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This chapter begins by describing what is unique about mathematics that has made it a central topic in the learning sciences. This research has historically been interdisciplinary, drawing on psychology, mathematics research and theory, and mathematics educators. It then describes two distinct approaches – the acquisitionist and the participationist. The acquisitionist approach considers learning to be what happens when an individual learner acquires mathematical knowledge. This part of the chapter reviews research on misconceptions and conceptual change that has been based in Piaget’s constructivist theories. The participationist approach views learning as originating in social interactions in diverse settings such as classrooms, homes and playgrounds, museums, and workplaces. This approach views learning as a collective sociocultural phenomenon, and uses methodologies such as interaction analysis and design-based research. This chapter concludes with a discussion of how teachers learn to teach mathematics.
The rest of this book is devoted to the project of minimizing the risks, perhaps the very presence, of the metaphor of object while trying to preserve most of its advantages. The task is admittedly ambitious and far from easy. To succeed, we will have to suspend disbelief and remain patient when we slip and falter, trying to bootstrap ourselves from our present discourse into the new one, yet to be invented.
Clearly, we are not going to be the first to undertake the task of disobjectification. The necessary initial step, then, is to survey the history of earlier attempts and try to understand the reasons for their insufficiency. The subsequent move will be to identify those developments that seem to have taken us “almost there” and can thus become a basis for our own trials. Finally, a specific proposal for a disobjectified discourse on thinking will be made. All along, we will need to keep in mind that our present task is not one of establishing empirical facts about thinking but rather of finding useful ways of talking about the phenomenon.
Monological and dialogical discourses on thinking
The numerous historical attempts to overcome the pitfalls of objectification took different forms, depending on whether they were undertaken as a part of monological or dialogical research. Let me elaborate.
It is so much a part of “thinking philosophically” to be impressed with the special character of mathematical truth that it is hard to shake off the grip of the Platonic Principle [according to which differences in certainty must correspond to differences in the objects known]. If, however, we think of “rational certainty” as a matter of victory in argument rather than of relation to an object known, we shall look toward our interlocutors rather than to our faculties for the explanation of the phenomenon. If we think of our certainty about the Pythagorean Theorem as our confidence, based on experience with arguments on such matters, that nobody will find an objection to the premises from which we infer it, then we shall not seek to explain it by the relation of reason to triangularity. Our certainty will be a matter of conversation between persons, rather than a matter of interaction with nonhuman reality.
The word did not exist in the beginning. In the beginning was the deed. … The word is the end that crowns the deed.
Lev Semionovitch Vygotsky
Rituals help us … to connect deeply with people. … The repetition that ritual always involves sets the present moment in a larger context and infuses it with wider meaning. It's difficult to invent rituals.
To use Walter Fisher's expression, humans are “storytelling animals” and mathematizing is just one special type of storytelling activity.
Without speech man would have no reason and no reason is possible without speech.
Johann Gottfried von Herder
[Development of mathematics is] enlargement of the mathematician's self-consciousness … a long, difficult, and extended exercise in which the human mind attempts to catch sight of itself catching sight of itself, and so without end.
In the last chapter, while defining thinking as individualized communication, I was careful to stress that all forms of communication need to be considered, not just verbal. Such an all-inclusive approach was necessary to ensure that the resulting definition of thinking does not leave out some of the phenomena that are commonly regarded as cases of thinking. This said, it is now time to give linguistic communication the attention it certainly deserves. This chapter is devoted to the conjecture that linguistic commognition is the primary source of perhaps the most human of our distinctively human properties: of our propensity for accumulation of complexity. More specifically, I will be arguing that this special human ability, along with many others, originates in our ability to “turn discourse on itself,” that is, to communicate about communication. This realization will eventually make me claim that what is traditionally called human development may now be considered as almost tantamount to the development of discourses.
Before addressing any of these all-important issues, however, let me take a closer look at the historical debate about relations between thinking and one particular form of communication called – speech.
The greatest magician … would be the one who would cast over himself a spell so complete that he would take his own phantasmagorias as autonomous appearances. … We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time.
Jorge Luis Borges
They recalled that all nouns … have only a metaphorical value.
Jorge Luis Borges
Our investigation is therefore a grammatical one. Such an investigation sheds light on our problem by clearing misunderstandings away. Misunderstandings concerning the use of words, caused, among other things, by certain analogies between the forms of expression in different regions of language.
The claim, made in the last chapter, that some foundational work may be needed before the resilient, long-standing quandaries can be resolved will now be reinforced by showing that these quandaries may, in fact, be the product of the way we speak. More specifically, it is posited that the source of the problem is in the way we think about human activities in general and, in particular, in the way we communicate with others and with ourselves about the activity of thinking, mathematical or otherwise. I am arguing that the different keywords around which the five quandaries revolve are not operational enough to ensure effective communication. I also show that most of them are metaphorical in nature, and metaphors are often like Trojan horses that enter discourses with hidden armies of unhelpful entailments.
We are what we repeatedly do. Excellence then, is not an act, but a habit.
The rules of formation operate not only in the mind or consciousness of individuals, but in discourse itself; they operate therefore, according to a sort of uniform anonymity, on all individuals who undertake to speak in this discursive field.
If the meaning of words is in their discursive use, Wittgenstein's exhortation “to let use teach us meaning” makes perfect sense and may even appear tautological. It is by reproducing familiar communicational moves in appropriate new situations that we become skillful discursants and develop a sense of meaningfulness of our actions. The all-important regularities to be found in any discourse are the focus in this chapter.
Meaningfulness from repetition
In chapter 4, communication was defined as a collectively implemented activity that, when observed over time in its diverse manifestations, displays repetitiveness, and thus patterns. The repetitiveness is the source of communicational effectiveness. If I know how to react to a given action of an interlocutor, it is because I was exposed to a similar situation before and am now able to implement an action quite similar to the one that was performed then.
Discursive patterns are multifaceted and intricately interrelated. Words and symbols are combined into utterances; the utterances, through their structural commonalities and through their recurrent coappearance in discourse, solidify into stable associations of communicational actions and re-actions; these latter associations, in turn, are coupled with sets of situations and practical deeds that, from now on, will occasion their use.
It is venturesome to think that a coordination of words (philosophies are nothing more than that) can resemble the universe very much. It is also venturesome to think that of all these illustrious coordinations, one of them – at least in an infinitesimal way – does not resemble the universe a bit more than the others.
Jorge Luis Borges
The fact that Newton's vocabulary lets us predict the world more easily than Aristotle's does not mean that the world speaks Newtonian.
We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time; but in its architecture we have allowed tenuous and eternal crevices of unreason which tell us it is false.
Jorge Luis Borges
We have come a long way since we first puzzled upon a bunch of persistent quandaries on human thinking. That early encounter led us to question the traditional acquisitionist discourse and resulted in the attempt to modify our thinking about thinking. It is now time to ask where we are at the end of this long journey. In this final chapter, after a brief summary of what has been done so far, I ponder about the implications of the shift to a commognitive outlook for research on human development and for educational practice. My first move, however, is to revisit the old quandaries, one by one, asking whether the change in our discourse has yielded the desired resolutions.
I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer – not-nine, not-eight, not-seven, not-six, not-five, etc. – is inconceivable. Ergo, God exists.
Luis Jorge Borges
I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, “134 divided by 29” meant a certain tedious chore, while 134/29 was an object with no implicit work. I went excitedly to my father to explain my discovery. He told me that of course this is so, a/b and a divided by b are just synonyms. To him, it was just a small variation in notation.
The “content” of mathematics does not exist in the material world; it is created by the activity of mathematics itself and consists of ideal objects like numbers, square roots and triangles.
The world for them is not a concourse of objects in space; it is a heterogeneous series of independent acts. … There are no nouns.
Jorge Luis Borges
To think is to forget differences.
Jorge Luis Borges
In this part of the book, I illustrate the workings of the commognitive approach by applying it to the special case of mathematical thinking. In so doing, my intention is to show what difference commognitive analysis makes in our interpretation of observed phenomena and in our practical decisions about teaching and learning. The discussion will eventually take me back to the dilemmas presented in chapter 1. The hope is that when scrutinized with the commognitive eye, at least some of the puzzles will be solved, whereas some others may disappear.
Being interested in learning, I focus in my analysis on the development of mathematical discourses of individuals, but I also refer to the historical development of mathematics whenever convinced that understanding this latter type of development may help in understanding the former. Considering the fact that communication is inherently collective, the term discourse of an individual or personal discourse may seem to be an oxymoron. Indeed, borrowing Ed Hutchins's words, one can say that those who equate human development with the development of discourses “move the boundaries of the cognitive analysis out beyond the skin of the individual person” and start speaking, instead, about teams of discourse participants as “commognitive systems.” Let me repeat then that thinking has been defined as self-communication.
This book is an attempt to change our thinking about thinking. Anna Sfard undertakes this task convinced that many long-standing, seemingly irresolvable quandaries regarding human development originate in ambiguities of the existing discourses on thinking. Standing on the shoulders of Vygotsky and Wittgenstein, the author defines thinking as a form of communication. The disappearance of the time-honoured thinking-communicating dichotomy is epitomised by Sfard's term, commognition, which combines communication with cognition. The commognitive tenet implies that verbal communication with its distinctive property of recursive self-reference may be the primary source of humans' unique ability to accumulate the complexity of their action from one generation to another. The explanatory power of the commognitive framework and the manner in which it contributes to our understanding of human development is illustrated through commognitive analysis of mathematical discourse accompanied by vignettes from mathematics classrooms.