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I'm tutoring a would-be Russian 7-grader who seems to have difficulties in understanding and application of formal rules (identities). I'm looking for a way to improve it, but I don't want him to try guessing the answer so I want to shut his arithmetic intuition off for this purpose.

Is there a good universal-algebraic theory that can be used for training how to apply known identities to simplify an expression or answer a question about it that doesn't allow for much intuition?

Is my general approach good? The guy has problems using identities like $\mathrm{gcd}(a, 0) = a, \quad \mathrm{gcd}(a, b) = \mathrm{gcd}(b, a \ \mathrm{mod} \ b)$ to compute things like $\gcd(1234, 58)$ without my constant supervision, but is it right to emphasize symbolic manipulation at this age, even the simple cases?

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    @Alexei: Understanding the use of symbolic variables seems to be a major obstacle for many. I would start by checking that. You may need to go back a grade or two to get to the root cause of the problem. I once managed to help my younger sister by leading her to the identity $a^2-b^2=(a+b)(a-b)$ with several examples: I asked questions like 8*8, 7*9, 6*6, 5*7,..., waited for the penny to drop. Then tried 7*7, 5*9, 10*10, 8*12,... and went from there. Your goal may be best served by starting with something simpler than Euclid - even if they are targeting that in the next grade.2011-06-29

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Ignoring the question of whether it is good/bad/otherwise to teach a student in this manner I would think that teaching him how to manipulate trigonometric identities would be an activity that satisfies your requirements and would be a good choice because:

  1. At this stage, according to your description, he has probably has never heard of $\sin$, $\cos$ etc. and so he has no intuition regarding these things. You can just define them as formal expressions that obey certain rules.

  2. You will actually be teaching him something intrinsically valuable that will help him in his later studies.

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    One piece of pedagogical advice that I do feel qualified to give is: be very wary of extrapolating from your own experiences as a student. Whenever you do this at all, you should counterbalance with "Now in what wa$ys$ are these students *different* from how I was -- or think or remember that I was -- as a student studying X?" I don't know the answer to that question here: do you?2011-06-29