Music educators have all said this at some point: “I already teach math in music class! We do fractions every time we talk about rhythms, meters, and time signatures.” However, if your experience is anything like mine, that argument eventually broke down when an administrator asked me to list what grade level math standards I was covering. It is assumed that students should already know the fractions involved in rhythm when they reach the secondary level. While our review of those concepts is valuable, it doesn’t meet any grade level standards.
I read the Indiana state standards for math in grades 6-12 multiple times and told two principals and countless other teachers that I thought meeting grade level math standards couldn’t be done within the secondary ensemble curriculum. There is too much algebra and other more abstract mathematics that aren’t involved in reading a music or playing an instrument. Or are they? After about five years of teaching I finally discovered that I was wrong.
Like any decent band director, I was losing sleep over junior high band intonation problems. Am I right?! I was writing a curriculum guide over the summer and incorporating some new ways of teaching intonation when it dawned on me that basic instrument tuning is a real world application of integers. I wrote several math problems based on Indiana math standards 6.1-3 and 7.NS.3, which are precursors to 8.NS.1-2. These address not only positive and negative integers, but their distance from zero in either direction. This is at the root of the questions we hear from our junior high band students about tuning every day: Am I sharp or flat? Do I push in or pull out? How much do I push in or pull out? Thanks to some help from the math, my students are better at answering these questions themselves, which helps us meet instrumental music standard 8.2.1 much more efficiently.
It doesn’t end there. When you get into intervals and just intonation, you can explore ratios (math standard 7.C.8, which is a precursor to 8.C.1). These help explain why some intervals are more resonant than others and why playing a major triad in equal temperament is still technically out of tune, but flattening the 3rd and raising the 5th sounds in tune. You are also teaching some serious science lessons here.
These intonation skills are crucial to helping our bands sound better, and the more the students internalize and own the material, the more effective they become at tuning. I have found that spending ten minutes here and there on the math has helped my students grasp these concepts much better. I have fewer students asking me whether they should push in or pull out. I have fewer students looking quizzically at me when I ask them to raise the third on that minor triad to get it in tune. Instead, I have students getting perfect intonation scores at solo and ensemble festival. Instead, I am looking forward to our upcoming pops concert because I really like how my band sounds. Instead, I am spending less time on tuning, yet listening to my bands play more in tune. I am losing a lot less sleep over intonation problems now than I was a year ago.
Adjusting Instrument Length with Integers
More materials are coming soon.