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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 Averchenko: Almost certainly very premature. Even for most mathematicians-to-be, intuition has been developed over a solid foundation of calculation. We just have forgotten! And $1234$, $58$ are fairly abstract for many at this age.2011-06-27
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    @user, the problem is: he can easily divide, multiply things etc., but has problem figuring out that $\gcd(1234, 58) = \gcd(58, 1234 \ \mathrm{mod} \ 58)$ given the general rule.2011-06-27
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    @Alexei Averchenko: Me too. But he is programmable, and after a while will be able to recognize a formal description of what he knows how to do.2011-06-27
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    How does the skill you are trying to inculcate fit into this student's curriculum? What kind of problems come up in his classes which require understanding and application of identities? In what country is this taking place? (And, probably not so important but: what is a "would-be 7-grader"?)2011-06-27
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    @Pete, this is taking place in Russia. He had problems with his 6th grade curriculum (divisibility, fractions, integers, rationals, proportions equations, text problems, things like that). He can perform $+,-,\times,\div$ very well by using a memorized process, but has has problems with even basic symbolic manipulation, e.g. he sometimes has problems with solving equations, decomposing numbers into primes, dealing with proportions or figuring out that if $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a \cdot b$, then $\operatorname{lcm}(a, b) = \frac{a, b}{\gcd(a, b)}$. ...2011-06-27
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    ...Yet I'm unsure if my approach is correct for him, that's why I'm asking.2011-06-27
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    @Alexei: in my opinion you should include the fact that the student is in Russia as part of the question. Presumably the educational system there is quite different in its curriculum, practices, expectations and so forth from those of, say, the United States. I for instance would not venture a response to your question because I know nothing about what is expected of 6th/7th grade Russian students. (In the US, a lot of the material you describe would be called "elementary number theory" and might not get taught until the college/university level.)2011-06-28
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    @Pete L. Clark: Done.2011-06-29
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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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