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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. – 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

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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I don't understand why you are "[i]gnoring the question of whether it is good/bad/otherwise to teach a student in this manner". Isn't that the most important pedagogical consideration: whether it will be good/bad/otherwise for the student? – Pete L. Clark Jun 29 '11 at 14:45
@Pete Because I don't really feel like I 'm qualified to answer that part of the question as it broadly applies. I honestly don' know the answer to it. Speaking for myself, I think I would have found the "game" of solving trig identities when I was, say a 7th grader, kinda fun. It's simple enough to be "played" with only a few fundamental rules, allows one to be creative and requires assembling a logical train of thought. Seems like good practice to me, but, I don't really have any empirical evidence to support that it would be good/bad or otherwise. – ItsNotObvious Jun 29 '11 at 17:19
well, I think one should be wary when giving out pedagogical advice, especially pedagogical advice motivated by "I think I would have enjoyed..." Not everyone is an expert in everything: I myself know nothing about teaching 7th graders, especially Russian 7th graders. – Pete L. Clark Jun 29 '11 at 19:16
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 ways 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? – Pete L. Clark Jun 29 '11 at 19:17

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