On Mathematical Learning and Understandingjwilson.coe.uga.edu/EMAT8990/FIRST/Papers2001/PaperRISE... · Web viewOn Mathematical Learning and Understanding A Constructivist’s Perspective

Seminar Discussion Paper. Do not cite without permission.

On Mathematical Learning and UnderstandingA Constructivist’s Perspective1

Leslie P. Steffe

Mathematical learning and understanding are currently being interpreted in social context (e.g., Cobb & Bauersfeld, 1995; Clarke, 2001; Lerman, 1996; Sfard, 2000; Wertsch & Toma, 1995; Voigt, 1996). In some of these interpretations, but not all, the claim that learning and understanding is inherently social is “very much in the limelight these days” (Wertsch & Toma, 1995). These claims involve not only the processes involved in learning and understanding, but in their products as well. For example, with regard to learning, Cobb (2000) commented that “learning is not merely social in the sense that interactions with others serve as a catalyst for otherwise autonomous conceptual development. Instead, the products of learning, increasingly sophisticated ways of knowing, are also social through and through” (p. 154). Cobb’s claim is quite interesting because it proceeded from an analysis of the development of increasingly sophisticated conceptual units among the Oksapmin people2 and the contribution of social interaction to that development (Cobb, 1996). Because the operations, unitizing and reprocessing, which produce the conceptual units to which Cobb referred are understood as emerging from within the individual as a result of individual-environment interaction3 (Steffe, von Glasersfeld, Richards, & Cobb, 1983), how they might be regarded as social through and through requires an explanation. The unitizing operation is used in the production of object concepts, and, when it is used to reprocess object concepts, the object concepts are reconstituted as conceptual units of the kind Cobb mentioned (Steffe, 1991). The work of Piaget (1954) indicates that the object concepts produced by unitizing are indeed based on child-environment interactions, and, further, that at least some of these interactions are interactions with others. But to claim that the construction of object concepts and the conceptual units that follow is inherently social or social through and through raises a fundamental question concerning the constitutive nature of mathematical thinking as well as mathematical learning. In its extreme form, the claim that learning is inherently social seems to eliminate the individual mind as does Restivo (1999) in his explanation what he refers to as The Social Construction Conjecture.

The basic claim relevant here is that all knowledge is socially constructed. This should be simple enough. For how else is it—how else could it be—that we humans come to know things, come to formulate words and sentences about things, except through our interactions with others? How else, indeed, is it that we are ourselves constructed” (p. 121)?

1 Writing this paper was supported by the College of Education, Deans Office, The University of Georgia. It is also written as part of the activities of NSF Project No. REC-9814853. All opinions are solely those of the author. 2 The conceptual units referred to are based on counting strategies Saxe identified in a study of the Oksapmin people in Papua, New guinea (Saxe, 1991). 3 The environment of the subject refers to “the experiential field in which the observer has isolated that organism” (von Glasersfeld, 1995, p. 123).

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Restivo’s understanding of thinking—the experience of “inner thought”—as internal conversation demonstrates just how strong his claim is: “Individuals are vehicles for expressing the thoughts of social worlds or ‘thought-collectives’. Or, to put it another way, minds are social structures” (p. 128). Restivo’s claim stands in stark contrast to Einstein’s reply to the questions set to him by Jacques Hadamard (1949, Appendix I). In his reply, Einstein (1949) made the following comment:

The words of the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined. (p. 142)

Einstein’s “psychical entities” and Restivo’s “inner thought” seem to be poles apart. But in a review of the Social Construction Conjecture, I made an appeal to Restivo that:

In the service of scientific discussion, it would be very helpful if Restivo would set forth conditions under which he would consider his understanding of inner thought falsified in Lakatos’ (1970) sense of sophisticated falsificationism. Knowing these conditions would be useful …in understanding a model which would include inner thought as internal conversation as well as other possibilities” (Steffe, 2000, p. 58).

My goal in this paper is to propose mechanisms that I use to account for children’s mathematical thinking and learning which cannot be accounted for by simply appealing to internal conversation, but which also include aspects which can be learned only by means of social interaction.

Sfard’s Discursive Approach to Cognition

Restivo’s Social Construction Conjecture seems compatible with Sfard’s discursive approach to cognition in which “thinking is nothing other than communication with oneself” (Sfard, 2000, p. ***). Given her definition of thinking, Sfard (2000) elaborated:

Investigating communication with others may be the best route to discovering the mechanisms of human thinking. From this conclusion, it follows that thinking is subordinated to, and informed by, the demands of communication. I wish to emphasize that most of these demands are basically the same whether the communication is with oneself or with others. (p. ***)

Sfard (In Press), however, sees what she considers as discourse analysis in her neo-Vygotskian research program as incommensurable with studies of the ontogenesis of mathematical concepts and operations in neo-Piagetian research programs. According to Sfard (In Press), two theories are incommensurable if they speak different languages rather than really conflicting with each other. In this case, “There is simply no set of common criteria with the help of which the apparent controversy could be rationally resolved” (Sfard, In Press).

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Following Sfard’s suggestion, one way to decide if two conceptual systems are incommensurable is to consider the languages used by proponents of each system. Toward this end, Sfard’s definition of discourse is a fundamental link between discourse analysis and the concept of interaction in radical constructivism. According to Sfard (2000), discourses are, “autopoietic systems, that is, systems that continuously generate themselves and that create their objects (see Maturana & Varela, 1987)” (Sfard, 2000, p. ***). Not only is this understanding of discourse a fundamental link between discourse analysis as elaborated by Sfard and the concept of interaction in radical constructivism, it is a fundamental link with Pirie & Kieren’s (1994) model of mathematical understanding and with their enactive perspectives on mathematical cognition.

In her analysis of discourse, Sfard (2000) focuses on communication and defines it as “an activity in which one is trying to make other people act or feel in a certain way” (p. ***). In elaboration, she cites Levinson (1983) as saying: “Communication consists of the ‘sender’ getting the ‘receiver’ to recognize that the ‘sender’ is trying to cause that thought or action” (p. 16). According to Sfard, what distinguishes her analysis of mathematical communication from previous studies conducted in the neo-Piagetian tradition concerns the relation between symbols and meaning. Essentially, she sees mathematical discourse and its objects as mutually constitutive: “It is the discursive activity, including its continuous production of symbols, that creates the need for mathematical objects; and these are mathematical objects … that, in turn, influence the discourse and push it into new directions” (p. …). So, it is no surprise that Sfard defined learning as an “initiation into a certain type of discourse” (Sfard, 2001), and postulated that “Children will not construct mathematical discourse without an expert participant” (Sfard, 2001).

Piagetian Interactionism

Piagetian interactionism and Sfard’s idea of communication, although compatible, are not identical. One difference is that, in Piagetian interactionism, there are two kinds of interaction that are taken into account in the construction of mathematical knowledge. These two kinds of interaction are implied by a quote from Piaget made by Piattelli-Palmarini (1980). This quote is important because Piattelli-Palmarini used it to establish interaction as a core hypothesis of Piaget’s genetic epistemology.

Cognitive processes seem, then, to be at one and the same time the outcome of organic autoregulation, reflecting its essential mechanisms, and the most highly differentiated organs of this regulation at the core of interactions with the environment” (p. 4; Piaget, 1974).

The first kind of interaction implied in this quotation is the basic sequence of action and perturbation implied by self-regulation at the core of interactions with the environment; that is, individual-environment interaction. In this, "perturbation" is not understood as the transferal of information from an environment outside of the experiential world of the individual to the individual. Rather, it is understood as a special kind of disturbance within the experiencing individual.

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Sociocultural critics of radical constructivism often interpret the phrase “interactions with the environment” as excluding interaction with others. But this criticism simply repeats what Smith (1982) referred to as the Social Objection to Piaget’s system:

Piaget’s theory fails because it treats social cognition as if it were physical cognition, because it takes a subject’s knowledge of the social world as being not essentially different from a subject’s knowledge of the physical world. Thus it is claimed … it takes the physical environment to be constitutive of a subject’s environment. (p. 174)

Even when both kinds of subject-environment are taken into consideration, the construction of mathematics cannot be accounted for subject-environment interaction. Another kind of interaction is essential, and it is implied by Piattelli-Palmarini’s quotation of Piaget. I refer to it as within-subject interaction because it consists of the interaction of constructs in the course of re-presentation or other operations that involve previously constructed items implied by organic autoregulation. Within-subject interaction can take place in the presence of subject-environment interaction as well as in its immediate absence4. The two kinds of interaction are reciprocal in the sense that subject-environment interaction can set within-subject interaction in motion and induce modifications in it which, in turn, modify subsequent subject-environment interaction.

Maturanaian Interactionism

Piaget’s interactionism is in many respects compatible with Maturana’s idea of interaction. According to Maturana (1980), an interaction occurs “Whenever two or more entities change their relative positions in their space of existence as the result of the interplay of their properties” (p. 27). This view of interaction is compatible with Piaget’s view for several reasons. First, the entities need not all be living organisms as indicated by Maturana’s (1980) paradigmatic example of a fly walking on a painting by Rembrandt. This is compatible with how Piaget (1964) regarded interaction, and it keeps open the possibility that the individual’s interactions in both its physical and its social environments contribute to the construction of mathematics.

Second, Maturana’s emphasis on the change of relative position in their space of existence of the interacting entities as a result of the interchange of their properties highlights the assumption in Piagetian interactionism that the concepts and operations of the interacting individuals are constitutive aspects of their interactions in their environments. Third, Maturana (1978) distinguished between two non-intersecting domains of interaction of the participants involved in an interaction. The first of these is the domain of interactions of the individual within his or her environment, and the second is the domain of interactions among components within the individual. As already noted, these two kinds of interaction are basic in Piaget’s view of interaction as well. Finally, in

4 I distinguish Piaget's regulation according to the two types of interaction. I use "self-regulation" in the case of regulation of individual-environment interaction, and "autoregulation" in the case of regulation of interaction of constructs within the individual.

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both Maturana’s and Piaget’s views on interaction, two individuals might interact but not communicate.

In order to communicate, Maturana & Varela (1980) suggested that one organism can affect the behavior of another organism by orienting the behavior of the other organism to some part of its domain of interactions comparable to those of the orienting organism: “This can take place only if the domains of interactions of the two organisms are widely coincident” (pp. 27-28). This notion of communication is in harmony with Piaget’s analysis of how a statement uttered by one person could be agreed to by another:

How could such a convergence be established? The two subjects necessarily have different, non-interchangeable perceptions: they exchange ideas, that is to say, judgments concerning perceptions but never the perceptions themselves! (Translated and quoted by von Glasersfeld, 2000, p. 7).

von Glasersfeld (2000) amplified on this quotation by saying: “He (Piaget) comes to the conclusion that meanings are a matter of ‘private symbolism’ and agreement cannot manifest itself except through reactions due to mutually compatible operations” (p. 7). These insights into communication justify emphasizing communication in constructive learning and establish a compatibility between Sfard’s approach to mathematical discourse and Piagetian interactionism. Still, there is a basic distinction to be made between the two approaches.

Self-Generative Activities in the Construction of Knowledge

That distinction concerns the role that von Glasersfeld (1995) attributes to re-presentation in his model of language.

The main addition to the Saussurian break-down is the introduction of ‘re-presentations’. In my view, this addition is essential, because the ability to call up re-presentations in listeners or readers is what gives language its enormous power and differentiates it from all forms of signalling. It is also the feature which distinguishes my model of language and communication5 from those of Humberto Maturana and Richard Rorty, many of whose other ideas are perfectly compatible with radical constructivism. (p. 132).

In von Glasersfeld’s model of language, the meaning of a word or a combination of words consists of the conceptual structure to which the word or a combination of words point. In the frame of reference of the individual, such meaning consists of those aspects of the conceptual structure of which the subject is aware in the course of re-presentation6. In the observer’s frame of reference, such meaning consists of whatever conceptual structure the observer can legitimately impute to the individual regardless of inferences made concerning the individual’s awareness. Because of these considerations of

5 See Schmidt (2000) for an account of von Glasersfeld’s model of language and communication.6 A re-presentation is a regeneration of a past experience when the relevant sensory material is not available.

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meaning, interaction cannot be taken as a given, explanatory construct in the construction of mathematical knowledge even though the individual is regarded constitutively as an interacting organism.

In learning how to interact mathematically with children, an adult must construct whatever concepts and operations are afforded by the language and actions of children as well as the images they produce using those concepts and operations as the children’s meaning of their language and actions. In this construction, the self-generative activities of children while they interact with each other or with an adult provide indication of their concepts and operations7. Children’s self-generative activities can and do occur in the context of their interactions, and should be regarded as a constitutive part of the interaction. In fact, the presence of self-generative activity in interaction provides reason to foreground the operation of assimilation in the context of interaction of all kinds.

An assimilatory apparatus is indispensable, according to Piaget (1980), for “deriving information from an object8” (p. 89). Further, Piaget expanded on his notion of these assimilatory apparatuses in a way that has greatly influenced my concept of the nature and origins of mathematics.

We understand by ‘endogenous’ only those structures which are developed by means of the regulations and operations of the subject … By serving as an assimilatory framework, then, these structures are added to the properties of the external object, but without being extracted from it. (Piaget, 1980, p. 90).

Following on from this comment, the hypothesis that mathematics is added to the individual’s experiential reality rather than being empirically abstracted from it can be constructed. This hypothesis leads to an important understanding that experience is insufficient for the construction of mathematics be it social or otherwise. Under this hypothesis, the sources of mathematical knowledge resides in what individuals make from their interactions. Viewing the sources of mathematical knowledge in this way again raises the question whether children’s constructed mathematical realities can be adequately accounted for by using Restivo’s Social Construction Conjecture. Moreover, it clarifies why I question whether mathematical thinking should be subordinated to the demands of communication or mathematical learning subordinated to the initiation of the neophyte into a mathematical discourse community9. What it does suggest to me is that we adults must learn to interact mathematically with our students and not take such interaction as unproblematic.

Interaction and Learning

7 The self-generative activities of the adult are also involved.8 This involves interaction between “the subject and the object”. I use quotation marks to indicate that I interpret Piaget as speaking in terms of observer language. 9 In saying this, I do not underestimate the need of human beings to communicate nor do I underestimate the advantages in establishing a mathematical discourse community. Both of these things are essential in mathematical learning.

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Cobb's (1990)view that learning is interactive as well as constructive could be interpreted as indicating that constructive learning can occur in the absence of interaction, a possibility that is often suggested by theorist's who work in social-cultural contexts. For example, Renshaw (1992), a neo-Vygotskian, made the following criticism concerning constructive learning.

In promulgating an active, constructive and creative view of learning, however, the constructivists painted the learner in a close-up as a solo player, a lone scientist, a solitary observer, a meaning maker in a vacuum (Renshaw, 1992, p. 91).

Renshaw's criticism of constructive learning does not make explicit the fundamental necessity of regarding cognitive processes as the outcome of self-regulation of individual-environment interaction as well as of autoregulation of the constitutive processes involved in the interaction. In his statement, he seems to view construction as a rarefied form of invention that occurs without individual-environment interaction. This view might stem in part from the necessity to regard accommodation, or constructive learning, as occurring as a result of within subject interaction. Even though constructive learning occurs in the context of subject-environment interaction or follows on from such interaction, I would like to keep open the possibility that constructive learning can also occur as an invention with only the most minimal preceding subject-environment interaction.

I can see no necessity to distinguish mathematical thinking and learning in a Piagetian framework and a Vygotskian framework on the basis of within-subject interaction, because Vygotsky accounted for this type of interaction as verbal thought. According to Vygotsky (1962), "in inner speech words die as they bring forth thought. Inner speech is to a large extent thinking in pure meanings. It is a dynamic, shifting, unstable thing, fluttering between word and thought. Its true nature and place can be understood only after examining the next plane of verbal thought, the one still more inward than inner speech" (p. 149). Vygotsky regarded this next plane as thought itself. The issue of whether verbal thought is sufficient to explain mathematical thought does not detract from the insight that Vygotsky’s notion of verbal thought can be considered as “within subject” interaction. In Vygotsky's social-cultural approach, then, a child may not appear to be interacting with any observable thing or item but yet be involved in thought. In this case, it is only from the observer's point of view that the child might not be involved in some form of purposeful mental activity.

In summary, in my understanding of constructive learning, with the possible exception of those rare cases of pure invention, accommodation follows on from subject-environment interaction and is understood as occurring within the instruments that are used in the subject-environment interaction. So, there is a reciprocal relationship between the two kinds of interaction in Piagetian interactionism: subject-environment interaction can occasion changes in the instruments used in that interaction, and these changes can in turn modify subject-environment interaction. In passing, it is important to note that subject-environment interaction can occur without inducing changes in the instruments of interaction. In fact, I make a distinction between active mathematical learning and

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mathematical activity that does not involve modification of the scheme involved in the activity. This distinction is important because engaging children in mathematical activity is not synonymous with engaging them in active mathematical learning.

Learning as a spontaneous process.

Piaget (1964) characterized learning in the following way:

In general, learning is provoked by situations--provoked by a psychological experimenter; or by a teacher with respect to some didactic point; or by an external situation. It is provoked as opposed to spontaneous. In addition, it is a limited process--limited to a single problem or to a single structure (p. 8).

Although I agree with Piaget that learning is not spontaneous in the sense that the provocations which occasion it may be intentional on the part of a teacher or some other person, I maintain that constructive learning should be regarded as a spontaneous process in a way similar to how Piaget (1980) regarded spontaneous development.

This brings us back to the child, since within the space of a few years he spontaneously reconstructs operations and basic structures of a logico-mathematical nature, without which he would understand nothing in school. … He reinvents for himself, around his seventh year, the concepts of reversibility, transitivity, recursion, reciprocity of relations, class inclusion, conservation of numerical sets, measurements, organization of spatial referents. (p. 26)

In regarding mathematical learning as a spontaneous process, two aspects of the spontaneity of development are of importance. The first is its unintentionality. Children certainly do not intend to construct the operations which Piaget mentioned, and it would be quite rare for others with whom young children interact to intentionally engender such operations.

It appears to be extremely difficult to define “mathematical contexts,” especially with reference to young children. Given the very general basis for construction of logical-mathematical operations … almost any situation that can be commented on, asked about, indicated as desirable, etc., can lead to actions, utterances, gestures, or other communicative acts that have something to do with logic or mathematics (Sinclair, 1990, p. 25).

Following on from Sinclair’s comment, it would seem that an observer would be hard pressed to identify causal communicative acts or situations that give rise to the operations Piaget identified. In fact, it would be a surprise if a child encountered a situation that, to an observer, would be obviously a transitive situation prior to transitive reason making its appearance in the child. This ubiquitousness of the constructive process is the second aspect of spontaneous reconstruction that is essential in regarding learning as a spontaneous process.

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In maintaining mathematical learning as a spontaneous process, it is essential to bring forth the schemes that children have constructed through spontaneous development in their mathematical education. It is by these means that mathematical learning need not be regarded as limited to a single problem or as a limited process. In fact, it should not be the intention of a mathematics teacher for children to learn to solve single, isolated problem even though the presented situation might be a problem for the children. Rather, the intention should be to understand the children’s schemes of assimilation and how to engender changes in those schemes as a result of children’s mathematical activity. So, I do not use “spontaneous” in the context of mathematical learning to indicate the absence of elements with which the children interact. Rather, I use the term to refer to the noncausality of teaching actions, to the self-regulation of the children when interacting, to a lack of awareness of the learning process, and to its unpredictibility. What children learn often is not what was intended even by the most perspicuous teacher, and children often learn when the teacher has no such intention. Even in those cases where the children seem to learn what the teacher intends, the event of learning cannot be said to be caused by the actions of the teacher. Teaching actions at most occasion children’s learning (Kieren, 1994).

Bringing Forth Children’s “Spontaneous Schemes”

Pirie & Kieren (1994) have elaborated a model of mathematical understanding in which they identified primitive knowing as the starting point in the process of coming to understand. Of primitive knowing, they commented:

Primitive here does not imply low level mathematics, but is rather the starting place for the growth of any particular mathematical understanding. It is what the observer, the teacher or researcher assumes the person doing the understanding can do initially. (p. 66)

I find it advantageous to interpret the idea of primitive knowing in terms of ways and means of operating that can be regarded as indicating children’s “spontaneous schemes”. One of these is the counting scheme of children that I refer to as the explicitly nested number sequence. Although this scheme may be constructed by children as a result of interacting in their social-cultural milieu10, it is my observation that their experiences in negotiating the traffic with numbers in school also plays a role (Steffe, Cobb, & von Glasersfeld, 1988). So, to acknowledge the role of children’s interaction in their social-cultural milieu outside of school, of embryogenesis, and of children’s interactions in their social-cultural milieu inside of school in the construction of this scheme, I consider the explicitly nested number sequence a “spontaneous scheme,” where the quotation marks are used to alert the reader to the possibility of all three factors being involved in the scheme’s construction.

The way in which Pirie & Kieren define primitive knowing assumes that not only is primitive knowing an observer’s concept, but it is also relative to some particular

10 I have come to believe that embryogenesis plays a part in children’s construction of this scheme, but I have no access to the contribution of this process.

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mathematical understanding. One doesn’t need to change topics and ask, say, how the explicitly nested number sequence can be regarded as primitive knowing with respect to multiplying and dividing schemes. Instead, Pirie & Kieren’s model can be used to differentiate understanding within a particular scheme when consideration is given to children’s construction of that scheme. For example, the perceptual counting scheme can be regarded as primitive knowing with respect to the explicitly nested number sequence. Counters of perceptual unit items can count things they can see, hear, or touch, but when the items to be counted are not in their perceptual field, they cannot count. Clearly, this is to be regarded as primitive knowing because children who are counters of perceptual unit items must establish items that are not in their visual field as countable items, which Pirie & Kieren call “image having”. But I will not pursue analysis of children’s construction of their number sequences using the Pirie & Kieren model because my interest here is in illustrating how the explicitly nested number sequence constitutes primitive knowing with respect to what I call the iterating units-coordinating scheme, which is a multiplying scheme. Rarely, if ever, are children’s ways and means of multiplying intentionally based on this “spontaneous scheme”.

The Explicitly Nested Number Sequence

Before considering how children can construct multiplying as an accommodation of their explicitly nested number sequence, I first illustrate how I brought this sequence forth in two children in their third grade in school and then explain what I mean be the phrase. The basic task that I used in a teaching episode11 with the two children, Jason and Patricia, was for them to cover part of 19 toys in the playground of TIMA: Toys12 and then to find how many toys were covered. The child who covered the toys did not know how many he or she covered except in the extreme case of covering only a few toys or all but a few toys. The child who was not covering toys shut his or her eyes so how many being covered could not be seen. In the protocol, “T” refers to the teacher, “P” to Patricia, and “J” to Jason.

Protocol I: Finding how many of a part of nineteen toys are covered.J: (Places 19 toys in the playground spaced so they do not overlap. Both

children count the toys as Jason places them.) T: (Asks Patricia to shut her eyes and Jason to cover part of the toys. After

Jason covers part of the toys quickly) Ok. P: OK. Let’s see. (Looks intently at the screen and subvocally utters number

words while counting the eight visible toys. She then looks away from the screen into space and continues to subvocally utter number words. She has her hands clasped together, but there were slight finger movements observable.) Whoops. (Starts over and counts the visible toys.)

J: (As Patricia initially started to count the visible toys, Jason rests his face in his hands and subvocally utters number words while looking at the screen. While Patricia is looking into space subvocally uttering number words,

11 This teaching episode was part of the activities of NSF project No. RED-8954678, “Children’s Construction of the Rational Numbers of Arithmetic”. 12 TIMA: Toys (TIMA: Tools for interactive mathematical activity) is a computer tool that was designed for use in the teaching experiment (cf. Biddlecomb, 1994).

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Jason completes his mental activity) I know it! (Waits for Patricia to complete her solution.) I forgot what the number was!

P: (When starting over, looks away from the screen, nods her head as she subvocally utters number words and almost imperceptibly sequentially moves her fingers) OK.

T: (Asks Patricia how many). P: (Counts the fingers she used to keep track). OK, eleven. J: (Agrees that it is eleven).

This was the first time that I had taught the children, so I had nothing to do with the way in which they found how many toys were covered. Nevertheless, I had asked the children to place twenty toys (rather than nineteen) in the playground with the expectation that they would place them one-by-one and count them as they were being placed. I also asked Jason to cover part of the toys and, by saying, “OK”, indicated to the children that they were to find how many toys were covered.

These are important considerations because I was obviously communicating with the children in the sense that Maturana explained. However, as noted, I had nothing to do with the way in which the children counted to find how many toys were covered. For me, this is a crucial point because it gives reason to Piaget’s notion of “assimilatory apparatuses” and to his notion that the children “added” their counting schemes to the situation of a partially covered collection of toys. Finally, it provides corroboration of the necessity to consider the children’s self-generative activities in interactive mathematical communication.

Number Words as Symbols

I claim that the children were at least at the level of image having13 in Pirie & Kieren’s model of mathematical understanding with respect to their number sequences (but not with respect to their multiplying schemes), because they could operate without actually counting the covered perceptual items. By all indications, “eight” was a symbol for counting the eight visible items just as “eleven” was a symbol for the counted items after the children counted. However, prior to counting, the hidden items seemed to be symbolized by the images the children generated in re-presentation. Although I will argue that these images were images created by using the auditory records in the items of their number sequence to nineteen, children can and do create images of discreteness that are not images of number words that I would argue are symbolic images. In any event, all of eight, nineteen, and the remainder of eight in nineteen seemed experientially real to the children, and I consider them as mathematical concepts that were symbolized by their number words. When number word symbolizes a mathematical concept, the number word brings forth14 the constitutive operations of the concept and images that can be produced using these operations.

13 Pirie & Kieren (1994, p. 66) defines “image having” as being able to operate without acting on physical objects. 14 “Stands in” is often used instead of “brings forth”. I use the weaker of the two phrases because children often cannot reason using the number words as stand ins for their images (cf. Steffe & Olive, 1996).

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The explicitly nested number sequence as a scheme.

I could argue that the children were even at higher levels of understanding in the Pirie & Kieren model than image having with respect to their number sequences, but rather than do that, I turn to explaining what I regard as the children’s counting schemes. In that a scheme can be partially understood as a goal-directed pattern of action or operation, I turn first to a consideration of the children’s goal. That their goal was to find the unknown numerosity of the covered toys is indicated by the fact that both children independently counted from “eight” up to and including “nineteen” and kept track of their counting acts. Patricia coordinated almost imperceptibly moving each finger with subvocally uttering the number words “nine, ten, ..., nineteen,” and, upon reaching “nineteen,” she counted her records of her counting acts and said, “OK, eleven” after the teacher asked her “how many?” Jason, on the other hand, counted his counting acts as he counted, because he said, “I know it” long before Patricia had completed her counting activity.

The scheme’s situation. A situation of a scheme is always interpreted as a constructed situation from the point of view of the child. To infer the situation the children constructed after Jason hid some of the nineteen toys, I look to how the children proceeded and to their inferred goal. The children had just counted the items as they were placed in the playground, and to count as they did means that they re-presented counting to nineteen. My inference is that the image created by the re-presentation consisted of what I call an image of the verbal number sequence; that is, an image of the auditory records of uttering number words. I make this inference because the children took the segments of their number sequence from “nine” up to and including “nineteen” as input for making countable items. So, what I regard as the explicitly nested number sequence was used to constitute the first part of their counting scheme; that is, the situation of counting. The number words the children counted from “nine” to “nineteen” referred to hidden perceptual items, but what the children were indeed counting were not perceptual items. Rather, they created the items they counted and those items were the number words that I have indicated.

The scheme’s activity. The counting activity of the children is an example of what von Glasersfeld (1980) called the activity of a scheme. In sensory-motor schemes, the scheme’s activity is triggered by the situation rather than being contained in the situation in the way that I infer that the activity of counting was contained in the children’s number sequences. When the children created their situation of counting, they produced an image of their verbal number sequences, and the items of this verbal number sequence symbolized counting acts. When they engaged in counting images of counting acts, they again used their number sequence both to create countable items in the activity of counting and to count those countable items. So, when coupled with the fact that the result, “eleven”, closed the children’s activity, it is legitimate to refer to the explicitly nested number sequence as a counting scheme because there was a constructed situation, an activity, and a result that closed the activity. In referring to it as a scheme, I again emphasize that the scheme’s activity was already recorded in the explicitly nested number sequence in the form of records of operating.

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My concept of a number sequence is a sequence of arithmetical unit items, where an arithmetical unit item is the record of unitizing an image of a counting act15. I do not regard arithmetical unit items as static objects stored somewhere in the mind that can be used to produce material for further operating. Rather, I think of them as records of unitizing operations—“slots”--that contain records of counting acts. An activated number sequence is somewhat like a resonating tuning fork with the stipulation that its resonating creates an anticipation of counting. In that case where the child produces an image of counting, the image might be simply some minimal re-presentation of the involved number words that symbolize counting, or it might be a plurality of flecks that symbolize countable perceptual items. As I have already explained, a child who has constructed the explicitly nested number sequence can take these images as material of using the number sequence. Another way of saying this is that the number sequence can be taken as material of its own operating, and in this sense, it is recursive. This permits the child to produce two number sequences, one to operate on and one to operate with.

One as an iterable unit. The ability to infer that a child takes a segment of his or her number sequence as input for creating countable items is necessary in making the further inference that the unit of one is an iterable unit, because only then can a child run through the sequence and abstract a unit that could be used in iteration to produce the sequence. The child is aware of the segment that it takes as input for counting, and keeping track of counting implies that the child is aware of operating—of how he or she is counting. That is, the child is aware that she or he will produce the next counted item of the sequence of counted items upon the next counting act.

When the child says, for example, “eleven is three”, the child is aware that it is going to produce the next item of the sequence and count that item as one more than the count of “eleven”. More generally, the child is aware that it will count the numerical item immediately following the currently counted item and this awareness is present prior to engaging in the activity. It is in this sense that I call the operations that the child uses to count its counting acts iterative operations. Not only is the unit of one abstracted from the sequence of units of which the child is aware (and thus implies that sequence), this abstracted unit is embedded in a program of operations that are used in iterating the unit.

Did the Children Engage in Mathematical Thinking?

As Patricia looked away from the screen into space while subvocally uttering number words, and as Jason rested his face in his hands while subvocally uttering number words, the children were engaging in what I regard as mathematical thinking, and rather intense mathematical thinking at that. Could these specimens of mathematical thinking be regarded as internal conversation as indicated by Restivo, or as the children communicating with themselves, as indicated by Sfard? I consider the children’s mathematical thinking as particularly relevant to these questions because they did use their number words in finding that there were eleven toys covered.

15 cf. von Glasersfeld, 1981 for a model of the unitizing operation.

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Although the children did use number words in thinking, both of them were caught up in the activity of counting, and there were no indications of what I consider as an internal conversation that were observable to me. Nor were there any indications that the children were communicating with themselves beyond Patricia saying, “Whoops” when she lost her place in counting. Indications of an internal conversation or communication would seem to involve the children assuming the roles of both speaker and listener in the counting episode or prior to the counting episode. The children were consumed with the activity of counting, and this seemed to exclude the reflective awareness that is involved in assuming the roles of both speaker and listener. In that case where such an inference could be made, the children would indeed be thinking using symbols that stood in for their mathematical schemes. Still, it would be necessary to specify those conceptual structures the symbols point to for a more complete account of their mathematical thinking.

The explanation of the children’s mathematical thinking in terms of their use of their numerical counting schemes provides justification for including the enactment of mathematical schemes in re-presentation in mathematical thinking. Social interaction is definitely involved in the construction of these schemes, for how else could children learn a sequence of number words, or learn how to coordinate the production of that sequence with the production of countable items at the sensory-motor level except by interacting with others? I have also indicated the role of social interaction in the construction of what I call the initial number sequence in that the self-regulation that emerged in social interaction set autoregulation of the operations that were involved in self-regulation in motion (Steffe, 1994).

The issue of whether the children were expressing the thoughts of social worlds turns on whether the images the children created prior to counting were learned by internalizing the operations and images of others. This would seem to be particularly implausible if for no other reason than the operations and images of others are inaccessible to observation. However, given that there are indeed ways of communicating that facilitate the production of images, social interaction could still be a prominent factor in the images the children produced after Jason covered part of the nineteen toys. But the independent way the children operated suggests that whatever images they made in constructing their situations were basically uninfluenced by whatever images the other child made. Of course, an historical analysis would need to be undertaken to make an account of the contribution of social interaction to the images the children made in re-presentation. If an analyst choose to undertake such an analysis without using the operation of re-presentation, a model would need to be advanced that explains how two individuals, who, in Piaget’s words, have different, non-interchangeable perceptions, could exchange such perceptions.

In the absence of such a model, to account for children’s construction of their initial number sequence in the earlier study that I already mentioned (Steffe; 1994), I had to appeal to the autoregulation of certain operations that were set in motion in individual-environment interaction to account for how the children’s counting scheme underwent a metamorphosis from what I call the figurative counting scheme to the initial number

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sequence—the children’s first numerical counting scheme. When speaking in this way, there is a danger of being interpreted as saying that children’s interaction with others carries no force (Lerman, 1996). Quite to the contrary, without subject-environment nothing else follows. But in saying this, I also want to be clear that I have never been able to explain children’s mathematical thinking without also appealing to their self-generative activities. In other words, I have never been able to establish a causal relation between subject-environment interaction and within-subject interaction. Rather, as Kieren (1994) suggested, subject-environment interaction occasions children’s mathematical thinking, which is by necessity subjective.

I also want to emphasize that I was able to communicate with Jason and Patricia primarily because I had a model—the explicitly nested number sequence--that I could use to orient the children to a domain of interactions involving counting that were compatible with the constitutive operations of my model. But their counting activity was initiated by them and emerged independently of anything that I said or did in the sense that it was they who choose to count. I did not suggest it aside from asking the children to place twenty toys in the playground of TIMA: Toys.

What Was Learned?

The issue of what Jason and Patricia might have learned when finding that eleven toys were covered is a very important one, because it illustrates the difference between engaging children in mathematical activity and engaging them in active mathematical learning. To put it simply, the children had learned nothing that was observable to me. But they did engage in intensive and regulated16 mathematical thinking.

One could object to my claim that the children had learned no observable thing if for no other reason than found that eleven toys were covered. Indeed they had, but this was simply the result of the children using their counting schemes in the particular situation at hand. Their way of counting was ready-at-hand for them, and they did not need to learn how to count in a way that was different from their already established ways. Lest the reader wonder how I can make such a claim if this was the first time I had taught the children, my claim is based on a model of children’s construction of number sequences that in turn was based on extensive experience teaching children (Steffe, Cobb, & von Glasersfeld, 1988).

The explicitly nested number sequence is an example of what Balacheff & Gaudin (Manuscript Copy) referred to as a knowing: “We use knowing as a noun to distinguish the students personal constructs from knowledge which refers to intellectual constructs recognised by a social body. This [is] intend[ed] to keep the distinction made in French between ‘connaissance’ and ‘savoir’ ” (p. 1). Further,

16 Self-regulation was especially apparent when Patricia started over after she lost her place in counting. It was also involved in the children keeping track of how many items they had counted each time they counted a number word. Autoregulation was involved in the children organizing counting in the way that they did as well as using their counting scheme in assimilation.

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A knowing is characterised as the state of dynamical equilibrium of a action/feedback loop between a subject and a milieu under proscriptive constraints of viability. (Balacheff & Gaudin, p. 3)

Rather than use “environment”, Balacheff & Gaudin choose to use the more restrictive term “milieu” to refer to a subset of the environment of a subject: “The milieu is a kind of projection of the environment onto its cognitive dimensions” (p. 3). I interpret “milieu” in the case of Jason and Patricia as what I construed as their constructed situation, which is in keeping with the notion of an environment of a subject as an experiential field of an observer that does not include the subject. In that both of Jason and Patricia formed a goal to find how many items were hidden, this goal can be thought of as a discrepancy between the current and the expected state of their counting scheme (as a perturbation), and the goal as eliminating that perturbation. The children could be said to be in a state of disequilibrium and that they counted to reestablish equilibrium. I interpret the action/feedback loop of a knowing as the activity of counting from “nine” up to “nineteen” and checking at each counting act to see if “nineteen” was uttered. In this way, there was a dynamic action/feedback system in play as the children counted. Finally, the children counted to restore the state of dynamical equilibrium in their milieu. Balacheff & Gaudin interpret viability as “a capacity to recover an equilibrium following some perturbation” (p. 3), so in their terms, children could be said to be viable in their milieu. Had the children made an accommodation in maintaining their viability (i.e., their adapted state) in their milieu, then I would have said the children learned.

After the children found that there were eleven items hidden, if they had reviewed the situation of there being eight of nineteen items visible along with the result of there being eleven of the nineteen items hidden and integrated17 eight and eleven to produce nineteen, this would have constituted setting the results of their counting scheme in relation to its situation and combining them into a structured whole. Such operating was possible for the children had they formed the new goal of finding how many is eight and eleven more. One might think that the children would have immediately known that eight and eleven more are nineteen, but the production of eleven closed the scheme. So, a new goal and further operating would be necessary for the children to produce nineteen. If the children reflected on the numerical relation they had established among eight, eleven, and nineteen for the purpose of fixing it in memory, the next time they encountered such a situation involving two of the three numbers, the children might have been able to simply re-present the three involved composite units in a structured relation without enacting counting in any way. This would constitute a modification of the children’s counting scheme and the children could be considered to have learned. It is in this way that I consider the learning of addition facts as a creative mathematical activity that is at the level of property noticing in the Pirie & Kieren model of mathematical understanding. Learning addition facts should not be regarded as the low level of mathematical activity that memorizing them implies.

17 To integrate two composite units together means to unite them together into a unit containing them and then to disunite their elements and unite these elements together into a composite unit. This is usually accompanied by the child finding the numerosity of the united elements.

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Did the Children Solve a Problem?

One might also object to my saying that the children had no problem to solve prior to counting, because they did not know how many of the nineteen toys were covered. I would agree that the situation was problematic for the children because I infer that they experienced perturbation prior to counting and because it was this perturbation which drove their counting acts. Prior to counting, the children were in a state of perturbation, but they didn’t have a problem because the operations of the explicitly nested number sequence permitted the children to count to find how many toys were hidden. They definitely eliminated the perturbation by counting, but there was no necessity for them to make an accommodation in their scheme in order to count. Because there was no change in their counting scheme that was observable to me, there was nothing that I would consider as having been learned. Learning is not driven by the successful use of schemes, but by the failure of a scheme to be used successfully or by using the results of a scheme in the service of a new goal18.

Perturbation is often interpreted as synonymous with cognitive conflict, but it is more appropriately interpreted using the metaphor of arousal. There are cases of arousal that I regard as experiential provocations rather than perturbations. Further, my saying “OK” after Jason covered some of the nineteen toys constituted an intentional provocation on my part. I infer that this intentional provocation led to what I consider as an experiential provocation in the children. Generally, an experiential provocation is a disturbance in a component of an interacting system created through the functioning of the system. For an experiential provocation to engender a cognitive perturbation, it must evoke a particular concept or scheme. For a cognitive perturbation to be created, a discrepancy must be produced between an expected state of the concept or scheme and the evoked state.

A cognitive perturbation is definitely a disturbance in a scheme, but it differs from an experiential provocation in that that the experiential provocation is relative to an element or to elements in the environment of the individual. I consider the children’s experience of the cover as an experiential provocation that was much more poignant for them than my saying, “OK”. In fact, I believe that my saying “OK” was not what led them to finding how many toys were covered. Rather, it was interpreted by them as an assent to go ahead and find how many toys were covered, which is what they intended to do without my saying so. The cover was a provoking element and it created a disturbance in their counting scheme (but not in their linguistic system). They had just counted the toys and found there to be nineteen, and the cover hiding part of these toys destroyed the children’s sense of closure of the scheme. This evoked the counting scheme again, and a perturbation was produced that I express as the children’s goal to count the hidden toys.

Using The Explicitly Nested Number Sequence In Learning

Balacheff & Gaudin (Manuscript Copy) define learning as “a process of reconstruction of equilibrium of the subject/milieu system which has been lost following perturbations of

18 cf. Konold & Johnson, 1991, p. 8 or von Glasersfeld, 1995, p. 68 for related interpretations.

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the milieu, or perturbations of the constraints on the system, or even perturbations of the subject itself” (p. 4). In this general orientation to learning, Patricia and Jason would be considered as learning because they did reestablish the equilibrium that had been lost when part of the nineteen counted toys were covered. My contention that they didn’t learn anything that was observable to me is based on the fact that I could not observe any alteration or adjustment that they made in their numerical counting scheme in order to reestablish equilibrium. So, I propose an alteration in Balacheff & Gaudin’s definition of learning by introducing the necessity of there being a change in the subject/milieu system in the process of reconstruction of equilibrium.

The learning theory that emerges from Piaget’s work can be summarized by saying that cognitive change and learning in a specific direction takes place when a scheme, instead of producing the expected result, leads to a perturbation, and perturbation, in turn, to an accommodation that maintains or reestablishes equilibrium. (von Glasersfeld, 1995, p. 6)

Although von Glasersfeld makes clear the necessity of introducing accommodation into the conception of learning, I will demonstrate in the following example a case of learning where an adjustment occurred in the establishment of a constructed situation that led to a change in the activity of the explicitly nested number sequence, but where the activity of the scheme led to its expected result. This example should not be considered as countermanding von Glasersfeld’s conception of learning any more than introducing accommodation into Balacheff & Gaudin’s definition countermanded it. I have experienced the events of mathematical learning that fit within von Glasersfeld’s definition and I consider the following example of mathematical learning as fitting Balacheff & Gaudin’s definition if they included accommodation in it. Nevertheless, the following example suggests sources of perturbation that can drive learning that von Glasersfeld’s definition does not account for. The example demonstrates the reciprocal relationship between models constructed in a research program using basic constructs of the program and the basic constructs. The basic constructs serve as tools in the construction of the models. But, because the models are experientially abstracted from interacting mathematically with children, the models contain conceptual material and ways of thinking that are not supplied by the basic constructs. So, it should be no surprise that the models can serve in modifying the basic constructs.

A key element in maintaining learning as a spontaneous process is to base it on children’s “spontaneous schemes”. Toward that end, shortly after the teaching episode which produced Protocol I, I engaged the children in a teaching episode in which I presented them with the following task.

Protocol II: Finding how many toys in six covered strings of toys with six toys in each.

T: Jason, you give Patricia a problem. J: (Makes a 6-string and a cover and copies six 6-strings under the cover.) T: (After clarifying that there were six 6-strings under the cover) You both

have to figure it out. Six rows with six in a row ....

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P: (Places her hands in her lap under the table, looks into space, and subvocally utters number words for approximately 15 seconds) I think I know (Whispers “thirty-six” to the teacher).

J: (Sequentially puts up his left thumb, forefinger, and middle finger and then his left forefinger, middle finger and ring finger. He then sequentially puts up his right forefinger and right middle finger and clasps all of his fingers to his head. Thereafter, he abandoned using his fingers and sat in deep concentration with his hands maintaining contact with his face. During this time, he did move his hands as well as his head, but these movements were apparently not a constitutive part of his thought. Rather, they were auxiliary movements of which he was not aware. The whole episode takes place within approximately 30 seconds.) I know. Thirty-five!

T: Let’s work it out and see who is right. Jason, can you tell us what you did? J: I just remembered and then counted on my fingers. I went six and that’s

one (putting up a thumb), and (shakes his head “no”) seven, eight, nine, ... ten, eleven, twelve.

T: (To Patricia) what did you do? P: Six plus six is twelve, and I kept going until I got twelve, and then I

counted by ones. T: How did you do that? Six plus six is twelve, and then what? P: And then I counted six more and then I kept on counting six more--six, six,

six, six (moves her hand each time she says “six” in a backward direction over her head).

T: Jason, what is twelve and six more? J: Eighteen. Thirty-six (indicating he had recounted while the teacher and

Patricia were talking.)

What was learned?

Just as was the case in Protocol I, I look to the children’s activity and the results of their activity in Protocol II to infer their situation. Both children again engaged in intense mathematical thinking prior to producing their answers, Patricia for about 15 seconds and Jason for about 30 seconds. There was a modification in how they kept track of counting that, from my perspective, involved the children using the unit of six in two ways—to keep track of counting the strings and to keep track of counting the toys in each string. Still, I infer that it was the goal of each child to find how many toys were hidden beneath the cover because Patricia said “thirty-six” and Jason said “thirty-five”. Moreover, when Jason was asked to explain how he proceeded, he made a record of uttering the number words “seven, eight, …, twelve,” and when Patricia was asked a similar question, she explicitly said that she counted by ones after she said, “Six plus six is twelve”. Based on their goal and on what they created as countable items, I infer that they used their explicitly nested number sequence in constructing their situation prior to counting.

The constructed situation. Based on how they counted, I also infer that there was a modification in their constructed situation in contrast to their constructed situation of

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Protocol I. For an indication of what the modification might have been, I look to how Patricia explained her activity. After the teacher asked her, “How did you do that? Six plus six is twelve, and then what?”, she answered, “And then I counted six more and then I kept on counting six more--six, six, six, six (moves her hand each time she says “six” in a backward direction over her head).” It was no accident that Patricia said “six” four times, because the teacher asked her what she said after saying “Six plus six is twelve”. She was aware that there were six strings hidden and she used her concept of six to keep track of “counting six more.” When she didn’t know the next number word in the counting sequence for sixes, she dropped down a level of unit and used her concept of six to keep track of counting six more beyond twelve. So, I infer that this dual function of her concept of six was already present in her constructed situation prior to her counting activity.

In explaining how the dual function was implied by her situation, it is important to note that children who have constructed the explicitly nested number sequence can take a composite unit as a given prior to operating. Moreover, their unit of one is iterable, so they can use their concept of six in re-presentation in such a way that they re-present an iterable unit of one that they intend to iterate six times. But there was a novelty that entered into the re-presented iterable unit of one that wasn’t present in the constructed situation of Protocol I in that it implied iterating enough times to produce six strings of toys, and enough times to produce six toys in each string. What made this dual implication of the iterable unit of one possible can be understood by using an analogy. In the case of the concept of, say, dog, children like Jason and Patricia can produce an image of a dog as a singular unit item, and then further articulate their image by producing, say, the legs of a dog. The concept, dog, carries with it records of experience that was used in constructing the concept. Likewise, in the case of a string of toys, a string could be taken as a singular unit item when considering that there were six strings without unpacking the concept and forming an image of six toys. This iterable unit of one, when focusing attention on its meaning as a string, could be also iterated enough times to produce an image of six toys. So, the re-presented iterable unit of one carried a dual implication for the children depending on how they construed the unit—as a unit item of the composite unit containing six strings, or as a composite unit comprising six toys. There was a third function of the re-presented iterable unit of one, and that was to iterate it enough times to count the toys. I point this out because the meanings of what was adjusted had to do with the iterable unit of one that could be used in iteration to produce the countable unit items, “one, two, …” when counting all of the toys

Generalizing assimilation. I consider the children’s constructed situation as a generalizing assimilation using their explicitly nested number sequence. An assimilation is generalizing if the scheme involved is used in situations that contain sensory material that is novel for the scheme from the point of view of the observer, but the scheme does not recognize it until possibly later as a consequence of the unrecognized difference, and if there is an adjustment in the scheme without the activity of the scheme being implemented (Steffe & Thompson, 2000, p. 289).

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From my point of view, the situation did contain sensory material that was novel for the explicitly nested number sequence in that the toys were organized into six strings of six. But this “sensory material” was not the raw material of perception. Rather, it was the raw material of the children’s operating19. They could take a string as a unit item and produce a composite unit of six such unit items, or they could take a toy as a unit item and produce a composite unit of six such unit items. What they were unaware of was the unit structure of a unit of units of units produced by the coordination of these two ways of operating20. It was the lack of the coordination of these two ways of operating prior to counting that I regard as the novelty which the scheme did not recognize. The adjustment in the scheme prior to operating concerned the triadic implication of the iterable unit of one that I have explained. It was this triadic function of the iterable unit of one that led to the children keeping track of counting by ones in the way that they did. Essentially, it was a modification of the recursive property of their counting schemes.

The activity and results of the scheme. A form of the units-coordination that I explained earlier in footnote 20 occurred in the activity of the children’s units-coordinating scheme and it was manifest in how they kept track of counting. Neither of Patricia nor Jason was aware that they were going to make a units-coordination prior to the activity, so they were not aware of a multiplicative operation. For them, the situation was simply one of those situations in which they counted. Their units-coordinating scheme was certainly an operative scheme, but the children were not aware of the operations that they carried out as they made a units-coordination. They had no access to why they counted as they did and counting just appeared to them as the thing to do. Nevertheless, they could explain how they counted.

The activity of Patricia’s units-coordination involved progressively integrating21 the items of each composite unit, six more, with the items of the results of prior integrations. She was aware of counting six more beyond the six more she had previously counted, and it is in this sense that I regard counting six more as iterating the composite unit six more. The progressive integration operations were perhaps only symbolized when Patricia, for example, counted, “13, 14, 15, 16, 17, 18” as six more than twelve, but that she could engage in these operations was documented several times as we taught her. Her progressive integration operations were constitutive operations of her units-coordinating scheme, but they were yet to become recorded in the first part of her scheme.

The results of the units-coordination scheme was not simply “thirty six”, although that was a result. The children also established an experiential sequence of six composite units of six whose elements were integrated together by means of progressive integration

19 I interpret “sensory material” here rather unconventionally as including the operative images children produce by using their numerical concepts in re-presentation. This broader notion of generalizing assimilation makes it useful throughout mathematical learning. 20 One way to think about coordination is to insert a unit of six into each slot of another unit of six to produce a sequence of six units of six. If the child then unites the elements of this sequence together, that produces a unit of units of units. 21 Progressive integration operations involve successively integrating a composite unit with the results of prior integrations. For example, “Six plus six is twelve; twelve plus six is eighteen; and eighteen plus six is twenty four”.

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operations. Realizing this as I taught the children, it became one of my major goals for the children to interiorize the records of making such experiential sequences and use them in the establishment of an image of the sequence of composite units prior to counting. In this case, the progressive integration operations would be recorded in the first part of her scheme, and serve in producing an image of the sequence of composite units prior to counting. Reprocessing the image of the sequence of composite units could lead to the establishment of an iterable composite unit that in turn could be used to re-present the sequence. Just as the case of the iterable unit of one, this would provide the children with great economy in multiplicative thinking. It would lead also to the establishment of strategic reasoning in multiplying just as the iterable unit of one leads to the establishment of strategic reasoning in adding (cf. Steffe, Cobb, & von Glasersfeld, 1988). It would be at this point in the constructive history of the children that they could establish the units-coordination as an object of reflection prior to counting, and it would be at this point that they could be said to have constructed a multiplicative operation. In the case of the iterative units-coordinating scheme that I have explained, the scheme as a whole constituted the children’s multiplicative concept, and the children could be said to be multiplying only upon enacting the scheme.

A modification of the units-coordinating scheme in its use. Engaging the children in making a continuation of a units-coordination using a different composite unit in the continuation served in confirming the above analysis of the architecture of the units-coordinating scheme. In a subsequent teaching episode, after Patricia had copied five 5-strings and five 4-strings under a cover, Jason operated in a way that we didn’t expect him to operate. In essence, as teachers, one might say that the way in which Jason operated allowed us to unexpectedly “strike it rich”. Jason explained how he arrived at “forty five” by saying, “I just counted up. 5, 10, 15, 20, 25, and then I counted the fours up.” The teacher, surprised about how Jason said he counted the fours up, exclaimed, “You didn’t do that for the fours! You didn’t say 3, 6, 9 for the four’s, did you?” Jason then commented, “Yeah, I did. And then I counted the ones!”.

Jason’s way of operating indicates that he broke a unit of four into a three and a one, then counted five threes and then five more ones onto the results of counting to twenty-five. It would be implausible that Jason broke each of five fours into a three and a one in re-presentation because he worked so quickly. It would be much more likely that he broke only one unit of four into a three and a one and then engaged in iterating the unit of three five times and then the unit of one five times. In fact, after Jason subvocally uttered “5, 10, 15, 20, 25,” it took him only twenty seconds to say “I know, forty-five!” During this twenty second period of time, the teacher interrupted Jason for approximately seven seconds inquiring how Jason was going to do the problem. Jason said, in reply to the teacher, “Just count them up!” So, there wasn’t enough time for Jason to keep track of breaking each of the five fours into a three and a one, and then to count the threes and the ones, because he only worked for a period of about 13 seconds, which is hardly enough time to continue on counting five more threes, and then five more ones. Moreover, he subvocally uttered number words immediately after counting by fives and prior to the interruption by the teacher, which solidly indicates that he was counting rather than breaking fours up into threes and ones. So, Jason could have broken only one four into a

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three and a one and then proceeded on, counting by three five times beyond “twenty-five” and then five more ones. In fact, he counted very quickly. So, Jason’s way of operating solidly corroborates the inference that the image he created of his iterable unit of one prior to operating implied five strings as well as four toys in each string. It also corroborates that he used only one such iterable unit in establishing his situation, because he first split a unit of four into two units, one of three and the other of one, and then iterated each of these units five times.

Jason’s accommodation22 in his way of operating in units-coordinating was spontaneous and intuitive and he was not aware of why he could operate in the way that he did. It seemed to just appear to him in the context of his activity and he didn’t seem aware of its source. Nevertheless, he was aware of counting by three and then by one, and he was aware that he counted in each way five times. But, this is different from explaining the operations that permitted him to count in this way.

Final Comments

Generalizing assimilation is the main event in learning in that case where children use their “spontaneous schemes” in solution of situations that contain elements that, from the point of view of the observer, would involve an adjustment of the scheme’s situation or activity to solve, where the adjustment does not involve the awareness of the child either before or after making it. I would not characterize this kind of learning as problem solving, because the involved accommodation did not produce a major reorganization in the scheme’s structure similar to the reorganization in the counting scheme that was apparent upon the construction of the initial number sequence. The construction of this numerical counting scheme involved the interiorization of the activity of counting so that it could be said that the activity of the scheme was contained in the situation prior to implementing the activity. It wasn’t the case that the units-coordination that was involved in the activity of the units-coordinating scheme of the children was contained in the situation even though the situation implied the units-coordination in the way that I have explained.

There is a distinct advantage in maintaining learning as a spontaneous activity in the mathematics education of children. Regarding mathematical learning as the capability of a “spontaneous scheme” to change to restore equilibrium has far-reaching consequences because it brings the history of the child's past interactions outside of school into the context of mathematical interactions. This may seem to be trivial, but a quotation from Piaget (1964) places it into its proper context.

If a structure develops spontaneously, once it has reached a state of equilibrium, it is lasting, it will continue throughout the child's entire life. (pp. 17-18)

22 It did constitute an accommodation because it recurred in other situations and in other teaching episodes. However, in that he did not intentionally operate in this way, the situations in which it did recur were situations in which he didn’t have a counting sequence ready-at-hand (such as counting by fours).

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However, rather than consider the products of spontaneous development as endpoints of the child's constructive possibilities in mathematics, I consider them as starting points of further constructions. Basing mathematical interactions with the children on those schemes we observe the children use independently is my fundamental way of acknowledging the children as self-organizing, living systems with their own history of interactions. But in contrast to Piaget's approach to studying spontaneous development, in which no agent intends for the children to construct reversibility or transitivity, for example, I do intend for children to construct mathematical concepts and operations of a particular kind.

So, rather than regard learning as being explained by development as did Piaget (1964, p. 8), I regard learning and development as reciprocally related. The difference is significant and is to be found in my understanding of learning as arising from individual-environment interactions in which I take into account my contributions to the interactions, intentional and unintentional.

Learning With Understanding

In view of what Jason and Patricia learned in Protocol II, it is possible to give an account of what I mean by the phrase, “learning with understanding”. I have characterized the children’s numerical counting scheme as primitive knowing for their learning of multiplying schemes and have indicated how children’s multiplying schemes emerge as accommodations of their explicitly nested number sequences. But the children’s number sequences can be gauged with reference to Pirie & Kieren’s (1994) model of knowing without considering that they are primitive knowing with respect to their multiplying schemes. This is important to note because what can be considered as primitive knowing is not always “primitive”. Other than those which I have already mentioned, there is one aspect of their model other than primitive knowing that I found salient in the way in which I think about the explicitly nested number sequence, and that is their notion of folding back.

When faced with a problem or question at any level, …, one needs to fold back to an inner level in order to extend one’s current, inadequate understanding. … This inner level action is part of a recursive reconstruction of knowledge, necessary to further build outer level understanding. (p. 69)

Although I don’t regard the children’s explicitly nested number sequence as being recursively reconstructed in their construction of units-coordinating schemes, the ability of the children to take their numerical counting schemes as input for further operating23 was the key element in the generalizing assimilation that produced the adjustment of the situation that in turn led to the units-coordination. After the construction of their units-coordinating schemes, I indicated ways in which they could use the scheme that were different and more sophisticated than prior to the accommodations they made, and I also indicated the possibility of the children modifying the scheme in its use. Nevertheless,

23 The reason the children’s numerical counting scheme was a recursive scheme was that the activity of the scheme was contained in the first part of their scheme.

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the iterative units-coordinating scheme should not be regarded as a scheme separate from their “spontaneous scheme”. Rather, it should be regarded as a modified numerical counting scheme that is itself a “spontaneous scheme” that could be used in learning many other things24. What I mean by “learning with understanding”, then, is the accommodation of “spontaneous schemes”.

Learning as Initiation into a Certain Type of Discourse

In contemplating what it might mean to initiate Jason and Patricia into a discourse concerning multiplication, what comes into the forefront immediately is the necessity to learn their “spontaneous schemes” and how to foster accommodations in those schemes. I certainly concur with Sfard concerning the necessity of an “expert participant” in children’s construction of mathematical discourse, and interpret “expert” as referring to an individual who can construct at least working models of children’s mathematical knowledge. The construction of such models is essential for an adult to intentionally affect the behavior of children by orienting them to some part of their potential to interact that is comparable to those of the adult. For the domain of mathematical interactions of an adult and a child to be “widely coincident”, it is the adult who must construct ways and means of operating that are harmonious with those of children rather than the other way around. In teaching children, I have always found it necessary to attribute mathematical knowledge to them that is independent of my own. I have also found that I cannot explain children’s mathematical knowledge by using only my own mathematical concepts and operations. This is an essential point, because even though I have portrayed the explicitly nested number sequence as a rather sophisticated mathematical counting scheme, it is a model of children’s mathematics, not of adult’s mathematics.

It has become apparent to me that my own mathematical concepts and operations serve an orienting function as I interact with children in that I experience echoes of my own mathematical knowledge as I experience children’s ways and means of operating. But to communicate with children in the way that Sfard envisions, I do need go beyond simply experiencing echoess, because in experiencing echoes the tendency is to act as if the children’s mathematical thinking is but a reflection of my ways of thinking. What is needed is to construct ways of thinking mathematically that seem harmonious with the way children think.

I regard children’s mathematics as a legitimate mathematics to the extent that I can find rational grounds for what they say and do. Finding rational grounds for what children say and do in an attempt to specify their mathematical schemes is an essential part of the work that I do and it is here that I become explicitly aware of creating a kind of mathematics that only children can teach me. This awareness is essential because my work is based on the necessity of providing an ontogenetic justification of mathematics; i.e., a justification based on the history of its generation in individuals. Ontogenetic justification is different from the impersonal, universal, and ahistorical justification often provided for the mathematical foundations of school mathematics. But I believe that

24 For example, I have indicated the possibility of the children constructing strategic reasoning in multiplication in Jason’s counting by threes and by ones to count by fours.

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specifying a school mathematics based on the functioning of children’s intelligence (Piaget, 1980; Skemp, 1979) is an appropriate way to regard school mathematics. My view of school mathematics as a product of the functioning of children’s intelligence defines it as a living subject rather than as a subject of being (Maturana, 1988). It is a dynamic mathematics that lives and grows in social and experiential contexts.

Mathematics of children. I find the proposition compelling that children have constructed mathematical knowledge that is rational and free from internal contradiction. The attribution of such knowledge to children is essential in communicating with them, but it does not provide models of what that knowledge might be like. Adults are obliged to construct models of children’s mathematical knowledge by means of interactive communication with them. I use the phrase “children’s mathematics” to mean whatever constitutes children’s mathematical knowledge and “mathematics of children” to mean my understanding of children’s mathematics. The knowledge that children construct to organize, comprehend, and control their experience (their first-order mathematical knowledge) is inaccessible to an observer in the sense that it can be only inferred based on what children say and do. A model that I construct of children’s mathematics is, in Maturana’s (1988) terms, “a mechanism that as a consequence of its operation would give rise in him or her [the observer]to the experience of the phenomenon to be explained” (p. 34). These models constitute second-order mathematical knowledge; that is, “models observers may construct of the subject’s knowledge in order to explain their observations (i.e., their experience) of the subject’s states and activities” (Steffe, von Glasersfeld, Richards, & Cobb, 1983). Second-order mathematical knowledge is constructed through social processes and I thereby refer it as social knowledge.

Mathematics for children. Regarding school mathematics as social knowledge in the sense that I have explained constitutes a fundamental shift and it is an implication of constructivism for school mathematics that is yet to be fully appreciated. In this shift, it might seem that what adults intend for children to learn remains unspecified. However, I use the phrase “mathematics for children” to refer to what children might learn. But rather than regard those concepts and operations as belonging to the mathematical foundations of current school mathematics programs, I base the mathematics for children on the mathematics that I have observed children actually learn. Mathematics of children becomes known through interpreting the language and actions of children and includes the modifications children make in their mathematical knowledge as a result of their interactions, social and otherwise. So, mathematics for children is at least as problematic as mathematics of children. As social knowledge, the only way it can be constructed is through interactive communication with children. Any a priori conjecture concerning mathematics for children has to be warranted experientially and adjusted and modified according to what children do learn.

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