Saturday, September 7, 2013

What Is Math? — Responses to an informal survey of young college students

For the past couple of years, I've been focused on teaching a particular course (whose title is the eponym for this blog, in fact). That course is meant to introduce students to mathematical proofs, transitioning them from computation-heavy, rote, follow-this-example style math into abstraction, problem-solving, and the communication of mathematical ideas. It's notoriously difficult, and rightly so. In particular, it's one of the first occasions where students have to face the truth that mathematics is even harder than they ever thought it was. As mathematicians, we constantly face our own lack of knowledge and the mysteries of the universe, and rarely is there "the right answer" that we just need to get to before circling it at the bottom of our paper. This course forces students to face these ideas head on, and it can be quite a shock. Now, this is true even though the course and its exercises are often structured so that there kind of is a "right answer" or some "correct method" to be used. But that just goes to show how even tossing a little bit of this uncertainty into the mix can be surprising and challenging for students.

On a larger scale, this course is a place where students' fundamental assumptions about what math is are challenged, questioned, reworked, and never really settled. Last summer, I had the idea to "pre-test" students on what they thought about math. Mostly, I was just curious, for myself. I thought that seeing their answers to the question "What Is Mathematics?" might reveal something about their backgrounds in math and their desires for the course; at least, the answers might confirm my suspicions that the course will be as shocking and difficult as it always is. It turns out that they were somewhat what I expected, and the course was, indeed, difficult for everyone (including me). But that was the point :-) (I'll have to go back and find these responses and analyze them as I do below…)

I'm now at another institution in a full-time teaching job. I'm teaching three different courses, none of which are this "transition to proofs" course that I've focused on for the last few years. In fact, two of the three courses I'm teaching now are completely populated with non-math-majors, students who are fulfilling distribution requirements/pre-requisites or getting extra preparation for their next requirement. Even more than ever, I felt the need to understand my students' backgrounds in math, their goals for learning, and what preconceptions about math they might bring with them. The results are both exactly what I expected and completely shocking at the same time.

I asked several questions of my students on the first day of class this week, and had them write their answers on an index card. Three of those questions were:
  • Why are you taking this course?
  • What do you hope to learn from this course?
  • What is math?
Again, I was hoping the answers to these questions might help me shape the course, the choice of material and examples and problems, my behavior in class, and so on. Unfortunately, I now feel rather daunted by these responses, but I'm hoping I can turn that into some powerful motivation (for both me and the students).

Let's talk about the third question: "What Is Math?". I knew I'd get a lot of strange responses here, and was expecting a general trend emphasizing computation and numbers, but the overwhelmingness of this trend is still surprising. I fed the 91 responses to Wordle to generate a word cloud:

Over half of the answers used the word "numbers". About a quarter used "problem(s)", and almost all of these were mentions of "problem solving" in some form. One sixth mentioned "equations". About one eighth mentioned "letters" or "symbols", and a handful put forth the idea that math is a "language".
I expected this. Most of my students are college freshman, and high school math emphasizes the idea that math is a system of useful rules for solving equations and problems. That's fine. I didn't necessarily expect these phrases to be so widespread and uniform, though. So many students seem to think that math is completely characterized by algebraic manipulations of letters, numbers, and symbols. I would like to dispel this rumor, but I really don't think I'll be able to in a course that is explicitly meant to teach these students algebraic techniques!

I find it interesting that many students pointed to the "usefulness" or "application" of math to the real world. I'm pleased that at least 20% of the responses indicated some positive usefulness of mathematics, even though many of the phrasings intimate that it's the quantitative/numerical aspect of math that makes it useful. Here are some example phrasings of this kind:
  • Math is how the world runs
  • Math is the concrete explanation to everything
  • Math is an extremely useful tool that can get very complicated
  • Math is difficult! It's the use of numbers to help explain different phenomena we see in our lives
  • Math is a problem solving tool involving numbers and symbols to form equations for real life applications
  • Math is a language which we use to describe and talk about our world
  • Math is the way numbers/measurements/data/etc is used and applied
  • Math is numbers, calculations, things that help justify why things are
  • Math is everything. Math is numbers, letters, shapes, etc. We need/use math daily.

Another aspect I expected is the idea that there's always a "right answer" in math. Admittedly, this is what drew me to math when I was young, and it took me far too many years (more than I care to admit) to realize that this is far from true. Luckily enough, I found that I also loved the difficulties of math no longer cared about the sureties of having a "right" and a "wrong". I'm not so sure that every student would respond favorably to this epiphany, though. And even still, I can't really disagree with a claim like "Math is a way of finding an answer a certain way with certain rules" when, in all likelihood, that is exactly what this student has been taught for years. I can only hope to relax and alter their view a little bit but, again, this will be difficult to do in a course designed to perpetuate those certainties. When their homework is completed online and the system immediately gives them a big green check mark or a big red "X" … what else can I really say/do to convince them otherwise?
All of that said, I marvel at the comfort inherent in the following two statements:

"Math is what you make of it. It is found everywhere and is one of the only truths in this world."
"Math is simple and always has a right or wrong answer."

Maybe I don't want to step in and shatter their worldview. Then again, maybe this is exactly what good teachers are supposed to do!
That last example also brings up the issue of difficulty. I would never have thought that the definition of a subject could have anything to say about how simple/complex it is; that is, shouldn't such a definition just say what the subject is, without making a claim about its difficulty? Several students incorporated the notion that "math is hard" into their definitions, though! Indeed, "Math makes numbers into difficult things, ha" was surely written in jest, so can't read too much into it. But, to receive an answer like "Math is annoying" on this kind of friendly, day-one survey is both incredibly disheartening and motivating. If I can make this one student reverse that thought over the course of this semester, I'll consider my entire efforts worthwhile.
While most of the students had some idea and shared it with conviction, some other students realized that either there isn't a good answer to this question or they didn't have one, themselves. One even told me, "That's an insane question" before going on to actually answer it, saying "Math is a universal language". Another hinted at the dependence on personal viewpoints, and answered "To me? Just numbers". Yet another was probably being cheeky, resorting to "2+2=4" as their complete answer, but I bet Whitehead and Russell would have a field day with that one …

Interestingly enough, one student seemed to answer the question from the point of a teacher, seeking to explain why we teach math and not just what it is: "Math is a subject taught in schools to students. It helps to benefit their knowledge with numbers and critical thinking." I don't disagree with this, and I hope this student keeps this in mind as he/she learns.

In the past, students have approached me looking for my own answer to this titular question. I haven't come up with a good, consistent answer, and I make sure to tell them as much. The real point in asking the question originally is to open the debate, not to indicate that there is a perfect answer that the students just don't have yet. Usually, though, any attempt I make at an answer mentions the search for and analysis of "patterns". I know other people use similar phrases, while others disagree with this characterization, for whatever reason. Regardless, not a single student in this informal survey mentioned anything like this. I wonder why, and I wonder whether I should try to share this view. Would that just be imposing my own beliefs about mathematics on them? Would that then make me a bad teacher, a selfish one? Or would it just be the way that teaching is done?

I'll conclude with my favorite answer, one that I might use myself in the future, and one that I hope will resonate with people who have devoted their entire lives to pursuing mathematics and people who have chosen to run far away from mathematics, alike:
"Math is one big puzzle"

(Next time, I'll talk about responses to the other survey questions and how they're potentially related to these responses.)

1 comment:

  1. Thank you Brendan. Interesting, very interesting. Looking forward to reading more.