The relation between finger gnosis and mathematical ability: why redeployment of neural circuits best explains the finding.

This paper elaborates a novel hypothesis regarding the observed predictive relation between finger gnosis and mathematical ability. In brief, we suggest that these two cognitive phenomena have overlapping neural substrates, as the result of the re-use ("redeployment") of part of the finger gnosis circuit for the purpose of representing numbers. We offer some background on the relation and current explanations for it; an outline of our alternate hypothesis; some evidence supporting redeployment over current views; and a plan for further research.

Neural reuse in the evolution and development of the brain: evidence for developmental homology?

This article lays out some of the empirical evidence for the importance of neural reuse-the reuse of existing (inherited and/or early developing) neural circuitry for multiple behavioral purposes-in defining the overall functional structure of the brain. We then discuss in some detail one particular instance of such reuse: the involvement of a local neural circuit in finger awareness, number representation, and other diverse functions. Finally, we consider whether and how the notion of a developmental homology can help us understand the relationships between the cognitive functions that develop out of shared neural supports.

Pathways to mathematics: longitudinal predictors of performance.

A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

Negative numbers in simple arithmetic.

Are negative numbers processed differently from positive numbers in arithmetic problems? In two experiments, adults (N = 66) solved standard addition and subtraction problems such as 3 + 4 and 7 - 4 and recasted versions that included explicit negative signs-that is, 3 - (-4), 7 + (-4), and (-4) + 7. Solution times on the recasted problems were slower than those on standard problems, but the effect was much larger for addition than subtraction. The negative sign may prime subtraction in both kinds of recasted problem. Problem size effects were the same or smaller in recasted than in standard problems, suggesting that the recasted formats did not interfere with mental calculation. These results suggest that the underlying conceptual structure of the problem (i.e., addition vs. subtraction) is more important for solution processes than the presence of negative numbers.

Knowledge of counting principles: how relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.

Selection of procedures in mental subtraction.

Adults solved simple subtraction problems (e.g., 16 - 9). Half of the 32 participants provided immediate self-reports of their solution processes on each problem. Performance was analyzed using both traditional descriptive statistics (i.e., means, standard deviations, and percentage of errors) and with statistics derived from fitting the ex-Gaussian distributions to latencies (i.e., mu and tau). The results support the view that ex-Gaussian analyses can be useful in exploring patterns of procedure selection that relate both to characteristics of the stimuli (e.g., problem size) and to characteristics of the participants (e.g., arithmetic skill). More generally, the results provide further evidence that adults use a variety of procedures to solve simple subtraction problems and that these choices are related to patterns of performance on more complex problems that require calculation.

Calculation latency: the mu of memory and the tau of transformation.

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N = 47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into mu (reflecting shift) and tau (reflecting skew) confirmed that retrieval trials contributed predominantly to mu whereas procedure trials contributed predominantly to tau. The format x problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems)was associated entirely with mu and not with tau. Reported use of procedures presented a corresponding format x size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.

What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2.

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.

Decomposing the problem-size effect: a comparison of response time distributions across cultures.

Is the locus of the problem-size effect in mental arithmetic different across cultures? In a novel approach to this question, the ex-Gaussian distributional model was applied to response times for large (e.g., 8 x 9) and small (e.g., 2 x 3) problems obtained from Chinese and Canadian graduate students in a multiplication production task (LeFevre & Liu, 1997). The problem-size effect for the Chinese group occurred in mu (the mean of the normal component), whereas the problem-size effect for the Canadian group occurred in both mu and tau (the mean of the exponential component). The results support the position that the problem-size effect for the Chinese group is purely a memory-retrieval effect, whereas for the Canadian group, it is an effect of both retrieval and the use of nonretrieval solution procedures.