Abstract concepts are better for teaching maths than real-world ones, according to a study reported on in the New York Times.
What does this mean for teaching maths and what does it mean for Maths-Whizz?
Rather than offer endless examples of trains setting off from stations, apples in baskets, bars of chocolate and slices of pizza, teachers might well be encouraged to have their students think of problems less in practical terms and more in terms of non-specific abstractions.
Abstract symbols vs. concrete examples when teaching maths
The study in question, conducted by researchers at Ohio State University, presented university students with a set of mathematical rules, expressed either with abstract symbols or with concrete examples (tennis balls, and so forth):
“Then the students were tested on a different situation — what they were told was a children’s game — that used the same math.”
“The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.”
The implication here is that abstract examples are transferable; students who study concrete examples are less able to ascertain the underlying principles, or apply those principles to other problems.
Teaching maths elementary or primary
The article goes on to say that this issue is as relevant for younger brains as those of university age, and that this has implications for how elementary, or primary, maths is taught:
“…researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.
“The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math.”
Whilst not all researchers in this field might agree with the implications for primary education, the principle seems sound. After all, it is quicker to write an equation than it is to count out apples or cut up loaves of bread.
But, and here’s the caveat from this corner, it is natural to analogise in education. In other words, we readily use specific examples from life to extrapolate to the general. In some disciplines (like economics or history) particular examples can be less instructive when looking at the general issue. But it is just as true that 2 and a half apples are equivalent to 5 half-apples as it is true that 12 quarter-loaves are equivalent to 3 whole loaves.