John Mighton writes the 9-times table on the board and asks the Grade 6 students to look for patterns.
The children stare at the numbers 18, 27, 36, 45, 54, 63, 72, 81 and 90 and make a few quick observations: The digits in each number add up to nine. One plus eight is nine, two plus seven is nine and so on.
Within minutes, every kid in the class is using the sum of the digits to determine if a number is divisible by nine. If it isn't, they learn to predict what the remainder will be.
They practise on half a dozen three-digit numbers. Then Mr. Mighton scrawls 121,252 on the board - a much bigger figure than they have tackled so far.
"What is your prediction?" he asks.
"Easy peasy lemon squeezy," one girl says as she copies the bonus question down in her notebook.
Mr. Mighton is not a certified teacher. He's a writer and mathematician who devotes most of his time to JUMP, a charity that helps youngsters learn to think mathematically.
Since he founded it as an extracurricular tutoring program in 1998, he has amassed compelling anecdotal evidence that it helps struggling students. The question now is whether it also works in regular classrooms, where kids have a wide range of mathematical abilities. It's under study, and the way Canadian students learn could ride on the answer.
"There is a perception that JUMP is only for weaker students and that it will hold faster kids back," Mr. Mighton says. This Grade 6 class at the private Mabin school in Toronto shows how JUMP can work for all children, he says.
The teacher, Mary Jane Moreau, started using JUMP last year, when the children were in Grade 5. Beforehand, she gave the students a standardized math-computation test: Their marks ranged from the 37th to the 75th percentile. After a year of JUMP, she says, all but one were in the 91st-to-the-99th percentile.
"It is like they are a gifted class," she says. "But they aren't a gifted class."
One thing that's obvious to an observer is that these kids now love math. Even the shyest children volunteer answers. Some seem giddy.
One class, of course, isn't proof. Ms. Moreau is an exceptional teacher, keen to use what brain scientists have found about learning to help her teach more effectively, which is why she was drawn to JUMP. She also has the luxury of teaching just half her class - nine students - at a time.
But Tracy Solomon, a researcher at the Toronto Hospital for Sick Children, is doing a study that should show if Ms. Moreau's class is an anomaly.
She is comparing 300 children in an Ontario school board: Half are being taught using JUMP methods and half in the regular manner, which stresses problem solving. Like the Mabin-school children, they started JUMP in Grade 5 and now are in Grade 6.
Dr. Solomon's hypothesis is that kids taught with JUMP will have better math fluency than the control group. If that proves correct, she wants to dig deeper and learn what makes JUMP work so well.
"It's an opportunity to marry two things separated for far too long - research and education," she says, "and draw on the substantial body of knowledge from psychology, cognitive science and neuroscience and infuse those findings into education."
(It may seem strange that a hospital researcher is studying math teaching, but Sick Kids takes the view that education is intertwined with long-term mental health.) Recent results from a smaller group of students in Britain suggest that children who are already at grade level can progress at a rapid speed with JUMP. In the Lambeth school district, all of the 74 students in Grades 5 and 6 who were close to meeting expectations for their age were able to perform beyond their grade levels after one or two years with JUMP. By the age of 11, 57 per cent of those students were three grade levels ahead.
Rising in the west
In Canada, JUMP has its strongest foothold in Vancouver, where Mr. Mighton has trained 400 teachers, and is also being used, although to a lesser extent, in Edmonton and Winnipeg. Teachers in Western Canada often have more autonomy to choose programs for their classes, he says.
The program has met with more resistance in Ontario, where it began. Many school board officials prefer the current approach, which helps students discover and understand mathematical concepts through problem solving. Still, JUMP is getting attention there: Mr. Mighton has trained more than 100 teachers in the public and Catholic boards since September.
The JUMP method breaks mathematical operations into small steps that kids practise and master before they move on. In learning subtraction, for example, children might do a whole page of identifying which questions require regrouping (what used to be known as borrowing).
This builds confidence to go on to the next step. Mr. Mighton says that's essential: "If kids feel inadequate, their brains aren't going to work well."
This approach taps into what brain scientists have discovered about learning: Practice helps to consolidate memories, and mastering small steps or skills can lead to big jumps in performance. Experiments have also shown that confidence can affect ability, and we may learn more from getting answers right than from getting them wrong.
JUMP has evolved, over the years, from a tutoring program to a charity that trains teachers and produces free teaching guides. It also sells workbooks that cover the curriculum from Grade 1 to 8 - the profits go back into the charity.
But many in the education field remain unconvinced that JUMP offers a better way for children to learn math.
Diane Muckleston, curriculum co-ordinator at the York Region District School Board outside Toronto, says research shows that the best way to help kids understand a concept is to come up with a rich, conceptual problem that everyone in the class can help solve.
Last year, for example, she visited the class of a primary-school teacher who had noticed that all the kids were wearing odd socks. The teacher came up with the concept of a sock factory, and the kids all brought in socks. Each child was given a different number of socks and their task, as a group, was to find a strategy that would combine them.
"But it wasn't, 'This is the way you would add the numbers,'" Ms. Muckleston says. "There is a fundamental pedagogical difference in [Mr. Mighton's]thinking on how a kid learns mathematics and what the worldwide research is about it."
But Dr. Solomon disagrees. She says research does not support the idea that the "discovery" approach to teaching math is the most effective. "It is a mystery why we ever went down that route."
There is whopping evidence that memories get laid down through meaningful practice, she says, and that has to come before putting that learning to novel and clever uses in problem solving.
It is like learning to speak a language or to play a musical instrument: "The creativity comes once you have mastered the basics."
Mr. Mighton describes JUMP as guided discovery: "You need a balanced program that recognizes the strengths of the brain, that kids can make discoveries. But you also need to recognize the brains' weaknesses: We have poor short-term memories. We are easily overwhelmed by too much information. Neurologists are now saying we need practice to consolidate ideas. Sometimes we need rigorous guidance."
The idea is to provide teachers with a spectrum of approaches to find the best ones for each student.
Teaching the teacher
Mr. Mighton starts the lesson at the Mabin school with a quick introduction: "If you don't get something today, it is my fault, so stop me."
The children quickly grasp that if the sum of the digits in a number is nine, then it is divisible by nine. The number 217, however, can't be divided by nine, because two plus one plus seven is 10.
What about the remainder? At first, they learn a simple approach. In many cases, it will be the difference between sum of the digits and nine. So when 217 is divided by nine, the remainder will be 10 minus nine - one.
But what happens next shocks even Mr. Mighton: The class discovers a better way, one that works with all numbers. The sum of the digits of 217 is 10 - and if you add those digits, 1 plus 0, you arrive at the remainder.
"I was stunned. I didn't know that," Mr. Mighton says.
After some further practice, it is on to the easy-peasy-lemon-squeezy bonus question.
Every hand in the class shoots up. The number 121,252 is not divisible by nine, one student tells him, and the remainder will be four.
"You are brilliant," he tells them. "You are all brilliant."
Anne McIlroy is The Globe and Mail's science reporter, specializing in learning and the brain.
Practice makes perfect - if it's the right kind.
A growing body of cognitive research suggests that practice and motivation may be more important that innate talent in developing expertise in chess, music, sports or other pursuits.
One expert on expertise, Florida State University's Anders Ericsson, argues that the key is repeatedly taking on challenges that are slightly beyond your competence level, or what he calls "effortful study."
And as mathematician John Mighton says, this suggests that virtually every student, given the right kind of instruction, can get an A in math. In which case, he adds, in every class "the bell curve would be tighter and shifted to higher levels of achievement."