My Reliance on Formulas

When I was a student, a large part of the reason why I didn't like math was because I viewed it as a set of very specific skills. Either you knew how to find the volume of a sphere, or you didn't. And if all that math was were such specific skills such as calculating the volume of a sphere and the like, and you didn't have a specific reason to use such particular skills, then math was useless. Looking back now, I do believe that I held this perspective because of the way that I was taught math in schools as a very formulaic process. Every single class followed the same pattern: my teachers would show us a particular formula at the beginning of class related to the specific area of math that we were learning about, do multiple examples on the backboard following this precise formula, and then make us practice the formula by answering the questions in the textbook. There was no explicit mention of the seven part mathematical process referenced in the Ontario Math Curriculum, nor was there focus placed on problem solving except for figuring out how to adapt a different problem to fit into the formula that we were taught.

"An information- and technology-based society requires individuals who are able to think critically about complex issues, analyse and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively." - Ontario Math Curriculum, page 4

Moving forward as a teacher, I will not be imitating the teachers who I had when I was a student. Instead, I am planning to follow the guidelines in the Ontario Math Curriculum about how to teach problem solving skills. The Curriculum recognizes that there is a general negative attitude towards math in our society. To counter this negativity, the Curriculum advises teachers to value various problem solving approaches, stating "Students need to understand that, for some mathematics problems, there may be several ways to arrive at the correct answer" (page 26). This is a direct contrast to the single formula that I was taught. I am also impressed with the way that the Curriculum is promoting cross-curricular and integrated learning, where math is taught together with another subject like science.

 I want my students to see math as a much wider set of skills than a list of specific formulas. I am considering using a journal approach with my students where they will have to reflect on the process that they used to solve a problem so that they become aware of the skills that they are developing as they work on solving these problems. I also plan to teach with cross-curricular lesson plans so that my students see the relevance of the specific skills that they are learning in a real life context.

However wonderful my pedagogical ideals are, they alone are not going to turn me into a fantastic math teacher. I am going to have to work on my content knowledge. If I want to teach my students to understand math logic as they develop their problem solving skills, I myself am going to have to understand this math logic myself.

"Teacher knowledge makes a substantial contribution to student achievement" - Toward a Practice-Based Theory of Mathematical Knowledge for Teaching, page 4

I believe that a good math teacher needs to have two types of knowledge: content knowledge, or an understanding of how math works, and pedagogical knowledge, or an understanding of how best to teach mathematical concepts. I felt that most of my teachers had good content knowledge, but needed to improve their pedagogical knowledge. I am concerned that while I will have fairly good pedagogical knowledge, that I will have a fairly terrible content knowledge. Reading a quote from Toward a Practice-Based Theory of Mathematical Knowledge for Teaching made me reflect on the state of my content knowledge. The article stated, "Second, looking at teaching as mathematical work highlights some essential features of knowing mathematics for teaching. One such feature is that mathematical knowledge needs to be unpacked. This may be a distinctive feature of knowledge for teaching. Consider, in contrast, that a powerful characteristic of mathematics is its capacity to compress information into abstract and highly usable forms" (page 11). The formulas that I learned as a child were not problematic in and of themselves, because they are the compressed form of math referenced in the article. The problem is when the formulas become the end-all and be-all, and aren't
ever unpacked to explain the math logic that created the formula. I fear that unless I develop a better understanding of mathematical concepts and logic, that I am going to slip into teaching only formulas because that is all that I understand myself.