Is There a ‘Right’ Way to Learn Math?

Because this seems wrong. (An Ode to Conceptual Calculus)

First Assignment Prompt for MAT270 — Conceptual Calculus:

Describe your previous experiences with math classes and after reading the Preface For Students in the texbook, explain your expectations for this class.

The first time I remember feeling genuinely excited about math was in the first grade. We were learning basic operations and after expressing some confusion about how multiplication worked, my teacher told me that multiplication was simply repeated addition; 6 x 3 was equivalent to 6 + 6 + 6. This information blew my little kid mind, and I soon realized that while my teacher had explained this concept in the context of one digit multiplication, I could apply it to any numbers. If I really wanted to, and I had enough time on my hands, I could multiply 650 and 250 using just addition. (Spoiler alert: I did. It took two and a half hours and it was totally worth it.)

I was no longer restricted to what my teacher could teach me. I didn’t need to wait for her to tell me all the little tricks, like dropping zeros and carrying digits. I understood why it worked, and I could use that information to figure it out myself. It was very freeing. And it wasn’t limited to just addition and multiplication, either. I soon discovered several other mathematical relationships, like how division was the inverse of multiplication. And while I could apply that information to solve the math problems in the textbook, that’s not what it was about. 5 x 5 didn’t equal 25 just because my textbook said it did. The math was already there, long before the textbook was written.

It was inarguably, demonstrably, universally true.

I couldn’t look at numbers the same way anymore. At recess the next day, I sat down at a picnic table and spent the whole time writing out numbers and creating what was essentially a 10 x 10 multiplication table.

I had never seen one before, and in that moment, I think I understood how Pythagoras felt when he discovered that a²+ b² = c². I understood why people devoted their lives to studying math, something that had previously seemed monotonous and boring.

It was beautiful. Really, truly beautiful. I felt like I had discovered an aspect of the world that I had never seen before; I became enchanted by numbers. Counting was my new favorite pastime. I would count the trees lining the streets, the words in my favorite songs, and the steps I took while I walked. The numbers didn’t just exist on the worksheets I was given in class, they were real.

My new favorite question in math class was “Why?” The answer to that first “Why?” had been eye opening in a way very few other things had been in my 7 year old life, and I desperately wanted more. Unfortunately, I found that as math became more and more complex, “Why?” was met with more and more contempt. I was dismissed and redirected and, at times, scolded for asking. In retrospect, I’m sure many of my teachers simply didn’t know “Why?” Perhaps they had once had the same questions I did, but had been rebuffed in the same way that they were rebuffing me.

By the time I was a sophomore in high school, the primary goal of math class had long ago shifted away from understanding and toward rote memorization. I held material in my brain just long enough to regurgitate it on tests, and then quickly forgot it. I did not feel challenged by any of the coursework because, regardless of the supposed difficulty of problems I was given, they all measured the same skill: plugging numbers into formulas. Some formulas were more complex than others… but they were still just formulas. There was no real learning taking place.

The math that I had once felt was inextricably linked to the real world, now seemed so contrived, like it existed in a vacuum separate from that world. If halfway through the year my Algebra II teacher had stopped teaching actual math and started just making things up, I wouldn’t have been able to tell. I was not free the way I was in first grade. I did not understand the purpose of the math I did or the concepts it was built upon. So, there was no way for me to solve the problems by myself. I relied completely on my teacher to tell me what to do. I didn’t know where I was going and, without her directions, I felt lost.

Despite feeling that I didn’t have a strong grasp of the material being taught, I consistently received good grades in class and high test scores on conventional standardized tests. I was, by every metric, considered to be a highly successful math student. The distinct disconnect between my understanding of math content and my performance on assessments was often distressing. I walked away from the ACT, for example, feeling like I got a higher score than I deserved. The result misrepresented my actual understanding, and I felt as though I had cheated somehow. And while I was able to get perfect scores on rote memorization questions, I failed miserably when I was finally confronted with conceptual, AP style questions (like when I got a 2 on my AP Physics Exam — after earning an A+ in the class.)

When registering for classes this summer for my first semester of my first year in college, my initial plan was to just take Brief Calculus and get it over with, despite having been placed into this higher level Conceptual Calculus class. I don’t anticipate my future career being very math heavy and figured any math class would just be more of the same kind of rote regurgitation. However, Conceptual Calc was recommended to me by another student who said that it, “…taught [her] how to think”. I found that commentary intriguing, and decided to give this class a shot.

After years of taking math in underfunded, understaffed public schools, I had resigned myself to the idea that the only thing higher level math classes could offer me was more complex formulas to plug numbers into. However, after reading the preface at the beginning of the textbook and the article linked with it, I am excited, for the first time in a long time, to take a math class.

I am very nervous. This class might be easy, but there is also a distinct chance that it will be unprecedentedly difficult. I would expect that the part of my brain associated with conceptual learning is severely underdeveloped, and that taking this class will be comparable to running a 5K on legs whose muscles have atrophied following years of disuse. I worry that I will no longer be able to hide my inadequate understanding of the way mathematical processes work behind easy multiple choice questions. I will be forced to learn the concepts and actually try…and in doing so, I risk being bad at it. I risk feeling stupid. There is a part of me that wants to cling to what I know, take a lower level class, and avoid challenging myself. It is tempting to sacrifice the kind of learning I have always instinctually craved for the security of “success”…because if there’s one thing I know how to do, it’s memorize. But, I’m not going to give into that. That’s not why I’m here.

I didn’t graduate high school early to subject myself to four more years of the same mind-numbing, contrived, busy-making autopilot that drove me away in the first place.

I’m going to stick it out, even though it scares me. Like, really scares me. I will struggle and do things wrong and get bad grades on assignments. I anticipate spending a lot of time at the Tutoring Center. I will have to overcome the compulsive need I have to be right. I will have to unlearn the reward system which has conditioned me to believe that failure is to be avoided at all costs, regardless of how unfulfilling their definition of ‘success’ is. I will have to learn to cope with the nausea-inducing, throat-tightening, eye-welling feeling that washes over me at the thought of being bad at something.

But… everything will be fine. It will be fine because I’m choosing the process over the outcome this time. I don’t want to be restricted to following the explicit instruction of my teachers. I want to gain the tools which will help me figure things out for myself. Not only will I learn things the way I have always craved to learn them, I will also learn to persevere in the face of confusion and frustration. I will learn to seek out help and let people teach me. In exchange for the struggle, I will acquire the ability to think in a way I have never have before and understand things I so want to understand, on a level that has never been offered to me — that I didn’t know existed. I will (hopefully) reactivate a part of my brain that has been locked away since first grade.