Today, what is 49 from 62? The University of Houston's
College of Engineering presents this series about the
machines that make our civilization run, and the people
whose ingenuity created them.
A New York Times piece by
Richard Rothstein got my attention. He talks about a
battle going on in math education. Look at a simple
second-grade problem, he says: "What's 62 minus 49?"
Since we have to take nine from the smaller number two in
the right-hand column, we borrow a one from the six and
turn the two into twelve.
Two schools of thought collide over how we should explain
this process to seven-year-olds. Most American teachers
"Two and six are neighbors. Just as a good neighbor lends you a cup of sugar, the six lends a one to the two."
The borrowing metaphor has a nice sound, but
how does it work?
Second-graders need to see that the top number, the 62,
can be regarded as either sixty and two or as fifty and
twelve. Replace that idea with the metaphor of borrowed
sugar, and the notion of moving a block of ten gets lost.
Worse yet, we create the impression that the six and the
two of 62 are unconnected. We've given the student
reliable means for getting an answer. But we've
undermined understanding. Short-term gain; long-term
In her fine book, Knowing and Teaching Elementary
Mathematics, Dr. Liping Ma uses her Chinese
background to help us sift through the problem of
subtracting 49 from 62. During China's Cultural
Revolution, she was taken from high school and sent off
to a poor farming region to be re-educated by peasants.
But they turned the tables and asked her to take on the
job of teaching their children. From that start, she went
on to become a professor at Berkeley.
Long before calculators, the Chinese used the abacus for
arithmetic. An abacus represents tens, hundreds, and
thousands explicitly. Chinese instruction in arithmetic
is accordingly more likely to keep the meanings of
decimal places clear.
So I asked friends to do the subtraction in their heads.
Then I asked how they did it. Almost none used the
conventional method. Most of us have created our own
means for breaking down calculations. They say things
like, "I take fifty from 62 and then add one." Many then
go on to tell stories of trouble with the teachers and
other students in second grade. Calling up their first
exposure to arithmetic unlocks demons. We do create
trouble when we let students get answers using means they
I have here an 1853 arithmetic book. In a time with far
fewer schools, learning had to be pretty independent. The
book uses arrays of blocks to explain the formalisms of
arithmetic. It shows how to move ten blocks from the pile
of sixty and add them to the two on the right. When you
do that, mystery is removed and the problem evaporates.
Anyone can see what's happening.
So trouble lurks in this matter. I found one thing very
suggestive as I put the question to friends: None
actually said the number they got when they took
forty-nine away from sixty-two. And I went away wondering
if, somehow, they felt it would be unlucky to speak that
particular number aloud.
I'm John Lienhard, at the University of Houston, where
we're interested in the way inventive minds work.
Ma, L., Knowing and Teaching Elementary Mathematics.
Mahway, NJ: Lawrence Erlbaum Associates, Publishers, 1999.
Rothstein, R., A Sane Position Amid Math's Battlefield.
New York Times, Wednesday, June 27, 2001, p. A16.
Fish, D. W., Fish's Arithmetic, Number One. New
York: American Book Company, 1853.
The Engines of Our Ingenuity is Copyright
© 1988-2001 by John H. Lienhard.