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How's teaching groups to teenagers going, then?

Monday, October 19, 2026

We’re six weeks into my experiment to teach basic group theory — the abstract theory of symmetry — to a self-selecting group of KS3 teenagers. This is a topic that is either introduced in A-level Further Maths, or else in the first year of university, so there are certainly some challenges.

My curriculum plan is to do some prerequisites and examples for this first term, then formally introduce the concept of a group at the beginning of next term. From then we’ll probably more-or-less follow Pinter’s treatment in A Book Of Abstract Algebra, with a few key milestones — I’m hoping that collaboratively they will be able to classify all groups up to order 10 during the spring term. By early summer I’m hoping we can do quotient groups, including the all-important homomorphism theorem. Any remaining time can be played by ear. Some of them are cubers, so maybe we could try and describe the Rubik’s Cube group, or we could look at some other application. Or maybe we’ll all be tired by then.

During the first week, I had three attendees. That has slowly grown to now seven students, and I think at this point I’ll have to halt the growth. For one, they’ve missed too much to reasonably catch up on now (it’s not actually that much, but there’s no way a 14-year-old is going to sit and self-study matrices, sets, and functions to catch up with an optional maths club). Moreover, we’re at about the point where they’re starting to behave a little more like a group of seven teenagers rather than a handful of dedicated scholars. They’re not being bad, by any means, but getting more distracted by the latest gossip and so on. It doesn’t help that the most recent couple of sign-ups seem to be here because their mums thought it would be good for their grades, rather than a passion for mathematics. I have emailed their mums to explain exactly what they’ve signed up for and that while their child is very welcome and I hope they enjoy it, any improvement to their grade will be a happy side-effect and not the goal of the club. To the credit of these students, they’re being very good sports and are making a real effort to participate and understand.

Why group theory? Well, it’s fun and interesting — in fact, it was reading a certain linear algebra textbook that started with group theory that gave me my passion for mathematics as an adult — and there aren’t too many formal prerequisites. You don’t need calculus or continuity or even linear algebra to understand it. In the end, a group is a simple and natural concept with many familiar examples. Speaking of examples, this term we’re doing modular arithmetic,^[1] 2x2 matrices, and dihedral groups.^[2] The other main example we’ll do next term is, of course, symmetric groups.^[3]

The main barriers to this topic are a) it’s more abstract than they’re used to b) there’s a lot of new notation and c) lots of new words. The concepts are all actually simple, at least so far, but they’re usually stated with precise technical language and notation. So I have to support this with informal descriptions, examples, and diagrams/pictures. Last week was the first session I think some of them really struggled, which was on composition of functions — a prerequisite for understanding the action of a group. They did well during my initial examples, but struggled to get the idea once I added the notation and introduced the identity function (even though nothing could be simpler, I think the notation scared them!)^[4]

Another barrier is time. I want them to be doing hands-on stuff, but I have to explain things very slowly and stop to answer questions a lot. That’s fine, and there are some points where I really expect it. But I wouldn’t have expected it for, say, the identity function. I have been supplying them with interesting^[5] problem sheets I wrote, but they’ve mostly not been done in the sessions or at home. It turns out children don’t like extra homework, especially if it’s hard. Who knew? But I have decided to trial running a second session at lunchtimes where they can come to my room and work through the problems with me over lunch. They may not like extra homework, but they sure like a warm room at lunchtime. Last week I did have a couple show up and actually get something out of it, so maybe they’ll tell their friends to come next time. This part is essential though — you’re not going to understand something until you’ve worked through some examples yourself, and that’s what the problems are for.

Am I doing proofs? The answer is yes. So far, we’ve got off pretty lightly and haven’t needed to prove many things because it’s mostly been setting up the definitions — a little proof of associativity of the operations we’ve encountered so far, and a couple of example proofs applying some definitions. An idea I have had that might help them, since the proofs are mostly very constructive, is to run the abstract proof alongside a concrete example of what the proof is describing. For example, I’d like to show them that modular arithmetic is well-defined. The proof is simple, but it does involve keeping track of several variables — more than they’re used to. I’m wondering if dividing the board/slide in two and doing the abstract proof with variables on the left, and on the right doing the same steps but with numbers instead of letters might make it clearer.

A final question is whether there should be any “assessment”. I am in two minds about this. For one, I want to keep it lighthearted and fun. But also, I do want them to come out of this having learned something, and assessment is a useful tool for that. After all, knowing that you will have to demonstrate your learning is a major motivator to actually learn something. Probably I won’t do it though, as more likely they’ll just stop coming if they perceive any pressure.

When we get back after the half-term break, we’ll hit dihedral groups, which means, of course, we’ll be cutting out shapes and studying the symmetries. This should be a nice confidence boost after the difficulties composition of functions gave them, and I’m hoping we can use the problem sessions to really get on top of function composition. I’ll post another update around Christmas time.

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1. For non-math folk, that’s arithmetic where the numbers “reset” after a certain point, like on a clock face — three hours after 11 o’clock, it’s 2 o’clock, because we “reset” at 12. ↩︎

2. Again, for the non-mathematical this is nothing more than the symmetries of regular polygons, like squares and equilateral triangles. When I said you don’t need much background knowledge for elementary group theory, I wasn’t joking! ↩︎

3. A symmetric group is all possible ways of re-arranging a set, like all possible ways of shuffling a deck of cards. These examples might seem unrelated, but that’s the beauty of an abstract theory like group theory — it highlights what unrelated-seeming concepts have in common. ↩︎

4. If functions are about ways we can transform an object, the identity function is “do nothing” — it leaves the object identical to how it started. We just give this operation a special symbol (id), which I think might have confused them into thinking it was more complicated than that (why do we need a special symbol for doing nothing!?) ↩︎

5. Some authors use the term “interesting” in a way that excludes mathematics. For our purposes, both here and in paragraph 3 we use the term in such a way that does not, as should be clear from context. If you’d prefer to insert a qualifier such as “relatively interesting”, you’re welcome to do so. ↩︎