[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 
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?

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 
is below.)

“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 Friday’s 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 
teaching methods. 

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 children’s 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. 

“It’s 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 
school students.

Other mathematicians called the findings interesting but warned against 
overgeneralizing. “One size can’t fit all,” said Douglas H. Clements, a 
professor of learning and instruction at the University of Buffalo. 

“That’s 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. 

“It’s 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, it’s 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.

Cheers, people
Stephen Loosley
Victoria, Australia

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