In Perfect Class Size-Prelude, I provided a question and an answer, but no explanation of that answer. For me, seeing such a statement is extremely unsatisfying. Even more so, I refuse to believe such an answer until a justification is given. It may seem that I don’t trust anyone or that I am overly skeptical. However, I try to keep this skepticism in order to keep my curiosity. That is, if you just accept an answer, there is nothing more to learn from a situation.
Right now, this topic seems especially fitting as we are seeing this situation play out on a global scale. In particular, Michael Atiyah has claimed to have given a proof of the Riemann Hypothesis. When he made this claim, did the math community just take his word for it? No, instead there is a healthy amount of skepticism about the validity of the proof. However, this leads most of us to be curious and want to investigate. On the other hand, if we just accepted the claim, we’d have no further need to think about the problem.
If you just believe something without evidence, you have given up on your curiosity of the subject. However, it is often the case that both the question and answer are significantly less important and interesting than the process and technique used to find the answer. That is, even if an answer to a question is known, there is still significant reason to further study the problem.
There are many examples of questions that, even if an answer was ‘known,’ the process of showing that the answer was correct has led to entirely new fields of study and methods of understanding. Some examples of these are
- Can you square a circle, double the volume of a cube or trident an angle with only straight edge and compass?
- Are there any nontrivial rational solutions to an +bn=cn for n >2? (Fermat’s last theorem).
- Can I walk over all of the bridges of Königsberg and return to where I started without walking over any bridge twice?
- Can you find the solution of a general polynomial using only roots, products, quotients, sums and differences.
In fact, each of these is impossible. The answer is quite simple and actually rather boring. However, providing a proof that these things were impossible all lead to extremely interesting mathematical insights.
As an educator, I often tackle with the thought, how can I instill this curiosity into my own students? That is, even though they can look an answer up, how do you get them to look more into a question? On the other hand, how do you teach material in such a way that you don’t stifle curiosity by giving answers in a way that doesn’t promote more thought?
Instead of tackling problems that stumped mathematicians for years or even millennia, I will look at how I attempt to do this in my Calculus class for a relatively simple question. That is, what is the derivative of xn with respect to x? While it seems there is always someone in the class that knows the answer, or the pattern presents itself quickly and we get nxn-1. However, why is it the case that this is true? The answer is much less simple than the students expect (or want) it to be.
I will explain both how I present this in Calculus and will talk about the needed results that I don’t always cover in Calculus in coming posts. If you’d like to see this, make sure to follow the blog so that you are updated as these come. If you have any examples of such questions you’ve worked with in class, let me know by commenting below. Conversely, if you’ve found that there are subjects you find yourself just giving answers to students because you can’t see how to incorporate the explanation, also let me know by commenting below.