The Benefits of Thinking Mathematically
“What subject do you study?”, they say. “Maths”, you say. This is probably the easiest way to end the conversation you were just having. Now I am not going to fruitlessly defend mathematics; I may take it as my degree, but it can often be the bane of my existence. Admittedly, a lot of mathematics at university is a mundane recital and derives various theorems and methods which we have to learn, understand and apply to the exams. However, the way someone thinks mathematically is an often overlooked part of mathematics that I feel should have more light shone onto it. I had been struggling to come up with ideas for this article, finding it very tempting to simply choose a mathematical problem which sounded interesting and then delve into the mathematics of the situation. Perhaps I don’t give myself much credit for my mathematical ability, but I can relate to the idea of thinking mathematically. It is a skill, like any other, that has become part of me over time through practice and endless days of staring blankly at my laptop screen.
In my view, mathematics is a language like any other. As a French student would learn to identify the niches of grammar, a mathematician learns to recognize patterns amongst awful-looking equations. To someone not knowing the languages, they both look unapproachable. In reality, most mathematical equations can be easily broken down into simple components that make sense to describe. However, the coding from real-life to mathematical lingo is a barrier that most mathematicians struggle with. This often gives mathematics the notoriety of being a ‘hard’ subject, which although merited, in my opinion should not be deemed harder than more creative arts subjects. The skills used are different, but one must use their creativity and tools to work through problems successfully.
I found GCSE maths very hard. It felt like we were bombarded with a mix of random methods for solving different problems with no real links made between them, like being a robot designed to complete a haphazard-many tasks with no idea of the relevance of them. Have no doubts, I am not a ‘mathematically-gifted’ student. The phrase is common, and many people have come to believe that some people have a seemingly natural inclination for mathematics. Writing this, I am unable to agree. I truly believe that everyone has mathematical ability, but that it shows itself as the person wants to express it. I have seen artist friends who are squeamish at the mention of mathematics who seemingly create drawings with the crispest colour tones and perfect proportions. Seems too good to be true without maths cropping up. Not convinced? You probably by now have calculated the rough time estimates to get from your house to lectures. And if you have not, I would not blame any lack of mathematical ability on this one, everyone loves a lie-in. Anyway, enough side-track. What I am trying to get at is that everyone has some amount of mathematical thinking, but it may show up unexpectedly in some scenarios.
I have been discussing mathematics, but what even is it? Google probably would not say 4 years of hell, but I can assure you it is. However, the term itself is very broad. I would say it contains three main areas: statistics (science students beware), pure mathematics, and applied mathematics. I am mainly an applied mathematician. I have now managed to mostly leave statistics behind and have only dipped my toes into pure mathematics at this point as I experienced many challenges trying to understand the concepts.
Pure mathematics builds abstract concepts from theorems you derive from others. You could see it as a giant Jenga tower where the top ‘new’ discoveries are built upon the foundations of previous theorems and at the very bottom, the so-called axioms of mathematics (involving how we define numbers). Now pure mathematics takes a lot of brain power and will to master, so I have mostly avoided it. However, briefly diving into the subject this year had challenged my traditional mathematical thinking.
Take Ergodic Theory, this is the study of dynamical systems by using a ‘measure theoretic’ approach. Forgive me for the nasty words, it is often best to describe it using an example. Perhaps one day you decide to let loose many gas molecules in a closed box. Maybe you ask yourself if the gas molecules might return to their original positions in time? With no mathematics, we can only guess at this. Maybe it will, but what are the chances of each molecule returning to its own position? Instead, we can take a measure theoretic approach. This involves breaking the box down into differing sets of numbers and applying ‘measures’ to these sets. All you really need to know is that we can use Ergodic theorems in processes to calculate how likely each molecule returns to each set. From my point of view, this was quite intriguing, because it was a case where the mathematics didn’t jump out at me first and the logic was hard to work through.
Ergodic Theory seems to involve a lot of very scary-looking symbols which do not shout out any meaning to me. This is scary because it is so different for me when it comes to applied mathematics. To overcome this difficulty, my strategy has been to rely on visualising in my head the process that is happening. For example, I may need to calculate the reverse of a function infinitely many times. Instead of full-out panic (most of the time that is) I try to draw what the function does to the sets of numbers I am interested in, and I try to slowly work backwards. I reverse it once, then again and again. By doing this, I have a good chance of seeing the behaviour in what is happening, and I can potentially extend this to infinite time. Indeed, the transition from looking at the mathematics on the page as a bunch of symbols to coming up with funny drawings to describe what is going on has much increased my understanding of the subject and given me at least a chance of braving next week’s tutorial questions.
Now, I have touched on how I seem to prefer applied mathematics more than pure. This is true, I see it much more gratifying and easier to visualise if I know how mathematics is used in the real-world context. Moreover, applied mathematics lends itself well to a few key topics, including calculus, analysis etc. Years of doing these has allowed me to develop a sixth sense for tackling these problems and I have a lot of tools in my arsenal. Thinking about it consciously, it is quite hard to pick apart my line of thinking as I try to tackle problems, but I do know that I have quite a few strategies I like to turn to, such as visualisation and estimating how my answer ought to look.
I would say studying mathematical oncology this year has been the most rewarding in terms of how thinking mathematically allows us to make huge conclusions from our work. Mathematical oncology is the use of mathematical modelling to study cancer growth and treatment. The field is greatly rewarding to study and has opened my eyes to the challenges of effectively modelling such a prolific disease. Recently, we investigated a model to try to see how the tumour would grow over time under certain assumptions and to which states the system will settle to. This process involved biology and involved repetitive algebra and analysis of the equations in hand. However, the outcomes proved to be incredibly insightful. Using common analytic techniques, we could prove that if there was no treatment given, then the tumour would proliferate entirely, killing all the healthy tissue. Also, the tumour would never tend to decrease in size unless there was a considerable decay term introduced. Then, with the treatment modelled, the system adjusted to this, and we saw that the best-case scenario was to apply a critical amount of treatment such that the cancer would settle to a controllable, although non-zero size. These conclusions may make logical sense, but the power in being able to convert our logical thinking to mathematics cannot be underestimated. By using a mathematical approach here, we can adjust the parameters in the equation and see how to change the level of treatment given depending on the environment of each individual tumour. Now these ideas are introductory but are relevant to the models which are used to try to beat cancer today.
Moving into the next year, I intend to explore more modules which involve mathematical modelling, such as fluid dynamics and mathematical biology, to broaden my applied mathematics skills (and ultimately fill up those credits). I hope that by challenging myself to explore new ways of thinking mathematically, it will allow me to better face those scary looking problems and perhaps appreciate more what the questions mean. Those symbols and smudges on the tutorial sheets really do mean something. That being said, I really should start the assignment I am yet to look at.
Illustration: Sarah Knight