One of the greatest things I took from today's reading on Mathematical Thinking by Barbara Ball was the idea of mathematical thinking vs. mathematical knowledge. From our previous class discussions, I think that many of us have been hung up on our lack of mathematical knowledge and we were concerned about how we can teach math without that strong foundation. Today's reading and class discussions have served to make me feel more comfortable with the not knowing. I am not saying that we as teachers should just "wing it" and not know how to do the math that we are trying to teach. What I am saying is that if we are trying to promote mathematical thinking, then we can worry less about the answer and more about the thinking and process to get us to the answer. In this digital age of Google and the internet, answers are but a few clicks away: If I want to know almost any formula, all I have to do is put it in a search engine and wait for the information to be displayed on a screen. However, the thinking that goes into knowing when and how to use that formula is more difficult to know. That is mathematical thinking: not just trying to get the answer, but thinking about ways to arrive at an answer.
I feel the the mathematical processes that we discussed today go hand in hand with mathematical thinking. It is through the use of the processes that we enhance our thinking and strengthen our understanding of math and how we become better able to use that thinking in larger contexts, which is the "independent thinking" that Polya speaks of.
My goal going forward is to not only challenge my students to think mathematically, but to also challenge myself, to learn to be comfortable with the discomfort of no one right answer and no one way of arriving at an answer.
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