Theory Building: The Missing Piece in Math Curricula

India's National Education Policy of 2020 says that education must develop "higher-order cognitive capacities, such as critical thinking and problem solving." The National Curriculum Framework for School Education, released in 2023, calls for a shift from rote learning to "discovery and discussion-based learning" in mathematics. It mentions the need for abstraction, structuration, generalisation, argumentation, and reasoning. It wants students to develop mathematical understanding and the ability to recognise the world through quantities, shapes, and measures.

In the United States, the Common Core State Standards list eight Mathematical Practices that students should develop. Practice 3 is "Construct viable arguments and critique the reasoning of others." Students should, the standards say, understand and use stated assumptions, definitions, and previously established results in constructing arguments. They should make conjectures and build a logical progression of statements to explore the truth of their conjectures. They should be able to analyse situations by breaking them into cases and can recognise and use counterexamples. NCTM's Principles to Actions goes further, calling for teaching practices that "promote reasoning and problem solving" and recognising reasoning and proof as "fundamental aspects of mathematics."

Singapore's Mathematics Curriculum Framework — the well-known Pentagon framework — places mathematical problem solving at the centre, surrounded by five components: concepts, skills, processes, attitudes, and metacognition. Under "processes," it specifically emphasises reasoning, communication, and connections — the ability to analyse mathematical situations, construct logical arguments, and see links between mathematical ideas and the real world.

The International Baccalaureate's Mathematics curriculum, particularly the Analysis and Approaches course, treats proof and abstract reasoning as central. The IB also has Theory of Knowledge as a required component, where mathematics is one of the areas of knowledge students examine — they are asked to consider how mathematical truths are constructed, what counts as proof, and whether mathematical knowledge is discovered or invented.

I am listing all of this to make a simple point: the curricula already say what I want them to say. The goals I care about — reasoning, argumentation, making and evaluating conjectures, understanding the structure of mathematical knowledge — are written into the policy documents of multiple countries and international frameworks. The problem is not that the goals are absent. The problem is that the actual teaching, in most places, does not deliver on them.

What happens in practice

Consider how geometry is taught in the NCERT textbooks used across most Indian schools. Euclidean geometry is introduced formally in the 9th grade. Students are given definitions, axioms, and proofs of the important theorems. They are asked to come up with proofs themselves, but only for less central results. Conjecturing is not explicitly encouraged. The textbooks present mathematics as a finished product — a set of results to be learned and reproduced — rather than as a process of inquiry.

This is not unique to India. In most school systems I am aware of, the experience of doing mathematics in a classroom looks roughly the same: the teacher presents a concept, demonstrates a procedure, and students practise the procedure on similar problems. The emphasis is on getting the right answer using the right method. Students rarely get the chance to define a concept themselves, to propose a conjecture and test it, to trace a chain of reasoning back to its assumptions, or to ask why we study one type of object and not another. The mathematical practices that the standards documents describe — constructing arguments, critiquing reasoning, making use of structure, attending to precision — remain largely aspirational.

I do not think this is because teachers are unaware of these goals or because they do not value them. It is, in large part, a structural problem. The curriculum is packed with content that must be covered. Assessments reward correct answers, not the quality of reasoning. Textbooks present finished mathematics, not the process by which mathematics is constructed. And teacher training, in most cases, does not prepare teachers for the kind of facilitation that genuine mathematical inquiry requires.

What theory building adds

Theory building, as I use the term, refers to a set of activities in which students construct mathematical knowledge rather than receiving it. This involves defining objects carefully, choosing which assumptions to start from, proving claims using those assumptions, classifying objects, extending definitions to new contexts, and putting everything together into a coherent structure. These are the activities that working mathematicians actually engage in, and they are also the activities that the various standards documents describe — even if the documents do not use the phrase "theory building."

Let me be specific about what theory building involves that most curricula do not.

The first is defining. Most mathematics classes treat definitions as given — the teacher or the textbook provides the definition, and students use it. But the process of choosing a good definition is itself a rich mathematical activity. When students have to decide whether to define even numbers by their last digit or by divisibility, they are engaging with questions about what makes a definition illuminating versus merely functional. When they have to decide whether a square should count as a rectangle, they are reasoning about the consequences of classificatory choices. This kind of work develops the capacity to ask "what do we mean by this?" — a question that is valuable in every domain, not just mathematics.

The second is assumption digging. Most proofs in school textbooks are presented as finished objects. Students may be asked to reproduce a proof or to prove a minor result, but they rarely get to trace a chain of reasoning all the way down to its foundations. Assumption digging — the process of asking "why should I believe that?" recursively — develops the habit of looking for hidden assumptions and testing whether each step of an argument actually follows. This is what Common Core Practice 3 describes when it talks about using "stated assumptions, definitions, and previously established results in constructing arguments." But in practice, most students never do it.

The third is conjecturing. The Common Core standards say students should "make conjectures and build a logical progression of statements to explore the truth of their conjectures." But in most classrooms, there is no space for conjecturing. The results are known in advance, and the student's job is to verify them, not to discover them. In the discrete geometry module I designed, students had to conjecture which worlds allow every straight line to be bisected, and then test their conjectures against specific cases. The process of forming a conjecture, testing it, finding it wrong, revising it, and testing again is central to mathematical practice — and almost entirely absent from most curricula.

The fourth is extending definitions. When a concept is moved from one context to another — from Euclidean geometry to discrete geometry, say, or from whole numbers to integer rings — definitions that were equivalent in the original context can come apart. Students have to choose which properties to preserve. This is listed nowhere, as far as I can tell, in any of the curriculum documents I have read. But it is one of the most important things mathematicians do, and it develops a sophisticated understanding of what definitions are and why they matter.

What the standards are missing

Even the best curriculum standards — Common Core, NCF, the IB — are missing some things that I think matter.

None of them, as far as I can tell, explicitly mentions the process of choosing between competing definitions. The Common Core mentions "attending to precision," which comes close, but the emphasis is on using precise language rather than on the process of deciding what language to use. The NCF mentions "abstraction, structuration, and generalisation," which is in the right spirit, but does not unpack what these look like in practice.

None of them mentions extending definitions to new contexts. This is understandable — it is a more advanced activity — but it is also one of the most revealing. The moment when students discover that two equivalent definitions come apart in a new context teaches them something deep about the nature of mathematical concepts.

And none of them, as far as I can tell, gives serious attention to the question of what makes a concept worth studying. Why do we care about parallelograms and not about "quadrilaterals with any two sides equal"? The philosopher Jamie Tappenden discusses this in terms of fruitfulness — we name and study objects that lead to significant consequences. This question — "is this concept worth paying attention to?" — is not just a mathematical question. It is a question about how we organise knowledge in any domain. And it is absent from the standards.

I want to be careful here. I am not saying that the standards are bad. They are, in many ways, remarkably good. The Common Core mathematical practices, for instance, describe habits of mind that genuinely matter. The NCF's emphasis on reasoning and argumentation is exactly right. The IB's inclusion of Theory of Knowledge alongside mathematics is a genuinely innovative structural choice. My point is that even these ambitious documents stop short of what a full theory building approach would involve.

There is precedent for this

In 1938, a mathematics educator named Harold Fawcett ran an experiment in which high school students constructed Euclidean geometry from the ground up. Students listed geometric terms they had heard of, identified properties, chose which statements to accept as axioms, defined the remaining terms, and proved theorems within the system they had built. The results were striking. Students who went through this course not only learned how to prove — they also learned the same amount of geometry content as students in traditional classes, as demonstrated by standardised tests administered afterwards. And the students judged the course to have had a significant impact on their lives many years later. Perhaps most interestingly, students began using the same thinking tools they had developed in the course to evaluate things outside of mathematics, including legislation.

Fawcett's experiment was published over eighty years ago. It demonstrated that students can construct axiomatic systems, that doing so does not come at the cost of content knowledge, and that the thinking habits developed through this process transfer to other domains. This is not a new idea. The question is why it has not been taken up more widely.

Part of the answer, I think, is practical. A theory building course requires a different kind of facilitation than a traditional mathematics class. The teacher cannot simply present material and check answers. They need to respond to student reasoning in real time, to spot interesting arguments (whether correct or not), to know when to push and when to step back. This is harder than lecturing, and most teacher training does not prepare for it.

Another part of the answer is structural. Curricula are packed with content. Adding theory building would mean removing something else, or at least reducing the emphasis on it. This is where I think the calculus question becomes relevant. There is no reason that calculus needs to be introduced in high school to everybody who does mathematics. For the vast majority of students — those who will not pursue STEM careers — the time spent on calculus could be spent on thinking abilities that would be useful in their personal, professional, and public lives. Theory building is one candidate for what could fill that space.

What this would require

I am not proposing that every mathematics class become a theory building class. There is value in learning procedures, in practising computation, in developing fluency with mathematical techniques. What I am proposing is that theory building — the process of constructing mathematical knowledge rather than receiving it — deserves a significant place in the curriculum. Not as an enrichment activity for advanced students, but as a core part of what it means to learn mathematics.

This would require changes at multiple levels. Curriculum documents would need to explicitly include theory building activities alongside content goals. Textbooks would need to present mathematics as a process, not just a product. Assessment would need to evaluate the quality of reasoning, not just the correctness of answers. And teacher education would need to prepare teachers for the kind of facilitation that theory building demands — a significant undertaking, but not an impossible one.

The irony is that the standards already call for much of this. The Common Core wants students to construct arguments and critique reasoning. The NCF wants abstraction, structuration, and argumentation. The IB wants students to understand how mathematical knowledge is built. The gap is not in what the standards prescribe but in what actually happens in classrooms. Theory building, as an approach, offers a concrete way to close that gap — not by adding new goals to the curriculum, but by taking the existing goals seriously and designing activities that actually achieve them.

I should acknowledge that my own experience is limited. I have designed and implemented one course, in two schools, with a specific population of students. Whether theory building can work at scale — in government schools, with large class sizes, with teachers who have not been trained in this approach — is an open question. But the precedent set by Fawcett, the alignment with existing standards, and the experience of my own students all point in the same direction: this is worth trying, and the curricula are already telling us so.