Note: This post originally appeared on my personal blog on January 24, 2014 and can be found here. At the end of this post you will find a question related to the following story–I would love to hear your thoughts, ideas and responses. –Malke
I’ve been thinking quite a lot lately about the role of physical objects in math education. Sometimes called manipulatives or, more generally, tools, I’ve discovered conflicting opinions and strategies around the use of such objects. In her book Young Children Reinvent Arithmetic, Constance Kamii helpfully sums up some of the issues with which I’ve been wrestling:
“Manipulatives are thus not useful or useless in themselves. Their utility depends on the relationships children can make…” p25
“Base-10 blocks and Unifix cubes are used on the assumption that they represent or embody the ‘ones,’ ‘tens,’ ‘hundreds,’ and so on. According to Piaget, however, objects, pictures and words do not represent. Representing is an action, and people can represent objects and ideas,but objects, pictures, and words cannot.” p31
So, it is not the object itself that holds the math, but rather the process in which the learner uses the tool that creates the meaning. But, of course, when we use this kind of language we are talking abstractly about hypothetical objects and generalized characteristics of ‘the child,’ not any specific object or individual learner in particular.
Too much generality and abstraction drives me crazy so imagine how pleasantly surprised I was when this showed up in my mailbox the other day:
What is it? Well…it’s an object. And a beautiful one, at that. An object that can be “manipulated” (the triangle comes out and can be turned). A thinking tool. It was designed and created by Christopher Danielson to investigate symmetry and group theory with his college students. Not only are parts of this tool moveable, but it also has the potential to help “facilitate [mathematical] conversations that might otherwise be impossible.” (Christopher on Twitter, Jan 17, 2014)
What was even better than getting a surprise package in my real life mailbox containing a real life manipulative (not a theoretical one) was my (real) eight year old’s interest in and reactions to said object.
She spotted the envelope and said, “Hey! What’s that?!” I told her that a math teacher friend of mine had sent me something he made for his students to use. I took it out of the envelope for her to look at.
First thing she noticed was the smell — lovely, smokey wood smell which we both loved. She investigated the burned edges, tried to draw with them (sort of like charcoal). This led to a discussion about laser cutters (heat, precision) and the fact Christopher had designed it.
I pointed out the labeled vertices on the triangle, showed her how you can turn it, and mentioned that the labels help us keep track of how far the shape has turned. She immediately took over this process.
She repeatedly asked if she could take it to school! I asked her, “What would you do with it?” She said, matter-of-factly: “Play around with the triangle…and discover new galaxies.”
Then, she turned the triangle 60° and said, “And make a Jewish star…” Then she put the triangle behind the the opening so it (sort of) made a hexagon. I asked, “What did you make there?” She said, “A diaper.” Ha!
I hope Christopher’s students are just as curious about and enthralled with the “object-ness” of this gorgeous thing as they are with the idea that it helped them talk and think about things that might otherwise be impossible to grasp. I know that the objects themselves hold no mathematical meaning but watching how intrigued my daughter was with Christopher’s gift, I am left thinking about what we miss out on if we consider a tool simply a bridge to the ‘real’ goal of mental abstraction.
Beautiful and intriguing objects, I think, have a role in inspiring the whole of us, all our senses, kinetics, and curiosities, not just our minds, to engage in the process of math learning. An object doesn’t necessarily have to be tangible; narrative contexts are highly motivating ‘tools’ when working with children. As I blend math, dance and basic art making I see over and over again how presenting the object (idea) first pulls my learners in — they are curious about what this dance is, how they might weave their own wonderful designs using math, what does she mean “growing triangles” and why are these pennies on the table?
“Beautiful and intriguing objects, I think, have a role in inspiring the whole of us, all our senses, kinetics, and curiosities, not just our minds, to engage in the process of math learning.”
There’s no better place to start than in a Montessori environment! “On the things you will see, feel, smell, hear, and taste!”
YEAH for “classrooms and homes filled with mathematical objects and things to wonder over.”
Great post, very thought provoking.
In high school I was quite interested in 3-D surfaces. I would graph various formulas on a pen printer they had at Indiana University. With that in mind I spent a weekend recently building http://formulatoy.net to graph surfaces. (I had in mind that this might help with some math tutoring in the future and that others would find it as compelling as I did.)
Now before you say: “wait that’s cheating – we’re talking about tangible physical objects”, take a look at FormulaToy. It has 2 settings to control the lighting of the object, you can define the objects shininess and transparency. It can look like metal or like plastic. The object spins in space. So it really looks and feels like an object, just one you can’t touch (or smell).
I should spend a weekend adding the ability to export the object as a STL file that could then be printed on a 3-D printer. Then we really would have an touch-able object.
Here is a gallery of math objects: http://www.pinterest.com/blwoodley/formula-toy/
BTW, Modeling surfaces as objects goes back a ways. See this paper on “Models of Surfaces and Abstract Art in the Early 20th Century”: http://bit.ly/13CYwPG
Hi Robert! Thanks for your comment and question. Here’s one back at you: What would you say to your formula toy app being called ‘an object to think with’? To me, the app is an object that creates other math objects to think about…which I think is VERY cool.
I suppose that perhaps ‘curiosity’ is what drives this conversation. A beautiful object can be anything (even a dance demonstration) which provokes curiosity, new questions, brings the learner toward it. It’ll be different for each individual, but it’s clear there is precedence for this phenomenon.
Thanks for sharing your thoughts!