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We're going to make a new math course for kids as intermediary between middle and high school with math profile (for preparation to entrance exams to high school), and before the main part (arithmetic, addition, multiplication, Minkowski addition, Fundamental theorem of arithmetic with proofs) we'd like to put some content about proofs tactics, how to proof theorems, formal logic and basic terms like ring, field, group, etc.

What would you recommend to look at among introductory courses as a preparation material to such classes?

I understand that it sounds like a broad question, but I'was trying to look through some materials like Coursera's "Mathematical thinking" and it seems not to be enough for this goal. Can you help me with collection of materials for kids, which are ready to take and systemized?

Thank you all!

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  • $\begingroup$ I give this presentation to kids of all levels middle school and up. The goal is to help them think about math in a different way. I'm not sure if it is exactly what you are looking for, but I thought it might be helpful. youtube.com/… $\endgroup$
    – johnnyb
    Commented Mar 2, 2020 at 18:14
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    $\begingroup$ This book is quite good, but assumes high-school math, and is aimed at college students. Still you might find it useful: Velleman, Daniel J. How to prove it: A structured approach. Cambridge University Press, 2019. $\endgroup$
    – Joseph O'Rourke
    Commented Mar 5, 2020 at 2:04
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    $\begingroup$ That is a VERY ambitious program. If your 12 and 13-year-olds can handle proofs and formal logic, you might have a look at my proof-checking freeware and the accompanying tutorial that are available at my homepage dcproof.com It introduces students to the basic methods of proof. Each line of proof that the student enters is verified immediately after it is entered. Feedback is instantaneous. It is impossible to write a invalid proof. $\endgroup$
    – Dan Christensen
    Commented Mar 5, 2020 at 3:15

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Not sure how to answer the question apart from suggesting a book about mathematical proof that can be found free online, and that is on the easier side of understanding. It's actually been adopted by a lot of universities and so I'm certain it is a quality textbook.

https://www.people.vcu.edu/~rhammack/BookOfProof/

I don't know if it's for kids or grade school but quite a lot of it should be useful to develop your own questions.

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In general, if you want good free books, here is a list: https://aimath.org/textbooks/approved-textbooks/

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  • $\begingroup$ Started to read the book on the first URL. It seems like a bit overkill for kids and too fast movement towards math logic in the beginning. $\endgroup$
    – paus
    Commented Mar 5, 2020 at 14:44
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Have them work through a few chunks of Euclid's Elements. Then they are doing proof, rather than reading about logic in a theoretical way, which is probably too abstract for kids in the age group you're targeting.

The theory of logic will make much more sense to them after they have had an opportunity to practice it, and the whole point of Euclid's text is to walk you step by step through the practice of geometry. In my experience children of that age respond extremely well to the actual text of Euclid.

I described how I learned Euclid here -- it was a college course but the general approach should be possible for younger students as long as the pace is adjusted.

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