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
“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 dis-
covery 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 min-
utes 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
“It’s not that discovery is a bad
memorization. We need to know our math facts – multiplication, addition,
subtraction – in order to continue on to do some discovery-based learn-
ing,” 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 provin-
cial leader in OECD student assessments (it ranked among the top
eight scores worldwide in the PISA study), uses a discovery-based curric-
ulum. 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
thing to do in elementary school,
but it should not become the
main course of your meal.”