Walking in the right direction on math

Opinion

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This article was published 25/11/2024 (289 days ago), so information in it may no longer be current.

Recently, the government of Manitoba has eliminated the teacher certification requirement of six credit hours-worth of university courses in mathematics. A professor of mathematics has recently argued in the Free Press (“Teachers need subject expertise,” Nov. 12 ) that this is a mistake because “common sense tells us that you can’t teach what you don’t understand” and that university-level mathematics courses provide a foundation for K-8 teachers to teach mathematics.

I have studied mathematics, worked for about 10 years as a mathematics high school teacher in B.C. (grades 8-12), have taught mathematics education for future early years, middle years, and high school teachers at the University of Manitoba for many years, and have worked and engaged in mathematics teaching and learning research with many K-8 teachers.

What I have learned is that university-level mathematics courses are irrelevant to the quality of the mathematics teaching and learning in K-8 school education.

First, let’s start with the unquestionable insight that you can’t teach what you don’t understand.

How is this, though, an argument for understanding university-level mathematics if the goal is to understand K-8 mathematical content? This is actually an argument against the need to understand university-level mathematics to teach K-8 mathematics. But don’t you need to know where the (K-8) mathematics that you are teaching your students is heading in high school and university?

This leads to my second point. The way in which K-8 teachers need to know K-8 mathematics is qualitatively different from the way in which the same mathematical topics are understood at university. For instance, that two times five is equal to five times two (commutative law) is in mathematics based on axiomatic properties of the number systems and their operations.

In school mathematics, on the other hand, understanding the commutative property of multiplication (a x b = b x a) is based in discovering that and why two rows of five markers result in the same total number of markers as does five rows of two markers. In school mathematics, symbols are given concrete and operational meanings.

There is a burgeoning area of research in mathematics education that focuses on mathe-matics-for-teaching, which is about the understanding of mathematical content as it is relevant to the teaching of mathematics. The hyphen in “mathematics-for-teaching” is intentionally used to express that the mathematics a teacher needs to know is qualitatively different from the way, for instance, a research mathematician or an engineer needs to know mathematics.

The notion that knowing pedagogy can be separated from knowing the subject to be taught does not hold water in light of this research. The idea that department of mathematics can help future or current K-8 teachers of mathematics understand K-8 mathematics as they need to in order to help their students learn mathematics is untenable. University level mathematics, thus, cannot provide any kind of “foundation” for the teaching of the subject in K-8. Rather, what provides that foundation is the mathematics-for-teaching that K-8 teachers need to know to engage students meaningfully in ways that help their students develop conceptual understanding and procedural fluency — and this mathematics-for teaching is the area of expertise that can be found in faculties of education, and that is also where future teachers are helped to develop this understanding.

Let’s illustrate this notion by drawing on the piano teacher example in the article I referenced earlier. While it is true that you would expect of the piano teacher of your child that they know how to play piano, you would not expect them to be a concert pianist. However, you would expect of the piano teacher that they have abilities and skills that you would not expect of a concert pianist, like the ability to modify or select sheet music to the appropriate level of your child, to be able to expose your child to different styles of music, to know how to modify piano playing skills to adapt to the smaller hands of children, or to know skill development progression relevant to your child.

A good and successful piano teacher for your child is not someone who is a good concert pianist (which can actually be in the way of your child’s learning to play the piano), but someone who knows about the playing of the piano in a way that is helpful to your child’s development as a piano player. Being a concert pianist is not a foundation for being a good piano teacher for your child.

Third, many years back I looked up mathematics education research that inquired into the question whether the taking of university-level mathematics courses by school teachers is linked to greater success of their students in the learning of mathematics. What I remember finding was that that was just not the case, but that there were studies that suggested that when a certain number of such courses was exceeded, the success of the teachers’ students actually declined.

As the arguments just presented suggest, common sense suggests that university-level mathematics courses are not helpful to prepare or support K-8 teachers in their teaching of mathematics.

Second, common sense is not sensible if it simply dismisses empirical research, including mathematics education research, relevant to the question. We not just need to walk the walk, we need to walk in the right direction.

» Thomas Falkenberg is a professor of education at the University of Manitoba. This column was previously published in the Winnipeg Free Press