And what happens, when the numbers don’t work so nicely, when you are multiplying not 25, but 37, to 44?
Both sides like to use a music analogy to make their case. The “basic skills” camp asks: Can someone become proficient musician without learning the scales and where the notes sit on the staff? The “progressive education” side counters: What’s the point of drilling young musicians on scales, if they want to give up the instrument as soon as their parents will allow?
Ultimately, the argument about basic skills versus conceptual thinking reduces math to a process – and often a grade to be made – rather than a complex exercise in thinking, that can be transferred to other areas of learning.
As Eric Muller points out, “The best mathematicians are the ones with self-reflection, who learn to ask themselves, ‘Am I doing this right? Am I on the right track?’ ”
So if our current binary debate is missing the point, what conversation should we be having about math in the classroom?
A national voice on math would be a start. Unlike the United States and many European countries, Canada lacks a central watchdog for math education. We have provincial associations for math teachers. And the little-known Canadian Mathematics Education Study Group brings together mathematicians and math educators once a year for discussions. But despite several attempts to change its mandate, the group doesn’t propose guidelines, or speak out on policy.
That means there is no group even to help build a consensus on what Canada can learn from the PISA tests, with their vast troves of data.
Canadian 15-year-olds, for example, actually performed pretty well when it came to the easy “basic skills” questions: The PISA test is broken into levels, and about 86 per cent of Canadian students mastered the first two, described as the math “a person needs to be a citizen in the modern world.”
Their weakness is higher-level, more conceptual math – only 4.3 per cent aced the highest level, compared with 30.8 per cent of those in Shanghai. Students in the Chinese city not only outscored everyone, according to OECD data, they spend barely half the time in math class as Canadians, who at five hours and 14 minutes a week, get more teacher time than in any other survey country.
How does Shanghai do so well? They devote an average of 14 hours a week to homework (versus three for the Canadians) and 70 per cent have parents willing to pay for extracurricular math classes (versus 28 per cent in Canada). And those students who seem to spend so little class time on math also have teachers trained more rigorously and subject to greater supervision.
The problem is that it is easier to teach math when it comes down to following step-by-step instructions. To be creative with math requires a teacher who understands the subject at a fundamental level, who loves it, and who is well trained on how to teach it. Currently, future elementary teachers (who may not even have taken math at university) receive less than 40 hours on how to teach math in some cases.
For students, this can translate as teachers who – regardless of official shifts in curricula – are not comfortable with abstract math. “I would bet my last dollar, what’s going on in elementary schools is anything other than [problem-solving, student-centred] math,” says David Robitaille, a professor emeritus at the University of British Columbia. “I would bet it’s a teacher giving a lesson, then seat work, followed by homework.”
In many of the high-ranking Asian countries, even the youngest students learn math from specialized teachers. In Shanghai, according to an OECD survey of principals conducted in conjunction with PISA, math teachers receive significantly more feedback from students, fellow teachers and external assessment than Canadian teachers, and their top performers get pay hikes and bonuses.
To make the grade in math, we need to properly instruct teachers, even in early grades, on how to balance “concept-based” math with basic arithmetic, so that the nobody spends math class colouring pretty pictures, and the time tables get their due.