Learning Math

Here is an article about America’s recent ascent to the top of world math competitions, with many points worth considering when we get to part 2 of the Blank Sheet of Paper process of outlining an appropriate model for our educational goals. Do read the whole thing, it is excellent.

A team of American teenagers won last year’s math olympiad. Some quotations and comments:

It also wasn’t an aberration. You wouldn’t see it in most classrooms, you wouldn’t know it by looking at slumping national test-score averages, but a cadre of American teenagers are reaching world-class heights in math—more of them, more regularly, than ever before. The phenomenon extends well beyond the handful of hopefuls for the Math Olympiad. The students are being produced by a new pedagogical ecosystem—almost entirely extracurricular—that has developed online and in the country’s rich coastal cities and tech meccas. In these places, accelerated students are learning more and learning faster than they were 10 years ago—tackling more-complex material than many people in the advanced-math community had thought possible.

Note: extra curricular. Not in classrooms during regular class hours. All over the place, it seems, parents and students are taking advantage of local and on-line resources for learning math. Some do this because math = money. But for the traditionally educated, math is hard:

Between 2003 and 2009, 48 percent of students pursuing a bachelor’s degree in a stem field switched to another major or dropped out—many found they simply didn’t have the quantitative background they needed to succeed.

Note that those even trying STEM fields are already the best of the best, almost always with advance placement math credits under their belts. And colleges routinely supply all sorts of remedial classes to help them catch up. But still nearly half fail.

The roots of this failure can usually be traced back to second or third grade, says Inessa Rifkin, a co-founder of the Russian School of Mathematics … In those grades… instruction—even at the best schools—is provided by poorly trained teachers who are themselves uncomfortable with math. In 1997, Rifkin… saw this firsthand. Her children, who attended public school in affluent Newton, Massachusetts, were being taught to solve problems by memorizing rules and then following them like steps in a recipe, without understanding the bigger picture. “I’d look over their homework, and what I was seeing, it didn’t look like they were being taught math,” recalls Rifkin, who speaks emphatically, with a heavy Russian accent. “I’d say to my children, ‘Forget the rules! Just think!’ And they’d say, ‘That’s not how they teach it here. That’s not what the teacher wants us to do.’ ” That year, she and Irina Khavinson, a gifted math teacher she knew, founded the Russian School around her dining-room table.

So people good at math see what is taught in schools isn’t even really math – what’s called math is really learning to follow orders – detailed, complex orders, but nothing the student is expected to understand.

Rifkin trains her teachers to expect challenging questions from students at every level, even from pupils as young as 5, so lessons toggle back and forth between the obvious and the mind-bendingly abstract. “The youngest ones, very naturally, their minds see math differently,” she told me. “It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn’t know enough mathematics, she will answer the simple question and shut down the other, more difficult one. We want children to ask difficult questions, to engage so it is not boring, to be able to do algebra at an early age, sure, but also to see it for what it is: a tool for critical thinking. If their teachers can’t help them do this, well—” Rifkin searched for the word that expressed her level of dismay. “It is a betrayal.”

“It is a betrayal.” Yes, if it were true that the schools wanted kids to learn math as “a tool for critical thinking.” But see what happens when mathematicians, who possess real critical thinking chops, get a look at what goes on in schools? They go do something else.

The new outside-of-school math programs like the Russian School vary in their curricula and teaching methods, but they have key elements in common. Perhaps the most salient is the emphasis on teaching students to think about math conceptually and then use that conceptual knowledge as a tool to predict, explore, and explain the world around them. …To keep pace with their classmates, students quickly learn their math facts and formulas, but that is more a by-product than the point.

So, if you are encouraged to think conceptually and understand the concepts, you can learn something deeply. More important, you learn how to learn – how to think, even. The formulas are a result, not a cause, of understanding.

So how do kids like it?

 

In my experience, a common emotion at New York Math Circle, at the Russian School, in the chat rooms of the Art of Problem Solving and similar Web site, is authentic excitement—among the students, but also among the teachers—about the subject itself.

So, if math education does not consist of having someone, often someone who doesn’t really even like or understand math, drill a room full of kids on how to follow orders to get the right answer, but instead consists of voluntary associations of people who love and understand math helping each other to learn, kids will be excited about it? Wow.

The pedagogical strategy at the heart of the classes is loosely referred to as “problem solving,” a pedestrian term that undersells just how different this approach to math can be. The problem-solving approach has long been a staple of math education in the countries of the former Soviet Union and at elite colleges such as MIT and Cal Tech. It works like this: Instructors present small clusters of students, usually grouped by ability, with a small number of open-ended, multifaceted situations that can be solved by using different approaches….

“Usually grouped by ability.” So, rather than grouping kids by something completely arbitrary and irrelevant to learning, such as age, for example, we have small clusters of student who are at similar levels of knowledge *about the subject of interest* learning together? Wow, again. One could carry this further, and suggest that different groups should be made any time a different subject comes up, so that the time and attention of the teacher and most especially the students for whose sake education is undertaken are not wasted.

Freewheeling collaboration across age, gender, and geography is a baseline value

Yep. Another way to put it: segregating kids by age doesn’t work for real education.

Anyway, the article is a good read, for the most part. Is does get bogged down in the inequality of opportunity these new extra-curricular methods entail, as only people who want their kids to learn math, know about these programs and have the time and money to do them take advantage of them. The solution, of course, is easy: if your neighbor has a cow and you have none, the government must take you neighbor’s cow. Oops! I mean, that’s a tough one! The toughest part is getting parents and kids to want to learn math. The rest is easy enough once that’s done. But I suspect that, just as the schools are not set up or motivated to teach critical thinking in any but the Newspeak meanings of the words, the focus will be on solving the access and money parts of the equation, until everybody is so sick of it that they just shoot the cow.