Math as a Three Legged Stool

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When I was in highschool math class, I generally swung between either thinking that mathematical concepts were extremely easy, or they were extremely difficult. So often I would follow along with the examples that my teacher wrote on the blackboard, understanding everything (or so I thought), correctly answer the first few homework questions, not do the rest of the homework questions (I clearly understood the concept because I answered the first ones correctly, and I had better things to do with my evenings), and then wonder why I got so many questions wrong when I received my unit test back. This repeated experience served to make me increasingly frustrated with math, and continuously doubt my math ability while doing nearly anything. My problem was that I was so focused on getting through my homework questions as fast as I could that I would use a single formula or process to answer every question, without thinking about whether there was anything else that I would need to do. I had zero adaptability when it came to answering a different type of question based on the same basic concept. Mind you, my teachers only ever taught by formulas and examples, so they did not shine as beacons of adaptability to me or my fellow classmates.

Looking back, I do believe that our teachers assumed that we would gather all of the information that we needed from their simpler examples to answer the more complicated questions. Whenever we had questions in class, they would direct us to review the example written on the board, or would provide yet another example following the same formula or procedure as before. They were effectively training me in procedural understanding (how to follow step by step instructions), while assuming that I would pick up conceptual understanding (the how and why of the concept) without ever explicitly teaching conceptual understanding. I was so used to procedural understanding training, that I specifically would request this from my math teachers, believing that the only way that I could do the math was if I had a formula handed to me.

As an adult, things have flipped for me. I have lost nearly all of the procedural understanding that I used to have, but instead I work through every problem using my conceptual understanding. This is a very slow process for me, since I have to work my way through each individual step before I can put all of the pieces together.

Neither of these situations are ideal.

Edugains has a fantastic poster resource that shows how instrumental (procedural) and relational (conceptual) understandings are both important to math education, and how they affect each other. As the poster states:

"Life-long learners of mathematics build new knowledge and skills in prior knowledge using the mathematical processes 

Confidence and flexibility results from conceptual understanding 

Accuracy and efficiency results from knowing facts and procedures" 

Some people like to argue about procedural versus conceptual, like they are two mountains separated by a deep valley. I do not see them as separate entities though, but rather as the two sides of a coin. As an adult, I do tend to place more importance on relational understanding, but this is more so due to a worry that it isn't being included in classrooms than that I devalue procedural understanding. I believe that when a person has a strong relational understanding, they develop procedural understanding naturally based on noticing repeated patterns while they engage in problem solving.

Procedural understanding, conceptual understanding, and problem solving are the three legs that support all of mathematical understanding.

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.

Changing Perspectives

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As a child, I was indifferent.
As a teenager, I hated it.
As an adult, I now see math's purpose, but still am utterly frustrated by it.

This is not the story that I want my students to be telling when they become adults.

So, a little bit about myself: I've always been a creative person. As a child I took an interest in stories, art, music, and history, and in high school, I developed interests in philosophy and psychology. I went to university for English literature. One of my life goals is to publish a fantasy novel, and my other hobbies include drawing, attempting to play guitar and flute, hiking, and writing poetry. I've always lived in the realm of the theoretical and the imaginative, where the focus was on creativity and beauty. Math never fit into this world of mine. Our relationship began as a tolerable dissonance that was easily forgotten while I learned basic concepts in elementary school, but developed into a seething hatred in high school when math homework required more time and effort, forcing its way into my evenings and interrupting my life. I deliberately went behind my guidance councilor's back to drop out of the grade 12 math course when I learned that I wasn't required by law to take it.

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Whenever the topic of math would come up in conversation, there were a couple of stories that I would always pull out to prove my point about how absolutely terrible math was. The first involved my grade 10 math teacher and parabolas. The setting was a beautiful autumn day, just after we had returned from spending our lunchtime freedom lazing on the grass outside, staring up at the changing leaves. Coming inside to the starkness of our white tiled classroom and staring at incomprehensible combinations of numbers was excruciating, and our teacher had difficulty in keeping our attention for longer than thirty seconds. "Guys, this is important!" she cried out in frustration. We stared at her blankly, then one of us asked her just when we were actually ever going to use parabolas in the future. "Well, if you ever need to drop a ball off of a roof..." she began, but I tuned out the rest of what she said immediately because I recognized those exact words from the word problems in our textbook. Despite how they tried, I couldn't relate to any of the situations the textbook writers wrote about, and my teacher never did anything but reiterate what they said.

The second story took place when I told one of my friend's mom about my teacher quoting our textbook word problem word for word, commenting that nobody I knew had ever used the math they had learned in high school when they became adults. "Well," she replied, "my husband actually used had to use some math that he learned in high school for his job the other week. I walked in on him Googling something about the quadratic formula." This was the moment that I realized just how pointless math class was, because even if I did understand a concept now, when I would actually run into a situation in my adult life where I would need to use it, I would be sure to have completely forgotten how to do it and would be resorting to relearning it through Google.

So, math class in school is completely pointless, right?

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Well, actually, adult me has come to understand that math is important, and is stealthily hidden in a lot of different places. I have a slight tinge of regret that I didn't take data management in grade 12 when I find myself in the middle of an argument with someone over the the skewed statistics involved in a political debate. Also, the strict dichotomy between fun creativity and boring analytical stuff broke down while I was in university writing English essays. I came to recognize the importance of logical thinking and problem solving skills, and I've become passionate about wanting the students graduating from my class to be proficient in these skills. And where are these skills commonly found? Math.

This TED talk discusses the issue of how math has been taught in our schools, and how this style of teaching has completely affected students. Dan Meyer is a high school math teacher who is focusing on teaching his students to learn the logic behind math rather than having them memorize formulas, and connecting everything that they learn about to real life situations. His students are more engaged, have a deeper understanding, and retain the information for much longer. This is the same kind of teacher that I want to be.

I've begun to change my perspective about math already, but I still have a long road ahead of me, and many layers of bias against math to rebuild.