Piaget’s cognitive behaviourist account of conservation in mathematics

Anoop Gupta

            After the Second World War existentialists and humanists like Sartre, by his emphasis of choice in directing our actions, required the discrediting of behaviourism. We were under the sway of behaviourism beginning in the 1920’s till the 1950’s. In the 1960’s, there was a cognitive revolution where we rediscovered the mind, so the story goes, which I dub the “standard view”.

Thanks to research programs like Piaget’s, to lines of study like Bartlett’s, and molar concepts like schemas, operations, and strategies, the concerns of Gestalt psychologists and of the Wϋrzburg predecessors remained alive in the English speaking world during the height of the behaviourist era. (Gardner, 1987, p. 118)

After the Second World War, when Piaget’s  theory appeared in a new guise to the English speaking community, it created a near sensation…[it] struck at the heart of the then dominant neo-behaviourism. (Beilin, 1992, p. 191)

[Piaget] indeed was one of the major figures responsible for the emergence of cognitivism from the earlier behaviourist era in psychology. (Anderson et al., 1999, p. 19)

Constructivism, originally developed by Piaget, views learning of mathematics as the construction of meaning and understanding based on the modeling of reality, the analysis of pattern, and the acquisition of a mathematical disposition. (Gales & Yan, 2001, p. 4)

The standard view is so pervasive that even Skinner identified Piaget with the notion of explaining what grows in the “mind” (Skinner, 1974, p. 67).

I shall attempt to argue that, on the contrary, Piaget’s account of conservation, that is, the notion that the number property stays the same despite changes in the objects to which it may be indexed, in mathematics, is describable in part in the terms of radical behaviourism of B.F. Skinner. Skinner’s idea is that behaviour can be explained in terms of the history of its contingencies. The notion that Piaget and Skinner are not juxtaposed I call the “non-standard view.” I am not just attempting to re-describe Piaget’s genetic epistemology, the attempt to uncover the universal structures involved in the acquisition of knowledge epistemology (1970, p. 1), but offer a more accurate account than has been passed down by scholars, who often explain what happened historically in 20^th century American psychology with less attention paid to what Piaget said. Piaget’s brand of behaviourism, I shall claim, gives rise to a new theory, henceforth, “cognitive behaviourism”.  

Since my aim is to show that behaviourists and cognitive theorists need not be enemies, I have offered only a working definition of the combined view.  In outline, cognitive behaviourism is the idea that we are shaped by our past experiences, explained in conjunction with some Skinnerian concepts, and what we think about them.  Behaviour is shaped by a history of contingencies (as Skinner held) that have a meaning for the subject (as cognitive theorists claim). Cognitive behaviourism is like the well-known social cognitive theory of Bandura (1997) but is imbued with both Skinnerian concepts, such as “history of contingencies” and “repertoire’s of behaviour”; as well as the nature of meaning in terms of core goals of the self, along the lines of Maslow (1970).

            First, I define the radical behaviourism of B.F. Skinner. Second, I consider Piaget’s account of conservation in mathematics, beginning with a general account of this cognitive theory. Third, I consider the virtues of the non-standard view for theory and pedagogical practice. Fourth, I discuss implications for viewing Piaget as a cognitive behaviourist for empiricist philosophers of mathematics, and briefly remark on Chomsky’s famous critique of Skinner. Finally, I consider the further development of cognitive behaviourism.

Scope

            Though I have chosen conservation in mathematics as the test case the account Piaget provided is assumed to apply more generally to his entire opus, and I believe is further unaffected by changes in his thought (discussed in  Section 4, “Advantages of the Non-Standard View”); I also draw upon other examples to make my case. I make no attempt to define the different stages of Piagetian development but make reference to them when necessary; that is not my focus. Further, I use Skinner as the exemplar of behaviourism even though there are other varieties. My reasons for doing so rest with his pre-eminence among academics of all disciplines and the prospective benefits to mitigating the wide-spread misunderstanding of his work, too.

My claim is not, I emphasize, that Piaget and Skinner hold identical positions but, rather, are similar on several points—concerning conservation; functionality of psychic mechanisms (e.g., automaticity); and the role of the environment (e.g., our dealings with everyday objects, in learning)—such that Piaget is best understood as a cognitive behaviourist. We may wish to note that I do not claim that the outline of cognitive behaviourism I develop provides a conclusive alternative other psychological theories; I claim only to provide a sketch that combines two of the seminal psychological traditions of the 20^th century. Nor do I aim to defend Piaget’s account of the origin of number or develop Skinner’s radical behaviourism in a cognitive direction. Rather, my purpose is more modest. I only aim to show that the cognitive behaviourist’s over-all account contributes to our understanding of mathematical teaching, learning, as well as the philosophy of that discipline, having more going for it than we are apt to think; and Piaget shows us why. At the very least, a cognitive behaviourist view must be considered.

Skinner and mathematical behaviour    

The most common misunderstanding of the behaviourist, according to Skinner, is that they deny we have feelings (the “no-feelings criticism”), and he lists twenty misconceptions, including: (1) the “no-feelings” criticism; (4) they do not account for cognitive processes; and (5), they provide no account of intention or purpose (Skinner, 1989; 1974).

            Skinner held humanistic psychology and cognitive science responsible for some of the caricature (Skinner, 1989). With some frustration, Skinner (1974) remarked that it is the cognitive scientist however, not the cognitive behaviourist, that said we are machines. Behaviourism got a bad reputation because of some of the extravagant contentions of John B. Watson, its founder, which however is different from Skinner’s programme also (Watson, 1913; Skinner, 1974).

            Skinner remarked that we often confuse methodological behaviourism, an attempt to confine ourselves to what is observable espoused by the Vienna Circle under the influence of Wittgenstein, with its cognitive Skinnerian version where feeling is a kind of sensory action like seeing or hearing.

            Nonetheless, Skinner noted that it is hard to explain feelings like embarrassment without antecedent consequent events; our feelings change based on antecedent environmental conditions; they are often metaphors in fact from experience (Skinner, 1989; 1974). “To mind” is to watch, and there is the “wandering mind”, “lost my mind”, and so on (1974, p. 86).  “Aiming at an object” in the future betrays its origins in the spatial metaphors we use: reaching for a goal (1974, p.  55). 

Learning is changing what we do (Skinner, 1989). A person is a body that does things; people do not respond to the world but possess it, and we only infer from what they do (1989, p. 28). Believing an apple is on my desk means I may pick it up. “[M]ind is what the body does”, as he puts it (1989, p. 67). Skinner’s (1974) strategy goes back to the genetic and personal histories responsible for our histories.

            It is often noted that Skinner claimed to get behaviourism from Bertrand Russell. He took from Russell the distinction between knowledge by description (“hit the nail with the hammer”) and acquaintance (“hitting the nail”), and elaborates it in his own terms.

Behaviour said to be due to knowledge by acquaintance (the product of immediate contingencies of reinforcement) is much more effectively executed than knowledge by description…Try [on the other hand] teaching your friend how to swim by telling her what she is to do with her head, arms, and legs or why she will then move smoothly through the water. (Skinner, 1989, p. 87)

Reinforcing events can be learned by imitation also, according to Skinner. Extending behaviourism to the formal sciences Skinner (1989) wrote, “Logic and mathematics presumably arose from simple contingencies of reinforcement” (p. 42). 

If an artificial organism could be designed to do what logicians and mathematicians do, or even more what they have done, it will be the best evidence we have that intuitive mathematical and logical thinking is only following rules, not matter how obscure. And following rules is behaving. [Emphasis mine.] (Skinner, 1989, p. 68)

For Skinner (1953), the rules of mathematics are learned.  In memorizing the multiplication table, for example, the stimulus ‘9 x 9’ is the occasion upon which the response ‘81’ is appropriately reinforced, either by an instructor or by the successful outcome of a calculation. (p. 109)

            Induction (or generalization) is not an activity of the organism; rather, control by a stimulus is applied to other stimuli with common properties according to him. He wrote:

What exists [and give rise to abstract objects] are the contingencies which bring behaviour under the control of properties of classes or objects identified as properties…A concept is simply a feature of a set of contingencies which exist in the world, and it is discovered simply in the sense that the contingencies bring behaviour under control. (Skinner, 1974, p. 94)

Responding to “6 + 3 = 9” is training. He said, “If we are content to speak of an arithmetical concept, we shall never find out what the child has actually learned” (Skinner, 1974, p. 107).

            He insisted there is a difference between following a rule and knowing it ([emphasis mine.] Skinner, 1974, pp. 124 – 5). Rules rest on contingencies. To know is to be in contact with and be able to do it. Yet knowing is more than just following rules.

The understanding we gain from moving from rule-governed to contingency shaped behaviour is usually reinforcing, in part because the reinforcers in the latter case are less likely to be contrived and hence less likely work in the interests of others. (Skinner, 1974, p. 142)

The contingencies of reinforcement are the situation in which the act occurs, the act, and subsequent consequences, according to Skinner.

            Cognitive behaviourists “solve [philosophical] problems by [reference to] assembling, classifying, arranging, and rearranging things” (Skinner, 1974, p. 223). We can avoid Plato’s dilemma—how can we search for what we do not know—by going back to how knowledge is acquired in our dealings with physical objects.

            Operant conditioning, learning behaviour strengthened or weakened by consequences or antecedents, is based on intention in fact ([emphasis mine.] Skinner, 1974, p. 55). For him, the meaning of an event is analyzed in terms of the “history of exposure to contingencies in which similar settings have played a part” (Skinner, 1974, p. 90). He wrote, “A self or personality is at best a repertoire of behaviour imported by an organized set of contingencies” (Skinner, 1974, p. 149). He noted that we do not seek freedom but the feeling of it ([emphasis mine.] Skinner, 1989, p. 113).

            Skinner did wish to study behaviour apart from reference from mental life or the nervous system, calling emotions “fictional causes”, and denied free will (Skinner, 1953, pp. 122, 160; 1974, p. 189). He offered operational definitions of emotions, for example anxiety would be the result of stimulus that proceeds after a strong negative response giving rise to anticipation.     

            “We may identify needs, wants and hungers as internal causes, that is, a condition resulting from deprivation and characterized by a special probability of response” (Skinner, 1953, p. 144 and pp. 143, 141). A negative reinforcer is “a stimulus the reduction or removal of which strengthens behaviour” (Skinner, 1953, p. 127). Reinforcement brings behaviour under the control of a given (appropriate) deprivation.

            The fact we describe our feelings with words already points to a social mechanism. Social behaviour takes the need mechanism to a group level, that is, the behaviour of two or more people with respect to one another or in concert with respect to a common environment. He explained, “The problem presented by group behaviour is to explain why many individuals behave together” (Skinner, 1953, p. 311). He conjectured that behaving as others do is likely to be reinforcing. In the group selfishness is controlled and altruism encouraged. In fact, we often perceive something to be valuable, that is, reinforcing because others do (Skinner, 1974).

We can trace a small part of human behaviour, and a much larger part of the behaviour of other species, to natural selection and the evolution of the species, but the greater part of human behaviour must be traced to contingencies of reinforcement, especially to the very complex social contingencies we call culture. (Skinner, 1989, p. 24)

Yet he said, “An organism cannot acquire a large repertoire of behaviour through operant conditioning alone in a non-social environment. Other organisms are important” (Skinner, 1989, p. 51).    

            If we ask however why some events do reinforce behaviour, we are led to an account, that is, consistent with and fills out Piaget’s (discussion in next section). For Skinner (1953) reflexes and other innate patterns of behaviour increase the survival of the species. To be interested in brain science and cultural evolution for him is to go back to human behaviour.  “The biological substratum itself is determined by prior events in a genetic process” as he explained (Skinner, 1953, p. 448; 1974, p. 44). Operant conditioning in short is a function of natural selection (Skinner, 1974; 1989).

Clinical psychology fails he tells us at some point because it does not consider the role of the environment (Skinner, 1989). He bemoans that education has not improved, criticizing the use of punitive action (discussed in “Pedagogical Reflections”, p.18).

Piaget’s account of the exact sciences                  

The key principle in our understanding of Piaget can orient us: when we encounter a novel event, cognitive disequilibrium occurs. The central mechanisms used to restore equilibrium are assimilation, fitting new information into a schema; and accommodation, altering our frame of reference. Piaget (1970) said, “To know is to assimilate reality into a system of transformations” (p. 15).

Arithmetic requires conservation as a number is identical with itself whatever the distribution of units of which it is composed.  According to Piaget, there are primitive forms of understanding upon which number conservation rely, negation (1 – 1 = 0), and reciprocity (A = B : : B = A). Number arises from being able to place objects, like bottles and flowers in a one-to-one correspondence.

Finally, the one-to-one correspondence becomes operational, and at that time there is conservation of number in the sense of the realization that number does not change just because the special arrangement does. In this instance, once the child has established one-to-one correspondence by taking a blue token for every red one, no matter how we change shapes, he will be able, without counting or even thinking hard, to say the number must still be the same because of the one-to-one correspondence established at the outset. ([Emphasis mine]. Piaget, 1970, p. 35)

                         ■      ■      ■      ■       ■           (blue)    

                         │      │      │       │      │                       

▲     ▲     ▲    ▲     ▲         (red)       

The logic of inclusion, ordering, and correspondence are the foundation of logical and mathematical structures according to Piaget. At the end of the first year of the infant, he said, “The conservation characteristic of sensory-motor intelligence takes the form of the notion of permanence of an object” (Piaget, 1970, p. 43). The notion of permanence is the sensory-motor equivalent of conservation at the operational stage. Piaget suggested (contra his colleague, Jerome Bruner) that identity is qualitative not quantitative. Identity is not sufficient for the conservation of number. With identity Piaget explained we only have what changes and what it is.  Ordering and classification are required for conservation.

 “Our contention is that conservation is necessary for all rational activity”, wrote Piaget (1941/1964, p. 3). A child that sees the same amount of liquid poured in a long narrow container and a short stout one will think the former (the long narrow container) has more at the first Piagetian stage, that of sensory-motor development. 

Our only problem is to discover by what means the mind succeeds in constructing the notion of constant quantity in spite of indications to the contrary provided by immediate perception. (1941/1964, p. 9)

That is, our thinking must become independent of perception (1941/1964, p. 89). Even in geometry, underlying any measurement is the idea that an object remains constant in size in any position (Piaget, 1960, p. 90).

                                                   ←                 ↑             ↓           →

Piaget distinguished two types of abstraction: simple and reflective. Simple reflection moves from an action to operation. Reflective abstraction reorganizes thought for Piaget. Mathematics is governed by laws and is self-regulatory.

            According to Piaget, every classificatory scheme occurs simultaneously with number. Assimilation requires the interiorization of action in the form of operations. According to him, language is a necessary, but not a sufficient condition for logico-mathematical knowledge. 

Piaget (1956) held, “[I]f every organ of a living body is organized, so also every element of an intellectual organization also constitutes an organization” (1956, p. 13). To track down the formation of cognitive structures, Piaget looks to behaviour. According to him, the organization of behaviour is rooted in needs facilitating the transitions from organic to psychic life. We repeat behaviours that serve a need according to Piaget.  Seeking out needs is his way to uncover the origin of “habit” ([emphasis mine]. Piaget, 1956, p. 134).

            According to Piaget, the application of familiar schemata to new situations is the first sign of intelligence. Repeated applications result in behaviour patterns (1956, p. 251). He said, “A complete continuity exists between the behaviour patterns characteristic of different stages” (Piaget, 1956, p. 383).  Empirical intelligence is explained by “progressive internalization and systematization of these [behavioural] processes which would properly account for intelligence properly so called” (Piaget, 1956, pp. 396 & 419). 

            Piaget wrote, “The kernel of these beliefs [that hold society together] is the feeling of the sacred, the source of all morality and religion [and Platonism]. Whatever offends against these powerful feelings is crime, and all crime is sacrilege” (Piaget, 1956, p. 330). According to Piaget (1965), in fact, the experience of regularity is the first step to a law, which illustrates the idea that abstract ideas are founded in interactions with physical objects.

In his work on logic with Inhelder (1964) in The Early Growth of Logic in the Child: Classification and Seriation, he noted, “Classification implies a relation of resemblance between members of the same class and one of dissimilarity between members of different classes” (1964, p. 5).  Yet “[P]erception is bound up with action schema of a higher order, and that these structures can influence those of perception” (Inheder & Piaget, 1964, p. 6). A class of oranges is based on a schema, both perceptual and motor. Sensory motor schema applies to stable patterns of movement with a perceptual component geared to the recognition of appropriate signals according to them.

            We see associations of objects with similar properties using “elementary classificatory behaviour” ([emphasis mine]. Inheder & Piaget, 1964, pp. 39, 44, 99).  However, they do admit the role of mental images in all stages of development (Inheder & Piaget, 1964, p. 295).

Mathematical Epistemology and Psychology is divided in two parts, with Beth (1966)  covering the history of the philosophy of mathematics and Piaget covering its psychological aspects. Piaget separated the logical issues he calls “normative” and the other “mental mechanism” (Beth & Piaget, 1966, pp. 132, 135). He sought to avoid psychologism, that is, attempting to settle “matters of validity” by appeals to “matters of fact” (Beth & Piaget, 1966, pp. 141, 311). We must separate what Piaget insisted were issues of logical foundations from psychology, which he calls “genetic” (Beth & Piaget, 1966, p. 143; Piaget, 1970, p. 2). He explained our constructions are explained by “these [mental] structures by a progressive construction due the subject’s activities” (Beth & Piaget, 1966, p. 142). 

            He hoped however for “improved collaboration between the two types of inquiry” (Beth & Piaget, 1966, p. 258). Conceiving normative and psychological laws to be on parallel tracks he calls this notion the “fundamental hypothesis” of genetic epistemology (Beth & Piaget, 1966, p. 142; Piaget, 1970, p. 13). He admitted, in fact, that the coordination of logic and empirical inquiry is required for scientific epistemology and suggests such activity as the foundation of mathematics (Beth & Piaget, 1966, pp. 193 – 4, 305, 308 -9).

            Using the example of chess, we may inquire into the origins of the normative rules. As Piaget said, “This amounts to asking how the rules of the game [of chess] are explained, not as valid or invalid, but as rules modifying the subjects behaviour” ([emphasis mine]. Beth & Piaget, 1966, p. 143).

            Yet, as he noted, Platonists’ normative considerations are rendered completely independent of fact. He pointed out that if Platonism is right however they cannot explain why only at certain stages of development we can do mathematics and logic. Experimental psychology forces us to explain how we apprehend mathematical truths: What would “constitute the necessary preliminary conditions of all cognitive activity [?]” he asked (Beth & Piaget, 1966, p. 147). “Nothing could be more true”, he said: Abstract objects like numbers are abstractions from the physical world (Beth & Piaget, 1966, p. 148). He said of logico-mathematical procedures, “There exist actions and operations concerned with the separation or recombination of the objects themselves as continuous wholes” (Beth & Piaget, 1966, pp. 183-4).

            What a type of organism does over an extended period of time are, according to Piaget, “acquisitions integrated into the framework of hereditary behaviours and the organic structure of the species” (Beth & Piaget, 1966, p. 194). All behaviour, even discovering abstract theorems, is affective and cognitive. Mathematical construction consists “of reconstructing an earlier structure but on a higher plane, where it is integrated in a larger structure” (Beth & Piaget, 1966, pp. 186 & 203). From the Piagetian point of view, mathematical constructions “are indefinite successions of combinations at once new and yet within a well-determined system of possibilities” (Beth & Piaget, 1966, pp. 207 & 301).

            Piaget said, “Nothing is harder for a psychologist to understand than what a mathematician means by intuition” (Beth & Piaget, 1966, pp. 208 & 212). He noted that every perception is marked by movement (Beth & Piaget, 1966, pp. 209 & 213).  We gain empirical intuitions as a function of the development of experimentation and at different stages: (i) action with objects, (ii) action internalized in the form of operations, and (iii) operations independent of all actions. Finally, we have symbolizing intuitions—like the law of commutation, “A + B = B + A”)—that evolve separately from operational ones—the later give rise to special images. Operational intuitions further have an unlimited development due to “reflective abstractions” (Beth & Piaget, 1966, pp. 224 – 5, 231).

“[L]ogico-mathematical experience has to do with the actions which the subject carries out on the objects”, according to Piaget ([emphasis mine.] Beth & Piaget, 1966, p. 232). To emphasize, he is concerned not with objects, but the actions we carry out on them; the dispensability of objects is what gives us pure mathematics. He explained that mathematics is a “coordination of actions on any objects whatever” (Beth & Piaget, 1966, p. 238). Understanding that “1” can be an apple or tomato is not an individual act, but “the most general coordination’s of every system of actions”, that is, what is common to all subjects (Beth & Piaget, 1966, p. 238). Piaget thought that the “system of actions” is rooted in the nervous and biological organization of the subject. Since sensory-motor coordination is rudimentary, psychology terminates in biology. He wrote, “The hypothesis of an epistemic subject is characterized by a logic of the general co-ordinations of actions” (Beth & Piaget, 1966, p. 309).

At the ages of seven and eight year olds, according to Piaget, sensory-motor intelligence is internalized as thought at the level of representation rather than the actual carrying out of actions. The result of conservation in a nutshell is that we can construct as Piaget (1972) put it, “[A]utonomous [mathematical] structures…arising from but not directed by experience” (pp. 76 -77).   

Our assimilation “confers meaning on the object it assimilates and assigns goals to the actions it organizes” (Piaget, 1985, p. 16). Yet, as he noted, the regulation by which we aim at equilibrium presupposes a regulatory system. He wrote, “A first interpretation might be to identify this guidance programme with the nature of things” (Piaget, 1985, p. 19).  He explained, “The play of assimilations and accommodations constantly brings into play reinforcements and corrections” ([emphasis mine]. Piaget, 1985, pp. 20-1). 

On a functional perspective, all mental activity is an attempt to satisfy needs: moving from momentary disequilibrium to re-equilibrium (Piaget, 1985). In fact Piaget criticizes Gestalt theorists for relying on the model of physical field theory, paying little attention to temporal construction of frames, that is, schemas that are well established and are not regulated per se. Piaget assured however that optimizing equilibrium does not lead to a static state as every improvement is oriented “in the direction of coherence or more developed form of internal necessity” (1985, p. 144).

Advantages of the non-standard view

Viewing Piaget as a cognitive behaviourist can be corroborated as plausible in several different ways that go beyond historical interpretation alone. That is, there are also theoretical and practical benefits from viewing him thus.  I shall attempt to highlight the advantages of viewing Piaget as a cognitive behaviourist by engaging his ideas in a dialogue with social psychologists, evolution theorists, and educationalists.

Kagan (1984), the Harvard child psychologist, summarized the central tenet of Piaget’s theory:

The central element in Piaget’s theory of the infant is the sensory-motor schema. It is a representation of the class of motor actions necessary to obtain a goal, and it is acquired through active manipulation of objects. (p. 48)

Kagan remarked, in an unpublished PhD dissertation submitted to Harvard University, that Vidal (1981) documented that in Piaget’s early study of snails where he argued that their morphology changes over time in commerce with the environment and then is genetically encoded, is an idea he would never abandon (Kagan, 1984). Kagan noted however that one of the key criticisms of Piaget is that he treated the principles of Western logic and mathematics as demanded by each person’s everyday experiences.

            For Piaget, the disequilibrium that occurs in novel situations is a negative reinforcement, which is removed by expanding our cognitive repertoire and accompanying behaviours. Learning, for both Piaget and Skinner, changes what we think and do. Though Piaget is reprimanded for not paying enough attention to the socio-cultural context, it can be accommodated within his theory, as we see when we respond to Kagan’s colleague, Gardner.

            There are two separate concerns, according to Gardner, that affect both behaviourism and cognitive science. According to Gardner (1987), cognitive psychology deemphasizes emotions and the socio-cultural context.  He wrote, “If, in the last analysis, anthropology proves to like largely outside of mainstream cognitive science, this will be an important (if somewhat disappointing) finding” (Gardner, 1987, p. 258). Gardner (1884) said regarding behaviourism: “Today the theoretical claims of behaviourism (though not its various applied achievements) are largely of historic interest”, though not its experimental methodology (pp. 110, 135).

            In fact, Skinner’s reference to the external environment is important as it addresses the very concern that it does not pay attention to the socio-cultural context, which troubles Gardner about cognitive science. Reading Skinner in light of Piaget, and vice versa, however, allows expanding the contingencies that shape behaviour, including the socio-cultural context. That is, behaviour that is rewarded with would be partly contingent on the socio-cultural context.

            Beilin (1992), in “Piaget’s Enduring Contribution to Developmental Psychology”, explained that all cognitive accounts of development must have a structural explanation, that is, functional mechanisms of mind. Beilin also pointed out that Piaget’s theory can be divided into four stages. There are invariant features among the stages, however. According to Beilin, the idea of moving from the egocentric to social reasoning pervades his account. Beilin (1992) wrote, “The child’s action was seen as the fundamental source of knowledge rather than the traditionally defined sources of perception and language” (p. 195).  Beilin (1992) noted, also, “As Piaget liked to put it, there is no form without function and no function without form” (p. 202).

I have already pointed out that the account of Piaget I develop is assumed to be unaffected by changes in his view; investigating the claim goes beyond the scope of this paper. Suffice it to say, however that to the extent that Piaget became more receptive to the role of the socio-cultural context, his view comes closer to the over-all cognitive behaviourist account provided. For Piaget and Skinner, it is the functionality of action in solving a problem that motivates learning which is at the heart of Piaget’s account of conservation in mathematics.

            Alessi (1992), writing in the American Psychologist, distinguishes three types of selection: (1) phylogenic, or evolutionary, (2) ontogenetic, the “development of behaviour patterns within an individual’s repertoire during its own lifetime though reinforcement”; and (3), cultural, the development of various coordinated patterns of behaviour (p. 1359). He separated proximate causes that deal with micro-explanations (DNA) from ultimate ones which concern macro-ones and allow us to understand “how come?” (Alessi, 1992, pp. 1359 – 1360). Phylogenic DNA is the bookkeeper of our problem solving histories, Alessi said, though it is the phenotype that negotiates the secretive pressures with the environment, not the genotype. All selection is based on function: certain rules will help a group survive. As Alessi noted, for Skinner, behaviour operates at all three levels, which will be made explicit when I consider Piaget.

Further aspects of Alessi’s account have been corroborated.  Donahoe (2002), in “Behaviour and Neuroscience”, noted that selection by reinforcement in our individual lives alters neural architecture, that is, neuroanatomy. Prior skills are required to learn new tasks according to him. Henrich and associates (2003) writing about genetic and cultural evolution in a different context have remarked on the ability of cultural selection to be embedded in genetic evolution also.  

Piaget’s desire to have a biological basis for his account resonates with Skinner, who had already mapped out the relationship between how ontogeny and behaviour, mediated through culture, affects our genetic evolution. According to Piaget, we do things based on needs, which are partly socially determined. The needs required for development are rooted in the evolution of the human brain, for Piaget. Culture can, in principle, lead to a change in the nature or substance in the selection of needs.

Bargh and Chartrand (1999), in “The Unbearable Automaticity of Being”, challenged a staple of psychology that caused aversion to cognitive behaviourism, the idea that we are consciously and systematically processing incoming information to construe, interpret the world, and to plan and engage in action. They noted, however, for the behaviourist, “Environmental events directed all behaviour in combination with the person’s reinforcement history” (Bargh & Chartrand, 1999, p. 462). We do not choose to see an orange on the desk they point out. Commenting, they wrote, “[T]he entire environment-perception-behaviour sequence is automatic with no role played by conscious choice in producing behaviour” (Bargh & Chartrand, 1999, p. 466).

            Making situations manageable, the environment-perception-behaviour sequence they claimed serves a functional role.  Much of what we do is in relation to goals is automated, they contended, at different levels. In a nutshell, general goals guide more specific ones. According to them, further, limited conscious choice they said is for novel situations. Some automatic responses are natural and others require experience to develop according to them.

The idea of automaticity reflects Piaget’s view that we attempt to assimilate and accommodate novel situations to reach a cognitive equilibrium. Cognitive equilibrium is the prior knowledge upon which we learn. Automaticity requires the repetition of behaviour.  Cognitive disequilibrium occurs when we are confronted with a novel problem, resulting possibly in assimilation or accommodation. We may wish to note that the Piagetian account, in fact, resonates with the famous neurologist Michael Gazzaniga (2005), in a different context, who explained how as we learn the brain whittles down neurons and simplifies the brain response making it automatic too. In a social context, we have evolved for efficiently interpreting our environment through repetition of behaviour patterns.

            Turning to practice, Anderson and associates (1999) compared situated learning, the idea that learning is maintained in the external social world and constructivism, the notion that knowledge lies in an individual internal state. Anderson and associates noted that both situated learning and constructivism share that knowledge cannot be decomposed and decontextualized. Yet for cognitive science all tasks are to be broken into sub-task, and context dependence of knowledge is contingent upon what kind of knowledge we are considering. They wrote, “Learning requires a change in the learner, which can only be brought about by what the learner does” (Anderson et al., 1999, p. 18).    

            Anderson and associates suggested that understanding the parts of a task, clear goals, and understanding are all important to the learning process. They noted that it appears constructivism aims at Skinner’s vision in Walden II (1948, pp. 119 – 120, cited in Anderson et al., 1999), which is puzzling.

            Piaget, however, advocated practice upon which to acquire skills. Practice is more effective when broken down into sub-tasks provided the time is appropriate. The confusion arises, in fact, from viewing Piaget and Skinner as opposites. Once we abandon the standard abhorrence for behaviorism, we are free to employ what works.

              Gales and Yan (2001), in a lecture to the Annual Meeting of the American Educational Research Association, however, pit behaviourism against Piagetian inspired constructivism. Their remark is typical of the standard view: “Teachers, who believe in constructivist approach to student learning, find it important for students to think creatively” (Gales & Yan, 2001, p. 12). Educationalists have been busy showing up the weaknesses in behaviourism in relation to constructivism.

The positive correlation of Piaget’s and Skinner’s work impacts practice. There are aspects of behaviourism and constructivism that are both useful such as breaking down tasks into parts and having clear goals, on the one hand, and appealing to the cognitive needs of the learner, on the other.

Smith (2003), a math educator, considered children’s reasoning by mathematical induction in the International Journal of Educational Research. He provides an example of mathematical induction:

                                          1 + 3 = 

                                1 + 3 + 5 = 

               1 + 3…+(2n – 1) =  

As Smith noted, however, Frege (1884/1953) held that at the base of mathematics is identity, “2 = 2”, established by one-to-one correspondence; Piaget too, wanted to explain conservation as, Smith noted. For Piaget, we acquired the conservation of objects, like a cookie. Subsequent to classification arises, for Piaget, the concept of number by one-to-one correspondence, “one” cookie is the same as “one” apple.

Lakoff and Núñez (2003), cognitive scientists, however, contended that very small numbers can be “subitized” arising with object permanence (pp. 15 -19, 99). It is at least plausible to think that understanding of the number concept must be seen on a continuum where grasp of small cardinalities leads to social interactions that include one-to-one correspondence activities. Suffice it to say, that Piaget’s account of conservation as the basis for acquiring the number concept requires further empirical investigation which goes beyond the scope of this paper. What I have attempted to bring out is the behaviourist themes in Piaget’s writings and the value for doing so.

An empiricist philosophy of mathematics  

Remarking on how surprising it is that Piaget has not influenced philosophical thinking, Mays (1972) the translator of The Principles of Genetic Epistemology attributed the oversight to the prevalence of the linguistic turn, the movement whose adherents attempted to solve ontological problems by an analysis of language. Ordinary language philosophers analyzed conceptual matters in terms of verbal statements. The ordinary language philosophers were antagonistic to naturalism, the notion that most questions about knowledge must be answered by the methods of the empirical sciences.

Reading Piaget in light of Skinner contributes to solving persistent problems in the philosophy of arithmetic and remains little remarked upon by their empiricist predecessors, like Kitcher (1983) who built on the work of J. S. Mill.

There are connections we can make between cognitive behaviourism and an empiricist philosophy of mathematics; specifically Kitcher’s work relies upon the naturalists turn of Quine. Piaget anticipated, but did not embrace, the idea of a naturalized epistemology developed by W.V. Quine, the idea, in this context, that the formal sciences must rest on psychology, and backs away, I think, under the weight of Frege’s (1884/1953) charge of “psychologism”. Psychologism, we may wish to recall, was the famous criticism Frege originally aimed at Husserl that we should not confuse the contexts of justification (i.e., logic) from that of discovery (i.e., psychology).  Piaget, however, explained the rationale for naturalized epistemology and anticipated why Quine (1969) thought we are forced to blur the distinctions between discovery and justification—the logical spade must turn somewhere, etcetera.

The only difference between genetic epistemology and  Quine’s naturalized version is the final step Piaget does not take but the contemporary naturalist does.  For Piaget, the acquisition of mathematical knowledge and its justification form parallel lines that do not meet. For Quine, however, the lines overlap: the psychology of the formal sciences can provide the foundations for mathematics. It is altogether reasonable to think that a Piagetian should embrace naturalism and with the good reasons that have already been elucidated by Quine; we need not rehearse them, as that goes beyond the scope of this paper. At the very least, the naturalist account of the origin of the formal sciences is not something to be scoffed at, as Mills’ views often have been; and Piaget had already helped us understand why. In fact, laying bare the origin of mathematics in our interactions with the physical world, Lakoff and Núñez (2003) have discussed the genesis of the discipline in detail, which adds substance to a Piagetian approach.

Also, Kitcher claimed that we must distinguish between the foundation so of the axioms of arithmetic that are acquired from experience from what we generate from them. Mays (1972) pointed out that the notion of equilibrium functions according to Piaget at lower and higher levels. At the lower level the child’s needs lead to disequilibrium. At the higher level, concrete classificatory schemes develop into formal ones.  In the jargon of the philosophy of mathematics, Piaget offered a two-tier epistemology, the idea that the axioms of a formal system like Peano number theory, are based in experience, but the theorems we generate from them are not. Pure mathematics, from an empiricist point-of-view, becomes intelligible for a cognitive behaviourist in that rules flow from experience. For Piaget, we have simple (first-tier) and reflective (second-tier) abstractions.

Piaget’s distinction between types of abstractions, anticipated Hilary Putnam’s (1983) modal account of the existence of abstract objects. According to Putnam, who has been influential in the philosophy of the formal sciences, mathematics is about what structures are and are not possible, not existence. In fact, the traditional mystery of intuition which Platonists are forced to—how do we grasp abstract objects like the truth-values—is  given content by the cognitive behaviourist, because, as I have configured it, she seeks out the origin of mathematical knowledge.

Nöe’s (2001) refrain in Action in Perception, though reiterating the standard caricature of behaviourism, is familiar to students of Piaget, “The root of our ability to think about the world is our ability to experience it; but experience is a mode of skilful encounter” (pp. 208 & 138). Along the lines of both Quinian naturalism and Putnam’s modal argument, Piaget (1972) offered the beginnings of an empirical, pragmatist account of mathematics. Before we embrace the cognitive behaviourist Piaget, we first must free ourselves from the sword of Damocles.

               That is, we may wish to pause to consider Chomsky’s (1956/1967) “A Book Review of B. F. Skinner’s Verbal Behavior”, which in some quarters has been taken as a refutation of that program. A few words are in order to show that my cognitive behaviorist view does not fall by the waist side with Chomsky’s critique. Putnam (1998), remarking on the downfall of behaviorism in a different context, summarized the gist of Chomsky’s critique:

Chomsky [1956/1967] pointed out that once the notion of a stimulus becomes so wide that World War II is a ‘stimulus’ (and the response takes place twenty years later), stimulus-response talk has become mere jargon with no real control. (p. 6)  

               We may wish to note that Chomsky characterized (1956/1967) his critique as a reduction ad absurdum, one that he said “I find little I would change [today]” and did “not see how his [Skinner’s view] could be improved upon” (p. 1).We must say something about Chomsky’s critique that has often been taken to be a knock down argument against behaviorism, though his tone verges on the polemical.   In Chomsky’s critique (1956/1967), we read of Skinner’s views with descriptors like: “no clear content” (p. 11), “pointlessness of these claims”, (p. 14),  “confused”, (p. 16), “an illusion” (p. 17), “extremely puzzling” (p. 18), “vagueness” (p. 21), “completely trivial or hopeless” (p. 23), and “futile” (p. 23).

               For Chomsky, Skinner just cannot show that all behavior is response to a stimulus in a way that is scientifically meaningful. Skinner, unwittingly perhaps, stacked against himself by aiming for a science of behavior. We may not be able to gauge that every behavior is response to a stimulus that can be measured.

               Getting away from the science of measured stimulus-response relations, however, the behaviorist model can be, I think, a potent psychology. As I have attempted to argue, Skinner helps us understand the mechanisms by which cognitive theories like Piaget’s works. The benefits of cognitive behaviorism are complimentary. Though my purpose has been to defend the behaviorist strain in Piaget and not Skinner, we have the following unexpected results. In Piaget’s thought, as I anticipated, we can better understand the sensory-motor mechanism by which cognitive change occurs by paying attention to the way behavior is conditioned. In Skinner’s thought, as I had not anticipated, there are cognitive dimensions, which he, on occasion, has gone to awkward troubles to avoid. He need not do so. Neural behavior and our behavior proper work in tandem. The slogan: mind and behavior work together.  

               Suffice it to say, Chomsky’s famous critique may be fatal to behaviorism as Skinner had envisioned it in Verbal Behavior where he was far more uncompromising towards the cognitive domain than his texts I have relied upon in this essay (e.g., Skinner 1953; 1974; 1989). Chomsky’s critique is unlikely to affect, however, the more general ideology of behaviorism utilizing the key idea that we are shaped by past experiences, which is indispensible, as well as some of his explanations about the processes involved. Further, once we emphasize the Skinnerian themes already present in Piaget’s writings, as I have done, we immunize ourselves from Chomsky’s critique. For Piagetians, talk of stimulus and response may be jargon, but ones that are useful in contributing to an explanation of behavior, nonetheless.

The further development of cognitive behaviourism

My interpretation of Piaget is based on the role he assigned to behaviour in an explanation of the acquisition of conservation in mathematics. Piaget and Skinner view mathematical knowledge as arising from our interaction and the manipulation of physical objects that itself is motivated by needs for Piaget, and deprivation in Skinner’s terms. Needs are formed in the context of conservation for Piaget as attempts to resolve cognitive disequilibrium. Skinner highlights the everyday origins of abstract objects, speaking of their metaphoric nature.

Skinner’s view is one-sided, however. Skinner, can benefit by paying attention to the cognitive dimensions of behaviour, and Piaget points the way. Rather than being taken by the “new turn” whereby behaviourism is just displaced, however, what we need is careful empirical studies of what works—on a case-by-case basis, which is what Piaget provided the ground for with observations on a scale “unrivalled in the history of developmental psychology”; and Skinner would concur (Beilin, 1992, p. 191; Skinner, 1953, p. 15).  There are also theoretical advantages for the philosophy of mathematics that come with cognitive behaviourism.

            Since I have emphasized, relying on Skinner, the social dimensions of learning, it is only natural to think of Vygotsky; and Bandura, often associated with the term “cognitive behaviourism”. When we read through the journals in the field, it is hard not to be struck by the number of fusions we see between, Piaget and Vygotsky, as well as others. The reason, ostensibly, is that the authors of such enterprises think that we gain more by putting psychological theories together than pitting them against each other as was often done in introductory textbooks. My aim, however, has been only to focus on Skinner and Piaget; but future work requires that we try as much as possible to bring a synthetic view to bear on understanding thinking and acting in a social context. The shape of future work will inevitably be guided by the debates that were forged between behaviourists and soldiers in the cognitive revolution.

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