Substantial effort has been put into documenting and analyzing students’ notoriously low
performance in the domain of rational numbers (e.g., Behr, Harel, Post, & Lesh,
1993)—performance which has often been typified as evidencing ‘additive reasoning’
where ‘multiplicative reasoning’ may have been more effective. For instance, a student
may reason that 3 : 4 is equal to 6 :7 because in both ratios there is a difference of 1
between the two numbers. It may be the case that students do not use multiplicative
reasoning because they lack coherent cognitive models for understanding and applying
multiplicativity in these cases. Plausibly, instruction should help students build such
models on the basis of existing knowledge. Insightful design research has called for
sensitive adaptation of instruction according to students’ intuitive understandings
(Confrey, 1998), and Vergnaud (1994) has suggested a broad view of the instructional
domain of multiplicativity as encompassing not only schooled but also everyday
knowledge. Some of this intuitive (Fischbein, Deri, Nello, & Marino, 1985), streetwise
(Schliemann & Carraher, 1992), and folk (Urton, 1997) knowledge has been described in
ways that illuminate prospective didactic solutions.