Learning Math by Comparison

Investigator: Lindsey Richland

Graduate Student Researcher: Janice Hansen

Funding: Institute of Education Sciences

The Learning Math by Comparison project assesses classroom teachers’ use of analogies and comparisons (potent reasoning opportunities) and explores novel ways to use these to promote students’ conceptual development in math.

One study done on this project examined cross-cultural differences in the everyday use of teaching comparisons in United States, Hong Kong, and Japanese mathematics classrooms using the TIMSS-R video dataset. The researchers found that teachers in all three countries frequently used comparisons (about twenty per lesson). However, there were striking differences in how teachers actually produced them instructionally. U.S. teachers were least likely to use pedagogical techniques that are thought to reduce processing demands and increase the likelihood that novices notice and benefit from relational comparisons. Specifically, they were less likely to use all of six identified strategies: using familiar sources, imagery, comparative gestures, and several kinds of visual cues to show and elaborate the representation of the sources. Teachers in Hong Kong and Japan were significantly more likely to use all of these strategies. U.S. students seriously underperform on international mathematics achievement tests compared with students from Hong Kong and Japan, and this may be related to U.S. teachers’ lower support for encouraging students to perform complex reasoning and problem solving (Gonzales et al., 2000). Click here to view a report from this research.

Current work on this project includes experimental studies that assess the math learning impact of cognitive support strategies identified in the cross-cultural video analysis and recommended by empirical research on analogy. Studies focus on teaching the mathematical concepts of permutation probabilities, operations with fractions, and eliminating misapplications of the linearity assumption. The studies use videotaped instruction to manipulate the nature of the teachers’ support for an instructional comparison, though the instructional content is always the same across conditions. Results to date suggest that systematically supporting learners’ comparative thinking leads to more robust, conceptual understanding. Conversely, less supported comparisons lead to adequate procedural knowledge but less robust understanding.