TPC2 PLC Spring, 2007 Plan
PI Pat Thompson developed the following guidelines for teachers working in professional learning communities in his NSF-funded project investigating a new model of professional development for pre-calculus math instructors.
Objective
We want to give you the freedom to decide your own activities, but at the same time we feel it is necessary to ask you to work toward a common objective. The objective we have in mind is that you develop a way of thinking about students’ mathematical thinking and students’ mathematical learning. We want you to see mathematical thinking and learning as byproducts of students coming to understanding a set of interrelated ideas coherently. From this perspective, instruction and curriculum become tools for affecting students’ thinking positively, and skills and procedures are the result of proficient reasoning instead of proficient remembering.
Your PLC’s task, then, is to decide what coherent body of ideas you want your students to understand in regard to some concept area and to create instruction and materials that will support students coming to understand them.
What Is an Idea?
An idea is something that has context and meaning, is generalizable, and has the potential of being foundational for other ideas and ways of thinking. Ideas are ways of thinking that can generate or motivate the procedures that have traditionally been the center of instruction.
Examples
Many textbooks confuse doing something with idea. For example, "graphing” is doing something, but “a graph” is an idea. To understand a graph means to develop meaning for it, like the idea of coordinate systems to locate points, the role of scale in creating axes, and the representation of quantity by putting measures on an axis.
Another example: “Factoring” is something you do, but “a factor” is an idea. To understand what a factor is is to develop meaning for the noun “factor” and to understand its relationships with other ideas, like product and quotient.
In general, ideas entail meanings; doing something entails … well, doing something. If you and your students are not discussing meanings, then you are not discussing ideas.
The Role of Activity in Learning Mathematics
Activity is at the heart of meaningful learning, but activity that matters is grounded in meaning and is directed toward a meaningful goal.
Textbooks
Textbooks are meant to be the backbone of a curriculum. But they sometimes are of little help when your aim is to teach ideas and idea-based reasoning.
Does your book contain any ideas? If so, what are they? Are they important? What makes them important? Are they coherent? We ask this because we recently reviewed Chapters 6-12 of an Algebra I book and concluded that it, really, did not contain any ideas. It contained lots of things to learn, but none of them was an idea.
The Importance of Attending to Student Thinking
How students think about an idea or an activity has consequences for the way they come to understand other ideas and learn other activities. For example, some students, when taught to use the Finger Tool*, think that they are tracing a graph that they can already envision. They see its main benefit as helping one to remember the graph’s shape. To them, the Finger Tool is of no use whatsoever when one doesn’t already know what a function’s graph looks like.
Instead, you want students to think that the Finger Tool is useful for envisioning the pattern of covariation between two quantities as their values vary simultaneously, especially when they do not know what the covariation’s graph looks like. For them to have this understanding, you must ensure that they understand that one begins by thinking about quantities that are covarying, and that, by using the Finger Tool, they can come to imagine that covariation. This shifts the learning from remembering what particular graphs look like to understanding how to interpret the meaning of any graph.
* The Finger Tool is a pedagogical invention of Pat Thompson. He and his teacher-partners encourage students to move their index fingers in tandem, modeling the tandem variation of two quantities.
Semester PLC Product
We ask that as a result of your semester-long discussions and collaboration that your PLC:
- Defines clearly the a particular body of ideas to be learned
- Finds out how students think about these ideas, including students who (ostensibly) have been taught them already. You decide on the approaches for finding this.
- Develops and/or adapts instruction and materials that have the promise of supporting students’ developing those ideas.
- Provides evidence of how instruction has impacted student thinking. You decide on the forms of evidence and how you will use them to determine changes in students’ thinking.
- Produces a report and presentations to communicate your findings to a wider audience (TPC banquet on Saturday, May 19; presentation to colleagues in your department and at NCTM state, regional, and/or national conferences). You will decide on the report and presentation format.