Mathematics and music have a long and storied history together. Extensive writing has been done on the relationship between these disciplines (see above for example). To this extensive writing I have contributed absolutely nothing, so the point of this post is not to offer any ground-breaking new ideas on the matter. The point of this post is to accomplish three things:
- To introduce, albeit briefly, the unique bond between these subjects for those who might not have been aware and provide links for further inquiry. Check.
- To offer some ideas (and hopefully get some back) for those interested in taking advantage of their students’ musical interests in order to engage them in mathematical thinking.
- To promote what I feel is some top-notch, musically creative, theologically deep and meaningful, Christian music that is currently being produced.
A Math Music Lesson Plan
If there are two things I can say with certainty about my students in general it is this: they don’t like math, but they do like music. They will quickly become bored with a worksheet and, as their mind drifts to other things, they will almost subconsciously start laying down beats of popular songs on their desks. So I ask myself this question: how do I tell my students that what they are doing has equal (and in reality probably greater) mathematical value than their worksheet without receiving a look back from them that implies I need serious mental help?
I offer up this lesson plan as a possible first step. I’m afraid I don’t know enough music to make this lesson as dynamic as it could potentially be. Hopefully in my free time (I’ve heard such a thing exists and I’m looking forward to experiencing it someday) I’ll be able to glean some insights from that extensive writing and make some improvements and additional lessons to exploit a connection that desperately needs exploiting. If there is anyone out there with their own suggestions, they would be greatly appreciated.
In case you are interested here is the typed lesson plan with student handouts.
Sample Lesson: Amplitude, Period, and Applications of Sinusoidal Functions (Pre-Cal)
Objectives/TEKS: The students will…
- …define the amplitude, period and frequency of a sinusoidal function.
- …discover how modifying the amplitude/period affects the graph of sine and cosine.
- …correlate their understanding of general transformations from Algebra II to the specific transformations of amplitude and period.
- …apply transformations of sinusoidal functions to sound waves and musical notes.
- …learn to value the influence mathematics has on discussions of beauty in music.
It is good to give thanks to the Lord, And to sing praises to Your name, O Most High; To declare Your lovingkindness in the morning, And Your faithfulness by night, With the ten-stringed lute and with the harp, With RESOUNDING MUSIC upon the lyre. For You, O LORD, have made me glad by what You have done, I will sing for joy at the works of Your hands.
- Unit Circle that the students have been assembling throughout the semester
- “Audacity” audio software, laptop, and projection device
- Handouts: student notes and worksheet
The students will be introduced to the “Audacity” audio software. This software displays a graphical representation of sound waves for any given piece of music. This software can also play a single note and display the corresponding sound wave, which results in a perfect sinusoid. The students will note that sound waves are modeled by a sinusoidal function. (In the previous lesson the students used their unit circles to derive the graphs of the sine and cosine functions).
Various notes will be played with the similarities and differences in their graphs being discussed. The question will be posed: how can we modify the way we write sinusoidal functions to model these changes that we observe in the sound waves of various notes?
Introduction of New Material:
The students will be presented with the terminology of amplitude and period as descriptions of sinusoid graphs. The students will then discover how amplitude and period are affected when constant coefficients A and B are introduced into the expression for the sinusoidal function: y = A sin (Bθ). The students will do this using pages one and two of the attached notes, completing a table of values and then sketching the graphs.
The students will then use their understanding of amplitude and period to graph several sinusoids without a chart of points – considering the graph purely as a transformation of the sinusoidal parent functions. This can be seen on the third page of the attached student notes.
The students will be given several problems on a worksheet similar to those presented in the guided practice section.
Returning to the “Audacity” audio software, the students will be asked to apply their knowledge of amplitude and period to their previous descriptions of how the sound waves varied between different notes.
For example, at the beginning of the lesson when the students were asked to describe the change in the sound wave that resulted from a move from an A to a G, answers will fall along the lines of: “The graph has more repetitions” or “the graph repeats itself sooner and more often” or perhaps even “the graph repeats itself more frequently.”
Now at the close of the lesson, the students’ answers should fall along the lines of: “The period is decreased/increased” and “the frequency is increased/decreased.”
Sample questions can include:
- If we modify the frequency (inverse of the period) of the sinusoid, how does that change affect the musical note that is produced?
- Based on the answer to (1), if we want to produce a higher note, how should we modify the period? If we want to produce a lower note, how should we modify the period?
- What can we glean from the fact that a perfectly played musical note produces a smooth and symmetrical sinusoid, while an off-key note does not?
- Can we classify certain music as good and certain music as not good? In other words, is there such a thing as objectively good music? How does math contribute to this discussion?
If you have never heard of Josh Garrels, do yourself a favor and check out his music. In a world of contemporary “Christian” music that produces a lot of fluff (for lack of a better word), Josh’s music stands out as being both theologically literate and musically creative. I’m sure you’ll catch me using phrases from his songs in various postings.
Other Lectures on Math and Music
I’m an aspiring music composer, and from what I find music isn’t just paired with math. More importantly, it is a bridge between pure linguistic arts and science. Just as math remind me of music, literature does too!
Yet whenever I compose, time and time again I come to the question of math. This isn’t as apparent when I write traditional, conventional music. But when I experiment with more radical and unconventional approaches, frequently I use math to make sense out of my work and thematically tie every bit together. Apparently many of minimalist and serial composers were in fact more at home as math professors.
But yes, thank you for such an inspirational and education-oriented blog! If only my math teachers are as goal-driven as you are…
Will regularly visit!