A student can get 12 × 7 correct by adding 12 seven times. Another student gets the same answer by reasoning that 12 × 7 is 12 × 10 minus 12 × 3. On paper, both answers look identical. Only one shows what researchers call multiplicative thinking, the ability to reason about the relationship between quantities, not just retrieve or construct a number.
That distinction turns out to matter more than almost anything else in a student’s mathematical development.
A study of nearly 7,000 Victorian students in Years 5 to 9 found a seven-year range in mathematics achievement within the same year level, a gap the researchers traced almost entirely to whether or not students had developed multiplicative thinking (Siemon & Virgona, 2001). Follow-up research across 32 secondary schools confirmed the same pattern in Years 7 to 9. Between 25% and 55% of Year 8 students don’t yet have access to it, and the gap isn’t evenly spread, students from lower socioeconomic backgrounds are far more likely to fall at the higher end of that range than students from wealthier backgrounds.
A separate 2024 study by Day, Siemon, Callingham and Seah, drawing on over 6,700 students across two large Australian research projects, tested whether this same capacity predicted performance in statistics, geometry and algebra, domains most people wouldn’t file under “multiplication.” It did, consistently. In one geometry task, correctly packing boxes into a carton, only 8.3% of the 432 students who attempted it succeeded.
The encouraging part
This is trainable, not fixed. Targeted teaching interventions built around a validated learning progression have produced some of the largest effect sizes reported in education research, ranging from 0.4 up to over 3.5 in individual school case studies. In one study, a group of struggling Year 6 students shifted four to five levels on the progression after just eighteen weeks of targeted intervention (Breed, 2011).
What separates a student who’s genuinely reasoning from one who’s arrived at the right answer through habit isn’t visible in a worksheet score. It shows up later, when the numbers get bigger, the context changes, or the problem stops looking like the one they practiced. That’s the same gap our piece on strategic fluency explores, from a different angle, “Strategic Fluency: The Missing Middle”.
Sources
Siemon, D. & Virgona, J. (2001). Roadmaps to numeracy, Middle Years Numeracy Research Project.
Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding Numeracy in the Middle Years, project findings and materials.
Day, L., Siemon, D., Callingham, R., & Seah, R. (2024). Multiplicative reasoning across the curriculum, Research in Mathematics Education.
Breed, M. (2011). Constructing paths to multiplicative thinking (PhD thesis, RMIT University).
Growing Mathematically project, AAMT in collaboration with RMIT, funded by the Australian Government Department of Education.
