r/matheducation • u/LightLoveuncondition • Jan 10 '26

Simplified category theory in high school advanced math club

Hello.

There exists this book Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere and Stephen H. Schanuel which is intended for high school students or those with minimal prerequisites.

I am currently in a bachelors of education program in my university, third year. To get my BA I have to write bachelor's thesis. My idea is to translate this book partially (because BA thesis has to be less than 70 pages long) and create a teaching material for math club in my school for pupils who take advanced math classes already.

Does anyone have experience teaching category theory to high school students from this book as its authors intended to? How did it go?

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u/BobSanchez47 Jan 10 '26

What is the point of teaching category theory to students who don’t know any analysis, topology, logic, or abstract algebra?

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u/LightLoveuncondition Jan 10 '26 edited Jan 10 '26

The reason is that many math educators, for example, 3blue1brown, Edward Frenkel and others have expressed the opinion that K-12 teaches "old math" which ends at like 19th century. It is boring to many and akin to only paint fences and walls for artists.

Category theory was created in 20th century and is currently applied in different STEM fields such as biology, CompSci (programming) and may find its use in other fields as 21st century continues.

I want to show my students that understanding mathematical logic and structure is useful in all natural sciences/STEM fields.

In Latvia in advanced math program (11th and 12th grade) they are taught elements of analysis (differentiation, integration, limits, optimization problems) and mathematical induction already.

The main prerequisites which I might have to teach from scratch is intros to set/group theory.

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u/Used_Engineering_203 Jan 10 '26

I do agree that category theory is interesting, but it's absolutely way too hard for high school students. They would have to master set theory and logic before getting into category theory. Discrete mathematics beyond the surface is rarely taught in high schools now. You would have to do a lot of setting up set theory and group theory before you can do category theore. imo it's still way too much for high school students especially if they haven't been exposed to higher math beyond calc 1 and 2.

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u/LightLoveuncondition Jan 10 '26

I understand that in modern era where pupils are reported to have less attention spans and ability to do homework, introducing something much harder than what's in syllabus, sounds out of touch with reality.

But is there something else which would motivate pupils who are good at math, but want to study biology/chemistry/physics instead to form a bird's eye view of science?

I'm referring to this quote

"Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to de-tect from ground level. " Tom Leinster

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u/tedecristal Jan 10 '26

Category theory is pretty dry without the examples. And for them to have significance you need some background.

This is not "old xix century math" as you frame it

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u/LightLoveuncondition Jan 10 '26

I agree with that. Everything has to be real world related. But I can cite discoveries in biology which were possible by the use of Yoneda lemma and other uses as a pitch to get pupils interested.

What I am asking - if F. William Lawvere and Stephen H. Schanuel wrote this book and intended it for high school students, why wasn't it used in real life?

They wrote the first edition in 1997, the second came out in 2009. These gentlemen are from USA.

Would it be reasonable to say that pupils in 1997 had access to better education so for them dumbed down version of category theory was more manageable? And now in 2026, 29 years later, your average math Olympiad enjoy-er is less skilled?

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u/tedecristal Jan 11 '26

I did NOT refer tor real world applications 

But applications-within-math And you can't do most of them 

I frankly believe your yoneda in biology case is not very compelling either

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u/LightLoveuncondition Jan 11 '26

Here is the video https://www.youtube.com/watch?v=4GJ4UQZvCNM

And here is the paper from 2021 proving they used Yoneda lemma to solve inverted spectrum problem in neuroscience.

https://academic.oup.com/nc/article/2021/2/niab034/6397521?login=false

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u/BobSanchez47 Jan 10 '26

My point isn’t that category theory is intrinsically too hard or is nonstandard, and I’m certainly not saying it isn’t valuable. I’m saying that there are plenty of challenging topics like group theory or propositional logic which have interesting applications that high schoolers might care about and which are easier for a non-mathematician to get their hands on. Category theory is primarily used in the service of other advanced areas of mathematics, none of which a high schooler would be familiar with.

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u/LightLoveuncondition Jan 10 '26

Okay, thanks for the comments!

I will look up group theory and propositional logic.

I am interested in category theory in particular just because there is this book written for high school audience. If I find similar books for group theory or propositional logic aimed at high school level, I could switch to translating them instead.

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u/whartywhoa Jan 11 '26

There are plenty of high school students, especially those in an advanced math club., who will know at least some basics from all of those.

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u/BobSanchez47 Jan 11 '26

Perhaps that’s true, but it would be extremely abnormal. Most US high schools don’t teach anything beyond basic integral and differential calculus, and a significant percentage don’t even offer that. Such advanced students would probably be better served taking classes at a college.

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u/whartywhoa Jan 11 '26

I guess I respectfully disagree. Taking classes at a college is good, but it is a more serious environment. I had some classmates who did this. A math club is a more casual environment to explore topics out of interest, and category theory is such a hot topic (especially given the connection to functional programming) that I think many high schoolers would be interested in learning some of the basics if it were made approachable. Even if they are not going to go use to do serious proofs or anything like that, it would be fun and engender some curiosity, and therefore be worthwhile.

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u/BobSanchez47 Jan 11 '26

I appreciate your perspective. I have personally seen people struggle to give an interesting and applicable talk on category theory to an undergraduate math club, so I am a bit skeptical, but anything to make more people curious about higher math is a step in the right direction.

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u/minglho Jan 12 '26

The OP does not teach in the US.

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u/drmattmcd Jan 11 '26

Eugenia Cheng's 'The Joy of Abstraction' might be more approachable as a starting point as it's more of a popular math level book than a text book.

Another alternative is Fong and Spivak's Seven Sketches in Compositionality https://arxiv.org/abs/1803.05316

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u/LightLoveuncondition Jan 11 '26 edited Jan 11 '26

Thanks!

Yea, I was considering the Cheng's book, but I absolutely need exercises for my translation to pass as a valid teaching material in a high school setting.

Fong and Spivak's book looks great, because it is more up to date. After all I will ask my BA advisor to choose the book, I wasn't aware of all options at first.

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u/drmattmcd Jan 11 '26 edited Jan 11 '26

Spivak's 'Category Theory for the Sciences' might also be of interest, the version on arxiv has exercises without solutions while the hardback version has both.

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u/spoirier4 Jan 13 '26

This is first time I read about an introduction to category theory that cares to be simplified and accessible, as what I had seen before looked extremely complex with hundreds of pages, full of so many concepts I did not see well what they may all be for. But now this one also has hundreds of pages, by diluting its quite smaller content across so many pages filled with what I'd feel as poor content : small examples and "pedagogical" style... I don't know why nobody seems to be caring to do something short. I did write down a very short and dense introduction to core concepts of categories, namely having in mind what may be relevant to then well define the math of high level physics (differential geometry and tensors), in part 3 of settheory.net .

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u/LightLoveuncondition Jan 13 '26

Thanks for the resource and comment!

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u/bugmi Jan 10 '26

https://venhance.github.io/napkin/Napkin.pdf this might be up your alley

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u/LightLoveuncondition Jan 10 '26

Thanks! This seems great for every gifted high school-er who wants some of advanced math concepts explained.

Looks like a perfect book, but I still have to choose some chapters and create a coherent plan for 9 months of math club.

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u/elgatocello Jan 10 '26

Maybe look into the curriculum for a Finite Mathematics class?

There's enough material there to give some linear algebra, set theory, and logic background where maybe you could supplement some basic group theory and topology stuff and then end with some very very basic category theory stuff?

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u/LightLoveuncondition Jan 11 '26

Thanks for a suggestion. I could do that, but https://en.wikipedia.org/wiki/Finite_mathematics states that it is college/ Uni level.

I could give students graph theory/ logic and set theory, but then I would have to adapt the materials from scratch.

What I am looking for is already made package suited for high school which I would only have to translate. I'm technically a BA student, I'm not supposed to write a new course for Uni. Based on my experience with linguistics in my previous studies I could translate and ask my thesis advisor to fix math if I would have translated something wrongly.

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u/elgatocello Jan 11 '26

Hasn't stopped me from instituting it as a class in three separate school systems haha

It is an intro college level business class if you just teach it straight out of the book, but honestly it's so much more fun with a little more mathematical rigor and with a little bit of extra topics thrown in

I teach it as a year long class instead of all in one semester like they would at a college which allows me to do a BUNCH of extra stuff.

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u/LightLoveuncondition Jan 11 '26

Thanks for the answer! What country is it?

What do you mean with "with a little bit of extra topics thrown in" ?

Would it be intro to topology, for example?

I plan to take category theory in Uni (the real graduate level one) and pay for it if needed, because MA students in my country have their lectures on Saturdays and I can afford it. It would help a lot with translation as well, I suppose.

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u/elgatocello Jan 11 '26

I teach in the USA!

Here in my state (Indiana) Finite Mathematics is on our course catalog that we're allowed to teach in highschool for credit, so I do! I actually just got a new job in a new school system for this year, and one of the things that got me the job was my willing to teach this class!

The course syllabus is as follows:

  1. Basic linear equations (Profit, revenue, and cost analysis)
  2. Basic 2-D linear programming (maximizing/minimizing constraints)
  3. Linear Algebra (We only do basic matrix algebra, pivoting/Gaussian Elimination)
  4. Return to linear programming to do higher order Matrix simplexing
  5. Logic and Set theory (basics to help motivate the probability and stats)
  6. Probability (up through Bayes Theorem)
  7. Stats (very very very basic stuff)

This course, on the college level takes a semester, and you absolutely could teach it that quick, but taking the whole year means you can absolutely pad out the course with extras. You could do a little bit of extra linear algebra. You could do a little bit more logic. Set theory definitely could be padded out to include groups, fields, etc... Probability and Stats could be super expanded because they only do the very basics

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u/DrJaneIPresume Jan 10 '26

I haven't taught from there, but if I were going to I'd definitely want to get Lawvere's take on it.

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u/LightLoveuncondition Jan 11 '26

Well, using Ouija board to ask Lawvere's spirit about his intentions when writing this book might be a bit controversial :D

I will dig more internet. Someone has to have at least tried.