Creating Contexts for Fraction Learning by Activating Relevant Prior Knowledge

• In Event: 35.069 - Fractions Learning: One Subject, Multiple Perspectives

Abstract

Learning fraction arithmetic is notoriously difficult for students. Children, and even adults, make errors that reflect incorrect transfer from prior knowledge (e.g., Siegler & Pyke, 2013; Ma, 1999). In this work, we apply an analogical transfer perspective (e.g., Gick & Holyoak, 1987) to create a supportive context for learning fraction division. From this perspective, children’s prior knowledge supports new learning when it is structurally similar to new problems, but negatively biases new learning when problems are perceptually, but not conceptually, similar. In this work, we identify domains of children’s prior knowledge that are most likely to support fraction division learning. The learning goal is understanding the conceptual structure for fraction division (quotative or partitive division). Thus, the relevant analogue in children’s repertoire is whole number division, which shares conceptual structure. This stands in contrast to children’s alluring, perceptually similar prior knowledge of other fraction operations. We investigated whether activating children’s whole number division concepts better supports fraction division than does activating children’s knowledge of other fraction operations (e.g., fraction subtraction) in two lab-based studies.
In Study 1, children were asked to “warm up” by solving problems on either whole number division or other fraction operations. Then, an experimenter gave a short fraction division lesson during which children were either asked about the “warm-up” problems or not. Changes in children’s fraction division concepts were measured using brief pre- and post-tests. Among children who were not asked to link between domains, children who warmed up with whole number division gained more conceptual knowledge than children who warmed up with other operations on fractions. Among children who were asked to link between domains, knowledge gains were lower (see Figure 1).
In Study 2, children were asked simply to demonstrate the conceptual structure of fraction division using manipulatives (see Figure 2), without a lesson. Some children demonstrated fraction division immediately after other fraction operations trials, while others demonstrated fraction division immediately following whole number division. Importantly, all children demonstrated all problems, but in two different orders: thus, we manipulated the recency of activating each analogue domain. Sixth graders were more likely to correctly model fraction division when it immediately followed whole number division, rather than other fraction operations (i.e. multiplication).
Although these experiments are not true educational interventions, they suggest that the analogical transfer framework may be useful for creating effective contexts for learning, by suggesting what kinds of prior knowledge are best activated before a new lesson. First, activating structurally similar, whole number division knowledge prepares students to learn about fraction division. Second, the recency of this activation matters: what a learner does immediately before tackling novel problems has consequences for learning.
These findings reveal advantages of drawing on children’s prior whole number knowledge in fraction instruction. Other research has suggested that children’s whole number understanding may negatively bias fraction learning (e.g., Ni & Zhou, 2005). Although some analogies to whole numbers may lead students towards fraction misconceptions, our findings demonstrate that analogies across key conceptual similarities frame students’ thinking in positive ways.

Authors

• Pooja Sidney, University of Wisconsin - Madison

• Martha W. Alibali, University of Wisconsin - Madison