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When I was a freshman in Mathematics we learned the usual integration techniques (lots of standard integrals, integration by parts, substitution, partial fractions,…). As homework we simple got a bunch of integrals we had to determine. Now I am teaching freshman Mathematics and also teach integration techniques. But given that you just type

integrate 'basically any function'

into Wolfram Alpha and you do not even need any software (besides a browser) or any knowledge of syntax whatsoever, I find it worthless to give homework of the form "Find antiderivatives of $\sin(x) e^x$, $1/\tan(x)$…" and so forth. Of course there will be students who take the challenge and go ahead and learn something but there will also be students who just use available technology and I really can understand them. Integration and especially subtechniques like partial fractions are things that computers can do much better than humans–at least for the standard homework integrals.

I don't want to argue about if one should still teach integration techniques or not (could be a good separate question) but ask under the premise that it is meaningful to learn integration techniques and that homework has to be done:

How should you design homework today to let students learn integration techniques?

Related question with a different focus: How to assign homework when answers are freely available or attainable online?

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    $\begingroup$ I had to look up "primitives." Perhaps that term is not common in the U.S. "Antiderivative" and "indefinite integral" seem to dominate. $\endgroup$ – Joseph O'Rourke Feb 17 '15 at 12:54
  • $\begingroup$ Good point, changed to "antiderivative"! $\endgroup$ – Dirk Feb 17 '15 at 15:37
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    $\begingroup$ I add as only a comment: Norton Starr's (2005) Estimating Definite Integrals from the MAA's College Mathematics Journal has some nice example problems for integration and ideas about how they could fit into a calculus course. (Side-note: I took several courses with Starr -- who earned his Ph.D. under Rota -- around the year that this note was published, including his course on Calculus III [having taken Calculus II with the professor named in example iv]. Excellent courses, in retrospect...) $\endgroup$ – Benjamin Dickman Oct 12 '15 at 18:28
  • $\begingroup$ See also matheducators.stackexchange.com/questions/7655/…. $\endgroup$ – J W May 28 '16 at 18:16
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I don't want to argue about if one should still teach integration techniques or not (could be a good separate question) but ask under the premise that it is meaningful to learn integration techniques and that homework has to be done: How should you design homework today to let students learn integration techniques?

One answer is trust, but verify. For example, you can assign low-stakes or zero-stakes homework on a certain technique of integration, then give a higher-stakes quiz or exam in which your students need to use the same technique in a situation where they don't have access to a computer algebra system.

Also, presumably you are assigning work that requires both computational skills and conceptual understanding. That is, you're not just assigning a list of 50 problems that all look like $\int f(x) dx$, where $f$ is some given function. When the problems are less stylized than this, the students who lack the computational skills are the same students who will not be able to figure out all the conceptual aspects of the problem, or even how to input the problem into a CAS. For example, it's very common to see students who can't handle any variable of integration besides $x$, or who can't understand how to handle symbolic constants, e.g., $\int dz/(z-a)$. If they don't understand what they're doing, then they aren't likely to be able to even put such a question into the form required by a CAS.

And of course you're doing applications, right? A CAS isn't going to be able to convert a word problem into an equation, or interpret the result of the calculation and draw a conclusion.

Below is another example of a type of problem that doesn't lend itself to solution by a CAS.

In problems 1 and 2, two indefinite integrals are given that involve functions which look similar to one of the following: \begin{equation*} e^{-x^2} \qquad x^x \qquad \frac{\sin x}{x} \qquad e^x \tan x \end{equation*} As discussed in section 9.3, the four functions given above can't be integrated in closed form. In each pair below, one can be integrated, while the other can be made into one of the above forms by a substitution, proving that it's impossible to integrate. Determine which is which, integrate the one that can be done, and check your answer to that one online.

1 (a) $\displaystyle \int x^{-3/4}e^{-\sqrt{x}}\: d x$ (b) $\displaystyle \int x^{-1/2}e^{-\sqrt{x}}\: d x$

2 (a) $\displaystyle \int x^{-2}\sin\frac{1}{x}\: d x$ (b) $\displaystyle \int x^{-1}\sin\frac{1}{x}\: d x$

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    $\begingroup$ +1 for the example at the end $\endgroup$ – Dirk Feb 17 '15 at 21:15
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    $\begingroup$ I wouldn't say "can be integrated" but "can be integrated with the functions we've used in this course." After all, this probably just follows the unit defining $ln(x)$ as a new function to handle an integral -- why deny that you can define $Si(x)$ or $erf(x)$ too if you want? $\endgroup$ – user173 Feb 18 '15 at 3:15
  • $\begingroup$ Do you know of a short discussion of $\frac{\sin x}{x}$ and why it "can't be integrated in closed form"? I'm not sure what your "section 9.3" refers to; a reference I know of offhand is provided at MSE here. $\endgroup$ – Benjamin Dickman Feb 18 '15 at 4:50
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    $\begingroup$ @BenjaminDickman: The reference is not to a proof that $\sin x/x$ can't be integrated in closed form, it's just a reference to a brief discussion of the fact that such examples exist, and a list of a few such examples. $\endgroup$ – Ben Crowell Feb 18 '15 at 6:00
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First of all, I try to be honest with my students by telling them directly and explicitly about the existence of such integration machines (It is silly of me assuming that they don't know that!). Then, I add, but we don't integrate for the sake of integration. For us, it is a practice of problem solving in which we need to choose the right techniques from the existing ones.

Having this change of perspective in mind, a good homework is the one that asks students to choose integration techniques. Consider this one, $x/(x^2-9)$, from an old paper of Alan Schoenfeld (see below). How should we evaluate the integral? By substitution, partial fractions, or by a trigonometric substitution!

I learned this "trick" years ago from a teacher who had to teach a meaningless algorithm (at least for students) for finding square root to her students. She changed the lesson from being a lesson about finding square root to a lesson about algorithms and the way they are written for a computer. My suggestion for integration techniques was just one way to use this trick. I am sure you can come up with some other ways. If yes, please let me know.

PS. These two papers of Schoenfeld Teaching Problem-Solving Skills and Presenting a Strategy for Indefinite Integration are quite related to the kind of change of perspective suggested above for integration.

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    $\begingroup$ Schonefield's second paper starts by saying "Students in a first-year calculus class should, with a reasonable amount of practice, be able to evaluate indefinite integrals with much the same facility as their instructor does." I disagree! $\endgroup$ – user173 Feb 18 '15 at 3:17
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    $\begingroup$ @MattF. That claim is somehow related to the ideas of "novice" and "expert" that were in use those days and the point was to help a novice to behave (think) like an expert. In this light, you can see the background and the reason of his claim. $\endgroup$ – Amir Asghari Feb 18 '15 at 18:44
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It seems to me that it is not the homework that needs any special design; it is only the way you grade it. Don't just ask for the answer, ask for a complete proof using the techniques taught in class. It is easy to check one's answer with Wolfram alpha, but I think one needs some kind of understanding to write a complete justification even when using a computer.

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  • $\begingroup$ Unfortunately, paying for W|A will often provide steps. $\endgroup$ – Opal E Oct 15 '15 at 22:15
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Give problems that a CAS won't be able to evaluate.

For example, look how Wolfram Alpha fails to give

$$\int x^{10^{10000}} \mathrm{d}x = \frac{x^{10^{10000}+1}}{10^{10000}+1}+C$$

Several other integral calculators I tried also fail to give it. Of course this is rather a simple integral, but one may also apply it to other integrals, like $\sin(10^{10000}x)$.

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  • $\begingroup$ I am not sure how long it will take until some CAS will have fixed this loophole. Anyway, I guess it can be cumbersome to figure out examples for which a CAS fails. Finally, this does not seems like a genuine approach to "find exercises that help students to train their integration skills and understanding, knowing that software exists that can do most integrals". $\endgroup$ – Dirk Oct 15 '15 at 7:40
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If you are collecting and grading homework, can't you also check that they used right technique to get the answer? NWNC (no work no credit, acronom from Navy nuke school).

If you are not collecting and grading homework (I wouldn't, just let the students drill themselves, try to pick texts with all the answers like Thomas Finney), than who cares? Just use frequent tests or quizzes to drive performance. When I took AP calculus, we had a full period test every Friday (first period and the teach had them graded during her practice period and available before day end).

Granted in college you don't have as many seat hours with the kids, but you could still do a test every second Friday (assuming MWF schedule). Really a lot of research shows that practice or tests (tests are emotionally intense practice episodes) are more helpful to learning than lecture anyhow. So don't feel bad about getting out of an hour of stage (or guide on the side) time.

Jaime Escalante did daily quizzes. (very short at beginning of class. No reason you can't do the same, even have the kids exchange papers and grade them so now work for you. Don't worry about cheating as it is minor and if you keep the point value low, it will still motivate kids but not be that much of a differential determiner.)

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I think there's an assumption here that as the instructor you are: (1) collecting homework, (2) correcting the homework, and (3) giving grades for the homework.

Honestly, I wrestled with doing that for about a decade and I decided that I simply had to give up on it. The overall process was so time-consuming, so distasteful, and so difficult for me to parse poorly-written student work that I was simply tearing my hair out. I think there was a Thanksgiving break where I had a quasi-breakdown over it and said to myself that it just had to stop.

So I think that a perfectly legitimate strategy at the college level is this: Don't collect, correct, or give points for homework. The homework exists as an exercise for the student to prove to themselves that they've mastered the skill. My students don't wait for me for confirmation: immediately check the answer at the back or, yes, type it into Wolfram Alpha (for, say, even-numbered problems). If they don't get it right immediately, then fix it. If they can't fix it on their own, have a study group or go to the math workshop. If they can't fix it then, ask about it in the next class.

I start every class with a call for any questions about the homework, and as long as a single student asks about something, I try to lavish them with praise and attention as we work through their problem. Hopefully this demonstrates the expected interaction and primes the pump for other future questions.

But grading is purely on the basis of quizzes and in-class tests (where of course, well-written justifications must be made in each case). As one example, my college algebra classes almost uniformly fail the first test, no matter how much I emphasize every day what the grading criteria will be. But generally after this "shock treatment" they do bounce back, mostly start doing their own homework, and asking questions in class about the best way to write or present results.

There's no way to verify that students are doing their own homework, so I think it's best to be honest about that fact and just say that the homework is for their own benefit to exercise and prove their knowledge to themselves. The graded tests are how we later assess that the skills and knowledge are really present.

Edit: To be clear, my point is that in discarding protocols 1-3 (the expectation that homework will be collected, corrected, and awarded grades), then the existence of automated math engines is non-problematic, and traditional math homework is seen to be as meaningful as ever.

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    $\begingroup$ You are right about assumptions 1 and 2 but not 3. No grades for homework, only pass or fail. Also, I make no attempt to prevent cheating. Finally, this answer is basically unrelated to the question. $\endgroup$ – Dirk Oct 11 '15 at 20:02
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    $\begingroup$ I add a twist, in that a "random" selection of students has to explain the homework to the TAs. I'm interested in them understanding, if they copied or got a CAS to solve it is irrelevant. $\endgroup$ – vonbrand Oct 12 '15 at 0:50
  • $\begingroup$ @Dirk: Pass or fail is a type of grade. $\endgroup$ – Daniel R. Collins Oct 12 '15 at 6:02
  • $\begingroup$ Regarding the edit: The question asks "How should you design homework today to let students learn integration techniques?" and not "How should you design homework today to grade students work on integration properly?" So I am looking for ways to make homework on integration techniques more inspiring and motivating. $\endgroup$ – Dirk Oct 12 '15 at 8:14
  • $\begingroup$ @Dirk: I have answered that question. Students learn from traditional homework as well as, or better than, anything else. The only point of confusion comes from "I get points for presenting the answer", which short-circuits the goal of the practice. $\endgroup$ – Daniel R. Collins Oct 12 '15 at 14:07
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You express a strong sentiment that technology gives students a reason to not need to integrate, but why do you feel like this particular topic in their mathematical careers would be different than any others prior? The same argument might be made in regards to simple arithmetic. 'Why bother with learning how to divide when a basic five function calculator can do the same thing?' I'm relatively certain that there is now a smartphone application that allows users to take a picture of a linear equation and it immediately outputs the answer. 'Why bother with learning Algebra if my phone can do it?'

Technology and Mathematics Education have had this type of relationship for (at the very least) the last century. I would not worry about students cheating if I were in your shoes if only because if a something is seeking answers from technology for integration solutions, they have probably been doing it for earlier topics.

Your question seems to knock on the door of 'why learn mathematics at all if technology can do it better than I ever can?' Of course, as mathematicians and mathematics educators, we understand the value in learning the content to help further the field and to apply the mathematical principles to other scenarios. To me, mathematics has always been about the journey, and not the destination. If you can instill this in your students, then you should feel comfortable giving them any problem set.

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    $\begingroup$ Actually this is not answering what I was asking. What you touch upon is also an important topic but should be discussed in a different question. $\endgroup$ – Dirk Feb 23 '15 at 6:34
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    $\begingroup$ I agree with Dirk, and for this reason I have downvoted. $\endgroup$ – Mark Fantini Feb 23 '15 at 7:39
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    $\begingroup$ I also agree, please retract the answer and ask a new question. Or several, there are different areas in which technology threatens to take over... $\endgroup$ – vonbrand Oct 12 '15 at 0:52

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