Loving and Hating Math

"We're devils and black sheep, really bad eggs"

In this post, I want to focus on mathematics education and what we really want to accomplish in teaching K-12 students about numbers, operations, variables, and the like. Note: quotes on this post are from the movie Pirates of the Caribbean: The Curse of the Black Pearl.

Currently, I do a lot of work around math education and I love math. However, I did not enjoy math when I was younger. In fact, I did not really like school at all. I was more interested in picking up rocks and seeing the creepy-crawlies scuttled away than listen to a teacher tell me something that I would not retain. It may be that science was my first obsession because I enjoyed my outdoor-explore time so much (outdoor life may have at some point transitioned to a pirate's life, it seems only natural that I would pick something that would allow me to both be outside by the sea and do math-- you know, compasses, charts, and non-Euclidean geometry). Science classes in my K-12 experience received the bulk of my engagement. I identified myself as a science person, though I bet my teachers did not know that. I did not always do well in my classes. Instead, I subscribed to the informal elements of learning about science, where pulling a thread about something interesting can lead down to a chasm of information about it, not the traditional structured school kind of science where we are told what to observe in controlled environments. I think this led to my misbehavior in the classroom at times. Still, I did try to participate.

In the 8th grade, I was doing very poorly in my science class. My teacher held multiple meetings with my parents about my low grades on tests and quizzes. I remember trying to do better because the threat of summer school meant less time exploring the outside world (see my interest in creepy-crawlies above). One lesson still sticks out particularly clearly in my mind. We were learning about friction and were watching a movie showing different activities and tasked with calling out whether we were witnessing rolling friction or sliding friction. Observing a ball or a wheel rotating was clearly rolling friction, pushing a box that grinds across the floor was clearly sliding friction. Then the video depicted someone ice-skating. I remember quickly noticing that rolling friction seemed to move more easily than sliding friction, and having been on ice, I know it is quite slippery, it couldn't be sliding friction. Could it be rolling friction? What happens between the ice-skate and the ice? Could water molecules be rolling between these surfaces? I decided it must be: "rolling friction," I said aloud. The teacher immediately admonished me, "where do you see wheels on the screen, Willy?" It was meant as a rhetorical question, I wasn't supposed to answer it, and the tone was not friendly.

I was embarrassed, not because I had been incorrect (as I had done that countless times before and my tolerance for embarrassment was high) but the reaction of the teacher felt to me as an intent to belittle, not to correct. I shut down for the rest of the class period, and I do not remember what, if anything we talked about after the video. Also, my science identity took a hit, not a great thing for a child who generally hates school except for science classes. My answer was wrong, but knowing what I know now, the response from the teacher was, at best, a missed opportunity to engage in a discussion about the different kinds of sliding friction. At worst, it sent the message to me and others in my class that wrong answers may be met with mockery and ridicule, not a good environment for encouraging participation. On top of that, she was conducting a formative assessment, this is the exact situation where a teacher should follow up in the moment on a student's answer.

I admit that it's a bit unfair to look back and dissect a decision a teacher made in the moment almost 30 years ago. Teachers make thousands of decisions each day, so one lapse in judgement is pretty good. Additionally, maybe her decision was correct in concept but not in execution. One could argue that there was little value in a deep dive into different kinds of friction, our time could be spent in other ways, and I respect that. However, the creators of the video included a variety of examples for a reason, I think that reason was to push students' thinking and their conceptual definition of friction, why not chat about it? I share this anecdote as an illustration of the emphasis on correct or incorrect answers.

"Wherever we want to go, we go. That's what a ship is, you know. It's not just a keel and hull and a deck and sails. That's what a ship needs."

I see the correct answer as the keel, hull, deck, and sail; important stuff, but not helpful for giving students the freedom to go where they want to go (I'd rather be the captain than crew). Instead of focusing only on the correct answer, we should focus on why someone gave that answer, at least during the stretches of time devoted to learning. While science education is nice, I feel less comfortable in that realm, I am going to shift to mathematics education, the topic of this website. While math and science share quite a few characteristics, they sometimes differ in terms of accepted truths. Much of science is based on how we test and draw conclusions about the environment of interest, and it is common to have scientists disagree with other scientists about conclusions. Mathematicians don't run experiments in the same way that scientists do. So disagreements in mathematics are often a result of contexts, special cases, and starting assumptions, because mathematicians use something called "proofs," which takes and uses other facts (i.e., definitions, previously proved theorems, etc.) to methodically prove that something is true. But that's the basis of mathematics... I mean, real mathematics. The thinking, the logic, the argument; those things are the math, not the conjecture or theorem alone.

I think a lot of people think back to their math learning experience and remember the emphasis on correct answers, and so they argue that the correct answer is the ultimate goal and therefore most important. I do not disagree that the correct answer in mathematics is "a fine goal, to be sure" (thank you, Jack Sparrow), "but it seems to me that [strong reasoning] makes this one here a bit superfluous." If one's reasoning is sound and connects well with other facts, then one will likely achieve the correct answer, no? If there are issues with the reasoning, then through an explanation, those issues will reveal themselves. Additionally, if someone's reasoning skills are strong, they can be persuaded by others who provide sound reasoning to counter incorrect answers.

Focusing on correct or incorrect answers devalues the thought process behind the answer. I would much rather know how a student arrived at an answer than what the answer should be. A numeric value is so much less interesting than where that value came from. I like to push preservice teachers to think about the following scenario: imagine paying $20 to walk into a movie theater to watch the 10 minute conclusion of a movie. That's not nearly as satisfying as watching from the beginning, understanding the story and the characters, the problems that come up and the journey to resolve those problems. Sometimes, like a numeric answer to a math problem, the ending of a movie is unsatisfying (I'm looking at you Pirates of the Caribbean: At World's End), and the only redeeming aspects of it were the action scenes, which we would have missed if we were only interested in the conclusion.