**“The essence of metaphor is understanding and experiencing one kind of thing in terms of another.”**

-George Lakoff and Mark Johnson, *Metaphors We Live By*

Why metaphors? What is "mathephorically teaching"?

Mathephorically Teaching is a name that I have given to the teaching and learning in my classroom. Over the last few years, I have discovered ways of facilitating authentic mathematical practice with my students. These ways began with a shift in my attitude on teaching and learning mathematics, and this adjustment was the result of having countless little experiences over the years while teaching. From these experiences, two repeating themes stuck out the most:

1) I realized that understanding is an experience, and since it is an experience, then I need to construct a learning environment where all students can access that experience;

2) That doing mathematics is a creative activity. I had been struggling with a latent frustration with the typical pedagogy--that mathematics is a boring lexicon of secretarial procedures to memorize rather than a practice that requires creativity in order to effectively do.

But, there's a real misunderstanding about what mathematics is and what is required to actually do it. There is also a real misunderstanding of what learning is, in general--though new discoveries in neuroscience are showing us what is required to effectively develop enduring understanding in any subject. Students need to struggle, and in order to understand a concept, they need to know where it came from. In other words, students need to be given opportunities to engage in rigorous discovery of mathematical concepts. But, lecture is the preferred method of teaching and learning in the classroom. And, there is a deep-rooted stigma about mathematics in our culture, which presents more challenges in positioning the subject, so that students believe that mathematics is something that they can do and engage in creatively. The fact that learning requires struggle through independent and personal involvement is a very tough sell. Even harder of a sell is that the content does not have to come directly from a lecture and a textbook--students can discover and author the theorems, themselves, and be part of the narrative of the curriculum.

As I was thinking of ways of changing the activities that students engage in and the overall culture of learning in my classroom, I started constructing metaphors in my mind--just to facilitate this transformation. In a way, I started pretending that I was not teaching mathematics, but instead, teaching other subjects that I enjoy studying. For instance, what if my class was actually a philosophy seminar? How would students be engaging with the ideas that we were learning in that environment? How might this perspective elicit student inquiry? What if my class was an art class? How would students be viewing the solutions they write each night? How would we be discussing them in class? How would we be valuing their problem solving process? What if my class was a Spanish class? Would they start owning the vocabulary, and use it to communicate their ideas, as they would if they had to communicate in a foreign language? Would my students start thinking in mathematics, just as they would be thinking in Spanish? What if my classroom was an improv theatre, and that they were told to tell a certain story and use certain props, but they could take whatever path they wanted to tell it? How would I, as a teacher, be engaging in the story? What would be my role in helping them construct the story that they are telling? How would my students view the activity of mathematizing? What if my classroom was a place where I was studying the theory of knowledge, where I was trying out different ways of engaging my students, and examining their work for authenticity and deep understanding? How would that affect my engagement as a teacher? How would that alter my perception of their unique and varied learning paths?

So I constructed the following metaphors:

1. My class is a philosophy seminar.

2. My classroom is an artists' studio.

3. My class is a language course.

4. My classroom is an improv theatre.

5. My classroom is an epistemology lab.

When I first started thinking of these metaphors, I did not realize the importance of what I was doing. But, now I understand that the structure of metaphor, itself, is a device. It is a mechanism through which a concept can be understood, as its relate-ability to another thing. As George Lakoff and Mark Johnson would suggest from the quote written above, I am viewing the subject of mathematics in terms of other subjects. I view it as philosophy. As art. As a language. And as a consequence, my students become philosophers, artists, and language learners. I view my classroom as other kinds of venues. When my classroom is an improv theatre, my teaching is responsive, in that the lessons may take various paths, depending on the questions my students have and their differing approaches to solving the problems I give them. When my classroom is an "epistemology lab", I am working to better the teaching and learning by habitually investigating the results of my students' engagement in various classroom activities.

The question is, if I think about the teaching and learning in my classroom in this way, will it affect the actual teaching and learning in the classroom? The answer is yes, which is why they are "the metaphors I live by" in my practice.