When and how Physical Models Benefit the Learning of Symbolic Multi-digit Numbers

• In Event: Examining the Roles of Patterning Knowledge, Spatial Assembly, and Analogic Reasoning in Early Mathematics Development

Abstract

How do children acquire human-invented symbol systems that are crucial for academic and life success? In mathematics, physical models (called manipulatives) are often used to perceptually ground the meaning of symbols. Efficacy studies yield mixed results; sometimes helping and sometimes not with no principled understanding about just how physical models are helpful. Borrowing from Gentner’s Structure-mapping theory, we suggest that the potential benefit of physical models does not lie in symbol-grounding, but rather in analogical reasoning. The benefit of this conceptualization is that there is a large body of empirical literature documenting principles for when and how analogies support the discovery of relational structure. Here, we sought to help preschoolers—through a relational mapping task—to discover two components about the place value system: the mapping between spoken names and written symbols, and the relative magnitudes of the written symbols. The focus of the mapping task was the discovery of the relational structure in the spoken names and written forms and their relative magnitudes (hundred > decade > unit): that 3-digit numbers are named from left to right as x-hundred, y-ty, z and the left-most position represents a larger magnitude.

In Study 1 (Fig. 1), 75 4- to 6-year-olds mapped components of heard number names to written digits, and to a third representation: base-10 blocks, abacus, or a repetition of the written symbols themselves (control). Post-test performance on mapping names to written symbols and magnitude comparisons provided no indication that training with base-10 blocks or abacus led to improvement. Adding these physical representations may have hurt the discovery of the relational structure as improvements only appeared in the control condition. Why weren’t physical models beneficial? The extant literature on relational learning suggests that learners need to be able to map the elements in one domain to the relationally same elements in the other and that complex, rich stimuli hinder this.

Study 2 (N = 59) contrasted three conditions (Fig. 2): the original abacus condition, a rich condition (using size to highlight the place value relations, e.g., a hundred bead is larger than a tens bead) and a sparse condition (using three sets of circles, but different from base-10 blocks, the size of the circles only represent relative sizes, not absolute sizes, e.g., the hundred circles are bigger than the tens, but not 10 times bigger). Our hypothesis was that the sparse condition would lead to better discovery of the relational structure. Children only improved in the sparse condition, not the rich nor the original condition. These initial studies on re-thinking manipulatives in terms of Structure-mapping theory provide a potential pathway to developing scalable principles for introducing children to place value and to symbol systems more generally.

Authors

• Lei Yuan, University of Colorado Boulder
  Presenting Author

• Richard Prather, University of Maryland - College Park
  Non-Presenting Author

• Kelly Sue Mix, University of Maryland - College Park
  Non-Presenting Author

• Linda B. Smith, Indiana University - Bloomington
  Non-Presenting Author