Teaching mathematics in intuitive and thought provoking way

Provided your student has a strength, you can play to that. In some sense that's what finding "applications" is about -- appealing to someone's immediate values. But there's also a danger in that, if all you do is appeal to someone's immediate values you may be forced to build up a rather contrived and limited notion of what mathematics is.

So in teaching it's also very useful to take students to foreign territory where they're weak, and build up their appreciation for what they may have found useless or "abstract". Finding applications isn't the only way to do this -- simply providing re-interpretations of things they know that seem complicated can be useful. If something they found hard becomes easier in a different setting, this will frequently be perceived as useful.

I'm teaching an introductory calculus class this semester. In the last few classes we've been differentiating trig functions. Recently students have been asking questions like "can we have a formula sheet on the exams?" The bulk of the number of things to remember is starting to appear heavy. Their concern is that things like the double-angle formula or the formula for the derivative of $\csc(x)$ are difficult to remember. So I spent a little bit of time on how you can get by, not remembering much at all about trig functions.

Here's the pencil-sketch of the idea that you can flesh-out:

I take it as an axiom people know what the graphs of $\sin(x)$ and $\cos(x)$ look like -- and students should be able to deduce these things from "$(\cos(x),\sin(x))$ is the point on the unit circle that you get to if you lay-out a string of length $x$ along the unit circle, starting from $(1,0)$ and going in a counter-clockwise direction".

So how do you remember what $\sin'(x)$ is? You sketch the graph of $\sin(x)$, lay down several tangent lines, and plot the slopes of the tangent lines. Observe that the slopes vary continuously, and fill in the gaps appropriately with your knowledge of the local maxima and minimal of $\sin(x)$. Then you observe the graph looks like $\cos(x)$, so you conclude $\sin'(x)$ must be $\cos(x)$. This isn't a formal conclusion, more of a seat-of-the-pants "well, how complicated can $\sin'(x)$ be?" type argument.

Similarly, the double-angle formula for $\cos(2x)$. How do you remember that? The idea is that we know there is a formula that relates $\cos(2x)$ to either $\cos(x)$ or $\sin(x)$ but people tend not to have the exact form on immediate recall. I sketch the graph of $\cos(x)$, then sketch the graph of $\cos(2x)$ as the previous graph with double the frequency. $\cos(2x)$ has twice the frequency -- you can also get twice the frequency by looking at $\cos^2(x)$ or $\sin^2(x)$ -- so spend some time sketching the graph of those functions. This amounts to knowing the graph of $\sin(x)$ and $\cos(x)$, plus the graph of $x^2$ -- to know that $\cos^2(x)$ for example not only has twice the frequency but that when $\cos^2(x)=0$ the function looks quadratic -- like $x^2$ around those points. From there it's just a scaling argument to say $2\cos^2(x)$ looks like $\cos(2x)+1$.

So the point of all this is that, with enough patience and geometric insight, all these formulas that students encounter you can cast them as things that, by their very nature they must be easily re-discoverable with very little probing. In particular, provided you're comfortable enough graphing functions it's less energy (and really far more informative) to remember the forms of these identities by intuiting them from your rough expectations. In that way you can re-cast identities like

$$\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$$

in a different light. The fact that this identity has such a simple form is somewhat remarkable -- but once you know it has a simple form, re-discovering it via non-rigorous arguments and heuristics is rather immediate.

Certainly, this line of reasoning isn't great for all students as it assumes a certain familiarity and an intuition with graphing that many students may not have. But once students have the knowledge that this is something they can think about, and that other people think this way -- that the discovery process is full of intuition and guesses before anything rigorous -- this sets them on a path.

There is no silver bullet - the hardest part of the project is getting the student curious about maths again. Once you do that, teaching them the skills to pass the exams is comparatively a breeze.

Unfortunately, one is usually hired by parents when the student has a problem, and not before, so motivating the student is a key part of initial sessions. It helps to remember that good students who have gone off maths were once good students who enjoyed maths; you need to try and find out what they liked about maths when they enjoyed it, and when they stopped enjoying it and why.

Find out why the student is there - often students need to pass maths to do something in the future that they are interested in (to get in to college perhaps); focussing on the goal can motivate some students.

I agree with J.M. and Ross that puzzles can be an effective way to engage good students whose interest in maths has been diverted. For example, a favourite of mine for kindling interest in trig is Table in Corner where elementary trig is used to solve a problem that appears to have too little information given (I drop the quarter circle condition when I state the problem - it provides an opportunity to discuss the reasonableness of the two possible solutions).

Elementary number theory is another rich source of good puzzles, and even tricks like divisibility tests can come in handy. I remember getting one student interested in practising the long division algorithm by coming up with exact multiples of horrible small primes like 13 and 17 for her to divide. She knew I couldn't possibly know the 17 times table to several thousand places, so where was I getting these numbers from? We discussed the plausibility of her suggestions and I kept her guessing for most of the session before I explained. She got the algorithm, and she got to understand essentially how it works in one go.

Also remember that abstraction means different things to different people. Students often say maths is abstract because they only see it employed in textbooks, where every problem is clearly contrived, absolutely divorced from the real world, and one of a collection of standard types (puzzles usually avoid the same problem by being contrived but non-standard). Since textbooks never give plausible real-world applications you can talk about applications often, in fact it is helpful to have a bank of non-contrived applications for most of the topics you want to teach. Books by the likes of Ian Stewart and Keith Devlin can help you here. Usually I prefer to say stuff like "who cares, it's fun" (and then give an application a bit later on), but then that's me - YMMV.

It depends upon the student's motivation. When you are hired for high school, is it by the parents when the student has a problem? And in pre-engineering is it the student? This is a big difference. If the student isn't motivated, I have no ideas. If s/he is motivated, showing what math is good for, as problems in the area of interest is a good approach. I've done a bit of this and what worked there was just to take their text and work exercises. The one-on-one work seemed to help understanding. For the general high school situation, again assuming motivation, there are many problem books that require creativity. Martin Gardner and Peter Winkler are the names I would first think of, but there are many more. Also the first problems at projecteuler.org give you a lot to work with.