Competence in math among Ontario public school students in grades 1 through 8 continues to decline and worried parents increasingly resort to private tuition, something virtually unknown 40 years ago. But such tuition is expensive. Fees can be $300 per child per month, so that children of poorer families are disadvantaged.
Ontario’s Ministry of Education literature reveals several reasons for this discrimination. These include: (1) a curriculum favoured by educators but unsuited to mathematics’ hierarchical structure; (2) an obsession with discovery/problem-based learning at the expense of direct instruction; (3) deficient textbooks; and (4) the failure to train elementary math teachers for the particular challenges of teaching children. The Ministry’s curriculum glossary also includes elementary errors, an example being the wrong definition of a rational number. The correct version is available in any good dictionary. These errors suggest that the officials responsible for curriculum development are ill-prepared for the task.
The structure of mathematics including algebra and calculus is based on five rules for adding and multiplying the natural numbers, these rules being the “chess-moves” of the subject. They govern subtraction and division together with operations on fractions, negative numbers, decimals and percentages. To advance learning, by grade 6 students should be able to automatically apply the five rules to all these operations. To free working memory for problem- solving, the 12 X 12 times table should also be memorized. Failure to achieve these goals strongly predicts under-performance in the higher school years and in college.
Concerned mathematicians distinguish between mastery curricular on the one hand and spiral curricula on the other. Progress in mathematics requires understanding thoroughly a particular concept before building on it. For example, division of a whole number by a fraction will be meaningless to the student if she does not understand multiplication by a fraction. Mastery curricula recognise this feature by intensely teaching each step in the hierarchy before moving up.
In contrast, in any given year spiral curricula cover many topics, which can range from basic arithmetic to probability and statistics. Each topic is treated only briefly, being revisited over several years. Although the intent is to treat concepts with increasing depth in successive grades, what tends to happen in mathematics is that students acquire only a superficial understanding of basic concepts. Ontario’s spiral curriculum is a major source of our problem.
Effective mathematics teaching in grades 1 to 8 requires a three-pronged strategy: (1) developing understanding of the basic principles; (2) teaching competence in using computational techniques; and (3) providing experience in applying mathematical ideas through problem- solving. As is typical of North-American educators, Ontario’s ministry literature focusses almost entirely on problem-solving, together with student-based discovery techniques. A recent Ministry “expert panel” report waxes lyrical about this technique by picturing students working in groups, with minimal guidance from the teacher. But a colleague who has observed such groups found that it is usually the brightest student who solves the problem on her own, with the others merely copying. As the cognitive scientist H. A. Simon, who, in 1978, received the Nobel Prize in Economics, observed in a discussion on teaching mathematics, “there is very little positive evidence for discovery learning, and it is often inferior” to direct instruction.
It is tragic that educators ignore half-century old evidence showing that direct instruction focusing on basic skills is the most effective in developing not only those skills, but also problem-solving ability. Moreover, direct instruction promotes students’ self-esteem. Australian education professor John Hattie’s 2009 publication of an analysis of 52,000 investigations confirms this. The most important teaching factors are, in order: (1) feedback from student to teacher; (2) instructional quality; and (3) direct instruction. All are part of competent teacher-directed instruction.
In relation to teacher training, recent ministry policy changes are likely to be ineffective. Although class hours have been increased, advisors evidently continue to push problem-based learning. Furthermore, there appears to be no provision for the specialist training that is needed to teach children the abstract concepts of mathematics. Traditional teacher training topped up with few university level math courses is not enough.
Four years ago, in an article in the Globe and Mail, University of Winnipeg mathematician Anna Stokke, who wrote the authoritative C. D. Howe Report on deficiencies in math teaching in Canada, identified the above problems, and laid out an easily implemented plan for reform. This past September, lamenting the lack of progress, she asked “how can we expect a change of direction when ministries of education tend to put the very people in charge who were responsible for choosing the wrong direction in the first place, and have staked careers on promoting ineffective math programs...”
I agree. In its Throne Speech, the Ford Government included math teaching reform as one of its aims. Let us hope that it has both the fortitude and wisdom to mandate effective change.