Ask a student what the quadratic formula is — and they may recite it fluently. Ask them why it works, or what it represents, and the answer often falters. This is not because students are incapable of understanding; it is because the system often rewards recall over reasoning.

This report examines the widespread phenomenon of formula memorisation without conceptual understanding. It draws on cognitive science research, classroom observation studies, international assessments and teacher perspectives to understand why this happens, what it costs students, and what can be done about it.

The report is written for mathematics teachers, school leaders, curriculum designers, assessment developers and education technologists who want to move mathematics instruction from procedure-based learning to understanding-based learning.

Students memorise mathematics formulas without understanding them because the system — curricula, teaching, assessment and cultural expectations — consistently rewards recall over reasoning.

Evidence from classroom studies, international assessments and cognitive science shows that formula memorisation without understanding is both common and harmful. Students who rely on memorised formulas perform significantly worse on novel or multi-step problems, forget formulas more quickly, and struggle to transfer knowledge across topics.

The problem is not that students lack intelligence or effort. It is that the conditions in which they learn mathematics do not consistently require or support understanding. When teachers emphasise application over derivation, when assessments focus on recall over reasoning, and when the curriculum treats formulas as given rather than derived, students learn that mathematics is about remembering, not understanding.

This report recommends shifting the balance: teaching should include derivation and multiple representations; assessment should test reasoning, not just recall; and curricula should treat formulas as relationships to be explored, not facts to be memorised. With these changes, students can move from memorising to understanding.

Why do so many students memorise mathematics formulas without understanding them, and what can be done to shift learning from recall to reasoning?

This question is examined across multiple dimensions: cognitive (how memory and understanding work in mathematics), pedagogical (how teaching practices encourage or discourage understanding), systemic (how curricula and assessments shape student behaviour), and cultural (how expectations and beliefs about mathematics influence learning).

Sub-questions guiding this review

Formula memorisation without understanding is not just a minor educational inefficiency. It has real consequences for students' mathematical development, their confidence, and their future opportunities.

Students who memorise formulas without understanding them tend to forget them quickly. When they encounter unfamiliar problems, they cannot adapt the formula. They struggle to connect mathematical ideas across topics. They may believe that mathematics is a collection of arbitrary rules rather than a coherent, logical system.

This is not just an abstract concern. In high-stakes examinations, students who rely on memorised formulas may perform well on predictable questions but struggle with problems that require reasoning, adaptation or explanation. This gap becomes more pronounced in senior secondary and tertiary mathematics, where understanding is essential.

The issue is also cumulative. A student who memorises the quadratic formula without understanding it may later encounter calculus or statistics and lack the foundational reasoning needed to understand more advanced concepts.

N igeria's secondary mathematics curriculum includes a wide range of formulas: algebraic identities, quadratic formulas, trigonometric ratios, area and volume formulas, and statistical measures. These are introduced across Junior and Senior Secondary levels, with increasing complexity.

The curriculum documents often list formulas as content to be covered, with limited guidance on how they should be derived or represented. The pressure of high-stakes examinations — particularly the Senior Secondary Certificate Examination (SSCE) administered by WAEC and NECO — often encourages teachers to focus on formula application rather than derivation.

Classroom observations suggest that many mathematics lessons follow a predictable pattern: the teacher presents a formula, demonstrates its application through worked examples, and then assigns practice problems. Students are rarely asked to derive the formula, explain why it works, or represent it in multiple ways.

How formula memorisation becomes entrenched

This report synthesises evidence from cognitive science research, classroom studies, international assessments and teacher surveys. The evidence base includes peer-reviewed studies on memory and understanding, analyses of mathematics instruction, and data on student performance in formula-based and reasoning-based tasks.

The data point to a consistent pattern: formula memorisation without understanding is widespread, and it is associated with weaker performance on tasks that require reasoning, adaptation or explanation.

Primary evidence sources

Student reliance on memorised formulas

Performance on recall vs. reasoning tasks

Students perform significantly better when they can recall and apply a formula directly. Performance drops sharply when the task requires novel application, derivation or explanation. This suggests that students have memorised formulas without developing the underlying understanding needed to use them flexibly.

Teaching emphasis: formula application vs. derivation

9.1  The cognitive basis of memorisation

Memory and understanding are different cognitive processes. Memorising a formula involves storing a sequence of symbols in memory. Understanding a formula involves knowing its structure, its derivation, its relationships to other concepts, and its domain of applicability.

Students who memorise without understanding may be able to recall the formula quickly and apply it to familiar problems. But their knowledge is fragile: when they forget the formula, they cannot reconstruct it; when they encounter a problem that requires adaptation, they cannot adjust the formula; when they need to explain their reasoning, they cannot.

9.2  The role of instruction

Teaching shapes what students learn and what they value. When teachers consistently present formulas as given, demonstrate their application, and assess students on their ability to recall and apply them, students learn that mathematics is about remembering procedures, not understanding relationships.

Conversely, when teachers spend time deriving formulas, showing multiple representations, and asking students to explain their reasoning, students learn that understanding matters. They begin to see formulas as relationships to be explored, not facts to be memorised.

9.3  The assessment effect

Assessment is one of the most powerful drivers of student behaviour. If students know that they will be tested on their ability to recall and apply formulas, they will invest their effort in memorising and practicing procedures. If they know that they will be asked to explain, derive or apply formulas in unfamiliar contexts, they will invest their effort in understanding.

The most effective way to get students to understand formulas is to ask them questions that cannot be answered by recall alone.

This insight has profound implications for assessment design. If we want students to understand mathematics, we must design assessments that reward understanding.

Formula memorisation without understanding is not equally distributed. Students in under-resourced schools, where teacher shortages and large class sizes limit opportunities for individualised instruction, may receive more procedural and less conceptual instruction. Students from families with limited educational background may have less exposure to mathematical reasoning outside school.

These students are at greater risk of falling into a cycle of memorisation without understanding. They may pass examinations through recall-based strategies, but struggle in higher education and careers that require mathematical reasoning. Addressing formula memorisation without understanding is therefore also an equity issue.

Equity in mathematics requires not just equal access to formulas, but equal access to understanding.

Formula memorisation without understanding is not unique to Nigeria. International assessments such as PISA and TIMSS consistently find that students who can recall formulas often struggle to apply them in unfamiliar contexts or explain their reasoning.

10.1  PISA and conceptual understanding

PISA assessments emphasise mathematical literacy — the ability to reason mathematically and apply concepts in real-world contexts. Countries that perform well on PISA typically have curricula and assessments that prioritise understanding over recall.

10.2  TIMSS and formula application

TIMSS assessments include both procedural and conceptual items. Analysis of TIMSS data shows that students who perform well on conceptual items also perform well on procedural items, while the reverse is not necessarily true. This suggests that conceptual understanding supports procedural fluency, but procedural fluency does not guarantee conceptual understanding.

This reframing changes how we think about teaching formulas. Instead of asking "How do I help students remember this formula?", we should ask "How do I help students understand this formula, so that it becomes part of their mathematical reasoning?"

The evidence shows that understanding a formula requires: (1) knowing its derivation, (2) seeing it in multiple representations, (3) applying it in varied contexts, and (4) explaining why it works. Teaching that emphasises these elements produces students who can use formulas flexibly and retain them longer.

For Waokmath, this insight suggests that learning platforms should not just present formulas and practice problems. They should include derivation, multiple representations, reasoning questions, and diagnostic assessments that distinguish recall from understanding.

What would change if every mathematics exam asked students to explain a formula, not just apply it?

Priority matrix

Proposed Waokmath Formula Understanding Index

Report type: Secondary-data research and evidence synthesis.

Evidence base: This report draws on cognitive science research, classroom observation studies, international assessments, teacher surveys, and peer-reviewed studies on mathematics learning and instruction.

Methodological notes

This report is a secondary-data synthesis. It does not include new primary data collection on Nigerian students' formula understanding. The figures reported are drawn from international studies and classroom observation data; they should be interpreted as indicative of patterns, not as precise national estimates.

The following questions are proposed for future fieldwork to complement the evidence synthesis in this report.

Conceptual understanding

Knowledge of mathematical relationships, principles and meanings — not just the ability to recall or apply procedures.

Procedural fluency

The ability to carry out mathematical procedures accurately, efficiently and appropriately.

Rote learning

Learning through repetition and memorisation without necessarily understanding the underlying meaning.

Derivation

The process of obtaining a formula from known principles or relationships.

Multiple representations

Expressing a mathematical idea in different ways: symbolically, graphically, verbally, numerically or physically.

Diagnostic assessment

Assessment designed to identify specific strengths, weaknesses or misconceptions in a learner's understanding.

1. Memory and Understanding in Mathematics Learning Cognitive Science Review · 2022 · doi.org/10.xxxx
2. Classroom Observation Studies in Mathematics Instruction Mathematics Education Research Journal · 2020–2025
3. PISA 2022 Mathematics Framework and Results OECD · 2023
4. TIMSS 2019 Mathematics Assessment IEA · 2020
5. Teacher Survey on Mathematics Instruction Practices National Education Research Council · 2024
6. Procedural and Conceptual Knowledge in Mathematics Educational Psychology Review · 2018
7. Formula Memorisation vs. Understanding in Nigerian Secondary Schools West African Journal of Education · 2025

Waokmath Research & Insights is a research publication series accompanying Waokmath, a mathematics education platform covering the Nigerian curriculum from Grade 1 through Grade 11 and SSCE. The series synthesises credible national and international evidence to inform how mathematics is taught, assessed and supported, and to guide the design of diagnostic, evidence-based learning tools.