In most of the years of my first decade of “regular” (school-year) teaching I also taught in a summer program for high schoolers at a local university. The folks who ran that program brought in some pretty interesting guests for our professional development. There was a particular presenter who said one thing that has always stuck with me. He suggested that the further you go in your education, the more the different fields (language, math, science, art, etc.) start to resemble each other. At the highest level all knowledge is one. Something like that. I remember feeling at the time like I was listening to a wise man from the top of a mountain trying to explain to us novices what the universe looks like from a state of enlightenment. I don’t think I’d respond quite that way these days but nevertheless his assertion about the similarity of all knowledge, once you reach a high enough level of understanding, continues to inform my educational worldview, as I hope the story below will illustrate.
Now, if you’ve been a math teacher for any length of time you’ve surely had students ask you The Question. It takes various forms but its most popular usage in my experience is, “When are we ever going to use this?” The speaker is almost invariably a student who has other ways to frustrate you as well (if you find this question frustrating, that is). So I was completely flummoxed when, one day in Algebra 1 in perhaps my fifth year, I got asked The Question by my very best student.
This caused me to (quickly) reassess what The Question was really about. I had always considered it simply an evasive move by the questioner, trying to get me off track. (This was often a successful tactic in my early years, and I sometimes allow it to be even now.) The students did seem to pay attention as I fumbled to try and give them a real-world application for rearranging an equation with four variables or what have you. But I knew my repertoire of answers to The Question was deeply unsatisfactory.
I knew that Susan (we’ll call her) would not ask The Question just to get me off-track. She was one of my Math Club kids who stayed after school twice a week to do more math. So there had to be some other reason, at least sometimes, for asking The Question. I’m sure I didn’t piece it all together right then but I came to realize at some point that students only ask The Question when they don’t understand what is going on. Learning is fun. I really believe that our brains are designed to make us feel good when we are learning. What students don’t find fun is when they are not learning.
I don’t recall for sure, but I suspect that Susan had already asked me other questions in order to help me clarify a lesson that I was apparently presenting particularly poorly. My responses must not have done the job, so Susan wasn’t learning and she resorted to The Question. Pulling one of my stock answers out of the toolbox wasn’t going to do. I needed to try and find an answer that really felt right to me, because that was the only type of answer that would satisfy Susan, I was sure. But what was The Answer?
We were indeed working on solving for different variables in equations that were made up of three or four variables. It occurred to me that the key to each algebraic step was that before and after you performed it, the relationships between all the variables were unchanged. Relationships. Let’s see…
Imagine that you have a decision to make and three different people you care about–say your mother, your best friend and your boyfriend–all want you to make a different decision. They are all pulling you in different ways, which we can think of as analogous to different operations: addition, multiplication, etc. Imagine that you are a variable in the middle of a somewhat messy expression on one side of an equation. You want to separate yourself out from everyone else but without affecting any of your relationships. This is what the algebraic equality properties allow you to do: separate one variable (person) out from all of the others that are interacting with it, but retaining all the essential relationships that were there in the original equation. So when you are learning how to use these particular tools in Algebra you are also learning tools for how to make better decisions in your own life.
I’d like to say that I received a standing ovation at that point but I’ll try to add that into the screenplay when they decide to film my story. What I do recall clearly is a great sense of satisfaction and relief. Susan appreciated my answer, and that was the real test. I felt like, at least that once, I had really found The Answer.
Since that day I’ve fielded The Question any number of times, including from some pretty strong students on occasion. I don’t always feel like I hit the answer out of the park (to completely reverse the metaphor) but I am much more comfortable with what I think the answer is. (Which is not to say that I think there is really only one Answer, but they all feel like shades of the same color to me.) My understanding of what math is for and about has evolved greatly over the years through reading, workshops, discussions with colleagues, and perhaps other methods as well, but I contend that the main reason why math is the second-most important subject for students to learn (language has to be number one–communication leads to all else) is that mathematical reasoning is transferable to every part of life in one way or another. As I told my class that day and have repeated innumerable times since: It’s all math.