College Math Teaching

December 21, 2013

Rant: please stop with the teaching of “gimmicks for calculation”

Filed under: calculus, editorial, integrals, student learning, Uncategorized — Tags: — collegemathteaching @ 1:05 pm

I finished teaching calculus II (our course: techniques for integration, applications of integrals and infinite sequences/series) and noticed that some of our freshmen students came in knowing how to do many of the calculations…did well on the first exam…then didn’t do so well in the rest of the course.

Evidently, they were well versed in calculation tricks learned in high school; give them \int x^3 sin(x) dx and they could whip out a table.

So here is my rant: we teach integration by parts not so much to calculate integrals like \int x^3 sin(x) dx (which can be rapidly done with a calculator) but rather so they can understand the technique of integration by parts.

Why? Well, there are many uses of integration by parts and I’ll just display a few uses of them:

1. Taylor Polynomials. How do we get these? If we assume that f has enough derivatives, we proceed in the following manner: calculate \int ^x_0 f'(t) dt in two different ways: use the Fundamental Theorem of calculus on one side (to obtain f(x) - f(0) and use integration by parts on the other side: u = f'(t), dv = dt, du = f''(t), v = t-x (yes, we are being choosy about which anti derivative of dv to use).
This means: -\int^x_0 f''(t)(t-x)dt +f'(t)(t-x)|^x_0 = f(x)-f(0) so f(x) = f(0) + f'(0)x -\int^x_0 f''(t)(t-x)dt =f(x) and one proceeds from there.

2. Differential equations: given y' + p(x)y = f(x) one seeks to find an integrating factor (which is e^{\int p(x)} so as to get:

e^{\int p(x)}y' + p(x)e^{\int p(x)}y = f(x)e^{\int p(x)} which can be written as \frac{d}{dx}(e^{\int p(x)}y) = f(x)e^{\int p(x)}. That is, the left hand side is just the product rule for derivatives, which, as you know (if you are a calculus teacher), is really all integration by parts is!

Sure, one can jazz it up (as we subtly did in the Taylor Polynomial calculation); the integration by parts formula is really \frac{d}{dx} (f(x)g(x)) = \frac{d}{dx}(f(x)+ C) g(x) + f(x)\frac{d}{dx}(g(x) + D) where C, D are arbitrary constants. But, my main point is that integration by parts should be UNDERSTOOD; short cuts to do tedious calculations are relatively unimportant, IMHO.

Now if you want to ask students “why does tabular integration work”, then….GREAT!


November 19, 2013

I hear you Secretary Duncan

Filed under: academia, editorial — Tags: — collegemathteaching @ 1:27 pm

I won’t weigh in on the merits of “common core standards” as I have no evidence one way or the other. But I’ll remark on this remark:

U.S. Education Secretary Arne Duncan told a group of state schools superintendents Friday that he found it “fascinating” that some of the opposition to the Common Core State Standards has come from “white suburban moms who — all of a sudden — their child isn’t as brilliant as they thought they were, and their school isn’t quite as good as they thought they were.”
Yes, he really said that. But he has said similar things before. What, exactly, is he talking about?
In his cheerleading for the controversial Common Core State Standards — which were approved by 45 states and the District of Columbia and are now being implemented across the country (though some states are reconsidering) — Duncan has repeatedly noted that the standards and the standardized testing that goes along with them are more difficult than students in most states have confronted.

Emphasis mine.

I don’t know about “white and suburban” but as far as parents having an inflated opinion of their kid’s abilities and the incoming students themselves: oh my goodness yes.

In some ways, university administration doesn’t help. I understand: to GET students to come to your university, you have to massage their egos a bit. No new freshmen means “no jobs for the professors”, at least at the smaller, “undergraduate education oriented” schools.

But…well, many of these students did well at their high schools and yes, they probably think that their high school is “one of the best” and they don’t seem to understand is that most of the incoming class has roughly the same level of talent.

Most have not been around a truly brilliant person and have no conception of what true brilliance is. Hey, if they…gasp…differentiated a polynomial in high school, they “know calculus”, right? 🙂

Rant to follow
It is my opinion (based only on my limited experience, not evidence), that most really don’t want an objective evaluation of their abilities and accomplishments; they want PRAISE, period.

And yes, there have been rare occasions when I’ve thought “so and so won’t make it” when, in fact, they had more talent that I’ve realized. But, surprise to surprise, most students have…uh…average talent. Judging from the boasting that I’ve seen on the social media sites, their respective parents will never believe that in a million years though. 🙂

Note: a reader pointed out that he thinks that the resistance to “common core” comes from the widespread belief that “local control is best”. That may well be true. This post does not mean that I think that Secretary Duncan is correct in the reasons that “common core” is resisted but rather that I agree with his assessment that many overestimate the intelligence of their kids.

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