Beyond the buzzwords

Ben always dreaded being asked for his “teaching philosophy,” he writes at Math with Bad Drawings. It seems to come out as:  “We buzzword to buzzword, not for the buzzword, but for the buzzword.”

Now he’s got it in 11 words:  Math is big ideas, approached from as many angles as possible.

For example, here’s a big idea:


He teaches from the historical, verbal and scientific angles. Then comes practice.

Math without any computational practice is a mushy math, a math with no spine. To understand what makes, say, linear equations tick, you’ve got to solve ‘em, graph ‘em, play with ‘em in a hundred different ways. You can’t grasp patterns until you’ve worked through examples. Without multiplication facts at your fingertips, you’re unlikely ever to apprehend deep truths about the distributive property. If you’ve never spent a day multiplying out products of the form (ax + b)(cx + d), then you’ll never internalize the methods for factoring quadratics.


Dance by numbers

Gifted fifth graders in Virginia are learning about patterns by dancing, reports PBS.

Carrie Lewis and Kelly Steele’s fifth grade students slide and spin across the classroom floor, doing the hustle, the robot and the running man. . . .

“Dances are patterns,” Lewis said. “We had identified that our students had trouble with patterns and this was a way to get them involved in it.”

On the Pillsbury Dough Boy website, students analyze the cartoon mascot’s six dance moves, assign each step a number and chart the patterns in his dance.

They also watch America’s Best Dance Crew to chart the repetition of dance moves.

Students then choreographed their own dance routines, repeating at least five moves.

Using stopwatches to clock the average time of their routine, students were asked to then calculate how many times their pattern would repeat throughout the course of the song, and then turn the resulting data into a graph.

“If your song is 100 seconds, how many repetitions will you do of your dance?,” Steele said. “If you use the extended version of your dance, 200 seconds, how many repetitions do you need to do? They are using their graph to figure that information out.”

When each group performs their routine for classmates, they freeze midway through the dance. Students must predict the next move in the pattern.

Brain calisthenics

Brain calisthenics” — such as computer-based exercises in quickly linking graphs to equations –  help students internalize abstract ideas and see patterns intuitively, say cognitive science researchers in a New York Times story.

Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of this same bottom-up ability, called perceptual learning. The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest.

In a 2010 study, UCLA and Penn researchers used perception training to teach fractions to  sixth graders in a Philadelphia public school.

On the computer module, a fraction appeared as a block. The students used a “slicer” to cut that block into fractions and a “cloner” to copy those slices. They used these pieces to build a new block from the original one — for example, cutting a block that represented the fraction 4/3 into four equal slices, then making three more copies to produce a block that represented 7/3. The program immediately displayed an ‘X’ next to wrong answers and “Correct!” next to correct ones, then moved to the next problem. It automatically adjusted to each student’s ability, advancing slowly for some and quickly for others. The students worked with the modules individually, for 15- to 30-minute intervals during the spring term, until they could perform most of the fraction exercises correctly.

In a test on the skills given afterward, on problems the students hadn’t seen before, the group got 73 percent correct. A comparison group of seventh graders, who’d been taught how to solve such problems as part of regular classes, scored just 25 percent on the test.

Notice how few students understand fractions.

Reading the comments reminded me of the parable of the six blind men and the elephant. Every reader seems to think the research proves their theory: Kids need more practice; kids need to construct knowledge, kids need real-world examples, kids need visuals.

I’m not doing well with abstract ideas this week, due to a horrible cold and a racking cough, but here’s UCLA’s graphs ‘n equations module.