Today, however, my focus is on the idea about how 'playing outside' an established structure (e.g. a limit) is a way of exploring the limitations themselves, a process which can often lead to new innovations, strategies and ideas. The article Playing outside: An introduction to the jazz metaphor in mathematics education is the inspiration for this post.
Stay with me here...
After debuting my blog in mid-October, an arts friend sent me a copy of an article, as referenced above, from the Australian Senior Mathematics Journal 18 authored by Jim Neyland, Victoria University, Wellington, New Zealand.
I read it and promptly penned (typed, actually) the post Building a Bridge which starts out "I am not a math teacher." Since then I've been revising that particular view, but that's really the subject of a future post.
Today I picked up the article again because I needed something to read while the kid was jumping over alligators in dance class and found some very interesting ideas! This excerpt from the article succinctly pulls together two of this blog's main topics -- connections between the arts (specifically music and dance) and mathematics education:
I read "rich mathematical activity" as something similar to a rich, productive creative process, one where ideas are experimented with, dropped, added, and revised in the process of becoming a fluent thinker within a discipline. I think that happens in my classes when children, after learning the basic vocabulary of percussive dance, are given the freedom to experiment and reach a level of comfort and fluency within a relatively short time frame of five days.
"It is becoming evident that experienced [math] teachers operate in a way that is better described by what is called a 'complexity' model, a radical alternative to the linear model...The teacher's role in the complexity model is that of an artist; but not any kind of artist. The teacher is not the sort of artist that turns lumps of clay into pottery, or a blank canvas into a painting. He or she is an improvisational artist who participates in the process of emergence, but in a special way. The improvisational teacher uses an 'attractor' -- that is a technical term used by mathematicians when referring to the way some chaotic systems eventually settle to an emergent order, and in teaching can be taken to mean what is called a 'rich mathematical activity'...and watches what happens when the students engage with it."
The author goes on to say,
"Much of what I have been referring to about the complexity model is evident in the way jazz improvisation occurs [...] In jazz, 'playing outside' refers to a radical form of improvisation that deliberately transcends the established [musical] structure [...] Why is playing outside done at all? It creates a high degree of tension. It is a way of exploring the limitations of the established structure. It is a way of keeping the structure secondary to creative improvisation...Playing outside, to put it differently, is playing with [emphasis mine] the structure, not within it as happens in normal improvisation. As such, playing outside is essential in the study of mathematics."In reference to my own creative process of developing the program Math in Your Feet, I'm pretty sure I was 'playing outside' the traditional view of my dance form as I asked questions and, over the course of a couple years, experimented with the best fit between a rhythm-based dance form and elementary math topics. There was actually quite a bit of tension as I did this -- a push and pull around what my role was as the artist in a math context, what kind of ratio of dance to math within my lessons, and whether or not what I was doing was 'real math' or not, etc. In the end, I think I remained true to the spirit and the structure of the traditional dance forms I know and love, but the context for exploring the dance changed.
Also, because of the limited amount of time I have with kids, I'm not sure if I really 'play outside' the structure of what I am trying to teach; at the very least I am 'playing inside' (improvising) the structure of my lesson plans as I respond to the varied skills, experiences, and motivations of my students. In terms of math education, however, the students' time with me may very well be the first time they have had the opportunity to 'play' with math ideas, free from procedural concerns, for a little while at least.
It is possible that I will pick up this article again in another month and see something completely different, but equally as thought provoking. For now, though, I'll leave it here as road marker on the path of this discussion about limits and creativity. As always, your thoughts and feedback are welcome!