Constructivist Alternatives to School Math

Some educators such as Magdalene Lampert (1990) and Deborah Ball (1991b), while at the same time trying to increase classroom safety, have exploited the riskiness of open conjecture in motivating students to defend their ideas in classroom arguments. Students who have a stake in maintaining their reputation as mathematical conjecturers are creative in finding clever arguments, examples, and counterexamples to support their own ideas and refute others'. In this way, the notion of mathematical proof is constructed as a limiting form of everyday classroom argument. A proof is a very convincing argument, and what counts as convincing depends very much on the sophistication of the community one is attempting to convince. Another illuminating example of large group mathematical activity is to be found in the doctoral work of Idit Harel (1988). In Harel’s project, a fourth grade class embarked on a project to author software for instructing third graders about fractions. Students were free to work alone or in groups, but in practice new ideas of how to present material circulated in the classroom in a "collaboration in the air" (Harel & Kafai, 1991). Even children who were most comfortable working alone ended up participating in the creation of a "fractions culture" and children who had been outside the social circle of the classroom enjoyed increased self-esteem as a result of the recognition of their contributions to that culture. In the past twenty years there has been some (though still not much) movement in the elementary and even high schools toward teaching children to be mathematicians instead of teaching them mathematics (see Papert, 1971). A considerable literature has grown up describing experiments in constructivist teaching in the schools (e.g. Hoyles & Noss, 1992; Steffe & Wood, 1990; Streefland, 1991; Von Glasserfeld, 1991). A plethora of creative approaches have begun to proliferate: Hoyles and Noss (1992) integrate the computer and the language LOGO into the mathematics classroom, providing new means of expressing mathematics, and encouraging a diversity of styles and approaches to learning/doing mathematics. Increasing availability of computers in classrooms has facilitated the development of computer microworlds for mathematical explorations (see e.g. Edwards, 1992; Schwartz & Yerushalmy, 1987; Leron & Zazkis, 1992) and emphasized dynamic mathematical processes as opposed to static structures. (see e.g. Vitale, 1992). Strohecker (1991) explores ideas of mathematical topology through classroom discussions of and experiments with tying knots. Cuoco & Goldenberg (1992) focus on geometry as providing a “natural” bridge between the visual intuitions of the high schooler and more symbolic areas of mathematics and, by so doing, “expands students’ conception of mathematics itself.” Confrey (1991) points out how easily we fall into the misconceptions trap and fail to listen to the sense, the novel meaning making of the mathematical learner. Schoenfeld (1991) moves from a college problem solving class using pre-decided problems to a focus on learners’ creation of related problems and, ultimately, to the development of classrooms that are “microcosms of mathematical sense making”. Konold (1991) works with beginning college students on probability and focuses on how students interpret probability questions. He concludes that effective probability instruction needs to encourage learners to pay attention to the fit between their intuitive beliefs and 1) the beliefs of others 2) their other related beliefs and 3) their empirical observations. Rubin & Goodman (1991) have developed video as a tool for analyzing statistical data and supporting statistical investigations. Nemirovsky (1993) & Kaput (in press) have developed curricular materials for exploring a qualitative calculus. They view themselves as part of a calculus reform movement that seeks to understand calculus as the mathematics of change and thus make calculus into a

2

Most of the above alternatives took place in the elementary school setting, and even those that took place in high school or college tended to focus on relatively elementary mathematics and relatively novice learners. As of yet, there has been very little written about the methods used at higher educational levels (late college through graduate school through life beyond school) which if anything are more rigidly instructionist (see Papert, 1990), provide less opportunity for self-directed exploration and conceptual understanding than do classes in the lower grades.

“No ideas but in things” - William Carlos Williams I will now advance a theoretical position which is the underlying force behind this study. This will involve a critical investigation of the concept of “concrete” and what it means to make mathematics concrete. Concepts that are said to be concrete are also said to be intuitive - we can reason about them informally. It follows that if we hold the standard view that concreteness is a property of objects which some objects have and some do not, we are likely to be led into the view that our intuitions are relatively fixed and unchangeable. This last conclusion, that of the relative unchangeability of intuitions, is the force behind the prescriptions to avoid intuitive probability judgments. When we replace the standard view of concrete with a new relational view, we also see the error in the static view of intuitions. In understanding the process of developing concrete relationships we learn how to support the development of mathematical intuitions. In the present chapter, I will expose the weakness of the standard view of concrete, and probe the very notion of “concreteness”. I propose a new definition of “concrete”. In the new model, concreteness is a property not an object itself but of our relationship to that object. A concept is “concrete” when - it is well-connected to many other concepts - we have multiple representations of it - we know how to interact with it in many modalities. As such, the focus is placed on the process of developing a concrete relationship with new objects, a process I call “concretion”. Implications for pedagogy of this new view of concrete will be explored, and a multi-faceted analogy will be constructed with conservation experiments of Piaget.

Seymour Papert has recently called for a “revaluation of the concrete”: a revolution in education and cognitive science that will overthrow logic from “on top and put it on tap.” Turkle and Papert (1991) situate the concrete thinking paradigm in a new “epistemological pluralism”-- an acceptance and valuation of multiple thinking styles, as opposed to their stratification into hierarchically valued stages. As evidence of an emerging trend towards the concrete, they cite feminist critics such as Gilligan's (1982) work on the contextual or relational mode of moral reasoning favored by most women, and Fox Keller's (1983) analysis of the Nobel-prize-winning biologist Barbara McClintock's proximal relationship with her maize plants, her “feeling for the organism.” They cite hermeneutic critics such as Lave (1988), whose studies of situated cognition suggest that all learning is highly specific and should be studied in real world contexts. For generations now we have viewed children's intellectual growth as proceeding from the concrete to the abstract, from Piaget's concrete operations stage to the more advanced stage of formal operations (e.g., Piaget, 1952). What is meant then by this call for revaluation of the concrete? And what are the implications of this revaluation for education? Are we being asked to restrict children's intellectual horizons, to limit the domain of inquiry in which we encourage the child to engage? Are we to give up on teaching general strategies and limit ourselves to very context specific practices? And what about mathematics education? Even if we were prepared to answer in the affirmative to all the above questions for education in general, surely in mathematics education we would want to make an exception? If there is any area of human knowledge that is abstract and formal, surely mathematics is. Are we to banish objects in the head from the study of mathematics? Should we confine ourselves to manipulatives such as Lego blocks and Cuisinaire Rods? Still more provocatively, shall we all go back to counting on our fingers? The phrases "concrete thinking", "concrete-example", "make it concrete" are often used when thinking about our own thinking as well as in our educational practice. To begin our investigation we will need to take a philosophical detour and examine the meaning of the word concrete. What do we mean when we say that something--a concept, idea, piece of knowledge (henceforward an object)--is concrete?

Our first associations with the word concrete often suggest something tangible, solid; you can touch it, smell it, kick it ; it is real. A closer look reveals some confusion in this intuitive notion. Among those objects we refer to as concrete there are words, ideas, feelings, stories, descriptions. None of those can actually be “kicked.” So what are these putative tangible objects we are referring to? One reply to the above objection is to say: No no, you misunderstand us, what we mean is that the object referred to by a concrete description has these tangible properties, not the description itself. The more the description allows us to visualize (or, if you will, sensorize) an object, to pick out, say, a particular scene or situation, the more concrete it is. The more specific the more concrete, the more general the less concrete. In line with this, Random House says concrete is “particular, relating to an instance of an object” not its class. Let us call the notion of concrete specified by the above the standard view. According to the standard view, concrete objects are specific instances of abstractions, which are generalities. Concrete objects are rich in detail, they are instances of larger classes; abstract objects are scant in detail, but very general. Given this view, it is natural for us to want our children to move away from the confining world of the concrete, where they can only learn things about relatively few objects, to the more expansive world of the abstract, where what they learn will apply widely and generally. Yet somehow our attempts at teaching abstractly leave our expectations unfulfilled. The more abstract our teaching in the school, the more alienated and bored are our students, and far from being able to apply their knowledge generally across domains, their knowledge displays a “brittle” character, usable only in the exact contexts in which it was learned. Numerous studies have shown that students are unable to solve standard math and physics problems when these problems are given without the textbook chapter context. Yet they are easily able to solve them when they are assigned as homework for a particular chapter of the textbook (e.g., DiSessa, 1983; Schoenfeld, 1985).

Upon closer examination there are serious problems with the standard view. For one thing, the concept of concrete is culturally relative. For us, the word “snow” connotes a specific, concrete idea. Yet for an Eskimo, it is a vast generalization, combining together twenty- world, depending on what kind and how many distinctions you make. There is no one universal ontology in an objective world. An even more radical critique of the notion of a specific object or individual entity comes out of recent research in artificial intelligence (AI). In a branch of AI called emergent AI, objects that are typically perceived as wholes are explained as emergent effects of large numbers of interacting smaller elements. Research in brain physiology as well as in machine vision indicate that the translation of light patterns on the retina into a “parsing” of the world into objects in a scene is an extremely complex task. It is also underdetermined; by no means is there just one unique parsing of the inputs. Objects that seem like single entities to us could just as easily be multiple and perceived as complex scenes, while apparently complex scenes could be grouped into single entities. In effect the brain constructs a theory of the world from the hints it receives from the information in the retina. It is because children share a common set of sensing apparatus (or a common way of obtaining feedback from the world, see Brandes & Wilensky, 1991) and a common set of experiences such as touching, grasping, banging, ingesting , that children come as close as they do to “concretizing” the same objects in the world. The critique of the notion of object connects also to the work of Piaget. In Piaget's view of development, the child actively constructs his/her world. Each object constructed is added to the personal ontology of the child. Thus we can no longer maintain a simple sensory criterion for concreteness, since virtually all objects, all concepts which we understand, are constructed, by an individual, assembled in that particular individual's way, from more primitive elements. Objects are not simply given to the senses; they are actively constructed. We have seen that when we talk about objects we can't leave out the person who constructs the object. It thus follows that it is futile to search for concreteness in the object -- we must look at a person's construction of the object, at the relationship between the person and the object.

I now offer a new perspective from which to expand our understanding of the concrete. The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that

4

representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object. Once we see this, it is not difficult to go further and see that any object/concept can be become concrete for someone. The pivotal point on which the determination of concreteness turns is not some intensive examination of the object, but rather an examination of the modes of interaction and the models which the person uses to understand the object. This view will lead us to allow objects not mediated by the senses, objects which are usually considered abstract—such as mathematical objects—to be concrete; provided that we have multiple modes of engagement with them and a sufficiently rich collection of models to represent them. When our relationship with an object is poor, our representations of it limited in number, and our modes of interacting with it few, the object becomes inaccessible to us. So, metaphorically, the abstract object is high above, as opposed to the concrete objects, Seymour Papert has said: “You can’t think about thinking without thinking which are down and hence reachable, “graspable.” We can dimly see it, touch it only with removed instruments, we have remote access, as opposed to the object in our hands that we can operate on in so many different modalities. Objects of thought which are given solely by definition, and operations given only by simple rules, are abstract in this sense. Like the word learned only by dictionary definition, it is accessible through the narrowest of channels and tenuously apprehended. It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her. As Minsky says in his Society of Mind: The secret of what anything means to us depends on how we've connected it to all the other things we know. That's why it's almost always wrong to seek the “real meaning” of anything. A thing with just one meaning has scarcely any meaning at all (Minsky, 1987 p. 64). This new definition of concrete as a relational property turns the old definition on its head. Now, thinking concretely is seen not to be a narrowing of the domain of intellectual discourse, but rather as opening it up to the whole world of relationship. What we strive for is a new kind of knowledge, not brittle and susceptible to breakage like the old, but in the words of Mary Belenky, “connected knowing“ (Belenky, Clinchy, Goldberger, & Tarule, 1986).

above reformulation of the concrete/abstract dichotomy: • It's a mistake to ask what is the "real meaning" of a concept. This "thing with only one meaning" is a characterization of a formal mathematical definition. An object of thought which is given solely by definition, and an operation given only by simple rules, is abstract in this sense. Like the word learned only by dictionary definition, it is accessible through the narrowest of channels and tenuously apprehended. It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her. • When concepts are very robust they become concrete. From Piaget's conservation experiments, we see that what appears to us to be a very concrete property of an object, like its very identity, is constructed by the child over a long developmental period. After a brief period of transition, when the child acquires the conserved quantity, she sees it as a concrete property of the object. Similarly, a machine vision program must build up notions of objects from more primitive notions in its vision ontology. Which things it adds to its ontology, making them concrete "wholes", depends on what primitive elements it started with and what objects it has already built up. As Quine has shown in his famous "rabbits vs. undetached rabbit-parts" example (Quine, 1960), there is no way unique way to slice up the world into objects. • There's no such thing as abstract concepts - there are only concepts that haven't yet been concretized by the learner. We mistakenly think that we should strive for abstract knowledge since this seems to be the most advanced kind of knowledge that we have. The fallacy here is that those objects which we have abstract knowledge of are those objects that we haven't yet concretized due to their newness or difficulty to apprehend. When we strive to gain knowledge of these, it seems like we strive for the abstract. But in fact, when we have finally come to know and understand these objects, they will become concrete for us. Thus Piaget is stood on his head: development proceeds from the abstract to the concrete.

Brouwer, L.E.J. (1981). Brouwer’s Cambridge lectures on Intuitionism.. New York: Cambridge University Press. Brown, S. (1983). The Art of Problem Posing. Philadelphia, Pa. Franklin Institute Press. Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard University Press. Carey, S. (1985). Conceptual Change in Childhood. Cambridge, MA: MIT Press. Carnap, R. (1939). The Foundations of Logic and Mathematics Carnap, R. (1962). The Logical Foundations of Probability Theory. Chicago: University of Chicago Press. Charischak, I. (1993). Personal Communication. Chen, D., & Stroup, W. (in press). General Systems Theory: Toward a Conceptual Framework for Science and Technology Education for All. Journal for Science Education and Technology. Cohen, B. (1990). Scientific Revolutions, Revolutions in Science, and a Probabilistic Revolution 1800-1930. in Kruger, L., Daston, L., & Heidelberger, M. (Eds.) The Probabilistic Revolution Vol. 1. Cambridge, MA: MIT Press. Cohen, P.J. (1966). Set Theory and the Continuum Hypothesis. New York. W.A. Benjamin. Collins, A. & Brown, J. S. (1985). The Computer as a Tool for Learning Through Reflection. Confrey, J. (1991). Learning to Listen: A Student’s Understanding of Powers of Ten. In Von Glaserfeld, E. (Ed.) Radical Constructivism in Mathematics Education. Netherlands: Kluwer Academic Press. Cuoco, A.& Goldenberg, E. P. (1992). Reconnecting Geometry: A Role for Technology. Proceedings of Computers in the Geometry Classroom conference. St. Olaf College, Northfield, MN, June 24-27, 1992. Davis P., & R. Hersh (1981). The Mathematical Experience. Boston: Houghton Mifflin. Davis, R. (1983). Complex Mathematical Cognition. in Ginsburg, H. (Ed.) The Development of Mathematical Thinking. Orlando, FL: Academic Press. Dennis, D. (1993). Personal communication. Dewey, J. (1902). The Child and the Curriculum. Chicago: University of Chicago Press. Dewey, J. (1916). Democracy and Education. Chicago: University of Chicago Press. diSessa, A. (1981). Unlearning Aristotelian Physics: A study of knowledge based learning. DSRE Working Papers 10, Division for Study and Research in Education. diSessa, A. (1983). Phenomenology and the Evolution of Intuition. In D. Gentner & A. Stevens (Eds.), Mental Models. Hillsdale, N.J.: Erlbaum. diSessa, A. (1985). Knowledge in Pieces. Paper presented at the 5th annual symposium of the Jean Piaget Society, " Constructivism in the Computer Age " Philadelphia, PA. Drescher, G. (1991). Made-up minds: A constructivist approach to artificial Intelligence. Cambridge, MA: MIT Press. Edwards, L. (1992). A LOGO Microworld for Transformational Geometry. In Hoyles, C. & Noss, R. (Eds.) Learning Mathematics and LOGO. London: MIT Press. Edwards, W. & von Winterfeldt, D. (1986). On Cognitive Illusions and their Implications. Southern California Law Review, 59(2), 401-451. Einstein, A., Podolsky, B. and Rosen, N. (1935). Can Quantum Mechanical Description of Reality be Considered Complete?, in Physical Review, 47:2. Evans, J. (1993). Bias and Rationality. In Mantkelow & Over (Eds.) Rationality (in press) London: Routledge. Falbel, A. (1991). The Computer as a Convivial Tool. In Harel, I. & Papert, S. Constructionism. Norwood N.J. Ablex Publishing Corp. Chapter 3. Feller, W. (1957). An Introduction to Probability Theory and its Applications., 1-2. Wiley. Feyerabend, P. K. (1975). Against Method: Outline of an Anarchistic Theory of Knowledge. London: New Left Books.

6

Fox Keller, E. (1983). A Feeling for the Organism: The Life and Work of Barbara McClintock. San Francisco, CA: W.H. Freeman. Fredkin, E. (1990). Digital Mechanics. Physica D. 45, 254-270. Frege, G. (1968). The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number. Evanston, Ill: Northwestern University Press. Geertz, C. (1973). The Interpretation of Cultures. New York: Basic Books. Gigerenzer, G. (1991). How to Make Cognitive Illusions Disappear: Beyond “Heuristics and Biases”. in Stroebe & Hewstone (Eds.) European Review of Social Psychology, Vol. 2, Chapter 4. Wiley and Sons. Gilligan, C. (1977). In a Different Voice: Psychological Theory and Women's Development. Cambridge, MA: Harvard University Press. Gilligan, C. (1986). Remappping the Moral Domain: New Images of Self in Relationship In T. Heller, M. Sosna, & D. Welber (Eds.), Reconstructing Individualism: Autonomy, Individualism, and the Self in Western Thought. Stanford: Stanford University Press. Gillman, L. (1992). The Car and the Goats. American Mathematical Monthly, volume 99, number 1, January, 1992. Gödel, K. (1931). On Formally Undecidable Propositions of Principia Mathematic. Goldenberg, E. P., Cuoco, A. & Mark, J. (1993). Connected Geometry. Proceedings of the Tenth International Conference on Technology and Education. Cambridge, MA, March 21-24, 1993. Goodman, N. (1983). Fact, Fiction and Forecast. Cambridge, MA: Harvard University Press. Granott, N. (1991). Puzzled Minds and Weird Creatures: Phases in the Spontaneous Process of Knowledge Construction. In Harel, I. & Papert, S. Constructionism. Norwood N.J. : Ablex Publishing Corp. Chapter 15. Greeno, J. G. (1983). Conceptual Entities. In D. Gentner & A. Stevens (Eds.), Mental Models. Hillsdale, N.J.: Erlbaum. Greeno, J. G. (1991). Number Sense as Situated Knowing in a Conceptual Domain. Journal for Research on Mathematics Education, 22, 170-218. Heidelberger, M. (Eds.) The Probabilistic Revolution. Vol.1. Cambridge, Ma: MIT Press. Hardy, G.H. (1940). A Mathematician’s Apology. New York: University of Cambridge Press. Harvey, B. (1992). Avoiding Recursion. In Hoyles, C. & Noss, R. (Eds.) Learning Mathematics and LOGO. London: MIT Press. Harel, I. (1988). Software Design for Learning: Children's Learning Fractions and Logo Programming Through Instructional Software Design. Unpublished Ph.D. Dissertation. Cambridge, MA: Media Laboratory, MIT. Harel, I. & Papert, S. (1990). Software Design as a Learning Environment. Interactive Learning Environments Journal. Vol.1 (1). Norwood, NJ: Ablex. Heyting, A. (1966). Intuitionism: An Introduction. Amsterdam: North Holland Publishing Co. Hofstadter, D.R. (1979). Godel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books. Hofstadter, D.R. (1982). Can Creativity be mechanized? Scientific American, 247, October 1982. Honey, M. (1987). The interview as text: Hermeneutics considered as a model for analyzing the clinically informed research interview. Human Development, 30, 69-82. Hoyles, C. (1991). Computer-based learning environments for Mathematics. In A. Bishop, S. Mellin-Olson, and J. Van Dormolen (Eds.), Mathematical Knowledge: Its growth Through Teaching. Dordrecht: Kluwer, 147-172. Hoyles, C. & Noss, R. (Eds.) (1992). Learning Mathematics and LOGO. London: MIT Press. Jackiw, N. (1990). Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press. Kahneman, D.. & Tversky, A. (1982). On the study of Statistical Intuitions. In D. Kahneman, A. Tversky, & D. Slovic (Eds.) Judgment under Uncertainty: Heuristics and Biases. Cambridge, England: Cambridge University Press. Kahneman, D. (1991). Judgment and Decision Making: A Personal View. Psychological Science. vol.2, no. 3, May 1991. Kafai, Y. & Harel, I. (1991). Learning through Design and Teaching: Exploring Social and Collaborative

Publishing Corp. Chapter 5. Kaput, J. (in press). Democratizing access to calculus: New routes to old roots. In A. Schoenfeld (Ed.) Mathematical Problem Solving and Thinking. Hillsdale, NJ.: Lawrence Erlbaum Associates. Kitcher, P. (1983). The Nature of Mathematical Knowledge. New York: Oxford University Press. Kleiner, I. (1990). Rigor and Proof in Mathematics: A Historical Perspective in Mathematics Magazine,64. Kline, M. (1953). Mathematics and Western Culture. Oxford University Press. Kline, M. (1972). Mathematical Thought from Ancient to Modern times. New York: Oxford University Press. Kolmogorov, A.N. (1950). Grundbegriffe der Wahsrscheinlichkeitsrechung . Chelsea. Knuth, D. (1974). Surreal Numbers. Addison-Wesley. Konold, C. (1989). Informal Conceptions of Probability. Cognition and Instruction, 6, 59-98. Konold, C. (1991). Understanding Students’ beliefs about Probability. In Von Glaserfeld, E. (Ed.) Radical Constructivism in Mathematics Education. Netherlands: Kluwer Academic Press. Kuhn, T. (1962). The Structure of Scientific Revolution. Chicago: University of Chicago Press. Kunda, Z. & Nisbett, R.E. (1986). The Psychometrics of Everyday Life. Cognitive Psychology, 18, 195-224. Lakatos, I. (1976). Proofs and Refutations. Cambridge: Cambridge University Press. Lakatos, I. (1978). The Methodology of Scientific Research Programmes. In Collected Papers, Vol. I. Cambridge, UK.: Cambridge University Press. Lampert, M. (1988). The Teacher’s role in reinventing the meaning of mathematical knowing in the classroom. The Institute for Research on Teaching, Michigan State University. Research Series No. 186. East Lansing, MI. Lampert, M. (1990.). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. In American Education Research Journal, spring, vol. 27, no. 1, pp. 29- 63.

Lampert, M. (1992). Practices and Problems in Teaching Authentic Mathematics. In Oser, F, Dick, A, & Patry, J. (Eds.) Effective and Responsible Teaching: the New Synthesis. San Francisco, CA: Jossey-Bass Publishers. Latour, B. (1987). Science in Action. Cambridge, MA: Harvard University Press. Leron, U. (1985). Logo today: Vision and Reality. The Computing Teacher 12:26-32. Leron, U. & Zazkis, R. (1992). Of Geometry, Turtles, and Groups. In Hoyles, C. & Noss, R. (Eds.) Learning Mathematics and LOGO. London: MIT Press. Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge, England: Cambridge University Press. Lopes, L. (1991). The Rhetoric of Irrationality. Theory and Psychology, vol 1(1): 65-82 Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.) Problems of representation in the Teaching and Learning of Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates. McCloskey, M. (1982). Naive theories of Motion. In D. Gentner & A. Stevens (Eds.), Mental Models. Hillsdale, N.J.: Erlbaum. McCulloch, W. (1965). Embodiments of Mind. Cambridge: MIT Press. McDougall, A. (1988). Children, Recursion and Logo Programming. Unpublished Ph.D. Dissertation. Melbourne, Australia: Monash University. Minsky, M. (1987). Form and Content in Computer Science. In ACM Turing Award Lectures: the First Twenty Years. New York: Addison-Wesley. Minsky, M. (1987). The Society of Mind. New York: Simon & Schuster Inc. Minsky, M. & Papert, S. (1969). Perceptrons: An Introduction to Computational Geometry. Cambridge, MA: MIT Press. Nemirovsky, R. (1993). Rethinking Calculus Education. Hands On! volume 16, no. 1, spring 1993. Nemirovsky, R. (in press). Don’t Tell me how things are, Tell me how you see them.

8

Melbourne: Collins Dove. Noss, R. (1988). The Computer as a Cultural influence in Mathematical Learning. Educational Studies in Mathematics 19 (2): 251 - 258. Noss, R. & Hoyles, C. (1992). Looking Back and Looking Forward. In Hoyles, C. & Noss, R. (Eds.) Learning Mathematics and LOGO. London: MIT Press. Packer, M. & Addison, R. (Eds.) (1989). Entering the Circle: Hermeneutic Investigation in Psychology. Albany, NY: State University of New York Press. Papert, S. (1972). Teaching Children to be Mathematicians Versus Teaching About Mathematics. Int. J. Math Educ. Sci Technology, vol 3, 249-262 Papert, S. (1980). Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books. Papert, S. (1987). A Critique of Technocentrism in Thinking about the School of the Future. Epistemology and Learning Memo #2, MIT. Papert, S. (1990). Personal Communication. Papert, S. (1991). Situating Constructionism. In I. Harel & S. Papert (Eds.)Constructionism. Norwood, N.J. Ablex Publishing Corp. (Chapter 1) Paulos, J. A. (1989). Innumeracy: Mathematical Illiteracy and its Consequences. Farrar, Straus & Giroux. Pea, R. & Kurland, M. (1984). On the Cognitive Effects of Learning Computer Programming. New Ideas in Psychology 2(1): 137-168. Perkins, D. (1986). Knowledge as Design. Hillsdale, NJ.: Lawrence Erlbaum. Perkins, Segal, & Voss (1991). Informal Reasoning and Education. Hillsdale, NJ: L. Erlbaum Associates. Phillips, J. (1988) How to Think About Statistics. New York: W.H. Freeman. Piaget, J. (1952). The Origins of Intelligence in Children. New York: International University Press. Piaget, J. (1954). The Construction of Reality in the Child. New York: Basic Books. Piaget, J. (1964). The Child’s Conception of Number. London: Routledge & Paul. Piaget, J. (1970). Genetic Epistemology. New York. Columbia University Press. Pimm, D. (1987). Speaking Mathematically. London: RKP. Poincare, H. (1908). The future of Mathematics, Revue generale des sciences pures et appliquees. Vol. 19, Paris. Polya, G. (1954). Mathematics and plausible reasoning. Princeton, N.J.: Princeton University Press. Polya, G. (1962). Mathematical Discovery. New York: John Wiley and sons. Popper, K. (1959). The Logic of Scientific Discovery. London: Hutchinson. Putnam, H. (1975) Mathematics, matter and method. London. Cambridge University Press. Quine, W. (1960). Word and Object. Cambridge, MA: MIT Press. Rabin, M. (1976). Probabilistic Algorithms. In Traub (ed.) Algorithms and Complexity: New Directions and Recent Results. New York. Academic Press. Rapoport, A. (1967). Escape from Paradox. Scientific American, July 1967. Resnick, M. (1991). Overcoming the Centralized Mindset: Towards an Understanding of Emergent Phenomena. In I. Harel & S. Papert (Eds.) Constructionism. Norwood, N.J.: Ablex Publishing Corp. (Chapter 11). Resnick, M. (1992). Beyond the Centralized Mindset: Explorations in Massively Parallel Microworlds, unpublished doctoral dissertation, MIT. Roberts, E. (1986). Thinking Recursively. New York: John Wiley and Sons. Rubin, A. & Goodman, B. (1991). TapeMeasure: Video as Data for Statistical Investigations. Presented at American Educational Research Association conference. Chicago, April, 1991. Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press. Sachter, J. (1991). Different Styles of Exploration and Construction of 3-D Spatial Knowledge in a 3-D Computer Graphics Microworld. In Harel, I. & Papert, S. Constructionism. Norwood, N.J.: Ablex Publishing Corp. Chapter 17. Savage, J. (1954). The Foundation of Statistics. New York. Schelling, T. (1978). Micromotives and Macrobehavior. New York: Norton.

Schoenfeld, A. ed. (1987). Cognitive Science and Mathematics Education. Hillsdale, NJ.: Lawrence Erlbaum. Schoenfeld, A. (1991). On Mathematics as Sense-Making: An Informal Attack on the Unfortunate Divorce of Formal and Informal Mathematics. In Perkins, Segal, & Voss (Eds.) Informal Reasoning and Education. Schwartz, J. & Yerushalmy, M. (1987). The geometric supposer: an intellectual prosthesis for making conjectures. The College Mathematics Journal, 18 (1): 58-65. Segall, R.G. (1991). Three Children, Three Styles: A Call for Opening the Curriculum. in Harel, I. & Papert, S. Constructionism . Norwood N.J. Ablex Publishing Corp. Chapter 13. Simon, J. (1991). Resampling Stats. Steedman, P. (1991). There is no more Safety in Numbers: A New Conception of Mathematics Teaching. In Von Glaserfeld, E. (Ed.) Radical Constructivism in Mathematics Education. Netherlands: Kluwer Academic Press. Steen, L. (1986). Living with a new Mathematical Species. Math Intell 8:2 33-49. Strohecker, C. (1991). Elucidating Styles of Thinking about Topology Through Thinking About Knots. In Harel, I. & Papert, S. Constructionism. Norwood N.J. Ablex Publishing Corp. Chapter12. Stroup, W. (1993). What Non-universal Understanding looks like: An investigation of results from a Qualitative-calculus assessment. Unpublished paper. Harvard Graduate School of Education. Tappan, M. (1987). Hermeneutics and Moral development: A developmental analysis of short-term change in moral functioning during late adolescence. Unpublished doctoral dissertation, Harvard University. Tarski, A. (1956). Logic, Semantics, Metamathematics. Oxford, England: Clarendon Press. Tobias, S. (1978). Overcoming Math Anxiety. New York: Norton. Toffoli, T. (1990). How cheap can mechanics’ first principles be? in W..Zurek (Ed.) Complexity, Entropy, and the Physics of Information. Redwood City, Ca: Addison-Wesley. 301-318. Turkle S., & Papert, S. (1991). Epistemological Pluralism and Revaluation of the Concrete. In I. Harel & S. Papert (Eds.) Constructionism. Norwood N.J. Ablex Publishing Corp. (Chapter 9). Tversky, A., & Kahneman, D. (1971). Belief in the Law of Small Numbers. Psychological Review, 76, 105-110. Tversky, A. (1973). Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 5, 207-232. Tymoczko, T. (1979). The Four Color Problem and its Philosophical Significance. The Journal of Philosophy. Volume LXXVI, no. 2, February. Tymoczko, T. (1985). New Directions in the Philosophy of Mathematics. Boston: Birkhauser. Vitale, B. (1992). Processes: A Dynamical Integration of Computer Science into Mathematical Education. In Hoyles, C. & Noss, R. (Eds.) Learning Mathematics and LOGO. London: MIT Press. Von Glaserfeld, E. (Ed.) (1991). Radical Constructivism in Mathematics Education. Netherlands: Kluwer Academic Press. Vygotsky, L. (1986). Thought and Language . MIT Press, Cambridge, MA. Wilensky, U. (1989). Fractional Division. Unpublished paper. Cambridge, MA: MIT Media Laboratory. Wilensky, U. (1991a). Abstract Meditations on the Concrete and Concrete Implications for Mathematics Education. In I. Harel & S. Papert (Eds.) Constructionism. Norwood N.J.: Ablex Publishing Corp. (Chapter 10) Wilensky, U. (1991b). Feedback, Information and Domain Construction. Unpublished paper. MIT Media Lab. Wilensky, U. (in preparation). Conservation of Worth: Agent Analyses of the Monty Hall Problem. MIT Media Lab. Wittgenstein, L. (1956). Remarks on the Foundations of Mathematics. New York: Macmillan. Wolfram, S. (1990). Mathematica: A System for Doing Mathematics by Computer. Redwood City, CA: Addison-Wesley Publishing Co.