Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures. These students were challenged to represent multiplicative relationships between unknowns: They changed the given relationship, did not think of the relationship as multiplicative until after concerted work, and used numerical values in lieu of unknowns. Our account for these challenges is that students needed to simplify the involved units coordinations. Ultimately students abstracted the relationship as multiplicative, but the exact relationship was not certain or had to be constituted in activity. Implications for teaching are explored.
Hackenberg, A. J., Jones, R., Eker, A., & Creager, M. (2017). “Approximate” multiplicative relationships between quantitative unknowns. Journal of Mathematical Behavior, 48, 38-61.