Mathematics as a Foreign Language: a Tale of Two Classrooms

Last Thursday's #mathchat topic was "Is the spirit of mathematical thinking being swamped by a focus on technique?". One of the things that caught my eye during this discussion was a comment by David Wees suggesting that we teach math more like programming. I've proposed something similar to this before, but as the conversation continued into the details of learning how to program I started to think of the process like learning a foreign language. While I quickly came to realize that there were differing views on how foreign languages should be taught, I think there might be something to this idea. The human brain has built-in hardware to assist in learning language. Can math education take advantage of it?

Mathematics has its something of its own written language. A "conventional mathematical notation" has emerged through a variety of social influences. Some of those notations "just make sense" in the context, while others are adopted for purely historical reasons. As an undergraduate, college mathematics was like learning a foreign language for me. I had no idea what " \forall n \in \mathbb{R} " meant. Aside from "n", those symbols were not used once in any of my previous courses! It was culture shock. I eventually adjusted, but I now understand why mathematical notation can have such an intimidating effect on people.

What follows are my experiences with learning two foreign languages and how I think the difference between the two methodologies relates to the "math wars". I had 2 years of Spanish in high school and 3 semesters of Russian in college. I'm going to refer to the teachers as Mrs. T and Mrs. R respectively, for reasons that I think will be obvious later.

Mrs. T's Spanish class was held in a portable classroom at the edge of the high school. The classroom held about 30 students and the air conditioning barely kept out the 100-120 degree desert heat. I must give Mrs. T some credit for being able to do her job under such conditions. The classes often started with practice reciting words and phrases, followed by worksheets in groups and ending in a quiz. "Capitones, vengan aqui", she would say while slamming her hand down on the table in front of her, indicating that the students in the front row of the class were to carry everyone's work up to her. Everyday she would do the same routine, and everyday I wished that table would snap in half. We had done so many 10 point worksheets that at the end of the semester I came to the mathematical conclusion that the 100 point Final was only 2% of my grade. Being the little smart-ass that I was, I pointed out that I could skip the Final and still get an A. I don't think she liked that very much, because she threatened to fail me if I didn't take it. Aye que pena!

Mrs. R's class was much smaller, with only about 8 students. It was more like a conference room than a classroom. There was a U-shaped table that opened towards the white board, so Mrs. R could walk up to each person and engage in conversation. There was some rote memorization at first, while we learned the alphabet and basic grammar, but after the first few weeks of class Mrs. R started refusing to speak English in class. Class started with everyone saying hello and talking about his/her day -- in Russian. We role-played different situations -- in Russian. If I needed to know a word, I had to ask about it -- in Russian -- and someone would explain it to me -- in Russian. We watched Russian films and listened to Russian rock music. It didn't feel like a class, but rather like 9 friends with similar interests hanging out for an hour each day.

In both of the classes I learned much about the respective languages, but what really stuck with me in each case was the culture. I might not remember enough of the vocabulary to consider myself fluent in either language, but I'll still find myself singing along with Santana or Mashina Vremeni.

In the "Math Wars", the Traditionalists follow something similar to Mrs. T's method while the Reformers want math to look more like Mrs. R's class. Both methods "work", if test scores are all you care about, but there's a very subtle difference between them. In Spanish class, I always felt like I was always translating to and from English in order to communicate. In Russian class, I felt like I was articulating ideas directly in Russian. There's something beautiful about just immersing yourself in a different language until you learn it. I learned how to program in C by installing GNU/Linux and reading other peoples' source code. Sure I read a few books on the matter, but it was immersing myself in "C culture" that really solidified my understanding.

For students to really learn math, they need to be immersed in the "culture of mathematical thinking". I might not agree with the term "spirit", but mathematicians seem to display a common pattern of asking very entertaining "what if?"s and seeking out the answers. You can find beautiful math in something as simple as drawing doodles in class. There's more mathematical thinking going on when two kids make up a game during recess than there is in a thousand worksheets. Our body of mathematical knowledge is formed through communication and peer-review. It's is such a shame to see math classes run like a dictatorship, built around memorizing a list of "techniques". Sure, mathematics is an essential skill in finance, data, and engineering, but lets not underestimate the importance of "asking questions" in our focus on "problem solving".

Proceeding with the question "what if we teach math like a foreign language?", what might we do differently?

Mrs. T might argue that repetition seems to work, and there's a substancial amount of evidence it does (at least in the short term). Math class already has its fair share of repetitious worksheets, but what if we shift the focus of the repetition to learning the "alphabet and grammar" of mathematics earlier like Mrs. R's class? We could start with "set theory" and "logic" then work up from a firm foundation. The benefits could be substantial.

Mrs. R might also argue that students need to be immersed in the culture of math. Students should learn about the history of math and be exposed to "mathematical pop culture". Let's laugh together at XKCD or collectively gasp in bewilderment at the arXiv. It's moments like those that make us human. Lets embrace them.

Embrace the "culture of math".

Of course, it would probably be a lot easier to do such a thing with a student-teacher ratio of 8:1. One can only dream...

Put a Former Dead Kennedy in the White House #istandfor #jello2012

Dear Mr. Stephen Colbert and Americans for a Better Tomorrow, Tomorrow,

I understand that you are looking for suggestions for what to do with that money that I gave you. I'm sorry it's not much, but now that Congress fucked up America's credit rating I'm expecting my students loan rates to go through the roof. While it is no longer my money to spend, I think the answer for what to do with it lies in the very same reason why I gave it to you in the first place:

I would like you, Stephen Colbert, to turn the 2012 Presidential Elections into a Media Circus.

If there's one person that can do it, it's you, and I'm pretty sure you were going to do it anyway.

In fact, Americans for a Better Tomorrow, Tomorrow has already started drawing much needed attention to campaign finance issues. As ABTT continues to create media headlines, it's benevolent ringleader can take this media coverage and convert it into Strategic Humorous Indoctrination Transmissions, or SHIT. Once the SHIT is produced, ABTT can seek volunteer monkeys to hurl the SHIT in every direction. The SHIT throwing will no doubt generate more media coverage, drawing new contributors to ABTT and providing the necessary resources to produce more SHIT. The next step is to find a presidential candidate that has what it takes to make our political leaders to look like a bunch of clowns. SHIT throwing monkeys are nice, but it just wouldn't be a circus without the clowns.

My first idea was to hijack the Republican primary with a write-in campaign. Put out a few advertisements suggesting that everyone write-in the name of their favorite Conservative pundit in the GOP primary. If the votes for rest of the Republican candidates are somewhat evenly split, a social media driven write-in campaign might be enough to make it "first past the post". Of course, there might be some potential legal and viability issues to consider with this approach. In the long run, perhaps this unnamed individual's talents might make him better suited towards influencing public opinion of the candidates than being one himself.

As an alternative, I'd like to suggest Jello Biafra (Eric Reed Boucher). Yes, that's right. The former Dead Kennedys front man: Jello Biafra. Jello has been involved in the Green Party presidential campaigns since 2000, and although he's hasn't announced any intention to run in 2012, there are some who think he should.

Now, hear me out here. The US Green Party is at a huge disadvantage in the upcoming election due to the Citizens United ruling. You see, like your pal Buddy Roemer, the Green Party takes issue with unlimited corporate contributions and candidates rely on small value donations from ordinary citizens. This means that Jello wouldn't be able to run on the Green Party ticket and still accept PAC contributions. This makes it practically impossible for any Green Party, or other 3rd party, candidates to compete with the corporate financed campaigns. For a 3rd party to influence the election at all, it needs to be the center of a media circus. This is where ABTT comes in. The ABTT doesn't have to support Jello, but rather it just needs to vilify him.

You see, Jello isn't really your "tip of hat" type. He's more of the "wag of the finger" type. He's the kind of guy that will suggest radical ideas like instituting a maximum wage, abolishing the military, or lowering the voting age to 5.. The media needs a crazy-socialist-liberal to keep the news interesting! Jello's political views should put him on the top of the "Threat Down", but most people have never heard of him or the Green Party. It's really too bad this is the case, because every superhero needs a supervillain and Jello would be a perfect arch nemesis for Steven Colbert. He even comes complete with built-in product placement and catchy potential campaign slogans like "Put a Former Dead Kennedy in the White House" or "There's always room for Jello".

If Jello were to run and selects a well-known Green Party figure as his running mate, like Ralph Nader, Cynthia McKinney or Howie Hawkins, the result could be a strong Green Party ticket. Jello's punk rock attitude and anarchistic tendencies could be just the thing to pick up the disenfranchised youth vote. Regardless of whether or not Jello can win the election, a voice like his is much needed to shift the course of debate in the 2012 elections.

The plan is simple. Musical guests are no stranger to the Colbert Report, so there would be nothing out of the ordinary about inviting Jello on to talk about "The Audacity of Hype". Ask him why he hates Obama and let the SHIT throwing monkeys do the rest.

For a little taste of Jello, check out the following videos:

Jello Biafra (public speech) - LIVE

Jello Biafra on Anarchism

Green Party 2000 Convention - Interviews with Key Figures 2

Jello Biafra on Net Neutrality & the COPE Act

Jello Biafra at Open The Debates Rally at DNC in Denver 2008