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."Report Typo/Error