I was blown away this summer when Brent Davis shared his research about teacher knowledge and that a teacher having high level mathematics courses doesn’t make them a better math teacher. When I heard that one of our courses over the summer was on metaphors in mathematics, I panicked. I didn’t see at all how metaphors could have anything to do with mathematics and so began the transformation of both my understanding of mathematics and my teaching.
When I started out as a teacher I was highly procedural and didn’t value visual representations of mathematics at all (and now really I understand that was actually me de-valuing conceptual understanding by my students). I think that this has to do with the two points brought up on page 303, (1) Few teachers seem to have been formally introduced to these aspects (including me) and (2) the prevailing belief is that mathematics is a purely logical enterprise. As I continued to read I was able to relate to the comment made by a teacher, “the images and analogies noted above were illustrations”, however through this program I can now relate to the comment, “whereas for us, they are the very essence of personal and collective understandings of multiplication” (p.303).
In the article, a teachers math course is discussed, which I think is brilliant. Reading Lakoff and Nunez (2000) this summer and gaining an understanding of grounding metaphors has allowed me to now be able to look at a students work and/or listen to how they describe their thinking and know how it is that they understand number. For example, if they understand number as count, but we are working on a concept that requires them to use measuring sticks or moving through space, we are speaking a different “number language” than the student is and as a result the student is not going to be able to grasp this concept until they are able to effectively see number through the correct metaphor.
We as teachers only work with a small portion of the program of studies, however the recursive nature of the program, really requires us to understand where students have been and where they need to go so that we are able to add to and transform the thinking of our students. Using VLT we need to be intentional of what we are keeping constant and what varies, so that our students are able to create competent generalizations. If the variety is too pronounced students may categorize, or fragment their learning, which is counter productive to our goal of promoting deep (connected) conceptual understanding.
I also looked at the nested system provided on page 314, through a different lens having read the Gerdes (1985) artcle this fall. Gerdes (1985) talks about the need to value the peoples mathematics, that it is valid mathematics and that we should not be oppressing the people. We can clearly see that culture plays a significant role in the nested diagram. I would argue that this comes into play in the Society, or the Body, which explicitly includes culture (4th nest), but also in the 3rd nest, Collectivities: Social Bodies, Bodies of Knowledge, and so on. Gerdes (1985) discussed this issue within a specific example and how a society should not be set up to value one mathematics over another, but the other key component for us as teachers is to value the diverse experiences that our students are coming into our classrooms with.
Until next time,
Davis, B. & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics teachers (need to) know. Educational Studies in Mathematics 61, 293-319.
Gerdes, P. (1985). Conditions and strategies for emancipatory mathematics education in underdeveloped countries. For the Learning of Mathematics, 5(1), 15-34.
Lakoff, G., & Núñez, R. (2000). Chapter 3: Embodied arithmetic: The grounding metaphors. In Where mathematics comes from (50-76). New York, NY: Basic Books.