What Does It Actually Look Like to Teach This Way?

In the middle of a session on definitions, a student named Anjali proposed testing whether a semicircle is a "podgon" — a made-up word whose definition the class was trying to figure out. Before I could respond, my colleague Mohanan, who was co-facilitating, stopped her. "Hold on," he said. "When you ask a question, you should know what it is you want to find out. So when you ask the question 'is a semicircle a podgon?', what do you want to find out?" Anjali thought for a moment and said, "Whether a curved side counts in a podgon."

That intervention — asking a student to articulate the purpose of their question before getting an answer — captures something important about what it means to facilitate theory building. Mohanan was not correcting Anjali. He was not telling her the answer. He was asking her to think about her own thinking. The semicircle was a perfectly reasonable thing to ask about, but asking about it without knowing why you are asking is a missed opportunity. Once Anjali knew she was testing whether curves matter, she could interpret the answer — yes or no — in a way that actually moved her understanding forward.

I have been running theory building activities with 12-to-15-year-old students in Pune, and the single hardest thing about it is not the mathematics. It is knowing when to intervene and when to stay quiet. When to push a student's thinking and when to let a confused idea sit for a while. When to give information and when to withhold it. I have not fully figured this out. But I have learned some things, and I want to share them honestly — including the parts I got wrong.

The facilitator as learner

In a traditional mathematics class, the teacher knows the answer and the student's job is to reach it. In a theory building class, the situation is different. The teacher knows some things the students do not — the definitions behind a game, or the structure of a proof — but the students will regularly say things the teacher has not anticipated. They will propose definitions the teacher has not considered, make arguments the teacher needs to think through in real time, and ask questions the teacher cannot immediately answer.

This means the facilitator's role is closer to what happens in problem-based learning than in a lecture. You are not delivering content. You are modelling what it looks like to think carefully — to ask for clarification, to notice when an argument has a gap, to say "I am not sure about that, let me think." The goal, eventually, is to become irrelevant. Once students are able to question each other's reasoning, propose their own definitions, and evaluate their own arguments, the facilitator can step back. But reaching that point takes time, and in the meantime, the facilitator has to do a lot of work that is easy to describe and hard to execute.

One thing I have come to believe is that a facilitator needs to be genuinely willing to learn from students and to be wrong. This is not a platitude. In several sessions, students said things that I initially dismissed or misinterpreted, and it was only later — sometimes while writing up my notes — that I realised they had been making a more interesting point than I had given them credit for. A student named Vivaan, during a discussion about why angle A equals angle C given that both equal angle B, said something along the lines of "if we work like that, then this follows." At the time, I moved on. In retrospect, he was identifying an axiom — saying that transitivity of equality is an assumption we choose to accept. I wish I had paused and explored that with the class. Without a facilitator who is willing to work with students to figure things out together, it is hard to see how a course like this could achieve its goals.

Productive struggle and its limits

The theory building activities are designed to create productive struggle — to place students in situations where they experience what Dewey calls "perplexity, confusion, or doubt" and have to work their way through it. Polya puts the tension well: the student should acquire as much experience of independent work as possible, but if left alone with no help, they may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little.

This sounds reasonable in the abstract. In practice, the line between productive and unproductive struggle is hard to see in real time. During the discrete geometry module, some students found the challenge engaging and creative. Devika said it made her "think outside the box" and that even the irritation of "never getting it right" was fun. Anjali said it encouraged her to find alternatives and loopholes. But other students found the same module opaque and frustrating. Arvind said he felt the material was too complicated. Arun said he could not understand the different types of circles in different worlds. Tushar raised a concern about group dynamics: "When a student does not understand something that is being discussed, he is completely left out of the discussion, and it makes it hard to understand things related to it later."

The challenge was the same in all these cases. What differed was how students experienced it — and that depended, I think, on factors I could not fully control: their comfort with ambiguity, their prior experience with open-ended mathematical work, and the dynamics within their group. This is an area where I think my implementation fell short. The discrete geometry module, in particular, builds on itself — if a student misses something in the first session, they can be lost for the rest. I did not have good strategies for catching students who had fallen behind without slowing down those who were engaged.

One thing I have taken from this is that theory building activities need to be accessible at multiple levels, as Sullivan (2016) discusses in the context of challenging mathematical tasks. A task should have a low floor — everyone should be able to start it — and a high ceiling — there should be depth for students who want to push further. The podgon module does this reasonably well: any student can propose a shape and get a yes or no answer, and the thinking gets more sophisticated as the game progresses. The discrete geometry module is less successful on this front. I think it needs more intermediate steps and more scaffolding for students who are struggling, without removing the challenge for those who are not.

Facilitation moves that matter

Over the course of teaching this material, certain facilitation moves came up repeatedly. I want to describe a few of them concretely.

The first is asking students to articulate the purpose of their actions. This is what Mohanan did with Anjali and the semicircle. In my own facilitation, I would ask things like "why are you asking about that shape?" or "what would you learn if the answer is yes? What would you learn if it is no?" When a student proposes a test case that would give the same information regardless of the answer, that is worth pointing out — gently. "If you learn the same thing whether the answer is yes or no, why ask?" This is not a criticism. It is an invitation to think more strategically.

The second is resisting the urge to interpret what students say. There were several times in my implementation where I said things like "you mean..." and then offered an interpretation that had only a tenuous relationship to what the student had actually said. Early in a course, when students are still figuring out how to express mathematical ideas, this is risky. They may accept your interpretation even though it is not what they meant, because you are the authority figure. A better strategy, I have come to think, is to probe rather than interpret — to ask "what do you mean by that?" and if the student cannot clarify, to offer multiple interpretations and let them choose. This preserves their ownership of the idea.

The third is knowing when to give information. In theory, a fully student-driven discovery process is ideal. In practice, there are moments where withholding information is counterproductive. When students in the triangle theory building module encountered circular reasoning — defining straight lines in terms of collinear points and collinear points in terms of straight lines — they could identify the circularity but could not see a way out. At that point, I gave them the concept of undefined terms and axioms. I did not derive it from their work. I told them. But I told them after they had experienced the need for it, which I think made the concept land differently than it would have if I had introduced it at the start.

The balance between following students' direction and steering toward learning outcomes is something I struggled with throughout. As a general principle, students ought to be the ones constructing knowledge, and it is valuable to follow the direction they take. But doing so can mean missing important ideas that they do not happen to stumble upon. And it requires the facilitator to react in real time to claims and arguments they may not have seen before. One approach that seems promising — though I have not tested it systematically — is to start with a facilitator-controlled discussion and gradually hand control to students as they become more comfortable with the process.

The language question

During the implementation, I avoided mathematical terminology as much as possible. I barely used words like "axiom," "undefined entity," "degenerate," or "proof by contradiction." The idea was to get students to understand the concepts before putting labels on them. If a student arrives at the idea that some terms must be left undefined without knowing the word "undefined term," they understand the concept in a way that someone who has merely memorised the word does not.

But there is a cost to this. Words are thinking tools. Having a label for an idea makes it easier to refer to, to build on, and to connect with other ideas. Imagine not having a word for "triangle" and constantly having to think in terms of the full definition whenever you want to use the concept. Both of these considerations — understanding before labelling, and labelling as a tool for thought — need to be balanced. In retrospect, I think I waited too long to introduce some terminology. Once a concept has been understood through experience, giving it a name reinforces and consolidates that understanding. The timing matters more than whether to use the terminology at all.

Who gets to participate

Mathematics education, as it exists in many places, excludes a lot of people. In India, the students who tend to do well in mathematics are those who can withstand uncreative, repetitive problem solving following pre-set rules they have to memorise. Those who can afford expensive tuitions have a significant advantage over the rest of society, especially those from historically marginalised groups who do not tend to have a history of formal education in their families. The system is focused on those who do well in it, largely ignoring those who do not.

Theory building, as an approach to mathematics education, has the potential to open up a different kind of mathematical experience — one that does not depend on memorisation or speed. When the task is to propose a definition, evaluate a conjecture, or trace an assumption, the skills involved are different from those that traditional math classes select for. This does not mean theory building is automatically equitable. Group dynamics can still be exclusionary. Students who are more articulate or more confident can dominate discussions. Those who are quieter or less sure of themselves can be sidelined, especially in the kinds of open-ended conversations that theory building involves.

I saw this in my implementation. Some students were consistently active participants — proposing ideas, challenging each other, building on previous contributions. Others were quieter. The group structure helped in some ways (smaller groups give more space to speak) and hurt in others (if the dominant students in a group are moving fast, a struggling student can be left behind entirely). I do not have a good solution to this. It is a problem that extends well beyond theory building and into the general question of how to run equitable discussions in any classroom. But it is worth naming, because the enthusiasm around student-led mathematical inquiry can obscure the fact that "student-led" does not automatically mean "led by all students."

Mindsets that matter

I have focused mostly on the intellectual tools of theory building — defining, proving, assumption digging, classification. But I have come to believe that these tools are not enough on their own. They require certain dispositions to be useful: a willingness to question your own beliefs, a tolerance for being wrong, and the habit of subjecting claims — including your own — to critical evaluation rather than accepting them because they come from an authority figure.

These mindsets are not something a single course can reliably develop. They require sustained practice and a culture that values them. And there is a further complication: the tools of knowledge construction can be used for good or bad ends. Someone trained in rigorous argumentation can use those skills to defend positions they know to be false, or to find flaws in others' arguments while refusing to examine their own. Without an ethical dimension — a genuine concern for truth and for the consequences of one's reasoning — the thinking tools on their own may not take us very far.

I mention this not because I have solutions but because I think it is dishonest to talk about teaching critical thinking without acknowledging that critical thinking alone is not enough. The intellectual tools need to sit alongside certain habits of character. Developing those habits is outside the scope of what I have worked on in depth, and it deserves serious attention from others.

What I would do differently

If I were to run this course again, there are several things I would change. I would build in more intermediate scaffolding for the discrete geometry module, so that students who are struggling have a way back in. I would introduce mathematical terminology earlier — not at the start, but sooner after students have grasped the underlying concepts. I would think more carefully about group composition, trying to ensure that each group has a mix of students who are likely to participate at different levels. And I would spend more time on the transition between modules, making explicit the connections between the different aspects of theory building rather than assuming students would see them on their own.

I would also pay more attention to the moments I missed. The facilitator's job is partly to notice when something important is happening and to create space for it. Vivaan's remark about transitivity was one such moment. There were others. A facilitator who is better prepared — who has thought through more of the possible directions a discussion could take — would catch more of these. But there is a limit to how much you can prepare for, since the whole point is that students are doing genuinely open-ended work.