## Learning Math: Symbols Beat the Real World

### May 10, 2008

There is a widely held belief (don’t ask me who the Widely’s are, but they include students, their parents, and even most of their teachers) that certain inscrutable subjects are easier to learn if we replace inscrutable symbols with concrete stories and pictures. Take algebra. Those *x*‘s and *y*‘s will never click with kids because symbols are too abstract, so replace them with real-world objects and – bingo – connections get made.

Maybe this point is too abstract? Fine. Here’s what I mean, you could teach algebra to students the “dry robot” way by teaching them rules for setting up and solving equations like 40(*t*-1)=400-50*t* or you could teach algebra the “friendly humanistic” way by giving them concrete examples like “One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?” (Hint to reader: the formula and the story are equivalent.) OK, the friendly humanistic approach requires a bit of story-telling, but its engaging, it can be pictured, it can be made into a movie. The formula? Bleah.

Yet it turns out the the widely held belief ain’t so. When it comes to learning math, the dry robot approach produces better results than the friendly humanistic approach (the latter appears to produce no learning at all!). A story in the New York Times (**“Study Suggests Math Teachers Scrap Balls and Slices,”** April 25, 2008 by Kenneth Chang) describes how scientists went about testing students ability to learn using the two approaches. (The **original research article** appeared in the April 25, 2008 issue of Science magazine.) Students who were taught with the concrete approach tended to remember the story, not the abstract principle that the story was supposed to convey. Students who were taught to process symbols, on the other hand, were more likely to remember and use the abstract principles.

Surprised? I’m not. I’ve seen this debate play itself out for decades in chemistry, the subject I teach. A chemistry teacher could make an even stronger case for the use of concrete examples (there really is a concrete world of phenomena underlying all of those symbols), but I wouldn’t be surprised if symbol-based learning actually produced better outcomes.