Studying a Mathematical Subject: An Oxford Physics Master's Graduate's View
Wondering whether to study a mathematic subject over a humanity? Sanjay (1st Class BA Physics graduate & Master's in Mathematical and Theoretical Physics with a Distinction, University of Oxford) gives his view on the Pros of Mathematical study & essential study skills!
In some ways, Maths and Physics, being the most mathematical of the sciences, are the easiest subjects to learn. Ideas are neatly and logically defined through the language of mathematics, and the scope of a course is also clearly laid out. To see what I mean, let's compare with a humanity.
Suppose you are learning about the First World War in History. There are countless things you can learn that will help you do well in the exam, and the relationships between these things are often subtle and up for debate. For instance, a syllabus may not mention any need to know about the Russian Revolution in detail, being a huge topic in its own right, but this would surely help you understand the War more thoroughly. Similarly, the question of whether certain events before 1914 contributed more or less to the outbreak of War can be answered in many different, but equally valid ways.
On the other hand, if you are studying calculus in Maths, you need to know relatively few particular things, such as what a derivative means, and how to differentiate certain functions. If you are studying mechanics in Physics, you need to know Newton's laws of motion, conservation of momentum, and know how to apply these to standard types of problem. Compared to huge and messy topics like the First World War, this is, I think, neat and nice.
It's worth taking a moment to ask why this is. It's slightly different between Maths and Physics, but the underlying point is that the language is mathematical.
In Maths, we study theories, such as calculus or algebra. These theories have particular starting points (unlike the First World War) such as axioms or definitions, and proceed logically (also unlike the First World War) through a series of well-defined steps from these to obtain results. Knowing the series of steps between the starting points and the results, and understanding why and how each step is taken, is basically all you need, together with some experience doing problems to solidify this understanding.
In Physics, objects of study are perhaps better described as models than as theories. For instance, in a course on mechanics, we might consider what happens when two perfectly spherical billiard balls on a frictionless table collide. Frictionless tables don't exist - or rather, they exist only in our model. Similarly, when you learn about nuclear physics, you are learning a particular model. There are many subtleties in more accurate models that are completely missed out. But in the model you are taught, there are certain rules, which you can learn, as well as a rationale behind them. The nucleus as you understand it only exists in this model. Models are what saves us from the hugely messy nature of reality, which is often extremely difficult to handle. It is only really in the second half of a degree in Physics that you start to look at the models that are actually used in research - or even later - I had to wait until my Masters to learn quantum field theory, the framework of particle physics.
What this all means is that as long as you know a certain prescribed and relatively short list of facts, and really understand the relationships between them, you will probably do well in the course. These two aspects, knowing facts and understanding their relations, are in fact symbiotic. On the one hand, you obviously can't understand relations between facts if you don't know the facts in the first place. On the other, understanding relations helps you understand why certain facts are important, or why we make certain definitions; the better you understand how a particular fact fits into the whole framework the more firmly it will be committed to your memory.
Of course, much of this is true of any subject, but again, it is because by nature of mathematical subjects the required facts and the relations between them are clear, rather than nebulous. Understanding that mathematical subjects are, in this sense, 'easy', is the key to changing your approach to studying them so as to exploit this feature.
Let's see how this can be done, starting with establishing knowledge, the base from which understanding is built. Textbooks and teachers will hopefully do most of this for you. To capitalise, it is extremely helpful to make this completely obvious in your own working. When it comes to revising, particularly if it's a subject you haven't looked at in a while, nothing is more repellent than an exercise book that is full of monochromatic, monotonous notes. Take it from someone who made this mistake for years: if you put effort into making important definitions and results stand out on the page you will make your life easier in the future. As you can hopefully see from the above discussion, this is particularly useful for Maths and Physics, where there are a relatively small number of facts you need to know. In fact, this will not just help you in the future, but also the present. Spider diagrams, which might be more used for humanities, are actually far more suited to the simpler structure of mathematical subjects. Moreover, the mere fact of taking the effort to write out a definition in a different colour, or to highlight a statement, forges a notion of its importance in your mind, and similarly the thought required to create a spider diagram will create associations in your mind that will concretise your grip of the subject. The influence of how you make notes on how ideas are committed to your memory and structured in your mind should not be overestimated.
The other side of a successful strategy for a mathematical subject is practice, practice, practice. Hopefully you already get a fair amount of this from homework, but you can always try more problems, and you will always benefit from it. If you can, try to answer every single exercise in your textbooks, and as many past papers as you can in the run up to exams. At some point everyone learns that it's impossible to 'get' everything just by reading definitions and a few examples - practice is integral to the learning process. You can't just passively receive information and expect to understand something - only through applying this information do you really get what you've been told.
Now, it may well be that in doing problems you are able to grasp something new, or in a better way than you have been taught. When this happens, I strongly advise incorporating this into notes you make yourself. The end product will therefore be notes that include everything important the textbook and teacher tells you, but enhanced and supplemented with what you have yourself gleaned from doing problems. In this way the utility of problems is twofold: in immediate terms, they solidify and elucidate concepts you have just learnt; and later on, they form part of an expanded body of ideas that you can draw on in the exam.
None of this is exactly revolutionary. Maybe this discussion has helped you appreciate how the mathematical nature of a subject makes it, at least in one important way, more approachable than others. But the rest of what I have said here is really about stress and emphasis. Hopefully you can now really appreciate the utility of creating your own learning materials and practicing problems, and implement them in your study.
u2 mentor, Sanjay, is looking to do a doctorate in theoretical physics in the future. He’s covered just about every branch Physics, and most major fields of Mathematics in his time at Oxford. Would you like your child to be mentored by someone like Sanjay? Enquire about tuition here and one of our consultants will be in touch.