Consider the following multiplication problem: 322 x 47. If you’re like me and schooled in a particular era (one that did not include calculators), you might tackle it like this:
But over the phone line from St. John’s, mathematician Sherry Mantyka describes a different way to do the problem. She has me multiply the top numbers by seven – except once I move into the tens column, the two becomes “20” (which it actually represents), and the three becomes “300.” When I return to the lower line to multiply by four, it becomes 40, which is what the four represents. The answer to each individual multiplication operation gets its own row; six in total. What’s more, it works:
The students who come to Dr. Mantyka for remedial help at Memorial University’s Math Learning Centre are often familiar with the second approach. It’s a discovery-based learning technique (sometimes called inquiry-based) for performing two- or three-digit multiplication. It’s transparent, if more complicated.
Trouble is, Dr. Mantyka believes it’s an approach that’s putting too many students on the math casualty list, unable to automatically recall math facts or use quick, efficient algorithms that are the underpinning of more complex mathematical concepts and skills. Students who come to her centre are among the 18-to-19 percent who annually fail a mandatory math test required for entry into any Memorial math course. In some cases students have attempted, unsuccessfully, to solve division questions using dots or successive subtraction, both discovery-based techniques. Most of these students are not math majors, but need at least one math course for their chosen program.
“Thirty years ago this only affected students who wanted to pursue degrees in science,” says Dr. Mantyka, who has been teaching at Memorial since 1978. “Now, it affects degrees in so many disciplines. Even a degree like social work has a fourth-year course in research methods, which is essentially statistics.”
Weak math skills among new students have been a longstanding complaint among Canadian university math departments. But over the last year a more widespread, acute math panic has settled into the popular Canadian imagination, with news stories painting a picture of a generation of students whose mathematical skills and futures are at risk. Last December, the Organisation for Economic Co-operation and Development’s Programme for International Student Assessment, or PISA, showed that while Canadian students were still performing above average in math, the country had fallen out of the top 10 list among 65 jurisdictions. Although some say the change is inconsequential, performance had declined by 14 points, from 532 in 2003, to 518 in 2012. Meanwhile, the OECD’s assessment of adult skills – the Programme for International Assessment of Adult Competencies – released in October, showed Canada was below average in math skills, with the millennial generation showing the worst performance.
The finger-pointing quickly turned to discovery-based teaching approaches in kindergarten to Grade 12 schools. These have grown in dominance across provincial school systems for more than a decade, supported by scholars in faculties of education. While education researchers have pointed to other sources of the mathematics deficit, such as a lack of teacher education, some mathematicians complain the current thrust of discovery-based learning has left students with no firm grasp of math fundamentals.
“You need to have the foundations in place in order for students to do any sophisticated problem-solving or discovery on their own,” says Robert Craigen, a math professor at the University of Manitoba. “It’s not that discovery is a bad thing to do [in elementary classrooms]. But it should not become the main course of your meal.”
The discovery approach, derived from constructivist theories of learning, argues that all students should know how a particular problem-solving approach works, to gain “deep understanding” of mathematics and be able to apply it to new problems. Students memorizing by rote takes a back seat to students constructing their own knowledge.
To develop this skill, students should be given enriched exploration activities, often using concrete models such as blocks or shapes that they can manipulate to discover the underlying mechanisms of mathematics. They are also encouraged to use a variety of strategies to solve a problem. “We want them to think, to develop their own reasoning,” says math education professor Annie Savard at McGill University.
Traditionalists say that unless students have automatic recall of math facts such as simple addition, subtraction and multiplication tables, and second-nature use of standard algorithms, their working memories will be too bogged down by complicated strategies to move to the next phase of deep understanding. Direct, explicit instruction, followed by lots of practice, is the way to go.
However, in her 15 years teaching elementary school in Quebec City before moving into academic research, Dr. Savard noticed that while some students were whizzes at memorizing multiplication tables, they weren’t necessarily good at applying that knowledge. Meanwhile, she had to give zeroes to students who showed sound reasoning in their problem-solving, yet arrived at the wrong answer through calculation errors.
“For students who don’t have a good memory, they’re done,” says Dr. Savard. “And we observe that for students who do have a good memory…they can do it in school, but in their personal life, there’s no [understanding] about it. It doesn’t make any sense.”
As in most intellectual debates, the discovery vs. traditional math-learning controversy does not shake down to either-or arguments. Talk to a pure math academic or an education professor and within a few minutes it’s common to hear that there’s a place for discovery and a place for mastery of hard skills. Disagreement comes over what the mix should look like and where the emphasis should be.
Discovery learning and traditional, direct instruction approaches are “both right,” says Doug McDougall, an associate professor at the University of Toronto’s Ontario Institute for Studies in Education and director of its centre for science, math and technology education.
“We need to have skills instruction because there has to be some memorization. We need to know our math facts – multiplication, addition, subtraction – in order to continue on to do some discovery-based learning,” says Dr. McDougall. “And we can also use discovery-based learning to reinforce and to learn our math facts, to build a better understanding of the big ideas in math.”
Even Quebec, now Canada’s mathematical powerhouse as the provincial leader in OECD student assessments (it ranked among the top eight scores worldwide in the PISA study), uses a discovery-based curriculum. However, it has a clear requirement that students must memorize certain math facts and algorithms by specific grades – unlike other provinces, which may require knowledge or understanding of those things, but not memorization. That is starting to change with public pressure: Manitoba brought back explicit instruction of math facts and standard algorithms last September. Alberta and Ontario have said they are looking at similar moves.
As an example, the standard, traditional algorithm for long division is sometimes left out of discovery-based math classrooms because it is considered too hard to represent through conceptual models. It’s also not always considered crucial for students to know.
Instead, students might be shown alternative strategies for accomplishing the same goal, through estimation or successive subtraction. But that neglects the fact that the standard algorithm is needed when students reach more advanced mathematics, such as dividing polynomials, says Dr. Craigen from University of Manitoba. He belongs to a group of mathematicians, educators and parents called the Western Initiative to Strengthen Education in Math (WISE Math) that helped pressure the Manitoba government into its change.
“Long division is an important systematic process that is mirrored in a lot of higher-level skills,” says Dr. Craigen. “Our entire society is built upon algorithms. People talk [about] 21st-century learning, and then one of the first things they do is they throw out algorithms.”
Robert Dawson, a professor in the department of mathematics and computing science at Saint Mary’s University, says the discovery method does provide a better conceptual framework than “simply learning a technique that someone else shows you.”
On the other hand, a high school student has to learn “a set of techniques that took some very bright people many hundreds of years to put together,” he continues. “To expect everyone to discover this themselves is unrealistic.”
What math and education professors generally agree on is that elementary teachers in particular – typically trained as generalists – need a much stronger foundation in math and mathematical concepts if their students are going to succeed. That’s especially true if teachers are using a discovery-based approach, where activities must be well-designed and where flexibility is key to properly responding to where students are at in their own conceptual understanding and explorations.
In Quebec, pre-service elementary teachers can receive up to 225 hours of in-class instruction in teaching math, whereas in Ontario teachers may receive just 30 hours. Beyond needing more time to learn how to teach the subject, teachers also need grounding in actual math content. Relatively few elementary teachers have any university-level math background, and it is common to encounter elementary teachers who lack comfort with and confidence in the subject, say researchers.
One cross-appointed professor says that faculties of education should require pre-service elementary teachers to have at least one math content course, and math departments need to think hard about how they can serve those students best. Math departments should “make courses accessible and ask questions about what is it future teachers need,” says Donna Kotsopoulos, an associate professor in the faculty of education and in the mathematics department at Wilfrid Laurier University. Some universities provide math-content courses for aspiring teachers, but not all. Dr. Kotsopoulos has advocated publicly for a “radical re-engineering of math education” at all levels, especially at the postsecondary level, where, she says, teaching approaches have remained static over time.
There’s no question that many elementary teachers are uncomfortable with math themselves and are unlikely to have the chance to go back and do a full math-content review. Recognizing that, the highly successful JUMP math instruction program is trying to fill the gap by developing step-by-step teaching resources and lesson plans.
Founded by John Mighton, a fellow at Toronto’s Fields Institute for Research in Mathematical Sciences, JUMP has been described as a “third way,” to teach math. It requires students to become adept with standard mathematical rules and procedures through repeated practice, yet it also embraces concrete modelling and the importance of students becoming creative problem-solvers.
“The things that [math and education professors] are both concerned about are genuine concerns,” says Dr. Mighton. While education profs want to ensure understanding, he says math profs believe facility with rules and procedures needn’t exclude that understanding. “There’s miscommunication between the sides. I think you can have a hybrid or a very effective combination of discovery with guidance that doesn’t draw these false dichotomies.”
Can we get beyond the current math impasse and move on, to a new era of excellence for students? Dr. Mighton thinks so. But ultimately, he says, teachers must be freed to experiment with what works best, along with a rigorous tracking of student results and appropriate adjustment of approach in response. Not only can students do better in math, they can go much farther than they’re being asked to now. “Even the advanced countries of the world who do better than [Canada] aren’t even coming close to realizing the potential of children,” says Dr. Mighton. “We are in a crisis – but every country is in a crisis.