Multiplicative Thinking: The Thread That Connects the Whole Maths Curriculum
A post from the Australian Association of Mathematics Teachers (AAMT) landed in our feed this week with a mind map worth pinning to the staffroom wall. It traced multiplicative thinking out from a single yellow circle into Number, Space, Measurement, Statistics and Algebra - nine, ten, sometimes a dozen sub-topics branching off each one.
Their point was simple, and it's one we've been making at Maths Australia for years.
Multiplicative thinking isn't a Year 3 topic. It isn't something you tick off once times tables are memorised. It's a thread that runs through mathematics from the first equal-groups activity in Foundation to exponential functions in senior secondary - and if a student hasn't made the shift from additive to multiplicative thinking, that thread snaps early, and everything built on top of it becomes shakier.
What the Research Actually Shows
AAMT pointed to a 2024 study by Lorraine Day, Dianne Siemon, Rosemary Callingham and Rebecca Seah, published in Research in Mathematics Education - and it's worth sitting with, because the findings are stark.
The researchers drew on two large Australian studies: the Scaffolding Numeracy in the Middle Years project (over 3,200 students) and the Reframing Mathematical Futures II project (3,500 secondary students across 32 schools). Using Rasch analysis, they built a Combined Multiplicative Reasoning Scale spanning eight zones, from little or no multiplicative awareness through to sophisticated, flexible reasoning.
Then they tested that scale against tasks in statistics, geometry and algebra – domains most teachers wouldn't automatically file under “multiplication.”
The results were consistent across all three. A sampling task about climate-change surveys, a packing-boxes geometry problem, a task matching graphs to real-world situations - in every case, the students who could not attend to the underlying multiplicative structure (as opposed to just getting an answer via repeated addition) hit a ceiling. Across the tasks studied, fewer than 20% of students reached the top zone of multiplicative reasoning, with one geometry task (correctly packing boxes into a carton) solved by only 8.3% of the 432 students who attempted it.
The authors' conclusion: multiplicative thinking isn't just a number skill sitting alongside statistics, geometry and algebra. It's a thread woven through all three, and a large majority of middle-years students haven't yet grasped it.
Why “Fast Adding” Only Gets You So Far
Many multiplicative situations that students meet in primary and middle school can be solved by repeated addition - and for very good reason, that's exactly how we introduce multiplication early on. Skip-counting, arrays built from repeated rows, the Australian Mathematical Sciences Institute (AMSI) TIMES (The Improving Mathematics Education in Schools) module on Multiplication of Whole Numbers even opens by describing multiplication as “equivalent to repeated addition” for whole numbers. That's the right starting point.
But Day, Siemon, Callingham and Seah make a crucial distinction: a correct answer reached by repeated addition is not the same as a response that shows multiplicative thinking. A student who solves “12 strawberry plants per row, 7 rows, how many plants?” by adding 12 seven times hasn't yet coordinated the two composite units the way multiplicative thinking requires. They're doing additive work that happens to arrive at the multiplicative answer.
This is precisely why the AMSI module and the research agree on where the real teaching work lies – in the array model and the area model. Turning a 3 × 5 array on its side to show 3 × 5 = 5 × 3. Decomposing 74 × 63 into an area diagram of (70+4)(60+3). These aren't just calculation shortcuts; they're the visual scaffolding that helps a student stop counting and start seeing structure – noticing factors, recognising commutativity and distributivity, and eventually working with proportion, rate and ratio.
The Evidence Behind Concrete, Multi-Sensory Teaching
This is exactly the gap our I-CRAVE Maths® Methodology was built to close - and the case for teaching this way doesn't rest on the AAMT post or our own experience alone. It draws on a well-established body of research into how students actually learn mathematics.
The starting point is Jerome Bruner's theory of representation, which proposed that learners move through enactive (physical, hands-on), iconic (visual/pictorial) and symbolic (abstract) modes of understanding - the theoretical foundation of what is now widely known as the Concrete-Representational-Abstract (CRA) sequence.
Since then, a substantial body of intervention research has tested CRA directly. Witzel, Mercer and Miller's (2003) study of algebra instruction found that students taught through an explicit CRA sequence significantly outperformed peers taught abstractly from the start, particularly students with learning difficulties. Agrawal and Morin's (2016) review of the evidence base similarly concluded that CRA is an effective, evidence-based practice for teaching mathematics to students with learning disabilities across number, operations and algebra. And in one of the largest analyses to date, Carbonneau, Marley and Selig's (2013) meta-analysis of concrete manipulatives found a genuine positive effect on mathematics learning - but only when manipulatives were used with clear instructional guidance connecting the concrete materials to the underlying mathematical idea, rather than left as free-play exploration.
That last finding matters. It's the reason multi-sensory teaching and sequential, explicit teaching have to go together. Sweller's (1988) Cognitive Load Theory explains why: working memory is limited, and concrete materials only support learning when they reduce the mental effort needed to grasp a new idea, not add another layer of complexity on top of it. Rosenshine's (2012) widely cited Principles of Instruction reach a similar conclusion from classroom-effectiveness research - the most successful teachers present new material in small steps, model thinking aloud, guide practice, and only move on once students reach a high success rate. The U.S. National Mathematics Advisory Panel's Foundations for Success (2008) reached a matching recommendation: struggling students benefit most from explicit, systematic instruction, with visual and concrete representations used deliberately rather than incidentally.
Put together, this research says something quite specific: concrete materials help, sequencing helps, and explicit teacher guidance connecting the two is what makes the difference between a manipulative being a genuine teaching tool and just an activity.
How I-CRAVE Maths® Applies This to Multiplication
This is precisely what the I-CRAVE Maths® Methodology, developed by Esther White, builds into every lesson. It takes the Concrete-Representational-Abstract sequence and extends it into a complete instructional framework: Identify what a student already understands and where the gaps are, build understanding Concretely with hands-on materials, move to a Representational drawing stage, connect this to Abstract notation, have students explain their reasoning Verbally in precise mathematical language, and deliver every step through clear, Explicit instruction - in that order, every time.
Applied to multiplication specifically, this looks like: identifying whether a student has secured the additive and place-value foundations multiplication depends on; building rectangles, factors and products physically with hands-on blocks so students see and touch the commutative property rather than being told about it; drawing the same array or area model on paper; connecting it to the written multiplication algorithm only once the structure is genuinely understood; having the student explain, in their own words, why turning an array on its side doesn't change the total; and sequencing all of this explicitly, one mastered step at a time, rather than moving on because the calendar says it's time for the next topic.
This is also where the AAMT research connects directly back to classroom practice. Day, Siemon, Callingham and Seah's findings show that a correct answer isn't proof of multiplicative thinking - a student can reach 12 × 7 through repeated addition and never coordinate the two composite units involved. The Concrete and Verbal components of I-CRAVE Maths® are designed to surface exactly that distinction: a student who can build the array, describe why it represents 12 groups of 7, and explain the commutative property in their own words has demonstrated something a worksheet answer alone cannot show.
The Takeaway for Teachers
The AAMT mind map - and the research behind it - points to the same conclusion the evidence on concrete, sequential teaching supports: multiplicative thinking cannot be assumed, and it cannot be rushed. It has to be explicitly identified, built concretely, represented visually, and tested verbally, with each step genuinely mastered before the next is introduced.
If your students are getting the right answers to multiplication problems but struggle the moment ratio, proportion, or a geometry or statistics task asks them to reason about a relationship between quantities, that's very likely the same thread the AAMT and the research team identified - multiplicative thinking that was never fully secured, because the concrete, sequential groundwork underneath it wasn't either.
Our Multiplication & Division training walks teachers, tutors and intervention specialists through exactly how to apply the I-CRAVE Maths® Methodology to identify and close this gap - using the same evidence-informed, multi-sensory approach behind every stage of the Maths Australia program.
Sources
The Australian Association of Mathematics Teachers (AAMT), Facebook, 3 July 2026 – “Multiplicative thinking isn't a Year 3 topic…”
Day, L., Siemon, D., Callingham, R., & Seah, R. (2024). Connecting the threads: the role of multiplicative thinking in algebraic, geometrical, and statistical reasoning. Research in Mathematics Education, 26(2), 325–347. https://doi.org/10.1080/14794802.2024.2372365
Australian Mathematical Sciences Institute (AMSI), TIMES Project: Multiplication of Whole Numbers, Number and Algebra Module 9, Years 4–7. https://www.amsi.org.au/teacher_modules/Multiplication_of_whole_numbers.html
Bruner, J. S. (1966). Toward a Theory of Instruction. Cambridge, MA: Harvard University Press (Belknap Press).
Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18(2), 121–131.
Agrawal, J., & Morin, L. L. (2016). Evidence-based practices: Applying concrete-representational-abstract model to teach mathematics to students with learning disabilities. Learning Disabilities Research & Practice, 31(1), 34–44.
Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.
Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12–19, 39.
National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education.
Maths Australia, I-CRAVE Maths® Methodology and Multi-Sensory Maths Specialist training pages.