[Offtopic] Maths teaching, concrete and abstract
stephen at melbpc.org.au
stephen at melbpc.org.au
Mon Apr 28 01:14:20 EST 2008
Study Suggests Math Teachers Scrap Balls and Slices
By KENNETH CHANG www.nytimes.com Pub: April 25, 2008
One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward
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?
Entranced, perhaps, by those infamous hypothetical trains, many educators
in recent years have incorporated more and more such examples from the
real world to teach abstract concepts.
The idea is that making math more relevant, makes it easier to learn.
That idea may be wrong, if researchers at Ohio State University are
An experiment by the researchers suggests that it might be better to let
the apples, oranges and locomotives stay in the real world and, in the
classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 -
50t, where t is the travel time in hours of the second train. (The answer
The motivation behind this research was to examine a very widespread
belief about the teaching of mathematics, namely that teaching students
multiple concrete examples will benefit learning, said Jennifer A.
Kaminski, a research scientist at the Center for Cognitive Science at Ohio
It was really just that, a belief.
Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler
did something relatively rare in education research: they performed a
randomized, controlled experiment. Their results appear in Fridays issue
of the journal Science.
Though the experiment tested college students, the 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
In the experiment, the college students learned a simple but unfamiliar
mathematical system, essentially a set of rules. Some learned the system
through purely abstract symbols, and others learned it through concrete
examples like combining liquids in measuring cups and tennis balls in a
Then the students were tested on a different situation what they were
told was a childrens game that used the same math. We told students
you can use the knowledge you just acquired to figure out these rules of
the game, Dr. Kaminski said.
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 problem with the real-world examples, Dr. Kaminski said, was that they
obscured the underlying math, and students were not able to transfer their
knowledge to new problems.
They tend to remember the superficial, the two trains passing in the
night, Dr. Kaminski said.
Its really a problem of our attention getting pulled to superficial
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.
But if the Ohio State findings also apply to more basic math lessons, then
teaching fractions with slices of pizza or statistics by pulling marbles
out of a bag might prove counterproductive. There are reasons to think it
could affect everyone, including young learners, Dr. Kaminski said.
Dr. Kaminski said even the effectiveness of using blocks and
other manipulatives, which have become more pervasive in preschool and
kindergarten, remained untested. It has not been shown that lessons in
which children learn to count by using blocks translate to a better
understanding of numbers than a more abstract approach would have achieved.
The Ohio State researchers have begun new experiments with elementary
Other mathematicians called the findings interesting but warned against
overgeneralizing. One size cant fit all, said Douglas H. Clements, a
professor of learning and instruction at the University of Buffalo.
Thats not denying what these guys have found, whatsoever.
Some children need manipulatives to learn math basics, Dr. Clements said,
but only as a starting point.
Its a fascinating article, said David Bressoud, a professor of
mathematics at Macalester College in St. Paul and president-elect of the
Mathematical Association of America.
In some respects, its not too surprising.
As for the answer to the math problem at the top of this article, the two
trains pass each other at 11 p.m. at the midway point between Stations A
and B. Or, using the abstract approach, t = 4.
More information about the offtopic