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by Annie and John Selden

Does it consist of competencies, such as being able to factor the difference of cubes, in knowing why certain results such as the Mean Value Theorem hold, or in seeing connections between mathematical concepts and results, or between various representations of a single concept like function? Often teachers' views of what it means to know mathematics are not explicitly expressed. Rather, how they teach, and especially how they test, provides clues to their mostly tacit views.

In this, our first Research Sampler column for MAA Online, we consider various answers to the above epistemological question. We take as our starting point the deliberations of the Working Group, Forms of Mathematical Knowledge, at this year's International Congress on Mathematical Education (ICME-8) in Seville. Organized by Dina Tirosh of Israel, these sessions brought together a variety of mathematics education researchers from around the world -- John Mason of the Open University, Paul Ernest of Exeter University, Eddie Gray and David Tall of Warwick University, Tommy Dreyfus of the Weizmann Institute, Anna Graeber of University of Maryland, Tom Cooney of University of Georgia, and MichÃ¨le Artigue of France. Although normally this column will report results of published research, here we describe various perspectives researchers are taking as they seek to clarify underlying ideas which may guide their future investigations.

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KINDS OF MATHEMATICAL KNOWING

At the first ICME session, John Mason began with the now customary procedural/conceptual distinction between *knowing how* and *knowing that* (discussed by scholars from various disciplines such as Polyani, Kuhn, Wittgenstein, and Hiebert), but went on to consider *knowing why*, *knowing to*, and *knowing through*.

By *knowing why*, he meant having "various stories in one's head" about why a mathematical result is so. For example, when partitioning an interval into *n* subintervals, one might recall that *n*+1 fenceposts are required to hold up a straight fence of *n* sections. *Knowing why* and proof are different -- in many cases, the proof doesn't reveal why. As an example, Mason suggested that when primary teachers ask why (-1)(-1)=1, they want images of temperature or depth, not a proof, or even a consistency argument that negative numbers work like positive numbers.

*Knowing to* means having access to one's knowledge in the moment -- knowing to do something when it's needed. For example, in evaluating a limit, a student might just *know to* multiply by a certain quantity divided by itself. This kind of enacted behavior is not the same as writing an essay explaining what one is doing -- it often occurs spontaneously in the form of schemas unsupported by reasons, whereas explanations require supported knowledge. [GÃ©rard Vergnaud, et al, speak of "theorems-in-action" when young children implicitly know the cardinality of the union of two non-overlapping sets can be obtained by adding. Cf. *Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education*, P. Nesher and J. Kilpatrick (eds.), Cambridge University Press (1990), pp. 23-28.] Often such knowing is tied closely to the situation and not easily explicated -- one just "knows to" do it. Studies like those of Jean Lave, on supermarket versus school math, indicate that *knowing to* (in the moment) is often separated from *knowing how*.

Mason stressed the roles of habituation, enculturation, imagery, and emotion. Students need confidence to act and to explain. He suggested that one of the weaknesses of problem-solving curricula is their concentration on *knowing how*, whereas the real issue is *knowing to*. Under pressure from a teacher, students can be prompted to shift their attention from doing to *knowing how*, *why*, or *that*. But it is one's experiential awareness of both social (taken-as-shared) and individual (constructed) mathematical practices that enables one to act.

Sometimes, after reflection, one may become explicitly aware of one's mathematical practices, and as a result, be able to formalize them into mathematical definitions or theorems. A variety of kinds of generalization might be involved -- inductive generalization, empirical generalization ("it's always been like that"), deductive generalization, abductive generalization ("it fits with what's known"), or intuitive generalization ("it just is") -- all of which John Mason included in *knowing through*. The power of generalization lies in making one's awareness explicit (in axioms or theorems).

When asked about implications for the curriculum and whether all teachers should have knowledge about knowing, John Mason responded that school is for educating one's awareness by harnessing one's emotions. Given the above complexity of mathematical knowing, he suggested that "teaching for understanding" may not be a useful slogan.

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A POSTMODERN VIEW

Paul Ernest suggested that in various disciplines from cognitive science, with its concern with the modular mind, to psychology, with its emphasis on situated learning, there has been a shift away from considering knowledge as abstract, to seeing it as embedded in human practices. He noted that most personal knowledge is tacit, rather than explicit. Following and expanding upon Philip Kitcher's views [*The Nature of Mathematical Knowledge*,Oxford University Press, 1983], Ernest classified various kinds of mathematical knowledge as either explicit or tacit. Theorems, proofs, problems and questions are explicit knowledge, whereas meta-mathematical views, symbolism, techniques, strategies, aesthetics, and values are tacit knowledge. To justify mathematical knowledge, one needs some kind of warrant. What one accepts as a justification varies, depending on the situation. He saw a parallel between justification and the assessment of students -- explicit recall is rated low, tacit knowledge is rated higher, and the production of warrants (proofs) and their critical evaluation is rated highest.

Ernest views mathematical knowledge as based on conversation. There is *intrapersonal conversation*, in which an individual's thoughts stem from past conversations (Ã la Vygotsky and Baktin), *interpersonal conversation*, in which mathematics is a "language game" (Ã la Wittgenstein), and *cultural conversation*, in which the reader interrogates the text as if engaged in conversation. He develops his "rhetoric of mathematics" in a forthcoming book, *Social Constructivism as a Philosophy of Mathematics* (SUNY Press), which parallels current work in the sociology of knowledge and the social studies of science.

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HOW DOES SYMBOLISM HELP ONE COPE?

There is too much information coming into people's minds, so they cope by filtering out most of it and concentrating on one or two things. In contrast to Paul Ernest's concentration on mathematics as rhetoric, David Tall thinks that mathematics is powerful because of its symbolism. For students, written symbols are a two-edged sword -- they can help them cope or they can overwhelm them. Interpreted flexibly as either processes or objects, symbols allow a duality of thought. As an example, in elementary school, the symbol 3 + 4 can be interpreted as either a procedure to be done or as a thing in itself, the sum. When interpreted as a procedure, the symbolism does not indicate whether, in obtaining the sum, one is to *count on* (4, 5, 6, 7) or *count all*, (1, 2, 3, 4, 5, 6, 7), or even whether to use one's fingers. When children act on objects in mathematics class, what do they take from their actions? For example, in counting five tokens, do they concentrate on the tokens as objects or do they come to regard 5 as an object? Children who focus their attention on properties of the set of objects, such as "fiveness," come to think flexibly with symbols. Those who listen to their teachers' admonitions to calculate, get overwhelmed in the details of carrying out procedures. There's just too much for them to hold in their heads at one time. Those who stick to procedural thinking eventually get hopelessly behind, whereas the flexible thinkers advance. Tall refers to this pons asinorum as the *proceptual divide* -- only those who come to think flexibly about processes and objects go on to success in mathematics.

Tall framed the problem encountered by mid-level university students as follows. School mathematics has two strands: (1) The arithmetic/algebraic, in which actions on objects predominate (e.g., counting). These actions eventually become encapsulated (or internalized) as abstract concepts such as 5 or number. (2) The geometric, in which objects are given visually and their properties are described. By middle-level university, however, teachers no longer ask students for answers about, or even have them think about, familiar mathematical objects. Students' previous, often successful, approaches to learning mathematics no longer work when they are confronted with formal definitions and axioms which specify properties of (unknown) objects. In this situation, their task becomes the construction of mathematical objects (examples) -- something quiet alien to their previous experiences. [Janet M. Duffin and Adrian P. Simpson have discussed three kinds of learning experiences: *natural*, which "fit" one's mental structures; *conflicting*, which are seen as inconsistent with one's mental structures; and *alien*, which one cannot connect with existing mental structures. Cf. "A Theory, a Story, Its Analysis, and Some Implications," *JMB* (June 1995), pp. 237-250.]

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WHAT TEACHERS SHOULD KNOW ABOUT MATHEMATICAL KNOWING

Anna Graeber listed some "big ideas" that preservice teachers should know. (1) Different forms of mathematical knowledge have been described. (2) Students possessing one form of mathematical knowledge do not necessarily possess others. (3) One needs to understanding students' current knowledge if one wants them to amend or extend what they know. (4) There is evidence that knowledge is more enduring when it is learned in a meaningful context, through reasoning from relatively primitive concepts, by explaining to others, and by reflecting on one's own knowledge growth. (5) Different logical and experiential paths can lead to the same mathematical ideas. (6) Intuitive knowledge is both an asset and a liability.

How does one help preservice teachers appreciate and use these ideas? Tom Cooney noted that teachers' beliefs greatly influence what they do in the classroom. While some beliefs are malleable, others are strongly held onto. It is often difficult for teachers to overcome the idea that the way they learned mathematics, which is often procedurally, is the best. He suggested that one approach might be to put preservice teachers in mathematical contexts they understand, but which cause perturbations. For example, one might give them a standard sheet of paper and ask, what's the largest pentagon one can draw on it?

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MATHEMATICAL DEFINITIONS

What is the best way to introduce a mathematical definition? What is a good definition? For scientists and philosophers, it is one that applies to what is defined and only to that. For those in education, it must also be understood by students. Thus, the mathematical correctness of a definition is a necessary, but not sufficient, condition for presenting it in the classroom. In order to be didactically suitable , a definition must consist of concepts known to the learner (Cobb). It should rely, as far as possible, on the intuition of the student (Fischbein). It should be within the grasp of the learner (inside the learner's zone of proximal development -- Vygotsky).

Roza Leikin discussed a workshop with Israeli teachers, in which various definitions of absolute value were presented. Most used only one definition with their students, preferring either |*x*| as the distance of *x* from the origin or |*x*| = *x*, when *x* > or = 0, and |*x*| = -*x*, when *x* x| = max {*x*, -*x*} or |*x*| = (*x*^2)^(1/2). They agreed that defining |*x -y*| as the distance between *x* and *y* implied |*x* - 0 | = |*x*| but did not regard these as equivalent. David Tall commented that the Advanced Mathematical Thinking Group of the International Group for the Psychology of Mathematics Education had considered this (psychological) problem, i.e., does the concept give rise to the definition or does the definition give rise to the concept? Definitions given in set language are inappropriate for young children who learn by first manipulating objects. Properties of objects only come later. When introducing a concept to students, one should consider whether they will see it as a description or a (mathematical) definition. For example, pupils often feel they know what rectangles are and that teachers are just describing their properties.

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SOME FINAL OBSERVATIONS

In summing up the ICME Working Group's discussions, MichÃ¨le Artigue noted that over time a shift has occurred from a hierarchical view of knowledge (Piaget's stages, van Hiele levels) to a more flexible, connected view. Now people are paying attention to the local and contextual characteristics of knowledge. She considered the most important part of mathematical knowledge to be "competences." While there is a dialectic (going back and forth) between tacit and explicit knowledge, explicitness alone is not compatible with economy of thought. While no one in the working group seemed able to clearly define or give explicit examples of tacit mathematical knowledge, there seemed a growing interest in investigating it. Artigue suggested that tacit knowledge consists of shared "stories" and experiences. New teachers at an institution don't have access to these, and hence can't rely on their own or their students' tacit knowledge. She asked, "What do we know about "didactic memory" (a term introduced by Brousseau) and the management of tacit knowledge?"

Artigue also suggested the procedural/conceptual distinction may be a dangerous opposition. Where is the place for technical knowledge? She saw a need for the rehabilitation of the technical dimension of mathematical activity. Mathematics is "cultural work" in which the manipulation of "ostensives" is guided by "non-ostensives" (ideas, concepts). In a seeming rebuttal to Paul Ernest, she said that mathematics cannot be only language games. Semiotics are also involved and "rhetoric is flat."

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CODA

Schemas enable one to act in familiar situations, but the question of whether mathematical schemas can be built automatically (tacitly) is open. Certainly, some everyday schemas, such as the "restaurant schema" are learned tacitly -- one knows from experience that events take a certain course -- one is seated, a menu is brought, one orders, one eats, one pays. [Cf. W. F. Brewer and G. V. Nakamura, "The nature and functions of schemas," in R. S. Wyler and T. K. Skrull, *Handbook of social cognition*, Hillsdale (1984).]

But what of mathematical schemas? How do they come about? Ed Dubinsky of Georgia State says, "An individual's mathematical knowledge is her or his tendency to respond to perceived mathematical problem situations by reflecting on problems and their solutions in a social context and by constructing or reconstructing mathematical actions, processes and objects and organizing these in schemas to use in dealing with the situations." [Cf. *Research in Collegiate Mathematics Education. I*, J. Kaput, A. Schoenfeld, and E. Dubinsky (eds.), CBMS Issues in Mathematics Education, Vol. 6 (1996), pp. 7-12.] His use of the term schema is close to Piaget's *schemata* and is similar in some ways to Tall and Vinner's *concept image*. [Cf. J. Piaget and R. Garcia, *Psychogenesis and the History of Science*, Routledge (1970) and *Educ. Stud. in Math.* 12 (1981), 151-169]. Dubinsky maintains that (mentally) constructing a concept involves interiorization, encapsulation, and reflective abstraction and speaks of an individual having "a function schema, a derivative schema, a group schema, etc." How does this view fit with John Mason's ideas on knowing how, knowing that, knowing to, knowing why, and knowing through? Do the above perspectives on mathematical knowing describe the same thing? Or, are we perhaps somewhat like the proverbial blind men attempting to describe the elephant? One thing seems clear. Thoughtful answers to the complex question, "Of what does mathematical knowledge consist?", would seem to require much more than a listing of behavioral objectives, such as factor a difference of squares or apply the Chain Rule to the composite of three functions.