Question about the process of creation of problems and exercises in Mathematics

Question

I have a question about formulating problems and exercises in Mathematics. When attempting to create a problem of Number Theory or Real Analysis, for example, in this process, is the problem first established completely and only then is its solution or do the solution and formulation of the problem happen simultaneously or even, is the solution first thought, and then the problem is established ? What is the order?

Answer 1

I will take a stab at an answer though clarifying what level of education we are talking about would help.

I have never created problems for things like Qual exams (essentially masters exams) so I can't really speak to those.

For almost everything else I would tend to start with a very well formed idea of what the problem would be.

You always know what it is you want to teach/test/train. From that you get the framework of your problem.

Testing if they know the quotient rule? I probably want them to differentiate something of the form $f(x)/g(x)$ and it would be a good idea if it didn't simplify, so no $\frac{x^2-1}{x+1}$ but $\frac{x^2}{x+1}$ might work, $\frac{\sin(x)}{x+1}$ is even better. There is very little more to do here since the problems are very easy. I don't need to do any backtracking since the answers will be reasonably "nice". I would check how long it takes and consider how I might split up the grading on it, but since it's a very straight forward problem I probably don't care much.

Let's take something more complex though still very easy.

I want to have the students train calc type graphing. The standard type of problem would be, "Graph function $f(x)$ using calculus techniques." It's a bit boring so I might consider wrapping it into a word problem, but that makes the problem a lot harder so let's leave it like that for the moment.

Here I want a reasonable function. $f(x)=x^2$ is too easy. On the other hand $f(x)=\log(\sin(x^3+e^x))$ is too hard. Depending on what we've seen in class chances are I will go for a rational function. $f(x)=\frac{x^3+x+1}{x^4-4x+5}$ maybe? This looks like it's gonna a be pain though. Neither the top nor the bottom factors nicely and taking (and simplifying enough to use) the first derivative will be awful, nevermind the second. So here I want to start much closer to the answer. I want both the top and the bottom to factor. Possibly even cancel partially so I might go for $\frac{(x-1)^2(x+2)}{(x^2-1)(x+3)}$. The second derivative will still be nasty though and even the first is nothing to write home about, I might be ok with that or do more work engineering the problem to get all the numbers to come out right.

This last example is quite common in my experience for common complexity problems. It works the same for polynomial factoring or root finding, elliptic curve point addition and Markov chain models. Once the problems are complex enough you usually want the students to get decent answers and so you tweak or work backwards from a chosen answer.

Then there are proof based problems. At that point you are usually back to the basic approach and don't need to bother tweaking anything much. That's due to the the fact that for most standard complexity proof based problems, (prove that an associative quasigroup has a unit element,prove that MA implies $\mathfrak{d}=\mathfrak{c}$) there are no numbers to bother about. As to how you come up with these very often it is something you thought about at some point or it's a "trivial result" noone has bothered to write up.

Of course here I'm discounting the most common way of assigning problems which is check the exercise section of the book you're teaching from.