Math is important

There have been many instances when people told me about how much they disliked math at school – especially when I speak highly of my studies and that I do abstract math frequently even today. On top of that, I have talked to many students (either B.Sc. students or teachers-to-be) who speak similarly about math at university-level. And finally, I know several math teachers who chime in to that tune as well. Among the most usual comments that I get are:

  • math at school was so boring and dry, it doesn’t have anything to do with what happens in real life
  • it would be sufficient if I had just learned the basic computations (addition, subtraction, multiplication and division) and to deal with percentages – that’s all I’ll ever need
  • math at school was so hard, I never understood why I should have come up with these particular tricks
  • the exercises at the university were so hard, I never managed any one of them on my own
  • it’s no use to learn so many hard topics in math that I will never use in my later work (and, oh boy, I had to go to so freaking weird lectures)

Here’s my random rant about why none of these comments is valid – or why any of these comments would apply just the same to any other high school subject. I just had to collect those arguments once in a single place, to get them off my heart. And for further reference. None of these arguments is originally mine, which might just show how deep this anti-math feeling is rooted.

Before I start, let me make something clear: I do not intend to say that the whole universe of mathematics is interesting in and of itself and that everybody should make themselves acquainted with that just for the sake of it. There is a lot of mathematics in which I have lost my interest as time went by (or in which I never had any interest in the first place). On top of that, most of what I have to say applies accordingly to other sciences or topics: I myself would never have become much of a biologist or theatre critic. In what follows, I will rather talk about things that should be standard knowledge to anybody who consider themselves an educated person: things that are part of what mankind has achieved, be it the Mona Lisa, be it Romeo and Juliet or be it Newton’s Law of Gravitation. In some way, many people tend to say that the first two are “must-knows” but the last one is not. You can live your life perfectly at ease if you’re unaware of Newton’s Law of Gravitation, but if you don’t recognize the Mona Lisa or have never seen it, some would think you’ve been living with eyes closed – how’s that? A painting may be easier to approach than a theorem, but both require some sort of education to properly grasp them and both will be (aesthetically) pleasing to the educated beholder. And there also is really deep beauty in maths, there is elegance, there are unexpected connections. But it is hard work, no doubt about that.

On top of that, there are many people who work in some shade of a scientific profession. That not only applies to scientists but also to engineers, programmers, actuaries, and – yes – especially (math) teachers. All these people must possess a certain way of thinking to do their job properly: an analytic, logical train of thought which allows them to cut through a problem and to make their reasoning clear to other people (it hardly matters whether you have to explain your work to your boss or to a bunch of children). Besides, these people have learnt to actually tackle a problem and not run away if they don’t understand it at first sight. In practical professional uses these trains of thought will be what people have learnt – this is what employers look for in scientists: they have learnt to think and they can’t be scared away by some hard and unsolved problem. This way of thinking is being taught by considering mathematical problems, but the actual skill is generally not mathematical. If the way of logical thinking could as well be taught by applying for a truck driver’s licence, then employers would also look for truck drivers, why not.

So, this is one of my first points that I’d like to make: in learning or studying math, you’re supposed to learn to think. The fact that you learn math is just incidental, if you will. You’re supposed to be suspicious about what people just tell you without proof – you’re supposed to think for yourself and to convince you of the truth for yourself. In social studies, in theology, in psychology, the teacher might shoot down any of your questions with a phrase like “I’m right because that’s the general opinion and you’re not yet aware of it”, but in mathematics and in the sciences any student is able to convince the teacher of mistakes that he’s made. I am aware that I’m over-simplifying here, since there is no such thing as an absolute truth in psychology and social sciences (theology is somewhat tricky in this respect) and there is a living culture of discussions – those professions will teach you a different, albeit similarly important way of thinking. Discussions are equally important in math, but in the end there is only one answer (apart from very special cases that usually aren’t taught) and the students are supposed to learn arguments towards that answer. In philosophy, no-one needs to be convinced by arguments since no-one is wrong; maybe different or on shaky ground, but not wrong. My point is not to diminish the non-mathematical sciences, my exact point is that you can learn argumentation and discussion on topics that actually have universal truth hidden inside them. This is how you can uncover false arguments, how to fix arguments gone wrong, how to think twice at crucial points – and how to wonder about unconvincing statements other people may make (and how to question these statements).

Another issue made about math is that it is considered hard. Well, that’s right. Math is hard and exercises are hard. But if your mind is to be trained to a purely logical way of thinking, then how else would you do the training? If you want to run a marathon, you also need to do your training (and yes, of course, running a marathon is hard as well). On top of that: problems in work are harder than the math exercises. Applications of mathematics are even harder than abstract math, since the abstraction is gone. There are no proper circles in the real world with a circumference of 2\pi r. In applications you always have to account for measurement errors, lack of data quality – and you have to stop dealing with those inaccuracies at some point because you couldn’t possibly get all of them right (but at least you should give an estimate then about how big your error is). Your math exercises are supposed to give indications about what can and what cannot be done, and how you can deal with that. About using math successfully, to discuss and to convince others by your logic. And without training, you can’t do anything properly. Learning is when you do things that you can’t do yet.

Now, some people will say they don’t have to deal with circles or anything similar in their everyday life, so this example doesn’t apply. Maybe so. But then they have to deal with something else and time and again something changes. People are not robots who receive their programming once and then do the same thing day after day – and every time something changes, you will have to adapt and to think about that. It might be useful to be able to think about that properly: what am I supposed to do now, how do I do that, how can I optimize that (i.e. have less trouble or effort with it). This doesn’t need to be mathematical thinking – it’s just that math and exercises may have taught you how to deal efficiently with “new things that you don’t yet understand”.

The other point of view to this is: math is universal because it’s abstract. Some things and techniques have been invented for a special purpose, but when you take the special situation away from it, the technique can be used more generally. I won’t start the long line of examples here, it would probably just distract from my point. Abstraction is hard, but it unveils the underlying principles and once you’ve understood those, many things will just fall into your hands like that. Or, to phrase it differently: you should make things as simple as possible – but not any simpler. Applications are not simple because they may be concrete – in this respect, math is simple because it is not concrete. And to give a quote from one of my high school math teachers: “The task of mathematics is to make life as simple as possible.”

About math exercises, some complain that, sometimes, tricks are required and you can’t come up with the trick in the first place. That’s right. But no-one expects you to invent a mesmerizing trick on your own. You’re supposed to look at the exercise and think about it, try to solve it. If you have spent some time with the exercise, sometimes you will come up with a solution (and sometimes a neater solution uses a dirty trick that you didn’t see). But there will always be exercises that you can’t solve. That’s not a problem. When you are given the solution, you are supposed to think about that, too. You wouldn’t understand it properly, if you hadn’t thought about the question in the first place – and you will understand the trick, so it will be part of your own quiver of tricks from now on. If there is a similar exercise later, you can use the trick since you’ve been taught it now. And, yes, there are many tricks. It takes experience and effort to know which can be used in a given situation and which can’t. This, in turn, enables you to deal with more complex problems that you couldn’t tackle without that quiver of tricks. That would bring us back to the training-argument from above: you can’t run a marathon, if you just walk around your house once a week. If you want to become better in something, this will take effort and time.

Much of what I said about some profession or science as a given example can be replaced by something else and still keep the same meaning. I actually intend to say, that a broad education is essential for anything an educated person does – you need to be really good in your special subject, but you also need to try and see the whole picture (or understand why you can’t see it). It’s rather my own point of view that made this posting start about mathematics. It actually applies to most other educated professions I can think of, and many people shun away from these “alien” educated professions when a discussion crosses those. I myself am often confronted with the mathematical angst of many people, which made me write this text.

Seeing the bigger picture (whatever your personal profession and specialty is), will also help you in another aspect. You always need to know a little more than you actually have to use. Thus, you will know the boundaries of what you’re doing (and the shortcuts that you may have taken). It will be immensely helpful to have an understanding of why the world works as it does, what the principles are – that doesn’t mean you have to properly understand each and every bit of it, your time is finite of course, but the basics should be standard knowledge. That makes it possible to appreciate the really hard stuff (e.g. it’s not an easy thing to encrypt your online banking access data; it’s not easy to write a song that will rank top in the charts; it’s not easy to understand why everyone actually puts any trust in paper-money instead of gold, bread or shells).

Especially teachers must know more than their students, and they must know more than what they actually teach their students. On top of that, teachers need to know how to deal with children and how to keep them awake and attentive in their classes. That’s why this job is so hard, after all.

Similar reasoning applies to languages. I can hardly overstate the need to speak foreign languages. You will understand the culture of a foreign country much better (if only at all) if you can make yourself familiar with its language. And that also applies to dead languages such as Latin or Ancient Greek. German educator Wilhelm von Humboldt once said “Having learnt Greek can be no less useful to a carpenter, than making tables is useful to a professor”. You learn to think analytically by learning the grammar and you learn about another culture. How can that be entirely useless, even though no-one will have a conversation with you in Latin? Of course, your time is finite and you’d rather deal with something else – but apart from very legitimate indulgent things like watching some football game or spending time with your friends: is a soap-opera really the better way to kill your time as opposed to learning a new language, playing chess, doing a painting or thinking about a science problem? What do you want to do with your time?

Finally, one paragraph about the boundaries of math. Mathematics is not a description of the world. Mathematics is a set of assumptions (called axioms) which are taken at their face value because no-one doubts them and you can’t prove anything without having a starting point. From those axioms, things are proven by logical conclusions. The construction is powerful enough that people have drawn useful and sensible conclusions from it to describe the world. But mathematics is not the world. It is a model (albeit quite a good one) and there are boundaries. Dealing with mathematics allows you to find out where those boundaries are and where your model assumptions break down (do they break down in your application?). You mustn’t use math without a sufficiently deep knowledge, and this is why economists, psychologists and many others have to deal with statistics, even though their computers will tell them about the significance of their test statistics. Still, you have to know the limits and deal with them.

That’s why people are supposed to deal with math.

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