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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?

John Clever
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    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) – Dave L Renfro Oct 11 '20 at 18:42
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    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. – Dave L Renfro Oct 11 '20 at 18:49
  • 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.
    • – Chaim Oct 30 '20 at 18:34