Numerical Estimation Under Supervision

Numerical Estimation Under Supervision John E. Opfer (opfer.7@osu.edu) Department of Psychology, The Ohio State University, 1835 Neil Ave., 245 Psychology Building Columbus, OH 43210, USA 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 Abstract numbers and match them across modalities (see Fig. 1) and (2) the linking between symbolic numbers and mental magnitudes is plastic and undergoes qualitative change (Opfer & Siegler, 2012). The psychophysical functions linking numbers and estimates of numerical value have emerged as practically important as well. Specifically, the functions generating children's numerical estimates correlate highly with real- Development of Quantitative Thinking world behavior, including children's memory for numbers, their ability to learn arithmetic facts, their math grades in school, and their math achievement scores (Booth & Siegler, 2006, 2008; Fazio, Bailey, Thompson, & Siegler, 2014; Siegler & Thompson, 2014; Siegler, Thompson, & Schneider, 2011). These findings suggest that children's representations of numerical magnitude play an important role in development of mathematical ability and should be a target for educational interventions. Children's number-line estimation has produced a lively debate about representational change, supported by apparently incompatible data regarding the descriptive adequacy of logarithmic (Opfer et al., 2011) and power models (Slusser et al., 2013). To test whether methodological differences might explain discrepant findings, we created a fully crossed 2x2 design and assigned 96 children to one of four cells. In the design, we crossed sampling (over-, even-) and supervision (with feedback, without feedback), which were candidate factors to explain discrepant findings. In three conditions (over-sampling/unsupervised-83%, even- sampling/unsupervised-67%, and over-sampling/supervised- 58%), the majority of children provided estimates better fit by the logarithmic than by the power function. In the last Fig. 1 was condition (even-sampling/supervised-30%), the reverse found. Overall, a reliable association (p < .0001) was found between proportion best fit by the power function and supervision. Results suggest that the fit of the power function to children's estimates is likely an artifact of supervision. Keywords: numerical cognitive development cognition; estimation; learning; Introduction In this paper, we attempt to reconcile seemingly incompatible data (Barth & Paladino, 2011; Opfer & Siegler, 2007; Opfer, Siegler, & Young, 2011; Slusser, Santiago, & Barth, 2012) regarding the psychophysical functions that link numbers to children's estimates of numerical magnitude. The psychophysical functions that link numbers to subjects' estimates of numerical magnitude are both theoretically and practically important. Of theoretical interest, the functions generating young children's numerical magnitude estimates have been observed in the non- symbolic number discrimination of a wide range of species (for review, see Nieder and Dehaene, 2009), to change abruptly with limited experience (Opfer & Siegler, 2007; Izard & Dehaene, 2008), and to closely track abilities to deal with numbers in other contexts (Booth & Siegler, 2006; Thompson & Siegler, 2010). Thus, just as an animal can better discriminate 1 and 10 objects than 101 and 110 objects, so too do children estimate the magnitudes of the symbols 1 and 10 to differ more than 101 and 110. These results suggest that (1) over the course of development, numerical symbols are linked to an innate mental number line that allows infants and other animals to discriminate Figure 1. Model of early numerical magnitude representations (from Opfer & Siegler, 2012). What psychophysical functions then are the most likely ones to generate estimates of numerical value? Across a