# A good training field for formal rule application?

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?

-
@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. – André Nicolas Jun 27 '11 at 7:10
@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. – Alexei Averchenko Jun 27 '11 at 7:22
@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. – André Nicolas Jun 27 '11 at 9:14
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"?) – Pete L. Clark Jun 27 '11 at 10:47
@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 $\operatorname{lcm}(a, b) = \frac{a, b}{\gcd(a, b)}$. ... – Alexei Averchenko Jun 27 '11 at 23:10

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.