Dissertation Critique Gwenanne Salkind George Mason University December 8, 2007 EDCI 858 & EDCI 726 Dr. Patricia Moyer-Packenham Dr. Margret Hjalmarson

Examining the Work of Constructing a Representational Context in Elementary Mathematics Teaching By Rhonda B. Cohen University of Michigan Doctoral Committee: Professor Deborah Loewenberg Ball, Chair Professor Hyman Bass Professor Magdalene Lampert Professor Elizabeth Yakel

Research Questions 1.What is the work of constructing a representational context in elementary mathematics teaching? 2.How does studying the work of constructing a representational context make more visible what teachers need to know and be able to do to use these kinds of instructional representations effectively in elementary mathematics teaching?

Central Questions What might teachers need to know in order to help students develop meaning for a representation? What might teachers need to be able to do to help students learn to use a representation? What might be some of the challenges or dilemmas in this work?

Methods Data Sources: Records of teaching (videotaped lessons, lesson transcripts, copies of student work, and teachers’ notes) of a third grade mathematics class taught by Deborah Ball during the school year. Qualitative Case Study: Analyzed 3 teaching episodes where Ball introduced a representation to help students solve a mathematics problem.

Theoretical Foundation of Analysis Mathematical Knowledge for Teaching Common content knowledge Specialized content knowledge Knowledge of content and students Knowledge of content and teaching

Representations Square Tiles – How does the teacher establish the language needed to deploy a representation? Elevator Model – How does the teacher make a representation usable to students? Number Line - What does the teacher do to make connections to other representations (especially representations that students introduce)?

Results The work of launching and preparing to use a representation involves knowing mathematics in ways that are special to the work of teaching. Demand for mathematical knowledge, skill, and sensibilities Need to be judicious in how language and mathematical symbols get used Importance of attending to the ways in which students’ prior knowledge and experience can both support and hinder the work of constructing a representational context

Establishing the Language Helping students record mathematical ideas in ways that emphasize the correspondences among the words, symbols, and materials Attending closely to the meaning of mathematical terms and the use of language Using transitional language Noticing the mathematical ideas for which a representation can be used and relating those ideas to what students need to learn

Making Representation Usable Attending closely to the meaning of the mathematical symbols and the use of language Piquing students’ interest Drawing students’ attention to key features and teaches students how to use the representation Comparing the relative merits of different representations

Making Connections Attending closely to what students mean by the terms “same” and “different” Drawing students’ attention to a structural elements that need to be the same for different representations Helping students give explanations Attending to task design considerations

Teaching Challenges Establishing the language needed to deploy a representation Making a representation usable for students Making connections to other representations

Ideas for Teacher Education Help teachers attend closely to how recording work with mathematical tools (e.g., base-ten blocks) can be used to emphasize the mathematics content being studied Have teachers explain the correspondences between a representation and the mathematics content

Ideas for Teacher Education Use caution in emphasizing the motivational purposes for using representations in teaching mathematics Help teachers develop criteria for discriminating among representations used in mathematics teacher – compare the relative merits of different representations

Critique Redundancy Focus Organization & Structure Omit the section on “Designing Introductory Tasks”

Questions I have Does this study generalize to other teachers’ work? If she had studied a different teacher, would she have found the same results?

Key Components – things I learned Her acknowledgement page was eloquently written She identified themes in her literature review Her argument for the study was both broad and specific – approached from many angles She defined terms and assumptions (tons of this!) She described her perspectives (lenses) She tells what she is doing and why she is doing it (over and over again!) She used stories and vignettes as examples and illustrations of her analyses