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From Numerosity to Numeral: Development of Mathematical Concepts

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2019, Doctor of Philosophy, Ohio State University, Psychology.
Despite the prevalence of quantity representation in everyday life, important questions regarding the ability remain unanswered, including how quantity representation is structured and develops with age, whether the representation is shared between numerical and non-numerical quantity, and how non-numerical factors can aid the quality of numerical representations. In this dissertation, I sought to answer these questions through a series of studies. The dissertation includes five studies that investigate quantity estimation and comparison. Study 1 examined children’s estimates of symbolic numbers on bounded and unbounded number lines. The study especially aimed to resolve a debate on whether number-line estimates mirror log-linear representation of number versus task- specific cognitive strategies. Compared to alternative models from the cognitive- strategy accounts, a single mixed log-linear model (MLLM) better described children’s estimates on bounded and unbounded number lines, predicting their actual math proficiency better. Results suggest that symbolic number—like numerosity—is represented on a log-linear scale. Study 2 and 3 compared children’s and adults’ estimates of numerical and non- numerical magnitude to examine developmental changes in representation for two dimensions. Inconsistent with the cognitive-strategy accounts, estimates of both nu- merical and non-numerical quantities were better predicted by the MLLM. Logarithmicity scores of MLLM in both dimensions were highly correlated with one another and with math proficiency. Furthermore, log-to-linear shifts in estimates were evident in both numerical and non-numerical tasks. These findings provide evidence that estimation of numerical and non-numerical magnitude is processed in part through a shared quantity representation that develops from log to linear with age and experience. The log-linear representation has been thought too noisy for exact numerical representation, but its actual limit is unknown. Study 4 and 5 examined the effects of non-numerical, visuospatial dimensions on numerosity and numeral comparison. Ratio effects in comparison decreased with visuospatial manipulations, as did Weber fractions. When all visuospatial dimensions supported the numerical dimension of numerosity, internal noise became as small as that in numerical comparison. The findings indicate that the noise in internal representation of numerosity comes largely from external, non-numerical properties that reduce signal-to-noise ratios, and that the analog representation is not as inherently fuzzy as has been believed. Together, the results point to the analog magnitude representation as the common representation for quantity. With major developmental changes (log-to-linear and noise-to-noiseless) and support from non-numerical properties, the analog magnitude representation for number improves, to the extent that it suffices to ground precise numerical representation.
John Opfer (Advisor)
Vladimir Sloutsky (Committee Member)
Stephen Petrill (Committee Member)
127 p.

Recommended Citations

Citations

  • Kim, D. (2019). From Numerosity to Numeral: Development of Mathematical Concepts [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1558020167528072

    APA Style (7th edition)

  • Kim, Dan. From Numerosity to Numeral: Development of Mathematical Concepts. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1558020167528072.

    MLA Style (8th edition)

  • Kim, Dan. "From Numerosity to Numeral: Development of Mathematical Concepts." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1558020167528072

    Chicago Manual of Style (17th edition)