Building Professional Community that Promotes
Teacher Learning
Through Sharing and Analysis of
Contrasting Cases
Sharon Derry, Margaret J. Wilsman & Alan J. Hackbarth
University of Wisconsin-Madison
Abstract
This is an interim report on a new STAAR TPD program at UW-Madison in which 20 middle school teachers are
enrolled for graduate credit. This course, “Understanding and Cultivating Students’ Algebraic Thinking,” began as a
four-day workshop in summer, 2004. It continues throughout the school year and beyond as a facilitated
professional learning community that incorporates both face-to-face and online interaction.
Community activities promote teacher learning through analysis and discussion of contrasting cases of teacher
problem solving, student problem solving, and teachers’ pedagogical practices. Cognitive research indicates that
contrasting cases is powerful for promoting idea abstraction, contextualization and transfer. Our work is evaluating
and perfecting ways of scaffolding contrasting cases discussions. Our approach involves building trust and
community norms so that teachers feel comfortable sharing and comparing their work, including videos of their
teaching and their own problem solutions.
Our objectives for teachers in this study include both content knowledge and professional process objectives. Using
a pre-post design with a matched comparison group, we will employ both content and process assessments to
determine the course’s impact on teacher practice.
This poster describes our four-day summer workshop, focusing on how we formatively evaluated and developed
our contrasting cases activities (called SAMs) in situ as the workshop progressed. Our workshop evolved around
three SAMs, each addressing a different foundation concept in the NCTM 2001 description of algebraic reasoning.
We collected participants’ (facilitators and teacher-learners’) reactions to SAM designs and used those to adjust the
design and scaffolding of each subsequent SAM. We also collected pre and posttest data as part of each SAM
activity, to help us assess the learning effects of the contrasting cases components. Although our data analyses are
not yet complete, in this poster we share what we have learned so far about designing and scaffolding contrasting
cases instruction in a way that builds trust and community.
We also outline our plans for next steps, which include creating online supports for the community and online
versions of the contrasting cases activities.
I. Purposes of Summer 2004 UW STAAR TPD Course
To design and evaluate a TPD course with the following objectives:
•
Content Outcomes—Teachers will:
1. Generate multiple representations.
2. Demonstrate representational fluency.
3. Generalize solution strategies.
4. Explain algebraic reasoning.
•
Process Outcomes—Teachers will:
1. Suspend their current teacher persona.
2. Actively participate in the learning community.
3. Demonstrate sincere concern for transitioning students from
arithmetic to algebraic reasoning.
4. Practice professional norms.
II. Features of 2004 UW STAAR TPD Course
• Facilitated Learning Community Approach (Wilsman, M. et al.
2002
“Online Professional Development: Sustained Learning with Friends.” Teaching
Children Mathematics Vol. 8, No. 9, pp. 505-509.)
• Practice-Based Approach (Smith, M. S.
2001. Practice-Based Professional
Development for Teachers of Mathematic. NCTM)
• Case-based Learning with Contrasting Cases Method (Beitzel, B. &
Derry, S. 2004; Catrambone & Holyoak, 1989; Gentner et al. 2003; Schwartz &
Bransford, 1998).
• Based in Cognitive Science Concepts, such as Learning & Transfer,
Expert-Novice Differences (Bransford et al. 2001 How People Learn).
• Hybrid Implementation
 Face-to-Face Classes
 Online Community Support
• Use of Metaphors to Build Awareness of the Course Outcomes
 High Tech Consultant (Practicing Professional Norms)
 Stone Soup (Active Participation)
 Karate Kid (Suspend Teacher Persona, Transfer)
 Apollo 13 (Situating Work in Practice)
III. Research Design
1.
2.
3.
N = 40 from the Madison Metropolitan School District
Grades 6-8 Mathematics Teachers and Special Education Teachers
TPD Treatment & Matched Comparison Group
Pretest and Posttest design
• Assessments of Student Work
Analyze videos of two student interviews collected by STAAR
researchers.
• Assessment of Mathematics Content Knowledge
Used instrument developed by L. Goldsmith, M. Driscoll and N. Seago
for use in their NSF ROLE Project: Turning to the Evidence:
Examining the impact of two professional development program focused
on algebraic thinking.
4.
Within-treatment studies of different designs for contrasting cases
instruction (SAM activities).
IV. Focus of this Presentation
•
•
Designing and evaluating SAMs (Sample Algebraic Modules), learning
activities for teachers with the following features:
•
Individual problem solution
•
Reading assignment(s)
•
Individual reflections
•
Contrasting cases of teacher problem solving, student work and
classroom teaching

Small group sharing

Whole group sharing

Individual and group comparisons
Theoretically-designed scaffolding to enhance learning from contrasting
cases
V. About SAMs (Sample Algebraic Module)
SAMs were designed to develop teachers’ understanding of the components of
algebraic thinking as defined by NCTM 2001 & 2004:
•
Recognizing and Analyzing Patterns, Relations and Functions
•
Representing and Analyzing Mathematical Situations &
Structures Using Algebraic Expressions & Symbols
•
Using Mathematical Models to Represent, Analyze and Understand
Quantitative Changes
•
Analyzing Change in Various Contexts
SAM designs allow evaluation of the impact of contrasting cases on
teacher learning. The steps in a SAM were:
•
Teacher problem solving.
•
Discussion of a journal article expected to enhance
teachers’ ability to analyze his/her problem solution
and generate alternative solutions.
•
A “pre assessment” in which teachers write a
reflection comparing their original solution and an
alternative solution.
•
Small group comparing of solutions.
•
Whole group examination of selected solution
strategies.
•
A second opportunity for teachers to generate an
alternative solution and compare original and new
solutions.
•
Whole group discussion.
•
Individual evaluations of the activities.
SAM 1: Based on NCTM Foundation of Algebraic Thinking Component
“Understanding Patterns, Relations, and Functions”.
The Problem:
“KEEP BUILDING”
This is a series of growing shapes. Imagine that the building pattern continues. How many blocks will it take to build:
1.
Shape 5? ________
2.
Shape 8? ________
3. Write a function rule to find the number of blocks in a shape. Let N represent the shape number. Let B represent the number
of blocks.
B=
4. Which shape would take exactly 107 blocks to build?
The Reading/Prompt: “Developing Algebraic Reasoning Through
Generalization” by John K. Lannin. 2003. Mathematics Teaching In The Middle
School. NCTM. How does the author suggest that teachers move students
between specific (numeric and geometric) strategies for generalization and
justification? And why?
Whole Group Discussion: Facilitator-led discussion of reading with
emphasis on the above reading prompt and problem.
Contrasting Cases (CC) Activities:
• Individual Reflections:
1. Reflect on your problem solution in light of the reading in terms of the
solution strategy you used and the elaborateness of your solution.
2. Show a strategy for solving the problem that is different than your
initial strategy. (You should produce a different function rule).
• Small & Whole Group Discussions:
1. Compare and contrast solutions in your small group.
2.
3.
Use one solution to share with the whole group and explain your reason
for choosing.
In the whole group compare and contrast the diverse and multiple
solutions.
• Individual Reflections:
1. Review your solution(s) and generate a (third) solution that is
different than your initial solution(s).
2. Use the reading to make comparisons between your solutions and
modify any solutions appropriately.
SAM 2: Based on NCTM Foundation of Algebraic Thinking Component
“Representing and Analyzing Mathematical Situations & Structures Using
Algebraic Symbols”.
The
Problem
“BALANCE IT”
1.
Which block (cylinder, sphere, or cube) – or combination of blocks must be added or removed in order to balance Scale C?
2.
List or draw the steps you followed to identify the block(s).
The Reading/Prompt: Gregg, Diana Underwood & Yackel, Erna. “Helping
Students Make Sense of Algebraic Expressions: The Candy Shop”. Mathematics
Teaching in the Middle School. Vol. 7. No. 9. Pp. 492-97. What do the authors
believe are the key benefits of this “bottom-up approach”?
Whole Group Discussion: Key ideas in the “Candy Shop” article were
posted during discussion so that they could later be seen, reviewed and used by
teachers as they reflected as individuals, in small or whole group discussions.
Teacher-Generated Ideas About the Value of Visual Versus Symbolic
Solution Representations:
• If students can have an object to relate a variable to, they understand the idea
of variable better.
• Seemed like a strength if students could make own notation (and translate to
standard notation). Moving students in this direction needs to be a
progression.
• Role of story—helps to establish what the variables will be.
• Could build a visual map that serves as a anchor—something to connect the
symbols and pictures (objects). The “bottom up approach” works because
students actually construct a map to anchor to pictures and symbols.
Contrasting Cases (CC) Activities:
• Individual Reflections:
1. Reflect on your problem solution in light of the reading. If you can, show a
strategy for solving the problem that is different than your initial strategy.
• Small & Whole Group Discussions:
1. Compare and contrast solutions in your small group.
2. Use one solution to share with the whole group explaining your
reason for choosing.
3. In your small group compare and contrast (how is it the same, how is
it different) your group’s solution strategy with each of the other
strategies displayed in the room. A different group member will
report on each comparison.
Small Group SAM 2 Solutions Representations
SAM 3: Based on NCTM Foundation of Algebraic Thinking Components:
“Recognize opportunities to generalize solution strategies to a mathematical
task”. “Demonstrate the ability to explain the algebraic reasoning that
occurs in solving a mathematical task”.
The Problem
There is no Guiness Book of World Records for ball bouncing. We think
bouncing these balls for an hour ought to get a person into the book.. Using a
the data presented in the table, predict the number of bounces in an hour (60
“BOUNCING BALL”
minutes) for each ball. Explain the process you used to determine your
answer. [Note: There two distinct representations for “average”.] Explain
which “average” you used and why you chose it.
If data is collected in a highly controlled way, the mean and median should be nearly identical. In this data set, what kinds of
variations in the way the data was collected might have affected the mean and/or median values? How might you vary your
use of the data in order to make the mean and median closer in value? Why might you want to do this?
The Readings/Prompt:
Lambdin, Diana V., Lynch, R. Kathleen, & McDaniel,
Heidi. 2000. “Algebra in the Middle Grades.” Mathematics Teaching in the Middle
School. Vol. 7. No. 9. Pp. 195-98. and Navigating through Algebra in Grades 6-8. 2001.
“Chapter 2 Analyzing Change in Various Contexts”. National Council of Teachers of
Mathematics. P. 19-20. How does collecting data in various contexts, plotting data into
graphs, and analyzing the shapes of those graphs help students to develop
understanding of change as an algebraic concept?
Contrasting Cases (CC) Activities:
• Individual Reflections:
1.
Reflect on your problem solution using the reading.
2.
If possible, show a strategy for solving the problem that is different
than your initial strategy.
• Small Group Sharing:
1.
Share solutions in your small group.
2.
Use the questions listed in “Guide For Comparing And Contrasting
Solutions” to make the explicit comparisons and contrasts (if you
haven’t already done so).
SAM 3: Guide for Comparing and Contrasting Solutions
Same or Different?
1.
2.
3.
4.
How is the task interpreted?
What representational forms are used?
What are assumptions about rate of change?
What can you guess about the learner?
1
Solutions
2
3
4
VI. Some Tentative Conclusions
•
SAMs were a successful activity if appropriately designed and scaffolded.
We learned how to use successful forms of scaffolding, such as:
•
To make more explicit mappings among the SAM elements—readings,
individual problem solving, comparing and contrasting of multiple
representations.
•
To use methods that requested detailed explicit comparisons:





Reflect on your problem solution using the reading.
Individually make a comparison during the whole group discussion.
Use the questions listed in “Guide For Comparing And Contrasting
Solutions” to make the explicit comparisons and contrasts.
In whole group compare and contrast the diverse and multiple
solutions using specifically selected questions.
Each small group member described the difference in algebraic
reasoning found in their group’s solution as compared to one other
solution strategy.
•
Teachers became metacognitive about their learning through the SAMs.
•
Feedback indicates that some teachers make immediate connections to
their classroom pedagogy.
VII. Next Steps
1. Analyze data—
• Effects of pre test post analysis
• Effects of teacher feedback on modification of SAMs
2. Analysis of the SAM designs
• Effects of different scaffolding methods on teacher ability to generate
multiple representations and discuss in terms of reading.
3. Design and beta test online versions of the SAMs using STELLAR
• This will permit the scaling up of the STAAR TPD.
4. For the current TPD course
• Facilitate community building within the online environment.
• Extend and continue the learning begun in the FTF meetings—
particularly analysis of their classroom practice.
• Focus on Pedagogical Outcomes.