4.2.1. Philosophical approaches

Philosophical approaches set goals for thinking, identifying criteria that critical thinking must meet. In this sense, they can be described as normative, insofar as they provide a list of characteristics that define the ideal critical thinker.

For example, it is important to follow the rules of formal logic, be fair and open to the ideas of others, use sound judgment in evaluating information, and conform to the principles of rationality in reasoning (Lai 2011). An attempt at a consensus approach to critical thinking – through a panel of experts brought together by the American Philosophical Associations – resulted in the following definition:

We understand critical thinking to be purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based. CT is essential as a tool of inquiry. As such, CT is a liberating force in education and a powerful resource in one’s personal and civic life. While not synonymous with good thinking, CT is a pervasive and self-rectifying human phenomenon. The ideal critical thinker is habitually inquisitive, well-informed, trustful of reason, open-minded, flexible, fair-minded in evaluation, honest in facing personal biases, prudent in making judgments, willing to reconsider, clear about issues, orderly in complex matters, diligent in seeking relevant information, reasonable in the selection of criteria, focused in inquiry, and persistent in seeking results which are as precise as the subject and the circumstances of inquiry permit […] (Facione 1990, p. 2).

Recognizing that such a definition may seem unrealistic, the aim is to highlight the ideal skills that education should aim to develop. However, the lack of explicit cognitive mechanisms that underlie such characteristics makes it difficult to operationalize this approach and to identify a set of shared, practical, clear indications for critical thinking education (Bailin et al. 1999).

Indeed, in the framework on philosophical approaches, the concept of critical thinking is often so broad that it encompasses all of the skills deemed necessary – or at least useful – to properly resolve a problem, articulate a clear argument and make the most sensible choices. The taxonomies of critical thinking skills and behaviors thus include a diverse range of elements, such as:

• – a good understanding of the issue;
• – proper analysis of the arguments;
• – the ability to identify logical errors;
• – understanding the context in which a question is asked;
• – evaluating the credibility of an information source according to a number of criteria;
• – evaluating the content of the information or explanation based on relevant evidence, seeking further information checking facts;
• – command of deductive and inductive reasoning and, more generally, the ability to make correct inferences;
• – analysis of the consequences of accepting a certain argument;
• – the ability to clearly express ourselves and detect unexplained assumptions.

This range of generic skills seems difficult to reconcile with the operationalization of critical thinking. Another pitfall is that it is difficult to assess whether a skill is being implemented correctly and thus to evaluate the effectiveness of pedagogical interventions designed to foster the development of critical thinking.

Consider the assertion that aspiring critical thinkers must, as Ennis (2011) suggests, seek out and open themselves up to alternative assumptions, explanations, conclusions, plans, sources, etc. In reality, alternative hypotheses are not always accessible, especially when relevant knowledge in the field is lacking. Thus, directing a person to look for alternatives can lead to them missing the point if there is no strategy associated with it. Moreover, the risk is not only that the crux of the matter is missed because of a lack of knowledge in the field, but also that too much openness can have undesirable consequences, such as extreme relativism.

The systematic search for alternative hypotheses when perfectly satisfactory hypotheses are available is characteristic of conspiracy theories. Therefore, a broad definition and a generic list of competences hinder the chances of defining the concept of critical thinking, and there is even a risk of counter-productive consequences broadening the list of competences associated with it.

4.2.2. Psychological approaches

In contrast to philosophical approaches, psychological approaches could be characterized as descriptive as they suggest identifying a set of decision-making and problem-solving skills (Lai 2011). However, with a few exceptions – such as Kuhn’s (1999) emphasis on metacognition and argumentation – a certain lack of specificity is noticeable in the proposed characterizations. For example, it can be difficult to distinguish manifestations of critical thinking from those of thinking in general, reasoning or rationality, notions which themselves remain loosely characterized.

Critical thinking can thus be presented as a higher order skill, the appropriate use of which increases the chances of “solving problems, formulating inferences, calculating likelihoods and making decisions. Critical thinkers use these skills appropriately, without prompting, and usually with conscious intent, in a variety of settings” (Halpern 1999, p. 70). The taxonomy of skills identified includes the ability to control common cognitive biases and implement algorithmic thinking (Stanovich and Stanovich 2010), metacognition, dialogic abilities (Kuhn 1999), strategies for evaluating evidence for a better assessment of probabilities (Nisbett 2015), to which discourse analysis, argument analysis, a structured approach to problem solving, and even creative thinking, are added.

A short taxonomy of critical thinking skills is proposed as a guide for instruction. a) Verbal reasoning skills: this category includes those skills needed to comprehend and defend against the persuasive techniques embedded in everyday language; b) Argument analysis skills: an argument is a set of statements with at least one conclusion and one reason that supports that conclusion. In real-life settings, arguments are complex, with reasons that run counter to the conclusion, stated and unstated assumptions, irrelevant information, and intermediate steps; c) Skills in thinking as hypothesis testing: the rationale for this category is that people function like intuitive scientists to explain, predict and control events. These skills include generalizability, recognition of the need for an adequately large sample size, accurate assessment, and validity, among others; d) Likelihood and uncertainty: because very few events in life can be known with certainty, the correct use of cumulative, exclusive, and contingent probability should play a critical role in almost every decision; e) Decision-making and problem-solving skills: in some sense, all of the critical-thinking skills are used to make decisions and solve problems, but the ones that are included here involve generating and selecting alternatives and judging them. Creative thinking is subsumed under this category because of its importance in generating alternatives and restating problems and goals. (p. 452)

Like philosophers, psychologists tend to limit critical thinking to an explicit, voluntary, goal-oriented activity, analogous to adult thinking, or at least requiring a certain level of cognitive development, which is neither spontaneous, nor automated, nor natural:

Critical thinking refers to the use of cognitive skills or strategies that increase the probability of a desirable outcome. Critical thinking is purposeful, reasoned, and goal-directed (Halpern 1999, p. 70).

In some cases (Stanovich and Stanovich 2010), the theorization of critical thinking is subsumed under the idea of cognitive functioning and cognitive structure, such as the heuristics and biases approach and the dual systems approach.

These dualist theories defend the idea that cognitive functioning depends on at least two types of processes: some that are quick and not particularly onerous (system 1, S1), which result from the slow evolution of our species, and are generally automated; and others that are slow and costly (system 2, S2), which must control the former and are voluntary and conscious.

Critical thinking is then identified with the implementation of S2 processes, and as a solution to the biases under S1 processes that affect our perceptions, memory, opinions and decisions (Stanovich and Stanovich 2010). However, dualist theories are debated in psychology (Mercier and Sperber 2017; Pennycook et al. 2018). In particular, some work invites a revaluation of the relationship between S1 and S2 by showing that S1 processes can lead to correct intuitions and that S2 responses provide a posteriori justifications for correct S1 intuitions (Bago and De Neys 2019).

In summary, while the definition of critical thinking is not very specific within current philosophical approaches and tends to focus on the characteristics of the ideal thinker, psychological approaches do not allow the separation of the concept of critical thinking from other general concepts, such as reasoning, reflection, rationality and decision. Both philosophical and psychological approaches thus identify critical thinking with such a diversity of skills that their combination can result in opposite and undesirable consequences. The fact that critical thinking is both associated with vigilance toward information and the opinions of others and with a greater openness of mind (Facione 1990) illustrates such a paradox. Systematically changing our minds is no more desirable than never doing so.

In some cases, the ambiguity surrounding the skills and behaviors associated with critical thinking could lead to an undesirable form of relativism (systematic doubt) or paralysis of action (suspension of judgment). It could become difficult to draw a line between critical thinking and attitudes encountered in the framework of conspiracy theories, which are characterized by a relativistic attitude and the uninterrupted construction of “argumentative mille-feuille” (Bronner 2013).

4.2.3. Forms of critical thinking education

Beyond the characterization of critical thinking, another question remains controversial: that of the domain-general or domain-specific nature of critical thinking. Is it possible to transfer critical thinking from one domain to another?

Glaser (1941) already underlined the difficulty of knowing whether the skills associated with critical thinking are general or domain specific. This debate remains relevant today, without the distinction between philosophical and psychological approaches.

Thus, Lipman (1988), Halpern (2001) and van Gelder (2005) support the idea of the genericity of critical thinking. Halpern (2001) justifies her position by calling attention to the positive results achieved through critical thinking teaching methods that do not rely on a specific disciplinary background. In this perspective, critical thinking is transferable to new contexts.

However, there is also support for the idea that critical thinking develops in a domain-specific way (Perkins and Salomon 1989; McPeck 1990; Bailin 2002; Willingham 2008). From this perspective, general skills have limited use (McPeck 1990) as they are overly generic (Bailin 2002). Spontaneous transfer to new contexts is extremely rare (Willingham 2008) and, furthermore, general skills could not be taught (Tricot and Sweller 2014).

However, Ennis (1989) considers that while it is impossible to think critically independently from content, it is still possible to teach general principles of critical thinking. He illustrates his position with the example of formal logic, which applies to all types of content. This connection justifies the implementation of a general approach to the teaching of critical thinking, though, at the same time, Ennis highlights the importance of drawing on examples from specific domains and a range of topics to foster generalization and transfer.

Ennis (1989) distinguishes four forms of critical thinking education:

• – first, it can be general, relying on the teaching of abstract principles;
• – second, immersion education is implicit and is based on disciplinary content from which the student will be able to develop critical thinking in the discipline concerned;
• – third, it can be similar to an infusion process, while the students work on domain-specific content, general principles are made explicit by the teacher;
• – the fourth form consists of a mixed education combining general and immersion education, or general and infusion education.

When addressing the question of whether critical thinking should be taught in a general or specific way, a difficulty also lies in the meaning of the terms “subject matter concepts” or “specific knowledge”, as domains vary in size and in the way they are broken down. According to Ennis (2018), the question of teaching critical thinking in relation to content – with a view to fostering transfer – can then only be addressed on a case-by-case basis.

A substantial number of educational programs have been designed with the goal of fostering critical thinking in the classroom. Most of them (Lehman and Nisbett 1990; Marin and Halpern 2011) assume that there is a set of multidisciplinary skills that characterize critical thinking. These programs are designed to enhance the traditional school curriculum, without being directly applied to any particular academic discipline.

These pedagogical devices are considered to be stand-alone forms of education, with the ultimate aim of achieving transfer in a variety of domains and disciplines (Willingham 2008). Some of these devices are based on general situations (Instrumental Enrichment Program; Feuerstein and Jensen 1980), stories containing mysteries (Productive Thinking Program; Covington 1972), or group discussions of everyday problems (Cognitive Research Trust Program (CoRT); de Bono 1985).

Willingham (2008) points out that most studies that measure the effects of these programs have methodological limitations. In particular, outcomes are most often assessed immediately after the program ends; long-term effects can therefore not be measured, given that no time has elapsed between the program and the evaluation. In addition, there is often no control group, and when such a group exists, it is generally passive as it does not carry out any alternative activity. This makes it impossible to distinguish between the effect produced by the content of the intervention and the effect generated by “teacher enthusiasm for a new method”.

Measures of transfer to real-world situations or to situations different from those used in instruction are also quite seldom, and only a small proportion of these studies have gone through the scientific publication processes involving peer validation (Willingham 2008).

Among critical thinking curricula, Halpern (1998) has proposed and tested an explicit learning model that has four components:

• – disposition or attitude development;
• – instruction and skilled practice;
• – teaching for transfer;
• – metacognition.

Marin and Halpern (2011) have assessed the impact of an intervention consistent with this model with high-school students. The experimental group consisted of volunteers who agreed to receive lessons outside of academic instruction. Six lessons were given over a three-week period that included both implicit, content-specific instruction on the psychology of biases and reasoning, and explicit instruction related to how to exploit such knowledge in order to think more critically. Two control groups were also constituted: an active control group which followed an implicit intervention (teaching of a cognitive psychology course with contents overlapping with the previous intervention, but in an implicit way) and a passive control group, without intervention. The experimental protocol followed the classic structure: pre-test, intervention or only usual program and post-test. It turned out that the experimental group (explicit + implicit method) and the active control group (implicit method) improved between the pre-test and post-test, unlike the passive group. It was also found that the experimental group improved more than the active control group.

This pilot study was then replicated in the form of six lessons given in a school setting, twice a week for six weeks. All classes took the pre-test and those in the experimental and active control conditions took an immediate post-test and then a delayed post-test.

The results indicated that more progress was made by the students in the explicit + implicit group compared to those in the implicit only group. The explicit nature of the learning method therefore seemed essential to promote transfer. However, the post-test – even when completed at a later date – was based on the course framework, which potentially contributed to students’ progress. The goal of critical thinking instruction is to ensure that transfer is also successful in everyday situations, and thus “the ideal learning assessment would occur naturally in the course of one’s life” (Halpern 1998, p. 451).

Fong et al. (1986) have attempted this kind of assessment of far transfer. At the end of the year, under the guise of a household survey about sports, Fong et al. (1986) asked students, during their first semester of a statistics course, everyday reasoning versions of statistics questions. The results indicated that in relation to this kind of content, transfer does happen.

Abrami et al. (2015) conducted a meta-analysis of 341 interventions designed to foster critical thinking skills. The lack of consensus on the notion of critical thinking confronted the authors with the diversity of the aims of the interventions, and they chose to identify those lasting at least three hours: to address one of the skills or dispositions defined by the American Philosophical Association (Facione 1990). Still, the interventions included in the meta-analysis present a multiplicity of objectives and means. Some aim to improve philosophical or argumentative skills, others aim to discriminate between facts and inferences, or solve problems in the framework of mathematics or science lessons. Some use graphic organizers, some ask students to conduct investigative reasoning and select newspaper articles.

Another difficulty combines the results of so-called critical thinking interventions represented by the heterogeneous structure of the programs – specific or general content, short or long term, aimed at children, teenagers or adults, etc. Also, assessment methods vary a lot, sometimes relying on standardized tests of critical thinking – such as the Watson Glaser Critical Thinking Assessment, the Cornell Critical Thinking Test, the California Critical Thinking Test – sometimes on non-standardized measures created by teachers or researchers, or sometimes on tests not designed to assess critical thinking, such as academic achievement. A final difficulty is represented by the quality of the tests, which rely most often on the quasi-experimental nature of the interventions: testing in classrooms without control groups or random assignment of participants.

Thus, a large proportion of the studies identified in this meta-analysis compared the results of the pre-tests with those of the post-tests for the same group of participants, making it impossible to control for “test” and “teacher” effects. However, the authors of the meta-analysis chose to take these studies into account, equating the pre-test result with that of a control group, which has the disadvantage of ignoring a possible developmental effect. Notwithstanding these limitations, the authors deduce from their meta-analysis that critical thinking skills can improve, with an effect size of 0.30 for generic skills and 0.57 for specific skills, for all school subjects (no effect of school subject) can be noticed for each grade level (no class impact).

Somewhat surprisingly, the duration of the intervention did not appear to influence the measured effect, while the latter depended on the type of instruction: methods based on teacher-led discussion with the teacher and with peers, mentoring and training in realistic situations (problem solving and role playing), having the greater impact.

Besides, there seems to be an emerging consensus that effective teaching must be explicit about its goals and methods, and rooted in different disciplines to promote transfer (van Gelder 2005; Willingham 2008).

4.3.1. The cognitive building blocks of critical thinking

Beyond the diversity of existing approaches to the teaching of critical thinking, the question arises of plausible psychological mechanisms that are solicited with both sufficient specificity to justify their relevance in the perspective of an intra-domain transfer, and sufficient genericity to transcend the domains. There are several candidates for the role of cognitive “building blocks” of critical thinking. The mechanisms known as “epistemic vigilance” or “selective confidence” (Sperber et al. 2010; Harris and Corriveau 2011) are among them, as are the mechanisms that allow us to adjust our confidence in a given representation, decision or choice.

Epistemic vigilance consists of a set of mechanisms specifically dedicated for assessing the reliability of sources and the credibility of information content. These mechanisms have been highlighted since childhood. Indeed, the 3-year-old child does not limit themselves to taking information in an indiscriminate way, but selects their informants on the basis of criteria – such as familiarity with them. Later, other criteria are added to their palette of source discrimination, such as the general or specific skill demonstrated by the informant and whether the informant is the subject of “popular” consensus or approval (Harris and Corriveau 2011).

These criteria are both discriminating and generic, and are implemented implicitly, involuntarily and at low cost. They, therefore, have an advantage in terms of speed of implementation and selection as a first approximation, but are limited and not very well adapted to complex contexts where more specific criteria would be required (for example, to finely assess the reputation of a source or its competence in the field under discussion). In terms of content, these criteria are also strongly dependent on the knowledge of the subject. Gauging the plausibility of information content means comparing it with one’s knowledge and checking its coherence with it. If the knowledge is inadequate, the plausibility judgment may give results that are not in line with reality.

We can thus affirm that “natural” epistemic vigilance constitutes both a natural cognitive basis for critical thinking, but that this basis is also “naturally” limited and therefore a dedicated education is required for someone to be duly equipped with more refined criteria, adapted to contemporary and complex contexts, regarding both the evaluation of the reliability of sources and the evaluation of the credibility of contents.

As for the specific confidence we have in representations (perceptions, opinions, beliefs) and in decisions or choices, this is also based on mechanisms that are early at work in children (Goupil and Kouider 2019), and that probably run through much of our cognitive functioning, even at very low levels of functioning (Meyniel et al. 2015). Confidence is a fundamental determinant of learning ability (as a challenge to our prior representations), as it is of decision making and of changing ideas, positions or choices (Grimaldi et al. 2015).

A third category of cognitive mechanisms called upon by the exercise of critical thinking is represented by cognitive flexibility as the ability to reconceptualize a situation and to place oneself in a different point of view than the one initially taken. Cognitive flexibility as a capacity to reconceptualize is particularly important in the case of choosing problem-solving strategies and in the capacity – in complex situations – not to dwell on the only information immediately available, but to think of alternative possibilities that deserve attention.

Critical thinking in these cases means being able to include several alternatives to be evaluated in our horizon – without stopping at the first one that presents itself (from the outside) or that comes to mind. For example, when an explanation for a complex phenomenon is given to us, we may evaluate the source of the explanation, the plausibility (according to our previous knowledge) and the relevance of the contents, without being able to think of alternative explanations – possibly more productive. This inability can hinder us when we are looking for the solution to a problem.

In the rest of this chapter (see section 4.3.2), we will therefore focus on this aspect of critical thinking, which concerns the ability to consider productive alternatives in explanations or problem-solving strategies. We will provide a practical example of how a pedagogical device oriented toward the development of this ability can have a positive impact on students’ scientific and mathematical reasoning.

4.3.2. Changing the perspective

Paul (1990, p. 17) argues that an essential component of critical thinking is dialogical thinking, or the ability to “test the strengths and weaknesses of opposing viewpoints”, which is similar to cognitive flexibility, seen as the ability to adapt to new situations. This involves the ability to choose among, and even switch between, several representations of an object or situation in response to environmental development (Jacques and Zelazo 2005; Chevalier and Blaye 2008; Cragg and Chevalier 2012).

This flexibility is then characterized by the voluntary shift of attention from one category of stimuli to another, or from one cognitive process to another (Miyake et al. 2000; Clément 2009); the ability to switch our attention between several tasks is referred to as switching or, alternatively, shifting. However, the form of flexibility that seems to play an even more prominent role in critical thinking is conceptual flexibility.

Indeed, the challenge in complex situations is to be able to change the point of view, to vary the prism through which a situation is analyzed; it is the ability to reconceptualize.

Consider a problem studied by Posner (1973):

Two stations are 50 miles apart. At 2:00 p.m. one Saturday afternoon, two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When it reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet?

As pointed out by Novick and Hmelo (1994), this problem, which we invite the reader to try to solve before continuing reading, seems complex and may require some equations. This is because the reading of the statement induces a certain point of view, that of the bird coming and going (see Figure 4.1(a)), which leads to trying to calculate the distance traveled by the bird when it first encounters a train, then when it encounters it for a second time, then for a third time, and so on, until the trains pass each other.

This strategy does not easily lead to the result. However, an alternative point of view radically simplifies the solution, that of the path traveled by the trains (see Figure 4.1(b)). The solution then becomes obvious: since the distance between the two stations is 50 miles and the trains travel at the same speed of 25 mph, they will pass each other after 1 h, when each has traveled 25 miles. Since the bird kept flying at 100 miles during that hour, it will have traveled 100 miles. When adopting this alternative point of view, elementary calculations are sufficient to solve the problem.

In this example, being flexible means being able to move from the point of view induced by the statement (that of the bird) to another point of view (that of the trains). Although this alternative point of view is less salient – almost no one considers it at first – it is much more fruitful for reaching the solution. The underlying question is to understand the forces that make this flexibility possible. What makes it possible to move from one point of view to another? What hinders it?

4.3.3. The role of metacognition

Is it possible to be conceptually flexible without requiring metacognitive skills? At first sight, being able to change our point of view requires us to be aware of our initial point of view, its components and its limits, in short, being aware of our own thinking. However, metacognition is precisely conceived as an individual’s faculty to appreciate, evaluate and estimate the validity of their own thinking (see Chapter 3).

Metacognition appears as an intentional state where a person reflexively takes into account their her own mental states. It is a form of second-order cognition, that is, composed of cognitive states that have as their object other cognitive states belonging to the same individual. Metacognition has the dual function of monitoring and controlling mental states (Dunlosky and Metcalfe 2008).

In addition, a distinction is also classically made between meta-cognitive knowledge (“I know what cognitive state I am in, I understand my cognitive abilities, I know what allows me to improve them”), meta-cognitive strategies (“I set up actions that will allow me to improve my performance, or at least to have control over them”) and meta-cognitive experiences or sensations (“I experience the sensation of being close to the goal, or on the contrary, of experiencing difficulty during a task”) (Flavell 1979; Efklides 2011).

However, the notion of metacognition can be considered beyond this reflective metacognition, which is based on awareness. Indeed, studies show that explicit, verbalizable metacognition gradually matures during childhood (Palmer et al. 2014). For example, the ability to estimate the difficulty of a task improves during the early years of schooling. The sense of knowing how to evaluate the quality of learning also improves, albeit in a less pronounced way, while the development of appropriate strategies in relation to the difficulty of the task is only observed at the end of elementary school.

However, much younger, preschool-aged children are also successful in estimating their level of knowledge, searching for missing information, or refraining from giving an answer when they are uncertain. For example, children aged 3 to 5 can learn to use a non-verbal confidence scale (pictures of confident or doubtful children) and express confidence in information in their possession. If they are asked to answer questions and express a confidence judgment about the answers given, a positive correlation between correct answers and confidence in those answers is observed, as reported by Goupil and Kouider (2019). The authors conclude that a metacognitive sensitivity is therefore already present at this age; it is an explicit level of evaluation, even if it is not verbal.

Other paradigms have been transposed from work in animal cognition to assess the preverbal implicit abilities of very young children. This involves comparing a situation of forced choice between options with another situation in which no choice is possible.

In the latter case, deciding not to choose is advantageous when there is uncertainty, as a small reward is given. It has been shown that when appropriate, the choice-denial option (when we are uncertain and know that we may make a mistake in choosing either option) is already appropriately chosen by 20-month-old children (Goupil and Kouider 2019). For example, children were asked to remember where a toy was hidden and the experimenter manipulated the time between when the toy was hidden and when the child had to find it. Children in the control group were required to indicate the location where the toy would be hidden. The children in the experimental group were allowed to ask for help from an adult and thus not decide for themselves where the toy was hidden.

The results show that children in the experimental group made fewer errors than the control group because they were able to ask for help when they anticipated their uncertainty. Thus, there are indicators that from the age of 20 months, children have varying degrees of confidence in decision making that are positively correlated with decision accuracy.

It has also been shown that at school age, differences in metacognitive strategies are correlated with grade level (Sangster-Jokic and Whitebread 2011). For example, Efklides et al. (1999) showed that students with lower grade levels had greater difficulty using metacognitive strategies than academically high-performing students. A correlation also exists between the metacognition level and academic level in general (Kuyper et al. 2000), as well as with the math level (Desoete and Veenman 2006; Stillman and Mevarech 2010).

4.3.4. Barriers to flexibility: the role of intuitive conceptions and inappropriate categorizations

Changing our perspective on a situation, however, can be extremely difficult. Indeed, in many situations, individuals remain fixed on their initial point of view and a phenomenon of insight, or sudden change of perspective, which only occurs after a long time of exploration, and is then still necessary for a minority of people.

Gestalt psychology has even placed this phenomenon at the heart of the problem-solving process. The candle problem, developed by Duncker (1945), highlights fixation effects. A candle, a matchbook and a box of tacks were presented to participants, and the experimenter asked them to fix the candle to the wall on a cork board without the wax leaking.

In the condition where the box of tacks was filled with said tacks, many adult participants failed to spontaneously use the box as a medium for attaching the candle to the wall, unlike the condition where the tacks were presented outside of the box. The box was thus more difficult to consider as a medium when its usual function as a container was salient. Knowledge about the uses of familiar objects can thus induce a point of view that hinders the resolution. The phenomenon of functional fixity consists of attributing only one prototypical function to an object. Numerous works show the difficulty of diverting an object from its usual function and making an atypical use of it.

In general, overcoming a spontaneous point of view is made all the more difficult when this point of view is based on intuitive conceptions. These intuitive conceptions are constructed on the basis of previous daily experiences (Vosniadou 1994; Clément and Richard 1997; Carey 2000; diSessa et al. 2004; Keil 2011).

Whether we speak of preconceptions (Ausubel 1968), misconceptions (Clement 1982), naive theories (Carey 1985), naive knowledge (Fischbein et al. 1989), tacit models (Fischbein 1989), naive reasoning (Reiner et al. 2000), conceptual schemas (Thompson and Saldanha 2003) or knowledge in piece (diSessa et al. 2004), it is common that a certain notion is conceptualized from prior knowledge. These result in a restrictive conception with respect to the notion concerned (Lautrey et al. 2008).

Thus, intuitive conceptions have a domain of validity that justifies their existence and persistence, but they prove to be limiting, so it is crucial to identify them in order to then implement reconceptualization work. These intuitive conceptions concern all of the notions that are taught in school (Sander 2017) and beyond. For example, an intuitive conception related to causality is that any correlation reflects a causal relationship. Therefore, as soon as two events follow one another, a person tends to see a causal relationship (Michotte 1946).

Going beyond such a conception is possible by developing the concepts of coincidence and of the confounding variable. Indeed, if one event precedes another, this can also be the result of a mere coincidence, or the consequence of the fact that a confounding variable causes both of these events.

In the field of elementary mathematical learning, it has been shown (Fischbein 1989; Sander 2008, 2016) that each notion is attached to an intuitive conception (e.g. “to subtract is to look for what remains after having taken away”, “to add is to look for the result of an addition”, “to multiply is to add repeatedly”, “to divide is to look for how much each one receives in the context of equitable sharing”), and that the compatibility of a situation introduced in class with the intuitive conception of the notion is an important factor in the difficulty of problem solving. Far from being eradicated by education, these intuitive conceptions are robust: their influence remains after schooling, as has been shown with all adults and even teachers (Tirosh and Graeber 1991; Gvozdic and Sander 2018).

The constraining character of intuitive conceptions on the points of view adopted on situations is to be compared to the phenomenon of miscategorization (Chi 2009), that is, to an inappropriate categorization. Indeed, when faced with a newly encountered situation, a novice who does not have an adequate category at their disposal will assign it to the spontaneously evoked category, which is potentially irrelevant to the needs of the task (Chi 2009), whereas, conversely, a relevant categorization proves fruitful: children between the ages of 4 and 7 who like dinosaurs can make appropriate inferences about unknown dinosaurs, once they have categorized them on the basis of surface features (Gobbo and Chi 1986).

Thus, assigning a certain object to a category allows one to infer properties that will be correct if that categorization is relevant, or inappropriate if it is inadequate (Vosniadou 2012). But this boundary between appropriate and inappropriate can be more blurred in some contexts, and sometimes a categorization that is wrong on some aspect can still allow for relevant inferences. This is observed in young children, through semantic approximations. For example, a 3-year-old child may say “undress the orange” instead of “peel the orange” (Duvignau et al. 2010). This choice of word may seem incongruous to an adult, as it reflects an idiosyncratic categorization of the child.

Indeed, the latter relies on a previously constructed category, that of undressing, and it is thanks to this category that the child gives meaning to this situation. This concept, which leads this child to consider that before eating a piece of fruit they must remove what covers the fruit, which is supposedly not a part of it, in the same way that clothes are not really part of a person, is quite useful from a food point of view, even if it is unfounded on a biological level.

On the other hand, certain categorizations can lead to profound misunderstandings: this is particularly true in the school context, where these categorizations result from the fact that a student assigns the concept to an inappropriate category. Therefore, to better understand a concept, they must make a categorical change.

Such a categorical change is made possible by the student’s awareness that the change is necessary and by the availability of the appropriate category. For example, whales are often categorized as fishes by students, and a category change can be easily introduced once the student has already constructed the mammal category and is told that whales belong in that category.

These categorical changes can be more difficult if they are miscategorizations that directly involve ontological categories, such as between entity and process (Henderson et al. 2017). For example, a human being is an entity that has attributes of mass, height, weight, etc.

In contrast, the biological evolution of human beings belongs more to the category of processes, which are characterized in particular by their temporality. When students encounter an instance of a concept with which they are not familiar, they tend to categorize it as an entity. For example, heat is often described as an entity – hot particles – by students and not as a process. Such miscategorization is difficult to overcome.

If students are presented with information that is contradictory to their categorization, they will reject that information (Chi 2013). A limitation then comes from a difficulty in switching categories to a lateral category. Overcoming miscategorization then relies on the ability to recategorize. However, this recategorization is difficult, because individuals have little awareness that their errors may be induced by inappropriate categorization, because of the reliability of categorizations used in everyday life and the irrepressible nature of spontaneous categorizations. It is then beneficial for the student to be aware that change is necessary and to have built the appropriate category for the situation (Chi 2009). In order to overcome the limitations of an inappropriate categorization, it is therefore usually necessary to recategorize in order to reach an alternative categorization. At the heart of this possibility lies the mechanism of multiple categorization.

4.4.1. Multiple categorization

Having multiple representations for the same object or situation makes it easier to move from one category to another in order to choose the most relevant point of view for a given situation (Chi 2009; Hofstadter and Sander 2013).

Being able to do this fluidly seems to be an indicator of expertise. Thus, the more diversified an individual’s repertoire of possible categorizations, the more they are able to adopt different perspectives. This means being able to both group or distinguish in a relevant way depending on the situation.

It is by articulating different points of view on the same situation that the individual can embrace its complexity, and this multiplicity of points of view guarantees adaptability: for example, the dining room chair can, depending on the situation, be categorized as a seat during a meal, as a stool to change a light bulb, as a wedge to prevent a door from slamming shut, as a den for a young child, as a clothes rack when a jacket is placed on its back, etc. A single chair can, therefore, be categorized in multiple ways. Although one of these categories dominates (seat), the chair ultimately has no fixed identity: depending on the perspective taken on an entity, its identity changes (Hofstadter and Sander 2013).

Work on categorization contrasts deep structure and surface features of a situation as two of the possible directions for categorizing it. Two situations have the same deep structure if they invoke the same relation, even if they are superficially dissimilar (Gentner and Kurtz 2006).

For example, the category “bridge” is vast since a shared property is that of connecting two things, which can be two places, two concepts, etc. In that respect, a wooden bridge over a stream has the same relational structure as a dental bridge. These two elements belong to the same category; they connect two locations, physical or symbolic, although their surface features are not the same (wide and made of wood for one; thin and made of metal for the other).

Therefore, each belongs to different categories in terms of their surface features (in this example the categories wooden objects or metal objects). Surface features are easily perceived, while deep features cannot be perceived directly (Chi and VanLehn 2012). One of the purposes of teaching is to help students succeed in transferring knowledge learned in one situation to another situation. To do this, they must be able to recognize that two situations with different surface features actually belong to the same category in terms of deep structure, for example, that two problems that involve shopping in a bakery in one case and counting points in a sports competition in the other, nevertheless, have a common mathematical structure. Transfer is thus largely dependent on the ability to perceive a common structure between a new situation – called “target” – and a previously encountered situation – called “source”, whereas the surface characteristics may be different (Richard and Sander 2000; Chi and VanLehn 2012).

Perkins and Salomon (1989) express this centrality of transfer as an educational goal:

The database students acquire in school ought to inform their thinking in other subjects and in life outside the school (p. 23).

And yet, teaching this ability to transfer is often found to be a failure. Halpern (1998) applies these considerations to critical thinking and explains that students should be taught transfer skills. When faced with a new situation, an expert, in contrast to a novice, is able to abstract from the surface features of a situation, in order to mobilize their repertoire of alternative categorizations.

For example, in physics, novices categorize problem statements according to the objects present in the statement and thus assign pulley and inclined plane problems to separate categories, whereas experts categorize the same statements according to the physics’ principle to be applied – Newton’s third law for example (Chi et al. 1981).

Chi and VanLehn (1991) also showed that the most successful students are those who categorize statements according to solution principles. Novices, unlike experts, construct their categories primarily on the basis of superficial information, such as specific objects, terms used, and question form (Schoenfeld and Herrmann 1982). To be an expert in a domain requires a fine and complex categorization, allowing for abstract schemes including solving strategies (Dupuch and Sander 2007).

In developmental psychology, many studies show that the level of expertise is a predictor of performance that goes well beyond general indicators of development. Indeed:

When the conceptual knowledge of participants is controlled – what is sometimes called the level of expertise – relative to specific domains – soccer, chess, dinosaurs – it becomes impossible to show any development: those who know more do better than others, at any age (Fayol and Monteil 1994, p. 91, author’s translation).

In addition, a second facet can be considered: one facet of expertise is the ability to vary categorizations. Being able to adopt a multiplicity of categorizations allows for a change of perspective depending on the needs of the task and the situation (Sander 2017).

For example, a physicist who sees a glass falling does not need to categorize the glass as a body subject to the law of gravitation and on which forces are exerted; a simple common sense categorization is sufficient. Categorizing the glass only as an object made of a fragile material is sufficient to act in the appropriate way, namely to catch the glass. In other contexts, however, the common sense categorization will be inadequate, and the same physicist will have to rely on these scientific categorizations to tell themselves, for example, that the perception they have of the sun they are seeing rising is in fact the result of a rotational phenomenon.

4.4.2. Operationalization through research in a school context

A study in an ecological setting – in this case a school setting – Rai’Flex (Scheibling-Sève 2019; Scheibling-Sève et al. 2019), was conducted to uncover the role of multiple categorization in the development of critical thinking. It aimed to introduce multiple categorization instructional activities and to assess the effect of these activities on students’ success in perceiving successful alternatives to their initial intuitions.

More specifically, the working hypothesis was that practicing categorizing in different ways allows us to better grasp the structure of a phenomenon or a problem, and thus not to restrict our perception to intuitive conceptions. To test this hypothesis, an experiment was carried out in a school environment, with 600 pupils in the fourth and fifth grades, in three types of schools (priority education, privileged, mainstream), divided in half between an experimental group and a control group. The pupils of the experimental group were involved in a pedagogical intervention designed to develop critical thinking in the field of proportional reasoning and causal reasoning, through the exercise of multiple categorization. The 300 students of the experimental group followed 12 sessions on proportional reasoning and 12 sessions on causal reasoning. The sessions, each lasting one hour, were included in the hours dedicated to mathematics and science by the teachers. Depending on the group of students, the sessions were conducted either exclusively or mostly by the experimenter. The students in the control group did not receive any intervention, but proportional and causal reasoning were also covered since it is a part of the curriculum. All students were given a pre-test in the first quarter and a post-test at the end of the school year.

The Rai’Flex intervention program was based on four principles. First, each session focused on one part of the curriculum and the associated intuitive conception. For example, one intuitive conception that plays a deleterious role in proportional reasoning is to conceive the fraction as a bipartite structure (a over b) and not as a magnitude or ratio (a/b) (Bonato et al. 2007).

For causal reasoning, a widely shared intuitive conception is the illusion of causality (Michotte 1946): as soon as two events follow one another, the tendency is to see a cause and effect relationship. The aim is to guide the students toward an awareness of their intuitive conception and the adoption of a conception more in line with the academic notion.

To do this, the second principle is to guide the student toward a recategorization of the situation: to adopt a coding detached from the salient but superficial features of the situation. During the sessions, the students were encouraged to recategorize the situation by making a change in point of view explicit. The students were thus led to adopt several points of view on the same situation.

For example, with regard to proportional reasoning, in order to promote mastery of the distinction between additive and multiplicative structures, students learned to distinguish the additive comparison “more” and the multiplicative comparison “times more”. Students were thus led to use the point of view “more”, “less”, “times more” and “times less”. For example, two points of view can be taken on the sentence: Jena has 15 marbles and Mateo has five marbles. Taking Jena’s point of view, we can conclude: Jena has three times more marbles than Mateo. From Mateo’s point of view, we can conclude: Mateo has three times less marbles than Jena. In this way, the students were led to work on the reciprocity of multiplication and division, which appears crucial in the construction of the concept of ratio. The same work was also done to distinguish “more” and “less”. Indeed, students have difficulty distinguishing between multiplicative and additive structures, due to intuitive conceptions of repeated addition for multiplication or fair sharing for division (Fischbein et al. 1989).

Regarding causal reasoning, an intuitive conception is that an event is either a cause or an effect. The aim of the sessions is to encourage the construction of the knowledge that a same event can be both cause and effect. In this perspective, students were led to adopt the point of view of the cause or the effect.

In a linear causal chain, A ➜ B ➜ C, the point of view “B is the effect of A” can be adopted just as the point of view “B is the cause of C”. Depending on the point of view adopted, B is therefore either a cause or an effect. Thus, to the question “Why did event C occur?” students were led to develop a more complex reasoning than the simple direct cause because of event B, and to propose an indirect causal chain: it is indirectly because of A and directly because of B.

The third principle is to diversify learning contexts in order to facilitate transfer. To be able to transfer autonomously to another learning context achieved in a given context is an essential learning objective, which demonstrates a certain mastery of a concept: regardless of the non-shared surface features, the student identifies a deep structure and refers to a concept learned in another context.

Like others (Gick and Holyoak 1983; Perkins and Salomon 1989; Bransford et al. 2000), Halpern (2013) proposes that to promote transfer, it is important to guide practice of the same reasoning across a variety of content. During the learning sessions, the analogy between the situations introduced in a same session was made explicit: students were asked to note the use of an identical reasoning in a new context. They were thus invited to transfer the same reasoning between problems of the same structure.

Finally, the last principle is based on explicitness. A number of works converge to conclude that explicit methods lead to more performance than implicit methods (Alfieri et al. 2011; Tricot 2017).

This principle was implemented in several ways. First, the points of view likely to be adopted by the students in the context of the tasks concerned were explicitly introduced and even lexicalized (“times more” point of view, “cause” point of view). Second, the objective of succeeding to transfer similar reasoning to different contexts was also made explicit during the sessions. Finally, the students were asked to give their own explanations at the end of the session. This was more of a metacognitive aspect, as the students were encouraged to explain their first understanding.

In math, students individually answered the question “what did I learn?” at the end of each session, and worked on their research journal. Inspired by the “number journal” (Sensevy et al. 2013), the purpose of the research journal was to allow students to write math at their level. They were given open-ended statements, such as “12 is 9 plus 3” (12 = 9 + 3), “12 is also 2 times 6” (12 = 2 × 6), etc., and then were asked to “observe and do the same thing with another number”. In science, the objective was to revisit a belief familiar to the students, again using the point of view adopted during the session (e.g. analyze the idiom “going under a ladder brings bad luck” through possible causal chains).

The results of the experiment were very favorable. The students who followed the Rai’Flex program obtained higher scores than the control students, both overall and for each form of reasoning, causal or proportional. Moreover, these results were confirmed at more detailed levels of analysis: students in each experimental subgroup – priority education, privileged, mainstream – obtained significantly higher scores than the corresponding control subgroups. These same differences were observed among both the fourth-grade and fifth-grade students. In addition, the fourth-grade students in the experimental group scored similarly or better than the fifth-grade students in the control group. The linear regression analysis also showed that the Rai’Flex program was a significant predictor of post-test performance: it allowed students to make more progress than control students. The effect sizes were intermediate to large and corresponded to a learning gain of between seven and nine months.

In addition, the differences between school types were reduced. In particular, the students in the experimental group increased their ability to solve problems that were incongruent with the intuitive conceptions. They thus developed their flexibility by being less dependent on the surface features of a problem for elaborating a solving strategy. Therefore, the Rai’Flex intervention seems to have had a favorable influence on the development of critical thinking in mathematics and science.