An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?

This might sound like a math problem. That’s because it is.

It was posed to 45,000 Grade 8 students. Most of them figured out that they were being asked to divide 1,128 by 36, but only one in four responded with the correct answer of 32.

Most of the other students answered either 31 or “31 remainder 12” – meaning they had done the division properly, but did not understand that a partially full 32nd bus would be needed.

Scenarios like this one have played out over and over again in elementary school classrooms across North America, with students telling teachers they believed there was a difference between the “school math” they were being taught and the real-world problems they were being asked to solve.

For some students, even the “school math” is a problem. Standardized test results are falling in most provinces, and the search for solutions is proving to be a contentious issue for parents and educators.

“Math touches every area of life,” Clive Packer, a father of two who lives in Ottawa, said Sept. 27 in an interview. “Kids need to know those things, because otherwise they’re going to struggle in later life.”

Packer has a daughter in Grade 9 and a son in Grade 7. He started paying attention to math education when his daughter – then in Grade 2 – was struggling with her homework.

“She was coming home with fairly simple addition problems … and she seemed very clueless as to how to begin. She would just kind of stare and sit at the paper,” he said.

Packer met with his daughter’s teacher, who said her role was not to give students answers but to expose them to mathematical strategies.

“What does it mean to expose a seven-year-old to a strategy? Like, what is that?” he said.

What it means, in a broad sense, is inquiry-based learning – a teaching method educators have gradually been pivoting toward in an attempt to help students understand the relevance of the concepts and ideas they were learning in math class.

Inquiry-based math lessons are typically broken into three parts, beginning with a teacher asking a question or posing a problem. Students then work – sometimes alone, sometimes in groups, sometimes with the teacher’s assistance – to figure out a solution. At the end of the lesson, the whole group comes back together to discuss how what they learned might connect to past lessons, prior knowledge or the world beyond the school’s walls.

This video shows how the same basic division question can be answered in four different ways. Some of them may look confusing to anyone who hasn’t been in a math class in decades. All of them are considered acceptable solutions.

While inquiry-based math is touted by many educators as allowing students to develop critical thinking skills, some parents and politicians have derided the approach as harmful to children’s math skills.

One of the most recent high-profile opponents of inquiry-based learning has been Ontario Premier Doug Ford, who said during the run-up to June’s election that he wanted to bring back previous methods of instruction.

"Kids used to learn math by doing things like memorizing a multiplication table and it worked,” he said.

Ford described the existing approach to the subject as “discovery math,” a term that is rarely uttered by anyone within the education system and does not appear anywhere in Ontario’s math curriculum. The phrase is often used by people outside the system seeking a return to the math-teaching approach they remember from decades past.

The issue with ‘discovery math’

Proponents of inquiry-based learning say students learn more and retain that knowledge better if they are able to discover a concept for themselves rather than have it explained it to them. Critics take issue with the idea that students should be free to use whichever methods they wish – even if their path to the solution is convoluted or inefficient.

“Some ideas are better than others, and some have better logic,” said Kim Langen, who co-founded the Spirit of Math after-school program for high-performing students and has advised school boards across Canada on math education.

What is often lost in the debate, math experts say, is that inquiry-based learning includes elements of memorization, just as a more traditional rote approach still makes use of problem-solving techniques. If a student realizes it will take 10 buses to transport 360 soldiers, and 20 to transport 720, and deduces the rest from there, they are using elements of both forms of teaching.

“There’s no such thing as one camp versus another camp. We shouldn’t look at them as dichotomists, but rather that they support each other,” Mary Reid, a program-co-ordinator at the University of Toronto’s Ontario Institute for Studies in Education, told CTVNews.ca in a telephone interview on Sept. 18.

As Reid sees it, the recall and numeracy skills Ford trumpets are necessary to build on the critical-thinking and problem-solving abilities sought by people who favour inquiry-based math.

Langen agrees that finding a “balanced approach” between the two methods is key.

“Parents are really frustrated, because they’re saying kids don’t have the skill sets,” she said.

“They’re asking for an understanding of what they’re doing and the ability to do something.”

Spirit of Math runs programs based on what Langen calls “disciplined discovery.” Its lessons include short, intense drills, followed by exercises in which students have to demonstrate an understanding of the logic behind the concepts in the drills and knowledge of how to apply them.

Ford has used declining test scores as justification for changing how math is taught in Ontario, saying other provinces are not seeing similar drops.

The Education Quality and Accountability Office administers standardized math tests in Ontario. Its latest data, released in August, stated that 61 per cent of Grade 3 students and 49 per cent of Grade 6 students met the provincial standard for math.

Both numbers have fallen steadily through the 2010s. Grade 6 students saw the bigger drop, peaking at 63 per cent in 2009, while Grade 3 scores topped out at 71 per cent in 2010. (No data is available for the 2014-15 school year, as the EQAO tests were cancelled due to teacher job action.)

Up until that point, EQAO math results had been increasing steadily since 2003. Fifty-seven per cent of Grade 3 students and 53 per cent of Grade 6 students met the provincial standard that year.

Most provinces issue standardized tests similar to EQAO’s, gauging the performance of students, teachers, and the education system as a whole. Because the tests are based on individual provinces’ own curricula, directly comparing results between provinces is considered difficult.

Still, a scan of junior-level results across the country shows nearly every province grappling with some form of decline in elementary school math scores over the last few years.

Perhaps the most noticeable exception is Manitoba, where small but gradual increases have been seen over the past decade.

A tool more commonly used to compare math education between provinces is the Programme for International Student Assessment (PISA), a test administered every three years. Although PISA surveys high school students, experts say math success at younger ages is a strong predictor of math success as students grow older.

Ontario has consistently finished below Quebec and Alberta in PISA reporting since the turn of the millennium, often also scoring lower marks than B.C.

The province’s score has fallen from a post-2000 high of 530 in 2003 to 509 in 2015. (The scores are all relative to an average of 500, meaning lower-ranking countries improving their performance hurts Ontario’s score as much as Ontario’s results worsening does.)

Most provinces saw their PISA scores fall between 2012 and 2015. Nova Scotia and B.C. held steady, while P.E.I. and Quebec increased their ratings – with Quebec hitting a 15-year high by scoring 544.

Despite the slippage from 2012, Canada was one of the top performers on the 2015 PISA test. Singapore and Hong Kong were the only jurisdictions to outperform Quebec, and only seven other nations scored higher than Canada as a whole.

It should be noted that inquiry-based learning makes up a significant part of all provinces’ math guidance for teachers, making it unlikely that differences between provinces can be chalked up to teaching methods alone.

Hopes for the future

Manitoba’s pattern of stable math scores doesn’t seem to have been affected significantly by the addition of basic formula instruction to the province’s largely inquiry-based curriculum in 2014.

Officials in Ontario hope to make a bigger impact by changing that province’s math curriculum. Math will be a major part of just-started public consultations on education.

Packer – who now advocates for math education reform and co-hosts a podcast about the subject – says he plans to take part in the consultations. He says the previous Liberal administration was “completely unreceptive” to listening to him, and hopes the new government will be more willing to make changes – after doing plenty of due diligence.

“I don’t want a knee-jerk reaction and this government, so far, has given us a bit of a cause for concern in that area,” he said.

Specifically, Packer says he wants to see younger students given a “slimmed-down” curriculum with more teaching of times tables and other numeracy skills – although not through repeated drills – and inquiry-based methods being introduced as children move into older grades. He also thinks fractions and other concepts should be taught all at once, rather than over a handful of weeks per grade.

“When I was growing up in England, I went ice skating once a year – and, of course, I never learned to skate,” he said.

Packer says both his children are now doing well in math, thanks in part to his help and that of several teachers he is quick to praise.