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?