5.2.2.1. The crucial role of prior knowledge

A large body of research shows that our experience and knowledge about the world play a crucial role in problem solving (Bassok et al. 1995). Indeed, the role of prior knowledge in conceptual acquisition and development has been demonstrated in different domains throughout its lifetime (see, for example, Vosniadou and Ortony (1989)). For example, in problem solving, Clément and Richard (1997) show that familiar knowledge about the actions that allow a change of state could explain the great differences in difficulty between isomorphic problems that share the same solution principle, but whose contexts require knowledge about the change of state that may or may not be compatible with the solution. To test the role of familiar knowledge evoked by the problem statements, different isomorphic versions of the same problem were used. In some versions (the change-of-place problems), the adequate point of view of the action was emphasized, while in others (the change-of-size problems), the point of view of the action evoked by the statements had to be abandoned in order to discover the most efficient solution path. Results have shown that in the change-of-size problems, in which change is conceived as a continuous process of growth, adults take longer to find the solution and/or perform more actions and rule violations than in the change-of-place versions. In the change-of-size versions, discovering the most efficient solution path has been interpreted by Clement (2009) as the expression of cognitive flexibility, because the first representation has to be abandoned and a new point of view on the problem has to be adopted.

Understanding a situation on the basis of prior knowledge is a general process of learning by analogy. It is defined by some authors as being at the heart of thinking (Hofstadter and Sander 2013). At school, for example, making use of students’ prior knowledge is one way to teach new concepts. In this way, teaching new concepts is often planned around this familiar knowledge, acquired through experience. Bassok et al. (1998) have demonstrated this by conducting an analysis of mathematics textbooks used in America across grades one to eight. The authors report that the semantic relationships between the objects in the problem statements are very strongly correlated with the arithmetic operations to be applied on these objects. They note that in 97% of addition problem statements, the objects to be added are objects belonging to the same category. For example, they would be red marbles and blue marbles, but never cookies and baskets. Similarly, 94% of problems that require division involve objects that are functionally related. In this case, the objects involved might be cookies and baskets, but never red marbles and blue marbles.

Furthermore, when students are asked to design simple addition or division problem statements involving pairs of objects – for example, tulips and daisies or tulips and vases – students tend to construct statements in which the mathematical structure and the semantic structure (i.e. the relationships between the objects) are compatible with the pairs involving objects of the same category. When tulips and daisies are involved, it evokes addition problem statements. With pairs of objects that have a functional relationship (e.g. containing, as with the pairing of tulips and vases), division problems are proposed by the majority of students (see Experiment 1 in Bassok et al. (1998)). These results suggest that participants who have been taught simple arithmetic problems throughout their school careers apply their mathematical knowledge in ways that are consistent with their familiar knowledge about the relationships between objects in daily life. This semantic alignment, which is guided by our knowledge about the world, is observed in both adults and children (see, for example, Martin et al. (2005)). In fact, in adults this process is highly automatic (Bassok et al. 2008).

Building on students’ familiar knowledge in order to enable them to understand new concepts is certainly a necessary step in learning. However, it may not be sufficient to lead students to a deep understanding and mastery of the concepts taught at school. In their experiment, Moss and Case (1999) illustrate this phenomenon. They demonstrate how educational initiatives that can be put into practice at school have a significant influence on the development of mathematical knowledge. In their study, the authors designed a program for Canadian fourth graders aged 9–10 years. In order to improve students’ understanding and performance in solving problems involving rational numbers, the authors conceived an experimental program by reorganizing the order in which rational numbers were traditionally taught. To this end, and in contrast to Canadian curricula which advocated the introduction of decimals as an alternative way of representing fractions, the learning sequence consisted of first introducing percentages, then decimals, and finally mathematical sequences on fractions. Moss and Case report that children who received this type of instruction were less likely to mistakenly transfer strategies that were relevant for whole numbers but ineffective for rational numbers.

This body of work suggests that the encoding of concepts in memory and the links that may be made between different domains are crucial steps in learning and mastering mathematical concepts. It shows how the design of learning sequences can affect the quality of learning. This also demonstrates the importance of considering the role of prior and familiar knowledge in curriculum design; besides it demonstrates that the knowledge transmitted and acquired at school can have a longer term impact on the scientific understanding of mathematical concepts.

5.2.2.2. Positive, negative or no transfer

As described, our knowledge about the world can facilitate, hinder or prevent the discovery of a successful solution to a problem. Indeed, the encoding of a new problem very often consists of trying to apply what we have learned when solving previous problems. This can lead to the transfer of inadequate solutions and strategies that are learned in previous situations and that, by analogy, are perceived by the solver as similar to the new situation. In fact, the way the situation is encoded and categorized determines if the transfer will be positive, negative or if there will be no transfer.

A large body of studies carried out in the field of transfer between analogs – that is, problems sharing the same solution principle, but not systematically the same surface similarities such as characters, places, temporality, and so on – shows that the ease or difficulty encountered by the solver lie in the way in which the new problem (the target) will be encoded and categorized on the basis of perceived similarities between the situations (see Chapters 7 and 8). Transfer will be positive when problems that share the same solution principle – that is to say the same abstract structure – are perceived by the solver as similar. In the same manner, transfer will be negative when problems that do not share the same abstract structure are perceived as similar. Finally, there will be no transfer when problems sharing the same solution principle are perceived as dissimilar. In other words, when the source problem and the target problem evoke similar structures, there will be transfer. When the source and the target evoke different structures, the situations are perceived as different and the solver does not use what they have previously learned to solve the new problem. To successfully transfer and solve a new problem, it is necessary to categorize the known problem and the new problem on the basis of their structural similarity, regardless of their surface similarity. In many cases, this means changing our point-of-view on the problem, in other words, being flexible.

Concerning the negative transfer between problems, the harmful effect of past experience has been well described since the beginning of the 20th century. This is what the Gestalt Theory described as a phenomenon of mechanization of thought. For example, the deleterious effects of experience on the solution discovery were highlighted in the famous jars problems created by Luchins (1939, 1942). In these problems, some quantities of liquid must be transferred from one jar to another to obtain a new quantity. Luchins reports that adult participants who solved an initial set of problems by applying the same sequence of operations between three jars do not spontaneously discover a simpler solution in which the transfer between only two jars is necessary and sufficient. Although this procedure is simpler than the first, the majority of adults continue to apply the known procedure involving the three jars at hand.

Similarly, studies with young children have shown how solving a first problem (the source problem) can negatively affect solving a new problem (the target problem). For example, Chen and Daehler (1989) report the difficulties of 6-year-olds in not blindly applying a previously taught solution to new problems whose taught solution principle proves ineffective. The authors used two problems with the following two solution principles. In one problem, the solution consists of recovering an object floating in a container of water, but which is not directly accessible. To recover the object, simply pour water into the container to raise the level and catch the object. The solution to the second problem of catching a distant object that is not directly accessible is to connect two objects together to extend the arm and catch the object. As previously noted, negative transfer occurs when the source and target problems do not share the same solution principle. That is, like the Luchins’ adults participants, children blindly apply the procedure they learned in the first problem. They try in vain to reproduce the learned solution and do not spontaneously discover the solution of the second problem. Furthermore, it turns out that training children to abstract the solution principle from the source problem is beneficial when the source problem shares the same solution principle as the new target problem. However, when faced with a new problem that does not share the same solution principle, this training does not allow children to identify the irrelevance of the solution principle they were taught.

Concerning the absence of transfer between problems – a finding often reported by teachers – it appears that transfer is not systematic, even in the case where several examples are offered to the solver in order to allow them to categorize the problems according to solution principle and thus transfer the solution from one problem to another. The simple exposition and resolution of isomorphic problems is not sufficient to categorize the examples as belonging to the same problem pattern. In addition, studies have shown that the transfer between problems belonging to different domains is not spontaneous if the time between learning and transfer is long or if the context changes (Spencer and Weisberg 1986; Holyoak 2005). Transfer only occurs by encouraging participants to identify the abstract structure of problems. For example, Catrambone and Holyoak (1989) asked students to study, compare and solve isomorphic problems involving a principle of division and convergence of forces on a target to be destroyed (e.g. a medical problem of destroying a tumor by preserving the patient’s healthy tissues, or a military problem of destroying a fortress by dividing troops into small groups, in order to avoid blowing up the mined roads leading to the fortress).

To encourage the students to abstract the deep structure of the problems, the experimenters asked them to answer a series of very detailed questions about the problems. Then, they were asked to solve a third problem. The solution principle common to all three problems was then presented to them. Finally, one week later, the same students were invited to participate in a new experiment before solving the tumor problem inspired by Duncker’s problem (1945). The authors report that more than 80% of the participants spontaneously discovered the solution of simultaneously delivering converging low-intensity radiation to destroy the tumor and preserve healthy tissue. According to Holyoak (2005), the expertise developed by students in this experiment allowed them to elaborate and access in memory, an abstract schema evoked by a new problem that shares the same structure. Similarly, encouraging students to compare the structure of problems, regardless of their superficial similarity, has also proven effective with young children (see, for example, Gamo et al. (2010)).

We can take advantage of such results in the field of academic learning. In this way, encouraging students to identify, beyond their superficial similarities, the abstract and deep structure of situations is certainly a teaching approach to be developed. It is one of the ways of stimulating cognitive flexibility that involves changing perspective on the problem and selecting the relevant cues in the situation. Thus, in these situations, stimulating cognitive flexibility consists of training the learners to abstract, beneath or beyond the form, the profound structure of the problem. In summary, work on the processes involved in transfer shows that effective and flexible knowledge transfer requires deep understanding and encoding of situations, but also recognition of the context in which that knowledge is applicable. Selecting the relevant cues in the situation and identifying the contexts in which the knowledge is applicable is evidence of cognitive flexibility as we define it.

As Brown and Campione (1984) point out:

A major barrier to flexible learning is often not a lack of transfer between different fields of knowledge, but rather inappropriate transfer from one field to another. Successful learning involves knowing when, where, and what to transfer rather than blindly applying knowledge (p. 185).

Thus, access to relevant knowledge in memory and its appropriate use to find the solution are heavily guided by the context of the new problem to be solved. It is what is called the “evocation processes” and the “problem content effects“. These aspects are developed in more detail in Chapters 7 and 8. Indeed, much more than the surface or perceptual objective similarities shared by the situations, the transfer between problems depends on the way these similitudes are evoked by the content of the problems and encoded by the solver (Kotovsky and Fallside 1989; Bassok et al. 1995). Indeed, it appears that perceiving similarities and analogies is one of the fundamental aspects of human cognition. It is in this sense that Vosniadou and Ortony (1989) describe this capacity as crucial in recognition, categorization and learning: perceiving a structural analogy between situations plays a determining role in the discovery of solutions, just as in the case of scientific discoveries or creative thinking. In a related area to problem solving, recent research on spontaneous analogies shows that when situations are familiar (in this case short scenarios describing everyday situations), they preferentially evoke in memory, situations that only share structural similarities and no surface similarities (Raynal et al. 2020). When situations are familiar, individuals tend to favor structural similarities and neglect surface similarities. On the other hand, when situations are unfamiliar, then individuals will tend to favor superficial cues. This result supports the idea that in the school domain, when new notions are sometimes difficult to understand, training students to categorize situations on the basis of their deep structure is a powerful way to allow them to construct new knowledge on the basis of relevant cues. In sum, this body of research shows how an individual’s knowledge, level of familiarity with the problem and problem context can contribute to the flexible expression of problem-solving behaviors and, more generally, are key determinants of successful learning and conceptual development.

So far, we have discussed the role played by the knowledge activated by the problem context and the expertise of the solver in the discovery and flexible transfer. A situation is a problem when solution discovery requires understanding the deep structure of the problem (the principles of the solution), while neglecting superficial similarities. In this context, the successful solution discovery consists of recoding the situation. It involves cognitive flexibility, which allows us to consider the situation from a new perspective, to modify our initial representation of the goal and to select the relevant information to achieve the goal (Clément 2009).

In order to introduce the topic of this chapter, we have focused on problem solving and the processes at work in the discovery and transfer of solutions. We have deliberately chosen to discuss the notion of cognitive flexibility from the perspective of the work carried out in cognitive psychology in the field of problem solving, as if the definition of flexibility was a consensus in the literature. In the following section, we will present the different meanings of this notion, the nuances and the differences that can be found according to the fields of research, although recently, links may be established between approaches developed in the field of education, neuropsychology, cognitive psychology or developmental psychology (for a review of the issues, see Clément (forthcoming)).

5.2.3.1. Research in mathematics education

In mathematics education research, flexibility is defined in a somewhat different sense than that defined by researchers in cognitive or developmental psychology. Thus, in studies conducted in mathematics didactics, flexibility is often understood as either the ability to change strategies flexibly in order to solve the problem as quickly as possible (Kilpatrick et al. 2001; Baroody and Dowker 2003; Verschaffel et al. 2007; Torbeyns et al. 2018), or the ability to choose the teacher-provided representational format that is the most appropriate for solving the problem (Acevedo Nistal et al. 2012).

Strategic flexibility, as defined by influential authors in mathematics teaching, is to be understood in the more general framework of the cognitive model developed by Lemaire and Siegler (1995). According to this model, four parameters contribute to strategic development and its efficiency:

• 1) the repertoire of strategies, which corresponds to the different strategies available to the individual for solving problems;
• 2) the distribution of strategies, which corresponds to the frequency with which each strategy is used;
• 3) the effectiveness of strategies, defined as the accuracy and speed of execution of the strategy;
• 4) the selection of strategies, which refers to the flexible choice of strategies.

The development of strategic skills would depend on the development of the four parameters thus defined. Furthermore, according to Lemaire and Siegler (1995), strategic flexibility depends on the effectiveness of each individual’s strategic skills in selecting the fastest strategy leading to the solution of the problem.

In line with Siegler’s model, Verschaffel’s work distinguishes between the notions of flexibility and adaptability, which may have different meanings depending on the authors. As noted by the author, for some researchers in the field, these notions are synonymous, while for others, these notions refer to two distinct competencies (e.g. Heinze et al. 2009). Thus, Verschaffel defines these notions as follows: strategic flexibility is the ability to choose flexibly between several strategies available in memory, but without necessarily selecting the most suitable one for the mathematical situation, whereas strategic adaptability also covers the ability to adopt the strategy most suitable for the problem.

More precisely, Verschaffel et al. (2009) introduce the notion of adaptability:

By an adaptive choice of a strategy, we mean the conscious or unconscious selection and use of the most appropriate solution strategy on a given mathematical item or problem, for a given individual, in a given sociocultural context (p. 343).

In addition, in this field of research, much work has investigated the effectiveness of different formats of external representations (e.g. graphs, diagrams, tables, etc.) that are provided to students to help them conceptualize and solve problems. In this case, representational flexibility refers to the ease with which the student uses multiple external representations and is able to switch from one to another to solve the problem. Acevedo Nistal et al. (2009) state that they use the term representational flexibility in the following way:

...We will use this term to speak about students’ disposition to make appropriate representational choices, taking into account the task, student and context characteristics that come into play in the resolution of the mathematical task at hand... (p. 629).

In this perspective, flexibility is defined as the student’s ability to switch between several external representations and select the most appropriate representation for solving the task. According to these authors, three factors influence this representational flexibility. The first refers to the characteristics of the task. The good fit between the external representations provided to solve the problem and the solution will facilitate the use and selection of the most appropriate external representation. The second factor concerns the characteristics of the student. Their conceptual and procedural knowledge of the conventional external representations provided in class, their knowledge of the problem-solving domain and their preferences and habits for using an external representation will determine the ease of flexible choice between several external representations. Finally, the context in which tasks are presented determines representational flexibility. Thus, devices that provide active support in the selection of external representations or that stimulate the comparison and evaluation of external representations play a significant role in students’ ability to use and choose the most relevant representation.

5.2.3.2. Research conducted in psychology

In psychology, research on the expression of flexibility in problem solving is still quite marginal. Indeed, in developmental psychology or neuropsychology, cognitive flexibility is usually assessed in the well-known switching task paradigm (see, for instance, Chapter 1, section 1.3.1.1).

In the cognitive approach of problem solving in which our work sits, we define cognitive flexibility to the individual’s ability to change mental representations of the situation, and not, as described previously, to the ability to switch from one external representation format to another. This nuance is important to emphasize, because often misunderstandings between researchers from different disciplinary fields are due to the semantics underlying the concepts used.

This approach is inherited from the pioneering work of Gestalt theory. Indeed, a classic example of mental rigidity, the corollary of flexibility, is what was described by the phenomenon of functional fixedness reported at the beginning of the 20th century by the gestaltists (Duncker 1945). Duncker reports that adult participants show great difficulty in spontaneously conceiving objects in an unconventional function and in using them in innovative ways to creatively solve problems. In some situations, knowledge about the conventional uses of objects prevents the discovery of creative solutions. The development of knowledge about the conventional use of objects due to the daily experiences of using familiar objects explains a result that may seem paradoxical at first sight: the development with age of a certain rigidity in the discovery of creative solutions.

In a study with children aged 5, 6, and 7, German and Defeyter (2000) only reported functional fixedness effects from age 6 onward, with 5-year-olds being much more creative than their elders. In this study, the authors created an adapted version of Duncker’s (1945) candle problem, the “Bobo the Bear’s House” problem, in which the child is asked to help a small teddy bear reach a toy sitting out of reach on a shelf. The children in the experiment were randomly assigned to two experimental conditions. In the first condition, a box filled with various objects constitutes a pre-use condition supposed to highlight the usual function of the box as a container. In the other condition, the objects are distributed around the box (condition without pre-use). The solution consists of using the box in an unusual function which serves as a stepping stone to grasp the toy. In the condition of no pre-use of the box, the 6- and 7-year-olds discovered the solution more quickly than their younger peers, while in the condition with pre-use of the box, the latter discovered the solution much more quickly than their elders. This result replicates the observations reported by gestaltists on the deleterious effects of past experiences and knowledge on the creative solutions discovery.

However, links between the development with age of cognitive flexibility in both linguistic and non-linguistic domains (see, for example, Deák (2003)) and the development of a certain rigidity that appears in the course of life experience remain to be understood and studied in more detail. The effect of schooling and the development of a certain conformism during adolescence are often evoked to explain this rigidity of behavior. However, it can be assumed that a principle of cognitive economy would be at the origin of such behaviors. In fact, it appears that behaviors that are certainly rigid are also effective most of the time in everyday life situations. Indeed, choosing whether or not to change one’s ways of doing and thinking can be cognitively costly. Nevertheless, this observation of rigid thinking linked to the school and cultural environment encourages future research on the development of educational programs stimulating cognitive flexibility to counteract this behavioral rigidity.

Concerning the distinction we make between strategic flexibility and representational flexibility that we defined as a change of viewpoint, we have been able to show that changing strategy or procedure is not sufficient to find the solution to a new problem that shares the same surface cues as some previously solved problems. This was empirically shown in an experiment with adults who were asked to solve Luchins’ jars problems (Clément 2006). In the terms used in transfer studies, the particularity of Luchins’ jars problems is that they share the same superficial indices (the number of available jars), but that the procedure to achieve the goal differs. In the first problems, the solution consists of using the three available jars, while in the following problems, it is possible to use two or three jars, and in the last problems, only two jars can be used among the three available.

The results show that, in the case of a solution with two jars, and even though the transfer strategy is simpler than with three jars, when some participants reach an impasse, that is, when the solution learned with three jars does not work, they change their strategy by testing new combinations, while continuing to use three jars without discovering the two jars solution. For these participants, a change in strategy or procedure may denote some form of strategic flexibility, but does not lead to solution discovery. Indeed, these participants do not show representational flexibility. They do not engage in processing the relevant properties to be considered, that is, the number of jars to be used. Thus, we defined the notion of representational, or conceptual, flexibility (see Chapter 4, section 4.3.2) as the ability to change our perspective, to re-represent the problem and the goal independently of the procedure; in other words, to engage in recoding and recategorizing the problem on the basis of its solution principle (Clément 2009).