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(This question developed from a comment in the thread "Revisiting the chain rule".)

Students know that WolframAlpha and other software/computational resources exist and will make use of them as they see fit. Is there anything we can change about the way we teach calculus (in particular, for this thread) to make use of these tools? Ideally, we would like to simultaneously:

  • Prevent students from committing academic dishonesty (however it be defined in your context) and, hopefully, discover when it occurs
  • Show students the immediate benefits of using such resources: save time in computation, visualize a graph, check our work
  • Show students the long-range benefits of using such resources: explore similar problems (e.g. sensitivity to initial conditions for diff eqs), experimenting with problems to identify patterns, getting a "solution" to discover a method that would yield such an answer

I'm interested in examples where you have made use of WolframAlpha (or others) in class or on an assignment, and whether the students seemed to gain anything from it. Did you address some of the points I listed above? Are there other potential positive/negative aspects that you noticed?

I'm also somewhat interested in how you address questions like, "Why do I need to learn this when WolframAlpha can do it for us?", but I'd prefer if the thread didn't devolve into only discussing this topic. Instead, have you addressed such a query by showing students the benefits of bringing mathematical knowledge and intuition to a problem so as to make it tractable for software?

Assume all of this is in the context of a college calculus course.

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    $\begingroup$ Incidentally I created the technology-in-education tag in paralle. I though it fits nicely and thus replaced software and computers. $\endgroup$ – quid Mar 19 '14 at 19:18
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    $\begingroup$ possible duplicate of On the use of calculators in elementary/high school? Computer algebra systems later on? $\endgroup$ – vonbrand Mar 19 '14 at 19:32
  • $\begingroup$ @vonbrand: I think the motivation of this question is similar, but I like to think I have a more focused view on what answers will be here. That other question is much broader. $\endgroup$ – Brendan W. Sullivan Mar 19 '14 at 19:34
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    $\begingroup$ "Why do I need to learn this when WolframAlpha can do it for us?" Because the point of a math class really isn't to learn how to do computations, it's to learn how to recognize patterns in order to understand when certain computations would be applicable. $\endgroup$ – Santiago Canez Mar 19 '14 at 21:54
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    $\begingroup$ I want to reiterate my point in asking the question, and state that this should not be closed. I'm specifically asking about college calculus courses, and looking for activities/assignments that make use of these systems. I don't want this to be a general discussion of technology in teaching, which is what that "similar thread" seems to be. $\endgroup$ – Brendan W. Sullivan Mar 20 '14 at 18:49
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I think it is better to highlight the limitations of WolframAlpha. For example using the free trial version can only get the basic calculations. Unless you open a pro account (which costs money), the "Show Steps" function is often not available for most calculations. WolframAlpha easily refuses the calculations when it finds that it cannot express the results in terms of built in functions (e.g. WolframAlpha cannot calculate the series form solution of $$\int x^x~\,\mathrm{d}x$$ thought it does in fact have a quite nice series form solution. WolframAlpha cannot calculate $$\int\dfrac{1}{ae^{bx^2}+c}\,\mathrm{d}x\qquad(a,b,c\neq0)$$ in terms of the incomplete polylogarithm function because incomplete polylogarithm function is not in the built in function list. WolframAlpha cannot guarantee that the output results are really in the simplest form (e.g. when calculating $$\int\sqrt{x^3+1}\,\mathrm{d}x,$$ WolframAlpha will give the quite awful elliptic functions result rather than gives the relatively simpler incomplete beta function or hypergeometric function results). WolframAlpha cannot perform calculations with restrictions on constant parameter inputs (e.g. when calculating $$\int\sin^nx\,\mathrm{d}x,$$ WolframAlpha will only treat $n$ as a complex number and does not have the wisdom to treat $n$ as e.g. an natural number). Advise students that WolframAlpha is not really as omnipotent as they feel. The sensible students will naturally not rely on WolframAlpha.

Always remember that the developments of the mathematical software by the mathematical software manufacturers often only care for the business, they don't care for the proper attitudes of studying mathematics.

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    $\begingroup$ How would you highlight these limitations in a calculus class, where students barely understand $\arcsin$, and do not understand the incomplete polylogarithm or beta functions at all? $\endgroup$ – user173 Mar 30 '14 at 10:37
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    $\begingroup$ There are other free options, such as Maxima and Yacas. Some of the limitations you refer to are specific to Wolfram's web-based system. $\endgroup$ – Ben Crowell May 14 '14 at 15:22
  • $\begingroup$ @BenCrowell there are certainly other free options but Wolfram Alpha makes it extremely easy due to its natural language processing. No longer does a student need to learn software specific syntax or computer programming. Often those things alone were enough of a barrier to prevent the use of Maxima (and others) by the types of students who would use them dishonestly. However your point is very important. $\endgroup$ – WetlabStudent May 14 '14 at 19:48
  • $\begingroup$ @Ben Crowell: This question focus about the issues of WolframAlpha, so it is better to give the answers also focusing about the issues of WolframAlpha, not other mathematical softwares. $\endgroup$ – doraemonpaul Aug 24 '14 at 4:22
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If use of is a blessing or a curse depends entirely on what you want them to learn. If you want them to find indeterminate integrals by hand, using a slew of tricky substitutions, it is clearly a curse (sort of, having something against which to check results is valuable in itself). If you want them to know when to compute an integral, and do something with the result, the mere computation better be taken over by the computer.

I contend we should see how to concentrate on the second aspect. Leave the "find/apply clever tricks to compute ..." to people interested in symbolic computation.

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    $\begingroup$ Indeed, and/but as many people (including myself) have commented other places, the more-conceptual stuff is actually more difficult than the silly formulaic and algorithmic or memorization-of-trick stuff. The fact that it's more appropriate intellectually, etc., is small consolation to the students who had trouble executing algorithms and now have comparable trouble with proper syntax... and then have to reason/discuss things at a higher level. $\endgroup$ – paul garrett Mar 21 '14 at 0:07
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    $\begingroup$ @paulgarrett, that isn't an excuse to not try to make people use their brain for more than mostly mechanical processes. $\endgroup$ – vonbrand Mar 21 '14 at 0:08
  • $\begingroup$ Indeed, but/and I think some care about "failure modes" is called-for, is my real point. Didn't mean to sound so negative. $\endgroup$ – paul garrett Mar 21 '14 at 0:18
  • $\begingroup$ @paul garrett: Although finding the "failure modes" seems to be negative, they give us the good intentions for not relying something. $\endgroup$ – doraemonpaul Mar 29 '14 at 22:10
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If you test them (and without any ability to reach the Internet, no phones, computers) than it will be an action that makes it hard to over-rely on WA. If you do graded homework, it is the opposite. Thus, I would suggest more tests, quizzes (in class) and less graded homework.

P.s. I think for a very strong student (or a working professional) there is nothing wrong with using WA or even just a handbook. After you are done with a course. But there is a disadvantage in being someone who only can use WA and can not do anything himself. I would include Wolfram, the actual man, and many others in this group. They learned it first analytically and it informs their use of WA, their development of it, their fluency reading physics or engineering derivations, and their analysis of problems in the real world.

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I think that the computer is an important thing to consider in an introductory mathematics class, and this may mean simply interacting with the internet and wolfram alpha, or using computational tools to solve mathematics problems.

If I were you, I would be incredibly excited to not have to spend weeks in the drudgery of differentiation techniques. I would ask myself what kind of calculus questions would you feel comfortable about presenting if you were to allow your students full access to wolfram alpha. WA cannot answer most important calculus questions, however it can do some basic computation and symbolics.

  • To eliminate dishonesty simply ask your own questions, and encourage students to use technology and the internet, including supporting the use of sites like stack overflow! Have them ask questions here, or use a class Piazza board. There is nothing wrong with talking about problems.

  • Farm the mechanical stuff out to a machine and let the students use WA or something else to answer it. Nothing wrong with software for problem delivery if used wisely.

  • Have students present their work and expect the work to be polished. This will show the students why to use technology. Demand that they incorporate words, tables, graphs, symbols, and other diagrams to make their arguments, and have them present these to others regularly. I'm amazed we still ask students in college to scribble work with paper and pencil simply because it's mathematics class. I begin my semester having students bring in scratch work and using short writing labs to work on turning these into more polished (typed!) solutions. I use $\LaTeX$, Python and Jupyter notebooks, but you could do this with google docs and wolfram alpha if you wanted.

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