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In today’s post I will address the three most common misconceptions I encounter about mathematics from otherwise knowledgeable and often well-educated people. I strongly believe that understanding why these misconceptions are erroneous will be an important step towards eliminating the math anxiety (which, in contrast with this blog’s name, I could also call mathophobia) so prevalent in North America, as documented in a recent psychological study. To be fair, such anxiety seems to be prevalent elsewhere as well, but I have heard “Oh, I hate/don’t understand/suck at math!” in response to questions about my occupation far more frequently from North Americans than from people from other continents (which could also mean that North Americans are simply more outspoken about their negative feelings towards math; in the case of some individuals I met it almost gets to the point of wearing these negative feelings as a badge of honor). My hope is for this post to challenge the misconceptions and generate discussion.
First, it never ceases to amaze me how many people think mathematics is about numbers. Sure, you start learning math by learning how to count, and then go on to learn about the operations you can do with numbers. But by the end of primary school, or perhaps early middle school, the emphasis changes. By the time you get to algebra, geometry and trigonometry, the numbers take a back seat. Mathematics is no more about numbers than writing is about letters or music is about notes. What mathematics is really about is patterns. These patterns describe relationships between different mathematical objects, which could be numbers (for instance, prime numbers, the subject of an earlier post, serving as building blocks for all natural numbers), but could just as easily be more abstract (for instance, irreducible polynomials serving as the building blocks for all polynomials). Numbers, in fact, play a less and less significant role in mathematics as one learns more of it. It is common for mathematicians to be the least reliable people when it comes to splitting the bill at restaurants: a source of great shame for many of us.
Second, if I had a dollar for every time someone told me math is hard, I’d be a rich man now. Although many math textbooks are based on the idea of repeating the same exact exercise with different data, the real core of math is the concepts, and there are not that many concepts you need to know to understand math. Let’s take the example of solving an equation. There is a single concept – that of isolating the variable of interest – that underlies the entire problem. Sure, there are many techniques for doing this – adding the same quantity to both sides, dividing out common factors, combining like terms – but they are all subservient to the single unifying goal: isolating the variable. The exact same concept carries over to systems of equations with multiple unknowns. Unlike biology, mathematics has very few facts that need to be memorized; but it does have a set of core concepts that must be understood. Everything else is techniques that can be perfected through practice, much like writing or music. Not everyone may be able to become Gauss (or Tolstoy or Mozart), but everyone is capable of developing the basic skills to the point of moderate proficiency, which is all that is required for math, at least through the end of freshman year at universities. Incidentally, Albert Einstein famously said: “It’s not that I’m so smart, it’s just that I stay with problems longer.” That’s why math is not hard; it’s simply a skill like many others.
Finally, I am baffled by the North American obsession with getting the right answer when doing math, which plagued me throughout high school – a careless arithmetic mistake (of which I made quite a few) was then just as bad as using a completely erroneous approach. This is inappropriate for a discipline where process, much more than the result, is critical; if you reduce 16⁄64 to 1⁄4 by “simplifying” the 6 on the top and bottom, you get the right answer… by the wrong method. I can only think of one reason for the popularity of the cult of right answers: grading is much easier when you only have to look at the answer (and as a grader, I did not always enjoy reading long, convoluted solutions to spot the mistakes in them). Just imagine if one tried teaching writing in this way: it wouldn’t matter what your thoughts were or how you expressed them, as long as there were no spelling mistakes. Math is about using the right approach, and the right approach means the right concept. Many of my friends, including myself, have written entire college-level math finals without even once opening the textbook (except to get the assigned problems), focusing instead on the concepts. We may not always have had enough time to derive everything from scratch in those finals to ace them, but we still remember these concepts now.
And if you were wondering whether getting the right answer matters in mathematical research, let me tell you that it doesn’t; it’s all about asking an interesting question. The mathematician and physicist Wolfgang Pauli once famously complained about a colleague’s work that it “wasn’t even wrong”. As for practical advice, one suggestion that I’ve repeatedly given to beginning mathematicians is to start with a small example. It’s surprising how concepts that might seem daunting at first encounter quickly become familiar once you play with them for a few minutes. There is really no reason to fear math if we approach it playfully. And the beautiful thing about math is that the same concept can usually be approached from so many different angles – one of my favorite branches of math, linear algebra, is the source of such great satisfaction to me because I can think about it geometrically (picturing planes or ellipsoids), algebraically (writing down equations or matrices), or abstractly (just using definitions and theorems). I truly believe that anyone can gain enough of an understanding and appreciation for math to enjoy it. And if it doesn’t work for you I’d much rather hear “I tried it, but I don’t really enjoy it” than any of the other negative statements about math that I commonly hear when I tell people what I do.