Dehaene Et Al (2008)

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures Stanislas Dehaene, et al. Science 320, 1217 (2008); DOI: 10.1126/science.1156540 The following resources related to this article are available online at www.sciencemag.org (this information is current as of June 6, 2008 ): Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/cgi/content/full/320/5880/1217 Supporting Online Material can be found at: http://www.sciencemag.org/cgi/content/full/320/5880/1217/DC1 This article cites 24 articles, 4 of which can be accessed for free: http://www.sciencemag.org/cgi/content/full/320/5880/1217#otherarticles Information about obtaining reprints of this article or about obtaining permission to reproduce this article in whole or in part can be found at: on June 6, 2008 http://www.sciencemag.org/about/permissions.dtl www.sciencemag.org Downloaded from Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright 2008 by the American Association for the Advancement of Science; all rights reserved. The title Science is a registered trademark of AAAS. REPORTS Before formal schooling, Western children may Log or Linear? Distinct Intuitions of the acquire the number-line concept from Arabic nu- merals seen on elevators, rulers, books, etc. Thus, Number Scale in Western and existing studies do not reveal which aspects of the number-space mapping constitute a basic intuition that would continue to exist in the absence Amazonian Indigene Cultures of a structured mathematical language and educa- 1,2,3,4 1,2,4,5 5 6 tion. In particular, we do not know whether the Stanislas Dehaene, * Véronique Izard, Elizabeth Spelke, Pierre Pica log-to-linear shift would occur spontaneously in the course of brain maturation or whether it re- The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this quires exposure to critical educational material or mapping a cultural invention or a universal intuition shared by all humans regardless of culture culture-specific devices such as rulers or graphs. and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene To address these issues, we gathered evi- group with a reduced numerical lexicon and little or no formal education. At all ages, the dence from psychological experimentation in the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western Mundurucu, an Amazonian indigene culture with adults used linear mapping with small or symbolic numbers and logarithmic mapping when little access to education (22, 23). Previous re- numbers were presented nonsymbolically under conditions that discouraged counting. This search has established that, although their lexicon indicates that the mapping of numbers onto space is a universal intuition and that this initial of number words is reduced and they have little or intuition of number is logarithmic. The concept of a linear number line appears to be a cultural no access to rulers, measurement devices, graphs, invention that fails to develop in the absence of formal education. or maps, the Mundurucu entertain sophisticated concepts of both number and space, albeit in an What then is mathematics if it is not a unique, early age and has a reproducible substrate in the approximate and nonverbal manner (22, 23). We rigorous, logical structure? It is a series of bilateral intraparietal sulcus (5–8). This region is therefore asked whether they conceive of these great intuitions carefully sifted, and organized remarkably close to or even overlapping with two domains as being related by a systematic by the logic men are willing and able to apply areas engaged in the coding of spatial dimensions mapping and, if so, what form this number-space – at any time. such as size, location, and gaze direction (9 11). mapping takes. on June 6, 2008 Interactions between numerical and spatial codes A total of 33 Mundurucu adults and children – Morris Kline, Mathematics: The Loss of Certainty in the parietal cortex may therefore occur at this were tested individually in a number-space task [(1), p. 312] level. Indeed, in human adults, the mere presen- (Fig. 1) (24). In each trial, a line segment was dis- tation of an Arabic numeral automatically elicits played on a computer screen, with 1 dot at left and he mapping of numbers onto space plays a spatial bias in both motor responding and atten- 10 dots at right (or, in a separate block, 10 and 100 an essential role in mathematics, from mea- tion orienting (11–13). Brain-lesioned patients dots, respectively). Then other numbers were Tsurement and geometry to the study of ir- show corresponding impairments in comparing presented in random order, in various forms (sets rational numbers, Cartesian coordinates, the real and bisecting line segments and numbers (14), of dots, sequences of tones, spoken Mundurucu number line, and the complex plane (1, 2). How and some people even report a vivid experience words, or spoken Portuguese words). For each www.sciencemag.org does the human mind gain access to such abstract of seeing numbers at fixed locations on an idio- number, the participant pointed to a screen lo- mathematical concepts? Constructivist theories syncratic spatially contiguous “number form” cation and this response was recorded by a mouse view mathematics as a set of cultural inventions (15, 16). click, without feedback. Only two training trials that are progressively refined in the history of Recent experiments document a remarkable were presented, with sets of dots whose numer- mathematics and are slowly acquired during child- shift in the child’s conception of how numbers osity corresponded to the ends of the scale (1 and hood and adolescence (3). However, the mental mapontospace(17–19). When asked to point 10). The participants were told that these two construction of mathematics may have deeper toward the correct location for a spoken number stimuli belonged to their respective ends, but that foundations. Mathematical objects may find their word onto a line segment labeled with 0 at left other stimuli could be placed at any location. Downloaded from ultimate origin in basic intuitions of space, time, and 100 at right, even kindergarteners understand Because training did not involve intermediate and number that have been internalized through the task and behave nonrandomly, systematically numbers, performance on all subsequent trials millions of years of evolution in a structured envi- placing smaller numbers at left and larger num- served to reveal whether the participants would ronment and that emerge early in ontogeny, inde- bers at right. They do not distribute the numbers spontaneously use systematic mapping, and if so, pendently of education (2, 4). Here we present evenly, however, and instead devote more space whether their mapping would be compressive or evidence that reconciles these two points of view: to small numbers, imposing a compressed loga- linear. Our results suggest that all humans share the intui- rithmic mapping. For instance, they might place The Mundurucu’s mean responses revealed tion that numbers map onto space, but that culture- number 10 near the middle of the 0-to-100 seg- that they understood the task. Although some specific experiences alter the form of this mapping. ment. This compressive response fits nicely with participants tended to use only the end points of Previous psychological and neuroimaging re- animal and infant studies that demonstrate that thescale(24), most used the full response con- search supports the view that a sense of number is numerical perception obeys Weber’s law, a ubiq- tinuum and adopted a consistent strategy of present in humans and many other species at an uitous psychophysical law whereby increasingly mapping consecutive numbers onto consecutive larger quantities are represented with proportion- locations (Fig. 2). There was a significant pos- ally greater imprecision, compatible with a loga- itive correlation between stimulus number and 1INSERM, Cognitive Neuro-imaging Unit, Institut Fédératif de Recherche (IFR) 49, Gif sur Yvette, France. 2Commissariat à rithmic internal representation with fixed noise mean response location, regardless of the modal- l’Energie Atomique, NeuroSpin center, IFR 49, Gif sur Yvette, (7, 20, 21). A shift from logarithmic to linear ity in which the numbers were presented. The task France. 3Collège de France, Paris, France. 4Université Paris-Sud, mapping occurs later in development, between first was easy when the stimuli were sets of dots sim- 5 IFR 49, F-91191 Gif sur Yvette, France. Department of Psy- and fourth grade, depending on experience and ilar to the reference labels placed at the end points chology, Harvard University, Cambridge, MA 02139, USA. – 2 6Unité Mixte de Recherche 7023 “Formal Structure of Lan- the range of numbers tested (17 19). [numbers 1 to 10, correlation coefficient (r )= 2 guage,” CNRS and Paris VIII University, Paris, France. All of these observations, however, were made 92.6%, 8 df; numbers 10 to 100, r = 91.9%, 8 df]. *To whom correspondence should be addressed. E-mail: in Western children who all had access to math- However, the Mundurucu continued to use sys- [email protected] ematical education and culture at an early age. tematic number-space mapping with stimuli they www.sciencemag.org SCIENCE VOL 320 30 MAY 2008 1217 REPORTS had not been trained with, which shared with the linear and log regressors. For this analysis, we ance was logarithmic, subdivision by education reference labels only an abstract concept of number: pooled the trials with 1 to 10 dots and number level indicated that logarithmic responding held sequences of tones 1 to 10 (r2 = 92.5%, 8 df), words, but excluded those with 10 to 100 dots for participants with 1 to 2 years of education spoken Mundurucu number words (r2 =91.8, and with tones, for which Western participants (t = 3.15, 16 df, P = 0.006; fig.

Load more