By: Neha Raheel

Neha Raheel, Director of Teaching and Learning, Luminos Fund

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)

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.

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.

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?

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.

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.

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.