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I always wanted to teach my siblings mathematics, and one, ten years of age, is particularly eager. For the purposes of specializing recommendations, I will add he can use arithmetic up to exponentiation. Over time, I have become increasingly afraid of his curiosity and love for mathematics being crushed by the school system.

Therefore, I am looking for a good way to teach him math without any of the terrible, A Mathematician's Lament sort of stuff present in schooling. I was thinking of using Art of Problem Solving, but I only have the Geometry textbook, which is predicated on a basic understanding of Algebra. I was thinking of teaching Knot Theory, seeing a Japanese textbook on teaching it to people his age, which he seems to be interested in after seeing multiple Numberphile videos on the topic.

To make this question as answerable as possible, I'll state it as this:

What topics (and textbooks on said topics) would be suitable for a proof/discovery oriented course with a 10-year-old?

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    $\begingroup$ The main concern I would have is that you don't make this too much of a chore, otherwise he may rebel and change interests to other things (which might happen anyway, and shouldn't be discouraged). Despite all my early school interest in math and science, I actively avoided anything having to do with electronics (this being late 1960s to early-mid 1970s) because my father was a huge ham operator (outdoor radio shack, we often traveled to ham fests, one of the top few dozen high speed code operators in the world, etc.) and old-style radio repair trouble-shooter, (continued) $\endgroup$ – Dave L Renfro Oct 11 at 18:42
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    $\begingroup$ and although he didn't really try to push it that much on me, I was just so saturated with it in my home life that I sought escape in other things (complex numbers, higher dimensions, basic special relativity, etc.). As for the effect of school mathematics on me, I don't recall that having any effect, positive or negative --- it was just something you had to do, like household chores, and had no connection with the math and science library books I read/looked-at, where I learned about square roots, negative numbers, complex numbers, and many other things long before I saw them in a class. $\endgroup$ – Dave L Renfro Oct 11 at 18:49
  • $\begingroup$ 1) There was a good book called Reviewing Mathematics, large pages and a soft Indian red cover, making it look like a workbook. Each chapter was about a certain style of solution -- making a table, sketching a picture, etc. I can't find it now on the web. 2) I would think that discrete math topics could appeal to kids, like word problems coming down to combinatorics. $\endgroup$ – Chaim Oct 30 at 18:34
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You mention Art of Problem Solving. They have a great curriculum for kids called Beast Academy. You could get that for him, and just be there for when he has questions. It is a lot of fun for many of the kids using it.

Also, you could play games with him. (You may have to squelch your desire to teach, just a bit.) I love Blokus and Katamino.

There are also two online geometric construction sites that gamify it. sciencevsmagic.net/geo and euclidthegame.com.

You might find other cool games and books to share with him at my blog: mathmamawrites.blogspot.com

One book he might enjoy reading and discussing with you is The Number Devil.

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I think the best thing for a beginner is to focus on concrete interesting mathematics. You definitely do not want to do any abstract stuff at this point unless it is motivated by some concrete problem. I recommend "Nets, Puzzles and Postmen" by Peter Higgins, and later on "How to Prove It" by Daniel Velleman. The first book is a really unique book, in that it is written at the layman level but does not handwave any of the core mathematical ideas, and furthermore it has notes for advanced readers!

I want to also note that playing combinatorial games (e.g. Hex, Go, ...) can be very helpful for honing intuition of quantifiers, because if you think about it a winning strategy of maximal length $k$ is nothing but a true sentence with $k$ alternating quantifiers. There are also many nice puzzles with very strong mathematical flavour such as Tatham's puzzles and Manufactoria. These can very easily inspire an investigation into reductions (i.e. if I can solve one puzzle easily I can solve another one too) and also encourage thinking about NP-hardness (i.e. it is possible that it is easy to verify a solution but hard to find one), which in some sense is similar to hardness of mathematical proofs (i.e. it is easy to verify a formal proof but may be hard to find one or even know whether one exists).

If your brother loves mechanical puzzles such as the Rubik's cube, you can happily explore various invariants with him as well. One very important invariant, namely parity of the pieces in each group (sides and corners), is crucial to a complete understanding of solving it and many other permutation puzzles.

It may also be of interest to you that my interest in mathematics grew out of playing around with such concrete mathematics, and one example I clearly remember from my childhood was investigating the relation between the forward difference iterates and the original sequence. Although at that time I did not know any of the terms, I knew about Pascal's triangle, which was enough to find and understand the decomposition of any given polynomial sequence into a weighted sum of (diagonal) columns of Pascal's triangle, where the weights can be easily obtained from the forward difference iterates. You can take a look at this post for a high-level explanation that I wrote up many many years later, but certainly the way to teach such stuff to your brother is via Pascal's triangle and not the abstract operator viewpoint. Well, at least not so soon. =)

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