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.
With numbers:
- 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
etc.
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.”
Suggestions, Anyone?
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?
For that one 4th grade boy, that afternoon, we spent the rest of our math club time practicing number bonds and playing games.
Denise Gaskins' Let's Play Math 
Great post. I am a math teacher, currently homeschooling my granddaughter. I subscribe to your blog and check it from time to time. I particularly liked this post and encounter similar problems from time to time. I wrote about such an experience in my blog. http://www.mathchique.com/2008/04/case-of-unknown-perimeter.html
Please visit mine. MathChique.com
You have a several interesting articles, MathChique. Thanks for dropping by and leaving a link!
Thanks for submitting this to the CoH!
Couldn’t agree more. You only have to look at the maths books available for kids nowadays. They’re just great long swathes of sums designed to test knowledge rather than understanding. But it’s not just the style of workbooks that’s the problem. It’s deeper than that.
There’s a pressure, certainly here in the UK, to ‘teach to the test’. In a constant race to measure progress, we’ve abandoned the target of long term learning and concentrated on the short term goal of passing the next government mandated test.
I honestly don’t know what the solution is since testing is such a valuable political tool and its detrimental effects are not felt until much later in the process of learning. I guess we need to educate parents about education as much as we need to educate children about maths.
We have that “teach to the test” problem here in the US, too. Check out this excellent post (quoting Rafe Esquith, author of Teach Like Your Hair’s On Fire.)
The irony is that kids instictively understand math concepts that we proceed to mess up for them. Two five year olds can easily tell you if one got more m&m’s than the other (division) or how many more (subtraction). Older kids can do it at a glance.
As much as word problems are villified, they are an attempt to connect the abstract number sentence with something in the world that kids have experienced.
One of the answers seems to be putting math back into the everyday. In a homeschool setting, I achieve that by talking about math while we do our daily activities. I don’t allow math to stay a subject that is bounded by math time and math books. I also cheerfully put math problems into terms that they identify with. If that means that I do an entire fractions lesson around different kids of pizza, fine. If I have to use Pokemon cards to make the point, that’s fine too.
I do something like what you describe, talking about math while we do other things — at least, with my younger students. As Andrei Toom writes in Word Problems in Russia and America, “I suggested that the main educative value of word problems is that they serve as mental manipulatives, paving children’s road to abstract thinking.”
Great post — and interesting question…. I’m an American mom living in Singapore. Yesterday I went to one of the big bookstores here, called Popular, where there’s a full floor devoted to workbooks and study aids for pre-university students, divided by subject and grade.
It was the first day of school holidays here, so the store was full of kids. I talked to several of them, asking “what’s your favorite subject?” Without exception, each kid said “Maths!”
The experience made me think that maybe it’s not the workbooks, themselves, that are the problem in the US, but the crummy way those workbooks present the material. The Singapore Math curriculum seems to be engaging and rigorous — but is also very much workbook-driven. Problems build logically on each other. And success, as a student, is exciting and motivating. These kids love math because they’re confident they can work through it and figure it out. And judging from the stuff I’ve seen on the math proficiency of students from different countries, Singapore is doing something right.
Anyway, just a thought…. (BTW, my kids go to the American school here, which doesn’t follow the Singapore Math curriculum. But I’ve noticed the school has begun to supplement with worksheets in K-5 …. hmmmm.)
I have always thought that math is the best class for giving students a taste of the joy of discovery — what I call the “Aha!” factor. It sounds like Singapore may do a better job at giving that to students than we do in America.
And of course, you are right. It’s not workbooks themselves that cause the problem. Rather, it is a combination of poorly-written books, teachers who don’t understand math themselves, and a culture that doesn’t expect students to learn or enjoy math.
My job as a teacher is to… teach. But along the way the math can impress the kids. And from time to time, it does.
High school. Pretty bright kids. And it is still the arithmetic that impresses them the most.
Once every two or three weeks I get a few oohs for doing some multiplication without a calculator, and I’ll stop and show them the “trick” I used. I swear, there’s a kid, it’s his favorite part of the course.
Jonathan
I felt the same as your students when I first read Liping Ma’s Knowing and Teaching Elementary Mathematics. I had never appreciated, or even really understood, arithmetic before that book.
I’m afraid it comes down to trust (in the teacher). I teach middle school math and I’ve come to understand the relationship between K-6 teachers that are comfortable with math and those that are not. It is even evident within certain strands of math. The problem is that elementary age students are teacher pleasers and will learn what their teacher is comfortable with. By the time they reach MS they are still counting (adding & multiplying) with their fingers because THAT is their level of comfort! I see too many “smart” kids come into my classroom each year only to realize their limitations because they’ve been able to manipulate the system as opposed to learn to be comfortable with math. Math is taught at the discretion of the elementary teacher and that is a fundamental step to correcting the “math problem” in our country!
It seems so backward to me as a parent/teacher, not to have the idea of “what makes ten” central to a child’s basic number understanding. With my three homeschooled children, I used Miquon Math, with Cuisenaire rods and a hundred chart (and other manipulatives like pennies and dimes) in the primary years. We spent a LOT of time working on concepts around ten, and also nine and eleven (what makes almost ten, what makes one more than ten, etc.). It was one of the things I liked best about the Miquon program, putting all that emphasis on tens. From what you’re saying, I don’t think it was overkill.
Thanks for the post – made for some great reading. Our son used to love mathematics in the first year of school. A year later he developed an aversion. The following year he had a wonderful teacher who got the kids playing mathematics in a variety of ways – puzzles(http://www.mensaforkids.org/play/games/), online games (http://www.jumpstart.com/parents/games/math-games), singing songs (http://www.songsforteaching.com/mathsongs.htm) and a bunch of other things. My son got back into ‘I love Math’ mode. In his case all it took to turn the tide was one wonderful teacher!
Puzzles and games can work wonders, providing positive motivation for math practice. I especially like games with a strategic element, rather than simple flashcard-style games. Have you explored the selection at Math Playground?
We’ll look it up Denise – thanks very much.