PROFESSIONAL NOTICING: HOW DO TEACHERS MAKE SENSE OF STUDENTS

MATHEMATICAL THINKING?

by

GINGER RHODES

(Under the Direction of Denise S. Mewborn)

ABSTRACT

The purpose of this study was to understand how teachers make sense of their students

mathematical thinking. Specifically, learning trajectories and professional noticing were used as

a way to examine how teachers understand and use students mathematical thinking in their

teaching practices. A qualitative research methodology was employed to address three research

questions that focused on teachers informal learning trajectories, what teachers notice about

students mathematical thinking during classroom interactions, and the ways that teachers

respond to that thinking during classroom interactions. Two high school geometry teachers were

observed and interviewed during one semester. In addition, the teachers attended biweekly

working-group meetings to discuss students mathematical thinking. I created two learning

trajectories for the teachers lessons to represent their thoughts about their students

mathematical thinking before and after lessons: the projected learning trajectory (PLT) and

enacted learning trajectory (ELT). The two learning trajectories were compared to identify

instances of teacher noticing. The PLT and ELT were similar in some instances but not others.

The teachers described and interpreted what they noticed in terms of their uncertainties and

surprises about students mathematical thinking. Furthermore, the teachers described and

interpreted what they noticed about students mathematical thinking in terms of the mathematics

tasks, their own mathematical knowledge, and their actions with students in the classroom. The

teachers typically responded to students mathematical thinking in five ways: posed a question,

asked students to share, told the students something about the mathematics, posed another task,

and used a pedagogical content tool. What the teachers noticed in the classroom interactions and

the ways that they responded to students affected their development of PLTs and ELTs, and

vice versa. When the students led the ELT, the teachers gave detailed descriptions for the learning

trajectories and gave more details for what they noticed during classroom interactions. In

contrast, when the teachers led the ELT the teachers struggled to describe learning trajectories and

what they noticed during classroom interactions. The reporting and analysis of the data reveal

implications for both research and teacher education.

INDEX WORDS: Teacher development, Learning trajectories, Professional noticing,

Teacher knowledge, Teacher listening, Instructional practices

PROFESSIONAL NOTICING: HOW DO TEACHERS MAKE SENSE OF STUDENTS

MATHEMATICAL THINKING?

by

GINGER ALAYNE RHODES

B.S., North Carolina State University, 1998

MAEd., East Carolina University, 2000

A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial

Fulfillment of the Requirements for the Degree

DOCTOR OF PHILSOPHY

ATHENS, GEORGIA

2007

2007

Ginger Alayne Rhodes

All Rights Reserved

PROFESSIONAL NOTICING: HOW DO TEACHERS MAKE SENSE OF STUDENTS

MATHEMATICAL THINKING?

by

GINGER ALAYNE RHODES

Major Professor: Denise S. Mewborn

Committee: Jeremy Kilpatrick George Stanic

Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia August 2007

iv

ACKNOWLEDGEMENTS

There are many people that I would like to thank. First, I would like to thank my

committee members Denise Mewborn, Jeremy Kilpatrick, George Stanic, and Paola Sztajn.

Each of them contributed greatly to my professional growth and the completion of this report by

supporting my efforts, challenging my thoughts and hypotheses, and providing thoughtful

feedback on my work. A special thanks goes to Denise for going above and beyond the call of

duty as a major professor. I have learned a wealth of knowledge about how to be an educator

from her patience, guidance, expertise, and her sincere dedication to students. Thank you!

I want to thank the two teachers who were the participants in this study. They will never

know the depth of my gratitude for their willingness to open their classrooms and share their

teaching practices with me. Like their students, I have learned so much from being in their

classrooms and talking with them.

My graduate work was supported in part by the National Science Foundation (NSF)

Grant ESI-0227586, the Center for Proficiency in Teaching Mathematics at the University of

Georgia and the University of Michigan. The opinions expressed in this report are those of the

author and do not necessarily reflect the views of the NSF.

The faculty, staff, and students in the mathematics education department at the University

of Georgia are wonderful people who have enriched my graduate school experience. I wish to

thank Jim Wilson for helping me have a unique and remarkable learning experience through

KSTF. He provided guidance and support as I learned to provide professional development to a

group of aspiring teachers. Also, thanks to Pat Wilson for providing opportunities to work with

v

prospective and practicing teachers. To Laurel, Dennis, Brian, Erin, Angel, Sarah, and many

more that I do not have room to mention, thank you for reading my work, listening to my

thoughts, and encouraging me to succeed. Above the collegial support, I want to thank you all

for your friendship.

With my deepest sincerity, I want to thank my family and friends. Dad, Mom, Grandma,

Tony, Daniel, Shelly, Julia, Emily, Shanaka, and SusanI am sure that I have not said thank you

enough for the love and understanding you have each shown me. You have all supported me,

kept me laughing, and encouraged me to do well throughout this experience and life in general.

Im very grateful and lucky to have such a wonderful family!

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TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .......................................................................................................iv

LIST OF TABLES.....................................................................................................................ix

LIST OF FIGURES ....................................................................................................................x

CHAPTER

1 Introduction...............................................................................................................1

My Personal and Professional Journey...................................................................3

Why Geometry? ....................................................................................................4

Why Focus on Teachers Understandings of Student Thinking? ............................5

Research Questions ...............................................................................................8

2 Literature.................................................................................................................12

Students Mathematical Thinking........................................................................12

The Mathematics Teaching Cycle........................................................................17

Theoretical Discussion ........................................................................................47

3 Method ....................................................................................................................49

Participant Selection............................................................................................49

Data Collection ...................................................................................................50

Participant and School Descriptions ....................................................................53

Working Group Sessions .....................................................................................61

Data Analysis ......................................................................................................68

vii

Strategies for Validity..........................................................................................70

Researcher Role and Subjectivities ......................................................................71

Limitations..........................................................................................................73

4 Results.....................................................................................................................76

Learning Trajectories: What do they Look Like for Teachers?.............................76

What do Teachers Notice?...................................................................................99

Teacher Responding .......................................................................................... 119

Concluding Remarks ......................................................................................... 127

5 Conclusions ........................................................................................................... 131

Findings ........................................