I love the fresh thought here on how to best present problems to students. I wonder how much work it takes to redefine each problem in the textbook.
Some problems won't be easily changed to suit his paradigm. For instance, how does one go about redefining an equation like, sqrt(2x+1) = sqrt(x) + 1? I'd like to know what this guy does for these types of problems.
I've been teaching community college mathematics for 10 years and we simply don't have the time to do what he says. Maybe I'm bad at motivating students but my anecdotal experience is that most of the students are solely interested in getting a degree and not in learning. It's understandable that their focus is no getting a degree but focusing on learning makes it easier to get the degree. It's very hard to get this point across.
While I see many problems with the current system of teaching mathematics there simply is no cure to apathy. At some point one has to be willing to sit down and learn to solve problems like, sqrt(2x+1) = sqrt(x) + 1.
EDIT: Ahh I thought he was saying that he had factorized/simplified the left side into the right and it wasnt making any sense.
2x + 1 = x + 2sqrt(x) + 1
x = 2sqrt(x)
x^2 = 4x
x^2 - 4x = 0
x(x - 4) = 0
x = 0 or 4By 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.
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.
I suspect that he'd also argue that an hour spent slowly and carefully exploring 1 problem like this is better than having them do 30 examples with no context and no motivation.
How about this for motivation, we solve such equations because we can. Solving equations is useful and the more equations we can solve the better.
It's not really motivating when we give a word problem whose model is a radical equation and then say solve the equation. Most equations can't be solved by algebraic methods and anyone who really needs to solve an equation and trust the answer is better off having a computer do the computation.
In solving the 30 problems one might get to a point in understanding why sqrt(2x+1) = sqrt(x) + 1 is slightly harder than solving sqrt(x+1) = sqrt(x) + 1. In solving 30 problems one might get to a point to discover why sqrt equations can be solved but why we don't solve cube root equations. Or why sqrt(2x+3) = x is fundamentally easier than sqrt(2x+3) = x + 1. You won't get this from solving word problems.
Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.