Development in the Estimation of Degree Measure: Integrating Analog and Discrete Representations

Development in the Estimation of Degree Measure: Integrating Analog and Discrete Representations Jonathan Michael Vitale (JMV2125@Columbia.Edu) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA John B. Black (Black@Exchange.TC.Columbia.Edu) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA Eric O. Carson (EOC2102@Columbia.Edu) Department of Mathematics, Science and Technology, Teachers College, 525 W. 120 th Street New York, NY 10027 USA Chun-Hao Chang (Seedmic@Gmail.com) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA Abstract We examined adult and child performance on two numerical, geometric estimation tasks. In both tasks adults demonstrated greater accuracy than children as well as more mature representations, in general. Furthermore, evidence from mouse tracking data demonstrates that adult strategy includes the application of discrete landmark values while child strategy, generally, does not. This evidence suggests that adults construct mental representations of landmark values and successfully integrate them into analog tasks. Implications for future intervention studies are discussed. Keywords: numerical estimation, embodied cognition, mathematical development, cognitive assessment Introduction Numerical estimation tasks provide researchers with a powerful means of assessing individuals’ mental representation for number. Evidence from brain scans demonstrates that approximate numerical tasks, such as less- than/greater-than judgments, activate cortical regions associated with spatial processes, while activities that rely exclusively upon recall, such as single-digit multiplication, do not (Dehaene, Piazza, Pinel, & Cohen, 2003). According to Dehaene (1997) our ability to map numerical values to spatial magnitudes is what is commonly referred to as “number sense,” and grounds all mathematical reasoning. Yet, the study of number sense extends beyond theoretical interest as recent evidence suggests a link between estimation and mathematical achievement. Along these lines Halberda, Mazzocco, and Feigenson (2008) discovered that 14-year-old’s ability to discriminate between dot displays of varying cardinalities was highly correlated with achievement scores extending back to kindergarten. Likewise, Siegler and Booth (2004) found that individual differences on a number line estimation task are correlated with standardized test scores. In the case of number line estimation individual differences may embody large, qualitative shifts in representation (Siegler & Opfer, 2003). Dehaene (1997) asserts that numerical symbols implicitly recruit a logarithmic representation that is more precise at smaller values. Siegler and Opfer (2003) found that young children, especially with larger numerical ranges, tend to apply this kind of logarithmic representation while estimating the position of a given value on a number line. Specifically, data of estimated magnitude over actual magnitude are best fit by a logarithmic function for these younger children. On the other hand, older children’s data, in many cases, is best fit by a straight line. The emergence of a linear representation has several possible causes and implications. In particular, Siegler and Opfer (2003) differentiate between two models of linear representation. In the accumulator model, adopted from Gibbon and Church (1981), noise in the mental representation for a numerical value increases in proportion to the mean. This representation implies increasing variability in the estimates as the magnitude increases. In the linear-ruler model – which was found to be a better fit for the data – variability in estimates has a constant relation to magnitude. The authors suggest that the mature, linear representation is developed through cultural, particularly school-based, experience. Furthermore, evidence of less variability near landmark values along the number line (e.g. quartiles) demonstrates a specific means for implementing the linear- ruler model. One may even speculate that at the lowest- level number representation may be logarithmic or accumulator in nature, but at the level of conscious-level processing number concepts are modulated for specific tasks. If the appeal of number line estimation tasks is due, in part, to its high ecological validity, one might then find it