Kamis, 26 September 2013

Mail Time -- Why is my algebra-based textbook using calculus?!?

One of my summer institute participants wrote in the other day with a question that I've been asked in similar
from this is NOT calculus!
form many, many times...

I discovered this morning that the textbook we have for the course, College Physics [by Serway and Vuille] has calculus-based content for 2D motion. I have resorted to using the textbook I have for general Physics, Physics: Principles and Problems by Glencoe, as an alternative for this section of material. Have any of you run into a similar issue or have any suggestions for other ways to communicate the necessary material at an AP level? I'm just wondering for this particular section, and if I run into a similar issue again down the road with content (since we're teaching algebra-based Physics).

Yeah, this is why I hate the standard-fare textbooks, written by Ph.D. physicists for Ph.D. physicists.  It's NOT calculus, even though it looks like it on first glance.  My correspondent pointed me to the section describing instantaneous velocity.  Serway uses a full page of equations, many of which look exactly alike to the untutored eye,* to make the simplest of conceptual points.

* My eye is tutored, but the 17 or 19 year old reading this text is not.  First rule of writing: know your audience.

The page in question uses mathematical notation that says the limit of a distance divided by a time as time goes to zero is the definition of instantaneous velocity.  Well, the eleventh grade student came to class angry: "We just started taking limits last week in my calculus class.  I've never seen this sort of thing before, and I don't understand it yet.  I thought you said we don't need calculus for this class!  How am I supposed to understand it?"

First, the general issue about algebra-based physics, calculus, and resources:  If students read texts or online information, they will often -- too often -- see mathematical explanations that resemble calculus.  See the picture above, which is using five rectangles to approximate the area under a curve.  "That's calculus, I know it!" says the novice.  Well, it's not.  The area of a rectangle is base times height -- that's 6th grade math, not calculus.  Even in algebra-based physics, we have to compare the slopes and areas of curved graphs... and you can expect someone to indignantly holler "Calculus!" when you draw the slope of a tangent line.  What "algebra-based" means is that we don't ever have to evaluate an integral or derivative of a function to make a numerical calculation or derivation -- it doesn't mean that we never look at curved graphs.

One of our tasks as a teacher of first-year physicists is to help them simplify tough ideas.  That means steering them away from poorly-written or misleading sources.  That means explaining in words, and minimizing mathematics.  A diligent but frustrated student must understand that yeah, the textbook is ridiculously confusing, but that doesn't mean you can get angry -- you simply have to find a different way to understand the topic.

Finally, the specific issue of Serway's presentation of instantaneous velocity:  All this math is just telling you that (1.) velocity is distance traveled per second, and (2.) "instantaneous" velocity means the velocity RIGHT NOW.  Distance divided by time is, generally, velocity.  If you use data over an hour, you get the average velocity for that hour.  But if you don't look at an hour, or a second, but at a fraction of a second, then you're looking at instantaneous velocity.  Serway is doing his highfalutin' physics professor best to say just that, but in mathematics.  I wish he'd speak English.


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