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?