Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division

Abstract

When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.

Implicit Analogies in Learning: Supporting Transfer by Warming Up

“…Importantly, because warm-up activities tend to reactivate very familiar prior knowledge, the specific characteristics of examples used in the initial learning context of that prior knowledge should be much less important than the characteristics of the warm-up activity that serves to reactivate prior knowledge. Finally, in line with Day and Goldstone’s (2012) conceptualization of transfer, we also suggest that warm-up activities may be especially powerful when they involve spatial representations (e.g., Thompson & Opfer, 2010) and physical or imagined action (e.g., Sidney & Alibali, 2017). Future work is needed to test these hypotheses for implementing implicit analogical instruction and the circumstances under which implicit analogy might (or might not) be effectively activated within and outside of the domain of math.…”

“…In our own work (Sidney & Alibali, 2015, 2017; Sidney, Thompson, & Rivera, 2019; Thompson & Opfer, 2010), we have demonstrated that children engage in more accurate problem solving and efficient learning in challenging domains when their highly practiced, relevant prior knowledge has been recently activated using warm-up activities designed to instantiate an implicit analogy. For example, we demonstrated that children’s conceptual understanding of fraction division was more likely to be accurate when structurally similar whole-number division concepts were recently practiced (Sidney & Alibali, 2017). When children had just modeled whole-number division problems using physical objects to show division by partitioning a given amount into groups, they were more likely to accurately model fraction division compared with children who modeled related but structurally dissimilar fraction multiplication immediately before fraction division.…”

Taking the relational structure of fractions seriously: Relational reasoning predicts fraction knowledge in elementary school children

“…For example, comparison, contrast, relational language, and structural alignment are known to improve relational learning (Christie & Gentner, 2010, 2014Gentner, 2010;Kurtz, Boukrina, & Gentner, 2013), and instructional methods that use these techniques may prove beneficial if adapted for use with fractions. However, there is much work yet to be done to determine how best to translate the findings from the lab to the classroom; for example, although introducing fraction division by analogy to whole number division has shown some benefits, some adjustments may need to be made to this approach before it is more widely effective (Sidney & Alibali, 2015;Sidney & Alibali, 2017).…”

Math predictors of numeric health and non-health decision-making problems

“…Participants were given three minutes to solve each set of six problems. Two problems in each section involved two fraction operands (Siegler et al, 2011), and four problems involved a whole number and a fraction (Sidney & Alibali, 2017). The variable was calculated as the number of correctly answered questions.…”