Math & Music – Harmony & Counterpoint*
by Cohen Math Prep Founder, Rob Cohen
*The following essay discusses what I perceive to be some of the deficiencies in math education, as that is my personal area of expertise. I’m of the opinion, however, that very similar arguments could be made for the other subjects that students are subjected to (pardon the pun) as part of a traditional K-12 education. In any subject, I advocate for a teaching style that allows students to find their own sparks of meaning and curiosity.
Mathematics and Music share extremely similar positions in the world of education due to popular perception of them. Ability in either field is seen by many as a “natural” talent that one is born with or not. I cannot count the number of times that somebody has told me that they are “not good at math,” “not a math person,” or “not good with numbers,” etc. Likewise, many people decide at some point that they are “not musicians”, “not musically inclined”, “don’t have an ear”, or “don’t have rhythm”. While it certainly is true that various mathematical and musical concepts come easier to some than others, it is rarely the case that someone’s learning difficulties are insurmountable. Desire, patience, and willingness to practice are the only prerequisites – not natural talent. I was certainly not a prodigy in music or in math, but my love for both subjects guided my study.
There is, however, one tremendous disadvantage to teaching someone math over music: even the uninitiated innately derive enjoyment from music in a way that most certainly do not from mathematics. I argue that most people’s general dislike of math comes more from the way it is taught than from something inherent in the subject itself. Despite what you may think, I find that math (as much as music) is an art-form, and there is such thing as “playing” with math. For me, one of the most articulate (and entertaining) presentations of this argument comes from Paul Lockhart, a Mathematics teacher at Brooklyn’s Saint Ann’s School. Check it out: A Mathematician’s Lament.
Our education system turns a lot of students off to math as a boring and impersonal subject devoid of creativity. On the contrary, the pattern finding, structural organization, and reasoning that comprise true mathematics make it one of humanity’s most original and creative pursuits.
Speaking of humanity, most people forget or fail to realize that the rules, definitions, patterns, and conclusions that we dub “Math” came from actual human beings. When possible, I try to engage students with math through its history. Very clever people all over the world have been doing math for thousands of years before things like Algebra or even our number system occurred to them. Schools dump hundreds of generations of difficult conclusions onto the heads of our students as nothing more than a load of obvious “facts”. Routine tasks for middle school math students such as: “find the slope of this line” are as recent as the 1600’s. Calculus is a little over 2 centuries old. Regarding his accomplishments (among them, inventing calculus), Isaac Newton humbly bragged, “If I have seen further, it is by standing on the shoulders of giants.” As math educators, we must ensure that our students are aware of the shoulders on which they stand so that they can see even further than the generations of mathematicians before them.
Many math teachers refute their students’ objections that most of the material is irrelevant or useless. That the material presented as math is largely a bunch of conclusions to be memorized, I agree with them! If students were never interested in the process of discovering these patterns, they will quickly forget these conclusions as adults and never give them a second thought. If, on the other hand, students gain an appreciation and an understanding of the process of arriving at these conclusions, they will stick with them forever. A calculator today can solve quadratic equations. It is much more important that a student appreciates why there must be a quadratic formula in the first place and what it means.
One of the ways in which students can be persuaded to “play” with math is through puzzles. I have been studying and collecting puzzles of several varieties for longer than I have been teaching math. Puzzles require of a solver the same sorts of patience, mental abstractions, and organization of thoughts that mathematics demands. Helping students with puzzles makes them better problem-solvers, which in turn makes them better math students.
As with the teaching of any subject, the interests of the student must guide the study. Some people may need to approach math as a tinkerer: simply learning how to use it for the time being and figuring out what it all means later. Others prefer to approach math philosophically in order to feed their insatiable “why???” with convincing proof. Some people simply enjoy math’s mechanics and rules their own sake. Some get whisked away by the applications of math in modeling things in the “real world”. Still others get intrigued by the historical human aspect to math and get turned on by thinking about the people who dreamed up these crazy things. None of these approaches is “more valid” than another, and different students require one or more of them to really get interested in the subject we drably call Math.
Having taught in the classroom myself, I can attest to the immense difficulty of making the math classroom as engaging and exciting as I have painted above. Still, it is not impossible, and I personally know several teachers who manage. Private tutoring, on the other hand, allows teachers to experience and guide the thinking of their students as they go. Tutoring makes it much easier to dislodge misconceptions, root out any confusion, and hopefully find the spark that lights the students’ interest once and for all.