Probing the mechanisms underlying numerosity-to-numeral mappings and their relation to math competence

Abstract

Numerosity estimation performance (e.g., how accurate, consistent, or proportionally spaced (linear) numerosity-numeral mappings are) has previously been associated with math competence. However, the specific mechanisms that underlie such a relation is unknown. One possible mechanism is the mapping process between numerical sets and symbolic numbers (e.g., Arabic numerals). The current study examined two hypothesized mechanisms of numerosity-numeral mappings (item-based “associative” and holistic “structural” mapping) and their roles in the estimation-and-math relation. Specifically, mappings for small numbers (e.g., 1–10) are thought to be associative and resistant to calibration (e.g., feedback on accuracy of estimates), whereas holistic “structural” mapping for larger numbers (e.g., beyond 10) may be supported by flexibly aligning a numeral “response grid” (akin to a ruler) to an analog “mental number line” upon calibration. In 57 adults, we used pre- and post-calibration estimates to measure the range of continuous associative mappings among small numbers (e.g., a base range of associative mappings from 1 to 10), and obtained measures of math competence and delayed multiple-choice strategy reports. Consistent with previous research, uncalibrated estimation performance correlated with calculation competence, controlling for reading fluency and working memory. However, having a higher base range of associative mappings was not related to estimation performance or any math competence measures. Critically, discontinuity in calibration effects was typical at the individual level, which calls into question the nature of “holistic structural mapping”. A parsimonious explanation to integrate previous and current findings is that estimation performance is likely optimized by dynamically constructing numerosity-numeral mappings through the use of multiple strategies from trial to trial.

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Data availability

The data set used for the current study is available on Open Science Framework https://osf.io/tas82/.

Notes

  1. 1.

    In Gandini, Lemaire, and Dufau's (2008) study, “anchoring” was defined as “Participants enumerated several dots (via counting), visually estimated the remaining dots based on the first enumeration, and then added the enumerated result and the estimated result” (e.g., “I first counted 3 dots, then 4 dots, added 3 and 4 = 7. Then, I estimated that there remained approximately twice as many dots, so I figured that there are 7 + 14 = 21 dots”). “Decomposition/recomposition” was defined as “Participants spotted one group of few dots, up to about four or five items, estimated the number of analogous groups, and then multiplied the number of items primarily subtilized [sic] by the estimated number of groups.” (e.g., “I saw a group of 3 dots, and I estimated that there were six other similar groups; so I multiplied 7 by 3, and thought there are approximately 21 dots”). In the current study, we focus on the fact that subgroups of dots were used to enumerate the whole collection, regardless of whether participants used counting or subitizing to enumerate the subgroups, or whether they used multiplication or addition strategies.

  2. 2.

    We also used the base range of associative mappings estimated from the “sliding window” analysis of 2 numerosities to assess the consistency of the original base range index. They were highly correlated [rs(54) = 0.74, p < 0.001], indicating that individual differences captured by the original index are consistency across estimation approaches.

  3. 3.

    Calibration method was either implicit (merely providing an upper bound in Yeo et al., 2019), or identical between two calibrated conditions (merely changing the calibration reference value in Izard & Dehaene, 2008).

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Acknowledgements

We would like to thank Olivia Lasala, Nathaniel Dorris, Jordann Lewis, and Reginald Wimbley for their assistance with data collection. DJY is supported by the Humanities, Arts, and Social Sciences International PhD Scholarship, co-funded by Nanyang Technological University and the Ministry of Education (Singapore). We would also like to thank the reviewers for their insightful feedback and analytical suggestions on a prior version of this manuscript.

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This work was supported in part by a National Science Foundation Grant (DRL 1660816) awarded to G.R.P.

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Yeo, D.J., Price, G.R. Probing the mechanisms underlying numerosity-to-numeral mappings and their relation to math competence. Psychological Research (2020). https://doi.org/10.1007/s00426-020-01299-z

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