A child who understands why a procedure works can rebuild it when they forget it. A child who has only memorized it cannot. This is the practical case for conceptual understanding, and it is supported by half a century of math education research. Memorization has its place — multiplication facts matter — but it works best when it sits on top of understanding, not in place of it.
The old debate
For more than a century, U.S. math education has cycled between two camps. One favors procedural fluency: memorize the algorithm, practice it, and move on. The other favors conceptual understanding: spend time on why the algorithm works, even if it slows the pace. The National Mathematics Advisory Panel concluded in 2008 that the binary is false. Students need both, and the relationship between them matters. Conceptual and procedural knowledge develop together, each reinforcing the other (NMAP, 2008).
What “conceptual” actually means
The cleanest distinction comes from the British mathematics educator Richard Skemp, whose 1976 paper “Relational understanding and instrumental understanding” remains a touchstone in the field. Instrumental understanding is knowing how — knowing the rule and being able to apply it. Relational understanding is knowing why — understanding the structure that produces the rule, so the rule can be reconstructed, adapted, and connected to other rules.
A child with instrumental understanding can compute 3/4 ÷ 1/2 by inverting and multiplying. A child with relational understanding can also explain that the question is “how many halves fit into three-quarters” and predict that the answer will be more than one and less than two before doing any calculation. The second child is far less likely to forget the rule, and far less likely to apply it where it does not belong.
What the research shows
The case for sequencing has strengthened over the past two decades. Rittle-Johnson, Siegler, and Alibali (2001) showed that conceptual and procedural knowledge develop iteratively: early conceptual understanding predicts later procedural skill, which in turn deepens conceptual understanding. Long-term retention is the clearest beneficiary. Children who learn procedures with understanding retain them across school breaks; children who memorize without understanding tend to relearn each year (Hiebert and Grouws, 2007).
International comparisons reinforce the pattern. Singapore consistently ranks at or near the top in international mathematics assessments (TIMSS, multiple years; PISA, OECD). A defining feature of the Singapore curriculum is that procedures are introduced only after the underlying concept is secure — what Singapore educators call mastery before pace (Ministry of Education, Singapore).
How Singapore Math handles this
The Concrete-Pictorial-Abstract progression is the operational answer to the conceptual-versus-procedural debate. A new idea enters through physical materials — counters, blocks, cuisenaire rods. It is then represented pictorially, often through bar models or arrays. Only then does the abstract symbol — the equation, the algorithm — appear. By the time the child meets the symbol, they have already met the structure three times. The symbol is a shorthand for something they understand, not a code they have to memorize.
Mastery before pace adds a second discipline: children move on when they have grasped the concept, not when the calendar says so. In practice, this means fewer topics covered in greater depth, with each topic built on a secure prior foundation. The research base for this kind of sequencing was examined recently by Pellegrini et al. (2025), which surveyed the evidence on Singapore-Math-aligned instruction.
The U.S. evidence
Singapore’s outcomes are sometimes dismissed as cultural — small country, selective system, high-pressure context. The more interesting evidence comes from U.S. schools using Singapore-Math-aligned curricula in ordinary American conditions. At Orange County Classical Academy, students reached 61% proficiency on California state assessments, against a 34% California state average (OCCA, 2023; California state assessment data). A single school does not settle a national debate. But it does suggest the method travels.
A parent’s view at the kitchen table
The difference between conceptual and procedural understanding is most visible at homework time. A child who has memorized “multiply by ten by adding a zero” will write 4.2 × 10 = 4.20. A child who understands that multiplication by ten shifts every digit one place to the left will write 42. The rule about adding a zero is a special case that breaks. The understanding does not.
Parents who notice their child reaching for the rulebook before reaching for the idea are seeing the limit of memorization in real time. The fix is rarely more drill. It is a step back to the concrete — to the cuisenaire rod, to the bar model, to the picture — and then forward again, with the structure intact.
Sources
- Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77.
- Rittle-Johnson, B., Siegler, R. S., and Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2).
- Hiebert, J., and Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning.
- National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education.
- Ministry of Education, Singapore — Mathematics Syllabus (Primary).
- Pellegrini et al. (2025). “Effects of the Enactive, Iconic, Symbolic (EIS) Intervention on Student Math Skills in Primary School”, J. of Educational, Cultural and Psychological Studies.
- Orange County Classical Academy state assessment outcomes, 2023; California state assessment data.
- TIMSS, IEA International Association for the Evaluation of Educational Achievement. PISA, OECD.