Studying Mathematics is Not About Getting the Right Answer!
Catchy title, I know. And yeah, maybe a little click-baity. But it's true. Studying math is not about getting the right answer. Never has been. Unfortunately, there have been many classes in computation that have been taught under the umbrella of mathematics, and now the two concepts have become conflated, much to the detriment of far too many students.
You probably want to know what in the world I am talking about. Most people do. I won't waste any more time. To illustrate the difference, consider this well-worn technique for doing multiplication by hand:
That is an example of a computation. The person completing it does not need to know any mathematics. All they need to know is their times tables (up to 9), addition, and of course the steps of the algorithm they used to hopefully get the correct answer to "237 multiplied by 421". As I type this I can actually hear the objections:
"But that IS math! I learned how to do that in math class!"
"You are just using wordplay. It's obvious that computation and math are the same thing!"
I've heard it before. Trust me. And I understand why people think this way, but humour me by continuing to read and I will explain.
First, the difference between the computation above and mathematics is that mathematics is what was used to create the algorithm that could then be employed by almost anyone to get the correct answer. The mathematics behind the algorithm require a much deeper understanding of how numbers work, what multiplication means, and also a proof that when employed correctly, the correct answer will always result. That proof is a critical part of the process. If we weren't sure that the algorithm would always deliver the correct answer there would be little to no point in teaching it. But none of this is required to use the algorithm. As an analogy, consider that successfully using a microwave to reheat last night's pizza does not make you an inventor or an engineer - it just means you know how to press the right buttons. So in the same way that many of us reap the benefits of the work of inventors and engineers when we use a microwave, so to have we historically reaped the benefits of the work of mathematicians when we use the algorithms they designed. The difference is that there are no classes in how to use a microwave that are called Engineering.
Now here's the funny thing about that multiplication example above. If you are over a certain age, you know how it was done, and you could have done it yourself. And if you are in a certain category of parent or other adult who has seen how some schools are teaching multiplication these days, you throw your hands in the air over how "complicated" they are making it, when it's so simple the old-fashioned way. But consider that the joke is on us. Because doing multiplication that way is an outdated and useless skill. Since the only purpose of learning that algorithm was to get the correct answer to a big multiplication question, that skill has been completely replaced by the computers we pretty much all carry around with us in our pockets or on our wrists these days. Your phone calculator can get the answer much faster than you. And if you have a smart watch, or some smart home device, you can literally just ask Google, or Alexa, or Siri the question without having to type or write a single thing. So not only is that not math, it's not even useful anymore.
I'll say that again: The algorithms for getting right answers to arithmetic that we learned when we were young are not math, and they are no longer useful. It's just a party trick now. And not even a fun one.
So why did they even teach it?
Well, historically, before we had computers and calculators, a Computer (uppercase C) was actually a person. You could pursue it as a career. One great example is during World War 1 and 2, code breakers required large calculations to be done accurately and quickly. Cryptography, which is the study of encoding and decoding, uses some very advanced mathematics, and so the cryptographers were generally mathematicians working to create unbreakable ciphers, and also to break the "unbreakable" ciphers created by the other side. This high level work requires intensive calculations, something the mathematicians themselves were not well-suited to do, for two reasons. First, they were probably not that good/fast at it, but second and more importantly, the time and energy it takes to perform the calculations are better spent solving the higher level problems. And so there would literally be rooms full of Computers who were assigned to do calculations needed for cryptographers to do their work. To be a Computer you had to be fast, accurate, and consistent. Although it is certainly not impossible to be a Computer and a mathematician, there is no requirement to be good at math to be a good Computer. And this is true even when applied to today's understanding of computer. Computers only follow instructions. They need to know exactly zero mathematics to calculate correct answers to even very complex problems. On the other hand, the programmers and mathematicians who write the code that gets executed do need to understand the mathematics of the algorithms they are coding. Modern computers are so fast and reliable, and so significantly more so than any human could ever be, that mathematicians are able to use them for complex calculations that could not previously have been done, and thus have been able to make great advances in mathematics and its application. But remember this: a computer understands no math! Well, perhaps now as Artificial Intelligence advances into new territories, that is changing. But the concept holds true. Computation and Mathematics are not the same thing.
Ok. You may be thinking that I forgot about the title of this article. I did not! I just wanted to help you see the difference between computation and mathematics. Now we're ready to pursue that idea from the title. Which is to say, math is not about getting the right answer. Sounds silly right? It isn't. Here is the official statement:
Mathematics is not about getting the correct answer. But math does give us the ability and insights to prove that "the answer" is correct.
Mathematics is a way of thinking. It is a philosophy, an art, a language and a science (maybe even in that order). Math is not "how do I do this", but "why does it work when I do this?" It's about using a foundational set of axioms (which are assumptions taken as true) and then using logical deduction and induction to study the blossoming conclusions. Which sounds really fancy right? But for example one axiom that many of us accept as true is that the shortest distance between two points is a straight line (this is one axiom from a set of axioms that make up what we call Euclidean Geometry, which most people think is the only geometry). And from this axiom we get all sorts of conclusions, one of which is the Pythagorean Theorem you probably learned in middle school and likely still remember. When you study math at the higher levels though, you investigate whether that axiom is actually useful (it is, sometimes, and other times it is incorrect), and what happens when you change the axiom. Sounds a bit out there, but consider that the shortest distance between your house in the suburbs and your cousin's house downtown is the one you get when you plug her address into Waze, and it is definitely not a straight line - so clearly Waze is using a different set of axioms for its geometry. And even if you could travel from your house to hers in a straight line, the curvature of the Earth means that that line isn't really straight, but is actually an arc. Which is why airplanes don't use Euclidean Geometry to plot flight paths, since they travel over a sphere, and all lines are arcs. And that's why a flight from Toronto, Canada to Beijing, China actually passes over the arctic, even though they are almost on the same latitude. And if that piques your interest, you may want to look up Spherical Geometry, which is an entirely different (and arguably more useful) geometry than what you were taught about in the early grades. Among many other cool features, the angles in a triangle in Spherical geometry do not always have to add up to 180 degrees, and in fact can be more or less than that by a significant amount.
Right. Ok. But that right answer thing. Surely mathematicians care about getting it?
Of course we do. And often, the right answer being something predetermined, it is a metric we can use to tell if a student knows the mathematics they are studying. But it's actually fraught with problems. The kind of math we teach in high school is now almost entirely possible to simulate using easily accessible technology. For example, you might remember from your algebra class that we like to solve equations like
and in Ontario, we spend quite a bit of time teaching our tenth graders how to do this. But check this out:
All I did there was go to https://www.wolframalpha.com/ and type in the equation. It took me about 7 seconds. The computer delivered the correct solution, along with a plethora of information about the question. I didn't have to do any math at all - although as a mathematician I can confirm the answer is correct, which I assumed anyway, since the clever people behind that site are all mathematicians who have put an unimaginable amount of work into creating that resource. So for a grade 10 students, delivering the right answer to the equation can be done without any knowledge of the mathematics behind why it is correct, how it was determined, or even what the question itself actually meant!
This is what I mean when I say mathematics is not about getting the right answer. Of course, if you are proficient in mathematics you will "get the right answer", but it is not the focus. The focus instead is on the process, the logical reasoning, and the deeper understanding of the beauty of it.
So next time someone asks you what math is good for, answer like I do, which is a paraphrase from a book I read a long time ago: "Math is not good for anything. It is simply good! Why does it have to have a reason? Nobody ever stood under the ceiling of the Sistine Chapel and bellyached about what it was good for. They just stand and marvel at the beauty and majesty of the work. So it is with the study of mathematics."
Thanks for reading,