External representations (ERs) are effective in reasoning due to their cognitive and semantic properties. We investigated subjects' use of ERs in their solutions to analytical reasoning problems. Two sources of data were analysed. The first consisted of a large corpus of ERs (`workscratchings') used by students in their solutions to problems administered via paper and pencil tests. The second source of data was collected using switchER, a computer-based system that administered the problems, provided a range of ER construction environments for the subject to choose between and which dynamically logged user--system interactions. SwitchER was developed in order to study the process and time-course of ER use and to investigate the mechanisms (such as ER switching) by which subjects resolve impasses in reasoning. The results showed great diversity of ER use across subjects, allowing the utility of various ERs under differing task conditions to be studied. The range of ERs used by subjects ...

Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of “core knowledge” associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.

Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)

1. The particular role of mathematical signs in the frame of epistemological conditions of mathematical knowledge In general, the importance of signs for human thinking is uncontested and fundamental. Without signs, no human thinking and no mental generalizations would exist. „We have no ability to think without signs “ (Peirce 1991, p. 42) reads the famous dictum by Charles Peirce (cf. also Radford 2001a). A general, essential identification of the role of the sign is the one that the sign stands for something else; Thomas of Aquin has described this feature with the famous definition „aliquid stat pro aliquo“. In the following it is tried under an explicitly epistemological perspective to work out the particular role of mathematical signs in certain essential points of view. This is in no case about the modeling of a general sign concept, but about a better understanding of the particularity of mathematical signs, partly referring to semiotic concepts and models of signs. This makes it necessary to concentrate on the epistemological nature of mathematical knowledge and of mathematical concepts, as mathematical signs serve mainly for „recording “ mathematical knowledge and mathematical concepts. Seen in this way, the particularity of mathematical

is a family of n + 2 spheres in R in which each n+1 spheres have a unique common point but all n+2 have empty intersection. A unit-sphere-system is a sphere-system consisting of all unit spheres. We prove that for every 2 3, there is a unit-spheresystem . The case n = 3 is open. We also prove that if there is a unit-sphere-system in R , then there is a tetrahedron in R one of whose "escribed" spheres lies completely inside the circumscribed sphere.

Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians’ creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity Mathematical creativity ensures the growth of the field of mathematics as a whole. The constant increase in the number of journals devoted to mathematics research bears evidence to the growth of mathematics. Yet what lies at the essence of this growth, the creativity of the mathematician, has not been the subject of much research. It is usually the case that most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation (Ervynck, 1991). The earliest known attempt to study mathematical creativity was an extensive questionnaire published in the French periodical L'Enseigement Mathematique (1902). This questionnaire and a lecture on creativity given by the renowned 20th century mathematician Henri Poincaré to the Societé de Psychologie inspired his colleague Jacques Hadamard, another prominent 20th century mathematician, to investigate the psychology of mathematical creativity

The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrized model, " Q", for the state vector evolution of spin-1/2 particles during measurement is developed. Q draws on recent work on so-called "riddled basins " in dynamical systems theory, and is local, deterministic, nonlinear and time asymmetric. Moreover, the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for considering this model arises from speculations about the (time asymmetric and uncomputable) nature of quantum gravity, and the (nonlinear) role of gravity in quantum state vector reduction. Although the evolution of Q’s state vector cannot be determined by a numerical algorithm, the probability that initial states in some given region of phase space will evolve to one of these attractors, is itself computable. These probabilities can be made to correspond to observed quantum spin probabilities. In an ensemble sense, the evolution of the state vector to an attractor can be described in by a diffusive random walk process, suggesting that deterministic dynamics may underlie recent attempts to model state vector evolution by stochastic equations. Bell’s theorem and a version of the Bell-Kochen-Specker paradox, as illustrated by Penrose’s "magic dodecahedra", are discussed using quantum entanglement Q as a model of quantum spin measurement. It is shown that in both cases, proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q, these deterministic propositions are physically uncomputable, and no non-algorithmic mathematical solution is either known or suspected. Adapting the mathematical formalist approach, the non-existence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. As a result, it is claimed that constrained by Bell’s inequality, locality and determinism notwithstanding. Q is not

Abstract: The goal of this paper is to suggest how an aesthetic image can be added to mathematics education. Calls for reform in mathematics education (e.g., National Council for Teachers of Mathematics, 1989, 2000) are premised on shifting teacher attention from an absolutist toward a social constructivist philosophy of mathematics and mathematics education. The goal of reform is success for all students of mathematics. In a previous paper, I used ideas from art education to suggest that success for all might be achieved by fostering an aesthetic image of mathematics (Betts & McNaughton, 2003). This paper continues the argument by shifting from questions of why to how.

. The logical symbol # (pronounced "for all") denotes universal quantification. For example, the formula "#x#B x # x 2 +1" reads "for all x a member of the real numbers, x is less than or equal to x-squared plus one" (i.e., no real number is greater than one more than its own square). The logical symbol # (pronounced "there 2 exists") denotes existential quantification. For example, the formula "#x#8 | x 2 =5x" states that there exists an integer whose square is equal to 5 times itself (i.e., x is either 5 or 0) . These connectives may be composed in more complicated formulae, as in the following example: "#x#8 #y#8#| y>x" which states that there is no largest integer. The logical connective (pronounced

The modern West believes in genius, but definitions vary widely when they can be come by at all. I think this