Hidden math masterpieces

We’re hiding math’s masterpieces from students, writes Edward Frenkel, a Berkeley math professor in the Los Angeles Times.

Imagine you had to take an art class in which you were taught how to paint a fence or a wall, but you were never shown the paintings of the great masters, and you weren’t even told that such paintings existed. Pretty soon you’d be asking, why study art?

That’s absurd, of course, but it’s surprisingly close to the way we teach children mathematics.

Math is the “language of abstraction,” writes Frenkel. To “gain proficiency with abstraction,” students need “mathematical knowledge plus conceptual thinking times logical reasoning.”

Frenkel talked about math ideas with students in fourth, fifth and sixth grades at a New York school. They were too young to be scared by math.

Using a Rubik’s Cube, he explained that every rotation of the cube is a “symmetry,” and these combine into what mathematicians call a group.  “I saw students’ eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group,” writes Frenkel.

We next studied the majestic harmony of Platonic solids using dice. And I told the kids about the curved shapes (such as Riemann surfaces) and the three-dimensional sphere that give us glimpses into the fabric of our universe.

The professor dreams of spending 20 percent of class time “opening students’ eyes to the power and exquisite harmony” of math.  The rest of the time they’d learn the multiplication tables, etc.

Would they be inspired? What percentage of math teachers are capable of doing this?

Fluent on facts, weak on abstraction

Fluency in addition and multiplication isn’t everything, writes Education Realist.

. . . plenty of solid math students don’t have fluency and—here is the important part—many exceptionally weak math students have strong fact fluency.

Ed Realist’s “math support” students, who are trying to pass the exit exam and graduate from high school, tend to be very literal and easily thrown by symbols. Ed Realist  asked students to read a simple equation as a sentence. When a student turned x + 6 = 14 into “what number do I add to six to get 14?” the answer was clear to most of the class.

One student, Gerry, still didn’t get it.  He said he could only do math if it doesn’t have letters.

“You need to look at these problems from a different part of your brain.”

. . . “X + 6 = 14. This is when you have to do stuff to both sides, right? I can’t do that.”

“What plus 6 = 14? 8.”

Gerry said he couldn’t do fractions. But when he turned x/5 = 9 into “what divided by 5 is 9?” he got 45 right away.  “I feel like a math genius,” he said.

“You know a lot more math than you think you do,” the teacher said. ” You just have to figure out how to ask the question in a way your brain understands.”

Not everyone is capable of understanding abstractions to the same degree, Education Realist concludes.

Some people do better learning the names of capitals and Presidents and the planets in the solar system. They’d learn confidence and competence through interesting, concrete math word problems and situations, and enjoy reading and writing about specific historic events, news, or scientific inventions that helped society. Instead, we shovel them into algebra, chemistry and literature analysis and make them feel stupid.

She quotes psychologist James Flynn on why IQ’s have risen steadily and significantly since the start of the 20th century (the “Flynn effect”).

Modern people . . .  are the first of our species to live in a world dominated by categories, hypotheticals, nonverbal symbols and visual images that paint alternative realities.

. . . A century ago, people mostly used their minds to manipulate the concrete world for advantage. They wore what I call “utilitarian spectacles.” Our minds now tend toward logical analysis of abstract symbols—what I call “scientific spectacles.” Today we tend to classify things rather than to be obsessed with their differences. We take the hypothetical seriously and easily discern symbolic relationships.

Well, some of us do. Flynn has a new book out, Are We Getting Smarter?