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Recently I asked a question on math.stackechange about the most ways to differentiate the same function and it didn't seem to generate any interest - rather, the reason why I'd ask such a question was the highest voted comment.

To me, there are many reasons why you would want to do the same question different ways, or to give questions to students that could be done in different ways - off the top of my head...

  • Checking your work since the more different ways you do the same question then the more support you have for the correct answer.
  • Different approaches highlight different topics, which then could provide insight into other questions of said topic. You'll see the pro's and con's of each approach.
  • Flexibility when it comes to doing future questions since you don't always have to do it the "expected" way, which could be laborious. With the intuition and the insight, "shortcuts" could be made.
  • Ability to accept that there are multiple ways to do the same question and that open-mindedness could help you interact with others who solve things differently from you.
  • Emphasizing that process is as important as final answer, and that an answer in itself doesn't mean we need to stop thinking about the problem.
  • Showcasing mathematical consistency in how seemingly separate methods of doing questions, or how different branches of math, actually relate to each other.

Unfortunately most of those points might be paraphrases of each other, but the fact remains that I've always tried different methods to do questions and I've always smiled it when my students give me approaches that I didn't think of. Even if said approach doesn't ultimately get the answer wanted, I love how the their thinking and knowledge wasn't restricted to current lesson.

So my question is, do you teach different proofs or calculations of same question? And if you do it in calculus or real analysis, what questions yield the most different methods of solving?

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    $\begingroup$ Regarding the specific Mathematics StackExchange question you mentioned (which I commented on), I actually would use some of the approaches mentioned in a first semester calculus course, including the kinds of things I suggested. For example, one can try using the quotient rule for $\frac{x^4}{x^2}$ and the product rule for $5x.$ Incidentally, when trying things like this, at some point one is likely to consider differentiating $x^2 = x\cdot x = x+x+\ldots x$ $(x$ many appearances of $x$ in the sum), which gives $1 + 1 + \ldots + 1$ $(x$ many $1$'s) $= x\cdot 1 = x,$ not $2x.$ What went wrong? $\endgroup$ – Dave L Renfro Apr 23 at 9:30
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    $\begingroup$ As for the apparent contradiction at the end of my last comment, see Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition). Also, see an explanation I originally gave in Fall 2003, which I was later asked permission to include it here (scroll down to "Another one of those ideas to emphasize"; regarding "Lin responds to Dave's post" at the end, see this answer). $\endgroup$ – Dave L Renfro Apr 23 at 9:55
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    $\begingroup$ As often as time allows. But I think the importance of showing multiple approaches is relatively modest. I haven’t taught real analysis. So I’m talking about calc.mainly. Integration is ripe for it sometimes, in both calc. II and diff.eq. Numerical analysis teaches multiple methods explicitly, but that probably doesn’t count. Unless it does, because having multiple methods is so important. $\endgroup$ – user1027 Apr 25 at 20:02
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My goal is not only for my students to recognize that there are multiple methods for solving a problem, but more importantly, for them to be able to identify the appropriate method(s) for solving each problem.

To me, once a proof of a particular method is learned, it becomes a tool that can be used.

Specific to differentiation, the reason why I would use different methods is because each method is an important technique on its own. At some point in the course, I expect my students to able to identify the tediousness or ridiculousness of using certain methods to answer a particular problem. I would not, for example, expect a student to check his work for differentiating a product of powers of three functions by expanding the product first and applying the power rule to each term. Rather, going through the work using the same method is sufficient (though, teaching how to properly check work is a different subject entirely).

There are a few cases where I require students to use a specific method. For example, one of my recent test questions say

Using the first principles definition of the derivative, show that $$\frac{d}{dx}\tan x=\sec^2x.$$

When the derivative of the tangent function was taught, we used the knowledge of trigonometric identities and other derivative rules (i.e. $\tan x=\sin x/\cos x$, use quotient rule). The reason why I would ask the question above is because it is an application of limits (one of the other topics taught in this course), and not because it is just a different method to use.

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Yes, whenever possible. Teaching different proofs or different solutions to the same problem has several benefits:

  • There is less cognitive load because the problem is already known and a solution has already been found;
  • You strengthen students' schemas by creating a connection between existing domains, possibly unconnected until then;
  • You exhibit an application of a very known result or a set of principles in an innovative way, expanding the set of tools students' have at their disposal.

Some care is needed: the new solutions should involve knowledge somewhat ready at the students' concept toolbox. If it's not you should be prepared to unwind some of the theory and prepare them to assimilate the new solution. Else you risk them simply overlooking the new ways, in which case you may get questions like "can I just know solution A?" and students' minds will be tuned off everything you do.

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