Being somewhat a part-time educator teaching (mostly adults) basic math I find myself troubled by questions I tend to get.
Being more specific, students ask me for conditions, algorithms and other ways to determine if some expression (basic algebra or some trig stuff, as example) can be simplified into something "shorter" and "better". Except for some cases, like dividing by (x-a) for polynomials, I can't actually answer anything decent.
It seems that at the beginning (pre-calculus algebra, trig equations, limits and series, integration) a lot of math depends on technical trickery — using various short multiplication forms, conjugates etc. The problem is, despite math being a pretty rigorous field, there's no way to determine if something can be simplified and, more broadly, represented in a suitable form.
The only way to guide someone would be telling "You just have to see that!", "Do another 1000 excercises and you'll get a better feel!", "Pay more attention to details" and so on — that doesn't seem either rigorous or intuitive and leaves students discouraged and demotivated.
I'd be extremely grateful if someone could point me to some algorithm, approach, idea or concept that is capable of presenting algebraic trasnformations as something besides a strange art only being mastered by extensive practice.
Thanks in advance!
Best Answer
In agree with you that most of the time, there isn't a step-by-step approach you can take to a problem—you have to be creative. However, there are a few things to note:
For example, here was a problem that I was struggling with today:
There are a number of things that make this problem interesting to me:
Here, the key to finding the solution was realising that
\begin{align} (\sin\theta+\cos\theta)^2&=\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta \\ &=1+2\sin\theta\cos\theta \end{align}
This links the concept of adding trigonometric functions with multiplying them—a key ingredient in finding the solution! When I saw that this was the key, it did feel very satisfying. At first, it seemed like the identity $\sin^2\theta+\cos^2\theta=1$ was unrelated to the product of $\sin\theta$ and $\cos\theta$, but it suddenly had become obvious what that relationship was. Rearranging the equation, we have
$$ \sin\theta\cos\theta=\frac{(\sin\theta+\cos\theta)^2-1}{2} $$
And, I got stuck again. I say this because most of the solutions given in textbooks make the process of finding the solution seem like a simple, mechanical process. Well it is if you already have the answer in front of you! For everyone else, though, you just have to keep on trying. There was actually an elegant solution that didn't require calculus, but here is what I did:
The maximum of $(\sin\theta+\cos\theta)^2$ is when the derivative of $\sin\theta+\cos\theta$ equals $0$:
$$ \cos\theta-\sin\theta=0 \implies \cos\theta=\sin\theta $$
This occurs when $\theta=\frac{\pi}{4}$. Hence, the maximum of $\sin\theta+\cos\theta=\sqrt{2}$, and substituting this back in gives the desired result. Using calculus, I got to the same answer that a clever rearrangement did. The fact that there were two equally valid approaches to this problem is also very reassuring. It shows that as long as we are doing good maths, and looking out for ways to simplify problems, then it is definitely possible to get to the answer, even if our solutions aren't mechanical. It's OK to get stuck, it's OK for your first attempt to be messy, and it's OK to not spot what you can do to simplify a problem. If you keep trying, the payoff is immense.