Keynote Part II

This video could obviously be used to launch a variety of pedagogical discussions: Did the teacher make good judgments? Why was time being used the way it was? We're more accustomed to that sort of use of video. What we're going to try to do right now is something that's a little less typical: We're going to use this video to focus specifically on mathematics. The argument for this focus is that one way to help teachers be prepared to use mathematics in their teaching is to engage them in solving problems of mathematics in the context of classroom teaching. This is not unlike the argument that children's mathematics learning can be enhanced by providing them with significant opportunities to apply and use mathematics in the context of specific, credible problems.

I'm going to propose six different tasks that could be used with teachers. Please work through as many as you would like. You might try them working together with a colleague at your location. Additionally, "discussion threads" have been set up for each task. I encourage you to spend some time working on the problem posed in the task with other conference participants who share your interest in exploring a particular task. Pay attention to the kind of mathematical work the problems entail, and the mathematical reasoning and ideas you are using as you try to address the questions. We'll conclude by talking about the sort of professional education this might provide if we did more problems of this type in the mathematical work we give teachers.

So here are the first three tasks -

Here are three more -

Before concluding, let's return from our separate discussion threads to a single discussion group and talk together about these various tasks and their potential for engaging teachers in learning the kind of mathematics required in the work of teaching.

What I'd like to do before wrapping up is to take these six tasks and to say a few things about the sort of mathematical problem solving that they require and then put a couple of caveats into what we've done together so far.

Let's just look through these six tasks. The following chart summarizes an initial analysis of the mathematical learning opportunities that may be embedded in work on each of these tasks:

If you think about the list I was offering earlier about problems of teaching, the first task about the graham cracker and the dollar can involve teachers in learning to compare representations and investigating the correspondences among representations. It's interesting: They both (the graham cracker and dollar bill) produce rectangles but they're not completely identical as contexts for representing of one fourth or three fourths. It's useful to think in detail about what the salient differences might be. Often, when teachers haven't had that opportunity, they seize one representational context without actually having noticed the subtle differences that there can be between a graham cracker that doesn't, for example, have the association with "quarters" which are inside of the dollar. And yet, dollars aren't something one literally breaks. It's interesting to think through the consequences of these differences even though graphically they look similar. And one can do some useful work thinking more closely about the nature of representation.

The second one, about "voting," may seem to be largely a pedagogical question and it could be difficult to orchestrate a discussion of this problem and keep it from being just that, but there are important mathematical questions here that arise in interpreting students' comments. What are the children actually saying when they talk about "voting"? What are the issues around mathematical reasoning, justification, and proof that might lead the teacher to think carefully about what to do in this situation? Many of us would say, voting is not what you do to decide mathematics and would want to shut it down quickly but the students' comments, if one were to listen carefully, suggests there may be more to hear in what they are saying and proposing to do. It suggests a teacher might be required to do some careful analysis about the nature of collective mathematical work and what it means to reason as a group.

The third task focuses on the question the teacher posed about "one half of six." This is an example of posing a question to scaffold student thinking and choosing specific numerical examples. We give far too little time and attention to the significance of the numbers we pick in the problems and examples we develop. Teachers are frequently in the position of needing to quickly pull together an example, when in the middle of something, and there are big consequences for those choices. More practice in becoming sensitive to and skillful at picking numbers - whether it's on the fly or in advance of class - could be useful. These on-the-fly decisions need to be made by teachers no matter how scripted the curriculum, therefore noticing that this is an important and consequential piece of mathematical judgment, having practice in doing this work, and talking carefully about these decisions seems worthwhile. In addition, it's not obvious that "one half of six" was a good choice so, if it wasn't, what were its problems? What would be a better alternative?

The fourth task, concerning the nature of the question the students were working on, is another example of thinking about specific numerical examples. The class had just worked on fractions of twenty-four the previous day so what was involved in talking three fourths of twelve that might have been different? How did it compare? Might there have been another question that would follow better from the children's work on fractions of twenty-four? If so, what is it? What opportunities and possibilities does it afford? This requires a lot of careful thought about the relationship between three-fourths and twelve - in particular, the relationship of the three and the four to the twelve and the three fourths is worth examining. Again, this task would provide practice with specific numerical reasoning.

The question of what problem to do next (i.e. the fifth task) is still in that same family but a little different. It has to do with thinking about the sequencing of mathematical ideas. "So, what's the next problem?" is slightly different than, "What's the question to ask during class?" It again might involve teachers in some detailed thought about numbers and examples. However, this also provides the opportunity to consider what might be the next mathematical question that follows from three fourths of twelve. What's the next kind of problem?

And, finally, looking closely at students (the sixth task) is also a way to work on mathematics for teaching. When you ask a question like, "Can you describe something that Keith seems to understand?" it requires the development of more insight into the mathematics: What is there to describe about Keith? What sorts of mathematical things did he say? How would you identify these things and label them? Learning to talk about the mathematics of students' thinking is also mathematical work. If one doesn't have this mathematical component, one is left with saying very vague things like, "Keith seems to be doing well" or, "Keith seems to understand the basic concept." This is not particularly helpful - it is neither helpful for talking to other professionals about Keith nor for working directly with Keith. Learning to discern what mathematics there is to notice in what a child does is mathematical work.

This is meant as an illustration of designing tasks for teachers as learners of mathematics, in and close to practice. These are a few examples of tasks that could be designed for use with teachers if one's goal was to work on mathematics for teaching.

A caveat is important here: Just as giving tasks to children doesn't necessarily help them learn mathematics, simply giving tasks such as these to teachers won't necessarily help them learn mathematics for teaching either. Still remaining is the need to consider is how one might scaffold teachers' work on these problems so that their work will lead to productive mathematical learning. How would one have to steer a professional development session so that the question and the discussion stayed on the mathematics and actually moved toward greater mathematical skill and understanding? We didn't do that here. All we have done so far is to look at some examples of problems that could be offered. The work that lies ahead of us is to talk about how would we use these kinds of problems and toward what ends.