Mathematical Swearing

If I ever get the chance to teach a proof writing course, I might try introducing the concept of mathematical swear words. I would penalize work that uses these words and keep a personal “swear jar” for when I use them myself with a class incentive for calling me out.  The basic premise of this idea is that in the same way that maturity is required to know when to use swear words appropriately, mathematical maturity is required to know when to use mathematical swear words appropriately. My list of words and terms isn’t very large so far and all fall in the same category. These swear words include: obvious, trivial (when used in a non-technical way), straightforward, and clear. Unlike swear words which fall into several categories (such as vulgar, obscene, blasphemous, offensive), all these mathematical swear words fall into one category: imprecision. In fact, in proof writing, that seems to be the main sin.

The analogy feels appropriate, especially since most students will pick up bad writing habits from textbooks (and if they’re not cautious, by the teachers) and it makes clear that though these words have a place in mathematical writing, it is (1) necessary to understand the context of when to use them and (2) best to use them sparingly. Both ideas translate well from this analogy.

Will We Ever Teach “Advanced” Topics in High School?

I once taught a little bit of group theory in a math for liberal arts class once. My pretense then was that I was teaching about symmetry. The class was too small to make an effective sample size. They were also largely uninterested; at one point, no one showed up to class. I was largely inexperienced, and I would like to think that if I ever had the chance to do something similar I would do a better job. But the point does remain that there was nothing dramatically difficult about the basics of the topic.

In fact, there are many mathematical ideas that seem to go unexplored in earlier mathematical education. My post-secondary mathematical education was basically a path to learning calculus. I learned a little more about conics while preparing for a competitive math test. I learned about vectors in physics (which seems standard). And I learned about Mobius strips in an English class! (In relation to Eliot’s Four Quartets.)

Back in high school, I had a very linear view of mathematics. One idea progresses to the next and so on. Even then, this view was false once you looked deeper. But I think that there would be a benefit to being exposed to some more variety in mathematical topics. Topology was never mentioned as far as I could tell, despite some of its ideas being novel and much more visually striking. I’ve already talked about how little analysis is in calculus.

Should We Replace Trigonometry with Basic Complex Geometry?

This is a follow-up to my previous post. I think it says quite a bit about a mathematician when you ask about what their initial definitions are. If you ask about the main trigonometric functions: sine and cosine, then there are plenty of options to choose from:

• classical geometry as corresponding to points on the unit circle,
• power series,
• solutions to the differential equation $y'' = y$ with appropriate initial conditions,
• real and imaginary parts of the function $e^{it}$,
• and plenty others (including one found in M. Spivak’s classic book Calculus, which strangely enough is not a formalization of the classical model).

Honestly, despite having difficulty dealing with it formally, my initial thoughts always land on the classical model. But the truth is that it’s easiest to deal with the objects directly in almost any other way. The problem with most of these definitions is that we usually want to define these functions before calculus, and most require at least a little knowledge of calculus to understand.

But there is an exception, namely, if you define cosine and sine as the real and imaginary parts of $e^{it}$, then there’s no calculus in sight. Of course, this requires introducing the complex numbers and you’re only pushing the calculus hiding to the exponential function. But my claim is that the latter is something we’re already comfortable doing. The former seems justified (at least to some extent), but introducing the complex numbers and their geometric picture seems to give an algebraic framework for understanding plane geometry.

And justifying the geometric picture seems no more hand-wavy to me than how we currently do things. And with this $pi$ is a constant with an actual definition (i.e. the first positive solution to $e^{it} = -1$), as opposed to the never quite defined number that most people think of it as. The addition formulas become extensions of exponential rules, which hopefully are well-known by that point.

I think that a major criticism is that Euler’s formula no longer becomes some special, but is just the definition. A fair criticism, but I would say it’s a pretty common phenomenon in math that a major result eventually becomes a definition.

No One Likes Trigonometry

I’ve never met a non-mathematician who likes the subject, and I’ve met only a few mathematicians who do. Even in the latter case, I suspect it has more to do with later learning of Fourier analysis and the like, rather than their first exposure to the class. I have no scientific data on the matter, but I suspect it’s rather high on the list of least liked mathematical topics, a list already full of disliked material.

The trigonometric functions seem enigmatic or even artificial. Among students, the belief that these functions were invented by mathematics teachers to torture students is probably not uncommon. It doesn’t help that when learning integrals, trigonometric functions are handy to make interesting (and therefore, difficult) integrals to solve later on.

My opinion on the matter is that the toolkit approach to algebra and trigonometry ironically makes the material seem less useful than if they came up more organically. The approach we seem to have in teaching trigonometry is to introduce the main functions, and then scrounge up some problems that can serve as applications. By coming up “organically”, I mean that it might be better served to teach a course on geometry which eventually leads to material that can’t be tackled without introducing the machinery of trigonometric functions. This approach might make trigonometry appear more natural in geometry and provide some context for how it is applied.

The Trouble with Pedagogy involving the Infinite

There’s a problem I have with teaching certain mathematical concepts, especially when they involve the word “infinity.” In algebra, one of the ways this pops up is during interval notation, where if you have an unbounded interval, you use the symbols $+\infty$ or $-\infty$ to denote that it is not bounded on whichever side. One of the problems with this is that now it seems as though we’re treating “infinity” as though it’s a number. People have some conceptual notion of infinity prior to any formal knowledge of how mathematicians use the idea Childhood number topping games trumped by crying out “Infinity,” only to be refuted with a “Infinity + 1” come to mind. Encouraging it seems like a bad idea as if one every gets to talking about when such concerns about infinite sets or real numbers arise the naive notion inclusion of infinity as a number. The usual way of explaining it away is the unsatisfying answer that “Infinity is a concept, not a number.” But that’s strange, what makes something a “concept” and what for that matter is a “number”?

Then again, I have tried the opposite approach of introducing the extended real line in a calculus course. Of course, this often fails, because of the subtleties involving not having all arithmetic operations available. So I ask non-rhetorically, is it better for people to treat $\pm \infty$ as part of an “extended” real number line, or to say “Infinity is a concept, but not a number”? If the latter, what are we trying to say with that?

As one final thought, yesterday, I talked about infinite sums. What is probably obvious to everyone who has dealt with analysis is that we do not in fact, add infinitely many number together. The entire point is that we can find finite sums that are within any margin of error to our value and therefore the infinite sum is that value. This is difficult to understand at the first exposure, since it’s a loaded concept with some much behind it. But the intuitive idea of adding infinitely many things together is probably how most calculus students think of the idea. Isn’t this why people have so much trouble with the equation $0.\bar{9}=1$?

My Favorite College Algebra Lecture

Well, it’s not an entire lecture, not even quite half a lecture. But my favorite part of a lecture in college algebra is the properties of exponentiation. Sounds strange, right? I basically list off some algebraic identities involving exponents, and I’m done. The value does not lie in the content, but in the presentation of the material, and how it comes the closest to what I enjoy about math.

We start with $b\geq 0$ and $m,n$ natural numbers. We notice that when looking at $b^{n}\cdot b^{m}$, this is first multiplying $b$ a total of $n$ times, and then multiplying $b$ another $m$ times. So this is the same as multiplying $b$ a total of $m+n$ times, or $b^{m+n}$ Here I might take a specific $m,n$ like $m=4$ and $n=3$ and write it out. So we get the equation $b^{m+n}=b^{m}\cdot b^{n}$.

Next if we take a look at $(b^m)^n$, we can expand it out as multiplying $b^m$ a total of $n$ times. So $b^m\cdot b^m\dotsb b^m$, then being a little suggestive, I expand the $b^m$ out vertically, creating an array of $b$‘s. Then it becomes clear by appealing to the area of a rectangle that this quantity is $b^{mn}$ the result of multiplying $b$ a total of $mn$ times.

So here’s the part I love. So far, we have appealed to the nature of exponents as repeated multiplication to deduce some properties about it. I then say that these are nice algebraic properties that will make our lives easier when we have to deal with exponentials, wouldn’t it be nice if they were still true even after we consider more general $m,n$? So let’s pretend they do still hold and see what happens.

We first take a look at $m=0, n=1$ and by looking at the addition property we get that $b^{1+0}=b^{1}\cdot b^{0}$, and since $1+0=1$ and $b^1=b$, we see that $b=b\cdot b^0$, and if we divide both sides by $b$, we see that $b^0=1$. So we get a very specific value for $b^0=1$. It wasn’t something handed down from the heavens, it wasn’t some convoluted aspect of $0$, or strange combinatorics. Just we want this addition property to be true, and for it to be so, $b^0$ must be $1$.

Then we take a look at negative numbers. So we let $m$ be any positive whole number and use the addition property and notice that $b^{m}\cdot b^{-m}=b^{m+(-m)}=b^{0}=1$. So erasing the middlemen, we have $b^{m}\cdot b^{-m}=1$, or by dividing both sides by $b^{m}$, we have $b^{-m}=1/b^{m}$. Now we have that negative exponents correspond to the reciprocals. Again something that pops out, just because we wanted to keep that addition property.

Finally we look at rational numbers. We first we deal with $1/n$. Finally the other property comes into play: $(b^{1/n})^n=b^{1/n\cdot n}=b^{1}=b$. Cutting out the middlemen again, we see that $(b^{1/n})^n=b$. But there is a special name for a number whose $n$th power is equal to $b$ (when $b>0$, and that is the $n$th root of $b$, written $\sqrt[n]{b}$. As for the arbitrary case, we see that since $m/n=(1/n)\cdot m$, and so $b^{m/n}=(b^{m})^{1/n}=\sqrt[n]{b^m}$.

Now, I may not have persuaded you that this is a good lecture, let alone one deserving of being called a favorite. The first thing to notice is that there are few hard quantities involved; it’s mostly algebra. When these classes are taught, the first thing a (pure) mathematician notices is that it’s deprived of any interesting abstract mathematical ideas. The teacher is forbidden (for good reason!) from straying too far into his own domain of general theory, and focus on the particular, lest the students become confused and bewildered. The more headstrong mathematician-teacher charges ahead anyway, and is met with sighing, head scratching and yawning. Proving the quadratic formula by completing the square of an arbitrary quadratic functions sounds like a great idea, but it’s a quick way to lose a room full of freshmen. This has a wonderful balance; easy enough that the algebra clarifies rather than obfuscate, yet complicated enough, that it says something actually interesting and not at first obvious.

Also take note that the progression of our ideas follows the same construction of the rationals from the natural numbers. We starting with a counting device; consider repeated counting, also called addition and then repeated addition, which we call multiplication. We consider 0 the neutral element in addition. Then we follow-up with negative numbers, or in more algebraic terms, the additive inverses we get from extending our monoid of natural numbers to the group of integers, which turns out to be a ring. We then form the ring of fractions, which give us multiplicative inverses. We follow the same chain of constructions without mentioning the fact that we are doing so, so we implicitly form a picture of the way numbers work without having to explicitly talk about a great deal of abstract algebra.

But what seems like the most important thing I love about this is the premise of the lecture. We begin with an observation that exponents work in a certain way, when we consider natural numbers. In particular, it’s very natural, almost completely obvious. Now rather than making rules for the other numbers and noticing that we get the same properties, making them more complicated by having so many cases, we say “wouldn’t it be nice if this still held?” And the point isn’t “This is how things are,” but “we want this, and we accept the consequences of our choice.” Too often, math is considered a bunch of strict rules, like a code of law. “You are allowed this action, but not this one.” And I admit I encourage this view, especially when dealing with things like dividing fractions, or adding fractions or… pretty much anything involving fractions. But this isn’t how math is, not the kind I love so much. It allows for so much creativity and invention. The only truly mathematical law is “Be consistent,” which is admittedly more restrictive than at first appears. But we allow every twist and turn of every possible mathematical rule or system and allow a complete freedom provided one thing: we must accept the logical consequences of our choices. “Everything not forbidden is permitted.”

a defense of boring math

There’s something wrong with the way we teach mathematics today.

So you will quickly hear if you talk to quite a few graduate students, professors, instructors, students, random people on the street… I do agree. There is something wrong with mathematical education (in the US, along with plenty of other places I would guess).

What exactly the problem is somewhat harder to say, but something that you will hear from graduate students and professors, people who have had exposure to “higher” mathematics, will often include the complaint what we teach is tedious, mundane, useless computations and manipulations. Questions have cookie-cutter forms that one presents multiple times throughout, demanding exacting details. Questions of any interest or requiring any independent though are not allowed in fear that students will complain or ignore them.

There is certainly truth to this. And I cannot say that I have anything resembling a solution to this problem or that I’m even a decent math instructor to have authority or experience to talk about what one should do. But despite this, I think that the demonizing of this mechanical understanding can be detrimental to the ultimate goal.

It’s been my understanding that mechanical understanding precedes deeper conceptual understanding. As far as I recall I memorized a list of numbers before I learned how to count things. When I first learned addition, I admit I sometimes leaped-frogged the number line, but with enough practice I just memorized what two arbitrary one-digit numbers added up to and then carried out an algorithm. Similarly I memorized the multiplication table without really caring about any deeper implication.

I think that in order to get a real grasp of calculus-track mathematics, one has to feel comfortable with numbers. And that one gets comfortable with numbers by familiarity with them, even in mundane and routine ways.