Teaching Mathematics by Comparison: Analog Visibility as a Double-Edged Sword

Comparing multiple solutions to a single problem is an important mode for developing flexible mathematical thinking, yet instructionally leading this activity is challenging (Stein, Engle, Smith, & Hughes, 2008). We test 1 decision teachers must make after having students solve a problem: whether to only verbally discuss students' solutions or make them visible to others. Fifth grade students were presented with a videotaped mathematics lesson on ratio in which students described a misconception and 2 correct strategies. The original lesson was manipulated via video editing to create 3 versions with constant audio but in which the compared solutions were a) presented only orally, b) visible sequentially in the order they were described, or 3) all solutions were visible after being described throughout the discussion. Posttest and delayed posttest measures revealed the greatest gains when all solutions were visible throughout the discussion, particularly better than only oral presentation for conceptual knowledge. Sequentially showing students visual representations of solutions led to the lowest gains overall, and the highest rates of misconceptions. These results suggest that visual representations of analogs can support learning and schema formation, but they can also be hurtful--in our case if presented as visible in sequence.