How to Have Mathematical Conversations at Home

A few years ago, I was at a dinner with friends and someone asked, half-jokingly, whether a hot dog is a sandwich. The conversation that followed was surprisingly engaged. People proposed definitions, found counterexamples, revised their definitions, and found new counterexamples. Someone pointed out that if a sandwich is anything between two pieces of bread, then an open-faced sandwich is not a sandwich, which seemed wrong. Someone else pointed out that if a sandwich is anything with bread around a filling, then a burrito is a sandwich, which also seemed wrong. The conversation went on for twenty minutes, and nobody resolved it.

What struck me was that the thinking involved — proposing a definition, testing it against cases, finding that it includes things it should not or excludes things it should, and revising — is exactly the kind of thinking that mathematicians do when they are defining objects. And nobody at the table thought of what they were doing as mathematics. They thought of it as a fun argument.

I think families can have conversations like this regularly, and I think they are genuinely valuable — not just as entertainment, but as practice in the kind of careful thinking that schools often do not develop. What follows are some conversations I have had with students that you could adapt for home, along with some thoughts about why they work.

Is a square a rectangle?

This is my favourite starting question for a mathematical conversation, because almost everyone has an opinion about it and almost no one has thought carefully about what their opinion implies.

Ask your child: is a square a rectangle? Some will say yes, some will say no. Do not tell them the answer. Instead, ask them to define "rectangle." They might say something like "a shape with four sides and four right angles." If that is their definition, then a square — which has four sides and four right angles — is a rectangle. But many children will resist this. A square is a square, they will say. It is not a rectangle.

This resistance is interesting, and it is worth exploring rather than dismissing. Ask them: if a square is not a rectangle, then what is the definition of rectangle? They will have to add something — "a shape with four sides and four right angles where not all sides are equal," or something like that. Point out that this is a choice. They are choosing to define rectangle in a way that excludes squares. The other choice — defining rectangle in a way that includes squares — is equally valid. But the two choices have different consequences.

If squares are rectangles, then anything you prove about rectangles is automatically true about squares. You do not have to prove it separately. This is what mathematicians call logical inheritance, and it is the main reason mathematicians prefer the inclusive definition. If squares are not rectangles, you lose that inheritance. Every theorem about rectangles has to be re-examined to see whether it also applies to squares.

You do not need to use the phrase "logical inheritance" with your child. But you can ask: if we know something about all rectangles, and a square is a rectangle, does the thing automatically apply to squares? They will see that it does. And then you can ask: is that useful? Is it worth including squares inside rectangles to get that benefit, even if it feels a bit strange?

The same question works with other categories. Is an equilateral triangle a special case of an isosceles triangle? Is a human an animal? Is a tomato a fruit? In each case, the answer depends on the definition, and the definition is a choice with consequences. I have often done exactly this move with students — starting with the square-rectangle question and then shifting to biological classification. Students notice the parallels without much prompting. The considerations are the same: what is the purpose of the classification, what do you gain from one system versus another, and what are you willing to give up.

The definition game

Here is a game you can play at home with almost no preparation. Pick an everyday object and ask everyone at the table to define it. A chair, for instance. What makes something a chair?

Someone will say: a chair is a piece of furniture you sit on. But you sit on a sofa. Is a sofa a chair? You sit on the floor. Is the floor a chair? They will revise: a chair is a piece of furniture with legs that you sit on. But a beanbag has no legs, and most people would call it a type of chair. Or would they? Is a stool a chair? What about a bench? Each proposed definition either includes something that does not feel right or excludes something that does.

This is not a trivial exercise. The difficulty of defining an everyday object reveals something important about how we use words: our intuitive categories are often fuzzy, and making them precise requires choices that are not obvious. Mathematicians deal with this constantly. When they define a concept — a triangle, a continuous function, a group — they have to decide which properties are essential and which are incidental. The same thinking applies to everyday words, and practising it with chairs and sandwiches makes the underlying skill accessible without any mathematical background.

You can make this harder by picking more abstract concepts. What is a game? Does it require competition? Does it require rules? Does solitaire count? Does a child making up rules as they go count? What is a sport? Is chess a sport? Is walking a sport? Each of these questions leads to the same process: propose a definition, test it against cases, find problems, revise.

The version of this I use in classrooms is called the podgon game, and it works like this. I invent a word — "podgon" — and hold two secret definitions. Students propose shapes and I tell them whether each shape satisfies Definition A, Definition B, both, or neither. Their job is to figure out the definitions. The game works because the two definitions overlap but are not identical, so students have to design test cases that discriminate between them. You can play a simplified version at home: one parent thinks of a secret rule for what counts as a "zibble" (or any made-up word), and the child proposes examples to figure out the rule. Is a dog a zibble? Is a cat a zibble? Is a fish a zibble? If the rule is "animals with four legs," the child has to figure this out by proposing examples and interpreting the responses. If you want to make it richer, have two people hold two different secret rules for the same word.

When I ran this with students in Pune, no student at either school listed it among their least favourite activities. The game has a natural appeal — it feels like a guessing game rather than a lesson — but the thinking it requires is genuinely mathematical. Students are proposing hypotheses, designing experiments, and revising based on evidence.

Why does the formula work?

If your child is doing geometry homework, there is a question you can ask that will almost certainly lead to an interesting conversation: why does the formula work?

Take the area of a triangle. Half base times height. Your child can probably use it. But ask them: where does the half come from? Why is it half and not, say, a third? If they are not sure, you can explore it together. Draw a rectangle. Its area is base times height. Now draw a diagonal. The diagonal splits the rectangle into two triangles. Each triangle has the same base and height as the rectangle, and each is half the rectangle. So the area of the triangle is half base times height.

This is a proof — a short one, but a real one. And it opens further questions. Does this work for all triangles, or only for right triangles? (The ones that come from splitting a rectangle are right triangles.) What about a triangle that is not right-angled? You would need to think about parallelograms, or about rearranging the triangle into a rectangle. Each step in the chain rests on something else, and tracing the chain is itself valuable.

You do not need to know the answers to these questions in advance. In fact, it is better if you do not. The conversation is more genuine when both of you are figuring it out together. When I work with students, some of the best moments come when I am genuinely uncertain about where a line of reasoning is going. The point is not to arrive at the right answer. The point is to practise the habit of asking why — of not being satisfied with knowing that something is true without understanding why it is true.

A student named Zoya, after going through a course I ran on this kind of thinking, said: "Earlier my mind used to believe everything that came in front of me. But now I ask questions like why." That shift — from accepting to questioning — is what these conversations develop.

How to ask good questions

The most useful question in any mathematical conversation is "what do you mean by that?" When your child says something that uses a word with a meaning they have not specified — and this happens constantly — asking them to clarify is not pedantic. It is the starting point for genuine thinking.

If your child says "a triangle has three sides," you can ask: what is a side? Does it have to be straight? If one side is curved, is it still a triangle? What if all three vertices are on the same straight line — is the resulting figure a triangle? These are not gotcha questions. They are the kinds of questions that arise naturally when you take definitions seriously. When I asked students to define a triangle in a world with finitely many points, three different students objected to a proposed definition for three different reasons — one said it was not closed, one said it only had one angle, one said it was really just a line segment. Each of them was grappling with what a triangle actually is, not just what it looks like.

The second most useful question is "why should I believe that?" Not as a challenge, but as a genuine request for reasoning. If your child says the angles of a triangle add up to 180 degrees, ask them why. If they say "because the teacher said so," that is an honest answer, and it tells you that they have been given a fact without a reason. If they say "because I measured some triangles," that is also honest, and it opens a conversation about whether measuring three triangles proves something about all triangles. If they actually try to explain the reasoning, you will both learn something about what the claim rests on.

The third useful question is "what if?" What if the sides did not have to be straight? What if there were only six points in the world? What if the rule were different? These questions sound playful, and they are, but they are also the questions that drive mathematical research. When mathematicians extend a concept to a new setting, they are asking "what if?" — what if we change this assumption and keep the others? The discrete geometry module I designed for students is built entirely on this question: what if the world had only finitely many points? Students found that familiar concepts — lines, circles, triangles — behaved differently, and that definitions that seemed equivalent in Euclidean geometry came apart.

What these conversations are not

I want to be honest about what I am proposing and what I am not. I am not proposing that you replace your child's mathematics education. Their school teaches procedures and content that matter, and getting fluent with computation is genuinely important. I am also not proposing that every dinner needs to become a seminar. These conversations work best when they are natural — when a question comes up and you follow it for a while, and then you stop when it stops being interesting.

I am also not claiming that having these conversations will make your child better at mathematics in the sense that schools measure. It might, but I do not know. What I do think it will make them better at is the kind of thinking that mathematics is supposed to develop but often does not: asking what words mean, asking why claims are true, testing ideas against cases, revising when things do not work. These are habits that matter in every domain — in evaluating news, in understanding disagreements, in making decisions. Mathematics just happens to be a particularly clean setting in which to practise them, because the claims are precise and you can check whether your reasoning holds.

Prerna, a student in my course, 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." Sanya said what she valued was that "we all would speak and not textbook knowledge but share different views and opinions about everything." These are not descriptions of mathematical achievement. They are descriptions of a way of engaging with ideas. And that way of engaging does not require a classroom or a teacher. It requires curiosity, a willingness to say "I do not know," and a question worth following. Most families already have all three.