Motivated by discussions on a Calculus Forum
Once upon a time I taught math at the University of Waterloo. Indeed, I did that for some 30 years!
Currently, there seems to be some discussion concerning whether students should be able to use all the techno-gadgets available.
I always thought that students should learn how to solve problems ... with any aid available.
Well, I mean symbolic algebra systems, like Maple, and/or graphing calculators and/or ...
For example, I might propose the following problem:
You are to build a circular, concrete wall about a swimming pool.
If a student could describe a valid procedure, as a sort of essay, I would be ecstatic.
It is to be 5 ft. thick at the base and 5 ft. high.
The inner radius of the circular base of the wall is 30 ft.
The cross-section of the wall has the shape of a cubic polynomial.
Describe how to solve for the shape that uses least concrete.
>What if the ground isn't level or ...?
Yes, but the student must make assumptions and that'd be part of the solution.
For example, it'd be great if the student began by saying:
Assume the ground is flat and level, that the x-axis lies along the ground with the origin at the inner base of the wall ... as in Figure 1.
Further, assume that y(x), the cubic, satisfies y(x) > 0 for 0 < x < 5
... to avoid something like this:
>Is it hard ... that problem? Can you solve it? What's the answer? How do you ...?
Let's continue with our imaginary student's solution:
The equation for a cubic is y = Ax3 + Bx2 + Cx + D ... with 4 unknown constants.
 y = 0 at x = 0
 y = 0 at x = 5
That gives two equations in the 4 unknowns.
At the location of the maximum height of the wall, at say x = M, we require:
 y = 5 at x = M
 dy/dx = 0 at x = M
That gives two more equations ...
>But what's M?
Now we have 4 equations in 5 unknowns: A, B, C, D and M.
Proceeding with our imaginary student's solution:
Solve for A, B, C and D in terms of M from equations  to .
The cubic is now expressed entirely in terms of M.
Determine V(M), the volume of the solid when the area beneath the cubic is revolved about x = - 30.
>And find the maximum value of V(M), right?
Determine the value of M which minimizes V(M) by setting
 dV/dM = 0.
>And that's it?
To see if the student's solution procedure works, we let her use something like Maple.
Note that there were 5 equations in 5 unknowns.
As we vary M, with  to  satisfied, the various cubics might look like Figure 2.
>But you said y > 0! One of those cubics look like it's got y < 0!
Yes, so the student would have to get the M-value which minimizes V(M) subject to y > 0 in 0 < x < 5.
>That's not an easy problem!
Aah, but if a student could describe a valid procedure, wouldn't you be happy?
Would it matter if the student forgot the formulas for volumes of solids of revolution?
Would it matter if the student had difficulty adding fractions?
Would it matter if the student ... ?
>If I could come up with the procedure I'd hire somebuddy to do the grunt work at $9.00 an hour.
Exactly my point!
A typical lecture has the teacher writing on the blacklboard (perhaps reading from notes) and students (hopefully) listening and taking notes.
>Yeah, that's what I figure ... but how would you know what's typical?
I spent a few years as
chairman chairperson of the Applied Math Dept. and had to determine annual salary increases, based upon Teaching, Research and Service
(serving on committees, organizing courses, reading/writing reports, etc.).
To assess teaching, I'd stand (unseen) outside of a classroom listening (and peeking) while a colleague gave a lecture.
Anyway, that one-way lecture is a lousy way for students to larn anything. Better is to have a conversation between teacher and students, with plenty of questions and ...
>I always felt my questions were stupid. Nobody else was asking questions. They all knew the answer, so I ...
And that discouraged you from asking, right? Yes. I know.
In my classes I always told students that, no matter the question, I'd make them look like Einstein if they asked it.
Then, when a student asked, I'd say: "Now that's a good question!"
>You're kidding, right? I mean, even the stoopid questions?
Well, when the first question of the year was asked (after I told them about the Einstein thing) I'd say: "That's the stupidest question I've heard all year."
Then I'd laugh ... and the class laughed ... and the ice was broken. Then I'd say: "Actually, that's the best question I've had all year."
In any case, classes were great fun with a forest of hands in response to a question I asked ... and many hands went up when I gave a lousy explanation.
>Which was often, eh?
Uh ... yes ... sometimes. When I first lectured in Math I was afraid to make a mistake.
When I was more "mature", I often made mistakes (sometimes purposefully) ... and encouraged students to identify them.
>Which they did in huge numbers, no doubt.
Well, not exactly HUGE.
Alas, it ain't easy for a textbook to have this type of conversation with students.
So, when I wrote up a set of lectures on Introductory Calculus, it was a conversation between a professor P and a student S.
The student would say things my students might have said. For example:
>Yeah? So where's this course? Is it online? Can I ...?
Yes, it's here.
>So why don't you write a textbook?
Funny you should ask. Once upon a time (long, long ago) a textbook salesman asked if his publisher could see my notes.
I agreed, and printed several lectures for him.
Eventually I got a letter from the publisher saying ...
>They wanted to publish it!
... saying it was unusual and non-standard ... and wouldn't sell.
>I've read your notes and ... uh, I agree.