a problem x y 1 y use these blocks to make a polygon with perimeter of 4y + 4
TRANSCRIPT
A Problem
x
y
1
y
Use these blocks to make a polygon with perimeter of
4y + 4
Prof. Ilana HornUniversity of [email protected]
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.