How to solve your problems

Anyone who peruses mathematical Q&A sites, or had students come to office hours or send questions via other means (email, designated forums, carrier pigeons, or written on a note tied to a brick tossed into your office) knows the following statement: "I don't know where to begin", or at least one of its variants.

Richard Feynman, who was this awesome guy who did a lot of cool things (and also some physics (but I won't hold it against him today)), has a famous three-steps algorithm for solving any problem.

1. Write the problem down.
2. Think. Real. Hard.
3. Write the solution down.

While Feynman's algorithm is quite simplistic, it really hits the nail on the head. But still, we seem to fail in solving a lot of our problems. Young students especially. Most of them might argue that the second step is really difficult, unless your name starts with Richard and ends with Feynman. While that's not entirely wrong, what most people miss most of the time is the first step.

Writing the problem down does not mean just writing down the actual question as given to you in the exercise sheet, or writing the theorem that you wish to prove. It means that you have to unwind the definitions, and unwind exactly what you have to verify, until you hit a sufficiently strong bedrock of understanding.

For example, if you have to prove that \(\aleph_1\leq2^{\aleph_0}\), then you need to first understand what all those symbols mean. \(\aleph_1\) is the cardinal of the least uncountable ordinal, \(\omega_1\); \(2^{\aleph_0}\) is the cardinal of \(\mathcal P(\Bbb N)\); \(\leq\) means that there is an injection from \(\omega_1\) into \(\mathcal P(\Bbb N)\). So we need to show that there is such injection. If you're unclear as to what do \(\mathcal P\), "uncountable" and "injection" mean, then you have more definitions to unfold here.

Now we think. We create this graph of theorems and associations. What do we know about a power set? We know that \(|X|\lt|\mathcal P(X)|\). Therefore the cardinal \(2^{\aleph_0}\) is uncountable. We know that the axiom of choice implies that every two cardinals are comparable. We know that \(\aleph_1\) is the smallest uncountable cardinal, therefore it cannot be strictly larger than \(2^{\aleph_0}\). Ah, so we know that \(\aleph_1\leq2^{\aleph_0}\).

The above description seems to include some redundancies. Students expect questions without any redundant details. Last year when I had a question with a minor additional detail (some function didn't need to be surjective), I got complaints from students that they didn't have to use that information. But that's actually a good thing. To know and understand that some details are not needed, but maybe they are there to guide you towards some nontrivial piece of information.

We didn't use the definition of \(\leq\), but rather some abstract theorem about it and the cardinals involved; and we didn't really use \(\mathcal P(\Bbb N)\) and \(\omega_1\); not to mention that we sort of skipped over a few trivial things. But was mentioning them really redundant information?

However there are two fine points here: the first being that these details help give us a more complete picture of the problem. Even if you don't use all these details; and the second point is that the theorems that we did cite, did rely on that information implicitly. So it is always good to refresh your memory with these things. Of course, when you're well trained in a particular topic, you have a rather comprehensive bedrock which is why your brain already made those connections and it was trivial to prove that \(\aleph_1\leq2^{\aleph_0}\).

Now you might wonder, we could have written so many more details. We could have appealed to a dozen other definitions and theorems. Why these ones? Well. That requires practice. You will probably write all the theorems and definitions at first. And with time, and solutions, your brain will train itself to find those quicker connections where you don't have to go all the way down to the turtles. Instead you only had to write the immediate definitions and one or two theorems.

And when you've got to that point, you know that you're ready for the next level of exercises.

In any case, this is why I always tell my students in introductory courses that the first thing to do when you read an exercise is to read it and understand it. And when we do a homework question from the week before on the board, I usually copy it on the blackboard and ask my students what is the first thing to do here. Many times someone will shout "You do this mathematical manipulation" to which I always say "No. We first read and understand the problem!" and then we review the question, the relevant definitions, and then we usually move to the earlier suggestion. I strongly suggest every TA, or a professor that solves problems on the board with students, to do. It keeps students involved and the break from "dive into the proof" is always refreshing to students, even if they won't admit it.

Finally, since we brought up Feynman and education. Here is a marvelous video: of him explaining why he does not want to explain magnets and magnetic force. It's taken from "Fun to Imagine" (see this page), and if you have the time, you should watch the entire thing or at least his explanation on fire which culminates in the poetic imagery that when you burn a piece of wood, the light and heat from the fire is the light that came from the sun, and was the energy that broke apart the carbon dioxide into carbon and oxygen which are reuniting in the flames. Awesome.