Algebra is necessary!

In a recent opinion piece on the New York Times website, Andrew Hacker raises the question “Is Algebra Necessary?

The short answer is: Yes.

The longer answer is that completing the square, or factoring quadratics is probably not going to be useful to most Americans when they enter the workforce, but that’s not (or at least should not be) the point of a math class. The point is to teach critical thinking and problem solving skills, so we don’t have a scarily irrational and illogical populace.

Hacker contends that mathematics is too difficult and that students who will never use it are held to unrealistically high standards regarding their mathematical aptitude.

Hacker cites the statistic that only 58 percent of those entering higher education end up with bachelor’s degrees. He claims that mathematics is the main stumbling block to graduation. Specifically at the City University of New York, 57 percent of the students did not pass the required algebra course. If these statistics are accurate, then one of two things is true:

  • Hacker is trying to pull a fast one on us, and lie with his statistics, or more likely
  • Hacker doesn’t realize that he’s proved nothing

These statistics, in tandem, are meaningless. Of course it’s impossible to tell from the single data point he gives us if these numbers are at all correlated. But even if they are (as I suspect), the implication is that mathematics is causing the students to drop out. I feel quite strongly that anyone who has earned the title “doctor” should better understand the difference between correlation and causation.

Hacker goes on to explain, to his apparent dismay, that many schools look for a 700 on the SAT math section (I’m a proud graduate of one of the schools he cites). On this page you can find the statistics for test scores of admitted students for WashU. I’ll summarize. The 25-75 percentile range for SAT scores in math in 2010 was 710-780. For verbal, it was 690-760, a negligible difference. The english and math ranges on the ACT are identical. It’s not that these schools are requiring abnormally high math scores. They’re as selective verbally as they are mathematically, a point which Hacker conveniently omits.

I would be remiss if I did not give Hacker the credit he deserves. He says “what is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.” Hacker is absolutely correct on this point. It is then almost laughable that in the very next paragraph, he states that “there is no evidence that being able to prove

(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2

leads to more credible political opinions or social analysis.” As Hacker said himself, mathematics is not about having memorized this result. What matters is what it conveys. I doubt Hacker knows this equation is exactly what one uses to generate Pythagorean triples.

To be fair, I don’t think we should expect non-mathematicians to recognize this formula. It’s beyond what I think is necessary to be a well-rounded citizen. But I do think we should expect the average American to be able to do the basic algebraic manipulations to prove this identity. This is the sort of mathematical literacy I would deem the rough equivalent of being able read, or being able to write in complete sentences.

To reiterate, being able to generate Pythagorean triples won’t help the average citizen become more informed, but that’s not really the point. Analogously, it’s not helpful to ensure that every student knows all of the president’s names and their dates of office, but that isn’t why we study history.

In fact, the problem with math education rears its ugly head exactly in this analogy. History teachers don’t care so much about our memory of dates, because whether the bill of rights was ratified in 1791 or 1792 isn’t super important. So why must math students memorize theorems and how to apply formulas? That’s the easy stuff. We should be testing them on their critical thinking and problem solving abilities. Admittedly, these sorts of tests are more difficult to write, but no one ever became a math teacher because it was easy.

Unfortunately we’re just now getting to the most appalling of Hacker’s ideas. He says that “instead of investing so much of our academic energy in a subject that blocks further attainment” we should start looking at alternatives. The implication here is that mathematics is the problem, rather than the manner in which it’s taught.

My favorite sentence in the piece is in the last paragraph: “…there is no reason to force them to grasp vectorial angles and discontinuous functions.” The implication here is that if the exact mathematical knowledge won’t be useful (and I’m not convinced that either of the things he mentioned are not useful) then it shouldn’t be taught. As a friend of mine pointed out: “most jobs do not require algebra just as most jobs do not require you to know that Barack Obama is president; that’s a really stupid argument against the value of knowing it.”

There is no doubt about it; math is difficult and often poorly taught. But that’s a pretty ridiculous reason to stop teaching it.

Yellow Pigs Day

We’re tantalizingly close to “the theorem,” but today we take a break from complex analysis to talk about something much more exciting: Yellow Pigs Day.

Yellow Pigs Day is a holiday celebrated at Hampshire College Summer Studies in Mathematics (HCSSiM) every July 17th. For those who don’t know, HCSSiM is a 6-week summer program for high school students. There’s a lot of time spent doing math, but not exclusively. In past summers there have been bands, frisbee teams, bridge clubs, and the like. Those familiar with the program will undoubtedly realize that this explanation doesn’t do HCSSiM justice. It’s so much more than “just a summer of math,” but we’ll get back to it.

Yellow Pigs day usually consists of classes (as usual) a talk on the social/historical significance of the number 17, a cake shaped/colored like a yellow pig, singing of math songs (a.k.a., yellow pig carols) and an ultimate frisbee game between the current students and the staff/alumni. That’s right many alumni come back to the program to celebrate. Alumni (such as yours truly) tend to be quite fond of the program.

No doubt you think this program, and in particular, Yellow Pigs Day, is quite odd. You are correct. One of the many things I love about HCSSiM is how readily everyone embraces these oddities.

Thus far, I’m sure you don’t really understand the pure joy HCSSiM can induce in it’s participants (students and staff). Not being particularly eloquent myself, I doubt I can give you a good explanation, but let me attempt anyway.

The focus of the program is not learning math, but creating it. The staff are around to guide the students in a right direction (notice “a,” not “the,” as there can be many correct directions to pursue). Of course, the students aren’t creating new theorems that no one has seen before, but that doesn’t take away from their excitement as they discover it for themselves.

Sometimes, math can seem quite dry, but not at HCSSiM. Staff (and often students) take pride in being able to explain the mathematics in an exciting manner. This might involve a motivating story, or a humorous analogy. Let me give you a few examples:

  • The proof for Euler’s formula (for planar graphs) is often described by pirates trying to attack the a castle displayed diagrammatically by the graph. How many walls must they destroy to be able to reach every room in the castle?
  • Just yesterday, one workshop described Halls Theorem in the following analogy: Ollivander has 100 wands he wants, and 100 incoming Hogwarts students to give them too. Of course, not every wand will work well with every student. Obviously, if there’s any hope at a matching, every collection of N students must be able to use at least N wands. Is this enough?
  • Students sometimes decide to use their own notation. This year, they wrote \binom{n}{k} as a pacman symbol with the n inside the packman, and the k in the mouth. They pronounced the symbol “n chews k.”
I’m really not explaining it well. The point is, HCSSiM has a way of making math fun and goofy without losing track of the mathematics. For your daily dose of HCSSiM, start reading Cathy O’Neil’s blog.

I can give nothing but praise to this program. As a student, it changed my outlook on mathematics. As a staff member, it challenged me to become a better teacher, to be creative (while still being accurate and appropriate) with my explanations. If you missed out on HCSSiM in high school, you should seriously consider applying to be staff. You don’t want to miss out on the best summer experience of your life.

Happy Yellow Pigs Day, all! Next year in Amherst!

Primes! Primes! Primes!

Wow, I haven’t posted in a really long time. I’m going to try to get back into it. Here’s an overview of my next series:
During the time I spent as a student and as junior staff at HCSSiM, I had the pleasure of seeing and participating in a talk entitled “infinitely many proofs that there are infinitely many primes.” The idea was to, in the course of an hour, give as many proofs as possible that there are infinitely many primes.

This has always been one of my favorite talks due to its level of accessibility. My goal with this series is to recreate much of that talk on this blog. The set-up will be as follows. Each post will contain exactly one proof of the infinitude of the primes. They may contain some fun history lessons too! Woohoo!

[Exeunt Categories]

There is certainly much more to be said about categories, but I’ve already said more than I feel qualified to talk about. As it did last time, the blog will quiet down for a while (probably until after quals in late March).

I think my next posts will be on number theory. My goal is to prove there are infinitely many primes 17 times (just to be sure it’s true). Should be exciting, and a bit more down to earth than all this category nonsense.

Philosophy and choice

Seriously, category theory is coming soon. I promise. In the meantime, a friend (who posts on this awesome blog) and I had a conversation about the axiom of choice from a philosophical standpoint. It started when I made the bold claim “whatever, the axiom of choice is obviously true.” Originally, this was meant half as a provocative joke, but really, I think it makes sense. My friend, who I’ll call Steve Neen for the sake of this post, declared, as many mathemematicians do, that he preferred constructive mathematics. After all, how can you say something exists without being able to construct it. This is why mathematicians prefer constructive proofs to non-constructive ones.

I had always sort of accepted this as a reasonable point of view (which I still think it is), but the following thought dawned on me: At their core, each of the axioms of ZFC simply say “there exists a something such that something else.” We may have notation for sets constructed from a specific axiom, but symbology shouldn’t have any bearing on whether or not an axiom is constructive. There is simply nothing intrinsically constructive or non-constructive about any of the axioms.

Of course, Steve then pointed out the difference may not be subjective. Look at what computers can do. They can represent unions, pairs, power sets (though not efficiently), etc. all in an effective manner. But look at the axiom of choice. Computers just aren’t built to do this constructively.

Fair point, Steve. One that had me thinking all of last night. One answer is that what computers can do is somewhat arbitrary, but that feels like a cop out. So here’s my two part answer, that I’m reasonably satisfied with.

  • Look at the Axiom of Infinity. It essentially says there is an infinite set. More precisely,

\exists x[\varnothing\in x\wedge\forall y(y\in x\to y\cup\{y\}\in x)]

Here’s an example of a set a computer can’t represent explicitly. One could argue that an implicit representation is possible, because in finite time one could decide whether or not a given set was an element of this one, but there is no bound on how long it would take to a computer to make this decision. Fair enough. But the power set of an infinite set (i.e., an uncountable set) cannot have such a decision procedure.
This is all to say, that if you discount the axiom of choice for it’s non-constructibility, you should also discount the axiom of infinity.
  • Jokingly, and to prove his point, Steve asked me to pick one element from each coset of \mathbb R/\mathbb Q. First of all, it’s important to note that one’s ability to do so is reliant on the axiom of choice. This is one of the reasons I think it should be true. Morally, I can pick out an element from each coset. I told Steve, all I need to do is produce a choice function on the coset. I claimed (again, jokingly) that I had such a function. In fact, I had uncountably many. If he named a coset, I’d consult my function and tell him the answer.
    Steve did point out, and is correct in doing so, that just because I claim it exists, and can verify as much as he asks doesn’t mean that I have such a function. But from a computational point of view, it sort of does. Any function that’s consistent with a choice function can be extended. If you accept the axiom of choice, then you can extend it to a choice function. If you don’t, then it can just be extended further, so long as the extension is countable. Regardless, all computations are going to be finite, so from a computational point of view, we never really need the axiom of choice, just as we don’t need the axiom of inifinty. We only need finite choice, which is provable from ZF, even without choice.

So that’s a recap of the discussion we had. I’m not entirely convinced anymore that the axiom of choice is “no different” from the other axioms, but it’s an interesting philosophical question I invite you to think about.

Time

So it turns out grad school is time consuming. I originally thought that this blog would be a good way to keep myself studying for the algebra qual, but it turns out to be less efficient than I expected. What this means is that, I have given up trying to post daily, or even weekly. From now on, posts will likely be sporadic, if existent at all. Sorry to disappoint.

Unfortunately this means that we’re sitting in the middle of Galois Theory without really finishing it off. Hopefully I’ll at least be able to finish this up within the next month or so.

After we’re done with Galois Theory, my posts will be more self-contained than a series on a specific topic. I do promise though, that if they continue, they will continue to be interesting.

Starting Galois Theory

Even though there is so much more cool stuff to do in linear algebra, people have been waiting patiently to see some Galois Theory. Now that we have some of the ideas of linear algebra, we are ready to grab the low-hanging fruit in field theory, which will lead us into Galois Theory. Tomorrow we’ll start talking about compass-and-straightedge constructions.