Paul Lacey begins his math lesson at Children’s Garden School by bringing up the equation 55 + 16 on the board. Immediately, half the students raise their hands, willing to give the answer.
When called upon, one boy correctly says the answer is 71, but Lacey doesn’t tell him he’s right.
“Prove it,” Lacey says without missing a beat.
The boy explains that he first extracted the six from 16, and added that six to the 55 in order to get 61. Then knowing there was another 10 left to add, the answer very simply became 71.
Lacey affirms the boy’s methodology.
“Right,” Lacey says. “So that’s a pretty standard algorithm.”
Every math class starts like this in Lacey’s class. The students are brought away from their desks, and have to use mental math to solve equations. Calculators don’t make an appearance.
Another question comes up: 575 + 75.
The boy who answers this one explains he knows the answer is 650 because he added the number 25 three times to 575, thus going 600, 625, and finally 650. He notes that he thought of it like money.
One more thing — these kids are just eight years old and finishing grade 3.
His secret to success? Finding the right balance between what is known as “old math” and “new math”.
“There seems to be the sense that there’s drill and rote memorization on one side and then problem solving and conceptual understanding on the other,” Lacey said. “And for some reason these have become two camps that are opposed to each other.”
He explained how the two sides have their benefits, but also have their drawbacks.
“Our adult generation when we were taught in school, we were generally taught the standard algorithm only, and by that I mean carry the one in addition, go next door get 10 more for subtraction — those kind of procedural rules,” he said. “Generally, we weren’t taught why they work.
“And a very good way of showing that would be to ask an adult from our generation ‘why does long division work?’ and see if they can explain it.”
It’s because of that style of learning, Lacey said, that it’s not always kids who are the hardest to teach, but it’s the parents.
“It’s still an uphill battle, it’s not easy,” he said. “Especially in a private school system where there’s more homework and parents want to help, but they’re scared of doing it wrong, and I understand that.”
So Lacey held two parent math nights this past year, where he had the parents doing some math problems, as well as watching their kids do the math, too. These turned out to be hugely successful.
“My parents are all on board, I’ve been very lucky,” he said. “They’ve all jumped on board because they’ve seen the results.”
One of those results, Lacey said, due to the way kids in his class learn, he doesn’t use textbooks.
“They usually come up with all the ideas in the textbook on their own,” he said. “And from there all I’m doing is solidifying their own ideas. So in that way, it’s student-centred.”
And student-centred learning is also the focus at Greenwood College. The school’s director of personalized learning, Heather Rigby, who also teaches math and science, says they’ve upped the ante on personalized learning with a new approach that embraces technology.
“I create online videos that the students watch, rather than watch the teacher at the front of the room. And I use that idea to create a personalized classroom,” she said. “So what I mean by that is the kids are given the whole unit at the beginning of the unit and can watch the videos at home, they can re-watch the lessons if they weren’t sure on a concept.”
Rigby says this allows kids to be able to learn at their own pace. She gave an example of a student who managed to complete both grade 11 and 12 math in one year.
“That’s sort of the key to what we’re trying to do at Greenwood is this idea of personalized learning, so meeting the needs of each individual student,” she said. “Technology really is playing the biggest role in the change in how we’re working with kids in math for sure.”
Helping kids teach themselves is also a key element of the learning process in Lacey’s class.
“The idea of problem solving as discovery is something that I think is really important,” he said. “It’s not a matter of I stand up there and teach them how to do something. For things like division, I start it with a word problem … I have remainders right away because I want them to get the idea that division means I’m sharing it with people.
“And yes, sometimes things are leftover, and I need to figure out what to do with them, it’s not just R2. In real life remainders mean something and you have to do something with them.”
So to help develop his students’ abilities to use mental math in real life, Lacey has taught them to think differently than the traditional methods. He gives a general addition question as an example.
“Even though there’s the rule you start at the ones, my kids know that no, you don’t have to start at the ones,” he said. “If I’m doing 29 + 36, I can turn that into 30 + 35. Start at the 10s, do 30 + 30, then add the five.”
Though there may be many different ways to solve an equation, Lacey says he’s happy about one main thing his lessons do.
“That’s the thing I’m most proud of is that they can manipulate numbers like that,” he said.
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