries that math and science education matters

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On the basis of a ‘problem solving as an educational outcome’ point of view, we analyse the contribution of math and science competence to analytical problem-solving competence and link the acquisition of problem solving competence to the coherence between math and science education. We propose the concept of math-science coherence and explore whether society-, curriculum-, and school-related factors confound with its relation to problem solving.

By using the PISA 2003 data set of 41 countries, we apply multilevel regression and confounder analyses to investigate these effects for each country.

Our results show that (1) math and science competence significantly contribute to problem solving across countries; (2) math-science coherence is significantly related to problem solving competence; (3) country-specific characteristics confound this relation; (4) math-science coherence is linked to capability under-utilisation based on science performance but less on math performance.

Conclusions

In sum, low problem solving scores seem a result of an impeded transfer of subjectspecific knowledge and skills (i.e., under-utilisation of science capabilities in the acquisition of problem solving competence), which is characterised by low levels of math-science coherence.

The ability to solve real-world problems and to transfer problem-solving strategies from domain-specific to domain-general contexts and vice versa has been regarded an important competence students should develop during their education in school (Greiff et al. [ 2013 ]; van Merriënboer [ 2013 ]). In the context of large-scale assessments such as the PISA study problem solving competence is defined as the ability to solve cross-disciplinary and real-world problems by applying cognitive skills such as reasoning and logical thinking (Jonassen [ 2011 ]; OECD [ 2004 ]). Since this competence is regarded a desirable educational outcome, especially math and science educators have focused on developing students’ problem solving and reasoning competence in their respective domain-specific contexts (e.g., Kind [ 2013 ]; Kuo et al. [ 2013 ]; Wu and Adams [ 2006 ]). Accordingly, different conceptual frameworks were proposed that describe the cognitive processes of problem solving such as understanding the problem, building adequate representations of the problem, developing hypotheses, conducting experiments, and evaluating the solution (Jonassen [ 2011 ]; OECD [ 2005 ]). In comparing these approaches in math and science, it seems apparent that there is a conceptual overlap between the problem solving models in these two domains. This overlap triggers the question regarding its contribution to the development of students’ cross-curricular problem-solving competence (Abd-El-Khalick et al. [ 2004 ]; Bassok and Holyoak [ 1993 ]; Hiebert et al. [ 1996 ]).

The operationalization and scaling of performance in PISA assessments enables direct contrasting of scores in students’ competences in math and problem solving. Leutner et al. ([ 2012 ]) suggest that discrepancies between math and problem solving scores are indicative of the relative effectiveness of math education (OECD [ 2004 ]). In line with a “Capability-Utilisation Hypothesis”, it is assumed that math scores that negatively deviate from their problem solving counterpart signify an under-utilisation of students’ problem-solving capabilities as indicated by their scores in generic problem solving.

We intend to extend this view in two ways: First, by introducing the concept of math-science coherence we draw the focus on the potential synergistic link between math and science education and its contribution to the acquisition of problem solving competence. Second, the introduction of a Capability Under-Utilisation Index will enable us to extend the focus of the Capability-Utilisation Hypothesis to both, math and science education. The combination of the concept of math-science coherence with the notion of capability-utilisation will help to further explore the facilitating processes involved in the transition of subject-specific knowledge and skills to the acquisition of problem solving competence. These insights are expected to contribute to a better understanding of meaningful strategies to improve and optimize educational systems in different countries.

Theoretical framework

Problem solving as an educational goal.

In the PISA 2003 framework, problem solving is referred to as “an individual’s capacity to use cognitive processes to resolve real, cross-disciplinary situations where the solution path is not immediately obvious” (OECD [ 2004 ], p. 156). This definition is based on the assumption of domain-general skills and strategies that can be employed in various situations and contexts. These skills and strategies involve cognitive processes such as: Understanding and characterizing the problem, representing the problem, solving the problem, reflecting and communicating the problem solution (OECD [ 2003 ]). Problem solving is often regarded a process rather than an educational outcome, particularly in research on the assessment and instruction of problem solving (e.g., Greiff et al. [ 2013 ]; Jonassen [ 2011 ]). This understanding of the construct is based on the assumption that problem solvers need to perform an adaptive sequence of cognitive steps in order to solve a specific problem (Jonassen [ 2011 ]). Although problem solving has also been regarded as a process skill in large-scale assessments such as the PISA 2003 study, these assessments mainly focus on problem solving performance as an outcome that can be used for international comparisons (OECD [ 2004 ]). However, problem solving competence was operationalized as a construct comprised of cognitive processes. In the context of the PISA 2003 study, these processes were referred to as analytical problem solving, which was assessed by static tasks presented in paper-and-pencil format. Analytical problem-solving competence is related to school achievement and the development of higher-order thinking skills (e.g., Baumert et al. [ 2009 ]; OECD [ 2004 ]; Zohar [ 2013 ]). Accordingly, teachers and educators have focused on models of problem solving as guidelines for structuring inquiry-based processes in their subject lessons (Oser and Baeriswyl [ 2001 ]). Van Merriënboer ([ 2013 ]) pointed out that problem solving should not only be regarded a mere instructional method but also as a major educational goal. Recent curricular reforms have therefore shifted towards the development of problem solving abilities in school (Gallagher et al. [ 2012 ]; Koeppen et al. [ 2008 ]). These reforms were coupled with attempts to strengthen the development of transferable skills that can be applied in real-life contexts (Pellegrino and Hilton [ 2012 ]). For instance, in the context of 21 st century skills, researchers and policy makers have agreed on putting emphasis on fostering skills such as critical thinking, digital competence, and problem solving (e.g., Griffin et al. [ 2012 ]). In light of the growing importance of lifelong learning and the increased complexity of work- and real-life problem situations, these skills are now regarded as essential (Griffin et al. [ 2012 ]; OECD [ 2004 ]). Hence, large-scale educational studies such as PISA have shifted towards the assessment and evaluation of problem solving competence as a 21 st century skill.

The PISA frameworks of math and science competence

In large-scale assessments such as the PISA studies, students’ achievement in the domains of science and mathematics play an important role. Moreover, scientific and mathematical literacy are now regarded essential to being a reflective citizen (Bybee [ 2004 ]; OECD [ 2003 ]). Generally, Baumert et al. ([ 2009 ]) have shown that students’ math and science achievements are highly related to domain-general ability constructs such as reasoning or intelligence. In this context, student achievement refers to “the result of domain-specific processes of knowledge acquisition and information processing” (cf. Baumert et al. [ 2009 ], p. 169). This line of argument is reflected in definitions and frameworks of scientific and mathematical literacy, which are conceptualized as domain-specific competences that are hierarchically organized and build upon abilities closely related to problem solving (Brunner et al. [ 2013 ]).

Scientific literacy has been defined within a multidimensional framework, differentiating between three main cognitive processes, namely describing, explaining, and predicting scientific phenomena, understanding scientific investigations, and interpreting scientific evidence and conclusions (OECD [ 2003 ]). In addition, various types of knowledge such as ‘knowledge about the nature of science’ are considered as factors influencing students’ achievements in this domain (Kind [ 2013 ]). We conclude that the concept of scientific literacy encompasses domain-general problem-solving processes, elements of scientific inquiry (Abd-El-Khalick et al. [ 2004 ]; Nentwig et al. [ 2009 ]), and domain-specific knowledge.

The definition of mathematical literacy refers to students’ competence to utilise mathematical modelling and mathematics in problem-solving situations (OECD [ 2003 ]). Here, we can also identify overlaps between cognitive processes involved in mathematical problem solving and problem solving in general: Structuring, mathematizing, processing, interpreting, and validating (Baumert et al. [ 2009 ]; Hiebert et al. [ 1996 ]; Kuo et al. [ 2013 ]; Polya [ 1945 ]). In short, mathematical literacy goes beyond computational skills (Hickendorff [ 2013 ]; Wu and Adams [ 2006 ]) and is conceptually linked to problem solving.

In the PISA 2003 framework, the three constructs of math, science, and problem solving competence overlap conceptually. For instance, solving the math items requires reasoning, which comprises analytical skills and information processing. Given the different dimensions of the scientific literacy framework, the abilities involved in solving the science items are also related to problem solving, since they refer to the application of knowledge and the performance of inquiry processes (OECD [ 2003 ]). This conceptual overlap is empirically supported by high correlations between math and problem solving ( r  = .89) and between science and problem solving ( r  = .80) obtained for the sample of 41 countries involved in PISA 2003 (OECD [ 2004 ]). The relation between math and science competence was also high ( r  = .83). On the one hand, the sizes of the inter-relationships, give rise to the question regarding the uniqueness of each of the competence measures. On the other hand, the high correlations indicate that problem-solving skills are relevant in math and science (Martin et al. [ 2012 ]). Although Baumert et al. ([ 2009 ]) suggest that the domain-specific competences in math and science require skills beyond problem solving (e.g., the application of domain-specific knowledge) we argue from an assessment perspective that the PISA 2003 tests in math, science, and problem solving measure predominantly basic academic skills relatively independent from academic knowledge (see also Bulle [ 2011 ]).

The concept of capability-utilisation

Discrepancies between students’ performance in math/science and problem solving were studied at country level (OECD [ 2004 ]) and were, for example for math and problem solving scores, interpreted in two ways: (1) If students’ perform better in math than in problem solving, they would “have a better grasp of mathematics content […] after accounting for the level of generic problem-solving skills…” (OECD [ 2004 ], p. 55); (2) If students’ estimated problem-solving competence is higher than their estimated math competence, “… this may suggest that students have the potential to achieve better results in mathematics than that reflected in their current performance…” (OECD [ 2004 ], p. 55). Whilst the latter discrepancy constitutes a capability under-utilisation in math, the former suggests challenges in utilising knowledge and skills acquired in domain-specific contexts in domain-unspecific contexts (i.e., transfer problem).

To quantify the degree to which students are able to transfer their problem solving capabilities from domain-specific problems in math or science to cross-curricular problems, we introduce the Capability Under-Utilisation Index (CUUI) as the relative difference between math or science and problem solving scores:

A positive CUUI indicates that the subject-specific education (i.e., math or science) in a country tends to under-utilise its students’ capabilities to problem solve. A negative CUUI indicates that a country’s educational system fails to fully utilise its students’ capabilities to acquire math and science literacy in the development of problem solving. The CUUI reflects the relative discrepancies between the achievement scores in different domains a .

The concept of math-science coherence

In light of the conceptual and empirical discussion on the relationship between math, science, and problem solving competence, we introduce the concept of math-science coherence as follows: First, math-science coherence refers to the set of cognitive processes involved in both subjects and thus represents processes which are related to reasoning and information processing, relatively independent from domain-specific knowledge. Second, math-science coherence reflects the degree to which math and science education is harmonized as a feature of the educational environment in a country. This interpretation is based on the premise that PISA measures students’ competence as educational outcomes (OECD [ 2004 ]). The operationalization of math-science coherence is realized by means of the correlation between math and science scores [ r (M,S)] at the country level. Low math-science coherence indicates that students who are successful in the acquisition of knowledge and skills in math are not necessarily successful in the acquisition of knowledge and skills in science and vice versa.

On the basis of this conceptualization of math-science coherence, we expect a significant and positive relation to problem solving scores, since the conceptual overlap between mathematical and scientific literacy refers to cognitive abilities such as reasoning and information processing that are also required in problem solving (Arts et al. [ 2006 ]; Beckmann [ 2001 ]; Wüstenberg et al. [ 2012 ]). Hence, we assert that math-science coherence facilitates the transfer of knowledge, skills, and insights across subjects resulting in better problem solving performance (OECD [ 2004 ]; Pellegrino and Hilton [ 2012 ]).

We also assume that math-science coherence as well as capability utilisation is linked to characteristics of the educational system of a country. For instance, as Janssen and Geiser ([ 2012 ]) and Blömeke et al. ([ 2011 ]) suggested, the developmental status of a country, measured by the Human Development Index (HDI; UNDP [ 2005 ]), is positively related to students’ academic achievements as well as to teachers’ quality of teaching. Furthermore, the socio-economic status of a country co-determines characteristics of its educational system, which ultimately affects a construct referred to as national intelligence (Lynn and Meisenberg [ 2010 ]). Research also indicated that curricular settings and educational objectives are related to school achievement in general (Bulle [ 2011 ]; Martin et al. [ 2004 ]). Besides these factors, school- and classroom-related characteristics might also confound the relation between math-science coherence and problem solving. For instance, the schools’ autonomy in developing curricula and managing educational resources might facilitate the incorporation of inquiry- and problem-based activities in science lessons (Chiu and Chow [ 2011 ]). These factors have been discussed as being influential to students’ competence development (OECD [ 2004 ], [ 2005 ]). Ewell ([ 2012 ]) implies that cross-national differences in problem solving competence might be related to differences in education and in using appropriate teaching material. These factors potentially confound the relation between math-science coherence and problem solving.

Discrepancies between math and problem solving scores are discussed in relation to quality of education. Although research has found that crossing the borders between STEM subjects positively affects students’ STEM competences (e.g., National Research Council NRC [ 2011 ]), we argue that the PISA analyses have fallen short in explaining cross-country differences in the development of problem solving competence, since they ignored the link between math and science competences and the synergistic effect of learning universally applicable problem-solving skills in diverse subject areas. Hence, we use the concept of math-science coherence to provide a more detailed description of the discrepancies between problem solving and domain-specific competences. In this regard, we argue that the coherence concept indicates the synergistic potential and students’ problem-solving competence the success of transfer.

The present study

The current study is based on the premise that in contrast to math and science competence problem solving competence is not explicitly taught as a subject at school. Problem solving competence, however, is an expected outcome of education (van Merriënboer [ 2013 ]). With the first step in our analyses, we seek to establish whether math and science education are in fact main contributors to the acquisition of problem solving competence. On the basis of this regression hypothesis, we subsequently focus on the question whether there are significant and systematic differences between countries ( Moderation-Hypothesis ). In light of the conceptual overlap due to cognitive processes involved in dealing with math, science and problem solving tasks and the shared item format employed in the assessments, we expect math and science competence scores to substantially predict scores in problem solving competence. Furthermore, since math and science education are differently organized across the 41 countries participating in the PISA 2003 study, differences in the contribution are also expected.

On the basis of these premises, we introduce the concept of math-science coherence, operationalised as the correlation between math and science scores [ r (M,S)], and analyse its relationship to problem solving and the effects of confounders (i.e., country characteristics) as a step of validation. Since math, science, and problem solving competence show a conceptual overlap, we expect problem solving and math-science coherence to be positively related. Countries’ educational systems differ in numerous aspects, their educational structure, and their educational objectives. Countries also differ with regard to the frequency of assessments, the autonomy of schools in setting up curricula and resources, and the educational resources available. Consequently, we expect the relation between math-science coherence and problem solving competence to be confounded by society-, curriculum-, and school-related factors ( Confounding-Hypothesis ).

In a final step, we aim to better understand the mechanisms with which math and science education contributes to the acquisition of problem-solving competence by exploring how math-science coherence, capability utilisation, and problem solving competence are related. We thus provide new insights into factors related to the transfer between students’ domain-specific and cross-curricular knowledge and skills ( Capability-Utilisation Hypothesis ).

In PISA 2003, a total sample of N  = 276,165 students (49.4% female) from 41 countries participated. The entire sample was randomly selected by applying a two-step sampling procedure: First, schools were chosen within a country. Second, students were chosen within these schools. This procedure consequently led to a clustered structure of the data set, as students were nested in 10,175 schools. On average, 27 students per school were chosen across schools within countries. Students’ mean age was 15.80 years ( SD  = 0.29 years) ranging from 15.17 to 16.25 years.

In the PISA 2003 study, different assessments were used in order to measure students’ competence in math, science, and problem solving. These assessments were administered as paper-and-pencil tests within a multi-matrix design (OECD [ 2005 ]). In this section, the assessments and further constructs are described that served as predictors of the contribution of math and science competence to problem solving at the country level.

Student achievement in math, science, and problem solving

In order to assess students’ competence to solve cross-curricular problems (i.e., analytical problem solving requiring information retrieval and reasoning), students had to work on an analytical problem-solving test. This test comprised a total of 19 items (7 items referred to trouble-shooting, 7 items referred to decision-making, and 5 items referred to system analysis and design; see OECD [ 2004 ]). Items were coded according to the PISA coding scheme, resulting in dichotomous and polytomous scores (OECD [ 2005 ]). Based on these scores, models of item response theory were specified in order to obtain person and item parameters (Leutner et al. [ 2012 ]). The resulting plausible values could be regarded as valid indicators of students’ abilities in problem solving (Wu [ 2005 ]). The problem solving test showed sufficient reliabilities between .71 and .89 for the 41 countries.

To assess mathematical literacy as an indicator of math competence , an 85-items test was administered (for details, refer to OECD [ 2003 ]). Responses were dichotomously or polytomously scored. Again, plausible values were obtained as person ability estimates and reliabilities were good (range: 0.83 – 0.93). In the context of mathematical literacy, students were asked to solve real-world problems by applying appropriate mathematical models. They were prompted to “identify and understand the role mathematics plays in the world, to make well-founded judgements and to use […] mathematics […]” (OECD [ 2003 ], p. 24).

Scientific literacy as a proxy for science competence was assessed by using problems referring to different content areas of science in life, health, and technology. The reliability estimates for the 35 items in this test ranged between .68 and .88. Again, plausible values served as indicators of this competence.

Country-specific characteristics

In our analyses, we incorporated a range of country-specific characteristics that can be subdivided into three main categories. These are: society-related factors, curriculum-related factors, and school-related factors. Country-specific estimates of National Intelligence as derived by Lynn and Meisenberg ([ 2010 ]) as well as the Human Development Index (HDI) were subsumed under society-related factors . The HDI incorporates indicators of a country’s health, education, and living standards (UNDP [ 2005 ]). Both variables are conceptualised as factors that contribute to country-specific differences in academic performance.

Holliday and Holliday ([ 2003 ]) emphasised the role of curricular differences in the understanding of between-country variance in test scores. We incorporated two curriculum-related factors in our analyses. First, we used Bulle’s ([ 2011 ]) classification of curricula into ‘progressive’ and ‘academic’. Bulle ([ 2011 ]) proposed this framework and classified the PISA 2003 countries according to their educational model. In her framework, she distinguishes between ‘academic models’ which are primarily geared towards teaching academic subjects (e.g., Latin, Germanic, and East-Asian countries) and ‘progressive models’ which focus on teaching students’ general competence in diverse contexts (e.g., Anglo-Saxon and Northern countries). In this regard, academic skills refer to the abilities of solving academic-type problems, whereas so called progressive skills are needed in solving real-life problems (Bulle [ 2011 ]). It can be assumed that educational systems that focus on fostering real-life and domain-general competence might be more conducive to successfully tackling the kind of problem solving tasks used in PISA (Kind [ 2013 ]). This classification of educational systems should be seen as the two extreme poles of a continuum rather than as a strict dichotomy. In line with the reflections above, we would argue that academic and progressive skills are not exclusively distinct, since both skills utilise sets of cognitive processes that largely overlap (Klahr and Dunbar [ 1988 ]). The fact that curricular objectives in some countries are shifting (e.g., in Eastern Asia) makes a clear distinction between both models even more difficult. Nonetheless, we will use this form of country-specific categorization based on Bulle’s model in our analyses.

Second, we considered whether countries’ science curricula were ‘integrated’ or ‘not integrated’ (Martin et al. [ 2004 ]). In this context, integration refers to linking multiple science subjects (biology, chemistry, earth science, and physics) to a unifying theme or issue (cf. Drake and Reid [ 2010 ], p. 1).

In terms of school-related factors, we used the PISA 2003 scales of ‘Frequency of assessments in schools’, ‘Schools’ educational resources’, and ‘School autonomy towards resources and curricula’ from the school questionnaire. Based on frequency and rating scales, weighted maximum likelihood estimates (WLE) indicated the degree to which schools performed in these scales (OECD [ 2005 ]).

The country-specific characteristics are summarized in the Table 1 .

The PISA 2003 assessments utilised a randomized incomplete block design to select different test booklets which covered the different content areas of math, science, and problem solving (Brunner et al. [ 2013 ]; OECD [ 2005 ]). The test administration took 120 minutes, and was managed for each participating country separately. It was established that quality standards of the assessment procedure were high.

Statistical analyses

In PISA 2003, different methods of obtaining person estimates with precise standard errors were applied. The most accurate procedure produced five plausible values, which were drawn from a person ability distribution (OECD [ 2005 ]). To avoid missing values in these parameters and to obtain accurate estimates, further background variables were used within the algorithms (Wu [ 2005 ]). The resulting plausible values were subsequently used as indicators of students’ competence in math, science, and problem solving. By applying Rubin’s combination rules (Bouhilila and Sellaouti [ 2013 ]; Enders [ 2010 ]), analyses were replicated with each of the five plausible values and then combined. In this multiple imputation procedure, standard errors were decomposed to the variability across and within the five imputations (Enders [ 2010 ]; OECD [ 2005 ]; Wu [ 2005 ]).

Within the multilevel regression analyses for each country, we specified the student level as level 1 and the school level as level 2. Since PISA 2003 applied a random sampling procedure at the student and the school level, we decided to control for the clustering of data at these two levels (OECD [ 2005 ]). In addition to this two-level procedure, we regarded the 41 countries as multiple groups (fixed effects). This decision was based on our assumption that the countries selected in PISA 2003 did not necessarily represent a sample of a larger population (Martin et al. [ 2012 ]). Moreover, we did not regard the effects of countries as interchangeable, because, given the specific characteristics of education and instruction within countries; we argue that the effects of competences in mathematics and science on problem solving have their own distinct interpretation in each country (Snijders and Bosker [ 2012 ]). The resulting models were compared by taking into account the Akaike’s information criteria ( AIC ), Bayesian information criteria ( BIC ), and the sample-size adjusted BIC . Also, a likelihood ratio test of the log-Likelihood values ( LogL ) was applied (Hox [ 2010 ]).

To test the Moderation-Hypothesis, we first specified a two-level regression model with problem solving scores as outcomes at the student level (level 1), which allowed variance in achievement scores across schools (level 2). In this model, math and science scores predicted problem solving scores at the student level. To account for differences in the probabilities of being selected as a student within the 41 countries and to adjust the standard errors of regression parameters, we used the robust maximum likelihood (MLR) estimator and students’ final weights (see also Brunner et al. [ 2013 ]; OECD [ 2005 ]). All analyses were conducted in Mplus 6.0 by using the TYPE = IMPUTATION option (Muthén and Muthén [ 2010 ]). As Hox ([ 2010 ]) suggested, using multilevel regression models without taking into account the clustering of data in schools often leads to biased estimates, since achievement variables often have substantial variance at the school level. Consequently, we allowed for level-2-variance within the scores.

After having established whether success in math and science education contributes to the development in problem solving competence across the 41 countries, we then tested whether cross-country differences in the unstandardized regression coefficients were statistically significant by using a multi-group regression model, in which the coefficients were constrained to be equal across countries. We compared this model with the freely estimated model.

Finally, we conceptualized the correlation between math and science scores as an indicator of the level of coherence in math and science education in a country. In relation to the Confounding-Hypothesis, we tested country-specific characteristics for their potentially confounding effects on the relation between math-science coherence and problem solving competence. Following the recommendations proposed by (MacKinnon et al. [ 2000 ]), the confounding analysis was conducted in two steps: (1) we estimated two regression equations. In the first equation, problem solving scores across the 41 countries were regressed on math-science coherence. In the second equation, the respective country characteristics were added as further predictors; (2) the difference between the regression coefficients for math-science coherence obtained in either equation represented the magnitude of a potential confounder effect.

Lastly, we tested the Capability-Utilisation Hypothesis by investigating the bivariate correlations among the CUU Indices and math-science coherence.

Regressing problem solving on math and science performance

To test the Moderation-Hypothesis, we specified regression models with students’ problem-solving score as the outcome and math and science scores as predictors for each of the 41 countries. Due to the clustering of data in schools, these models allowed for between-level variance. Intraclass correlations (ICC-1) for math, science, and problem solving performance ranged between .03 and .61 for the school level ( M  = .33, SD  = .16).

We specified multilevel regression models for each country separately. These results are reported in Table  2 . The regression coefficients for math on problem solving ranged from .53 to .82 with an average of M( β Math )  = .67 ( SD  = .06). The average contribution of science towards problem solving was M( β Science )  = .16 ( SD  = .09, Min  = -.06, Max  = .30). The combination of the distributions of both parameters resulted in substantial differences in the variance explanations of the problem solving scores across the 41 countries ( M[R 2 ]  = .65, SD  = .15, Min  = .27, Max  = .86). To test whether these differences were statistically significant, we constrained the regression coefficients of math and science competence within the multi-group regression model to be equal across the 41 countries. Compared to the freely estimated model ( LogL  = -4,561,273.3, df  = 492, AIC  = 9,123,530.5, BIC  = 9,128,410.7), the restricted model was empirically not preferred LogL  = -4,564,877.9, df  = 412, AIC  = 9,130,579.8, BIC  = 9,134,917.6; Δχ 2 [80] = 7,209.2, p  < .001. These findings lend evidence for the Moderation-Hypothesis.

From a slightly different perspective, the country-specific amount of variance in problem solving scores that is explained by the variation in math and science performance scores ( R 2 ) is strongly associated with the country’s problem solving score ( r  = .77, p  < .001), which suggests that the contribution of science and math competence to the acquisition of problem solving competence was significantly lower in low-performing countries.

As shown in Table  2 , the regression weights of math and science were significant for all but two countries. Across countries the regression weight for math tended to be higher than the regression weight for science when predicting problem solving competence. This finding indicates a stronger overlap between students’ competences in mathematics and problem solving on the one hand and similarities between the assessments in both domains on the other hand.

Validating the concept of math-science coherence

In order to validate the concept of math-science coherence, which is operationalised as the correlation between math and science scores [ r (M,S)], we explored its relation to problem solving and country characteristics.

Regarding the regression outcomes shown in Table  2 , it is apparent that math-science coherence varied considerably across countries, ranging from .39 to .88 with an average of M(r)  = .70 ( SD  = .13). Interestingly, countries’ level of coherence in math-science education was substantially related to their problem solving scores ( r  = .76, p  < .001). An inspection of Figure  1 reveals that this effect was mainly due to countries that both achieve low problem solving scores and show relatively low levels of math-science coherence (see bottom left quadrant in Figure  1 ), whilst amongst the remaining countries the correlational link between math-science coherence and problem solving score was almost zero ( r  = -.08, p  = .71) b . This pattern extends the moderation perspective on the presumed dependency of problem solving competence from math and science competences.

figure 1

The relation between math-science coherence and problem solving performance across the 41 countries.

As a result of the moderator analysis, we know that countries not only differ in regard to their average problem-solving scores and level of coherence between math and science, countries also differ in the strengths with which math-science coherence predicts problem solving scores. To better understand the conceptual nature of the link between math-science coherence and problem solving, we now attempt to adjust this relationship for potential confounding effects that country-specific characteristics might have. To this end, we employed linear regression and path analysis with students’ problem-solving scores as outcomes, math-science coherence (i.e., r [M,S]) as predictor, and country characteristics as potential confounders.

To establish whether any of the country characteristics had a confounding effect on the link between math-science coherence and problem solving competence, two criteria had to be met: (1) a reduction of the direct effect of math-science coherence on problem solving scores, and (2) testing the difference between the direct effect within the baseline Model M0 and the effect with the confounding Model M1 (Table  3 ).

Regarding the society-related factors, both the countries’ HDI and their national intelligence were confounders with a positive effect. Furthermore, the countries’ integration of the science curriculum was also positively related to the problem solving performance. Finally, the degree of schools’ autonomy towards educational resources and the implementation of curricula and the frequency of assessments were school-related confounders, the former with a positive effect whilst the latter represents a negative confounder. The direct effect of math-science coherence to problem solving decreased and thus indicated that confounding was present (MacKinnon et al. [ 2000 ]).

These findings provide evidence on the Confounding-Hypothesis and support our expectations on the relation between math-science coherence, problem solving, and country characteristics. We regard these results as evidence for the validity of the math-science coherence measure.

Relating math-science coherence to the capability under-utilisation indices

To advance our understanding of the link between math-science coherence and problem solving scores, we tested the Capability-Utilisation Hypothesis. To this end, we explored the relationship between math-science coherence and the CUU Indices for math and science, respectively. For math competence the average Capability Under-Utilisation Index was rather neutral with M CUUI-Math  = -0.001 ( SD  = 0.02). This suggests that, on average, all countries sufficiently utilise their students’ math capabilities in facilitating the development of problem solving competence (i.e., transfer). It also suggests that math education across participating countries tends to sufficiently utilise generic problem-solving skills (Figure  2 ). The picture is different for science education. Here, the Capability Under-Utilisation Indices and their variation across the participating countries ( M CUUI-Science  = -0.01, SD  = 0.04) suggest that in a range of countries knowledge and skills taught in science education tend to be under-utilised in the facilitation of the acquisition of problem solving competence (Figure  3 ).

figure 2

The relation between math-science coherence and the capability under-utilisation index for math and problem solving scores across the 41 countries.

figure 3

The relation between math-science coherence and the capability under-utilisation index for science and problem solving scores across the 41 countries.

For math competence, the relative difference to problem solving was not related to math-science coherence ( r  = .02, p  = .89; Figure  2 ). In contrast, the Capability Under-Utilisation Index for science showed a strong positive correlation with math-science coherence ( r  = .76, p  < .001; Figure  3 ), indicating that low levels of coherence between math and science education were associated with a less effective transfer of domain-specific knowledge and skills to problem solving.

The present study was aimed at investigating the differences in the contribution of math and science competence to problem solving competence across the 41 countries that participated in the PISA 2003 study (Moderation-Hypothesis). To this end, we proposed the concept of math-science coherence and explored its relationship to problem solving competence and how this relationship is confounded by country characteristics (Confounding-Hypothesis). To further extend our understanding of the link between math-science coherence and problem solving, we introduced the concept of capability-utilisation. Testing the Capability-Utilisation Hypothesis enabled us to identify what may contribute to varying levels of math-science coherence and ultimately the development of problem solving competence.

The contribution of math and science competence across countries

Regarding the prediction of problem solving competence, we found that in most countries, math and science competence significantly contributed to students’ performance in analytical problem solving. This finding was expected based on the conceptualizations of mathematical and scientific literacy within the PISA framework referring to shared cognitive processes such as information processing and reasoning (Kind [ 2013 ]; OECD [ 2005 ]), which are regarded as components of problem solving (Bybee [ 2004 ]; Klahr and Dunbar [ 1988 ]; Mayer [ 2010 ]).

It is noteworthy that, for some of the below-average performing countries, science competence did not significantly contribute to the prediction of problem solving competence. It can be speculated that education in these countries is more geared towards math education and modelling processes in mathematical scenarios, whilst the aspect of problem solving in science is less emphasised (Janssen and Geiser [ 2012 ]). The results of multilevel regression analyses supported this interpretation by showing that math competence was a stronger predictor of problem solving competence. On the one hand, this finding could be due to the design of the PISA tests (Adams [ 2005 ]), since math and problem solving items are designed in such a way that modelling real-life problems is required, whereas science items are mostly domain-specific and linked to science knowledge (Nentwig et al. [ 2009 ]; OECD [ 2004 ]). Moreover, one may argue that math and problem solving items allow students to employ different solution strategies, whereas science items offer fewer degrees of freedom for test takers (Nentwig et al. [ 2009 ]). In particular, the shared format of items in math, science, and problem solving may explain an overlap between their cognitive demands. For instance, most of the items were designed in such a way that students had to extract and identify relevant information from given tables or figures in order to solve specific problems. Hence, these items were static and did not require knowledge generation by interaction or exploration but rather the use of given information in problem situations (Wood et al. [ 2009 ]). In contrast to the domain-specific items in math and science, problem solving items did not require the use of prior knowledge in math and science (OECD [ 2004 ]). In addition, some of the math and science items involved cognitive operations that were specific to these domains. For instance, students had to solve a number of math items by applying arithmetic and combinatorial operations (OECD [ 2005 ]). Finally, since items referred to contextual stimuli, which were presented in textual formats, reading ability can be regarded as another, shared demand of solving the items. Furthermore, Rindermann ([ 2007 ]) clearly showed that the shared demands of the achievement tests in large-scale assessments such as PISA were strongly related to students’ general reasoning skills. This finding is in line with the strong relations between math, science, and problem solving competence, found in our study. The interpretation of the overlap between the three competences can also be interpreted from a conceptual point of view. In light of the competence frameworks in PISA, we argue that there are a number of skills that can be found in math, science, and problem solving: information retrieval and processing, knowledge application, and evaluation of results (Griffin et al. [ 2012 ]; OECD [ 2004 ], [ 2005 ]). These skills point out to the importance of reasoning in the three domains (Rindermann [ 2007 ]). Thus, the empirical overlap between math and problem solving can be explained by shared processes of, what Mayer ([ 2010 ]) refers to as, informal reasoning. On the other hand, the stronger effect of math competence could be an effect of the quality of math education. Hiebert et al. ([ 1996 ]) and Kuo et al. ([ 2013 ]) suggested that math education is more based on problem solving skills than other subjects in school (e.g., Polya [ 1945 ]). Science lessons, in contrast, are often not necessarily problem-based, despite the fact that they often start with a set problem. Risch ([ 2010 ]) showed in a cross-national review that science learning was more related to contents and contexts rather than to generic problem-solving skills. These tendencies might lead to a weaker contribution of science education to the development of problem solving competence (Abd-El-Khalick et al. [ 2004 ]).

In sum, we found support on the Moderation-Hypothesis, which assumed systematic differences in the contribution of math and science competence to problem solving competence across the 41 PISA 2003 countries.

The relation to problem solving

In our study, we introduced the concept of math-science coherence, which reflects the degree to which math and science education are harmonized. Since mathematical and scientific literacy show a conceptual overlap, which refers to a set of cognitive processes that are linked to reasoning and information processing (Fensham and Bellocchi [ 2013 ]; Mayer [ 2010 ]), a significant relation between math-science coherence and problem solving was expected. In our analyses, we found a significant and positive effect of math-science coherence on performance scores in problem solving. In this finding we see evidence for the validity of this newly introduced concept of math-science coherence and its focus on the synergistic effect of math and science education on problem solving. The results further suggest that higher levels of coordination between math and science education has beneficial effects on the development of cross-curricular problem-solving competence (as measured within the PISA framework).

Confounding effects of country characteristics

As another step of validating the concept of math-science coherence, we investigated whether country-specific characteristics that are linked to society-, curriculum-, and school-related factors confounded its relation to problem solving. Our results showed that national intelligence, the Human Development Index, the integration of the science curriculum, and schools’ autonomy were positively linked to math-science coherence and problem solving, whilst a schools’ frequency of assessment had a negative confounding effect.

The findings regarding the positive confounders are in line with and also extend a number of studies on cross-country differences in education (e.g., Blömeke et al. [ 2011 ]; Dronkers et al. [ 2014 ]; Janssen and Geiser [ 2012 ]; Risch [ 2010 ]). Ross and Hogaboam-Gray ([ 1998 ]), for instance, found that students benefit from an integrated curriculum, particularly in terms of motivation and the development of their abilities. In the context of our confounder analysis, the integration of the science curriculum as well as the autonomy to allocate resources is expected to positively affect math-science coherence. At the same time, an integrated science curriculum with a coordinated allocation of resources may promote inquiry-based experiments in science courses, which is assumed to be beneficial for the development of problem solving within and across domains. Teaching science as an integrated subject is often regarded a challenge for teachers, particularly when developing conceptual structures in science lessons (Lang and Olson, [ 2000 ]), leading to teaching practices in which cross-curricular competence is rarely taken into account (Mansour [ 2013 ]; van Merriënboer [ 2013 ]).

The negative confounding effect of assessment frequency suggests that high frequencies of assessment, as it presumably applies to both math and science subjects, contribute positively to math-science coherence. However, the intended or unintended engagement in educational activities associated with assessment preparation tends not to be conducive to effectively developing domain-general problem solving competence (see also Neumann et al. [ 2012 ]).

The positive confounder effect of HDI is not surprising as HDI reflects a country’s capability to distribute resources and to enable certain levels of autonomy (Reich et al. [ 2013 ]). To find national intelligence as a positive confounder is also to be expected as the basis for its estimation are often students’ educational outcome measures (e.g., Rindermann [ 2008 ]) and, as discussed earlier, academic achievement measures share the involvement of a set of cognitive processes (Baumert et al. [ 2009 ]; OECD [ 2004 ]).

In summary, the synergistic effect of a coherent math and science education on the development of problem solving competence is substantially linked to characteristics of a country’s educational system with respect to curricula and school organization in the context of its socio-economic capabilities. Math-science coherence, however, also is linked to the extent to which math or science education is able to utilise students’ educational capabilities.

Math-science coherence and capability-utilisation

So far, discrepancies between students’ performance in math and problem solving or science and problem solving have been discussed as indicators of students’ capability utilisation in math or science (Leutner et al. [ 2012 ]; OECD [ 2004 ]). We have extended this perspective by introducing Capability Under-Utilisation Indices for math and science to investigate the effectiveness with which knowledge and skills acquired in the context of math or science education are transferred into cross-curricular problem-solving competence. The Capability Under-Utilisation Indices for math and science reflect a potential quantitative imbalance between math, science, and problem solving performance within a country, whilst the also introduced concept of math-science coherence reflects a potential qualitative imbalance between math and science education.

The results of our analyses suggest that an under-utilisation of problem solving capabilities in the acquisition of science literacy is linked to lower levels of math-science coherence, which ultimately leads to lower scores in problem solving competence. This interpretation finds resonance in Ross and Hogaboam-Gray’s ([ 1998 ]) argumentation for integrating math and science education and supports the attempts of math and science educators to incorporate higher-order thinking skills in teaching STEM subjects (e.g., Gallagher et al. [ 2012 ]; Zohar [ 2013 ]).

In contrast, the CUU Index for math was not related to math-science coherence in our analyses. This might be due to the conceptualizations and assessments of mathematical literacy and problem solving competence. Both constructs share cognitive processes of reasoning and information processing, resulting in quite similar items. Consequently, the transfer from math-related knowledge and skills to cross-curricular problems does not necessarily depend on how math and science education are harmonised, since the conceptual and operational discrepancy between math and problem solving is rather small.

Math and science education do matter to the development of students’ problem-solving skills. This argumentation is based on the assumption that the PISA assessments in math, science, and problem solving are able to measure students’ competence as outcomes, which are directly linked to their education (Bulle [ 2011 ]; Kind [ 2013 ]). In contrast to math and science competence, problem solving competence is not explicitly taught as a subject. Problem solving competence requires the utilisation of knowledge and reasoning skills acquired in specific domains (Pellegrino and Hilton [ 2012 ]). In agreement with Kuhn ([ 2009 ]), we point out that this transfer does not happen automatically but needs to be actively facilitated. In fact, Mayer and Wittrock ([ 2006 ]) stressed that the development of transferable skills such as problem solving competence needs to be fostered within specific domains rather than taught in dedicated, distinct courses. Moreover, they suggested that students should develop a “repertoire of cognitive and metacognitive strategies that can be applied in specific problem-solving situations” (p. 299). Beyond this domain-specific teaching principle, research also proposes to train the transfer of problem solving competence in domains that are closely related (e.g., math and science; Pellegrino and Hilton [ 2012 ]). In light of the effects of aligned curricula (as represented by the concept of math-science coherence), we argue that educational efforts to increase students’ problem solving competence may focus on a coordinated improvement of math and science literacy and fostering problem solving competence within math and science. The emphasis is on coordinated, as the results of our analyses indicated that the coherence between math and science education, as a qualitative characteristic of a country’s educational system, is a strong predictor of problem solving competence. This harmonisation of math and science education may be achieved by better enabling the utilisation of capabilities, especially in science education. Sufficiently high levels of math-science coherence could facilitate the emergence of educational synergisms, which positively affect the development of problem solving competence. In other words, we argue for quantitative changes (i.e., improve science attainment) in order to achieve qualitative changes (i.e., higher levels of curriculum coherence), which are expected to create effective transitions of subject-specific knowledge and skills into subject-unspecific competences to solve real-life problems (Pellegrino and Hilton [ 2012 ]; van Merriënboer [ 2013 ]).

Finally, we encourage research that is concerned with the validation of the proposed indices for different forms of problem solving. In particular, we suggest studying the facilities of the capability-under-utilisation indices for analytical and dynamic problem solving, as assessed in the PISA 2012 study (OECD [ 2014 ]). Due to the different cognitive demands in analytical and dynamic problems (e.g., using existing knowledge vs. generating knowledge; OECD [ 2014 ]), we suspect differences in capability utilisation in math and science. This research could provide further insights into the role of 21 st century skills as educational goals.

a The differences between students’ achievement in mathematics and problem solving, and science and problem solving have to be interpreted relative to the OECD average, since the achievement scales were scaled with a mean of 500 and a standard deviation of 100 for the OECD countries (OECD [ 2004 ], p. 55). Although alternative indices such as country residuals may also be used in cross-country comparisons (e.g., Olsen [ 2005 ]), we decided to use CUU indices, as they reflect the actual differences in achievement scores.

b In addition, we checked whether this result was due to the restricted variances in low-performing countries and found that neither ceiling nor floor effects in the problem solving scores existed. The problem solving scale differentiated sufficiently reliably in the regions below and above the OECD mean of 500.

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Scherer, R., Beckmann, J.F. The acquisition of problem solving competence: evidence from 41 countries that math and science education matters. Large-scale Assess Educ 2 , 10 (2014). https://doi.org/10.1186/s40536-014-0010-7

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  • Capability under-utilisation
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Problematizing teaching and learning mathematics as “given” in STEM education

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Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

figure 1

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

figure 2

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

Not applicable

Interestingly, Lakatos was advised by both Popper and Pólya—his ideas being in some ways a unification of Pólya’s emphasis on mathematics as an empirical discipline and Popper’s reflections on the nature of the scientific endeavor.

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Teaching Creativity and Inventive Problem Solving in Science

  • Robert L. DeHaan

Division of Educational Studies, Emory University, Atlanta, GA 30322

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Engaging learners in the excitement of science, helping them discover the value of evidence-based reasoning and higher-order cognitive skills, and teaching them to become creative problem solvers have long been goals of science education reformers. But the means to achieve these goals, especially methods to promote creative thinking in scientific problem solving, have not become widely known or used. In this essay, I review the evidence that creativity is not a single hard-to-measure property. The creative process can be explained by reference to increasingly well-understood cognitive skills such as cognitive flexibility and inhibitory control that are widely distributed in the population. I explore the relationship between creativity and the higher-order cognitive skills, review assessment methods, and describe several instructional strategies for enhancing creative problem solving in the college classroom. Evidence suggests that instruction to support the development of creativity requires inquiry-based teaching that includes explicit strategies to promote cognitive flexibility. Students need to be repeatedly reminded and shown how to be creative, to integrate material across subject areas, to question their own assumptions, and to imagine other viewpoints and possibilities. Further research is required to determine whether college students' learning will be enhanced by these measures.

INTRODUCTION

Dr. Dunne paces in front of his section of first-year college students, today not as their Bio 110 teacher but in the role of facilitator in their monthly “invention session.” For this meeting, the topic is stem cell therapy in heart disease. Members of each team of four students have primed themselves on the topic by reading selected articles from accessible sources such as Science, Nature, and Scientific American, and searching the World Wide Web, triangulating for up-to-date, accurate, background information. Each team knows that their first goal is to define a set of problems or limitations to overcome within the topic and to begin to think of possible solutions. Dr. Dunne starts the conversation by reminding the group of the few ground rules: one speaker at a time, listen carefully and have respect for others' ideas, question your own and others' assumptions, focus on alternative paths or solutions, maintain an atmosphere of collaboration and mutual support. He then sparks the discussion by asking one of the teams to describe a problem in need of solution.

Science in the United States is widely credited as a major source of discovery and economic development. According to the 2005 TAP Report produced by a prominent group of corporate leaders, “To maintain our country's competitiveness in the twenty-first century, we must cultivate the skilled scientists and engineers needed to create tomorrow's innovations.” ( www.tap2015.org/about/TAP_report2.pdf ). A panel of scientists, engineers, educators, and policy makers convened by the National Research Council (NRC) concurred with this view, reporting that the vitality of the nation “is derived in large part from the productivity of well-trained people and the steady stream of scientific and technical innovations they produce” ( NRC, 2007 ).

For many decades, science education reformers have promoted the idea that learners should be engaged in the excitement of science; they should be helped to discover the value of evidence-based reasoning and higher-order cognitive skills, and be taught to become innovative problem solvers (for reviews, see DeHaan, 2005 ; Hake, 2005 ; Nelson, 2008 ; Perkins and Wieman, 2008 ). But the means to achieve these goals, especially methods to promote creative thinking in scientific problem solving, are not widely known or used. An invention session such as that led by the fictional Dr. Dunne, described above, may seem fanciful as a means of teaching students to think about science as something more than a body of facts and terms to memorize. In recent years, however, models for promoting creative problem solving were developed for classroom use, as detailed by Treffinger and Isaksen (2005) , and such techniques are often used in the real world of high technology. To promote imaginative thinking, the advertising executive Alex F. Osborn invented brainstorming ( Osborn, 1948 , 1979 ), a technique that has since been successful in stimulating inventiveness among engineers and scientists. Could such strategies be transferred to a class for college students? Could they serve as a supplement to a high-quality, scientific teaching curriculum that helps students learn the facts and conceptual frameworks of science and make progress along the novice–expert continuum? Could brainstorming or other instructional strategies that are specifically designed to promote creativity teach students to be more adaptive in their growing expertise, more innovative in their problem-solving abilities? To begin to answer those questions, we first need to understand what is meant by “creativity.”

What Is Creativity? Big-C versus Mini-C Creativity

How to define creativity is an age-old question. Justice Potter Stewart's famous dictum regarding obscenity “I know it when I see it” has also long been an accepted test of creativity. But this is not an adequate criterion for developing an instructional approach. A scientist colleague of mine recently noted that “Many of us [in the scientific community] rarely give the creative process a second thought, imagining one either ‘has it’ or doesn't.” We often think of inventiveness or creativity in scientific fields as the kind of gift associated with a Michelangelo or Einstein. This is what Kaufman and Beghetto (2008) call big-C creativity, borrowing the term that earlier workers applied to the talents of experts in various fields who were identified as particularly creative by their expert colleagues ( MacKinnon, 1978 ). In this sense, creativity is seen as the ability of individuals to generate new ideas that contribute substantially to an intellectual domain. Howard Gardner defined such a creative person as one who “regularly solves problems, fashions products, or defines new questions in a domain in a way that is initially considered novel but that ultimately comes to be accepted in a particular cultural setting” ( Gardner, 1993 , p. 35).

But there is another level of inventiveness termed by various authors as “little-c” ( Craft, 2000 ) or “mini-c” ( Kaufman and Beghetto, 2008 ) creativity that is widespread among all populations. This would be consistent with the workplace definition of creativity offered by Amabile and her coworkers: “coming up with fresh ideas for changing products, services and processes so as to better achieve the organization's goals” ( Amabile et al. , 2005 ). Mini-c creativity is based on what Craft calls “possibility thinking” ( Craft, 2000 , pp. 3–4), as experienced when a worker suddenly has the insight to visualize a new, improved way to accomplish a task; it is represented by the “aha” moment when a student first sees two previously disparate concepts or facts in a new relationship, an example of what Arthur Koestler identified as bisociation: “perceiving a situation or event in two habitually incompatible associative contexts” ( Koestler, 1964 , p. 95).

In this essay, I maintain that mini-c creativity is not a mysterious, innate endowment of rare individuals. Instead, I argue that creative thinking is a multicomponent process, mediated through social interactions, that can be explained by reference to increasingly well-understood mental abilities such as cognitive flexibility and cognitive control that are widely distributed in the population. Moreover, I explore some of the recent research evidence (though with no effort at a comprehensive literature review) showing that these mental abilities are teachable; like other higher-order cognitive skills (HOCS), they can be enhanced by explicit instruction.

Creativity Is a Multicomponent Process

Efforts to define creativity in psychological terms go back to J. P. Guilford ( Guilford, 1950 ) and E. P. Torrance ( Torrance, 1974 ), both of whom recognized that underlying the construct were other cognitive variables such as ideational fluency, originality of ideas, and sensitivity to missing elements. Many authors since then have extended the argument that a creative act is not a singular event but a process, an interplay among several interactive cognitive and affective elements. In this view, the creative act has two phases, a generative and an exploratory or evaluative phase ( Finke et al. , 1996 ). During the generative process, the creative mind pictures a set of novel mental models as potential solutions to a problem. In the exploratory phase, we evaluate the multiple options and select the best one. Early scholars of creativity, such as J. P. Guilford, characterized the two phases as divergent thinking and convergent thinking ( Guilford, 1950 ). Guilford defined divergent thinking as the ability to produce a broad range of associations to a given stimulus or to arrive at many solutions to a problem (for overviews of the field from different perspectives, see Amabile, 1996 ; Banaji et al. , 2006 ; Sawyer, 2006 ). In neurocognitive terms, divergent thinking is referred to as associative richness ( Gabora, 2002 ; Simonton, 2004 ), which is often measured experimentally by comparing the number of words that an individual generates from memory in response to stimulus words on a word association test. In contrast, convergent thinking refers to the capacity to quickly focus on the one best solution to a problem.

The idea that there are two stages to the creative process is consistent with results from cognition research indicating that there are two distinct modes of thought, associative and analytical ( Neisser, 1963 ; Sloman, 1996 ). In the associative mode, thinking is defocused, suggestive, and intuitive, revealing remote or subtle connections between items that may be correlated, or may not, and are usually not causally related ( Burton, 2008 ). In the analytical mode, thought is focused and evaluative, more conducive to analyzing relationships of cause and effect (for a review of other cognitive aspects of creativity, see Runco, 2004 ). Science educators associate the analytical mode with the upper levels (analysis, synthesis, and evaluation) of Bloom's taxonomy (e.g., Crowe et al. , 2008 ), or with “critical thinking,” the process that underlies the “purposeful, self-regulatory judgment that drives problem-solving and decision-making” ( Quitadamo et al. , 2008 , p. 328). These modes of thinking are under cognitive control through the executive functions of the brain. The core executive functions, which are thought to underlie all planning, problem solving, and reasoning, are defined ( Blair and Razza, 2007 ) as working memory control (mentally holding and retrieving information), cognitive flexibility (considering multiple ideas and seeing different perspectives), and inhibitory control (resisting several thoughts or actions to focus on one). Readers wishing to delve further into the neuroscience of the creative process can refer to the cerebrocerebellar theory of creativity ( Vandervert et al. , 2007 ) in which these mental activities are described neurophysiologically as arising through interactions among different parts of the brain.

The main point from all of these works is that creativity is not some single hard-to-measure property or act. There is ample evidence that the creative process requires both divergent and convergent thinking and that it can be explained by reference to increasingly well-understood underlying mental abilities ( Haring-Smith, 2006 ; Kim, 2006 ; Sawyer, 2006 ; Kaufman and Sternberg, 2007 ) and cognitive processes ( Simonton, 2004 ; Diamond et al. , 2007 ; Vandervert et al. , 2007 ).

Creativity Is Widely Distributed and Occurs in a Social Context

Although it is understandable to speak of an aha moment as a creative act by the person who experiences it, authorities in the field have long recognized (e.g., Simonton, 1975 ) that creative thinking is not so much an individual trait but rather a social phenomenon involving interactions among people within their specific group or cultural settings. “Creativity isn't just a property of individuals, it is also a property of social groups” ( Sawyer, 2006 , p. 305). Indeed, Osborn introduced his brainstorming method because he was convinced that group creativity is always superior to individual creativity. He drew evidence for this conclusion from activities that demand collaborative output, for example, the improvisations of a jazz ensemble. Although each musician is individually creative during a performance, the novelty and inventiveness of each performer's playing is clearly influenced, and often enhanced, by “social and interactional processes” among the musicians ( Sawyer, 2006 , p. 120). Recently, Brophy (2006) offered evidence that for problem solving, the situation may be more nuanced. He confirmed that groups of interacting individuals were better at solving complex, multipart problems than single individuals. However, when dealing with certain kinds of single-issue problems, individual problem solvers produced a greater number of solutions than interacting groups, and those solutions were judged to be more original and useful.

Consistent with the findings of Brophy (2006) , many scholars acknowledge that creative discoveries in the real world such as solving the problems of cutting-edge science—which are usually complex and multipart—are influenced or even stimulated by social interaction among experts. The common image of the lone scientist in the laboratory experiencing a flash of creative inspiration is probably a myth from earlier days. As a case in point, the science historian Mara Beller analyzed the social processes that underlay some of the major discoveries of early twentieth-century quantum physics. Close examination of successive drafts of publications by members of the Copenhagen group revealed a remarkable degree of influence and collaboration among 10 or more colleagues, although many of these papers were published under the name of a single author ( Beller, 1999 ). Sociologists Bruno Latour and Steve Woolgar's study ( Latour and Woolgar, 1986 ) of a neuroendocrinology laboratory at the Salk Institute for Biological Studies make the related point that social interactions among the participating scientists determined to a remarkable degree what discoveries were made and how they were interpreted. In the laboratory, researchers studied the chemical structure of substances released by the brain. By analysis of the Salk scientists' verbalizations of concepts, theories, formulas, and results of their investigations, Latour and Woolgar showed that the structures and interpretations that were agreed upon, that is, the discoveries announced by the laboratory, were mediated by social interactions and power relationships among members of the laboratory group. By studying the discovery process in other fields of the natural sciences, sociologists and anthropologists have provided more cases that further illustrate how social and cultural dimensions affect scientific insights (for a thoughtful review, see Knorr Cetina, 1995 ).

In sum, when an individual experiences an aha moment that feels like a singular creative act, it may rather have resulted from a multicomponent process, under the influence of group interactions and social context. The process that led up to what may be sensed as a sudden insight will probably have included at least three diverse, but testable elements: 1) divergent thinking, including ideational fluency or cognitive flexibility, which is the cognitive executive function that underlies the ability to visualize and accept many ideas related to a problem; 2) convergent thinking or the application of inhibitory control to focus and mentally evaluate ideas; and 3) analogical thinking, the ability to understand a novel idea in terms of one that is already familiar.

LITERATURE REVIEW

What do we know about how to teach creativity.

The possibility of teaching for creative problem solving gained credence in the 1960s with the studies of Jerome Bruner, who argued that children should be encouraged to “treat a task as a problem for which one invents an answer, rather than finding one out there in a book or on the blackboard” ( Bruner, 1965 , pp. 1013–1014). Since that time, educators and psychologists have devised programs of instruction designed to promote creativity and inventiveness in virtually every student population: pre–K, elementary, high school, and college, as well as in disadvantaged students, athletes, and students in a variety of specific disciplines (for review, see Scott et al. , 2004 ). Smith (1998) identified 172 instructional approaches that have been applied at one time or another to develop divergent thinking skills.

Some of the most convincing evidence that elements of creativity can be enhanced by instruction comes from work with young children. Bodrova and Leong (2001) developed the Tools of the Mind (Tools) curriculum to improve all of the three core mental executive functions involved in creative problem solving: cognitive flexibility, working memory, and inhibitory control. In a year-long randomized study of 5-yr-olds from low-income families in 21 preschool classrooms, half of the teachers applied the districts' balanced literacy curriculum (literacy), whereas the experimenters trained the other half to teach the same academic content by using the Tools curriculum ( Diamond et al. , 2007 ). At the end of the year, when the children were tested with a battery of neurocognitive tests including a test for cognitive flexibility ( Durston et al. , 2003 ; Davidson et al. , 2006 ), those exposed to the Tools curriculum outperformed the literacy children by as much as 25% ( Diamond et al. , 2007 ). Although the Tools curriculum and literacy program were similar in academic content and in many other ways, they differed primarily in that Tools teachers spent 80% of their time explicitly reminding the children to think of alternative ways to solve a problem and building their executive function skills.

Teaching older students to be innovative also demands instruction that explicitly promotes creativity but is rigorously content-rich as well. A large body of research on the differences between novice and expert cognition indicates that creative thinking requires at least a minimal level of expertise and fluency within a knowledge domain ( Bransford et al. , 2000 ; Crawford and Brophy, 2006 ). What distinguishes experts from novices, in addition to their deeper knowledge of the subject, is their recognition of patterns in information, their ability to see relationships among disparate facts and concepts, and their capacity for organizing content into conceptual frameworks or schemata ( Bransford et al. , 2000 ; Sawyer, 2005 ).

Such expertise is often lacking in the traditional classroom. For students attempting to grapple with new subject matter, many kinds of problems that are presented in high school or college courses or that arise in the real world can be solved merely by applying newly learned algorithms or procedural knowledge. With practice, problem solving of this kind can become routine and is often considered to represent mastery of a subject, producing what Sternberg refers to as “pseudoexperts” ( Sternberg, 2003 ). But beyond such routine use of content knowledge the instructor's goal must be to produce students who have gained the HOCS needed to apply, analyze, synthesize, and evaluate knowledge ( Crowe et al. , 2008 ). The aim is to produce students who know enough about a field to grasp meaningful patterns of information, who can readily retrieve relevant knowledge from memory, and who can apply such knowledge effectively to novel problems. This condition is referred to as adaptive expertise ( Hatano and Ouro, 2003 ; Schwartz et al. , 2005 ). Instead of applying already mastered procedures, adaptive experts are able to draw on their knowledge to invent or adapt strategies for solving unique or novel problems within a knowledge domain. They are also able, ideally, to transfer conceptual frameworks and schemata from one domain to another (e.g., Schwartz et al. , 2005 ). Such flexible, innovative application of knowledge is what results in inventive or creative solutions to problems ( Crawford and Brophy, 2006 ; Crawford, 2007 ).

Promoting Creative Problem Solving in the College Classroom

In most college courses, instructors teach science primarily through lectures and textbooks that are dominated by facts and algorithmic processing rather than by concepts, principles, and evidence-based ways of thinking. This is despite ample evidence that many students gain little new knowledge from traditional lectures ( Hrepic et al. , 2007 ). Moreover, it is well documented that these methods engender passive learning rather than active engagement, boredom instead of intellectual excitement, and linear thinking rather than cognitive flexibility (e.g., Halpern and Hakel, 2003 ; Nelson, 2008 ; Perkins and Wieman, 2008 ). Cognitive flexibility, as noted, is one of the three core mental executive functions involved in creative problem solving ( Ausubel, 1963 , 2000 ). The capacity to apply ideas creatively in new contexts, referred to as the ability to “transfer” knowledge (see Mestre, 2005 ), requires that learners have opportunities to actively develop their own representations of information to convert it to a usable form. Especially when a knowledge domain is complex and fraught with ill-structured information, as in a typical introductory college biology course, instruction that emphasizes active-learning strategies is demonstrably more effective than traditional linear teaching in reducing failure rates and in promoting learning and transfer (e.g., Freeman et al. , 2007 ). Furthermore, there is already some evidence that inclusion of creativity training as part of a college curriculum can have positive effects. Hunsaker (2005) has reviewed a number of such studies. He cites work by McGregor (2001) , for example, showing that various creativity training programs including brainstorming and creative problem solving increase student scores on tests of creative-thinking abilities.

Model creativity—students develop creativity when instructors model creative thinking and inventiveness.

Repeatedly encourage idea generation—students need to be reminded to generate their own ideas and solutions in an environment free of criticism.

Cross-fertilize ideas—where possible, avoid teaching in subject-area boxes: a math box, a social studies box, etc; students' creative ideas and insights often result from learning to integrate material across subject areas.

Build self-efficacy—all students have the capacity to create and to experience the joy of having new ideas, but they must be helped to believe in their own capacity to be creative.

Constantly question assumptions—make questioning a part of the daily classroom exchange; it is more important for students to learn what questions to ask and how to ask them than to learn the answers.

Imagine other viewpoints—students broaden their perspectives by learning to reflect upon ideas and concepts from different points of view.

How Is Creativity Related to Critical Thinking and the Higher-Order Cognitive Skills?

It is not uncommon to associate creativity and ingenuity with scientific reasoning ( Sawyer, 2005 ; 2006 ). When instructors apply scientific teaching strategies ( Handelsman et al. , 2004 ; DeHaan, 2005 ; Wood, 2009 ) by using instructional methods based on learning research, according to Ebert-May and Hodder ( 2008 ), “we see students actively engaged in the thinking, creativity, rigor, and experimentation we associate with the practice of science—in much the same way we see students learn in the field and in laboratories” (p. 2). Perkins and Wieman (2008) note that “To be successful innovators in science and engineering, students must develop a deep conceptual understanding of the underlying science ideas, an ability to apply these ideas and concepts broadly in different contexts, and a vision to see their relevance and usefulness in real-world applications … An innovator is able to perceive and realize potential connections and opportunities better than others” (pp. 181–182). The results of Scott et al. (2004) suggest that nontraditional courses in science that are based on constructivist principles and that use strategies of scientific teaching to promote the HOCS and enhance content mastery and dexterity in scientific thinking ( Handelsman et al. , 2007 ; Nelson, 2008 ) also should be effective in promoting creativity and cognitive flexibility if students are explicitly guided to learn these skills.

Creativity is an essential element of problem solving ( Mumford et al. , 1991 ; Runco, 2004 ) and of critical thinking ( Abrami et al. , 2008 ). As such, it is common to think of applications of creativity such as inventiveness and ingenuity among the HOCS as defined in Bloom's taxonomy ( Crowe et al. , 2008 ). Thus, it should come as no surprise that creativity, like other elements of the HOCS, can be taught most effectively through inquiry-based instruction, informed by constructivist theory ( Ausubel, 1963 , 2000 ; Duch et al. , 2001 ; Nelson, 2008 ). In a survey of 103 instructors who taught college courses that included creativity instruction, Bull et al. (1995) asked respondents to rate the importance of various course characteristics for enhancing student creativity. Items ranking high on the list were: providing a social climate in which students feels safe, an open classroom environment that promotes tolerance for ambiguity and independence, the use of humor, metaphorical thinking, and problem defining. Many of the responses emphasized the same strategies as those advanced to promote creative problem solving (e.g., Mumford et al. , 1991 ; McFadzean, 2002 ; Treffinger and Isaksen, 2005 ) and critical thinking ( Abrami et al. , 2008 ).

In a careful meta-analysis, Scott et al. (2004) examined 70 instructional interventions designed to enhance and measure creative performance. The results were striking. Courses that stressed techniques such as critical thinking, convergent thinking, and constraint identification produced the largest positive effect sizes. More open techniques that provided less guidance in strategic approaches had less impact on the instructional outcomes. A striking finding was the effectiveness of being explicit; approaches that clearly informed students about the nature of creativity and offered clear strategies for creative thinking were most effective. Approaches such as social modeling, cooperative learning, and case-based (project-based) techniques that required the application of newly acquired knowledge were found to be positively correlated to high effect sizes. The most clear-cut result to emerge from the Scott et al. (2004) study was simply to confirm that creativity instruction can be highly successful in enhancing divergent thinking, problem solving, and imaginative performance. Most importantly, of the various cognitive processes examined, those linked to the generation of new ideas such as problem finding, conceptual combination, and idea generation showed the greatest improvement. The success of creativity instruction, the authors concluded, can be attributed to “developing and providing guidance concerning the application of requisite cognitive capacities … [and] a set of heuristics or strategies for working with already available knowledge” (p. 382).

Many of the scientific teaching practices that have been shown by research to foster content mastery and HOCS, and that are coming more widely into use, also would be consistent with promoting creativity. Wood (2009) has recently reviewed examples of such practices and how to apply them. These include relatively small modifications of the traditional lecture to engender more active learning, such as the use of concept tests and peer instruction ( Mazur, 1996 ), Just-in-Time-Teaching techniques ( Novak et al. , 1999 ), and student response systems known as “clickers” ( Knight and Wood, 2005 ; Crossgrove and Curran, 2008 ), all designed to allow the instructor to frequently and effortlessly elicit and respond to student thinking. Other strategies can transform the lecture hall into a workshop or studio classroom ( Gaffney et al. , 2008 ) where the teaching curriculum may emphasize problem-based (also known as project-based or case-based) learning strategies ( Duch et al. , 2001 ; Ebert-May and Hodder, 2008 ) or “community-based inquiry” in which students engage in research that enhances their critical-thinking skills ( Quitadamo et al. , 2008 ).

Another important approach that could readily subserve explicit creativity instruction is the use of computer-based interactive simulations, or “sims” ( Perkins and Wieman, 2008 ) to facilitate inquiry learning and effective, easy self-assessment. An example in the biological sciences would be Neurons in Action ( http://neuronsinaction.com/home/main ). In such educational environments, students gain conceptual understanding of scientific ideas through interactive engagement with materials (real or virtual), with each other, and with instructors. Following the tenets of scientific teaching, students are encouraged to pose and answer their own questions, to make sense of the materials, and to construct their own understanding. The question I pose here is whether an additional focus—guiding students to meet these challenges in a context that explicitly promotes creativity—would enhance learning and advance students' progress toward adaptive expertise?

Assessment of Creativity

To teach creativity, there must be measurable indicators to judge how much students have gained from instruction. Educational programs intended to teach creativity became popular after the Torrance Tests of Creative Thinking (TTCT) was introduced in the 1960s ( Torrance, 1974 ). But it soon became apparent that there were major problems in devising tests for creativity, both because of the difficulty of defining the construct and because of the number and complexity of elements that underlie it. Tests of intelligence and other personality characteristics on creative individuals revealed a host of related traits such as verbal fluency, metaphorical thinking, flexible decision making, tolerance of ambiguity, willingness to take risks, autonomy, divergent thinking, self-confidence, problem finding, ideational fluency, and belief in oneself as being “creative” ( Barron and Harrington, 1981 ; Tardif and Sternberg, 1988 ; Runco and Nemiro, 1994 ; Snyder et al. , 2004 ). Many of these traits have been the focus of extensive research of recent decades, but, as noted above, creativity is not defined by any one trait; there is now reason to believe that it is the interplay among the cognitive and affective processes that underlie inventiveness and the ability to find novel solutions to a problem.

Although the early creativity researchers recognized that assessing divergent thinking as a measure of creativity required tests for other underlying capacities ( Guilford, 1950 ; Torrance, 1974 ), these workers and their colleagues nonetheless believed that a high score for divergent thinking alone would correlate with real creative output. Unfortunately, no such correlation was shown ( Barron and Harrington, 1981 ). Results produced by many of the instruments initially designed to measure various aspects of creative thinking proved to be highly dependent on the test itself. A review of several hundred early studies showed that an individual's creativity score could be affected by simple test variables, for example, how the verbal pretest instructions were worded ( Barron and Harrington, 1981 , pp. 442–443). Most scholars now agree that divergent thinking, as originally defined, was not an adequate measure of creativity. The process of creative thinking requires a complex combination of elements that include cognitive flexibility, memory control, inhibitory control, and analogical thinking, enabling the mind to free-range and analogize, as well as to focus and test.

More recently, numerous psychometric measures have been developed and empirically tested (see Plucker and Renzulli, 1999 ) that allow more reliable and valid assessment of specific aspects of creativity. For example, the creativity quotient devised by Snyder et al. (2004) tests the ability of individuals to link different ideas and different categories of ideas into a novel synthesis. The Wallach–Kogan creativity test ( Wallach and Kogan, 1965 ) explores the uniqueness of ideas associated with a stimulus. For a more complete list and discussion, see the Creativity Tests website ( www.indiana.edu/∼bobweb/Handout/cretv_6.html ).

The most widely used measure of creativity is the TTCT, which has been modified four times since its original version in 1966 to take into account subsequent research. The TTCT-Verbal and the TTCT-Figural are two versions ( Torrance, 1998 ; see http://ststesting.com/2005giftttct.html ). The TTCT-Verbal consists of five tasks; the “stimulus” for each task is a picture to which the test-taker responds briefly in writing. A sample task that can be viewed from the TTCT Demonstrator website asks, “Suppose that people could transport themselves from place to place with just a wink of the eye or a twitch of the nose. What might be some things that would happen as a result? You have 3 min.” ( www.indiana.edu/∼bobweb/Handout/d3.ttct.htm ).

In the TTCT-Figural, participants are asked to construct a picture from a stimulus in the form of a partial line drawing given on the test sheet (see example below; Figure 1 ). Specific instructions are to “Add lines to the incomplete figures below to make pictures out of them. Try to tell complete stories with your pictures. Give your pictures titles. You have 3 min.” In the introductory materials, test-takers are urged to “… think of a picture or object that no one else will think of. Try to make it tell as complete and as interesting a story as you can …” ( Torrance et al. , 2008 , p. 2).

Figure 1.

Figure 1. Sample figural test item from the TTCT Demonstrator website ( www.indiana.edu/∼bobweb/Handout/d3.ttct.htm ).

How would an instructor in a biology course judge the creativity of students' responses to such an item? To assist in this task, the TTCT has scoring and norming guides ( Torrance, 1998 ; Torrance et al. , 2008 ) with numerous samples and responses representing different levels of creativity. The guides show sample evaluations based upon specific indicators such as fluency, originality, elaboration (or complexity), unusual visualization, extending or breaking boundaries, humor, and imagery. These examples are easy to use and provide a high degree of validity and generalizability to the tests. The TTCT has been more intensively researched and analyzed than any other creativity instrument, and the norming samples have longitudinal validations and high predictive validity over a wide age range. In addition to global creativity scores, the TTCT is designed to provide outcome measures in various domains and thematic areas to allow for more insightful analysis ( Kaufman and Baer, 2006 ). Kim (2006) has examined the characteristics of the TTCT, including norms, reliability, and validity, and concludes that the test is an accurate measure of creativity. When properly used, it has been shown to be fair in terms of gender, race, community status, and language background. According to Kim (2006) and other authorities in the field ( McIntyre et al. , 2003 ; Scott et al. , 2004 ), Torrance's research and the development of the TTCT have provided groundwork for the idea that creative levels can be measured and then increased through instruction and practice.

SCIENTIFIC TEACHING TO PROMOTE CREATIVITY

How could creativity instruction be integrated into scientific teaching.

Guidelines for designing specific course units that emphasize HOCS by using strategies of scientific teaching are now available from the current literature. As an example, Karen Cloud-Hansen and colleagues ( Cloud-Hansen et al. , 2008 ) describe a course titled, “Ciprofloxacin Resistance in Neisseria gonorrhoeae .” They developed this undergraduate seminar to introduce college freshmen to important concepts in biology within a real-world context and to increase their content knowledge and critical-thinking skills. The centerpiece of the unit is a case study in which teams of students are challenged to take the role of a director of a local public health clinic. One of the county commissioners overseeing the clinic is an epidemiologist who wants to know “how you plan to address the emergence of ciprofloxacin resistance in Neisseria gonorrhoeae ” (p. 304). State budget cuts limit availability of expensive antibiotics and some laboratory tests to patients. Student teams are challenged to 1) develop a plan to address the medical, economic, and political questions such a clinic director would face in dealing with ciprofloxacin-resistant N. gonorrhoeae ; 2) provide scientific data to support their conclusions; and 3) describe their clinic plan in a one- to two-page referenced written report.

Throughout the 3-wk unit, in accordance with the principles of problem-based instruction ( Duch et al. , 2001 ), course instructors encourage students to seek, interpret, and synthesize their own information to the extent possible. Students have access to a variety of instructional formats, and active-learning experiences are incorporated throughout the unit. These activities are interspersed among minilectures and give the students opportunities to apply new information to their existing base of knowledge. The active-learning activities emphasize the key concepts of the minilectures and directly confront common misconceptions about antibiotic resistance, gene expression, and evolution. Weekly classes include question/answer/discussion sessions to address student misconceptions and 20-min minilectures on such topics as antibiotic resistance, evolution, and the central dogma of molecular biology. Students gather information about antibiotic resistance in N. gonorrhoeae , epidemiology of gonorrhea, and treatment options for the disease, and each team is expected to formulate a plan to address ciprofloxacin resistance in N. gonorrhoeae .

In this project, the authors assessed student gains in terms of content knowledge regarding topics covered such as the role of evolution in antibiotic resistance, mechanisms of gene expression, and the role of oncogenes in human disease. They also measured HOCS as gains in problem solving, according to a rubric that assessed self-reported abilities to communicate ideas logically, solve difficult problems about microbiology, propose hypotheses, analyze data, and draw conclusions. Comparing the pre- and posttests, students reported significant learning of scientific content. Among the thinking skill categories, students demonstrated measurable gains in their ability to solve problems about microbiology but the unit seemed to have little impact on their more general perceived problem-solving skills ( Cloud-Hansen et al. , 2008 ).

What would such a class look like with the addition of explicit creativity-promoting approaches? Would the gains in problem-solving abilities have been greater if during the minilectures and other activities, students had been introduced explicitly to elements of creative thinking from the Sternberg and Williams (1998) list described above? Would the students have reported greater gains if their instructors had encouraged idea generation with weekly brainstorming sessions; if they had reminded students to cross-fertilize ideas by integrating material across subject areas; built self-efficacy by helping students believe in their own capacity to be creative; helped students question their own assumptions; and encouraged students to imagine other viewpoints and possibilities? Of most relevance, could the authors have been more explicit in assessing the originality of the student plans? In an experiment that required college students to develop plans of a different, but comparable, type, Osborn and Mumford (2006) created an originality rubric ( Figure 2 ) that could apply equally to assist instructors in judging student plans in any course. With such modifications, would student gains in problem-solving abilities or other HOCS have been greater? Would their plans have been measurably more imaginative?

Figure 2.

Figure 2. Originality rubric (adapted from Osburn and Mumford, 2006 , p. 183).

Answers to these questions can only be obtained when a course like that described by Cloud-Hansen et al. (2008) is taught with explicit instruction in creativity of the type I described above. But, such answers could be based upon more than subjective impressions of the course instructors. For example, students could be pretested with items from the TTCT-Verbal or TTCT-Figural like those shown. If, during minilectures and at every contact with instructors, students were repeatedly reminded and shown how to be as creative as possible, to integrate material across subject areas, to question their own assumptions and imagine other viewpoints and possibilities, would their scores on TTCT posttest items improve? Would the plans they formulated to address ciprofloxacin resistance become more imaginative?

Recall that in their meta-analysis, Scott et al. (2004) found that explicitly informing students about the nature of creativity and offering strategies for creative thinking were the most effective components of instruction. From their careful examination of 70 experimental studies, they concluded that approaches such as social modeling, cooperative learning, and case-based (project-based) techniques that required the application of newly acquired knowledge were positively correlated with high effect sizes. The study was clear in confirming that explicit creativity instruction can be successful in enhancing divergent thinking and problem solving. Would the same strategies work for courses in ecology and environmental biology, as detailed by Ebert-May and Hodder (2008) , or for a unit elaborated by Knight and Wood (2005) that applies classroom response clickers?

Finally, I return to my opening question with the fictional Dr. Dunne. Could a weekly brainstorming “invention session” included in a course like those described here serve as the site where students are introduced to concepts and strategies of creative problem solving? As frequently applied in schools of engineering ( Paulus and Nijstad, 2003 ), brainstorming provides an opportunity for the instructor to pose a problem and to ask the students to suggest as many solutions as possible in a brief period, thus enhancing ideational fluency. Here, students can be encouraged explicitly to build on the ideas of others and to think flexibly. Would brainstorming enhance students' divergent thinking or creative abilities as measured by TTCT items or an originality rubric? Many studies have demonstrated that group interactions such as brainstorming, under the right conditions, can indeed enhance creativity ( Paulus and Nijstad, 2003 ; Scott et al. , 2004 ), but there is little information from an undergraduate science classroom setting. Intellectual Ventures, a firm founded by Nathan Myhrvold, the creator of Microsoft's Research Division, has gathered groups of engineers and scientists around a table for day-long sessions to brainstorm about a prearranged topic. Here, the method seems to work. Since it was founded in 2000, Intellectual Ventures has filed hundreds of patent applications in more than 30 technology areas, applying the “invention session” strategy ( Gladwell, 2008 ). Currently, the company ranks among the top 50 worldwide in number of patent applications filed annually. Whether such a technique could be applied successfully in a college science course will only be revealed by future research.

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  • Cognitive flexibility and academic performance: Individual and cross-national patterns among adolescents in 57 countries Personality and Individual Differences, Vol. 217
  • Are we teaching novice instructional designers to be creative? A qualitative case study 16 January 2024 | Instructional Science, Vol. 4
  • What's in a word? Student beliefs and understanding about green chemistry 1 January 2024 | Chemistry Education Research and Practice, Vol. 25, No. 1
  • Preservice Physical Education Teachers’ Resistance to Change: The Importance of Occupational Socialization Experiences 26 October 2023 | Trends in Higher Education, Vol. 2, No. 4
  • Searching for creativity: How people search to generate new ideas 1 December 2023 | Journal of the Association for Information Science and Technology, Vol. 10
  • How can we measure metacognition in creative problem-solving? Standardization of the MCPS scale Thinking Skills and Creativity, Vol. 49
  • TOWARDS ENHANCING CREATIVITY AND INNOVATION IN EDUCATION SYSTEM FOR YOUTH IN HAIL REGION 1 August 2023 | Advanced Education, Vol. 10, No. 22
  • Investigating the impact of innovation competence instruction in higher engineering education 12 June 2023 | European Journal of Engineering Education, Vol. 10
  • Regaining creativity in science: insights from conversation 17 May 2023 | Royal Society Open Science, Vol. 10, No. 5
  • How transdisciplinary integration, creativity and student motivation interact in three STEAM projects for gifted education? 29 March 2023 | Gifted Education International, Vol. 39, No. 2
  • Does creative coursework predict educational, career, and community engagement outcomes for arts alumni? 15 November 2022 | Creativity Research Journal, Vol. 35, No. 2
  • Make science disruptive again 27 March 2023 | Nature Biotechnology, Vol. 41, No. 4
  • Promoting Creativity in Undergraduate Recreation and Leisure Services Classrooms: An Overview 19 March 2021 | SCHOLE: A Journal of Leisure Studies and Recreation Education, Vol. 38, No. 1
  • A Positive Association between Working Memory Capacity and Human Creativity: A Meta-Analytic Evidence 13 January 2023 | Journal of Intelligence, Vol. 11, No. 1
  • Students Creativity Through Digital Mind Map 26 July 2023
  • Pedagogical and School Practices to Foster Key Competences and Domain-General Literacy 23 August 2023
  • Coping with Challenges and Uncertainty in Scientific Research Asia-Pacific Science Education, Vol. 8, No. 2
  • Teaching design thinking as a tool to address complex public health challenges in public health students: a case study 12 April 2022 | BMC Medical Education, Vol. 22, No. 1
  • ‘Allowing them to dream’: fostering creativity in mathematics undergraduates 26 May 2022 | Journal of Further and Higher Education, Vol. 46, No. 10
  • Interaction with metaphors enhances creative potential 1 July 2022 | Journal of Poetry Therapy, Vol. 35, No. 4
  • Creative problem solving in knowledge-rich contexts Trends in Cognitive Sciences, Vol. 26, No. 10
  • Teaching Protein–Ligand Interactions Using a Case Study on Tau in Alzheimer’s Disease 1 July 2022 | Journal of Chemical Education, Vol. 99, No. 8
  • Student approaches to creative processes when participating in an open-ended project in science 30 June 2022 | International Journal of Science Education, Vol. 44, No. 10
  • Arts, Machines, and Creative Education
  • Perceived Learning Effectiveness and Student Satisfaction
  • A Contribution to Scientific Creativity: A Validation Study Measuring Divergent Problem Solving Ability 3 September 2021 | Creativity Research Journal, Vol. 34, No. 2
  • Problem Solving and Digital Transformation: Acquiring Skills through Pretend Play in Kindergarten 28 January 2022 | Education Sciences, Vol. 12, No. 2
  • Growing Innovation and Collaboration Through Assessment and Feedback: A Toolkit for Assessing and Developing Students’ Soft Skills in Biological Experimentation 12 May 2022
  • The Role of Creativity in Teaching Mathematics Online 1 December 2022
  • Breathing Life Into Marketing Scholarship Through Creativity Learning and Teaching
  • Mobilizing Research-Based Learning (RBL) in Higher Education
  • Trends and opportunities by fostering creativity in science and engineering: a systematic review 2 September 2021 | European Journal of Engineering Education, Vol. 46, No. 6
  • The effect of a scientific board game on improving creative problem solving skills Thinking Skills and Creativity, Vol. 41
  • Create Teaching Creativity through Training Management, Effectiveness Training, and Teacher Quality in the Covid-19 Pandemic 6 August 2021 | Journal of Ethnic and Cultural Studies, Vol. 8, No. 4
  • Creativity and technology in teaching and learning: a literature review of the uneasy space of implementation  11 January 2021 | Educational Technology Research and Development, Vol. 69, No. 4
  • Üstün Yetenekli Öğrencilerin Bilimsel Yaratıcılık ve Bilimsel Problem Çözme ile İlgili Öz Değerlendirmeleri 15 July 2021 | Yuzunci Yil Universitesi Egitim Fakultesi Dergisi
  • Cultivating creative thinking in engineering student teams: Can a computer‐mediated virtual laboratory help? 23 November 2020 | Journal of Computer Assisted Learning, Vol. 37, No. 2
  • Entrepreneurial competencies of undergraduate students: The case of universities in Nigeria The International Journal of Management Education, Vol. 19, No. 1
  • Promoting Creativity in General Education Mathematics Courses 5 August 2019 | PRIMUS, Vol. 31, No. 1
  • Methodological Considerations for Understanding Students’ Problem Solving Processes and Affective Trajectories During Game-Based Learning: A Data Fusion Approach 3 July 2021
  • The main trends in the process of building the creative potential of engineering students 18 June 2021 | E3S Web of Conferences, Vol. 274
  • Research on the Present Situation and Countermeasures of Cultivating Graduate Students’ Innovation Ability under Cooperative Innovation Environment Creative Education Studies, Vol. 09, No. 02
  • Innovation Centers and the Information Schools: The Influence of LIS Faculty Journal of Education for Library and Information Science, Vol. 61, No. 4
  • Biomimetics: teaching the tools of the trade 28 September 2020 | FEBS Open Bio, Vol. 10, No. 11
  • STEM academic teachers’ experiences of undertaking authentic assessment-led reform: a mixed method approach 21 March 2019 | Studies in Higher Education, Vol. 45, No. 9
  • Problems of forming marketing competencies in the digital economy IOP Conference Series: Materials Science and Engineering, Vol. 940, No. 1
  • Culturally Responsive Assessment of Physical Science Skills and Abilities: Development, Field Testing, Implementation, and Results 2 June 2020 | Journal of Advanced Academics, Vol. 31, No. 3
  • Culturally Responsive Assessment of Life Science Skills and Abilities: Development, Field Testing, Implementation, and Results 3 June 2020 | Journal of Advanced Academics, Vol. 31, No. 3
  • Curriculum Differentiation’s Capacity to Extend Gifted Students in Secondary Mixed-ability Science Classes 27 June 2020 | Talent, Vol. 10, No. 1
  • Student Motivation from and Resistance to Active Learning Rooted in Essential Science Practices 23 December 2017 | Research in Science Education, Vol. 50, No. 1
  • Integrating Entrepreneurship and Art to Improve Creative Problem Solving in Fisheries Education 27 February 2020 | Fisheries, Vol. 45, No. 2
  • Concepts Re-imagined: Relational Signs Beyond Definitional Rigidity 15 August 2020
  • Promoting Student Creativity and Inventiveness in Science and Engineering 24 October 2020
  • Teaching for Leadership, Innovation, and Creativity
  • Problem Çözme Becerileri Eğitim Programının Çocukların Karar Verme Becerileri Üzerindeki Etkisi 30 December 2019 | Erzincan Üniversitesi Eğitim Fakültesi Dergisi, Vol. 21, No. 3
  • Mento’s change model in teaching competency-based medical education 27 December 2019 | BMC Medical Education, Vol. 19, No. 1
  • The Effects of Individual Preparations on Group Creativity 21 January 2020
  • The Value of Creativity for Enhancing Translational Ecologies, Insights, and Discoveries 9 July 2019 | Frontiers in Psychology, Vol. 10
  • Using the International Classification of Functioning, Disability, and Health to Guide Students' Clinical Approach to Aging With Pathology Topics in Geriatric Rehabilitation, Vol. 35, No. 3
  • Impact of Brainstorming Strategy in Dealing With Knowledge Retention Skill: An Insight Into Special Learners' Needs In Saudi Arabia 1 January 2021 | MIER Journal of Educational Studies Trends & Practices
  • The potential of students’ creative disposition as a perspective to develop creative teaching and learning for senior high school biological science 12 March 2019 | Journal of Physics: Conference Series, Vol. 1157
  • Evaluating Remote Experiment from a Divergent Thinking Point of View 25 July 2018
  • Exploring Creative Education Practices and Implications: A Case study of National Chengchi University, Taiwan 4 April 2022 | Journal of Business and Economic Analysis, Vol. 02, No. 02
  • Exploring Creative Education Practices and Implications: A Case study of National Chengchi University, Taiwan 1 January 2020 | Journal of Business and Economic Analysis, Vol. 02, No. 02
  • Diverging from the Dogma: A Call to Train Creative Thinkers in Science 14 September 2018 | The Bulletin of the Ecological Society of America, Vol. 100, No. 1
  • Comparison of German and Japanese student teachers’ views on creativity in chemistry class 16 May 2018 | Asia-Pacific Science Education, Vol. 4, No. 1
  • The use of humour during a collaborative inquiry 27 August 2018 | International Journal of Science Education, Vol. 40, No. 14
  • Connecting creative coursework exposure and college student engagement across academic disciplines 29 August 2019 | Gifted and Talented International, Vol. 33, No. 1-2
  • Views of German chemistry teachers on creativity in chemistry classes and in general 1 January 2018 | Chemistry Education Research and Practice, Vol. 19, No. 3
  • Embedding Critical and Creative Thinking in Chemical Engineering Practice
  • Introducing storytelling to educational robotic activities
  • Investigating Undergraduates’ Perceptions of Science in Courses Taught Using the CREATE Strategy Journal of Microbiology & Biology Education, Vol. 19, No. 1
  • Teachers’ learning on the workshop of STS approach as a way of enhancing inventive thinking skills
  • Creativity Development Through Inquiry-Based Learning in Biomedical Sciences
  • The right tool for the right task: Structured techniques prove less effective on an ill-defined problem finding task Thinking Skills and Creativity, Vol. 26
  • The influential factors and hierarchical structure of college students’ creative capabilities—An empirical study in Taiwan Thinking Skills and Creativity, Vol. 26
  • Teacher perceptions of professional role and innovative teaching at elementary schools in Taiwan 10 November 2017 | Educational Research and Reviews, Vol. 12, No. 21
  • Evaluation of creative problem-solving abilities in undergraduate structural engineers through interdisciplinary problem-based learning 28 July 2016 | European Journal of Engineering Education, Vol. 42, No. 6
  • What Shall I Write Next? 19 September 2017
  • Inquiry-based Laboratory Activities on Drugs Analysis for High School Chemistry Learning 3 October 2017 | Journal of Physics: Conference Series, Vol. 895
  • BARRIERS TO STUDENTS’ CREATIVE EVALUATION OF UNEXPECTED EXPERIMENTAL FINDINGS 25 June 2017 | Journal of Baltic Science Education, Vol. 16, No. 3
  • Kyle J. Frantz ,
  • Melissa K. Demetrikopoulos ,
  • Shari L. Britner ,
  • Laura L. Carruth ,
  • Brian A. Williams ,
  • John L. Pecore ,
  • Robert L. DeHaan , and
  • Christopher T. Goode
  • Elizabeth Ambos, Monitoring Editor
  • A present absence: undergraduate course outlines and the development of student creativity across disciplines 3 October 2016 | Teaching in Higher Education, Vol. 22, No. 2
  • Exploring differences in creativity across academic majors for high-ability college students 16 February 2018 | Gifted and Talented International, Vol. 32, No. 1
  • Creativity in chemistry class and in general – German student teachers’ views 1 January 2017 | Chemistry Education Research and Practice, Vol. 18, No. 2
  • IMPORTANCE OF CREATIVITY IN ENTREPRENEURSHIP 1 January 2017
  • Accessing the Finest Minds
  • Science and Innovative Thinking for Technical and Organizational Development
  • Learning High School Biology in a Social Context Creative Education, Vol. 08, No. 15
  • Possibilities and limitations of integrating peer instruction into technical creativity education 6 September 2016 | Instructional Science, Vol. 44, No. 6
  • Creative Cognitive Processes in Higher Education 20 November 2014 | The Journal of Creative Behavior, Vol. 50, No. 4
  • An Evidence-Based Review of Creative Problem Solving Tools 6 April 2016 | Human Resource Development Review, Vol. 15, No. 2
  • Case-based exams for learning and assessment: Experiences in an information systems course
  • Case exams for assessing higher order learning: A comparative social media analytics usage exam
  • Beyond belief: Structured techniques prove more effective than a placebo intervention in a problem construction task Thinking Skills and Creativity, Vol. 19
  • A Belief System at the Core of Learning Science
  • Student Research Work and Modeled Situations in Order to Bridge the Gap between Basic Science Concepts and Those from Preventive and Clinical Practice. Meaningful Learning and Informed beneficience Creative Education, Vol. 07, No. 07
  • FOSTERING FIFTH GRADERS’ SCIENTIFIC CREATIVITY THROUGH PROBLEM-BASED LEARNING 25 October 2015 | Journal of Baltic Science Education, Vol. 14, No. 5
  • Scaffolding for Creative Product Possibilities in a Design-Based STEM Activity 16 November 2014 | Research in Science Education, Vol. 45, No. 5
  • Intuition and insight: two concepts that illuminate the tacit in science education 18 June 2015 | Studies in Science Education, Vol. 51, No. 2
  • Arts and crafts as adjuncts to STEM education to foster creativity in gifted and talented students 28 March 2015 | Asia Pacific Education Review, Vol. 16, No. 2
  • Initiatives Towards an Education for Creativity Procedia - Social and Behavioral Sciences, Vol. 180
  • Brian A. Couch ,
  • Tanya L. Brown ,
  • Tyler J. Schelpat ,
  • Mark J. Graham , and
  • Jennifer K. Knight
  • Michèle Shuster, Monitoring Editor
  • Kim Quillin , and
  • Stephen Thomas
  • Mary Lee Ledbetter, Monitoring Editor
  • The Design of IdeaWorks: Applying Social Learning Networks to Support Tertiary Education 21 July 2015
  • Video Games and Malevolent Creativity
  • Modelling a Laboratory for Ideas as a New Tool for Fostering Engineering Creativity Procedia Engineering, Vol. 100
  • “Development of Thinking Skills” Course: Teaching TRIZ in Academic Setting Procedia Engineering, Vol. 131
  • Leadership in the Future Experts’ Creativity Development with Scientific Research Activities 4 November 2014
  • Developing Deaf Children's Conceptual Understanding and Scientific Argumentation Skills: A Literature Review 3 January 2014 | Deafness & Education International, Vol. 16, No. 3
  • Leslie M. Stevens , and
  • Sally G. Hoskins
  • Nancy Pelaez, Monitoring Editor
  • Cortex, Vol. 51
  • A Sociotechnological Theory of Discursive Change and Entrepreneurial Capacity: Novelty and Networks SSRN Electronic Journal, Vol. 3
  • GEOverse: An Undergraduate Research Journal: Research Dissemination Within and Beyond the Curriculum 1 August 2013
  • Learning by Practice, High-Pressure Student Ateliers 2 August 2013
  • Relating Inter-Individual Differences in Verbal Creative Thinking to Cerebral Structures: An Optimal Voxel-Based Morphometry Study 5 November 2013 | PLoS ONE, Vol. 8, No. 11
  • 21st Century Biology: An Interdisciplinary Approach of Biology, Technology, Engineering and Mathematics Education Procedia - Social and Behavioral Sciences, Vol. 102
  • Reclaiming creativity in the era of impact: exploring ideas about creative research in science and engineering Studies in Higher Education, Vol. 38, No. 9
  • An Evaluation of Alternative Ways of Computing the Creativity Quotient in a Design School Sample Creativity Research Journal, Vol. 25, No. 3
  • A.-M. Hoskinson ,
  • M. D. Caballero , and
  • J. K. Knight
  • Eric Brewe, Monitoring Editor
  • Understanding, attitude and environment International Journal for Researcher Development, Vol. 4, No. 1
  • Promoting Student Creativity and Inventiveness in Science and Engineering
  • Building creative thinking in the classroom: From research to practice International Journal of Educational Research, Vol. 62
  • A Demonstration of a Mastery Goal Driven Learning Environment to Foster Creativity in Engineering Design SSRN Electronic Journal, Vol. 111
  • The development of creative cognition across adolescence: distinct trajectories for insight and divergent thinking 8 October 2012 | Developmental Science, Vol. 16, No. 1
  • A CROSS-NATIONAL STUDY OF PROSPECTIVE ELEMENTARY AND SCIENCE TEACHERS’ CREATIVITY STYLES 10 September 2012 | Journal of Baltic Science Education, Vol. 11, No. 3
  • Evaluation of fostering students' creativity in preparing aided recalls for revision courses using electronic revision and recapitulation tools 2.0 Behaviour & Information Technology, Vol. 31, No. 8
  • Scientific Creativity: The Missing Ingredient in Slovenian Science Education 15 April 2012 | European Journal of Educational Research, Vol. 1, No. 2
  • Could the ‘evolution’ from biology to life sciences prevent ‘extinction’ of the subject field? 6 March 2012 | Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, Vol. 31, No. 1
  • Mobile innovations, executive functions, and educational developments in conflict zones: a case study from Palestine 1 October 2011 | Educational Technology Research and Development, Vol. 60, No. 1
  • Informing Pedagogy Through the Brain-Targeted Teaching Model Journal of Microbiology & Biology Education, Vol. 13, No. 1
  • Teaching Creative Science Thinking Science, Vol. 334, No. 6062
  • Sally G. Hoskins ,
  • David Lopatto , and
  • Leslie M. Stevens
  • Diane K. O'Dowd, Monitoring Editor
  • Embedding Research-Based Learning Early in the Undergraduate Geography Curriculum Journal of Geography in Higher Education, Vol. 35, No. 3
  • Jared L. Taylor ,
  • Karen M. Smith ,
  • Adrian P. van Stolk , and
  • George B. Spiegelman
  • Debra Tomanek, Monitoring Editor
  • IFAC Proceedings Volumes, Vol. 43, No. 17
  • Critical and Creative Thinking Activities for Engaged Learning in Graphics and Visualization Course
  • Creativity Development through Inquiry-Based Learning in Biomedical Sciences

Submitted: 31 December 2008 Revised: 14 May 2009 Accepted: 28 May 2009

© 2009 by The American Society for Cell Biology

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STEM Education

STEM Education image 1 (name what is stem education)

What is STEM education?

STEM Education image 2 (name what is stem education)

STEM stands for Science, Technology, Engineering, and Mathematics. STEM education focuses on incorporating these four disciplines with an emphasis on problem solving and project-based learning. Introducing students to STEM educational experiences at an early age can help develop logic and reasoning skills as well as foster life-long learning that supports overall academic growth. STEAM is an evolved version of STEM, where the “A” stands for Arts. The addition of this discipline seeks to expand the limits of STEM education, promote inclusivity, and encourage problem-solving, independent thinking, and collaboration.

How can your student benefit from STEM education?

Careers in science, technology, engineering, and mathematics are on the rise. A K12-powered online STEM education can help your student build a strong and well-rounded foundation that fosters active learning, curiosity, problem-solving skills, and teamwork — essential skills that help them throughout their academic careers and support them in the professional world.

Introducing your students to STEM education promotes their personal development and academic growth while developing the mindset and skills to help them become better learners. K12’s online STEM education is an overarching, interdisciplinary approach that engages your student in hands-on learning with implications for the real world.

How can your student use STEM in the real world?

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STEM extends far beyond the classroom. There are many places where STEM shows up in the real world, and many students already unknowingly engage with it in their day-to-day routines. Highlighting how everyday experiences can relate to their STEM education is another way of augmenting an online STEM program. Simple activities like making DIY cleaning products, cooking, or baking are examples of how STEM principles can be applied to things we already do. Naturally integrating STEM education into your student’s life can empower them to be curious and explore imaginative solutions.

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Careers in STEM

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Students who follow a STEM-based education program gain experiences that help prepare them for careers in science, technology, engineering, and mathematics. Plus, the critical thinking and problem-solving skills they develop are widely applicable across career paths and industries. Students studying in a STEM field can expect job security, job growth, a competitive salary, and cutting-edge positions where they can make a difference. For K12-powered students interested in exploring career possibilities, there are a number of career and college prep pathways they can pursue while still in school. Pathways such as Agricultural, Food & Natural Resources, Engineering & Manufacturing, Health Sciences, and Information Technology can introduce STEM students to the opportunities awaiting them after graduation. Some Exciting Careers in STEM:

  • Marine biologist
  • Statistician
  • Software developer
  • Biomedical engineer
  • Environmental scientist
  • and so many more!

STEM in K12’s Curriculum

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K12’s curriculum is grounded in the principles of how students learn. Our online learning platform uses the power and convenience of technology to make learning exciting, compelling, and accessible. K12’s online curriculum offers math and science courses as well as STEM activities for  middle school  and  high school students . Many other courses include exercises, labs, and projects that involve various STEM disciplines. Our lesson content focuses on integrating all four STEM disciplines to engage students and prepare them for the possibility of applying these principles to careers or other real-life situations.

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STEM in K12’s Extracurricular Activities

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At K12, we know that a student’s education extends beyond the classroom, and we understand the importance of  student socialization . When it comes to STEM education, we offer several STEM clubs* and activities as opportunities for students to connect with peers and teachers over a shared educational interest. Our clubs, activities, projects, and challenges supplement an online STEM program. They provide enrichment and encourage learners of all ages to make new friends, explore common passions, discover new ideas, and expand their online education experience. Examples of STEM Clubs:

  • Engineering
  • Caring for Animals

Student Spotlights

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“I’ve always been a hands-on person. So that’s one reason I’ve always liked science.”✝︎

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“I’ve always wanted to help people.”**

More stories.

For more stories of K12-powered students finding success with online schooling, explore our Day in the Life features.

Empower Your Student with STEM Education at K12-Powered Schools

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A STEM education can spark new passions and interests in your student as well as teach them skills they need to succeed in school and beyond. At K12, we incorporate STEM into our curriculum to provide a well-rounded education and support students in their academic growth. Explore our programs to see what might be the best fit for your student.

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Learn More About STEM Education

STEM and STEAM focused education approaches are a great way to get students excited about learning. We’ve put together lots of information and resources here so you can learn about and navigate through the many possibilities.

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STEM in Curriculum

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STEM vs. STEM

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STEM in the Real World

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STEM Projects, Activities, and Challenges

*Clubs vary yearly and by school. **Kierra is a 2020 student at Destinations Career Academy of Colorado and her statement reflects her experience at her school. ✝︎  Joshua is a 2019 student at a K12-powered school in Washington, D.C. and his statement reflects his experience at his school.

STEM Education Guide

Beyond the Basics: Exploring The Intersection of STEM and Science Learning

STEM is a way of engaging with a child's natural curiosity about the world.

My kids are constantly asking questions. Some I don't even know the answer to. Kids are so present with their surroundings and are incredibly observant. Imagine how pure and powerful that sense of curiosity is in those little bodies.

So how do we use STEM to engage with their endless questions? We can use it as a learning opportunity, and using concepts from science can be straightforward.

Hence, why science is the first umbrella in the STEM acronym.

Scientific core concepts like the scientific method to understand cause and effect provide structure to finding an answer. If you view STEM as a house, science is the pillar that holds it all together. However, It goes much deeper than what is at the surface here. Let's dive in to find out how!

What Do STEM Students Study

STEM breaks down into four specific categories: Science, Technology, Engineering, and Math. More recently, some programs have expanded their STEM programs into STEAM. The “A” is representative of the arts. This allows for students to implement creative expression and artistic exploration.

STEM is structured to bring real-world issues and problems into the classroom. This makes the opportunities endless. Students use the concepts and manipulatives from each STEM subject to study, research, experiment, and find solutions to real-life problems.

They can be given a specific topic to then explore and study, or they can find answers to their own questions. For example, a student might ask, why do we throw away so much leftover food at lunch? This can lead to discussing compost and creating a school-wide compost system.

  • Science : They can experiment with the different ways composting works. Notice the various processes and tests which would work best for their school. 
  • Technology : Students can research the different benefits of composting. What type of composting best fits their school?
  • Engineering : What materials would they need to compost? Do they build an outdoor compost station? What does that look like?
  • Mathematics Talk to the school to discuss a budget. How much money would they need each school year? What are the measurements they need to build a compost station?

It is so empowering for kids to find solutions that relate to their lives. They are able to utilize their skills and apply them directly. It is an authentic way of learning that is known to increase engagement. It silences that question of, "when are we ever gonna use this in the real world?"

What does it look like when STEM is being used in a classroom? The classroom is busy, active, engaging, and full of energy. Kids are working in groups, or are doing their assigned tasks for the team.

See the importance of STEM education in early childhood development in our article here.

Why Science is Important

We all need guide posts when it comes to learning, and science provides that. Science helps categorize an understanding of the physical and natural world.

Then, using the scientific method, we can experiment and engage with our kid's questions that relate to these fields. It helps them further engage with the world around them, and what better way to find an answer to your little's question than by creating an active learning process using science?

Within that process, kids practice interpersonal skills like forming opinions, collaboration, communication, and perseverance. It also supports analytical skills like problem-solving and reasoning. As you see, the benefits are so applicable, and it paves the way for future Thomas Edisons and Marie Curies.

How to Build a Science Education

Building in science education might seem overwhelming. Once you understand what it might look like, then it will come intuitively. Every question you hear will spark an opportunity to bring in science. When it comes to STEM, teachers are there to support and guide.

Use Your Students’ Ideas: Whether it's a comment about the weather, their pet fish, or reading about clean water supply. Use their ideas and questions to spark active engagement. 

What do They Already Know? Make it a group conversation to explore what they already know about their question or topic. Help them pull in previously learned concepts.

Help Them Create a Plan: What steps do they need to take to experiment or find an answer to their question? How can they use the scientific method? What materials do they need? A plan gives them focus and keeps them on track.

Ask Questions & Monitor: They have a plan, and it is in action. Ask open-ended questions to tie in a scientific concept. Ask questions to help them work together. Monitor what they are thinking and where they are at in their plan.

Help Build Scientific Vocabulary: Students may not know how this applies to science. It is up to teachers to help them build their scientific vocabulary. Rather than giving them the word and definition, throw out the word and see if they understand how it might apply in context. Help them break apart the word to understand what it means.

Support Reflection : What did they learn? What was a road bump? How did they move through it? Help them understand their own process and how they got to the solution.

Careers with a STEM education

STEM education helps keep up with the fast-paced growth of technology. With the development of a digital economy. Major tech companies like Google, Microsoft, and Facebook are investing in grants, resources, and learning programs to support the next generation of innovators.

You can find our HUGE list of interesting STEM Education Facts here.

When your child learns in a STEM-based environment, they can thrive in over 300 types of careers, from agriculture, nursing, engineering, food science, neurology, and computer science.

The options are endless, and while these require a STEM education. Every career requires problem-solving, critical thinking, teamwork, and communication. Skills you gain within a STEM classroom.

The first step toward a STEM-related career is understanding what type of major you need to study to prepare for your future career. Figuring out a life path can be stressful with the limitless options out there.

Check out our article STEM Programs : How to Choose the Right Major to get some insight.

We think STEM education can shape the future , and we know science is at the root of that. It gives hope for making learning fun, and it can happen inside and outside the classroom!

The post How Does STEM Education Fit With Science Education appeared first on STEM Education Guide .

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Beyond the Basics: Exploring The Intersection of STEM and Science Learning

Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews     feb 14, 2024.

Elevating Math Education Through Problem-Based Learning

Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

problem solving in science education

Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

problem solving in science education

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

problem solving in science education

Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

Imagine Learning

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COMMENTS

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  5. Teaching Creativity and Inventive Problem Solving in Science

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  12. Problematizing teaching and learning mathematics as ...

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    result of this, (FME 2004 & 2008) stresses that science education shall lay emphases on the teaching and learning of science process and product so as to inculcate problem-solving abilities on the students. Problem-solving was said to be a special case of meaningful learning (Ausubel, 1968). This can perhaps be associated to a

  15. Full article: Understanding and explaining pedagogical problem solving

    Understanding and explaining pedagogical problem solving: a video-based grounded theory study of classroom pedagogy John-Paul Riordan , Lynn Revell , Bob Bowie , Mary Woolley , Sabina Hulbert & Caroline Thomas Pages 1309-1329 | Published online: 16 Nov 2021 Cite this article https://doi.org/10.1080/02635143.2021.2001450 In this article Full Article

  16. Teaching Creativity and Inventive Problem Solving in Science

    Engaging learners in the excitement of science, helping them discover the value of evidence-based reasoning and higher-order cognitive skills, and teaching them to become creative problem solvers have long been goals of science education reformers.

  17. PDF An Overview of Problem Solving Studies in Physics Education

    2. Studies on Problem Solving in Physics Education The aim of this study is to present an overview problem solving studies in Physics Education. For this purpose, the results of the extensive research dating back to 1980's have been presented chronologically. In addition, student level, methodology used for the studies were also presented.

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    Improving students' problem-solving skills in mathematics and science education has always been given special attention; however, the problem-posing approach which plays a key role in mathematics education has not been commonly utilized in science education.

  19. A rationale for and the development of a problem solving model of

    A rationale for and the development of a problem solving model of instruction in science education. Edward L. Pizzini. Science Education Center, The University of Iowa, Iowa City, Iowa 52242 ... Science Education Center, The University of Iowa, Iowa City, Iowa 52242. Search for more papers by this author.

  20. Laboratory Activities, Science Education and Problem-solving Skills

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    This paper presents a typology of field activities named after their main educational goal and categorized with regards to a set of educational criteria. Afterwards, it discusses the extent to which each type of field activity is consistent with problem-solving, namely with a problem-based learning approach. Previous.

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  23. Problem solving in science and technology education

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  24. Beyond the Basics: Exploring The Intersection of STEM and Science ...

    The post How Does STEM Education Fit With Science Education appeared first on STEM Education Guide. ... Every career requires problem-solving, critical thinking, teamwork, and communication. ...

  25. Elevating Math Education Through Problem-Based Learning

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