Kevin Houston

In Cal Newport’s Study Hacks blog he recently had a post about a piano student called Jeremy.

Jeremy’s Strategies for Becoming Excellent…

Strategy #1: Avoid Flow. Do What Does Not Come Easy.
“The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.”

Strategy #2: To Master a Skill, Master Something Harder.
“Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.”

Strategy #3: Systematically Eliminate Weakness.
“Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.”

Strategy #4: Create Beauty, Don’t Avoid Ugliness.
“Weak pianists make music a reactive task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.”

Of course as we are talking about studying for a public performance of music these strategies don’t translate immediately or perfectly to the study of mathematics. Nonetheless, I’ll give it a go.

1. Avoid Flow. Do What Does Not Come Easy.
The second part is crucial and is where many students make a big mistake. They get good at half of the course and hope that that will get them through the exam. I saw it in contemporaries at university and I see it in some of my students. They do they easy parts, for example focussing on doing the calculations rather than the concepts.

Hence the advice is to do the deeper work. I.e., become competent in not just the superficial calculations but have deeper understanding. For example, don’t just know what the definition is, know why it is the way it is and — hardest of all — know why it is not some other way.

2. To Master a Skill, Master Something Harder.
When I was first studying group theory I went further by studying why equations of degree 5 do not have solutions via radicals – a topic which was not part of the course. I found that doing this not only helped me grasp the basics of the material such as the definition of simple group but also helped with the harder stuff such as Sylow p-groups.

Mathematics is a linear subject in the sense that one concept is built upon another. Despite this we don’t have to — and probably don’t — learn it in a linear way. When learning proofs I would often try to learn the proofs from the end of the course first. One, this told me what was important in the course — if Lemma 2.1 kept coming in the later parts of the course, then it was a good idea to pay attention to it. Two, when it came to study the earlier parts in depth these seemed much easier. (An important point: you have to at least have some idea of what is in the start of the course, you can’t start in the middle, what I am saying is you don’t necessarily need to learn the start in depth to begin with.)

There is another good reason for studying the last half of the course. The harder parts of a course are often there because they are needed as prerequisites for later courses.

3. Systematically Eliminate Weakness.
When studying vector calculus I found that my weak point was calculating accurately the various types of integrals – I could quote the definitions and theorems such as Stokes’ Theorem without problem – but I kept messing up the calculations by forgetting terms, eg, missing the side of a cube. So I got Schaum’s outline series book on vector calculus and did many examples from there until I felt my accuracy had improved.

4. Create Beauty, Don’t Avoid Ugliness.
Hardy famously said there is no room for ugliness in mathematics (I rather less famously said, “Then why do they write so many ugly papers”). But how to make point 4 relevant to mathematics? Well, although it can be seen above that I took action to avoid mistakes (ugliness), my main ambition was to have an overall understanding of the course so that everything fitted together perfectly for me. One method I used to achieve this was to rewrite the lecturer’s notes. I don’t mean word-for-word rewriting but trying to make major improvements. Lecturers rarely provide perfect notes (you try writing over thirty hours of material) and many theorems can be generalized with only a little bit of effort. I sometimes found that when I had a generalized theorem, then I could deduce some of the theorems from the course with ease.

For me, that was where the beauty was! Making it simple.

I haven’t linked to a TED video in a while so here is a very interesting one that is not very mathematical but I’m sure is of interest to mathematicians – even if it tells us that mathematics does not lead to fame.

Of particular interest is one of the speakers, Erez Lieberman Aiden, who will be familiar to reader of Cal Newport’s Study Hacks Blog as he has written about him here and here. If you don’t have time to read those (and if you are a student I strongly suggest that you do), then the short story is that Lieberman Aiden has published only six papers but has had an enormous impact because they have been good papers. And I mean good. All have been in Science or Nature and two have been cover articles. The talk in the video is about one of those, the hunt for cultural shifts using the data from Google’s controversial book digitization programme. You can read about it here in the New York Times.

Unusually for TED the talk is a two-hander with Lieberman Aiden sharing the stage with Jean-Baptiste Michel. You’ve got to feel a bit sorry for Michel. He’s obviously a smart guy – he holds a post-doc position at Harvard and is a Visiting Fellow at Google – and yet is overshadowed in the media’s reception of the work.

Judging by site stats my posts on getting a good degree are quite popular, so since this is the start of the academic year I’ll talk about one of the most important elements of getting a good degree. And this is an important one:

Sleep is vital for humans. No-one knows why we do it. It seems strange and wasteful to spend about a third of our lives asleep when obviously we could be out enjoying life. What is known is that lack of sleep leads to poor health and poor concentration. And concentration is vital to getting a good degree.

Hence, don’t stay up mindlessly watching TV (how many repeats of Friends or Two and Half Men do you need to see?), surfing the internet (the internet will still be there tomorrow and most of it’s rubbish) or texting friends (are your texts of an Oscar Wilde standard of repartee? No? Then cut out the late night lols and omgs).

How to get up in the morning
Of course, a bigger problem is getting out of bed in the morning. Students are stereotypically famous for not getting up. Current popular research suggests this is due to the hormones of young people and we shouldn’t try to fight it; the school day should start later.

However, if you have a large number of those unpopular 9am lectures, then here are some simple tips on getting up:

• Take it easy: Try 15 minutes earlier for a few days. Next try an extra 15 minutes earlier and so on until you get to the goal time.
• Don’t put the alarm clock within reach. The action of moving to reach the alarm will wake you up a bit.
• Never use snooze. You know why.
• Get out of the bedroom as soon as possible. This helps wake you up and avoid the temptation of staying in bed.
• Have a morning ritual. This helps you get going. Whether it is breakfast, exercise or reading notes for the day’s lectures, doing something gives you a reason to get up.

Note that the middle three can be achieved very easily: put your alarm clock near your bedroom door!

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 is a contour. The length of the contour is .”

Instead many students slapped down . They lost a mark because they didn’t tell me what , and 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 , and 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 . Instead of stating that it is the coefficient of in the Laurent expansion, a number of students gave a procedure for calculating it. Eg.
or .

The former will calculate the residue for a simple pole and the latter for a pole of order . 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 to get speed and we then integrate this over time:
.

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:

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.

It is the dream of any student: work less but be more successful.

However, this dream can be a reality according to Cal Newport’s philosophy. Actually, he isn’t saying that you shouldn’t work hard – he points out that successful people often work hard – he says that you shouldn’t work so hard that your life is miserable. You should be able study hard and enjoy life.

His latest blog post is relevant to maths students and probably comes a bit late for my students about to do their exams. Maybe it will provide motivation for the next academic year:
http://calnewport.com/blog/2011/04/28/on-becoming-a-math-whiz-my-advice-to-a-new-mit-student/

There’s plenty of other good stuff on his blog so have a good root around if you aren’t already familiar with his work.