Math as a Three Legged Stool

Memecenter

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.