This week, I read the article "Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know" by Davis and Simmt. The article develops a theoretical description of what mathematics teachers (need to) know from various workshops they conducted with in-service mathematics teachers. The teacher workshops were conducted to help teachers gain a more nuanced understanding of the mathematical knowledge at stake within the mathematics curriculum, but also acted as a research opportunity for the authors, to see how the teachers learned "new" mathematics. The authors used complexity theory to interpret their results, although the discussion of this theory as a method of interpretation in research would require a completely different discussion.
The authors identified four major aspects of teachers' mathematics-for teaching. Namely, "mathematical objects," "curriculum structures," "classroom collectivity," and "subjective understanding." (Sidenote: It was unclear to me how these aspects were uncovered and why there were not any others. I was generally unsatisfied in the authors' description of how these themes emerged in their research.) In the discussion surrounding "mathematical objects" as an aspect of mathematics-for-teaching, the authors note that many of the teachers' were unaware of a figurative notion of multiplication. That is, multiplication as something besides "repeated addition" or "groups of." They were simply unaware that alternative images and metaphors for a core concept of the mathematics curriculum existed.
My question is, how are teachers expected to come to know this? Is making such connections not a key component of building mathematical understanding? Should teacher education programs include deep, mathematical discussions of core concepts?
Short answer: I don't know. I'll express some further ignorance that I am not sure about teacher education requirements, but I do know that `math for teachers' courses are taught in math departments - at least in some cases. I haven't had as much contact with this as I'd like, but I do know students are expected to learn multiple ways of thinking about many `elementary' mathematics topics. But assuming as the authors assert, that most teachers do not have deep understanding, but do have a reasonable degree of competency: how easy is it to learn on the go as new ideas get incorporated into the curriculum? By knowing some well, will that not aid in gaining depth later? After all, we are talking about adding richness to `understanding' that already exists, not some entirely new concept.
ReplyDeleteI suppose it comes down to what grade level we're talking about. When multiplication is first introduced in grade 3, I fully believe that students only need to understand it as "groups of" or "repeated addition". I am curious what other deeper understanding of the definition of multiplication the researchers are alluding to. Perhaps in grades 6 and 7 when students begin to learn about multiplying negatives by negatives, there is some room for the development of a deeper understanding.
ReplyDeleteFrom my experience, the math ed courses were taught in the math classrooms of the education building. I only received one math class in my teacher training, and it was only for half a year. We really only had time to pick up a few interesting ways to introduce topics, and we were able to practice teaching a sample lesson here or there. In order to develop a deeper understanding of any topic, teachers must take it upon themselves. They must sign up for workshops during pro-d days. Unless it becomes mandatory, the only people who are going to possibly look into developing these deeper understandings are the teachers who have a real passion for teaching math. In high school, this can be expected as it may be the only subject a teacher teaches. However, in elementary school a teacher is expected to teach 8 subjects. I can think of many teachers who would not necessarily place math at the top of their priority list.
I believe that the only way we can ensure deeper understandings of mathematical core concepts is to integrate them into the curriculum, and invest in resources (printed/on-line textbooks, software, manipulatives, etc) that support this curriculum.