A student can produce the correct answer to a mathematics question and still misunderstand the mathematics behind it. Conversely, an incorrect answer can sometimes reveal something far more useful than a mark of zero: it can expose the student's underlying model of the concept.

This distinction is central to mathematics education. Misconceptions are not simply mistakes to be crossed out and replaced with the correct procedure. They can become stable ways of interpreting mathematical objects, symbols and relationships. Once established, they may reappear in unfamiliar problems, even when a student has successfully memorised a procedure.

This Waokmath Research & Insights report examines common misconceptions found across secondary mathematics, with particular attention to fractions, algebra, equations, variables, proportional reasoning and geometry. It places international research alongside Nigeria's learning context and asks a practical question: how can schools move from correcting answers to diagnosing mathematical thinking?

Mathematical misconceptions are persistent patterns of reasoning that can survive repeated instruction, particularly when students learn procedures without developing sufficiently connected conceptual knowledge.

Research across secondary mathematics identifies recurring difficulties with variables, equality, fractions, proportional reasoning, algebraic transformations, geometry and the interpretation of mathematical symbols. Studies of algebra learners, for example, have found that students can perform algebraic procedures while holding incomplete or incorrect conceptions of what variables represent.

The Nigerian context makes this issue especially important. NALABE 2022, conducted nationally by UBEC with UNICEF support, assessed mathematics performance among Primary 3, Primary 5 and JS2 learners. In mathematics, 50% of P3 learners reached minimum proficiency, compared with 34% of P5 learners and only 14% of JS2 learners. The assessment therefore points to substantial difficulty by the time learners reach junior secondary education.

The evidence does not support treating every incorrect answer as a misconception. Some errors are procedural slips, computational mistakes or misunderstandings of the wording of a question. The important distinction is whether an incorrect response is random or reflects a coherent but mathematically inappropriate way of reasoning.

The central recommendation of this report is therefore diagnostic: mathematics instruction should routinely seek evidence of how students are thinking, not only whether they have reached the correct answer. This requires carefully designed questions, error analysis, explanation, multiple representations and deliberate opportunities for students to confront and revise incomplete conceptions.

What are the most common mathematical misconceptions among secondary school students, why do they persist, and how can teaching and assessment identify and address them?

The question is deliberately broader than asking which topics students find difficult. A difficult topic is not necessarily associated with a misconception. This review therefore distinguishes between lack of knowledge, procedural error and systematic misunderstanding.

Sub-questions guiding this review

Secondary mathematics is cumulative. New concepts frequently depend on structures developed earlier. A student who has an unstable understanding of fractions may subsequently encounter difficulty with algebraic fractions, ratios, percentages, probability and equations. Similarly, weak understanding of the meaning of the equals sign can affect the interpretation of equations and algebraic transformations.

This means that misconceptions can behave like structural faults in a mathematical knowledge system. They may remain invisible while questions resemble familiar examples, then become obvious when the student encounters an unfamiliar representation or multi-step problem.

Research examining misconceptions across algebra, geometry, rational numbers and probability suggests that some underlying knowledge structures cut across topic boundaries. This reinforces the importance of addressing foundational reasoning rather than treating every error as an isolated topic-specific problem.

N igeria's basic education system provides the context in which mathematical misconceptions develop, persist and become visible. The National Assessment of Learning Achievement in Basic Education (NALABE) 2022 assessed learners across all 36 states and the Federal Capital Territory, including 23,564 JS2 learners from 788 junior secondary schools.

Mathematics assessment in NALABE covered numbers and numeration, basic operations, algebraic processes, mensuration and geometry, and everyday statistics. The assessment found a marked decline in the proportion of learners reaching minimum proficiency as grade level increased.

At JS2, only 14% of learners reached minimum proficiency in mathematics, while 86% did not. This does not by itself demonstrate that misconceptions caused the performance gap. It does, however, establish the scale of the learning challenge and the importance of diagnosing the specific knowledge structures underlying learner errors.

From foundational knowledge to secondary mathematics

This report synthesises national assessment evidence and peer-reviewed research on mathematical misconceptions. The evidence base includes large-scale assessment, systematic review, classroom-based error analysis and qualitative investigations of students' reasoning.

The international studies should not be interpreted as direct estimates of Nigerian students' misconception rates. They are used to identify recurring mathematical structures and mechanisms that can inform questions for Nigerian research and classroom diagnosis.

Primary evidence sources

National proficiency signal

Selected misconception domains

JS2 mathematics proficiency in NALABE 2022

The national pattern is important because it shows that mathematics difficulty becomes more pronounced as learners progress into increasingly demanding mathematical content. The data do not identify specific misconceptions, but they establish a strong rationale for diagnostic research at the junior secondary level.

9.1  From arithmetic rules to algebraic misconceptions

One of the most important transitions in secondary mathematics is the move from arithmetic to algebra. Arithmetic primarily involves known quantities and operations; algebra introduces variables, expressions, equations and relationships that must be interpreted symbolically.

Research on Year 8 learners has found that students may successfully execute algebraic procedures while misunderstanding variables. Some learners can manipulate expressions but struggle to translate verbal statements into algebraic representations.

9.2  Why procedural rules become misconceptions

Rules are essential to mathematics, but rules detached from meaning are vulnerable to overgeneralisation. A student may remember that a fraction involves two numbers and then apply whole-number reasoning to each component. Similarly, a student may learn a cancellation procedure and attempt to apply it whenever identical symbols appear.

9.3  The role of representation

Mathematics is represented through symbols, diagrams, words, tables, graphs, physical models and numerical examples. A student may understand one representation while failing to connect it with another.

Fraction misconceptions provide a clear example. Understanding a fraction requires more than manipulating numerator and denominator symbols. Students also need to understand part-whole relationships, magnitude and position on the number line.

When representations are disconnected, students can learn procedures without learning the mathematical object those procedures describe.

Effective diagnosis therefore requires teachers to vary representations and ask students to move between them rather than relying exclusively on symbolic manipulation.

Misconceptions do not develop in a vacuum. Learners encounter mathematics through different teachers, school environments, instructional resources, languages, prior educational experiences and opportunities for practice.

NALABE 2022 found that mathematics performance varied across learner and school contexts. The assessment reported differences associated with school location and school type, although gender differences were not significant across the assessed classes.

These findings reinforce the need to avoid interpreting misconceptions solely as individual learner deficits. A diagnostic mathematics system should examine the interaction between the learner, the mathematical task and the instructional environment.

Equity in mathematics is not only about who gets the correct answer; it is also about who gets the opportunity to understand why the answer is correct.

International research shows that misconceptions are not unique to Nigeria. Studies across different education systems repeatedly identify difficulties involving rational numbers, algebraic variables, equations, geometry and proportional reasoning.

10.1  Foundational knowledge structures

Rakes and Ronau's study of 1,133 students found relationships between misconceptions across algebra, geometry, rational numbers and probability. Their analysis identified foundational structures including variable, measurement, spatial reasoning, additive and multiplicative structures, and absolute versus relative comparison.

10.2  Fractions and secondary mathematics

Cross-national evidence using TIMSS data has shown that fraction magnitude remains challenging at Grade 8. This is significant because understanding fraction magnitude is not merely a computational skill; it supports later reasoning about rational numbers, ratios and algebra.

10.3  Absolute value

A 2025 systematic literature review of secondary students' errors in absolute value identified two major areas: difficulties transitioning from arithmetic to algebraic thinking and difficulties understanding absolute value as a piecewise-defined function.

This reframing changes the purpose of assessment. Instead of using an incorrect answer only to assign a mark, teachers can use it as diagnostic evidence. If several students make the same structurally similar error, the pattern may reveal a shared misconception that requires instructional attention.

For Waokmath, this creates an important opportunity: mathematical assessment can be designed not merely to classify students as correct or incorrect but to classify the reasoning patterns associated with their responses.

A future Waokmath diagnostic system could therefore connect question design, misconception classification, remediation and reassessment into a continuous learning cycle.

What if a mathematics test could tell a teacher not only that a student is wrong, but exactly what the student's wrong reasoning is?

Priority matrix

Proposed Waokmath Mathematics Misconception Index

Report type: Secondary-data research and evidence synthesis.

Evidence base: National assessment data, peer-reviewed research, systematic reviews and research-indexed studies concerning mathematical misconceptions and error patterns.

Methodological notes

This report is an evidence synthesis rather than a new national study of Nigerian students' misconceptions. Consequently, it does not estimate the national prevalence of individual misconceptions such as fraction, variable or equality misconceptions.

The following questions are proposed for future Waokmath fieldwork with teachers, students, school leaders, curriculum specialists and assessment practitioners. They are intended to move the research from secondary evidence toward direct evidence of mathematical thinking in Nigerian classrooms.

Mathematical misconception

A stable but mathematically inappropriate conception or way of reasoning about a mathematical idea.

Error

An incorrect response that may result from misunderstanding, faulty procedure, calculation, interpretation or other causes.

Error analysis

The systematic examination of students' errors to identify patterns and possible underlying reasoning.

Conceptual understanding

Understanding mathematical concepts, relationships and meanings rather than merely recalling procedures.

Procedural fluency

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

Diagnostic assessment

Assessment designed to reveal what a learner understands, misunderstands or is ready to learn.

Proportional reasoning

Reasoning about relationships in which quantities change in a consistent multiplicative or proportional manner.

Representation

A way of expressing a mathematical idea, such as symbols, diagrams, graphs, tables, words or physical models.

1. National Assessment of Learning Achievement in Basic Education (NALABE) 2022 Report Universal Basic Education Commission (UBEC) & UNICEF · 2025 · National assessment of P3, P5 and JS2 learning achievement. UNICEF Nigeria
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3. Cognitive Map: Diagnosing and Exploring Students' Misconceptions in Algebra Fitria, A., Subanji, Susiswo & Susanto, H. · Mathematics Teaching Research Journal · 2023 · ERIC EJ1412238. ERIC
4. Error Analysis in Algebra Learning: Exploring Misconceptions and Cognitive Levels Mathaba, P. N., Bayaga, A., Tîrnovan, D. & Bossé, M. J. · Journal on Mathematics Education · 2024 · ERIC EJ1428049. ERIC
5. Errors and Misconceptions of Secondary School Students in Absolute Values: A Systematic Literature Review Papadouris, J. P., Komis, V. & Lavidas, K. · Mathematics Education Research Journal · 2025 · ERIC EJ1488603. ERIC
6. Probing Grade 10 Students about Their Mathematical Errors on Simplifying Algebraic Fractions Makonye, J. P. & Khanyile, D. W. · Research in Education · 2015 · ERIC EJ1119811. ERIC
7. Misconceptions of Sixth Grade Secondary School Students on Fractions International Electronic Journal of Elementary Education · 2018 · ERIC EJ1185072. ERIC
8. A Cross-national Comparison of Fourth and Eighth Grade Students' Understanding of Fraction Magnitude TIMSS-based comparative research · 2022 · ERIC EJ1366202. ERIC Full Text
9. Underprepared College Students' Understanding of and Misconceptions with Fractions Lee, H. J. & Boyadzhiev, I. · International Electronic Journal of Mathematics Education · 2020 · ERIC EJ1249568. ERIC

Waokmath Research & Insights is a research publication series accompanying Waokmath, a mathematics education platform covering the Nigerian curriculum and supporting mathematics teaching, learning and assessment.

The series examines important questions in mathematics education using credible national and international evidence. Its purpose is not simply to report educational problems, but to identify patterns, interpret evidence and develop practical insights that can inform teachers, schools, policymakers, researchers and education technology developers.

The series also provides an evidence base for the development of future Waokmath diagnostic tools, learning resources, assessment systems and mathematics education research projects.