How feedback improves children’s numerical estimation

Abstract

Developmental change in children's number-line estimation has been thought to reveal a categorical logarithmic-to-linear shift in mental representations of number. Some have claimed that the broad and rapid change in estimation patterns that occurs with corrective feedback provides strong evidence for this shift. However, quantitative models of proportion judgment may provide a better account of children's estimation patterns while also predicting broad and rapid change following feedback. Here we test the hypothesis that local corrective feedback provides children with additional reference points, rather than catalyzing a shift to a different mental representation of number. We tested 117 children from several second-grade classrooms in a number-line feedback study. Data indicate that the proportion-judgment framework accounts for individual differences in estimation patterns, and that the effects of feedback are consistent with the unique quantitative predictions of the framework. They do not provide evidence supporting the representational shift hypothesis or, more broadly, for the proposal that cognitive change can occur rapidly at the level of entire mental representations.

Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks

“…When this model is applied to a typical 0-1000 number line task, This model was later generalized to account for cases in which the observer makes use of additional reference points (for example, estimating the proportion of a cylinder that is partially filled with liquid by judging the liquid's level relative to a halfway point, rather than relative to the entire height of the cylinder; Hollands & Dyre, 2000). This work showed that the use of such reference points 1) produces a pattern of estimates with multiple S-shaped or inverse S-shaped curves or "cycles" (hence the term "cyclical power model" of proportion estimation; see Figure with this model are observed in children's number-line estimates Barth et al, 2016;Rouder & Geary, 2014;Slusser et al, 2013), as they are in adults' estimates of proportions using various continua (Cohen & Blanc-Goldhammer, 2011;Hollands & Dyre, 2000).…”

“…Converging evidence from multiple research groups, however, has recently shown that a different theoretical explanation offers a better explanation of performance on cognitive tasks, including number-line estimation tasks, that are commonly used to assess learning and development in numerical thinking Barth, Slusser, Kanjlia, Garcia, Taggart, & Chase, 2016;Cohen & Blanc-Goldhammer, 2011;Cohen & Sarnecka, 2014;Rouder & Geary, 2014;Slusser et al, 2013;Sullivan, Juhasz, Slattery, & Barth, 2011;see also Chesney & Matthews, 2013;Hurst, Monahan, Heller, & Cordes, 2014). These and other findings have fostered an ongoing debate, calling into question the hypothesis that a shift from logarithmic to linear mental representations of number underlies developmental change in numerical estimation and, in turn, observed improvements in formal math (see Barth, Slusser, Cohen, & Paladino, 2011;Opfer, Siegler, & Young, 2011).…”

“…The proportion estimation framework also explains performance differences across more-vs. less-familiar numerical ranges (Slusser et al, 2013), the effects of feedback on children's number-line estimates (Barth et al, 2016;cf. Opfer & Siegler, 2007), and adults' numerical estimation performance (Cohen & Blanc-Goldhammer, 2011;Sullivan et al, 2011).…”

“…The proportion estimation framework predicts that this information may make children more likely to produce performance patterns resembling two-cycle, rather than one-cycle, models (e.g. Barth et al, 2016) while the representational shift hypothesis predicts that children may be more likely to produce linear, rather than logarithmically, spaced estimates (Opfer & Siegler, 2007). While the analysis below does not explicitly address these hypotheses, interested readers are encouraged to review the papers and articles cited above for a more detailed examination of the role of corrective feedback and explicit provision of reference points in children's number line estimates.…”

Evaluating the Effect of Labeled Benchmarks on Children’s Number Line Estimation Performance and Strategy Use

“…Evidence in favor of this benchmark use is based on several sources, such as participants’ error rates and estimation latencies (Ashcraft and Moore, 2012), the superior fittings of one- and two-cycle power functions compared to logarithmic and linear functions on individuals’ estimation patterns (Barth and Paladino, 2011; Cohen and Blanc-Goldhammer, 2011; Slusser et al, 2013; Rouder and Geary, 2014; Barth et al, 2015; Reinert et al, 2015, though see Opfer et al, 2016), verbal reports of participants’ solution behavior (Newman and Berger, 1984; Peeters et al, 2016, 2017, unpublished), and, finally, eye-movement data (Schneider et al, 2008; Heine et al, 2010; Sullivan et al, 2011). Based on the evidence coming from all these sources, the development in children’s benchmark use can be depicted as follows.…”