Despite the thrill of discovering how this kind of thinking is used in my program Math in Your Feet, I was left with lingering questions about the differences between identifying attributes and using attributes in a design process. I suppose they are two sides of the same coin, but I can't help thinking that being able to choose from an inventory of possibilities is the preferable skill-building activity in the long run.
I tested my theory on myself and my six year old daughter by using the Symmetry Artist at Math is Fun. Maybe you've seen this before? Check out all the attributes you can choose from to make your symmetry designs:
You can choose between reflection or rotation symmetry, with six to eight choices in each category.
Your pen has five choices.
And many, many colors.
There's a lot to choose from here, but the tool is easy to use, so you can easily change your mind and start over, or take out your last move or series of moves.
While you experiment and play around you are also noticing relationships between the lines and shapes you make and observing the structure of the final design. The addition of color serves to increase the complexity and interest in both the process and the product.
I suppose the difference between identifying attributes and using attributes is that one is a more closed process than the other. I'm thinking that with a set of attribute blocks, for example, although you are learning to discern differences and similarities, there are really only right and wrong answers. The benefit of this kind of activity is that you are actively using math vocabulary: thick/thin, large/small, circle, square, triangle, edges, etc. This is all very useful but, as I said, a somewhat closed process in terms of inquiry.
In contrast to 'compare/contrast/identify' there's the process like the one you use in Symmetry Artist. Instead of simply identifying attributes, you are using this skill in context while thinking mathematically in an active way -- you are actually 'doing' mathematics. By this I mean you are asking questions ("What would happen if I started my circle here? What would the same design look like with nine iterations instead of four? How'd that pentagon get there?"), experimenting with and analyzing your 'answers' (designs), erasing your answers, starting over, printing out the answers you like, asking more questions.... This is the creative process in action; it is also mathematics in action.
I love Symmetry Artist because it is a beautiful and fun way to play around with, learn about, and compare how lines, shapes, and iterations interact within these two symmetries. Below I've put just a few of the designs my daughter and I recently made.
As a first grader, my kid uses this tool primarily for exploration. I started by explaining the different categories, but not much more, and she jumped in from there. I, on the other hand, being aware of just how many choices I had, jumped in but soon got overwhelmed with too many questions which led to too much erasing and/or starting over, resulting in pretty much nothing to show for my efforts! That didn't stop me from noticing a few things, however. Take a look at what we did, see what you think and then go try it yourself!
Here's my daughter's first design. Notice that she set it for rotation, 'eight', pen, one color, and a medium thickness. She 'drew' with abandon, and I was thrilled with what resulted, mostly because I would have never thought to do it that way! As you'll soon see, my initial approach was a bit more measured.
Her second design was inspired by the first. "I want to draw a sun!" she said. I don't know if you can see it, but she started with 'four' red, then 'nine' red, then 'nine' yellow.
Here's the one I made as an example to show the kid what you can do with the tool. I used 'five', circle, and some variations in thickness and color. I love the almost-pentagon in the center where the five big circles cross.
The one thing during my experimentation that really thrilled me was that I noticed a difference between the process of drawing a design with a line of symmetry compared with what happened while using 'two' rotation. Can you see it?
I know what it looks like, and how to describe it, when a two-person team transforms their percussive patterns using both kinds of symmetries, but there was something about using Symmetry Artist that made it stand out to me in a different way. You are actually drawing one design and the multiples show up automatically. There was something about seeing it happen in real time (moving patterns!) that I would not have noticed if I had been drawing it by hand, one reflection or rotation iteration at a time. If I had been doing it that way (pencil and paper) I probably would have noticed something else altogether.
This is all just more evidence that you really do need multiple opportunities to observe and work with a math concept in a number of different situations to really understand it. In the end, I think that both the act of identifying attributes and and the act of using attributes in a design process (choreography, visual art, tangrams, etc.) have something to offer each other.
Because I am a teaching artist, dancer and musician, I am coming at math education from a different direction than many. I am also a big picture learner, so putting math in context makes a great deal of sense to me. What I am still thinking about are things like: When is it important to just teach math as math? When do you move on to reflection on and representation of the math you've discovered during an open-ended exploration? These are the questions that quite literally keep me up at night, and one reason that it's taken me so long to get this post the way I wanted it. I suppose I may be overthinking things, but I really am curious about all this.
What's your experience with attributes? I'd love to hear what you think about all this!