Both rhyme and reason: toward design that goes beyond what meets the eye.

Abrahamson, D.
In T. Lamberg & L. Wiest (Eds.), Proceedings of the Twenty Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 287 – 295). Stateline (Lake Tahoe), NV: University of Nevada, Reno.

Drawing on design-based studies where students worked with learning tools for proportionality, probability, and statistics, I appraise whether students had opportunities to construct conceptual understanding of the targeted mathematical content. I conclude that visualizations of perceptually privileged mathematical constructs support effective pedagogical activity only to the extent that they enable students to coordinate perceptual conviction with mathematical operations—intuiting that, and not how, two representations are related constitutes perceptually powerful yet conceptually weak situatedness. In constructivist learning, as in empirical research, regularity apprehended in observed phenomena is measured, expressed, and schematized. Students should articulate or corroborate visual thinking with step-by-step procedures, e.g., synoptic views of multiplicative constructs should include tools for distributed-addition handles.