# How to get a good maths degree

I recently taught complex analysis to our second year students. One particular problem jumped out whilst marking the exam.

One question was “Define the length of a contour.” This was only worth about 2 marks and the bulk of the students got it mostly right. My point is that students’ responses can tell us something about how they see mathematics and perhaps how they do mathematics.

The main mistakes were
1. Not giving enough information.
2. Not being mathematical.
3. Giving the procedure.
4. Trying to memorize without understanding.

Let’s deal with these in turn (the most important is number 4!).

Number 1. Not giving enough information.
A good answer to “Define the length of a contour” is “Suppose that $\phi :[a,b]\to C$ is a contour. The length of the contour is $\int _a^b | \phi \prime (t) | \, dt$.”

Instead many students slapped down $\int _a^b | \phi \prime (t) | \, dt$. They lost a mark because they didn’t tell me what $a$, $b$ and $\phi$ were. This happens a lot, students focus on the equation and forget about the surrounding information. If I did not know the definition of a contour, then the equation doesn’t tell me enough. I wouldn’t know where the $a$, $b$ and $\phi$ were coming from and their relevance.

2. Not being mathematical.
Another problem with definitions in general, not just this one, is that students give a hand waving definition, e.g. “It’s the actual distance that the curve moves.” This is not very mathematical and would not help anyone understand length except in an intuitive way. (In this case you could probably guess from the name that length is to do with distance!)

3. Giving the procedure.
Another very common mistake with definitions is confusing the definition with a procedure used to calculate the object defined.
For example, “Define the order of a pole” is often incorrectly answered by “It is the multiplicity of the zero of the polynomial in the denominator”. (This is my tidied up version, the students giving this answer usually say something like “It is the power of the thing on the bottom”.)
Of course, in practice when we have a quotient of polynomials and the numerator is non-zero at the pole, then the multiplicity of the zero of the denominator is the order of the pole. The point is that this is not a definition (unless you only deal with poles given by quotients of polynomials).

Another example: one question in the exam asked for the definition of residue of a complex singularity at the point $p$. Instead of stating that it is the coefficient of $(z-p)^{-1}$ in the Laurent expansion, a number of students gave a procedure for calculating it. Eg.
$res\, (f,p) = \lim_{z\to w} (z-p) f(z)$ or $res\, (f,w) = \frac{1}{(N-1)!} \lim_{z\to w} \frac{d^{N-1}}{dz^{N-1}} \left\{ (z-w)^N f(z) \right\}$.

The former will calculate the residue for a simple pole and the latter for a pole of order $N$. That is both can be used for calculating. (In fact the latter can be used as serviceable definition of residue. But who on earth would give such a convoluted and unclear definition in a course?)

I think this problem goes back to A-level where procedures are the important thing. Students are taught how to find the derivative of a function and tested on reproducing the procedure in the exam so that is what a derivative becomes in the mind of the student – it is the process of finding the derivative. (This isn’t a criticism of the teachers, just a criticism of the way they are forced to teach – but that’s another blog post.)

4. Trying to memorize without understanding.
Now we come to the important mistake when asked for a definition. Students try to memorize. One student gave a handwaving definition of the length “The length of the contour is the actual length of the contour” and added “What that is mathematically I forget”.

My question is why would you need to remember the precise definition? Mathematics is great, if you understand the concept, then you can recreate the definition.

For the length of a contour we are looking for the length of a curve in a plane (i.e., the image of the contour). One could visualize this as the distance travelled by a point moving along the curve. And how do we measure distance? Well, if I am driving a car I know my speed. If it is 50 miles an hour, and I’ve driven for half an hour, then I’ve done 25 miles. In general what I do is take my speed and integrate over time.

Given a curve finding its speed is easy, we differentiate the position vector (in this case the contour) to get the velocity vector and the length of the velocity vector is the speed. Hence we take $| \phi \prime (t)|$ to get speed and we then integrate this over time:
$\int _a^b | \phi \prime (t) | \, dt$.

There is no need to memorize this formula. Just remember the concepts: we want distance so we integrate the speed over time. Too many students spend their time memorizing all the examples they have been given (again this is partly due to what they are forced to do to succeed in A Level).

Now, I think that memorizing stuff to speed up giving exam answers is ok as you don’t want to waste time working out everything in an exam but memorizing when you don’t understand the concept is a bad thing. In fact, it is easier to memorize when you understand. For example, memorizing the following string of letters is quite hard:

cdglbagigjfklhr

However, it is easier to memorize if I point out the pattern. These are all examples of three letter airport codes:

cdg lba gig jfk lhr

So cdg is Charles De Gaulle in Paris, lba is Leeds/Bradford, gig is Galeão International Airport in Rio de Janeiro, jfk is John F Kennedy airport and lhr is London Heathrow.

Here’s the key idea to doing mathematics: Understand the concepts and the relations between them, don’t try to memorize all the worked examples that have been given.