The Error That Isn’t Random: What Students’ Mistakes Reveal About How We Teach
By: Neha Raheel
Across sub-Saharan Africa, the learning crisis in foundational education is well-documented, but the conversation has largely centered on literacy. Although that focus is warranted—with roughly 70 percent of children in low- and middle-income countries unable to read a simple text by the end of primary school—numeracy deserves equal urgency.
Research consistently shows that early math skills are among the strongest predictors of long-term academic achievement. Children who leave primary school without secure number sense and arithmetic fluency face compounding disadvantages at every stage of schooling that follows.
Neha Raheel, Director of Teaching and Learning, Luminos Fund
Paying close attention to the common mistakes students make is one of the most powerful ways to improve how we teach.
At Luminos, we have spent years developing and refining structured numeracy materials for vulnerable and out-of-school children in Ethiopia, Ghana, Liberia, and The Gambia. Our programs build on proven teaching methods: explicit, systematic instruction; moving from concrete to pictorial to abstract representations to build understanding in deliberate stages before working with symbols; and regular assessments to keep learning on track.
Yet good frameworks alone do not guarantee learning. The challenge is ensuring that students truly grasp foundational concepts—and paying close attention to the common mistakes they make is one of the most powerful ways to improve how we teach.
Students in Ghana work out a math problem using stones and bottle caps during group work. (Photo by Mo Scarpelli for the Luminos Fund)
What Real-Time Data Can Reveal About Teaching
Our team routinely reviews classroom observation and student assessment data, analyzing patterns and asking what they reveal about learning progress. During one of these reviews in Liberia, a pattern emerged that stopped us in our tracks.
Internal data showed students performed well on foundational tasks but lost ground as arithmetic problems grew more complex; particularly addition and subtraction with three-digit numbers, regrouping numbers across place values, and understanding number sequences.
The Luminos Method: Real-Time Data
Real-time data drives classroom- and program-level decision-making for individual student impact, and is an essential component of the Luminos Fund’s accelerated learning program.
Taken individually, each of these gaps looks like a different instructional problem. Viewed together, however, they point to a common root: students needed a stronger understanding of place value.
Research on how children learn math is clear: lasting fluency isn’t just about knowing the steps to solve a problem, it depends on truly understanding the concepts behind them.
Understanding the Root Cause
Research on how children learn math is clear: lasting fluency isn’t just about knowing the steps to solve a problem, it depends on truly understanding the concepts behind them. A child may learn the steps for ‘carrying’ in an addition problem, but if they do not understand what those steps mean, the process can quickly fall apart. Carrying is not just moving a digit. It means exchanging ten ones for one ten because our number system is built around groups of ten. When students only memorize a procedure, their knowledge can be fragile. One unfamiliar problem format, or one forgotten step, and they may no longer know what to do.
Place value is one of the building blocks of early math. It helps children understand what a digit means based on where it appears in a number. It explains why a zero matters, why we carry or borrow when doing arithmetic, and how larger numbers work.
Without this foundation, students often rely on memorized rules rather than truly understanding the problem they are solving. Yet it is one of the most commonly underestimated gaps in early numeracy instruction. This is not because teachers fail to teach place value, but because students often encounter more complex arithmetic problems before they have had enough hands-on practice working with physical objects or pictures to really understand what each digit in a number means.
Our analysis in Liberia showed that the issue was not the lesson content itself, but how effectively teachers helped students grasp the underlying concepts.
Excerpt from the Luminos Foundational Numeracy Misconceptions Guide, 2026
Turning a Pattern Into a Resource
Our response was practical: we created a guide for teachers on common numeracy misconceptions and evidence-based strategies for addressing them. We designed it as a classroom tool rather than a theory document, to help teachers look beyond the wrong answer and understand the thinking behind it.
For each misconception, it asks three questions: What does the student actually believe? What misunderstanding is driving this error? And what can a teacher do, today, to address it?
A student who makes the same mistake repeatedly is showing you where their understanding broke down — not how little effort they are making. Diagnosing the specific misconception is the first step to helping them.
Luminos Foundational Numeracy Misconceptions Guide, 2026
We introduced the Foundational Numeracy Misconceptions Guide during teacher training in Liberia and the impact was immediate. We saw it in the quality of discussions during training sessions and, importantly, in how supervisors later described what they observed in the classroom. This is a mature program with years of implementation and iteration behind it. The fact that careful data analysis could still uncover opportunities for improvement is not a failure—it is exactly what a responsive system should do. Just as importantly, the solution was not redesigning our materials, but sharpening our focus on how we support teachers.
In Ethiopia and Ghana, classroom data revealed a similar pattern with missing number identification. In response, we adapted and translated the guide for use there, and we have already seen improvements in math scores across the board.
The guide will become part of our Teachers’ Guides for the upcoming academic year, signaling that what began as a targeted response to a data pattern has become a core part of how we train and support teachers across our programs.
Why This Matters
A growing body of research points to the importance of foundational numeracy. However, the sector has yet to develop the same shared practitioner vocabulary around diagnosing numeracy challenges that the science of reading brought to literacy instruction.
In many low-resource contexts, teachers receive training in what to teach but less support in recognizing the common misconceptions that predictably emerge, understanding what those errors reveal, and responding to them in the moment.
Access Our Materials
Great tools should be in as many hands as possible. Our classroom materials—including Teachers’ Guides, Student Books, and supplementary reading materials (Decodable Readers and Read Aloud stories)—are freely available.
This matters because misconceptions in numeracy compound. A student who doesn’t understand place value will struggle with regrouping. A student who doesn’t understand regrouping will struggle with multi-digit operations. And a student who has only ever learned procedures, without understanding the ideas behind them, will find every new topic harder than the last; not because they are incapable of learning, but because they are building on a foundation that was never solid.
Sharing what we are learning is one way Luminos hopes to contribute to this conversation. We do not have all the answers, but we have found that the most useful insights come from paying close attention to what the data tells us and using it to ask better questions.
A student who has only ever learned procedures, without understanding the ideas behind them, will find every new topic harder than the last; not because they are incapable of learning, but because they are building on a foundation that was never solid.
What We Owe Every Child in the Classroom
I keep coming back to a particular kind of student: one who has been sitting in a classroom, trying their best, but making the same mistake week after week. Quietly, that student is concluding that math is not for them. That they are just not good at numbers. That something is wrong with them.
More often than not, the problem is not ability. It is understanding. The student was taught a procedure without being shown what it means. They were asked to memorize before they were allowed to understand. Over time, the gap between what they understood and what they were being asked to do became wide enough that they stopped trusting themselves.
Our job—as program designers, as trainers, and practitioners—is to close that gap. Not by moving faster or repeating instructions louder, but by asking: What does this child actually believe? Where did the understanding break down? And what, concretely, can we do about it today?
Those are the questions that guide our work. Behind every data point is a child sitting with a problem they cannot yet solve. What we owe them is not just better teaching materials: it is the curiosity to ask why, and the humility to let the answer change what we do next.
Neha Raheel is Director of Teaching and Learning at the Luminos Fund where she leads the development of high-quality teaching and learning materials grounded in the latest evidence from the science of teaching. She also designs and delivers impactful training on foundational learning and strengthens the capacity of colleagues and partners to help every child learn. Prior to joining Luminos, Neha worked at the World Bank, where she supported the Education Global Practice team on curriculum, instruction, and learning, focusing on foundational learning.
