[SlyShy]

Math major here: I think actually he is arguing for a cure to the apathy. The reason people are apathetic about mathematics is it is presented like a dry and idiotic subject. A common complaint about word problems is "this is contrived, why does the water tank have exactly that parabolic curve, pfft math sucks, I quit" whereas the common complaint about questions that are pure symbol manipulation is "this doesn't matter to me, I quit."

By introducing questions more open endedly (Could you kick a door down?) you avoid making the question sound contrived. Open ended questions also lead to generalization. (Side note, I've worked as a tutor and coach, so yes, I have classroom experience). The students get to saying, "well, we can't solve this question until you give us numbers." They get several sets of numbers, and they begin to realize that the method for solving the problem is generalizable. They develop the formulas from the examples, and learn they could have answered the open ended question all along.

That's in stark contrast to the current model, where the formula is taught, and then specific examples of close ended questions are presented. The key to teaching mathematics is fostering exploration.

The very best illustration of this idea is the Ross Mathematics Program in Columbus, Ohio. It is a two month long ground up rigorous exploration of number theory and abstract algebra. And here's the best part, there is no background necessary. If you know arithmetic, you have enough to begin, because all you start with the axioms of the natural numbers. And yet, the students exit the program having proved that groups of order p are cyclic, the Quadratic Reciprocity Law, etc. Everything is learned through problem sets, which the students do at their own pace. Every theorem used is proved, and every new proposition is discovered.

Yes, it is a great deal of work, but nobody ever said teaching was easy work. Passion should be the #1 hiring criterion.

[yequalsx]

Using word problems is one likely to discover why sqrt(2x+1) = sqrt(x) + 1 is fundamentally different than cuberoot(2x+1) = cuberoot(x) + 1? Give this problem to calc one students and many will do both problems incorrectly. They'll incorrectly cube both sides in the second equation. They don't understand why we solve sqrt problems but not cube root problems but they know how to compute doubling time for bacteria growing in a petri dish.

Your fourth paragraph is an argument in favor of what I've been saying. The applications we teach ought to be applications to math; not to physics or biology.

But back to the apathy problem. People aren't going to be motivated by number theory or bacteria in a petri dish if they possess too great a disdain for knowledge.