E-Book, Englisch, 400 Seiten, Format (B × H): 152 mm x 229 mm
Geary / Berch / Mann Koepke Evolutionary Origins and Early Development of Number Processing
1. Auflage 2014
ISBN: 978-0-12-800888-1
Verlag: William Andrew Publishing
Format: EPUB
Kopierschutz: 6 - ePub Watermark
E-Book, Englisch, 400 Seiten, Format (B × H): 152 mm x 229 mm
Reihe: Mathematical Cognition and Learning (Print)
ISBN: 978-0-12-800888-1
Verlag: William Andrew Publishing
Format: EPUB
Kopierschutz: 6 - ePub Watermark
The first volume in this ground-breaking series focuses on the origins and early development of numerical cognition in non-human primates, lower vertebrates, human infants, and preschool children. The text will help readers understand the nature and complexity of these foundational quantitative concepts and skills along with evolutionary precursors and early developmental trajectories.
- Brings together and focuses the efforts and research of multiple disciplines working in math cognition.
- The contributors bring vast knowledge and experience to bear on resolving extant substantive and methodological challenges to help advance the field of basic number processing.
- Introductory sections and summaries will be included to provide background for non-specialist readers.
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Number and Magnitude in Non-Human Animals
Chapter 1 - At the Root of Math: Numerical Abilities in Fish
Christian Agrillo, Maria Elena Miletto Petrazzini, and Angelo Bisazza
Chapter 2 - Foundations of Number and Space Representations in Precocial Species
Giorgio Vallortigara
Chapter 3 - Numerical Concepts: Grey Parrot Capacities
Irene M. Pepperberg
Chapter 4 - Numerical Cognition and Quantitative Abilities in Nonhuman Primates
Michael J. Beran, Audrey E. Parrish, and Theodore A. Evans
Number and Magnitude in Infants and Young Children
Chapter 5 - Evolutionary and Developmental Continuities in Numerical Cognition
Ariel Starr and Elizabeth M. Brannon
Chapter 6 - On the Relation Between Numerical and Non-Numerical Magnitudes: Evidence for a General Magnitude System
Stella Lourenco
Chapter 7 - Foundations of the Formal Number Concept: How Preverbal Mechanisms Contribute to the Development of Cardinal Knowledge
Kristy vanMarle
Chapter 8 - Intuitive Non-Symbolic Arithmetic
Koleen McCrink
Chapter 9 - Analog Origins of Numerical Concepts
Jessica F. Cantlon
Chapter 10 - The Small-Large Divide: A Case of Incompatible Numerical Representations in Infancy
Tasha Posid and Sara Cordes
Chapter 11 - The Influence of Multisensory Cues on Representation of Quantity in Children
Joseph M. Baker and Kerry E. Jordan
Chapter 12 - What is a Weber Fraction?
Justin Halberda
Chapter 13 - Multiple Evolved Functions of Systems for Number and Magnitude
David C. Geary, Daniel B. Berch, and Kathleen Mann Koepke
Foreword
Rochel Gelman; C.R. Gallistel, Rutgers University It is a pleasure to introduce what is sure to become a prominent mark of progress in the scientific understanding of the origins of numerical cognition. This volume provides a comprehensive view of that progress, revealing the emergence of a broad consensus on the answers to several foundational questions, a consensus far removed from the views commonly entertained half a century ago. A half century or so ago it was commonly assumed that numerical cognition depended on language for its development and that it developed, therefore, only in humans. We now understand that numerical cognition is evolutionarily ancient. Many of the chapters in this volume review aspects of the empirical basis for this new understanding, but the reader is directed most particularly to the first four chapters and to the final chapter. We also now understand that the pre-linguistic system of numerical estimation and reasoning, which we share with most if not all vertebrates (and possibly with invertebrates), plays an important role in the development of verbalized mathematical reasoning. A majority of the chapters touch on the evidence for this latter conclusion, but the reader is referred most particularly to Chapters 5 through 10. In this introduction, we begin with a brief history of contrasting ideas about what a number is. We do so to clarify for the reader some important distinctions, most importantly the distinction between numbers as symbols for discrete (countable) quantities and numbers as the players in the system of arithmetic reasoning. Gelman (1972) referred to this as the estimator-operator distinction. Throughout most of the history of mathematics, numbers were defined by what they referred to, which is to say countable quantities. The traditional view is memorably summed up by a well known quote attributed to Kronecker, “God made the integers; all else is the work of man.” (Quoted in an obituary by Weber, 1893.) Only with the extensive formalization of mathematics in the latter part of the 19th century did an alternative view gain currency, the view that now dominates among mathematicians. This view is articulated by Knopp (1952, p. 5), who gives an axiomatization of arithmetic followed by: “Once the validity of these fundamental laws has been established, it is unnecessary, in all further work with the literal quantities a, b,…, to make use again of the fact that these symbols denote rational numbers. … From the important fact that the meaning [that is, the reference] of the literal symbols need not be considered at all…, there results immediately the following extraordinarily significant consequence: If one has any other entities whatsoever besides the rational numbers, …, but which obey the same fundamental laws, one can operate with them as with the rational numbers, according to exactly the same rules. Every system of objects for which this is true is called a number system, because, in a few words, it is customary to call all those objects numbers with which one can operate according to the fundamental laws we have listed.” (italics in original). On this formalist view, numbers are defined in the same way that chess pieces are, that is, by what can be done with them. The empirical reasons for regarding the brain's non-verbal representation of discrete (countable) quantity as part of a more general system of arithmetic reasoning that represents both continuous and discrete quantity is the focus of Chapter 6 (see also Walsh, 2003; Gallistel, 2011). The fact that a system of symbols as simple as arithmetic can so effectively represent so much of experienced reality is astonishing (Wigner, 1960). The representational power of arithmetic reasoning probably explains why it emerged so early in evolution. Our intuitive identification of number with countable quantity is evident already in the troublesome ambiguity about what exactly an author may mean when using ‘number’ in many contexts. Do they mean to refer to the numerosity of a set? Or do they mean to refer to a conventional symbol, such as ‘two’ or ‘11’ or ‘IV’ or (5-2), that may (or may not) represent the numerosity of a set? The distinction here corresponds to the distinction in computer programming between, on the one hand, the bit-code representation of a number that enters into the arithmetic operations that a computer performs and, on the other hand, the bit-code representation of the textual symbol for that number. This distinction (between x = 7 and x = ‘7’) often flummoxes newcomers to the programming craft. It is now common in the technical literature to use ‘numerosity’ when the first meaning is intended. Gelman and Gallistel (1978) suggest ‘numerlog’ when reference to the conventional symbols is instead intended, but that coinage has not come into common use. In any event, failing to distinguish between the numerosities themselves, on the one hand, and the symbols for quantities in an arithmetic system, on the other, may lead some readers into confusion. Some of the current authors use ‘symbolic’ to refer to conventional symbolizations of quantity, particularly when asserting that the evolutionarily ancient system is “non-symbolic” or that only humans have a “symbolic” representation of number. We would suggest that this usage invites confusion. It seems to imply that the evolutionarily ancient system does not employ symbols in the sense in which a computer scientist understands this term (see Gallistel & King, 2010). Insofar as someone subscribes to a radical connectionist view of the brain, a view in which brain activity is “sub-symbolic” and does not in any true sense represent the experienced world, this might be the intended meaning, but we do not believe it is the meaning intended by any current author. To distinguish between conventional symbols and the brain's symbols, we suggested ‘numerons’ to refer to the brain's symbols for quantity (Gelman & Gallistel, 1978), but again that coinage has not gained currency. Symbols are essential elements of representations (Gallistel & King, 2010). A representation has two fundamental components, a component (set of processes) that map from aspects of the represented system (for example, from numerosities) to the symbols that thereby refer to them and a component (set of processes) that operates on the symbols (ordering them, adding or subtracting them, multiplying or dividing them, concatenating or extracting them, and so on). Symbols are the stuff of computation, and representations are constructed by computation. That is the essence of the computational theory of mind, which is the central doctrine of cognitive science. If the brain represents quantities—and, in our view, the chapters in this volume show beyond reasonable argument that it does—then it does so by means of symbols. Because the meanings of numerical symbols were once commonly thought to derive from their reference, there were doubts about whether negative numbers and irrational numbers really existed, even among professional mathematicians living into the 19th century (Kline, 1972). However, by then, the use of numerical coordinates to represent different aspects of the physical and financial world was rapidly advancing. Because the origin of a useful coordinate system is almost always arbitrary (the Greenwich meridian, for example) and/or because many of the quantities represented by numbers are inherently directed quantity (distance north and south, for example), it is all but impossible to avoid using negative numbers in practice. Moreover, in algebraic reasoning, one often does not know until it is, so to speak, “too late,” whether one is or is not trafficking with negative quantities. Thus, some of what Kronecker regarded as the works of man (the negative numbers and fractions) seemed unavoidable if one were to have a generally useful system for representing quantities, as even Kronecker conceded. Kronecker agreed that negative numbers and fractions could be admitted into the system made by God (the counting numbers), provided that it could be proven that they could be made to abide by the manipulation rules that God's creatures (the so-called natural numbers) abided by. However, the enlargement of the system people took to be the “natural” numbers to make it more generally useful led most mathematicians to include among the numbers the so-called transcendental numbers like p, whose very existence Kronecker disputed. (He insisted that there was no such number!) One begins to see why the question whether a symbol did or did not abide by the rules of arithmetic came to be an ever more important consideration, emerging eventually as definitive of number itself. Theories of the development of verbalized numerical reasoning divide along similar lines. Some authorities urge that the preschool child understands the meaning of ‘one’ if and only if s/he knows that this word refers to the numerosity of sets that have only one member, and similarly for the meaning of ‘two’ and ‘three’ (Carey, 2010). This view is similar to (and perhaps partially motivated by) Gottlob Frege's attempt to found mathematics on logic and set...
