Students Speak: From Failing High School Algebra to Acing College Calculus: Embracing Mistakes
By Dayne Patillo
Dayne Patillo
“It’s okay, Dayne, some people simply don’t have math brains.”
“You’re just more right-brained and creative.”
“There’s nothing wrong with sticking to literature and writing!”
“No. It’s not that you ‘don’t have a math brain.’ You’re just stuck in a Russian poetry class where you don’t understand Russian. And, even if you did speak Russian, you don’t have the technical poetic vocabulary to even speak about the poems.”
Since the third grade, many teachers have told me that I simply “don’t have a math brain.” Some teachers suggested to me that maybe I would never understand the mathematical concepts they were teaching. Two university calculus courses later (as well as a Linear Point Set Theory class, a Linear Algebra class, and a Number Theory class), I think I can say that the theory of left-brain or right-brain people is not as exclusive as my teachers let on; in reality, the “or” in that phrase is an inclusive or.
However, these teachers are not to blame. They were deceived into believing that this pervasive, yet false notion was true: Either you have a natural aptitude for math, or you do not, and if you do not, there is no striving to understand or discover more math than is necessary. I am so thankful that I had teachers who recognized the falsity of this statement. Now, this is not to say that there is no such thing as a natural aptitude for mathematics. Many students have a natural aptitude for math, and teachers should foster that. However, telling a student who struggles in math that there is a definite limit to the amount of information that she can understand is ludicrous; the student may not fully understand the concept before the chapter test, but that does not mean she should stop trying to understand it. If teachers had continued to perpetuate that lie throughout high school, I would have never overcome the mental block that was inhibiting me from reaching my full potential, and I definitely would not be willingly studying mathematics at a university level.
Now, I’ll just come out and say it; I don’t have a natural aptitude for math. Nevertheless, that doesn’t mean I don’t have a “math brain,” and it certainly doesn’t mean that I’m not good at mathematical thinking. In the classroom, especially the mathematics classroom, it’s all about mindset. The first piece of advice that helped me overcome my mental block with math was given to me in the form of a metaphor. My Linear Algebra professor told me that math is a lot like a Russian poetry class. To be able to talk about and understand Russian poetry, you have to have two things: first, there’s the actual Russian language that you need to learn. Secondly, there’s the poetic structures and concepts that you need to learn. Whenever I’m having trouble in a math class, I always troubleshoot in this way. I ask myself if 1) I am getting tripped up on my Russian (that is, the language of symbols or the algebra that accompanies big picture ideas), or 2) if I’m not engaging with the poetic structures and concepts correctly (that is, is there a fundamental concept I’m not understanding?). Breaking down the unknowns of new math like this makes the discomfort of confusion much easier to bear because it gives me a chance to recognize what I do know. For example, maybe I actually understand the big picture concept, but I’m just getting tripped up on how to convey it with the correct mathematical logic, symbols, or algebraic expressions.
The second piece of pivotal advice that helped to foster my love of mathematics came from my Linear Point Set Theory professor. He taught me how to “play” with mathematics. I remember being shocked to realize just how absolutely petrified I was to make a mistake in my math classes. One day in office hours, this same professor told me to physically write down “2+2=5.” I definitely didn’t want to do this. First of all, it had nothing to do with what we were studying. Secondly, it’s just plain wrong! However, I did it, and he told me “See, nothing happened. Your paper didn’t catch on fire, the world didn’t implode, nothing happened. All you have to do is skip a line, try a different approach, and continue with the problem.” While mistakes are still wrong, that doesn’t mean they’re bad. Now, some mistakes may be silly, like 2+2=5, but just as there exist smart questions, there exist smart mistakes. My professor treated conceptual mistakes as insights, and it really fostered my love of mathematics.
Although it may seem counter-intuitive, focusing on why various common mistakes are wrong is crucial to creating a positive classroom atmosphere. Not only does this lead to an active discussion among students as opposed to a traditional lecture, it also makes students feel more comfortable voicing their confusion. If mistakes are seen as insights, students feel that they are contributing to the class discussion in a meaningful way instead of feeling that they are holding the class back because they aren’t grasping the concepts as quickly as others. Moreover, framing mistakes as insights is key to developing a willingness to “play” and better engage with math, which only further fosters mathematical and problem-solving instincts. The more a student lets go of her fear of making mistakes, the more that student can embrace the problem-solving process and, in turn, the more she can learn to love mathematics.
For me, learning to play with mathematics was a bit of a challenge; it required a shift in the way I approached math problems. However, one class in particular made this shift easier. During my junior year of college, I took a math elective called Problem Solving. This class met once a week and had no formal assignments and no formal tests. Each week, the class would be given a Problem Set consisting of five problems from previous Virginia Tech Regional Mathematics Contests or past Putnam Exams. The students had to work on the problems for a minimum of two hours each week. We were not allowed to break up the two hours into smaller increments of time either, and we could not move on to another problem until we had spent at least an hour on it. Every week, I carved two hours out of my schedule to work on these math problems. The first week I tackled them, I spent about 15 minutes on one problem before I felt like giving up; I was stunned to discover just how little brain stamina I had for these problems, not to mention how unwilling I was to take risks and to try creative strategies; I didn’t want to try to solve a problem unless I knew for certain I was taking the correct approach. As a result, I definitely felt a little defeated walking into class the next week. I had worked for the better part of three hours on these problems and had no correct answers. Luckily, that wasn’t the point of this class. The point of this class was to give my brain enough time to make connections that aren’t necessarily obvious or instinctual and to then think creatively about how I could solve the given problem. But, even if I was unable to get to the correct answer on a given problem, by following the mathematical instincts I had developed over the years in conjunction with integrating new problem solving strategies, I was able to develop even better mathematical and problem solving instincts and gain key insights into solving many of these problems. To be honest, I don’t think I ever got a problem “correct.” Nevertheless, this class was invaluable to my positive experience with math; it really made me appreciate insights and creative solutions, helping me break my perfectionist habit of only attempting problems that I felt I could solve quickly and formulaically.
As previously stated, I don’t have a natural aptitude for math. But, as an English major pursuing a Pure Mathematics minor, I’m keeping up with my peers in my university math classes while having taken half as many math classes as them. I do have to put in a little bit more effort and make use of office hours a bit more, but unlike my grade school days, putting in the effort no longer feels like pulling teeth. I enjoy asking questions, making mistakes (which is really just equivalent to learning a new insight), and knowing that, even if I don’t arrive at the correct answer, it doesn’t mean I “can’t do math.” The problem-solving process required in mathematics is beautiful and, most importantly, creative, which is what makes math so appealing to my brain.
You don’t have to have a natural talent for numbers to love math; you just need a positive environment, a passion for creativity, and the ability to embrace mistakes!
Dayne Patillo is a rising senior at the University of Dallas, and in addition to a B.A. in English, she is pursuing a minor in Pure Mathematics. She hopes to one day become a math teacher so she can give future generations of students the same gift she received: the opportunity to experience the beauty of mathematics.