Learning Linear Spatial-Numeric Associations Improves Memory for Numbers

Learning Linear Spatial-Numeric Associations Improves Memory for Numbers Clarissa A. Thompson (cat3@ou.edu) Department of Psychology, The University of Oklahoma, 455 W. Lindsey St., 727 Dale Hall Tower Norman, OK 73019, USA John E. Opfer (opfer.7@osu.edu) Department of Psychology, The Ohio State University, 1835 Neil Ave., 245 Psychology Building Columbus, OH 43210, USA Memory for numbers improves with age and experience. We tested the hypothesis that one source of this improvement is a logarithmic-to-linear shift in children’s representations of numeric magnitude. In Experiment 1, we found that linearity of representations improved with age and that the more linear children’s magnitude representations were, the more closely their memory of the numbers approximated the numbers presented. In Experiment 2, we trained children on a linear spatial-numeric association, and we found that children who learned to represent numbers as increasing linearly with numeric magnitude also improved their memory for numbers. These results suggest that linear spatial-numeric associations are both correlated with and causally related to development of numeric memory. Keywords: spatial-numeric associations; representations; numerical estimation; memory number Introduction Remembering numeric information is an important part of daily life. Sometimes it is necessary to remember numeric information exactly (e.g., social security numbers, phone numbers, flight numbers, street addresses), whereas other times remembering the general gist of numeric information will suffice (e.g., savings account balances, temperatures, number of students in a lecture hall). Across both types of memory, children’s memory for numbers improves greatly with age and experience (Dempster, 1981; Brainerd & Gordon, 1994). Here we tested the hypothesis that children’s numerical memory improves with age due to changes in how children represent numerical magnitudes. Development of Numerical Representations. Children’s representations of the magnitudes of symbolic numbers appears to develop iteratively, with parallel developmental changes occurring over many years and across many contexts (Opfer & Siegler, in press). Early in the learning process, numerical symbols are meaningless stimuli for young preschoolers. For example, 2- and 3-year-olds who count flawlessly from 1-10 have no idea that 6 > 4, nor do children of these ages know how many objects to give an adult who asks for 4 or more (Le Corre et al., 2006). As young children gain experience with the symbols in a given numerical range and associate them with non-verbal quantities in that range, they initially map them to a logarithmically-compressed mental number line. Over a period that typically lasts 1-3 years for a given numerical range (0-10, 0-100, or 0-1,000), children’s mapping of symbolically expressed numbers to non-verbal representations changes from a logarithmically-compressed form to a linear form, where subjective and objective numerical values increase in a 1:1 fashion (Bertelletti et al., 2010; Opfer, Thompson, & Furlong, 2010; Siegler & Opfer, 2003; Siegler & Booth, 2004; Thompson & Opfer, 2010). Use of linear magnitude representations occurs earliest for the numerals that are most frequent in the environment, that is the smallest whole numbers, and is gradually extended to increasingly larger numbers (Thompson & Opfer, 2010). Changes in numerical representations occur not only with increasing age, but also with specific experiences designed to train linear spatial-numeric associations. For instance, Opfer and Siegler (2007) provided second graders with corrective feedback on the location of numbers near 150, the maximally discrepant point between a logarithmic and linear function forced to pass through 0 and 1,000 (see Figure 1). After receiving feedback, children adopted a linear representation that spanned the entire 0-1,000 numeric range. 3RVLWLRQRQ1XPEHU/LQH Abstract 1XPEHU3UHVHQWHG Figure 1: Logarithmic and linear functions. Distance between representations is greatest at 150 (725 vs. 150); this means that the logarithmic function increases more than the linear representation between each successive pair of numbers up to 150, but increases less than the linear function above 150. Thus, numbers below 150 are more discriminable in the logarithmic representation, and numbers above 150 are more discriminable in the linear representation.