These days I am reading Imre Lakatos's Proofs and Refutations and I can't stop thinking how one could utilize it in the classroom (mostly high school). Some stray half-baked ideas I have had so far are:

  • To use parts of the text as a basis for a theatrical play in classroom, providing thus the opportunity to discuss (before and/or after) the ideas presented in it.
  • To hand out fragments of the text and start a discussion based on those.
  • Given the difficulty of Lakatos' text for high schoolers, to initiate a dialogue in classroom of a similar fashion in a more appropriate setting, e.g., some (even naive) conjecture about the naturals.

Regarding the first two ideas or similar ones, the text itself, albeit a source of inspiration, is also prohibitive. It is quite abstract in its most part and probably not the best way to introduce students to concepts around proofs (I am more than happy to be proven wrong here, however). Regarding the latter, any reference to similar approaches that might have been tested in the past would be welcome.

So, to wrap up, my question is how could one bring Lakatos' work in the classroom, either (and preferably) by utilizing some genuine parts of the text in some way or even by drawing strong inspiration from it?

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    $\begingroup$ I have this book. It probes some deep issues and in my opinion is far too sophisticated for most high school students. If your goal is to teach robust proof-writing techniques, there are quite a few more suitable free online books such as "Book of Proof" (Hammack), Proof (Martin Day) - and others, I am sure. I wouldn't introduce Lakatos until I felt the student could write a fairly "clean" proof of a simple number theory problem - and maybe not even then. $\endgroup$
    – Clive Long
    Sep 12, 2023 at 17:41
  • $\begingroup$ @CliveLong Yes, that is my major concern as well; that most ideas in the book are way too advanced. i don't know, I may consider adding it as an optional group project throughout the year so anyone interested might work on it. I have not read the other two books you mentioned, so I will add them to my reading queue. :D $\endgroup$ Sep 13, 2023 at 7:25

2 Answers 2


I've used Proofs and Refutations in the classroom (college students in the US), but not in the direct ways you suggested. When leading up to Euler's formula $V-E+F=2$, I try to get the students to see that it is not obvious what is a vertex, an edge, a face, by showing them examples of interpenetrating faces, coplanar faces, etc. I have them experimenting with polydrons, making their own counts, trying to discover Euler's formula. This can require them to revise their definitions (one of Lakatos' primary lessons), e.g., to require an "edge" to have a dihedral angle different from $\pi$.

(On this specific example, it turns out that allowing an "edge" to be flat, as in deltahedra, still satisfies Euler's formula, but with different counts of $V,E,F$.)

Are the five coplanar yellow triangles one face?

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    $\begingroup$ I see your point. It is more of following Lakatos' spirit rather than bring the text itself into the classroom. Which, of course, has its own merits. Thanks for the suggestion! $\endgroup$ Sep 14, 2023 at 8:21

Frame-challenge answer:

It's better to start with the educational goal you have in mind, and from there decide the best way to achieve it. Starting with a book that you like, and trying to find a way to use it, is like the tail wagging the dog.

In general, there is a temptation while teaching to try to make the material interesting for YOU, the teacher. I've certainly experienced this. But you can't give in to that temptation. The stuff you find interesting is mostly too advanced and will just confuse the students.

To be clear, you do want to make the material interesting, but for them, not necessarily for you.

  • $\begingroup$ It might not be clear in the statement of the question, but my intention was to utilize it, if possible, in a discussion around proofs. Yet, as pointed by @Clive Long, it is indeed a hard text to parse without any formal background, thus it should be left for more advanced audience. Regarding teaching being interesting, an educators most interesting pursuit is teaching per se, in my viewpoint, so the lesson is always interesting. My question is thus more of an exploration of Lakatos's book direct usefulness in the classroom. Anyway, thanks for your contribution! :) $\endgroup$ Sep 14, 2023 at 8:28

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