Symbolic Number Representations Research Articles

Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats.

Numerical ratio effects are a hallmark of numerical comparison tasks. Moreover, ratio effects have been used to draw strong conclusions about the nature of numerical representations, how these representations develop, and the degree to which they generalize across stimulus formats. Here, we compute ratio effects for 1,719 children from Grades K-6 for each individual separately by computing not just the average ratio effect for each person, but also the variability and statistical magnitude (effect-size) of their ratio effect. We find that individuals' ratio effect-sizes in fact increase over development, calling into question the view that decreasing ratio effects over development indicate increasing representational precision. Our data also strongly caution against the use of ratio effects in inferring the nature of symbolic number representation. While 75% of children showed a statistically significant ratio effect for nonsymbolic comparisons, only 30% did so for symbolic comparisons. Furthermore, whether a child's nonsymbolic ratio effect was significant did not predict whether the same was true of their symbolic ratio effect. These results undercut the notions (a) that individuals' ratio effects are indicative of representational precision in symbolic numbers, and (b) that a common process generates ratio effects in symbolic and nonsymbolic formats. Finally, for both formats, it was the variability of an individual child's ratio effect (not its slope or even effect-size) that correlated with arithmetic ability. Taken together, these results call into question many of the long-held tenets regarding the interpretation of ratio effects-especially with respect to symbolic numbers.

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3–4–5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1–3–5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol–symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia

BackgroundAdaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations.Methods"The Number Race" software trains children on an entertaining numerical comparison task, by presenting problems adapted to the performance level of the individual child. We report full mathematical specifications of the algorithm used, which relies on an internal model of the child's knowledge in a multidimensional "learning space" consisting of three difficulty dimensions: numerical distance, response deadline, and conceptual complexity (from non-symbolic numerosity processing to increasingly complex symbolic operations).ResultsThe performance of the software was evaluated both by mathematical simulations and by five weeks of use by nine children with mathematical learning difficulties. The results indicate that the software adapts well to varying levels of initial knowledge and learning speeds. Feedback from children, parents and teachers was positive. A companion article [1] describes the evolution of number sense and arithmetic scores before and after training.ConclusionThe software, open-source and freely available online, is designed for learning disabled children aged 5–8, and may also be useful for general instruction of normal preschool children. The learning algorithm reported is highly general, and may be applied in other domains.

Overcoming response bias using symbolic representations of number by chimpanzees (Pan troglodytes)

We previously reported that chimpanzees were unable to optimally select the smaller of two candy arrays in order to receive a larger reward. When Arabic numerals were substituted for the candy arrays, animals who had had prior training with numerical symbols showed an immediate and significant improvement in performance and were able to select reliably the smaller numeric representation in order to obtain a larger reward. Poor performance with candy arrays was interpreted as reflecting a response bias toward the intrinsic incentive and/or perceptual features of the larger array. In contrast, the Arabic numerals represent numerosity symbolically and appear to promote response choice on the basis of abstract processing of numerosity, with minimal interference from the inherent properties of the choice stimuli. The present study tested the hypothesis that, for mixed symbol-candy choice pairs, the requisite processing of the abstract numeral may foster a mode of numerical judgment that diminishes the interfering incentive/perceptual effects of the candy stimuli. The results were consistent with this hypothesis. Whereas performance on candy-candy arrays was significantly below chance levels, performance on numeral-candy choice pairs was significantly above chance and comparable with performance on numeral-numeral pairs.