It is not Symbol-Grounding; it is Analogy! Whether and How Physical Models Promote Number Learning

• In Event: Development of Number Sense

Abstract

Do human-invented symbol systems—such as letters of the alphabet or digits of the Arabic numbers—have to be grounded in physical objects to gain meanings? This question has strongly influenced research and practice in education. However, evidence on the effectiveness of physical models is mixed. In the context of children’s early learning of multi-digit numbers, we propose a rethinking of physical models—not as a path to grounding but as analogies that help learners discover inherently abstract relations.

In science, physical models are often used not to ground meaning but as analogies to distill the skeleton of an idea: for example, an atom is like the solar system in that each has smaller elements rotating around a larger one. Gentner’s Structure Mapping theory (Gentner, 1983, 2010) proposes that analogies work because they support the alignment of two relational systems that enable the relations—independent of the elements in those relations—to be extracted. Traditional mathematical manipulatives—such as base-10 blocks—require children to map multiple representations: number words, written symbols, and physical quantities that are meant to provide a complete grounding of base-10 principles (see Fig. 1 top). This approach can be overwhelming to young learners. In contrast, reducing the complexity of physical models and highlighting the to-be-learned relational structure may render them more beneficial to learning.

Experiment 1 shows that kindergarteners failed to learn and generalize the multi-digit naming system and their represented magnitudes with traditional mathematical manipulatives (i.e., base-10 blocks, abacus). In contrast, the alignment between number words and written symbols is sufficient for this early learning. As shown in Fig. 1 top, 5-year-olds imitated an experimenter to make multi-digit numbers upon hearing number words using different materials: number cards only (the Symbols-to-Symbols condition, N = 27), number cards with an abacus (the Symbols-to-Abacus condition, N = 23), or number cards with base-10 blocks (the Symbols-to-Blocks condition, N = 25). Pre- and post-test assessed children’s ability to map written multi-digit numbers to their names and to compare the magnitudes represented by written multi-digit numbers. Only the Symbols-to-Symbols condition led to significant learning (Fig. 1 bottom).

Experiment 2 reduced the complexity of traditional manipulatives and highlighted the to-be-learned relational structure, resulting in significant learning. As shown in Fig. 2 top, we contrasted a Standard abacus condition (N = 21) to a Sized abacus condition (discs varied in size to reflect the relative magnitudes of places, N = 19) and a Discs only condition (a deconstructed “abacus” that used only discs varying in size and set on the table in separate groups, N = 19). The Discs only condition was the simplest and highlighted the differentiation of the three places and their relative magnitudes. As predicted, the Discs only condition was the only one with significant improvement from pre- to post-test (Fig. 2 bottom).

Overall, this study suggests that grounding mathematical concepts in physical models is not inherently good or bad. Highlighting the to-be-learned relational structure—through either physical models or symbols themselves—is the key to significant learning.

Authors

• Lei Yuan, University of Colorado Boulder
  Presenting Author

• Richard Prather, University of Maryland - College Park

• Kelly Sue Mix, University of Maryland - College Park

• Linda B. Smith, Indiana University - Bloomington