I found a brief article (actually the transcript of an introduction to a faculty discussion about humanistic mathematics) by Gizem Karaali, an Assistant Professor of Mathematics at Pomona College. This particular faculty discussion defines humanistic mathematics as "the human face of mathematics". Karaali makes many interesting points during his introduction, such as: those in different areas of mathematics "really do not talk to one another", the terms that groups of mathematicians use to discuss their respective specialties often creates a language barrier, and "our goal is nothing short of breaking the barriers between these allied disciplines". The goal of the faculty discussion is to create ways to "encourage scholarly work that transcends disciplinary bounds" and to brainstorm about "how can we integrate the arts and the humanities into a capstone mathematical experience for our students? Conversely, how can we integrate mathematical experiences into a humanistic classroom?"
This article can be found at www.aacu.org/meetings/ah11/documents/CS34.pdf
I love music and welcome any opportunity to integrate it into mathematics instruction. I already discussed this a bit in this week's gathering, so I apologize if I'm repeating myself (and also apologize for being rather talkative to begin with). Also, this is only a rough idea of something I hope to do someday; any suggestions about my 'fantasy lesson' would be appreciated.
For younger students, I would explore fractions by integrating music into a music lesson called "Is it possible to hear a fraction?" Students who are already studying an instrument might find this initially redundant, so I would encourage them to help me lead the discussion and/or bring their instruments in to provide examples. To begin, the lesson would demonstrate the relationship of the 'beat' (time signature) to fractions (hopefully there would be a percussionist in the class to demonstrate as rhythm isn't my strong point). We might write out beats of songs in fractional form (express music as a function of time). As students become familiar with this concept, we will discuss how musicians who play a song together intuitively use math to make their fractions (notes) 'match up' though each musician is working with a different 'set' of fractions (percussion vs. keyboard vs. vocals, etc., this is much easier to hear than write about; listen to your favorite music and you'll know what I mean). Again, students would be encouraged to either perform or provide their favorite music as examples.
The lesson will take an interdisciplinary approach when we explore how fractions (in the form of time signatures) are often associated with particular dances (3/4 for waltzes, 4/4 for the nightclub dance scene, 6/8 for an Irish jig); dance students (or instructors) would be encouraged to comment. Following that, we will discuss common mathematical 'fraction relationships' that characterize particular genres of music. At this point, I would love to co-teach with language and/or history teachers to further explore how culture, the spoken words of language, mathematics (via fractions), socioeconomic position (similarities/differences between those who composed assisted by instruments vs. those who didn't have that choice) and the desire for aesthetic enjoyment combine to form what we call 'music'.
For older students, I would use this framework (and encourage student contributions), but also include a discussion about how wavelength influences the sounds that humans perceive and explore the basic mathematical principles of sound, which would be great co-taught with with their science teachers. The discussion of fractions would be expanded to include an introduction to harmonics and how these fractions influence the sounds we perceive (guitarists and bassists would probably be happy to give a demonstration). Musical intervals have a mathematical basis; this and their effect on humans' aesthetic enjoyment will be explored and combined with all of the topics of my 'younger student' interdisciplinary unit.
I know this is WAY wordy, thanks for bearing with me and reading through.
I'm going to answer this question as a teacher whose curriculum is largely geared toward producing 'acceptable' standardized testing results (such as those required by NCLB or state agencies).
We need to teach students not only how mathematics is integrated with other disciplines, but also how 'society' might separate it, as this is what often happens with standardized tests. Therefore, students need to be able to apply - and demonstrate - their math learning in both 'forward' and 'reverse'.
Parents are often quite anxious about standardized testing - it begins in elementary school and continues right up to the SAT's (or ACT's). Teachers need to take the time to teach (and reassure) parents about the benefits of a nontraditional, interdisciplinary approach. However, this reassurance can only be validated by the student's ability to demonstrate their learning in both traditional- and non-traditional settings.
Therefore, I think it's essential to ensure that students can demonstrate their learning in either 'forward' or 'reverse'. At this time, I don't know of any way to accomplish that other than having students practice mathematics in both settings, and assess them in both settings so that they will get an idea of what the 'rest of the world' is looking for as far as future employees.
Students who are able to experiment with both approaches may find they enjoy one approach more than the other; this realization may be very valuable when the student must choose a career path.
My 'fantasy lesson' could use tech help in many ways. A metronome is essential. Rhythm tracks can be computer-generated; such software is readily available. Software that compares audio input to a mathematical ratio (time signature) is also readily available, though this might create a high-pressure situation if students are involved, so I probably wouldn't ask students to demonstrate this (they'd probably get a kick out of seeing how their music teacher measures up, though). Software that translates audio input into graphic depictions is also available; students can actually visualize sound, which reinforces the connection between what we hear and mathematical functions that we can see). The user can also alter sound by manipulating these graphics. Sounds from different audio inputs (saxophone, drums, anything) can also be synced (either if the musicians don't/can't or if the user (such as a dj) wants to create something new). Students can explore different outcomes without actually having to perform the piece (or ask someone else to perform it). Students would also be taught about software such as Auto-Tune, which combines math and programming to produce a desired sound.
In closing, I want to thank my 8th grade General Music teacher, Ms. Engler, whose classes were the inspiration for my lesson. She motivated us to practice good behavior by offering the reward of devoting one 45-minute class period at the end of each month to discussing whatever music students brought in (or wished to play on their instruments). I'll never forget how surprised I was when she demonstrated that some songs by Pink Floyd and Teena Marie had the same time signature (I was a classically-taught violin player who was taught that 'popular' music had nothing in common with classical (wrong!) and that thinking outside the box would certainly result in musical destruction). Despite those 'words of wisdom', I'm still here, learning ragtime, bluegrass, and Irish, and I'm not ashamed to admit that making fractions add up to a whole on paper is always easier than making your two hands cooperate and 'add up' when playing against a metronome. I would never ask a student to perform in class unless they were completely comfortable with the possibility of failure; as teachers, we need to reassure all students that failure (or possibility of) is an essential part of learning. As an education student, I look back on how Ms. Engler integrated her unit on jazz history with Black History Month (and what we were learning about in English and American History) and I am in awe of her organizational skills and commitment to making sure that her students learned about an essential part of music history and its relationship to the socio-economic-political climate of the time. She gave us 100% and then some, and related it all to math by means of fractions (time and intervals).