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Is the current state of math teaching in undergrad college courses struggling with the availability of easy cheap access to Wolfram Alpha?

The homework problem below, one of 40 assigned from one community college lecture, took my son and me each several hours to solve. (He struggled because he's new at this; I struggled because I was rusty at it.) He is enrolled in a second-quarter calculus course which he's already failed once after a lifetime of not struggling in school.

Find the arclength between 2 and 5 of the function:

$f(x)$ $=$ $x^5 \over 10$ $+$ $1 \over 6x^3$

The answer is $618639 \over 2000$, taken through the arclength integral

$\int_{a}^{b} \sqrt{1+(\frac{dy}{dx})^2} dx$

That is:

$\int_{2}^{5} \sqrt{1+\frac{(-1 + x^8)^2}{4x^8}} = {618639 \over 2000}$

His question is: Why bother with the tedious symbol manipulation given WolframAlpha as a cheap tool? The answer I'm looking for is a description for why current college maths teaching still sets problem sets like this.

I can order him to buckle down for a good result and already have; no need for those "it's good for you to struggle" answers. One analogy that justifies him is the fact that teachers no longer require students to learn how to cipher a square root. So, why would we require them to go through algebra contortions like this if the tool to do the "monkey work" exists and he can spend his time reviewing its output to make sure he set the problem up right?

I disagreed: This problem exposed me to insights about the fundamental theorem of calculus and the shortcuts possible with definite integral problems. The tedious problems teach you the shortcuts Wolfram Alpha is using between steps. You're using these kinds of problems to hone your thinking skills.

Is there anything salient I could add to that argument? Is this a controversy in maths teaching, or is my son just plain wrong and needs to drill for other reasons?

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    $\begingroup$ You are correct that this is an issue of contention among mathematics educators: How important is it for students to work through calculations when they could instead be learning ideas and concepts? What is the right balance between these two activities?. $\endgroup$ – Mike Pierce May 2 '18 at 23:07
  • $\begingroup$ I learned a lot just working the problem. Remembered what I'd forgotten. $\endgroup$ – Rob Perkins May 2 '18 at 23:37
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    $\begingroup$ Being curious as to how tedious the computations were, I worked this out. While working this, it became obvious this an “old style problem” specifically designed for hand-computations, which it appears you recognize and which I’m sure most others here also recognize, but for the record . . . $\frac{dy}{dx} = \frac{5}{10}x^4 - \frac{3}{6}x^{-4} = \frac{1}{2}(x^4 + x^{-4}),$ so the radical part becomes $\sqrt{1 + (\frac{1}{2})^2(x^4 + x^{-4})^2} = \frac{1}{2}\sqrt{4 + (x^4 – x^{-4})^2},$ or $\frac{1}{2}\sqrt{(x^4 + x^{-4})^2} = \frac{1}{2}x^4 + \frac{1}{2}x^{-4},$ (continued) $\endgroup$ – Dave L Renfro May 3 '18 at 11:52
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    $\begingroup$ so the indefinite integral is $\frac{1}{10}x^5 - \frac{1}{6}x^{-3},$ so the definite integral is $\frac{1}{10}(5^5 - 2^5) - \frac{1}{6}(\frac{1}{125} - \frac{1}{8}).$ Are calculators allowed? I spent more time rewriting this as a single reduced fraction (I'm omitting the details here) than I spent with the algebraic manipulations. In my opinion, having to do a lot of these can be tedious, but if doing this with a calculator took more than $10$ minutes (and I’m being pretty generous), then I think more practice is probably needed, because nothing I did was all that tricky. $\endgroup$ – Dave L Renfro May 3 '18 at 11:52
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    $\begingroup$ I remember learning the process for extracting a square root in elementary school in the 70s. It was very powerful concept--not the specific tool, but the type of tool. It was called an "algorithm" and we discussed the etymology and the concept--"like a computer program". It was a whole door that was opened. Has always stuck in my head as a mental image of a computer program, now. A set of steps. Then again, see the article "Why Johnny can't code" by David Brin. Having some intuition about things, versus a black box can be helpful, even in CS. $\endgroup$ – guest May 3 '18 at 13:10
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  1. Why have it at all: You need to know it to solve physics E&M problems. (Not even junior level E&M, just calculus-based E&M.)

  2. In terms of why ever learn to do it by scratch, it's really part of a much bigger question (could ask the same about a lot of math content). My personal take is that if you are going to do engineering or a physical science, that this is reasonable to learn how to do, explicitly. Sure, if you have a computer and you are solving these problems all the time, you automate it. But if you are reading papers, following derivations on the blackboard, etc., you should be able to do this sort of problem. If you only have a black box knowledge (plug it into WA), you'll struggle in standard physics and engineering classes.

  3. I don't think it is really that much symbol manipulation. Lot less than a series solution to a diffyQ or the like. Just memorize the formula wih the dy/dx squared or how to derive it (think it is a Pythagorean relation). Or both. The only real hard part with arc length problems is remembering the formula. The work itself is not that intricate (not some double integral washers thingy.)

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  • $\begingroup$ Yeah, it's an application of the Pythagorean theorem, and compared to the CFD and FDM stuff I do in industry, not at all that much symbol manipulation. There's irony: that same day he was bored with his beginning electronics course professor, one of the career exploration 101's, (Ohm's Law, etc) for not diving into the physics more. $\endgroup$ – Rob Perkins May 2 '18 at 23:36
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    $\begingroup$ Do you have a good source of E&M problems that would need this kind of computation? I would love to have this as a resource. $\endgroup$ – Chris Cunningham May 3 '18 at 15:26
  • $\begingroup$ I think I was thinking of line integrals. $\endgroup$ – guest May 3 '18 at 15:55
  • $\begingroup$ Just looked and can't find much on physical applications of arc length other than suspension cables. $\endgroup$ – guest May 3 '18 at 16:08
  • $\begingroup$ That makes the application of arc length reducible to a specific rule of thumb... if you're not engineering the bridge, that is... $\endgroup$ – Rob Perkins May 3 '18 at 22:13
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I think I agree with the sentiment of the OP which is why in my classes (which do not allow technology on exams) I only ask that they set up the integral correctly (including correct limits, an integrand where derivatives have been performed, and the correct differential). The hard part about this section of the content is remembering the right formula when you start to have lots of properties that are being computed: arc length, area below, area to left, and surface area and volume when being revolved. (And this is most easily achieved by remembering the geometrical reason behind it, e.g. Pythagorean Theorem.)

The great thing about this approach is that you’re free to give the students more realistic and challenging scenarios; you don’t have to worry whether the algebra in the anti-derivative will be doable. (Heck, you don’t even have to worry if the function is algebraically integrable.) And, yes, on homework and projects (or in their science classes) I expect that, once they have the correct integral, they will use Wolfram Alpha to find what they need from there.

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    $\begingroup$ Interesting answer. Do you by chance have an example of an exam that you have given, available online to look at? $\endgroup$ – mickep May 3 '18 at 18:03
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    $\begingroup$ I did this very often on calculus 2 tests for applications of integration (centers of mass, areas, volumes of revolution, moments of inertia, etc.), but almost none are online. One example (this for a business calculus final exam) is problem #12 on p. 5 of this test. Incidentally, some of the graphs did not fully render correctly in the .pdf format that was posted. As for actually evaluating such integrals, this was done on homework, which I tried to avoid having to grade, (continued) $\endgroup$ – Dave L Renfro May 4 '18 at 8:20
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    $\begingroup$ so I gave lots of short/medium/long in-class single topic quizzes, often allowing them to use their notes and homework while taking the quiz. For applications of integration I sometimes had "group quizzes", usually 2 students per group, this being mostly a graded classroom activity for learning purposes than a formal assessment, and one favorite of mine was to have them calculate areas, center of mass, etc. using TWO methods $(x$-integration and $y$-integration), with the intent that one student does it one way, the other the other way, (continued) $\endgroup$ – Dave L Renfro May 4 '18 at 8:26
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    $\begingroup$ @mickep: and they check that they get the same numerical result, trouble-shooting their work until they do (sometimes with my help --- I wandered around the room helping as needed during this). For example, this 23 February 2003 sci.math post gives a lot of such worked examples. In fact, that sci.math post was originally an email to my students that I thought worth archiving in sci.math for others to use. $\endgroup$ – Dave L Renfro May 4 '18 at 8:28

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