Hardly a master I: What is maths?*Baby don't hurt me *

Don't hurt me

No moreFor many people, perhaps the answer is

*something I want to stay away from*. Of course, I can't invalidate opinions, and, realistically speaking, maths is something people can manage to avoid for most of their lives after finishing their high school education.

For the scientifically-minded, maths is about understanding the world, just like any field of science is. For the layman, maths is an extreme form of risk management — it tells us how things [are expected to] behave or not behave, without needing constant testing, monitoring, or scrutiny. For this reason, maths is about

*models* or

*theorems* (which may mean the same thing to you).

Spend enough time studying/on the internet/"both", and you will eventually run into this comic:

Okay, not trying to stroke my own ego here, but, going from right to left, the fields appearing in the comic are bound to feel more and more accessible, in the sense that you probably have a better and better idea of what the field is about (perhaps insert economics and finance on the very left and overlay engineering over the middle or something). You may have heard maths enthusiasts yell out

*maths is all around us* as they desperately try to convince you that their field of interest is a serious academic area as well as the source of various party tricks, the latter of which is more likely to make an impression.

In recent [Australian] times, we have seen some blips of maths appearing on the media radar. Issues like falling STEM standards, shortage of STEM teachers, gender representation in STEM, as well as the superstar teacher Eddie Woo… Unfortunately maths still needs a far better [long-term] marketing department until the situation in Australia will improve.

For now, let's hope you're convinced that maths is something useful, despite it having been processed by developments in all the other fields appearing in the comic above before actually reaching you.

...what is the point of this post?Well, lest something terrible happen, I will be graduating from my masters degree in maths at the University of Melbourne at the end of this year. The last 4 semesters have been more or less ferociously intense, so it would be good to reflect somewhat (yes, again, instead of studying for my exams, as usual).

I hope to spend some posts introducing what studying maths as a university major/degree is like [in Australia/Melbourne] as well as various related issues, e.g. a sustainable student mindset and career prospects. I'm also fairly sure that I'll expose along the way some of my own opinions on the state of maths/science in Australia (spoiler: mostly negative).

**Please remain sceptical of them when they appear**.

Why maths?Let's say you have just finished high school somewhere in Australia or are in the early years of your undergraduate degree. I would say there are two reasons why you should choose maths for a university major/degree in Australia:

**You enjoy maths.** This is a no-brainer. As is the case for many (but not all) fields of tertiary study, dedicating yourself to something that you don't particularly enjoy is usually a recipe for mediocrity or, even worse, disaster. For doing maths as a major/degree, I would say disaster is imminent if you don't enjoy maths. Most importantly, your idea of what maths is will change as you do more maths at university, and if at any point you no longer enjoy it then you should seriously reconsider.**You are in it for the long haul.** You believe in lifelong learning. Most importantly, you have the resources to sustain an education in maths (money, time, dedication). This is a bit of a strange reason, but not as important as the previous one.

Now, some serious qualification of these two reasons needs to happen, because different people will have their own interpretations, and unfortunately some of those may be slightly misleading. Let's debunk some myths (I'm sure that there may be some ATAR Notes post doing roughly the same thing already… but I didn't search hard enough). Just to be on the safe side, each point will start off with the

**[more] appropriate view** rather than the statement of the myth.

Not just about using formulae

Maths is not just about using formulae, which may be hard to believe if you've just come out of the Australian high school education. An even more appropriate statement would be that maths is about the journey that people took to come up with those formulae. Equally, this means that maths is absolutely not meant to be some mechanical procedure of doing calculations — at university, either the calculation is done by computers or the calculation is in fact not very mechanical at all. This leads me to say that the **flavour** of maths at university is very different from what it is at Australian high schools.

Different approach to mastery at university compared to at high school

Your maths grades in high school aren't great predictors of how well you may do in maths at university. This is naturally a hard pill to swallow, but the reasoning is similar to the point above. At least from a Victorian point of view, past a certain point of understanding, extreme peak performance in maths at high school revolves around completing ludicrous amounts of practice questions to achieve a desirable speed as well as knowing some small library of shortcuts.

Now I can't make blanket statements about all the maths subjects at Australian universities, since I only studied at UoM and even then only around a third of the maths subjects on offer, but generally peak performance in maths at university is about attaining a level of intuition about the material to a point where you almost couldn't explain your own perspective to someone else if you tried.

*Okay, but then do you really understand it?* This sort of response annoys me to no end. Think about someone who's just finished a painting. They start trying to explain to you the different choices of texture and technique used in different parts. You don't understand. *Okay, so they didn't explain very well.* Sure. Are you going to claim they don't understand the structure of their own painting? On a side note, this is also roughly why great mathematical thinkers don't necessarily make great teachers.

The crucial point is that doing well in maths at university is about letting the ideas you've come across incubate in your head for **so damn long** that the mechanics behind them start to feel second-nature. In this sense, the flavour of maths at university is more creative than it is mechanical.

Lack of jobs needing the precise technical skills

Very few office jobs use the maths taught at universities, especially in Australia. This is related to the second reason behind why you should study maths and is somewhat of a lesson I learned the hard way. If you believe that you will sit down at a desk at your office job and start using and **analysing** complicated mathematical models that you saw at university then I have some bad news for you... These office jobs exist but are few and far between, and, more often than not, will require extensive academic research experience (read: PhD or higher).

Using complicated mathematical models would be far more likely in a research lab (but probably not a lab for maths research since Australia doesn't have those). If you are using and **analysing** models (not just running empirical tests but actually developing theoretical results) then I daresay you must be part of some maths faculty somewhere.

As a result, if you're not aiming to become an academic, you need to have a strong sense of career direction, because it's unlikely that the precise technical skills you gain doing maths for a major/degree will lead you anywhere obvious for an office job. University marketing departments may try to convince you otherwise by giving you cherry-picked fairytale examples, but, compared to the number of obvious career options for, say, accounting graduates, there is almost nothing out there specifically for maths graduates in Australia that makes use of the technical expertise.

*What about statistics?* Well... there are certainly jobs out there where you'd be handling data, but the technical side, which may not be present in a lot of the work, would mostly be limited to statistical computing. Whether you would call this a use of the maths taught at universities is an individual judgment call.

My personal view is that [in-house] software development often provides similar challenges to those in maths, in the sense that you really need a holistic understanding of the underlying systems (which includes, in software development, the programming language syntax) in order to produce an efficient solution. However, I have limited experience with software development, and I would not be surprised if frontend development was totally void of what I'm thinking of.

Even if you are not aiming for pure software development as a career option, programming skills are undoubtedly one of the most useful skills that a technically-minded graduate could have. In the off chance that you're offered a job that deals with mathematical models, you would definitely need to be familiar with programming.

Words are used in mathematical work

When you complete assignment work for maths subjects at university, you are going to need to use words to explain the thought process and reasoning throughout. Well, technically this should already be happening at the high-school level, but it is far easier there to get away with wordless responses. This is especially important when you are writing a mathematical proof, which you will familiarise yourself with as you study more maths subjects. Mathematical proof is the core format of communication in formal mathematical work, and every step in a proof needs to be sufficiently well reasoned. In particular, it's pretty much impossible for a mathematical proof to have no words.

There are different types of maths

At the high-school level, there is more or less a single sequence of maths subjects. At any given school, everyone does the same sequence, and some take it further than others. (Okay, not strictly accurate, even for VCE…)

At university, a similar thing happens for the first year. Maths subjects in the first year set you up with the foundational knowledge that every single field of maths eventually uses, and after the first year you begin to choose subjects from different areas of maths. In high-school level maths subjects, there would probably have been different topics in the syllabus. At university, some of these topics will constitute entire subjects themselves (and possibly more than one).

It is pretty much impossible to study all the maths subjects at, say, any Go8 university in Australia, simply because you have a limit on how many [maths] subjects you are allowed to take for a bachelors degree. Most students in a maths major/degree usually take all the maths subjects offered in one or two fields.

In the same vein, at university, there is usually no concept of "good at maths", simply because you don't get to find out how good you are at all the different maths areas. The reality is that you will have an affinity for some areas and be weaker at others. You may even find inconsistency between how well you understand different subtopics within the same subject.

An appropriate mindsetI will finish off this post with some thoughts on what a suitable mindset to approaching maths at university is like. This will be more philosophical and perhaps somewhat more generally related to studying any field of science at university. Most of these are views I've developed from my own experience as a maths student so far as well as from seeing the journeys of my peers.

Let's not sweep it under the rug: maths is

**hard**. Maths is not something that you ever finish learning, and the hot milestones that appear in international media or cool applications that you've heard of in computer science or engineering can sometimes be the culmination of decades of collaborative research.

Learning maths (and doing research in maths) takes immense patience and has many ups and downs. If you are only motivated to learn maths by the prospect of being able to discover cool things (in the sense of a research discovery) then, at the risk of sounding a bit cruel, unless you mean cool things

**in a decade or two**, I would say your view is a bit short-sighted. Throughout my high school education, my bachelors degree, and now my masters degree, I have seen my peers' interest in maths wax and wane. I would say my UoM classmates whose interest in maths has been sustained until the end of a masters degree have been those who've maintained a deep respect for understanding the mechanics in maths, sometimes in how it's applied to other fields, but mostly in an intrinsic sense.

This brings me to the issue of

**maths haters** (bit of a hyperbole; read on). No, I don't mean the people who have hated maths since high school and never studied it at university. I mean the university students in a maths major/degree who vehemently avoid the maths outside of their own area of study or, worse, even treat with disdain the people studying those areas.

This concept was first planted in my head during an orientation session held at the beginning of my current masters degree. A professor gave an analogy with coffee. (Forgive me if what follows makes no sense; I don't drink coffee.) The professor said that most self-proclaimed coffee enthusiasts probably have in their mind very specific ways in which the coffee they consume is to be made, probably due to certain preferences in where the beans are sourced or other things like timing, milk, temperature, and whatever else that might affect the taste or texture. What ends up happening is that, for these self-proclaimed coffee enthusiasts, there exist far more ways in which they would hate a coffee than ways in which they would like one. So then… "coffee haters" doesn't sound so far off? (Forgive the specific example of coffee; I'm sure the same could apply to other hobbies or indulgences.)

Applying the same reasoning to studying maths, it would seem sensible that anyone seriously involved in studying maths should reconsider how serious they are exactly if they seem to close themselves off (either in fear or in contempt) from areas of maths outside their own focus. Whether an area of maths is applied, theoretical, old, new, rigorous, heuristic, the theme of Fields medal research, or secretly the mechanics behind

party tricks, there will always be valuable insight within and researchers with an advanced understanding. It is important to remain

**observant and respectful of all areas of mathematics**. Maths has the privilege of being a science which can avoid almost all the political roadblocks prohibiting progress in other sciences — it would be counterproductive to introduce any friction into the scene ourselves.

[Well, okay, that was characteristically rambly. To be continued in

*Hardly a master II*.]