In my job, this question is one I get asked very often. To be honest, it usually comes in a slightly different form …

#### “When am I ever going to use this? What is it good for?”

As a high school math teacher for 15 years, this is one of the most common questions I received. When I began lecturing at university, I was surprised to find that I still sometimes get asked variations of this question. I suppose it’s a good question, if the idea is that at some point someone will say to you

“Determine

and have your answer on my desk by 5pm today. And don’t get any funny ideas about using WolframAlpha!”

Because the truth is, that rarely happens.

I often give a joking answer, and say
flat out, “You won’t,” and then go on a rant about how math doesn’t need to be
good *for* anything, because it is just
*good*. Nobody ever stood in the Sistine
Chapel, staring at the ceiling, asking what it was good for! They just
appreciate the inherent beauty, because it speaks to their soul. Math is the
same.

I think that’s a perfectly good answer,
to be honest. But in a more serious light, I find the answer to the question is
actually another question: “When are you *not*
going to use this?”

Of course, there are direct applications
of many branches of math. But those tend to be very specific, and these days
depend heavily on software to do the heavy lifting, so I tend not to think of
those. Instead, consider that football players perform bench press as part of
their training, to the point that the ability to bench press 225 pounds for as
many reps as possible is tested at the NFL combines. Yet not once have I ever
seen a football player perform the bench press during a game. Why do they do it
then? Couldn’t they just practice the skills they will *actually use in a game*? I can promise you that at no point during a
football game does a player think “oh, this situation is just like bench
pressing 225 pounds – I will apply that same skill now.” And I imagine there
are very few football players who complain while lifting weights that they will
“never use this in real life”. Of course, we know that the reason they train
the bench press is that it increases strength and power, so that when the time
comes that they need it, it will be there without consciously calling upon it.

Studying mathematics is the same. Math teaches so much if we are awake to the lessons. Here are some things I have learned, continue to learn, and apply regularly from my math studies, along with some examples of how they have impacted me personally.

#### Scale simple solutions to solve large problems

It is almost always the case that large problems can be effectively solved by breaking them into smaller problems, or by developing scalable solutions to simpler problems. For example, about 3.5 years ago I decided I wanted to learn to draw, so I took a piece of white printer paper and a mechanical pencil and drew a superhero-esque muscle man. It sucked. Like a lot. But I was not discouraged in the least by that. I was fueled by it. Why does this suck so much? I know how I want it to look, why can’t I make it look that way? I was excited by the fact that I could recognize how much it sucked, and by the prospect of working to slowly strip away the suckness. I spent hundreds of hours, solving small problems that were contributing to the suckyessence, and slowly scaling them up. Want to draw a heavily muscled arm? Learn to draw a cylinder. Then learn to draw little cylinders that lie on the main one. Then learn to draw “twisted” cylinders and tubing that changes diameter as it twists. Learn anatomy. Now put it all together. I intuitively understood platonic solids and how they interact with and reflect light. I applied these understandings to understand the types of skills I needed to hone with the way I held and manipulated pencils. I started looking closely at things I never paid attention to before. I still do this, and at no point during this process do I ever consciously say “Oh, that’s just like <fill in math course here>”, but at every point I feel *exactly the way I feel when I am working on difficult math problems*.

#### Being right also means proving you are

Math is really never about just “getting the right answer”. It’s about *proving* that an answer – or a result – is correct. The emphasis on proof is critical. In the real world, being right is rarely enough if you can’t convince others that you are. Careful, methodical, and audience-appropriate explanations are invaluable in this regard. Developing and writing proofs in mathematics is as much an art form as it is a science (perhaps even more so), and my studies in mathematics immeasurably improved my approach to constructing an audience-appropriate argument or explanation. This has had a profound impact on my communication skills, as well as my approach to confrontation. I have used this skill in more ways than I can list, but some examples are: when I have been in contract negotiations, when I deal with sales people when buying big-ticket items (and even when I bargain at markets), when I find myself moderating arguments between friends, family, colleagues or students, and when I used to work as a personal trainer and had to motivate and justify the kinds of exercise and diet choices I wanted my clients to make. In every single one of these situations, and more, I am really constructing proof. In fact, I would say that proof dominates almost all my communication.

#### Emotional attachment to a belief is irrelevant

Not wanting to be wrong about a belief, especially if it has been long-held, is normal. It is, however, illogical and possibly even dangerous in the face of proof to the contrary. Mathematics trains us to seek, understand and ultimately accept proof on its own merit, and not on any emotional yearning. It also trains us to be grateful when proven wrong, since it makes little sense to want to be wrong for even one moment longer than necessary. My training in math has led to a much more open-minded approach to new thoughts and ideas, and after careful consideration – which involves listening to argument dispassionately, asking relevant questions and weighing evidence – I find myself either happily embracing a new thought, or else more confident in the one I already had, having had the opportunity to test it rationally against a differing viewpoint.

#### Creativity and math are NOT mutually exclusive.

Not even close. Deep study of mathematics reveals and refines a strong creativity that aligns with and is mutually supportive of logic. This fusion is relatively rare, and people who have it are prone to what seem to be exceptional accomplishments. In truth, the exceptionality of it is not the accomplishment itself but the relative scarcity of people who can do it. One of my favourite examples is Leonardo da Vinci, who most people think of as a great artist, but who was also an accomplished mathematician and scientist, and who did not consider these as separate pursuits or modes of thinking. I find the same is true in my own life, although there are many people who wonder how a mathematician could be artistic.

#### Clarity lives just on the other side of contemplation

The journey math students regularly take from being
completely mystified and often intimidated, to understanding and comfort is a
lesson in overcoming that serves us well in all the challenges the future can
bring. It instills a confidence that says, “I may not understand this right
now, or even feel like I ever could, but I know I can do it.” General wisdom
suggests that “easy” might seem gratifying in the moment, but true satisfaction
comes from overcoming a challenge. Many people shy away from challenge for fear
of failure, but studying mathematics teaches us that we can tackle large
problems, even if they seem overwhelmingly daunting at the outset. An example
that makes me laugh is the time I purchased a large and intricate piece of
exercise equipment for my home gym (a functional trainer/smith machine combo).
I bought it used, so it did not come with any assembly instructions, and perhaps
embarrassingly, it didn’t occur to me to use Google. When I picked it up the
seller had already “helpfully” disassembled it into *n* pieces, where *n* is
large. I was completely baffled at how to reassemble it when I got it home. But
I was not daunted. I laid all the pieces out on the floor, shuffled them around
into sensible groups, and slowly assembled sections that made sense. I made
mistakes and discovered them when they led to chaos. I backed up, took a
different approach, and eventually put it together. The process was not “clean”
– I hurt my hand trying to brace a nut while tightening a bolt, and cursed
myself for not taking the time to get a wrench to hold it in place. But the
result looks like it was assembled by a pro. I’ve had it for many years now,
and it still works perfectly. I am fully aware that my engineer friends would
consider this a trivial exercise, but for me it was a hard-fought and
well-earned victory. This type of approach has stood me well time after time.

#### You don’t always have to see the whole path to the goal

How often have you been working on a difficult proof or problem, not really knowing if you were getting anywhere good, nevertheless continuing to take small, logical steps – always forward, occasionally pausing to reorient yourself to see if the direction made sense – when suddenly you found yourself having completed the entire thing? Some mathematicians call this the “follow-your-nose” principle of proof. *A* leads to *B* which leads to *C* etc. This might be the most important lesson of all. If you have a long-term goal that seems incredibly distant and perhaps overly ambitious, consider that if you just point yourself in the right direction and take small steps, occasionally reorienting yourself, you do eventually get where you want to go. Plus, the journey is so rewarding. In my life I have used this principle I learned from proof over and over and is in fact how I ended up lecturing at university, something that has been a dream of mine since the 12^{th} grade.

And that concludes my very long answer to the common question! I hope you found something of value.

Thanks for reading!

Rich