Comments on: Why do we overcomplicate calculus like this?

By: Rohedi

Rohedi — Sat, 30 May 2009 10:33:14 +0000

Mr.Robert, Rohedi believes after reading the above explanation your college students as soon as understand to distinguish between average velocity and instantaneously velocity. My question, how to prove that dS/dt=0 is correct for S=constant. Thanks.

By: amca01

amca01 — Thu, 21 May 2009 00:49:24 +0000

A very good point indeed – calculus is indeed made much harder than it needs to be, and elephantine books like Stewart (and all the others) don’t help. What the world needs is a sequence of short, straight-forward calculus texts, which start from what the students know (invariably not very much) and build up carefully from there. I use no books – can’t stand any of them – but my own notes instead.

By: Keeping things in context « Casting Out Nines

Keeping things in context « Casting Out Nines — Sun, 01 Mar 2009 21:41:04 +0000

[...] been thinking that my post about the, er, pedagogically challenged way that Stewart Calculus does its examples about instantaneous velocity is really about the idea that you need to make sure that a person learning a new idea has some [...]

By: Robert

Robert — Sun, 01 Mar 2009 16:05:45 +0000

They had to repeat this for several time intervals of decreasing length. So I can see the utility of an average velocity formula. But still I think it’s more instructive and confidence-building to do several straight calculations by hand without algebra first, and then fold it in so there is some real motivation to make the investment (namely, the algebra automates the calculations and therefore makes life easier) — rather then just pop the algebra on them at the outset as though the reasons for doing so were obvious to all.

By: Kevin

Kevin — Sun, 01 Mar 2009 14:33:04 +0000

I sent you a PM about the comments.

By: Clint H

Clint H — Sat, 28 Feb 2009 06:51:57 +0000

I’m just curious: In the problem you mentioned, is the student expected to find the average velocity over several different intervals, or only on the interval [2, 2.5]? If s/he were asked to find multiple velocities, I could understand why you would want to find the complex algebraic expression. If s/he only needs to find one average velocity, I agree that the book is just being ridiculous.

This does, however, highlight what I tell my students whenever we start calculus: the hardest thing about calculus is usually the algebra.

By: jd2718

jd2718 — Sat, 28 Feb 2009 02:27:38 +0000

I think it’s great to use position as a function of time – but I recall hating finding average velocities. There was always something that was clearly not very useful about the answer…

Different from what you are seeing and saying, but you reminded me.

Jonathan

By: jedward706

jedward706 — Fri, 27 Feb 2009 19:44:16 +0000

Ah, well –
I suppose when you fret over enough of these issues with a particular text, you’ll find the motivation to write your own text — or — develop lecture notes sufficiently and find online resources to make the text unnecessary

By: Robert

Robert — Fri, 27 Feb 2009 16:16:18 +0000

The thing is, the actual book does this problem the same way. Use a ton of algebra and then plug in h = 0.5. I think the manual was just following the book, but oddly it doesn’t *explain* the book but merely repeats it.

By: jedward706

jedward706 — Fri, 27 Feb 2009 13:40:26 +0000

My suggestion:
If the “problem” is primarily the “solutions” manual, then eliminate it from the textbook package and focus efforts on improving/developing peer recitation groups — you can always post answers to even questions (those not in the back of the text) if students need the confidence boost.

For those problems, which you expect are difficult enough to warrant some problem solving “assist” — have a sharp student or TA post hints.

One big problem with solutions manuals is that students assume the solution is THE way to solve the problem — this may or may not be the case — and tends to squelch sharp students from creative and “class comment worthy” approaches.