Challenging a Literal Approach: Learning Fractions 'through' Music

The last week or so I’ve been thinking quite a bit about what makes real connections between math and dance in a learning setting.  I’ve come to the conclusion that none of those connections can even be attempted without first looking at our assumptions about what math is and what dance is. 

I’ve been sharing my thinking on this blog, to an audience which (I assume) is more familiar with math learning than with arts learning and over at ALT/space, an online writing project I edit and curate for the Teaching Artist Journal.  I’ve not had a huge amount of responses, but the quality of the feedback I have received has been incredibly helpful in moving me forward on this line of inquiry.

Most recently I had a great conversation with friend and colleague Nick Jaffe.  Nick is, among many things, an incredible musician, teacher, thinker, writer and editor.  He’s not a math educator, but he’s definitely not math averse. 

We were talking about how people often try to teach fractions by connecting it to musical notation, which I have always seen as a very literal approach to what people perceive as music (written notation).  Over the course of the conversation I finally understood why I had always viewed that as a very shallow, un-meaningful activity.  Specifically, Nick has a lot of interesting and helpful things to say about discrete and relative quantities when playing music so I thought I’d share our discussion here in case it is helpful to you in any way.
………………………………… 

Me:  I have seen very little evidence that making the correlation between fractions and music have had had any impact. Maybe one study, but they didn't let the kids actually play around with ideas or make their own music -- they just taught them how to play the fractions. And, seriously, do you think like that when you're *playing* music? 

Nick: When people ask me about music and fractions, the first thing we need to establish is that the analogy between musical notation (or the musical reality it describes) is multi-dimensional. Notation denotes both the note as a discrete duration, but also as a division of a larger whole. Which, as I understand it, is also in the dual nature of fractions and an important thing to grasp, and not entirely intuitive.

What seems essential, at least in elementary and middle school math, to understand about fractions is that they have a dual function mathematically--they are a way of signifying two things at once: a discrete quantity, and a relative one. Musical notation functions the same way rhythmically.  In both the abstract, mathematical context, and in the context of music, the same problems arise with regard to this dual nature.  And it is somewhat counterintuitive to students at first to consider this--it's never explained clearly but it's the crux of everything one does with fractions as far as I can see.

I like the question you raise about how one thinks when making music. I think the answer is yes and no. Let's leave aside conscious methods of fractional or theoretically driven music making. Let's consider improvisation which would seem to be the most spontaneous, least theoretical approach to music making. And let's just stick to fractions as a rhythmic concept for this particular argument.

When I improvise I do not consciously think about the duality of a note as a time duration (discreet and part of a larger repeating whole). However I absolutely have to manipulate time, consciously and/or unconsciously as both things.

I think that duality is at the heart of choices about phrasing, for instance. You have to feel a note as a discreet duration, an almost word-like gesture. But you also have to feel it in relation to a pulse, even if there is no pulse, or you are trying to erase any implied pulse--it's still in reference. One does not have to consciously think in terms of fractions to do any of that. However, learning to read notation, and practicing manipulating that duality (playing behind, on, or in front of the beat for instance) can often increase one's control over it.

Finally, analyzing, generalizing and theorizing about that same duality can open up new options, or allow the translation of one dynamic application (actual playing) from one musical context to another.  In that sense theory is both a stimulus to new ideas and means of translating an idea for different contexts. I think theory plays those roles in most disciplines--a generalization that makes possible the prediction of as yet unknown dynamics. 

Me: The problem with using music notation to teach fractions is precisely what you just explained.  The written notation is the quantity, and the actual musicality/music playing is the quality.  What people don't get about mathematics is that 1) it’s not just about the notation (same for music) and 2) it's really about the quality of the quantities.  If you focus just on "how much" you get a dead language or, at the very least, information devoid of meaning. Which is why so many people hate math – for many reasons, and I’m speaking very generally, math education has essentially been bleached of its meaning.  

Nick: I’m actually not that interested in the idea of using musical notation to teach fractions. That seems boring and highly inefficient. What I am interested in, and I think kids might be as well, is working with music and fractional ideas at the same time to make interesting things. Perhaps those things would be musical, perhaps mathematical, perhaps visual, perhaps both.   Undoubtedly one would learn various things in doing such work, but I think it is the making that is what is appealing to me (and perhaps students) and that any real insights and learning depend on the impulse to explore in order to make. 
…………………
 To sum things up, Nick’s thinking about the nature of a mathematical idea (in this case fractions) in relation to his art form is exactly the kind of thinking process I engaged in to find a meaningful overlap between math and percussive dance in Math in Your Feet. It is also a perfect example of how it is possible to teach parts of math starting with your own experiences inside another discipline.  

Where to go next? One of the best music, math and movement resources I’ve run across is from Ellen Booth Church. The activities described in this article describe perfectly how the core concepts of mathematical activity can be connected to musical and movement activity.  Ellen is an early childhood specialist, so her ideas are based on that age group and applicable up to about grade 2, but I think this is the perfect starting place for anyone interested in making these kinds of connections in their own classrooms.