Children’s acquisition of fraction knowledge from concrete versus generic instantiations

Children’s acquisition of fraction knowledge from concrete versus generic instantiations Jennifer A. Kaminski (kaminski.16@osu.edu) Center for Cognitive Science, Ohio State University 210A Ohio Stadium East, 1961 Tuttle Park Place, Columbus, OH 43210, USA Vladimir M. Sloutsky (sloutsky.1@osu.edu) Center for Cognitive Science, Ohio State University 208C Ohio Stadium East, 1961 Tuttle Park Place, Columbus, OH 43210, USA Abstract The goal of this experiment was to investigate elementary school children’s ability to acquire basic fraction knowledge. The degree of concreteness of instantiations of proportions was varied between subjects. First-grade children learned to label proportions of objects with fraction. Proportions were presented either as concrete, colorful flowers or as generic black and white circles. Following instruction, participants were given a test of learning and an immediate or delayed test of transfer involving proportions of novel objects. Those who learned with the generic materials scored higher on learning and transfer than those who learned with the concrete materials. Differences between learning conditions were attenuated for the delayed transfer test. These findings suggest that concrete, perceptually rich instantiations of fractions may hinder children’s acquisition of basic fraction knowledge in comparison to simple generic instantiations of fractions. Keywords: Psychology; Education; Relations, Mathematics Education. Learning, Transfer; Introduction Mathematical concepts are often difficult for children to acquire. One response to this challenge is to introduce concepts to students through concrete instantiations which include perceptually rich, familiar material. The use of concrete material is widespread in education (see McNeil & Uttal, 2009 for discussion). Concrete instantiations of mathematics may involve familiar contexts and can be visually appealing and engaging. For example, simple arithmetic concepts are often instantiated through sets of familiar objects, such as two apples plus three apples equals five apples. Such material may spark interest in the learning task and maintain attention on the learning material. However, a primary goal of learning mathematics is the ability to apply mathematics to new situations. Therefore, successful acquisition of mathematical knowledge implies that the learner has not only acquired knowledge of the mathematical relations in the context of learning, but also has the ability to transfer the mathematical knowledge to novel isomorphic situations. There is evidence that concrete instantiations can hinder transfer of learning. Adults who learned a novel mathematical concept from a generic, perceptually sparse instantiation were better able to transfer this knowledge to a novel isomorph than those who learned the same concept from a concrete instantiation (Kaminski, Sloutsky, & Heckler, 2008; Sloutsky, Kaminski, & Heckler, 2005; see also Goldstone & Sakamoto, 2003; Goldstone & son, 2005 for related findings). In comparison to more abstract, generic instantiations, concrete instantiations of a given concept communicate more extraneous information. For example, a photograph of a person communicates more nonessential information than a simple stick figure drawing (see Kaminski & Sloutsky, 2011 for a discussion). Similarly, instantiating addition as the sum of apples communicates more information than instantiating it as the sum of tally marks. This additional information (e.g. the appearance, taste, etc. of apples) is extraneous to the mathematics and may present an obstacle for learning for the following reason. Mathematical concepts are defined by relational structure. Relations are less salient than objects (e.g. Gentner, 1988). Instantiating mathematical concepts through concrete material, in comparison to a more generic format, may increase the salience of superficial aspects of the learning material and consequently divert the learner’s attention from the to-be- learned relational structure (see Goldstone, Medin, & Gentner, 1991 for a similar argument regarding similarity judgments), thus hindering learning. Therefore, it appears that generic instantiations of mathematical concepts have an advantage over concrete instantiations with respect to transfer. However it could be argued that this advantage is limited to older learners. After all, in educational practice, older students are expected to learn and reason with abstract instantiations including symbols, equations, and other standard notation. It may be that younger learners (e.g. elementary school students) may need concrete instantiations to begin to acquire abstract knowledge. In addition to the notion that concrete material may be more engaging for the learner, support for the use of concrete instantiations to teach abstract concepts to young children is often tied to theories of learning and development. Some developmental theories (Montessori, 1917; Piaget, 1970) posit that young children’s thinking is inherently concrete and that they are not capable of reasoning about abstract concepts using symbols. According