I’ll just come out and say it I guess; I’m a math lover. I can’t say that it has always been this way. If you were to talk to me when I was in junior high or high school I would probably have told you that I wasn’t good at it, and that it was hard. I would have also told you that it was useful (with my dad being a physicist it’s hard to not know it has some good uses). Yet, you would never hear me say that I loved it.
When I looked at what I was learning in class and seeing my dad take the math and use it in ways that I didn’t understand, as he would talk about how it was useful, I would experience a disconnect. If it is so useful, why can’t I find a way to make it useful? Why do some people just “get it” and others not? Why does it seem like such a drudge to me when I know that it can solve amazingly difficult problems?
I gave up on my dream of becoming an engineer my first year of college when I took a college algebra class and basically bombed the final. (I got an A in that class, somehow, which means everyone probably did poorly. Not a great indication of how well it was understood, and also taught, by people who obviously needed a little help.)
When I chose to start my major with education I was not too worried about taking my math classes, after all, I thought that I knew all of the necessary steps, and it would just be a brush up of my knowledge. The day I walked into the classroom for Basic Concepts of Math, I had no idea that my perspective of math was soon to change. The change started small. At first I was amused by the tasks given to prove why basic math concepts worked. Then, when it seemed like we were just about to enter into a rut, my instructor gave us a task that changed my life.
I am awed somewhat by the subtlety that my instructor used to build me up to the point so that, when that lesson came, I would be able to change my perspective. She had built up a classroom culture and structure in such a way that when the lesson happened, I was able to take the most of it and run with it.
The first task given was to write a formula to find the area of a square, then a rectangle, and finally, a simple parallelogram. The instructor told us that we couldn’t look up any of the formulas and that if we remembered it, we would have to justify it. For the square and rectangle it took only a few minutes to write the formula and justify it – way too easy. However, when it came to the parallelogram I was stumped. So I drew a picture, and just stared at it. After all, pictures are for dumb people, or people who can’t follow the formulas or steps well enough and therefore need remediation. Yet, there I was trying to figure it out, something so elementary that I shouldn’t have been struggling with.
It was a common practice of my instructor to let us talk and share our ideas with a neighbor as a way of processing our thoughts. Although I didn’t share, I did listen to a couple of my classmates. They were also working on it. I heard one of them say that the formula was the same as that of a rectangle, just length of the bottom multiplied by the height. So, I drew that on my picture. In that one spectacular moment as I looked at the picture, my mind stretched in a way that I hadn’t done for years, and in a way that I had rarely connected to math. I had a creative moment where I could see in my model “cutting” off the ends of the parallelogram and moving one side to the other to make a rectangle, where the formula matched what my classmate had said. I had realized that I could cut apart a shape and move it around in anyway I found useful!
At this moment I excitedly started sharing my picture with my classmates and seeking to justify my reasoning and seeking to actually answer their questions. My mind was flush with the joy and fun that I had just figured out. When we moved on to looking at the area of a trapezoid I was more than ready for the challenge and excited to get started. I was also surprised by the different ways that my classmates had solved it. It didn’t take long before I was recording what they were doing and drawing pictures to model and express what they had done, to explain it to a confused classmate or just to make sure to have it clear myself!
How I wish that I could take you through that wonderful process. It wasn’t always that way. I still had times where it was less than exciting. There were times where I would struggle harder than I would like to admit for concepts about basic math. More importantly, however, I found myself enjoying the challenge more, getting into the reasoning and process, and enjoying listening to my peers as we discussed the ideas.
In essence, I was surprised by how much about basic math I didn’t know how to explain (but could “use”) and how much more the math that I learned before made more sense. I learned that math could be a way to creatively express ideas and logic. I learned that it was more than just simply the steps behind the computation. I learned that math was actually creative, logical, applied, relevant, complicated yet simple, flexible and had some give in how it could be thought of, and a way of easily expressing my ideas.
The interesting part is that as I continue to deepen my understanding of math I find that I crave that deeper knowledge; and just being given a basic “here’s how” as being rather unfulfilling. Now, I must admit that my understanding of math still has many holes, especially with some algebra. But, I’ve gotten really good at the basics and I’m growing in my understanding of other math ideas.
Without actually having an opportunity to go deeper, I would never have learned to love math. I found that for me to really remember the “steps” of math, I actually needed the why also. I have found that I don’t forget the “steps” anymore, because I have a deeper understanding than I’ve ever had before. I can also generalize ideas about mathematical principles, without doing computation, and make judgments based on those. This has helped me be more confident and fluent when I work with any mathematical idea I come across.
This is why I am for reforming how we teach math. I have been on the side where it failed, yet I have also started down on the side where it succeeded. The side that failed was traditional math teaching. The side that helped me succeed is the one that is “confusing,” “illogical,” and “teaching to the lowest common denominator.” (Which, in many times in math, a lowest common denominator actually makes things clear and can be useful.)
I’ve had people tell me that we shouldn’t try to go deeper first because the kids aren’t ready for it, much like what happened to me, go deeper later. I can’t agree with that. I’ve seen kids go deep and make amazing, real growth that they are completely ready for. I’ve also seen some kids who are on the edge do amazingly well, until someone tells them that isn’t how you do math because getting the right answer quickly is the only important part. It isn’t by the way, but I’ve seen this happen.
So, the next time you hear someone complain about the way math is being taught, remember my experience. Think that there may be may be more to the story than what the infuriated person is saying.
If nothing else, remember that we want more than simple, easy “solutions.” We want what works best because it really works, even if that means a little more struggle and challenge.
The Math Dragons 