MATH MATTERS PART 1

“Let’s get rid of long division all together. It’s too difficult for 11 year olds.”

That kind of conversation is currently going on at teachers’ colleges at U of T, Queens and the University of Ottawa. I certainly agree that long division is too hard for the Grade 6 student who can’t multiply three digit numbers or subtract fractions. It’s also too demanding for the student who’s been spared from memorizing times tables because that’s considered ‘drill and kill’ by some of today’s experts. On the other hand, I’ve never encountered the need to divide 57 by 3.615 and if I did, I certainly would use a calculator.

It seems conversations about math concepts and pedagogy have been going on ever since followers of Pythagorus tried to cover up their discovery of the existence of irrational numbers.

At the high school level, the first major revision to the modern day math curriculum came in the 1930’s. An emphasis on arithmetic and geometry was replaced by a greater emphasis on trigonometry. The prospects of a second world war waged by aircraft and submarines sparked a need to infuse math curricula with tools that would assist the war effort.

In the early 1960’s, a second wave of math reform came in response to the Russians launching a satellite in space. Dutifully following the Americans lead, Canadian high school students were now taught the ‘new math’ which featured more algebra, transformations and functions. Calculus became a part of the high school curriculum in the early 1970’s. Euclidean geometry was phased out during the 1980’s and vectors were given more prominence.

The reduction of the high school program from five to four years necessitated more changes in the late 1990’s. The question as to whether Calculus should be dropped from the high school curriculum altogether was a most contentious issue.

Recent curriculum changes were accompanied with the introduction of new math terminology and new textbooks at both the elementary and high school level in Ontario. Many elementary school teachers are hard pressed to deliver this investigation-based curriculum that features lots of open-ended problem solving. These changes have also left many parents feeling disengaged from their children’s learning. A 2012 Ipsos Reid poll this month showed that only 43% of parents in Ontario felt that they could assist their elementary school children with math homework. No wonder many kids have math anxiety....their parents and some of their teachers have it too!

In addition to a new curriculum has come a new approach to math instruction: teaching math with understanding. Again, this is a wonderful initiative if used skillfully and strategically. However, columnist Margaret Wente of the Globe and Mail, expresses her cynicism for this change in approach in the following excerpt from her column dated December 15, 2011.

“The common methods used to add and subtract are known as standard algorithms. They are efficient and foolproof. But, instead of being taught these methods, students are encouraged to find “strategies,” such as breaking numbers into units of thousands, hundreds, tens and ones and working horizontally. It works, but it’s not efficient. And every time a student sees a new problem, he has to start from scratch – and pick his “strategy.” It’s like playing the piano without ever learning scales, or hockey without basic drills.”

Wente’s remarks probably strike a chord with anyone who has become impatient with a teenage cashier struggling to make change for a purchase at McDonald’s. However, the standard algorithms that seem so bulletproof according to Wente also have their drawbacks. Daniel Capozzi, a Grade 8 teacher and Brebeuf alumnus, wrote me with these insightful comments.

“Wente is correct when she says that common algorithms are proven; I used them all the time. However, we must be sure they are being used correctly and that they make sense when we find our result. And, sometimes, the standard algorithm is not as efficient. I remember working with the Grade 4 problem of making change from a ten dollar bill. Crossing out all the zeros and replacing them with borrowed ones and nines makes little sense if we look for patterns first. For instance, 10.00 - 4.76 happens to be the exact same thing if we change it to 9.99 - 4.75. The resulting three subtractions are far easier than keeping track of the ones, nines and zeros. This is using the concept of subtraction as “difference” as opposed to “take away.” Both are valid and both are algorithms but which will give the right “answer” in a more efficient way?”

In a similar way, teaching a child that to divide by a fraction requires that you invert and multiply reminds me of performing a magic trick. Far better to teach the concept initially with at least some degree of understanding and then follow-up with the Houdini stuff.

I strongly feel that teaching math with understanding is the only way to go. Doesn’t it make more sense to have a child discover the value of pi by making measurements of the radius and circumference of cylinders rather than simply telling him or her that pi equals 3.14? And don’t you think that building ramps with linking cubes is a much more insightful way to learn about slope than memorizing that slope equals rise over run? We learn concepts best by experience, by doing, not by listening or watching.

However, I do think what’s missing is the consolidation piece. Kids need practice whether it’s riding a bike or finding the length of the hypotenuse of a right triangle. But that’s the subject for another blog.

Going back to my original question about getting rid of long division, I’m still undecided. I’d love to hear what you think!