Perceptual Routes to Rational Number

• In Event: 2-082 - External Representations in Mathematical Thinking and Learning

Integrative Statement

Fractions knowledge is considered to be an important building block of future mathematical competence (e.g. Siegler, Thompson, & Schneider, 2011). Even though the importance of fraction ability is well documented, there is still a dearth of knowledge regarding how fractions magnitudes are processed. Moreover, there have been few psychological studies examining what sorts of external representations might best help foster fractions processing abilities. In this talk, we will discuss some recent work investigating how basic perception of nonsymbolic quantities might relate to processing fractions magnitudes.

A growing body of empirical research suggests that human beings have intuitive, perceptually-based access to primitive fractions concepts when they are instantiated with continuous (i.e., uncountable) nonsymbolic graphical representations (e.g., Bonn & Cantlon, 2017; Jacob, Vallentin & Nieder, 2012, Matthews, Lewis & Hubbard, 2016; Figure 1). Lewis Matthews and Hubbard (2015) dubbed this basic perceptual apparatus the Ratio Processing System (RPS) and have called for new research exploring its limits and how it might be used to support reasoning with formal mathematical symbols (see also Jacob et al., 2012). One question of interest is whether RPS-processed representations may be a privileged class of representations when it comes to providing access to rational number magnitudes (Rau & Matthews, 2017).

With this research, we used a series of fraction comparison tasks among three different age groups (2nd graders, n = 85; 5th graders, n = 53; and adults n = 24) to investigate (1) whether nonsymbolic representations might provide more efficient access to rational number magnitudes than formal symbolic representations and 2) whether comparisons between nonsymbolic and symbolic stimuli required significant translation costs relative to within format comparisons. Participants were asked to indicate the larger of two fractions in three different conditions: paired symbolic fractions, paired nonsymbolic fractions made from line segments (Figure 1) and mixed symbolic/nonsymbolic cross-notation pairs.

We found that all participants could complete nonsymbolic tasks quickly and accurately (Table 1). Indeed, participants in each age group were consistently faster comparing nonsymbolic fractions relative to comparing symbolic fractions. Furthermore, adults were equivalently as fast and accurate when making cross-notation comparisons as they were when making symbolic fraction comparisons within format, perhaps suggesting a common magnitude code can be accessed without substantial translation costs. We found similar results among children – nonsymbolic comparisons were the fastest and most accurate, and children paralleled adults by showing similar patterns regarding the similarity of cross-notation and within format symbolic notations.

These results stand in stark contrast with previous conclusions from studies examining symbolic vs. nonsymbolic processing with whole-numbers (e.g., Lyons, Ansari & Beilock, 2012), and leaves open the possibility that extracting symbolic fraction magnitudes and perceiving the magnitudes of nonsymbolic fractions may rely in part on shared mechanisms.

Authors

• Percival Matthews, University of Wisconsin-Madison
  Presenting Author

• John Binzak, University of Wisconsin-Madison
  Non-Presenting Author

• Priya Kalra, University of Wisconsin-Madison
  Non-Presenting Author

• Yunji Park, University of Wisconsin-Madison
  Non-Presenting Author

• Edward Hubbard, University of Wisconsin-Madison
  Non-Presenting Author