# mixedmath

Explorations in math and programming
David Lowry-Duda

Every year a bit before Christmas, a particular 8th grade science teacher from my old middle school holds a Christmas Party. Funny enough, she was not my science teacher (but she was my science olympiad coach), even though I learned much science from her. She has a reputation for being strict and rigorous (which she deserves), and in general she was one of those teachers that people know are simply good.

Many other of the good middle school 8th grade teachers come to this party every year as well. This is always particularly interesting to me, because although I've changed a whole lot since my 8th grade year, there is a clear line connecting me to then. It was my 8th grade math teacher, teaching me geometry, that made me first like learning math in school. Back then, my school was not afraid of letting the highest-level students learn more and deeper material than other, and it was 8th grade where this difference really became pronounced for me.

Every now and then, I reminisce about my primary and secondary education. Was mine good? I think it was better than most, but certainly not the best. A bigger question to me is always: what is the purpose of public education? Is it to establish a certain minimum level of knowledge, or to create a basic universal level of civic virtue? Most of my close friends and I were bored throughout primary and secondary (and a lot of college, for that matter) education. I was thinking of these sorts of things when I saw my old middle school teachers at this Christmas Party.

It turns out my math teacher, whom I should perhaps credit as directing me towards math as a career, no longer teaches math. He hasn't retired or anything - he simply doesn't teach math anymore. Why not? He loved teaching math, and he tells me he plans on going back "once the curriculum settles." The problem is that right now, the math curriculum is changing too often, and each time the teachers have to go back and pass some sort of certification. It used to be simple - there was a three-track system (i.e. an advanced, intermediate, and basic level for each grade). The advanced had pre-algebra in 6th grade, algebra in 7th, and geometry in 8th. In high school, this would continue to algebra II, AP Statistics, analysis (fancy word for pre-calc), AP Calc. So in 8th grade, students would be taking geometry on the advanced track, algebra on the intermediate, or pre-algebra on the basic track. (The three-track course broke down in high schools - seniors might take AP Calc AB or BC, or AP Stat, or pre-calc, or a version of albebra III, or perhaps something else).

A few changes later brings us to the current 'integrated curriculum.' So in 6th grade, one learns either 'advanced 6th grade math' or 'basic 6th grade math.' It's 2-track and fully integrated (whatever that really means). As my teacher tells it, some thrive under the system and some flounder, as with every system. One of the greater problems is that the advanced track is more or less equivalently hard as before (let's not mess with the imprecision of this statement), while it is more challenging to teach interesting classes to the lower track. In his words: "There's not much difference between an upper algebra student and a geometry student. Perhaps the algebra student just moved into town and so wasn't yet able to transfer up to an appropriate level. Now, he's stuck at the lowest class, and he's bored. How are we to make it at all interesting to him?"

Behind this story lies the idea that schools should keep their students interested. Is that something that public schools should do? Almost every mathematician or math PhD candidate I have talked to says that they became mathematicians 'in spite of' the math of their schooling. I frequently say that I learned 13 years of arithmetic before I learned any math. I firmly believe that the current public education system allows innumerate students to unabashedly enter the 'real world.' As Paulos says in his books on Innumeracy, people can publicly admit that they are 'not numbers people,' i.e. that they are bad at math and arithmetic, and not feel ashamed. It also means that dinner conversation is a lot harder - so many people are poisoned against math.

Another first-year graduate student at Brown named Paul and I talk about some of the basic ideas of math education and testing frequently. Usually, this conversation centers around the GRE and other ETS tests. Paul and I once talked with a foreign grad student who spoke English as a second language, and his GRE strategy. The Verbal GRE section is often formidable to English speakers, let alone foreigners. He revealed to us that his strategy was to memorize a list of 10000 or so words so that he could answer all the vocab-based questions, and answered 'c' to everything else. In fact, he and his friends practiced determining whether a question was vocab and coloring the bubble 'c' as quickly as possible if it was not. And he did very well.

Paul and I have a theory, or at least an interesting experiment. If we were to read application materials of students, leaving out test scores, how well could we predict general GRE and GRE Math subject scores? Now we just need to get access to those materials... that's a bit challenging. But it sounds like a great experiment. How worthwhile are those scores, really?

But these are just brief musings on math education. A good friend of mine named John Kosh and I had a great conversation about education once. We talked about a thought experiment: what would happen if for the first 5 years of school, reading, writing, and critical thinking were the only educational goals? No explicit math, history, geography, etc. We weren't saying this because we thought it should be done. Instead, we thought that critical thinking skills are poorly under-emphasized, and it's an interesting idea. Would it be so bad for a mathematical education? I sort of suspect not. There is the problem that even basic reading and critical thinking skills require a certain amount of math, so it's not as though all math is forgone. I'm getting beside myself. The point is that we were able to come up with a system that addressed each of the shortcomings that we identified of the current system, but were of course unable to examine any new problems. That is the way with these - it's much easier to identify errors than to fix problems.

And so it is with the ETS and the GRE, in my opinion.

Anyhow, I wish anyone who read this far a Merry Christmas and, if I don't update before, a Happy New Year.

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