Photo by otisarchives3.
I discovered a case of MWS (Math Workbook Syndrome) one afternoon, as I was playing Multiplication War with a pair of 4th grade boys. They did fine with the small numbers and knew many of the math facts by heart, but they consistently tried to count out the times-9 problems on their fingers. Most of the time, they lost track of what they were counting and gave wildly wrong answers.
The Times-9 Trick
We stopped the game in mid-turn to teach a multiplication trick based on the distributive property…
Multiplying by nine is the same as multiplying by “ten minus one.”
Nine of anything is the same as ten of that thing, take away one of them.
- Nine books is ten books, take away a book.
- Nine horses is ten horses, take away one of them.
- Nine pizzas is ten pizzas, after you eat one of them.
- 6 × 9
is the same as 6 × (10 – 1),
ten 6s take away one 6,
or 60 – 6 = 54.
- 8 × 9
is the same as 8 × (10 – 1),
ten 8s take away one 8,
or 80 – 8 = 72.
- 25 × 9
is the same as 25 × (10 – 1),
ten 25s take away one 25,
or 250 – 25 = 225.
By reducing the multiplication to a simple subtraction, this trick makes the times-9 table a cinch. We spent a few minutes going through the times-9 facts together, just to practice the pattern:
1 × 9 = 10 – 1
2 × 9 = 20 – 2
3 × 9 = 30 – 3
4 × 9 = 40 – 4
5 × 9 = 50 – 5
A Bigger Problem
To my surprise, the older boy (a first-year homeschool student) could not do the subtraction without counting on his fingers. In several years of classroom math, he had not learned the number bonds, the pairs of numbers that make 10.
No, that can’t be true. I am sure he had learned them, probably in kindergarten, but his teachers had never led him to see how these simple facts could help him solve problems. So he had just forgotten them. When I probed further, I found he could not mentally add 10 to a 2-digit number.
This boy was not stupid. Far from it! When I put the numbers on paper and gave him a pencil, he knew how to follow the rules for adding and subtracting multi-digit numbers just fine. He could calculate into the thousands and beyond.
How could he do that when he knew so little about how numbers worked?
He suffered from Math Workbook Syndrome, the ability to crank through textbook calculations without any understanding of real mathematics. He had been shafted by several years of systematically poor instruction dished out by teachers who themselves did not understand math.
Yes, It’s Contagious
[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it.
— Time magazine, June 18, 1956
Quoted by George Polya in How to Solve It.
Many of us adults grew up afraid of math. We got by in school, understanding a bit here and a bit there, but never seeing how the framework fit together. Finally we crashed in a blaze of confusion, some of us in high school algebra, some in college calculus. Now that we are parents and teachers, we see the danger for our children.
How can we help them learn something that never made sense to us?
Parents dread helping with their children’s math homework. Some teachers follow the manual faithfully and hope for the best. Others scour the Internet for enrichment activities to supplement the curriculum, hoping something will catch the students’ imagination, but always wondering whether the anti-reform traditionalists might be right after all. Home school teachers switch from one math program to another looking for a “magic bullet.”
Our children must learn to handle numbers accurately and with confidence. They need to reason logically and make deductions. And we hope they come to appreciate the “Aha!” factor. How can we make it happen?