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Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education
 Educational Studies in Mathematics
, 1999
"... ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cogniti ..."
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ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to “represent ” events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a “virtual ” culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills.
An Infinite Set Of Heron Triangles With Two Rational Medians
 American Mathematical Monthly
, 1997
"... Introduction If we denote the sides of a triangle by (a; b; c) then the area is given by (1) = p s(s a)(s b)(s c) where s = (a + b + c)=2 is the semiperimeter. This formula is usually attributed to Heron of Alexandria circa 100 BC  100 AD. However, it was already known to Archimedes prior to 2 ..."
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Introduction If we denote the sides of a triangle by (a; b; c) then the area is given by (1) = p s(s a)(s b)(s c) where s = (a + b + c)=2 is the semiperimeter. This formula is usually attributed to Heron of Alexandria circa 100 BC  100 AD. However, it was already known to Archimedes prior to 212 BC [5, p. 105]. Our investigation is limited to triangles with rational sides. Even with sides of rational length, \Heron's" formula shows that the area need not be rational; any triangle with three rational sides and rational area is called a Heron triangle. The smallest such triangle with integer sides is the familiar (5; 4; 3) right triangle (with area 6) shown in Figure 1. B C a = 5<F
Olga TausskyTodd/s Influence on Matrix Theory and Matrix Theorists  A Discursive Personal Tribute
, 1977
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Evolution of the function concept: A brief survey
 The College Mathematics Journal
, 1989
"... received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the histo ..."
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received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the history of mathematics and its use in the teaching of mathematics. Introduction. The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A onesentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of
Is probability the only coherent approach to uncertainty?, Risk Analysis, forthcoming
"... In this paper I discuss an argument that purports to prove that probability theory is the only sensible means of dealing with uncertainty. I show that this argument can succeed only if some rather controversial assumptions about the nature of uncertainty are accepted. I discuss these assumptions and ..."
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In this paper I discuss an argument that purports to prove that probability theory is the only sensible means of dealing with uncertainty. I show that this argument can succeed only if some rather controversial assumptions about the nature of uncertainty are accepted. I discuss these assumptions and provide reasons for rejecting them. I also present examples of what I take to be nonprobabilistic uncertainty. Key Words: Cox’s Theorem, NonClassical Logic, Probability, Uncertainty, Vagueness Uncertainties are ubiquitous in risk analysis and on the face of it, we must contend with a number of quite distinct sorts of uncertainty. There are, of course, many methods on hand to deal with uncertainty, so it is important to select the method best suited to the uncertainty in question. There is, however, a growing push towards dealing with all uncertainty in one fell swoop. That is, it is thought to be desirable to employ a single method capable of quantifying all sources of uncertainty. One candidate for this task is probability theory. For such a program to succeed, a demonstration that all uncertainty is,
Between laws and models: Some philosophical morals of Lagrangian mechanics
, 2004
"... I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and HamiltonJacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been ..."
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I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and HamiltonJacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels—“laws of nature ” and “models”. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. Newton’s fundamental discovery, the one which he considered necessary to keep secret and published only in the form of an anagram, consists of the following: Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. In contemporary mathematical language, this means: “It is useful to solve differential equations”.
Fisher’s Instrumental Approach to Index Numbers. Supplement to the History of Political Economy
"... ‘If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. ’ (Hilbert 1902, 448) ‘Sometimes control with a single lens is impossible since some incompatible features are required and a compromise becomes necessary calling for further judgement on th ..."
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‘If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. ’ (Hilbert 1902, 448) ‘Sometimes control with a single lens is impossible since some incompatible features are required and a compromise becomes necessary calling for further judgement on the part of the designer as to which error should be reduced and to what degree. ’ (Bracey 1960, 18) 1.
Compass and Straightedge in the Poincaré Disk
, 2001
"... The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Eucli ..."
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The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Euclidean straightedge, and can be carried out by hand. Indeed, M.C. Escher used something like the methods we give here to produce his well known Circle Limit I, II, III, and IV prints [9]. In [6], H.S.M. Coxeter describes a remarkable correspondence with Escher. Having met at the 1954 International Congress of Mathematics in Amsterdam, Coxeter apparently sent Escher a paper in which a drawing of part of a tiling of the Poincaré disk appeared. Coxeter must have been quite pleased and surprised to find a print of Circle Limit I in his mail in December 1958. It is quite remarkable that the drawing in the paper Coxeter had sent Escher was not even as detailed as our Figure 1, and did not show the “scaffolding ” Coxeter had used in its construction. Nonetheless Escher deduced and generalized the technique of its construction, producing incredibly fine tesselations of the Poincarédisk.
On HamiltonJacobi theory: its geometry and relation to pilotwave theory
, 2002
"... This paper is an exposition, mostly following Rund, of some aspects of classical HamiltonJacobi theory, especially in relation to the calculus of variations. The last part of the paper briefly describes the application to geometric optics and the opticomechanicsal analogy; and reports recent work ..."
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This paper is an exposition, mostly following Rund, of some aspects of classical HamiltonJacobi theory, especially in relation to the calculus of variations. The last part of the paper briefly describes the application to geometric optics and the opticomechanicsal analogy; and reports recent work of Holland providing a Hamiltonian formulation of the pilotwave theory. The treatment is technically elementary, and provides an introduction to the subject for newcomers. Submitted to Contemporary Physics ‘Dont worry, young man: in mathematics, none of us really understands any idea—we just get used to them.’ John von Neumann, after explaining (no doubt very quickly!) the method of characteristics (i.e. HamiltonJacobi theory) to a young physicist, as a way to solve his problem; to which the physicist had replied ‘Thank you very much; but I’m afraid I still don’t understand this method.’ 1