Mathematical abilities in Preschoolers: Potential Diagnostic Probes for Developmental Dyscalculia Elena Rusconi and Brian Butterworth Institute of Cognitive Neuroscience, UCL, London, UK Recent empirical evidence suggests that preschoolers possess a starter kit of numerical abilities enabling them to abstract numerosity from sets of visual and auditory objects/events and perform approximate arithmetic operations. Formal education in mathematics would then extend and refine their capacity by providing additional (symbolic) tools. Developmental dyscalculia is a severe handicap that affects as many people as dyslexia [1]. The UK Department of Education and Skills defines developmental dyscalculia as A condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence. ([2], p.2). In Box 1, we give an example of a dyscalculic adult struggling with problems most people would find trivial. Unfortunately, there are no agreed criteria for dyscalculia, and hence experimental investigations have led to inconsistent results, depending on how the sample was classified (e.g., the proportion of population performing below an arbitrary cut-off on a standardized arithmetic test). One problem is that poor arithmetical attainment may be caused not only by a congenital cognitive deficit, but also by the quality of the educational experience both within and outside school [1]. What is needed ideally is a specific test of the capacity to acquire arithmetic that is, as far as possible, independent of educational experience. Studies on numerical abilities in infants and preschoolers can be useful in developing diagnostic tests since they focus on the capacities that children have before they are exposed to formal schooling. One important proposal as to what these capacities might be has been puit forward by Feigenson, Dehaene and Spelke [3]. They suggest that learners are born with what we might call a Address correspondence to: Elena Rusconi, PhD, University College London, Institute of Cognitive Neuroscience, 17 Queen Square, London WC1N 3AR United Kingdom Phone: +44 (0)20 76795431; Fax: +44 (0)20 79168517; E-mail: e.rusconi@ucl.ac.uk 1
starter kit of two core systems of numerical knowledge: 1. Precise representations of up to about three distinct individuals and 2. Approximate representations of numerical magnitude (plus the support of the language system for the later development of mental representations of exact number and arithmetic). One critical issue is whether these types of representation can enter into arithmetical operations. A recent study by Barth, La Mont, Lipton and Spelke [4] shows that preschoolers (5 to 6 years old) can not only abstract approximate numerosity from large sets of visual and auditory objects or events (between 10 and 58) but also carry out additions on those representations. (See Figure 1) However, when confronted with homologous verbal problems (e.g., If your mum gave you 27 marshmallows, and then she gave you 31 more, how many would you have? Would it be more like 58 or 33?), participants were not even able to produce, or to choose between two given alternatives, the approximate result. Crucially, Barth et al. s participants were able to compare and add sets when presented either in the same (visual) or in a different (visual and auditory) modality implying that the children can generate abstract representations of approximate numerosities and can carry out mental operations that seem to be logically equivalent to addition and comparison of exact numbers. Moreover, they are doing this without the aid of linguistic support or support from visual continuous variables such as the perimeter of a virtual enclosing rectangle. One puzzling feature of this study is this: the final stage in both the comparison and the addition tasks is the same (a comparison between two numerosities); while the addition task requires a prior computation (the mental addition of two numerosities). One would therefore expect the addition task to be more difficult because it involves an extra step. However, both tasks show similar levels of accuracy. Feigenson et al s conception of the starter kit comprises both a precise representation of distinct individuals and an approximate representation of numerical magnitude (with a limit of about 3) [3], where the latter enables children to exploit the verbal and symbolic tools provided by formal and informal education to develop exact representations of larger numbers and to acquire arithmetical skills. Barth et al s [4] study may point the way to new diagnostic tests for selective deficits in the number domain. For example, it suggests that dyscalculic like BO would have difficulty not only with symbolic arithmetic, but also the approximate arithmetic. 2
References 1. Butterworth B (2004). Developmental dyscalculia. In J. Campbell (Ed.) Handbook of Mathematical Cognition. Psychology Press, New York. 2. DfES (2001). Guidance to support pupils with dyslexia and dyscalculia (No. DfES 0512/2001). London: Department of Education and Skills. 3. Feigenson L, Dehaene S, Spelke E (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307-314. 4. Barth H, La Mont K, Lipton J, Spelke ES. (2005) Abstract number and arithmetic in preschool children. Proceedings of the National Academy of Sciences, 102, 14116-21. 5. Delazer M, Girelli L, Granà A., Domahs F (2003). Number processing and calculation Normative data from healthy adults. The Clinical Neuropsychologist, 17, 331-350. 6. Gelman, R. & Butterworth, B. (2005) Number and language: How are they related?. Trends in Cognitive Sciences, 9, 6-10 7. Landerl K, Bevan A, Butterworth B (2004). Developmental dyscalculia and basic numerical capacities: a study of 8-9-year-old students. Cognition, 93, 99-125. 8. Barth H, La Mont K, Lipton J, Dehaene S, Kanwisher N, Spelke E. (2006). Non-symbolic arithmetic in adults and young children. Cognition, 98, 199-222. 3
Figure 1 Figure 1 Caption In the visual comparison experiment (a), children were presented with an array of blue dots that was covered by a rectangular occluder before an array of red dots appeared on the screen. In each case, the dots were presented for a time too short for counting and, when all the stimuli disappeared, a guess was requested on whether more blue or red dots had been presented. Three ratio levels between blue and red arrays were chosen and the correlation between the size of the virtual enclosing rectangle (that was likely to be exploited as a cue [8]) and numerosity was positive in half of the trials and negative in the other half. Children s performance in the comparison task was significantly above chance (mean level of accuracy: 67%) and declined as the ratio of the two sets approached 1. Accuracy was also significantly higher when the size of the enclosing virtual rectangle was positively correlated with numerosity (i.e., 4
children did not base their judgment on abstract numerosity alone but also on physical cues), however, this variable did not account for total performance. In the visual addition experiment (b), blue arrays from the previous task were divided in two unequal subsets that were each less numerous than the successive red array. On half of the trials the red array was less numerous than the sum of the two blue arrays, thus if children based their judgments on a comparison between all the visible arrays, they would have performed at chance. In fact, mean accuracy was 66% (significantly above chance), it declined as the ratio approached 1 and, once again, total array size exerted an influence on performance. A tendency to choose the sum as the larger numerosity was also found. In the following two experiments, comparison (c) and comparisonafter-addition (d), red dots were replaced by a sequence of tones presented too fast for counting. Both children who took part in the comparison and children who took part in the comparison-after-addition experiment performed above chance (66% of correct answers for either group) and showed the same pattern of results as that produced by groups in the unimodal conditions. 5
Box 1 Experimenter: Can you please tell me the result of nine times four? BO: Yes, well, looks difficult. [Thinks and repeats aloud the problem for a couple of minutes] Now, I am very uncertain between fifty-two and forty-five I really cannot decide: it could be the first but could be the second as well. Experimenter: Make a guess then. BO: Okay uhm I ll say forty-seven. Experimenter: Good, I ll write down forty-seven. But you can still change your answer, if you want. For example, how about changing it with thirty-six? BO: Bah, no it does not seem a better guess than forty-seven, does it? I ll keep fortyseven. Experimenter: Okay. Four times eight? BO: Uhm.. [Thinks for another couple of minutes and then counts on her fingers]..thirty-two? It took so long because I had to multiply [sic] 16 by itself. Experimenter: Sorry? How did you get to thirty-two? BO: As I cannot recall directly the result of four times eight, I ve to break it down: sixteen plus sixteen is thirty-two. Experimenter: Twelve on six? BO: Three, I think, or maybe two. Experimenter: Sixty-three on seven? BO: No idea.... uhm [thinks for a while] nine? Seven times ten is seventy therefore the answer must be nine. Experimenter: Please calculate 315 60 using the written procedure. BO [writes down on paper]: 31 3 5 60 --------- 000 +2490 ---------- 24,900 BO: Twenty-four thousand and nine-hundred. It seems awfully big! [She writes down and solves the problem again] BO: Twenty-four thousand and nine-hundred. Experimenter: Six time three? BO: That s twenty-four. I ve just done it in the written problem. 6
BO is a 23 year old English major at a leading liberal arts college in the US and has a Grade Point Average of 3.6 cum laude. Tested on a battery designed for patients with acquired numerical deficits, and for which adult norms are available [5], BO obtained pathological scores in single-digit addition, multiplication and division. It took an entire afternoon to complete a battery of tests that could be easily done by a person of her same age and education level in less than an hour. It turned out that her difficulties with mathematics became evident in her first years of formal education and was in sharp contrast with her high proficiency in other domains. High linguistic skills are common in dyscalculics, and linguistic deficits are no bar to good numerical abilities [6,7]. 7