Dehaene (this volume) articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the `number line' (analog magnitude) system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene's naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural number. Rather, the developmental source of explicit integer list representations of number are more likely to be systems such as the object--file representations that articulate mid--level object based attention, systems that build parallel representations of small sets of individuals.

What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of non-human animals will tell us little about these abilities, because only educated, enculturated human adults engage in organized games, formal mathematics, gourmet cooking, or map-reading. In this chapter, we argue against this seemingly sensible conclusion. When human adults exhibit complex, uniquely human, culture-specific skills, they draw on a set of psychological and neural mechanisms with two distinctive properties: they evolved before humanity and thus are shared with other animals, and they emerge early in human development and thus are common to infants, children, and adults. These core knowledge systems form the building blocks for uniquely human skills. Without them we wouldn’t be able to learn about different kinds of games, mathematics, cooking, or maps. To understand what is special about human intelligence, therefore, we must study both the core knowledge systems on which it rests and the mechanisms by which these systems are orchestrated to permit new kinds of concepts and cognitive processes. What is core knowledge? A wealth of research on non-human primates and on human

While I will begin with my own evidence for the decline of geometry in this century and my own description on how such a decline has proceeded, my basic theme is hopeful. Geometry has not died because it is essential to many other human activities and because it is so deeply embodied in how humans think. With the introduction of computers with rich graphical capacities and the

This article gives a brief introduction to a new discipline called the cognitive science of mathematics (Lakoff & Núñez, 2000), that is, the empirical and multidisciplinary study of mathematics (itself) as a scientific subject matter. The theoretical background of the arguments is based on embodied cognition, and on relatively recent findings in cognitive linguistics. The article discusses Mathematical Idea Analysis—the set of techniques for studying implicit (largely unconscious) conceptual structures in mathematics. Particular attention is paid to everyday cognitive mechanisms such as image schemas and conceptual metaphors, showing how they play a fundamental role in constituting the very fabric of mathematics. The analyses, illustrated with a discussion of some issues of set and hyperset theory, show that it is (human) meaning what makes mathematics what it is: Mathematics is not transcendentally objective, but it is not arbitrary either (not the result of pure social conventions). Some implications for mathematics education are suggested. Have you ever thought why (I mean, really why) the multiplication of two negative numbers yields a positive one? Or why the empty class is a subclass of all

We propose that the unique ability of humans to have separate mental representations for each finger and to move them in different sequential orders were redeployed for arithmetic. We tested our hypothesis with a behavioral dual-task experiment, where subjects (N=46) solved addition problems (primary task) and performed a sentence comprehension task (control task), while concurrently tapping their fingers (secondary task). We examined two sequential finger tapping tasks: one that was more automatic and followed the anatomical finger order (simple) and one that relied heavily on sequence processing (complex). The results revealed that both simple and complex finger tapping differentially interfered with addition compared to sentence comprehension. These results provide support for a finger-based representation of numbers and shared use of sequence processing resources for finger movements and addition.

Developmental cognitive neuroscience necessar- ily begins with a characterization of the developing mind. One cannot discover the neural underpinnings of cognition with- out detailed understanding of the representational Capacities that underlie thought. Characterizing the developing mind nvolves specifying the evolutionarily given building blocks from which human conceptual abilities are constructed. de- scribing what develops, and discovering the computational mechanisms that underlie the process of change. Here, 1 pres- ent the current state of the art with respect to one example of conceptual understanding: the representation of number.

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SUMMARY Many doctors, patients, journalists, and politicians alike do not understand what health statistics mean or draw wrong conclusions without noticing. Collective statistical illiteracy refers to the widespread inability to understand the meaning of numbers. For instance, many citizens are unaware that higher survival rates with cancer screening do not imply longer life, or that the statement that mammography screening reduces the risk of dying from breast cancer by 25 % in fact means that 1 less woman out of 1,000 will die of the disease. We provide evidence that statistical illiteracy (a) is common to patients, journalists, and physicians; (b) is created by nontransparent framing of information that is sometimes an unintentional result of lack of understanding but can also be a result of intentional efforts to manipulate or persuade people; and

We outline two theories of mathematical language acquisition and development, and discuss how a computational model of these theories may help to bridge the gap between automated theory formation and situated embodied agents. Finally, we briefly describe a simple theoretical case study of how such a model could work in the arithmetic domain.

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