Conceptual Change and Education

Problems of Comprehension

Problems of Knowledge Mastery

Other difficulties in learning knowledge requiring CC are not related to difficulties of understanding but of mastery. New academic knowledge requiring CC can rarely be mastered without first becoming acquainted with the modes of reasoning it requires (Amin & Levrini, 2018; Chinn, 2018). Research has identified the difficulties novice learners face compared to domain experts and often highlights the additional learning difficulties children face. In contrast to novices, experts have a high level of prior knowledge, experience, and mastery of the reasoning patterns necessary to adequately interpret the knowledge involved. Interpretations of the fundamental differences between novices and experts regarding the issues of CC vary between theories. Fragmentation theorists interpret them primarily as a difference between experts’ stable and systematic mastery of a highly coherent body of academic knowledge as opposed to novices’ more unsystematic and highly context-sensitive reliance on self-explanatory principles (i.e., fragmented knowledge; diSessa, 2014, 2018a). Coherence theorists highlight the ontological or epistemological presuppositions providing different explanatory frameworks between novices and experts, which are systematically activated. Framework theory and ontological theory state that novices, especially children, tend to be unaware of the assumptions that constrain their reasoning, including their ontological or epistemological presuppositions, or they lack the required modes of reasoning. These deficits concern ontological categories, according to ontological theorists (Chi, 2005; Chi et al., 1981), and the thinking patterns required by academic knowledge, such as systematic and intentional hypothesis testing, according to framework theorists (Kyriakopoulou & Vosniadou, 2020; Vosniadou & Skopeliti, 2014). Fragmentation theorists, by contrast, have criticized claims of radical differences between novices and experts or between children and adults (diSessa, 2017, 2018a; Smith et al., 1994). They have documented many situations where preinstructional, self-explanatory principles are included in or share significant similarities to expert-type reasoning and play a facilitating rather than blocking role in their realization.

Research has also shown that even when students, especially children, understand new knowledge that requires CC, they often face difficulties in mastering it. They may mix adequate modes of explanation with familiar but inadequate prior modes of explanation or may alternately and unstably rely on one and then the other depending on the context (diSessa, 2017, 2018a; Kyriakopoulou & Vosniadou, 2020). diSessa’s microgenetic studies have particularly revealed that students can switch from one explanatory principle to another depending on the context. For example, in an interview, a student was asked, “What happens when you toss a ball into the air?” (diSessa, 2018a, p. 70). She first offered a correct normative response: there is only one force, gravity, which slows the ball down. Once the interviewer asked, “What happens at the peak of the toss?” she began to reformulate her model and introduced an upward force equal to gravity that comes from the hand that overbalances gravity and reaches an equilibrium at the top, and then gravity takes over on the way down. In the next sessions, which included instructional sequences, the subject gave an almost perfect physical explanation but, after being asked subsequent questions, returned to the erroneous two-force model (diSessa, 2018a, p. 70–71). Table 1 summarizes the CC learning difficulties imposed on education.

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Educational Recommendations to Overcome Comprehension Difficulties

To address the challenges teachers face in facilitating CC, proponents of the different CC research trends have provided a series of educational recommendations. One of the overarching educational goals, which inspires the first set of recommendations, is to help students appropriately relate their prior knowledge to new knowledge to be learned. To this end, research converges on the need to first identify the student’s relevant prior knowledge and, according to coherence theorists, its implication for coherently interconnected causal attributions and categorization relationships. The recommendations for relating prior knowledge to new knowledge differ depending on the approach (Mayer, 2002). Fragmentation theorists favor the identification of prior knowledge and familiar patterns of reasoning that could be used productively by being refined and reorganized through their integration into new, more systematic, and coherent academic knowledge (diSessa, 2017, 2018a). In other words, this line of research focuses on building on prior knowledge to foster CC by providing continuity with the new knowledge to be learned. Returning to the abovementioned incorrect explanations of force, e.g., students see balancing at the peak of the toss. But “balancing is a rough version of an incredibly important principle in physics, conservation of energy. Similarly, the upward ‘force’ in the incorrect explanation is not absent, but it is what physicists call momentum” (diSessa, 2014, p. 89–90).

By contrast, coherence theorists have focused on making students aware of and correctly understand the differences between prior and new knowledge to further highlight their discontinuity (Chi, 2005, 2013; Henderson et al., 2018; Vosniadou, 1994, 2019). Framework theorists and ontological theorists have also insisted that education should be adapted to first change prior knowledge that constrains the CC in connected or subordinate bodies of knowledge. For example, according to ontological theory, when children make a category error, such as confusing whales with fish, education must first make them aware of this error at the categorical level (e.g., by pointing out that whales do not breathe through gills but through a blowhole) to induce a categorical change through which the conceptual and causal attributes associated with the other category can then be attributed to the concept (Chi, 2013). In science learning, categorical shifts often require helping students to overlook superficial perceptual similarities at the pattern level (Chi, 2013). As for framework theory research, to build on the example of the astronomical concept of the Earth, to make this concept understandable, education should at least start by prioritizing children’s understanding of how people can stand on Earth’s bottom and side without falling off (i.e., explain gravity) or why Earth is round even though they perceive it as flat (e.g., by demonstrating with a large round object that it can appear flat from the perspective of someone standing on it; Vosniadou et al., 2001).

Educational Recommendations to Overcome Problems of Knowledge Mastery

Mastery of new knowledge that requires CC often requires mastery of specific modes of reasoning, and students, especially children, often struggle with this. Therefore, another overarching goal, which is accompanied by another set of educational recommendations, is to develop mastery of reasoning. Two general recommendations for achieving this mastery are shared in CC research. The first is to develop students’ abilities to differentiate between, relate, and coordinate their prior habitual modes of reasoning or explanatory principles and the new academic modes (Chi, 2013; diSessa, 2017, 2018a; Kyriakopoulou & Vosniadou, 2020; Treagust et al., 2018; Vosniadou & Skopeliti, 2014). The second recommendation is to increase the consistency and systematicity of use by students (Chi, 2013; diSessa, 2017, 2018a; Kyriakopoulou & Vosniadou, 2020; Treagust et al., 2018; Vosniadou & Skopeliti, 2014). Satisfying these two recommendations for achieving mastery of reasoning contributes to refining students’ epistemologies, i.e., their conceptions of the difference of nature between their own preinstructional knowledge and the new academic knowledge. To meet these recommendations, specific strategies differ between approaches.

To increase students’ abilities to differentiate between, relate, and coordinate their prior and new ways of thinking, diSessa’s (2017, 2018a) knowledge-in-pieces theory emphasizes students confronting a variety of diverse contexts in which they learn different ways of interpreting the same concepts. This develops knowledge transfers that account for – rather than neglect – contextual variation (Wagner, 2006). Knowledge-in-pieces theory particularly emphasizes the need to develop coordinated, consistent, and systematic use across and within diverse contexts, as it considers that it is the very medium that induces CC. Framework theorists rather emphasize the development of the intentional (i.e., conscious and deliberate) metacognitive mastery of executive functions (working memory, shifting, inhibition) to increase conscious mastery of scientific reasoning and to inhibit the spontaneous reliance on everyday knowledge (Kyriakopoulou & Vosniadou, 2020; Vosniadou, 2019). Framework theorists also emphasize the importance of students understanding that their prior modes of reasoning are not false but are only one perspective (e.g., an empirical perspective) among others (e.g., scientific perspectives; Vosniadou, 2019; Vosniadou & Skopeliti, 2014). Ontological theorists focus on students confronting (directly or indirectly) their misconceptions to help them revise and replace them (Chi, 1992, 2005, 2013). They assume that most or all preinstructional knowledge impedes CC because it is misconceived or lacks accurate categories. Thus, they privilege directly teaching the new categories to master the type of reasoning required. Table 1 summarizes the general educational recommendations from the three CC research trends.

Theory of CC in Education from a DT Perspective

According to Vygotsky (1934/1987), academic or scientific concepts are understood in a broad sense, not limited to the natural sciences, as a set of abstract, precisely defined features, and relations linked systematically to one another and to other concepts by objective and causal-logical relations, especially relations of generality. In contrast to academic concepts, concepts formed in everyday life during recurrent cultural and empirical experiences – i.e., “everyday” or “empirical” concepts – tend to be rich in personal and concrete experiences, less precise and fixed, used spontaneously and without reflection, and mostly used without being based on systematic causal-logical relationships. Vygotsky hypothesized that the mutual discrepancies between academic and everyday knowledge make it difficult for students to establish appropriate relationships for mastering academic knowledge, especially for children; however, these discrepancies also allow for important CCs through systematic education. Vygotsky and DT researchers rejected the views that preinstructional empirical knowledge and academic knowledge are antagonists and that academic knowledge must replace preinstructional empirical knowledge. Indeed, they considered that preinstructional empirical knowledge is not wrong or misconceived (contrary to ontological theory) but valid in certain contexts and for certain cognitive goals and types of understanding (Arievitch & Stetsenko, 2000; Davydov, 1972, 1999; Gal’perin, 1969/1989a), in line with framework theorists and especially fragmentation theorists.

Vygotsky assumed that academic concepts become meaningful and concretized when one reflects on everyday concepts and integrates them into their well-defined system of explanations and generalization. The CCs that result, for Vygotsky (1934/1987) and DT researchers, range from significant changes in systematicity, coherence, and precision, including higher levels of generalization and abstraction, to radical changes in understanding (Davydov, 1972/1999; Gal’perin, 1969/1989a). The latter include: (a) discovering systematically interrelated essential properties of a series of objects, phenomena, or events and (b) thinking systematically and independently on the basis of these essential properties. This independent thinking is related to an increase in intentional metacognitive mastery when learning academic knowledge (Davydov, 1972/1999; de Araujo et al., 2020; Vygotsky, 1934/1987). Metacognitive mastery is considered important for reasons similar to those of framework theorists: because it increases awareness of one’s own thinking processes and the control over spontaneous reliance on everyday knowledge. Thus, Vygotsky opened a line of research focusing on the potential of instruction to foster significant CCs, especially in children, which DT researchers continued.

DT researchers consider that the specificity of academic concepts, which they call theoretical concepts, is the theoretical mode of thinking their content requires: to explain – via specific causal-logical relationships – the origin, genesis, and functioning of various concrete phenomena or objects based on a core set of properties systematically interrelated (Arievitch & Stetsenko, 2000; Davydov, 1972/1999). For instance, the theoretical concept of life in biology is based on the core interrelation between an organism and an environment, which enables one to explain theoretically the diversity of biological phenomena (Ilyenkov, 1960/1982). This view of academic knowledge contrasts with Vygotsky’s emphasis on abstract classificatory relations of generality and his lack of attention to theoretical mental methods of analysis (Davydov, 1972/1999; Karpov, 2003). DT researchers developed educational approaches that aim to develop theoretical thinking in every school subject to foster students’ conceptual development, including by overcoming CC challenges (Davydov, 1972/1999; Gal’perin, 1969/1989a).

Vygotsky is particularly known for his hypothesis that higher psychological processes (e.g., the ability to think with academic knowledge) emerge from social interactions in societal practices (for details, see Chaiklin, 2015). However, the sociocultural and interactional levels were of interest to Vygotsky (1934/1987) and DT researchers to the extent that they are translated, via an internalization process, into individual mental processes and capacities (Davydov, 1972/1999). For instance, DT researchers emphasized that specific types of school education (types of learning and teaching, cognitive operations performed, contents and materials involved, educational interactions) inform specific cognitive developments. Yet, many research trends appropriated Vygotskian theories by prioritizing the sociocultural or interactional levels over – and often at the expense of – the individual mental level, including: (a) social constructivist education (Amineh & Asl, 2015; Palincsar, 1998), (b) situated cognition approaches of Lave and Wenger’s (1991) community of practice theory or of Rogoff’s (1990) learning by apprenticeship theory, and (c) the self-proclaimed third and fourth generations of cultural-historical activity theory (Engeström & Sannino, 2020).

The Reasons for the CC Learning Difficulties from a DT Perspective

DT approaches were significantly designed on the basis of analyzing students’ typical learning difficulties in school, especially those of children, including CC problems (Arievitch & Stetsenko, 2000; Davydov, 1972/1999; Gal’perin, 1969/1989a, 1978/1992; Karpov, 2003). Although DT researchers did not use the following terms, they found evidence about the persistence of misconceptions and the formation of new misconceptions or inconsistent understandings, including sequences of progressive CCs and what framework theorists call synthetic misconceptions, as well as miscategorizations of the types studied by ontological theorists (Davydov, 1972/1999, Gal’perin, 1969/1989a). DT researchers also documented the unstable or insufficient mastery of modes of thinking that require CC in different subject domains, including mixing preinstructional and academic modes of explanation, alternately relying on one and then the other, or relying unsystematically on the adequate mode of reasoning depending on the context (Davydov, 1972/1999).

DT researchers considered that these learning problems arise because of educational approaches that fail to develop theoretical thinking and because preinstructional empirical knowledge negatively influences learning (Davydov, 1972/1999; Gal’perin, 1969/1989a, 1978/1992). They assumed, like ontological theorists, that one empirical thinking tendency, in particular, impairs CC: the spontaneous tendency of students, especially children, to rely on superficial perceptual features, particularly salient and visual features, in contrast to relying on essential properties (Davydov, 1972/1999; Gal’perin, 1969/1989a). Thus, DT researchers emphasized: (a) the qualitative differences between prior and new knowledge, (b) a systematic and negative influence of a few preinstructional thinking tendencies on CC, and (c) students’ lack of adequate modes of thinking. DT researchers’ interpretations of the difference between prior and new knowledge and of the resulting CC problems consequently align with coherence theories.

DT criticisms of the failures of other educational approaches to overcoming CC-related problems highlight approaches that insufficiently help students bridge prior and new knowledge (e.g., conventional transmissive approaches; Arievitch & Stetsenko, 2000; Davydov 1972/1999) and approaches that insufficiently make understandable the discontinuity between prior and new knowledge by relying primarily on students’ prior knowledge or self-directed reasoning, including constructivist educational approaches (Cobb et al., 1996; Davydov, 1972/1999; Giest & Lompscher, 2003; Karpov, 2003; Schmittau, 2004). I will show that DT approaches provide strong educational guidance through specific learning tasks, which are designed and sequenced to make students understand unambiguously one specific theoretical meaning (Davydov, 1972/1999; El’konin, 1975; Gal’perin, 1974/1989c). This contrasts with the tendency of constructivist educational approaches, which are not necessarily related to CC research, to stimulate primarily students’ own interpretations and negotiations of meaning by relying primarily on their self-directed reasoning, own experience, or familiar preinstructional modes of thinking (Cobb et al., 1996; Giest & Lompscher, 2003; Karpov, 2003; Schmittau, 2004).

Educational Recommendations to Foster CC from a DT Perspective

Methods for Mastering the New Knowledge

Modeling the Theoretical Knowledge. DT approaches require the use of visual aids that reflect and model the core conceptual interrelations to help their generalization, abstraction, and mastery and to develop systematic model-based theoretical thinking. In El’konin-Davydov’s approach, a visual model is elaborated through learning tasks and represents the most fundamental conceptual interrelation or germ cell in a subject area (Davydov, 1972/1999). When the germ cell is modeled, students learn to derive its general and particular forms and instantiations and the theoretical principles that derive from it so that they are all integrated into a unified conceptual system. This learning process, which moves from the most general conceptual interrelation to various general and particular instantiations (an ascent from the abstract to the concrete, in Davydov’s terminology), helps students further master the germ cell, its explanatory potential, and limitations. The emphasis on developing knowledge transfers that account for contextual variations in order to develop systematic uses of knowledge is shared by the knowledge-in-pieces theory.

Vary the Types of Instantiation and Representation of the Core Conceptual Interrelations. The elaboration and use of the model take place through varying and contrasting the types of instantiation and representation of the core conceptual interrelations to strengthen their abstraction and generalization (Davydov, 1972/1999; Gal’perin, 1978/1992). In Gal’perin’s method of stage-by-stage formation of mental actions and concepts, learning progresses systematically by following specific stages. They are: (1) learning tasks that imply either hands-on manipulation of physical objects or their visual representations, which must, in either case, help reflect the core conceptual interrelations; (2) reasoning verbally and then internally without external support; and finally (3) mastering “purely” mental reasoning. Table 2 summarizes the CC challenges in education and educational recommendations to overcome them from a DT perspective and the divergences and convergences with the three CC research trends.

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Example 2: The Concept of Real Number

This example illustrates the aforementioned educational principles in practice in mathematics. Conventional mathematics curricula in primary school usually begin with concrete teaching of arithmetic based on counting discrete objects (e.g., “How many crayons are there?”) and, through this, the properties of natural numbers. On this basis, algebra, with its abstract symbols, formulas, and unfamiliar properties of numbers (e.g., rational numbers are represented by negative numbers; decimals; or the ratio of two integers, such as −3, 0.5, or 1/2), is taught in later years. However, DT researchers and CC researchers, especially framework theorists (Christou et al., 2007; Merenluoto & Lehtinen, 2004; Vamvakoussi & Vosniadou, 2004, 2007), documented the misconceptions that result from that way of learning mathematics by enriching children’s preinstructional concepts of numbers developed in the act of counting discrete objects in everyday life (Christou et al., 2007; Vamvakoussi & Vosniadou, 2007). A misconception typically formed when learning mathematics that way is the idea that numbers have a discrete nature (i.e., there is a definite successor for each number), which interferes with the learning of rational numbers (that have a continuous nature) and algebra (with its abstract symbols and formulas and unfamiliar properties). DT mathematics lessons differ radically. They were designed to overcome these CC difficulties and to develop, by contrast, early mastery of mathematical types of thinking (Kilpatrick et al., 1975; Venenciano et al., 2021).

DT mathematics is rooted first in an analysis of the most fundamental concepts of mathematics, such as the concept of real number and its core conceptual interrelation that is at its origin: a certain quantity expressed by a unit of measurement (Arievitch & Stetsenko, 2000; Davydov, 1975a, 1975b; Gal’perin, 1969, 1989a; Karpov, 1995; Kilpatrick et al., 1975). Accordingly, from the outset, students become familiar with this core interrelation and the type of theoretical thinking that produced it. To illustrate how DT researchers designed mathematics education by relating this content analysis to students’ existing cognitive capacities and knowledge, I describe Gal’perin’s kindergarten mathematics lessons and El’konin-Davydov’s mathematics curriculum for first graders.

To introduce the concept of quantity, Gal’perin’s (1969/1989a) lessons for six-year-olds started with a demonstration of the importance of measurements in different areas of life, such as weighing objects in shops, trying on shoes, and measuring cloth. Back in the classroom, the students were invited to measure various objects: tables, windows, one another, and so on. They were asked, “What shall we measure with?” (p. 34). They subsequently realized that objects have different properties and that each object must be measured with its own measure, such as a spoon to measure water, something long to measure length, and some sort of weight to measure weight. The teacher ensured that students paid special attention to discriminating the parameter of the question asked: what is bigger, smaller, or equal? They were also taught how to measure. Initially, they measured carelessly, e.g., by leaving a space between two measurements or by measuring a segment they had already measured. When asked for their opinion about these ways of measuring, they – or the teacher – pointed out that their result would be incorrect and showed why and how. These learning tasks illustrate the DT principles of the following:

In addition to divisible things, the students would measure indivisible things (e.g., the length of a door), thus eliminating the discreteness of measured quantities (Gal’perin, 1969/1989a). The students, who were measuring enthusiastically, often did not notice the result they were obtaining. They, or the teacher, would point out that by using the same small object to measure, a mark could be made on the thing being measured, and the number of marks corresponded to the number of times the measurement had been applied. Students, therefore, began to “see a thing as a set of ‘the number of times a measurement was applied’” (Gal’perin, 1969/1989a, p. 35). The students then learned to compare the sets they had obtained. They were presented with a problem in this respect:

“Two rather large (15–20 pieces) and unordered groups of markers differing in only one or two elements were presented (the difference was imperceptible to the eye, and the children were unable to count). The children would then be asked whether the groups were identical or different from the others. Initially, they gave arbitrary and, of course, varied responses. But the experimenter asked, ‘How can we show who is right so that all can see it?’ If the children experienced difficulties, the experimenter would demonstrate a way to line up two (horizontal) series in a reciprocally unambiguous relation. This became the principal means for quantitative comparison of sets and, through them, magnitudes. By comparing series in this way, it was easy to teach them the ideas of ‘as much as,’ ‘equal,’ ‘more and less,’ ‘more by so much and less by so much’” (Gal’perin, 1969/1989a, p. 35–36).

On this basis, the children gradually formed an understanding and mastery of the core theoretical aspects of a real number. They understood that a unit is an abstraction, something equal to its own measure, and a number is a quantity expressed by a unit of measurement (e.g., 2 means something that is equal to two measures; Gal’perin, 1969/1989a). These learning tasks illustrate DT principles of the following:

Similarly, in El’konin-Davydov’s curriculum for first graders, students were first asked to identify the physical attributes of objects that can be compared (Davydov, 1975a, 1975b). The intention was to begin lessons by appealing to children’s spontaneous curiosity about the relationships between quantities in everyday life, including comparisons, seriation, and classifications of objects, as their “practical methods of distinguishing and designating certain mathematical relationships” (Davydov, 1975a, p. 90).

Six-year-olds typically compared lengths, volumes, and masses, such as the length of one book compared to another, as Davydov (1975b) assumed. Then, they were asked to describe the relationship between the attributes of the objects being compared, e.g., the relationship between the lengths of two books placed next to each other. The students or the teacher suggested several possibilities. They named one length A and another B, e.g., then verbally said that the two lengths were not the same (in this case, they were unequal) – they used paper strips to show that one was longer. They then represented the relationships by drawing line segments (see the left side of Fig. 1).

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Finally, they could write statements such as B > A. The students were then tasked with making unequal quantities equal, or the reverse, by adding or subtracting a length. By identifying this length, say C, they understood that to achieve equality, they must either add C to A or subtract it from B. They again represented these relationships by line segments (see the right side of Fig. 1).

Thereafter, they began to write formal equations such as B = A + C, A + C = B, B − C = A, and B − A = C. The students were subsequently asked to compare two quantities that cannot be compared directly. They were thus put in a situation where their usual representations of quantity were ineffective and misleading in representing relationships between quantities to induce the need for the concept of real number. The following quote illustrates Dougherty’s (2010) application of Davydov’s curriculum in this regard:

“To motivate that discussion (about the problem), children were given two lengths, such as the length of a bookcase and the height of a bookcase across the room. They could not move the two bookcases to make a direct comparison. They were challenged to find a way to decide if the lengths are the same, or if one is longer than the other. They decided to use an intermediary unit. That is, they made the length of one bookcase using string. They then used the string to compare it to the other length (or height). This led quite naturally to an introduction of the transitive property of equality. The children symbolically represented this as A = P, P = R, and A = R. What the children said is that if length A and length P are the same, and length P is the same as length R, then length A has to be (their emphasis) equal to length R. In this example, length P represented the intermediary unit that was used to compare one length to the other” (Dougherty, 2010, p. 66).

The lessons progressed in the same vein, with children understanding that x symbolizes an unknown quantity thanks to which “one can pass from the inequality to the equality [in equations]:

A < B

A = B − x.

x = B − A” (Davydov, 1972/1999, p. 168–169)

Finally, the most fundamental conceptual interrelation or germ cell of the concept of real number was represented in a visual symbolic model under the form of a formula, providing its explicit theorization: “A/C = N, where N is any number, A is an object represented as a quantity, and C is a unit of measure” (Davydov, 1972/1999, p. 169). Davydov’s mathematics curriculum progresses by following the DT principle that each learning step establishes, as seamlessly as possible, a systematic interrelationship with what has been previously learned so that it fits into a single conceptual system. For instance, students learn multiplication and division in direct relation to their prior familiarization with the construction and measurement of quantities, which introduces the concept of multiplicity in terms of how many times a quantity fits into what is being measured. They learn to use schematic forms such as U ||| ----→ A, which indicates that three units are used to construct or measure quantity A, or U ? ----→ A, which indicates that the student should measure the quantity A using the unit U and determine the value of? (Davydov, 1992).

Example 3: The Theory of Evolution

This example further illustrates some of the aforementioned teaching principles. Hedegaard (1996) designed lessons on the theory of evolution and collaborated with a teacher who gave these lessons to two classes of third graders (33 children aged eight to nine; with one control group of 21 children) in Denmark. Hedegaard collaborated with academic specialists to delineate the germ cell of animal evolution: the interrelation between an animal species, a population, and an ecological niche. Key examples that illustrate this most fundamental conceptual interrelation and that are relatable to the student’s prior knowledge were used as introductory real-world problems. For instance, the students were told that the Arctic hare was introduced from Norway to the Faroe Islands, where there is no snow, and after some generations, the population of Arctic hares changed color from white to brown. Different tasks and exploratory activities (e.g., identifying the living conditions of the polar bear in the Arctic after watching a film about the life of the polar bear in Greenland) and classroom discussions (e.g., students were asked “What would happen if desert animals were moved to Greenland?”) followed. Living in Denmark, the children likely already had some geographical knowledge with the examples, namely, Greenland and the Faroe Islands (both Danish protectorates). Engaged in such learning tasks, the students modeled in three steps, under the teacher’s guidance, the germ cell of the theory of animal evolution (see Fig. 2).

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“Stage I depicts the relationship between an animal species and its environmental conditions. Stage II depicts whether a species adapts. A species will survive if members of the population can adapt to changes in their living conditions. Stage III depicts changes in a species. New species will evolve under the following circumstances: (a) segregation and isolation of part of a population that entails change in the biotope, which involves change in the way(s) of living (ways of survival and cohabitation are important in this relation); (b) variation in the genetic material both through gene mutation and gene combination (cohabitation, parental care, and relation to other species are important for this relation); and (c) selection of offspring that have changed characteristics that are optimal relative to the environmental change (ways of survival, parental care, and defense against other species are important in this relation)” (Hedegaard, 1996, p. 14).

Hedegaard’s lessons on evolution continued in fourth-grade history courses such that the core conceptual interrelations of the animal evolution model were specified in a subsumed visual model of humans’ origin; this model itself was specified in fifth grade in a visual model of the historical change of societies (Hedegaard, 1996). This learning progression illustrates Davydov’s principle of further developing the initial germ cell and its limitations and explanatory potentials by deriving submodels, principles, and instantiations to integrate them all into a single conceptual system.

Evidence of DT Approaches’ Ability to Address Issues of CC

The available evidence on the learning outcomes of DT approaches since the 1950s shows their consistency (Arievitch & Stetsenko, 2000; Chudinova, 2019; Davydov, 1972/1999, 1988/2012; de Araujo et al., 2020; Ferreira, 2005; Gordeeva, 2020; Hedegaard & Lompscher, 1999; Karpov, 1995; Karpov & Bransford, 1995; Karpov & Haywood, 1998; Lantolf et al., 2020; Rubtsov & Ulanovskaya, 2020; Venenciano et al., 2021). It was collected in different national contexts and covered different age groups and subjects ranging from mathematics to science. This evidence has been obtained either from systematic comparative studies with control groups, from longitudinal studies (sometimes spread over several years), or from case studies. The results revealed that students consistently demonstrated a high degree of understanding, mastery of theoretical knowledge and thinking, and higher degrees of understanding than in conventional teaching. Their understanding and mastery included deeper levels of understanding, high levels of accuracy, intentional and systematic use of knowledge, far knowledge transfer, and flexible application of knowledge to new situations. Among others, model-based reasoning skills and intentional metacognitive mastery were generally emphasized. Furthermore, students’ understanding and mastery were developed earlier than in conventional education, including successful CCs, as students learned the foundations of academic knowledge earlier in different school subjects. Thus, in several areas, DT researchers have invoked this type of findings to challenge assumptions that a range of limitations in children’s comprehension abilities regarding certain CCs are inherent to developmental characteristics related to their age and experience rather than depending on the type of education received.

I briefly illustrate these claims with some of the examples described above. Hedegaard’s (1996) lessons on evolution enabled 8- and 9-year-olds to develop a high degree of mastery and systematic understanding of the core conceptual interrelation between species and ecological niches and of the temporal dimension of evolution. Other DT studies on the topic of adaptation of the living organism for 11- and 12-year-olds have found similar results (e.g., Hintz, 1984; Lompscher, 1984). This mastery included transfer to new situations directly or indirectly related to biological evolution, including historical human conditions (which were studied with systematic integration into the general model of evolution), thus indicating an integrated, flexible, and systematic model-based understanding. These degrees of systematic understanding and mastery are generally found in older educated students and are known to be difficult to achieve because of the CCs they require (Rosengren, 2012).

As another example, I explained that conventional learning of mathematics, which begins with the concrete teaching of arithmetic based on counting discrete objects, typically forms the misconception that numbers have a discrete nature. CC researchers, especially framework theorists, showed that this misconception leads to a series of related misconceptions (Christou et al., 2007; Merenluoto & Lehtinen, 2004; Vamvakoussi & Vosniadou, 2004, 2007). For example, the more digits a number has, the higher the number is; multiplication always makes the number larger; addition and subtraction are conceptualized in terms of counting; multiplication is a form of repeated addition (Vamvakoussi & Vosniadou, 2007). These misconceptions usually interfere with the learning of rational numbers and algebra, making them particularly difficult to learn, especially for children, because of the CCs it requires (rational numbers, e.g., have a continuous nature; Christou et al., 2007; Vamvakoussi & Vosniadou, 2004, 2007).

By contrast, these CC difficulties are not encountered when using DT approaches because, from the beginning, they familiarize young students with the most general and fundamental conceptual interrelations underlying real numbers, such as quantity relationships and the concept of a unit of measurement (Karpov, 1995; Kilpatrick et al., 1975; Ni & Zhou, 2005; Venenciano et al., 2021). They avoid reinforcing preconceptions about numbers as discrete. From the beginning, they also accustom students to the logic of algebraic equations. Therefore, students tend to develop an early mastery and understanding of complex mathematical concepts such as negative numbers, decimal fractions, and some algebraic reasoning (Arievitch & Stetsenko, 2000; Davydov, 1972/1999; Karpov, 1995; Venenciano et al., 2020, 2021). Specifically, after completing the mathematics lessons described above, students in kindergarten and first grade overcame the typical tendency to rely on salient superficial features of objects to evaluate quantities (a tendency that Piaget considered to be inherent in a general preoperational cognitive stage from approximately two to 6 years of age; Arievitch & Stetsenko, 2000; Davydov, 1972/1999; Gal’perin, 1978/1992; Karpov, 1995). Students indeed developed an early orientation toward discrimination and reliance on the mathematical properties of objects, including mastery of the conservation of quantity principle. More generally, DT researchers have contributed to challenging a series of supposed limitations of young children’s mathematical abilities by showing their potential capabilities in a way that has impressed the mathematical community (Venenciano et al., 2021).