How do you remember mathematical rules?

[JeffM]

I am probably trivialising this conversation, but I hate memorization. It is a burden and is just amother (although relatively minor) one of the many impediments to creative thought.

A thread or two ago, pka railed against integral tables. I disagree. Looking something up may trigger a thought. Putting something into a black box cannot ever do so.

Nothing that normally engages my attention causes me to use trigonomtric functions with any frequency. Consequently, I remember almost nothing about trigonometry except the unit circle, the definition of sine, and

[MATH]sin^2( \alpha ) + cos^2( \alpha ) = 1.[/MATH]
When I need to tutor someone in trigonometry, I spend an hour refreshing myself on the basics and then can derive whatever else I need. When students see me figuring things out, they learn that what is critical is the ability to think rather than to memorize. So many students say, "But that [specific thing] was not in the book." If it is of interest to you, it is not already in a book you know.

OK. I admit that I do have some tricks to avoid looking absolutely everything up.

The quotient rule is an example. If I must use it, then I remember the power rule and go:

[MATH]f(x) = \dfrac{1}{x} = x^{-1} \implies f'(x) = - x^{-2} = \dfrac{0 * x - 1 * 1}{x^2} \implies[/MATH]
[MATH]f(x) = \dfrac{u(x)}{v(x)} \implies f'(x) = \dfrac{u'v - uv'}{v^2}.[/MATH]
But the only reason I ever need the quotient rule is that some student has been told to use it

 

[Harry_the_cat]

I can never remember the identities which go with sin²(x) + cos²(x) = 1.

Is it tan²(x) + 1 = sec²(x) or cot²(x) = 1 - sec²(x) , etc ???

And I don't expect my students to remember them either, but expect them to use sin²(x) + cos²(x) = 1 to derive them by quickly dividing both sides by sin²(x) or cos²(x).

 

[Harry_the_cat]

When looking at and discussing graphing trig functions in the form y = a sin[b(x+c)] +d, and interpreting the different transformations, I usually pick on a kid in my class whose name begins with D, by saying "Danny's a cool bloke". The class looks at me as if I am mad, until I explain that the best way, I believe, to (manually) draw these graphs is to interpret the transformations in that order ie d(axis of oscillation), a(amplitude), c (phase shift) and b(to calculate period). They never have problems with drawing trig graphs after that and it stays in their memory! And Danny becomes a maths hero!

 

[Dr.Peterson]

I can never remember the identities which go with sin²(x) + cos²(x) = 1.

Is it tan²(x) + 1 = sec²(x) or cot²(x) = 1 - sec²(x) , etc ???

And I don't expect my students to remember them either, but expect them to use sin²(x) + cos²(x) = 1 to derive them by quickly dividing both sides by sin²(x) or cos²(x).

I have the same difficulty, and recover the identities by drawing a right triangle with one of the legs labeled as 1 and the other tan or cot. I also point out your method (which is the textbook's) for those who are less visual.