a problem x y 1 y use these blocks to make a polygon with perimeter of 4y + 4

A Problem

x

y

1

y

Use these blocks to make a polygon with perimeter of

4y + 4

Prof. Ilana HornUniversity of Washingtonlanihorn@u.washington.edu

Teacher Collaborations around Mathematics

Transitions Math ProjectAugust 5, 2008

Overview of Session Studying teacher communities

Communities as… contexts delimiting possibilities places to learn

Questions & discussion

The Significance of Teacher Community

Research finding:

In innovative and equitable departments, teacher collaboration is a regular feature

of teachers’ work.

The Significance of Teacher Community

Research finding:

In innovative and equitable departments, teacher collaboration is a regular feature

of teachers’ work.

Studying Teacher Community

Observational Study: Collaborative math teacher group

Comparative Study: Comparing math teachers and English teachers in same high school

“Replication” Study: Current project working with math teachers in Puget Sound High Schools

Studying Teacher Community

Groups of high school teachers gathering to work on improving teaching and learning, mostly of mathematics.

Most groups met weekly, either during or after school.

Studying Teacher Community

Over 300 hours of recorded teacher conversations

Create transcripts, use other records of teacher activity, observations of teaching, data about student achievement

Analyze differences and similarities within and across groups.

The Significance of Teacher Community

Teacher communities provide an important emotional and social context for teachers’ work.

Teacher communities lay out a vision of what is possible in the classrooms of particular schools.

Both of these implicate teachers’ goals and commitments in their teaching --> learning.

Overview of Session Studying teacher communities

Communities as… contexts delimiting possibilities places to learn

Questions & discussion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Communities as Contexts

Bundled Bonded

• Low Commitment • High Commitment

• High Inclusion • High Inclusion

Fragmented Split

• Low Commitment • High Commitment

• Low Inclusion • Low Inclusion

Overview of Session Studying teacher communities

Communities as… contexts delimiting possibilities places to learn

Questions & discussion

Possibility in the classroom

Setting: A diverse public high school A progressive staff

Community: A “bundled” departmentA top-down chair

Event: A meeting discussing the elimination of remedial courses.

Possibility in the classroom

I don’t think we can fly geometry.

Possibility in the classroom

Our kids can’t get through geometry as we teach it now.

Possibility in the classroom

What they won’t be able to handle is the logic in the two column proofs.

Possibility in the classroom

Setting: A diverse public high school A progressive staff

Community: A “bonded” departmentCollaborative co-chairs

Event: A meeting discussing their de-tracked algebra classes.

Possibility in the classroom

There’s kids that know a lot, and there’s kids that feel like slow learners

Possibility in the classroom

And I’m trying to come up with activities that help close that gap.

Possibility in the classroom

I wonder if it’s not just the activities, but status.

Possibility in the classroom

The kids who feel low status will continue to play that role.

Possibility in the classroom

Some of us can go in and give you feedback.

Possibility in the classroom

Both groups identified a root of the problem.

Both groups provided an “appropriate” response to the problem.

The elaboration of the problem implicated students, subject, and teachers.

Overview of Session Studying teacher communities

Communities as… contexts delimiting possibilities places to learn

Questions & discussion

Communities as places to learn

Emotional support

New ideas or information

“Reality checks”

Coordination of expectations/pacing

Communities as places to learn

Understanding of complex teaching situations

Ideas about the application of professional development

Communities as places to learn

Formal PD

Classroom practice

Communities as places to learn

Formal PD

Classroom practice

Communities as places to learn

Over time… Shared ideas/language about teaching

and learning with their colleagues

Trust in colleague’s judgments

Shared expectations for students

Consequential conversations

Representation of practice

Consequential conversations

Principled statement or question

Consequential conversations

Revised representation of practice

Consequential conversations

The Problem

x

y

1

y

Use these blocks to make a polygon with perimeter of

4y + 4

The Problem

x

y

1

y

Total perimeter of blocks:6y + 2x + 4

The Teachers

Carrie

Charlie

Alice

Annie

A Solution

yx

yy-x2

2 + y

Another Solution

xy-x

y-x

x

Alice’s Solution

xy-x

Alice: Wait, so this one (3 s). This is positive and this is a minus…

Annie: Uh-huh. And the y’s are (all positive).

Alice’s Solution

xy-x

Alice: This is a y minus x plus xAnnie: Right.

Alice’s Solution

yx

yy-x2

2 + y

Alice: So I have 1, 2 –– 2, 3, 4 y’s

Reflecting on the Rehearsal

Carrie: But we were talking about a strategy they might use. You really should write that down.

Reflecting on the Rehearsal

Alice: So the strategy that I’m saying if they’re again, so, if the x’s cancel out?

Sharing Pedagogical Reasoning

Annie: “Making zeroes.” That’s the way I like to put it. [….]. Because canceling, you know, “canceling,” they think that they just throw it away.

Sharing Pedagogical Reasoning

Alice: Do you use “canceling?” What words do you use?

Sharing Pedagogical Reasoning

Carrie: So “making ones,” “making zeroes.”Annie: Making ones, making zeroes.

Sharing the Classroom

Representations of the classroom provide a means for teachers to consider both student and teacher perspectives on an activity.

Sharing the Classroom

Polygon building activity

Sharing the Classroom

Polygon building activity

“Canceling” vs. “Making Zeroes”

Communities as places to learn

By bringing representations of the classroom in their conversations, the teachers are able to reason carefully about both the mathematics and the pedagogy involved in an activity.

Your questions, reflections, and

observations

lanihorn@u.washington.edu