What If Math Class Taught You How to Think, Not Just How to Calculate?
How many fingers do you have? If you said ten, some people will disagree. They will insist you have eight fingers and two thumbs. The debate can get heated, but there is no real disagreement here. The answer depends on what you mean by the word "finger." If thumbs count as fingers, you have ten. If they don't, you have eight. There is nothing to fight about.
This sort of pseudo-disagreement comes up all the time. In family arguments, in political debates, in conversations online. People talk past each other because they are using the same word to mean different things, and nobody stops to notice. Is a person who eats fish a vegetarian? Is taxation theft? Is pluto a planet? Lurking beneath many of these disputes is a question that sounds simple but turns out to be surprisingly rich: what exactly do we mean by this word?
Noticing this is, at its core, a mathematical skill. It is also one that most math classes never teach.
In a typical math class, students learn formulas. The area of a triangle is half base times height. They practise procedures — solve for x, apply the quadratic formula. They memorise theorems and reproduce proofs they have been given. What they rarely do is ask why. Why is the area formula what it is? Why do we define triangles the way we do? Why should we believe that the angles of a triangle add up to 180 degrees, and what are we assuming when we say that?
These are not obscure philosophical questions. They are the questions that working mathematicians actually spend their time on. And I have come to believe that they develop ways of thinking that are useful to everyone, not just future mathematicians.
In How We Think, Dewey says that the role of education is to "cultivate deep-seated and effective habits of discriminating tested beliefs from mere assertions, guesses, and opinions." If we accept Dewey's vision, we need to ask what role learning mathematics can play in developing those habits. One possible answer lies in what I call theory building.
Theory building and what it involves
When mathematicians build a theory, they are not just solving individual problems. They are constructing a coherent body of knowledge — defining objects carefully, figuring out what can be proved from what, choosing which assumptions to start from, and organising everything into a structure that makes sense. This involves a set of activities that most students never encounter in school, and I want to walk through some of them.
Consider the number 6. Is it even? Obviously. But why? You could define "even" as "a number whose last digit is 0, 2, 4, 6, or 8." Or you could define it as "a number divisible by 2." In ordinary arithmetic, both definitions give the same results. But the second one tells you something deeper about what evenness actually is. It does not depend on the arbitrary choice of base 10. It gets at the heart of the concept. This distinction — between a superficial characterisation and a definition that reveals the nature of a thing — matters far beyond mathematics. When we debate whether artificial intelligence is "creative" or whether a particular policy is "fair," we are dealing with competing definitions, whether we realise it or not.
Or take a different sort of question. Why do mathematicians care about parallelograms? Parallelograms have a rich set of interesting properties: their opposite sides are equal, their opposite sides are parallel, and their diagonals bisect each other. These are three different characterisations of the same object, and that richness is what makes parallelograms worth studying. Compare that with "quadrilaterals with any two sides equal." You probably cannot say much about those that is not already true of quadrilaterals in general. The ability to ask "is this concept worth paying attention to?" is valuable whenever you are trying to make sense of a complex domain, whether that is biology, law, or your own experience.
Then there is the question of assumptions. Every claim rests on other claims. The area of a triangle is half base times height — but why? Well, it has to do with the relationship between triangles and parallelograms. And why is the area of a parallelogram base times height? Because of how parallelograms relate to rectangles. Keep asking why and at some point you hit bedrock — the assumptions you are choosing to start from. I call this process "assumption digging," borrowing a metaphor from archaeology. The ability to trace a chain of reasoning down to its foundations, and to notice when a link in that chain is missing or flawed, matters everywhere. It matters when evaluating a news article, when reading a scientific paper, and when trying to understand why you believe something.
There is also the matter of classification. Is a square a rectangle? It depends on your definition of rectangle. If a rectangle is any quadrilateral with four right angles, then a square is a rectangle. If you require that the sides not all be equal, then it is not. This is not trivial. If squares count as rectangles, then anything you prove about rectangles automatically applies to squares. You get what mathematicians call logical inheritance, which saves you work and reveals structure. The choice of classificatory system shapes what you can see and what you can prove. The same sort of thing comes up in biology — are humans animals? — and in everyday reasoning.
None of these activities feature prominently in most mathematics classrooms. Students learn to solve problems and reproduce proofs, but they rarely get the chance to define, classify, choose assumptions, or construct a theory from the ground up. I think this is a significant gap.
What this looked like with actual students
I designed a course on theory building and taught it to 12-to-15-year-old students in two schools in Pune, India. The course had three modules, each working through a different aspect of theory building, all within geometry.
In the first module, I invented a word — "podgon" — and gave students examples of shapes that were and were not podgons according to two different secret definitions I held. Their job was to figure out the definitions by proposing test cases. It sounds like a game, and it is, but the thinking it requires is not trivial. Students had to propose definitions consistent with the evidence, then invent shapes that would help them distinguish between competing possibilities. They had to understand what a counterexample does — it eliminates a definition — and what it does not do. One student saw that a single open shape was not a podgon and immediately concluded that podgons must be closed. But that does not follow. All he had shown was that one particular open shape was not a podgon. Another open shape might have been. This is a subtle but important distinction, and the session gave students a chance to bump into it.
In the second module, students listed everything they believed to be true about triangles and then tried to prove those beliefs using each other's claims. This is assumption digging in action: tracing statements back through their justifications to the assumptions they rest on. Students discovered that some of their "obvious" beliefs were difficult to justify. They encountered circular reasoning — defining straight lines in terms of collinear points and collinear points in terms of straight lines — and had to figure out how to escape it. They bumped into the need for undefined terms and axioms, not because I told them about these concepts, but because they needed them.
In the third module, I asked: "In a world with exactly six points, can every straight line segment be bisected?" The question is deliberately ambiguous. Students had to decide what "straight line" and "bisection" even mean in such a world. They had to extend familiar definitions into unfamiliar territory, discovering along the way that some properties of Euclidean geometry carry over while others do not. They were, in effect, creating new mathematical worlds and exploring the consequences.
What students said
When asked what they found valuable about the course, every student mentioned that it made them think. But many went further. Zoya from one of the schools said, "Earlier my mind used to believe everything that came in front of me. But now I ask questions like why." Tanya said, "The course taught us not to assume the given facts to be true. We should cross-check and go to the root of things to understand them better." Another student observed, "We understood how mathematicians build up chains of derivations and conclusions from basic knowledge or axioms."
These students were not learning to compute faster. They were learning to question assumptions, to demand reasons, and to see that knowledge has structure. Whether they end up in mathematics or not, these seem like valuable habits of mind.
Of course, I should be careful about drawing too much from this. The course was implemented once, in two schools that are part of the same group, with a single facilitator — me — who also designed it. Many of the students had worked with me before. It is impossible to separate their relationship with me from their engagement with the materials. At best, this serves as something like an existence proof: it is possible for 12-to-15-year-olds to engage meaningfully with these ideas. Whether the same would hold in other contexts is something that needs further exploration.
Beyond mathematics
The thinking tools developed through theory building are not confined to mathematics. Classification in mathematics shares significant similarities with classification in biology. Reasoning from axioms in mathematics bears a structural resemblance to reasoning from principles in ethics. The process of deducing consequences from a scientific hypothesis mirrors the process of deducing theorems from mathematical assumptions, with the important difference that scientific theories must also answer to observation.
I have previously done activities that make these connections explicit. One involves asking students to classify biological organisms after classifying shapes, noticing how the considerations are similar — purpose of classification, logical inheritance, simplicity. Another involves what I call "Theoretical Anatomy," where students try to figure out how the human body works purely from external observation, mirroring the process of scientific theory construction. A third involves ethical reasoning: given two competing ethical principles, deducing their consequences in a particular situation is structurally similar to deducing consequences from mathematical axioms, even though the process of resolving conflicts between principles is very different.
A single course that involves scientific, mathematical, and ethical theory building would allow students to see both the connections and the differences between these endeavours. This is what I mean by transdisciplinarity — not the intersection of two fields, but a concern for the tools of knowledge construction that cut across disciplines at some level of abstraction. Assumption digging, for instance, is useful everywhere: asking why we believe something, and then asking why we believe the assumptions made in the justification, is a major part of any discipline and of activities as ordinary as reading a newspaper.
What this asks of us
If we want mathematics education to be valuable to everyone — including the vast majority who will not become mathematicians or engineers — we need to rethink what we are teaching and why. There is no reason calculus needs to be introduced in high school to everybody who does mathematics. Rather, the focus ought to be on thinking abilities that would be useful to everyone in their personal, professional, and public lives.
Mathematics education, as it exists in many places, excludes a lot of people. The students who tend to do well are those who can withstand uncreative, repetitive problem solving following pre-set rules they have to memorise. That is not the same thing as being good at thinking. A course focused on theory building — on defining carefully, questioning assumptions, building and evaluating arguments — opens up a different kind of mathematical experience, one that does not depend on memorisation or speed.
I do not want to overstate what this course accomplishes. As I mentioned, the broader goal requires developing certain mindsets alongside the thinking tools: the habit of intellectual scepticism (not the blind rejection of authority, but subjecting beliefs to adequate critical evaluation), and the willingness to be wrong. Without these mindsets, the thinking tools on their own may not take us very far. And there is the matter of ethical sense — the tools of knowledge construction can be used for good or bad ends. These are areas I have not worked on in depth, and they deserve serious attention from others.
But I do think the basic observation stands. Mathematics has the potential to train ways of thinking that are useful to everybody, not just those pursuing mathematical careers. Theory building, as an approach to mathematics education, is one way to try to realise that potential. The question is not whether young students can engage with these ideas — they can, and I have watched them do it. The question is whether we are willing to make room for this in our curricula, alongside the formulas and procedures that currently dominate.