What Happened When We Let Students Build Their Own Math

In the winter of 2019, I taught a course on theory building to thirty students across two schools in Pune. The students were between 12 and 15 years old. The course ran for nine sessions of about ninety minutes each, spread over a few weeks. Each school had about fifteen students. The schools — I will call them Ganga and Indus — are relatively low-cost private schools affiliated to the CBSE board, owned by the same non-profit. Both use NCERT textbooks. Euclidean geometry is introduced formally in the 9th grade. The students had been given definitions, axioms, and proofs to all the important theorems in their regular classes, but conjecturing was not explicitly encouraged.

I had worked with many of these students before. Most of the Ganga students had been through a course I had run in 6th grade, and three had participated in a workshop connected to my Master's thesis. At Indus, eight students had done the 6th-grade course and all but five had been in the earlier workshop. This matters because I cannot separate the students' engagement with the material from their relationship with me as a facilitator. They knew me, they knew roughly what to expect, and they may have been more willing to engage with unfamiliar activities because of that prior trust. This is one of the limitations of the implementation, and I want to name it upfront.

The course had three modules. The first was a definition-guessing game called Podgons, which served as an icebreaker. The second was Triangle Theory Building, where students listed everything they believed about triangles and tried to justify those beliefs. The third was Discrete Geometry, where students worked in worlds with finitely many points and had to extend familiar geometric concepts into unfamiliar territory. The modules were interleaved — we did not finish one before starting the next — and I was joined by a friend, Mohanan, for the first session at Ganga, and by Sabareesh, an undergraduate, for several sessions at both schools.

What follows is an account of what happened — what worked, what did not, and what surprised me.

The podgon game

The podgon module was the most straightforwardly successful part of the course. I invented a word — "podgon" — and held two secret definitions. Students proposed shapes and I told them whether each was a podgon under each definition. Their job was to figure out the definitions.

Seven of sixteen Indus students and four of ten Ganga students listed this session among their favourite parts of the course. No student at either school listed it among their least favourite. The activity has a natural game-like quality that draws students in, and the thinking it requires — proposing definitions, designing test cases, interpreting results — sneaks up on them. They are doing something mathematically substantive without it feeling like a maths lesson.

Two moments stand out. At Ganga, Mohanan stopped a student named Anjali who had proposed testing a semicircle: "When you ask a question, you should know what it is you want to find out." Anjali thought about it and said she wanted to know whether curved sides count. That intervention — asking students to articulate the purpose of their question before getting an answer — shaped the rest of the session. At Indus, a student named Imran drew an open figure, learned it was not a podgon, and immediately concluded that all podgons must be closed. That does not follow — he had only shown that one particular open shape was not a podgon. But his error was productive. It gave us something concrete to discuss about the gap between a single example and a general conclusion.

Triangle theory building: the surprise

The triangle theory building module produced the biggest surprise of the course — not in terms of what students learned, but in how differently the two schools reacted to it.

The activity was straightforward in concept. Students listed claims they believed to be true about triangles — things like "the sum of angles is 180 degrees," "angles opposite equal sides are equal," "the side opposite the greater angle is greater." Then they tried to justify these claims using each other. Which claims can you prove from which others? What do you need to assume? Where do the chains of reasoning end?

At Indus, twelve of sixteen students mentioned this module among their favourites. Not a single student mentioned it among their least favourites. Uday said it "intrigued us to prove things we normally would not question." Meghna said she "enjoyed being asked to define everything." Tushar said he "liked the way we figured out the interdependency between properties." Students at Indus seemed genuinely engaged by the process of connecting claims to each other — of seeing that the area formula for a triangle rests on claims about parallelograms, which rest on claims about rectangles, which rest on assumptions about area.

At Ganga, the picture was very different. Only five of fifteen students mentioned the module among their favourites, while five listed it among their least favourites. Gauri said it was "a repetition and too easy for my grade." Anuj said "we kept going endlessly." Anjali said it "gave minimal opportunity for finding alternatives and doing something different/new."

I did not fully understand this asymmetry then, and I am not sure I do now. There were pedagogical differences between the two schools — the sessions were not identical, because I was responding to what students gave me. The initial claims students proposed were different, which led the discussions in different directions. The time of day may have mattered. It is even possible that one or two influential students shaped the overall mood. But I think the most likely explanation is that the two groups had different thresholds for what they found engaging. The Ganga students, some of whom had more prior experience with this kind of work, wanted something novel. The Indus students, many of whom were encountering this kind of thinking for the first time, found the process of proving and connecting claims genuinely eye-opening.

The most interesting student reaction came from Anya at Ganga. When asked what she valued about the module, she said: "finding proofs, proving them wrong, again finding another proof." Anuj, who listed the module among his least favourites, said the problem was that "we kept going endlessly." These are opposite reactions to the same experience. Anya found the iterative process rewarding. Anuj found it exhausting. I do not think either of them was wrong about what the experience felt like. The question for me is how to design activities that are more reliably engaging across different student temperaments — and I do not have a good answer to that yet.

Discrete geometry: the mirror image

The discrete geometry module — where students worked in worlds with exactly six points and had to define straight lines, bisection, circles, and triangles — produced almost exactly the opposite pattern.

At Ganga, eight of fifteen students listed it among their favourites and only one among their least favourites. Gauri, who had found the triangle module too easy, said discrete geometry was "very different in terms of Euclidean geometry and challenging for the mind." Anjali, who had found the triangle module lacking in alternatives, said discrete geometry "encouraged us to find alternatives and loopholes and also use our creativity." The students who were bored by the triangle module came alive when given the chance to create something new.

At Indus, only two students listed discrete geometry among their favourites, while seven listed it among their least favourites. Arun said he "was not able to understand the different types of circles in different worlds." Aryan said he could not help his group because he "least knew about discrete geometry." Arvind said "we were digging a lot into stuff which made me feel that it was complicated."

The challenge was the same at both schools. What differed was how students experienced it. And here is the thing that struck me: the students who had a positive reaction and those who had a negative reaction often described the same feature of the module. Devika at Indus said, "It made me think outside the box. When we used to define what lines, points, etc. are, I used to feel kinda irritated because we never got it right, but it was really fun." The irritation and the enjoyment came from the same source — the fact that familiar concepts behaved differently in these new worlds.

The discrete geometry module builds on itself more than the other modules. If a student missed something in the first session — say, the idea that distance in these worlds is counted by connections — they were lost for the rest. I did not have good scaffolding for students who fell behind. This is the biggest pedagogical weakness of the module, and it is something I would need to address before running it again.

What students found valuable

At the end of the course, I asked students what they found valuable. Four themes emerged.

The first was that the course made them think. Every student who responded mentioned this in some form. Imran said the most valuable lesson was "to never stop thinking and a question may have many answers." Zoya said she liked "how they taught us to be curious and try to comprehend what, how, and why for everything."

The second was intellectual scepticism — not taking things for granted. Tanya said the course taught her "how we shouldn't stick with textbook knowledge and we should prove or justify the things we know to understand it deeper." Zoya said, "Earlier my mind used to believe everything that came in front of me. But now I ask questions like why." Prerna 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." Eight of fifteen Indus students explicitly mentioned not taking textbook knowledge for granted.

The third was understanding mathematics differently. Vandana said, "Because of this workshop I was able to understand the real depth of mathematical concepts and its real meaning." Tanya said, "We understood how mathematicians build up chains of derivations and conclusions from basic knowledge or axioms." Pankaj said, "I actually came to know that no one proof or theorem is enough — we need more and more."

The fourth — and this was not a learning outcome I had designed for — was teamwork. Several students mentioned the value of working in groups. Sanya said, "We all would speak and not textbook knowledge but share different views and opinions about everything." Aryan said, "When we together solve a problem we look at the thing from different angles which is really valuable." I do not want to overstate this, since students may say positive things about group work in any course survey. But it came up frequently enough that I think the collaborative aspect of the activities mattered to students in ways I had not fully anticipated.

What I would change

The course had a pre-test and a post-test, though the instrument had not been validated and the sample size is too small to draw strong conclusions. There was a clear increase in scores, especially on a question about definitions. But I am cautious about reading too much into this.

What I am less cautious about is the list of things I would change. I would pay more attention to group composition — several students raised concerns about uneven participation, and allowing students to form their own groups led to some groups where a few dominant students did most of the talking. I would add more scaffolding to the discrete geometry module, particularly intermediate steps for students who fall behind. I would think more carefully about pacing — some students wanted more time ("increased time, up to three hours, increased frequency, up to four to five times a week," as Karan put it), while others felt discussions went on too long without resolution.

I would also be more deliberate about the things I missed in the moment. There were instances where students said something interesting that I did not explore — Vivaan's remark about transitivity being an assumption, or moments where a student's error could have been turned into a richer discussion if I had caught it in time. A facilitator who has done this more times, or who has a clearer mental map of the possible directions a conversation can take, would catch more of these. I caught some. I missed others.

What I think I can say

I do not want to generalise from a single implementation in two schools with a facilitator who designed the materials and knew many of the students personally. The conditions were about as favourable as they could be. Whether the same activities would work with different students, different teachers, and different institutional constraints is an open question.

But a few things seem clear to me. First, 12-to-15-year-old students can engage meaningfully with theory building activities — with defining, classifying, proving, conjecturing, and extending concepts. This is not beyond them. Second, different students respond very differently to different types of mathematical activity. The module that one group found boring, another found revelatory. This suggests that a course on theory building needs variety — different kinds of activities that appeal to different kinds of thinkers. Third, students value being asked to think rather than to reproduce. The most common theme in the feedback was that the course made them think, and several students contrasted this with their regular mathematics classes, where they "use theorems blindly." Fourth, the facilitation is hard. Harder than lecturing, harder than running problem sets. It requires responding to student reasoning in real time, knowing when to push and when to let things sit, and being willing to learn from the students.

These are modest claims. But they are honest ones, and I think they are enough to suggest that the approach is worth pursuing further — in different contexts, with different students, and by different facilitators. What happened in those nine sessions in Pune was imperfect and preliminary. But it was real, and the students knew it.