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Showcase August 2014: Mechanisms of Spatial Learning: Using Comparison and Contrast to Teach Children Geometric Categories

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Mechanisms of Spatial Learning: Using Comparison and Contrast to Teach Children Geometric Categories

Linsey Smith1, Raedy M. Ping2, Bryan J. Matlen1,3, Micah B. Goldwater4, Dedre Gentner (Co-PI)1, and Susan Levine (Co-PI)2

1Northwestern University, 2The University of Chicago, 3WestEd and 4The University of Sydney (Australia)

Geometric shape categories are formally defined by rules. For example, a triangle is ‘an enclosed shape with three straight sides.” National standards in mathematics emphasize that students master geometric shape rules such as triangle or rectangle by kindergarten (NCTM, 2006, CCSSM, 2010). These basic categories form the basis for later understanding of more complex geometric concepts and procedures.

Unfortunately, young children’s early understandings of shape categories often do not include defining rules. Instead, categories like triangle are centered on a prototypical member, such as an equilateral triangle (Figure 1a). When deciding if a new case is a member of the geometric category, children make their judgments based on whether the new case is perceptually similar to the prototype. Such a strategy leads to predictable errors (Satlow & Newcombe, 1998). Specifically, children (1) reject shapes that are perceptually dissimilar from the prototype, e.g., a ‘skinny’ or rotated triangle (Figure 1b), and (2) accept non-shapes that are perceptually similar to the prototype, e.g., an equilateral ‘triangle’ with an incomplete side (Figure 1c).

Figure 1
Figure 1. Children’s triangle category is centered on a prototypical member—the equilateral triangle (a). Children frequently reject (b) exemplars that are perceptually dissimilar from the prototype and accept (c) non-exemplars that are perceptually similar to the prototype.

Creating a correct shape category requires children to overcome reliance on perceptual similarity as a cue for membership and instead use abstract defining rules. The current project explores how comparison of category exemplars (and non-exemplars) can help children make this leap. Prior work demonstrates that analogical comparison can support children’s category acquisition (e.g., Childers, 2011; Gentner, Anggoro, & Klibanoff, 2011; Gentner & Namy, 1999; Waxman & Klibanoff, 2001). In this research, we test whether a brief instructional experience using comparison can help three- and four-year-old children learn the triangle category. Further, we explore whether different comparison types support learning in different ways.

During instruction (Figure 2), half the children saw a series of within-category comparisons (two triangles) and half saw a series of between-category comparisons (triangle vs. non-triangle; see Figure 3). Compared cases also varied in whether they were perceptually similar to or perceptually dissimilar from one another. For within-category comparisons, children were invited to consider why both cases were triangles. For between-category comparisons, children were invited to consider why one was a triangle and the other was not. All children completed a categorization task (classifying triangles and non-triangles) before and after instruction so learning could be assessed.

Figure 2
Figure 2. Instructional Comparison Task.

Figure 3
Figure 3. (a) within-category comparison (two triangles) and (b) between-category comparison (triangle and non-triangle).

All instructional groups improved in categorizing triangles and non-triangles from pre-test to post-test (Figure 4). However, within-category and between-category comparisons fostered learning in different ways. Children who compared two triangles successfully extended the category to more triangles at post-test than at pre-test, whereas children who saw between-category comparisons did not (Figure 5). Conversely, children who compared triangles with non-triangles reduced their tendency to (incorrectly) accept non-triangles at post-test, whereas children who saw within-category comparisons did not (Figure 6). Further, the positive effects of between-category comparison were only found for perceptually similar pairs.

Figure 4
Figure 4. Average category task performance (d’) at pre- and post-test, by Comparison Type. Higher d’ Rates indicate better performance.

Figure 5
Figure 5. Triangle Acceptance Rate (Hit Rate) at pre- and post-test, by Comparison Type. Higher Hit Rates indicate better performance.

Figure 6
Figure 6. Non-Triangle Acceptance Errors (False Alarm Rate) at pre-test and post-test, by Comparison Type. Lower False Alarm Rates indicate better performance.

Overall, this study found that both within-category and between-category comparisons can support children’s learning of geometric categories. Our data also reveal important differences in the benefits learners can derive from these different comparisons. Currently, these findings are being used in a follow up study assessing how to best incorporate comparison into lesson plans for pre-K teachers, to help them teach shapes in the classroom.


  • ♦ Childers, J. B. (2011). Attention to multiple events helps 21⁄2-year-olds extend new verbs. First Language, 31, 3-22.
  • ♦ Common Core State Standards Initiative (2010). Common core state standards for mathematics. Washington: National Governors Association Center for Best Practices and the Council of Chief State School Officers.
  • ♦ Gentner, D., Anggoro, F. K., & Klibanoff, R. S. (2011). Structure-mapping and Relational Language Support Children’s Learning of Relational Categories. Child Development, 82(4), 1173-1188.
  • ♦ Gentner, D., & Namy, L. (1999). Comparison in the Development of Categories. Cognitive Development, 14, 487–513.
  • ♦ National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston: Author.
  • ♦ Satlow, E., & Newcombe, N. S. (1998). When is a triangle not a triangle? Young children's conceptions of geometric shapes. Cognitive Development, 13, 547–559.
  • ♦ Waxman, S. R., & Klibanoff, R.S. (2000). The role of comparison in the extension of novel adjectives. Developmental Psychology, 36(5), 571–581.
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