As math teachers, we need to stay vigilant about how we represent our subject to our students.
At her recent workshop for teaching artists here in the Twin Cities, Malke Rosenfeld said to the gathered group,
It would be fair to say that most of think about math inside a textbook context.
She paused.
Heads nodded, eyes wide with recognition. Malke prepared to demonstrate that this is not the full story.
Math comes from, and lives within, textbooks. I am not OK with this.
So what can we do in every lesson every day to represent mathematics as a subject that comes from, and lives within, the minds (and bodies) of our students?
Overthinking my teaching 
I have been teaching for over 20 years. It wasn’t until about four years ago that I finally escaped from the textbook. I began to think about what each subject (algebra, calculus, etc) means to me, and how I think it should be organized. I still use textbooks – it seems necessary at college level – but mainly as repositories of homework problems. I explain in a way that makes sense to me, and sometimes we compare my version versus the book author’s version.
Today I shared a poem I wrote about imaginary numbers with my pre-calc students. (Actually I’m not sure poem is the right word. Would you call this a poem or not? I’m curious.) I thought it was a good way to introduce a concept that often leaves students cold.
Have I reached Malke’s beautiful standard of embodied math? Nah. But it’s in my head as a goal, whenever I can make sense of it.
Malke, here’s one body thing I do that probably doesn’t count. The graph of y=x^2 is a parabola. I like to think of it as having both arms up. I like to think of the graph of y=x^3 as having one arm (left) down and the other (right) up. We are working with graphing more complicated polynomial functions, and I ask them to show me (what I call) the big picture with their arms. I’m hoping to get them to hold this visual information more firmly in place while they work through the detailed parts, so that their graph will reflect their thinking on both the big picture and the details. I’m using body movement just as a cuing device, really.
Stop teaching the totally crappy way the textbooks and our educational materials present it. Throw the danged things out and create our own. I’m sitting here dragging a student through utterly incomprehensible problems on a 50-problem assignment that is a mess of assorted creations, most of which really belong in the next semester.
Sue, I think making sense of what math is in the body is a lengthy ongoing proposition for most of us, even the dancers. My goal is to at least raise some questions: Is the graph the math? Are the arms doing the graph the math? My answers: The graph is the representation of the idea and the arms are the representation of the graph. This is why my blog is called ‘The map is not the territory’. Those teachers Christopher mentioned are part of the legions of people who, by no fault of their own, mistake the textbook (map) for the math (actual experience of thinking and doing math) because they’ve never had a chance to travel the actual terrain of math land. Personally, I think there’s no problem with a ‘cuing device’ or a mnemonic as long as you are aware of where it lies in the continuum of math learning and your teaching goals 🙂
As a physics and chemistry teacher for the past 29 years I am amazed at how few students can solve a mathematical chemistry or physics problem with out being told what to do. “Give me the equation to use and give me numbers to plug-in” Science teachers hate the term “plug-in.” Even if the textbook has an equation they still aren’t sure what to do because in science we rarely use “x” and instead use variables such as m for mass or v for velocity. My students come into my class as juniors and seniors and still believe that algebra is all about solving for 6 letters of the alphabet (a, b, c, x, y and occasionally z) and nothing else. It is my belief that this lack of practical application, and only teaching the theory throughout much of the U.S., is the primary reason why we lag behind much of the developed world in mathematics. I know of many math teachers who are proactive to this change and practice applied math in their classes with their students, but unfortunately it does not appear that the majority of math teachers are.
I am a math education fresh graduate..
The first time I went for internship in a school near by my campus, I found most of the students experienced math anxiety..
As I discussed more with them, a student asked me, “what does actually the math concept for? I mean, logarithm, integral, derivative, and any other complicated materials.”
Here, I realized how context should be there in our teaching math. At least, to emphasize that we’ll give something useful for students through this learning math.