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### Dissertation Critique

### Examining the Work of Constructing a Representational Context in Elementary Mathematics Teaching

Gwenanne Salkind

George Mason University

December 8, 2007

EDCI 858 & EDCI 726

Dr. Patricia Moyer-Packenham

Dr. Margret Hjalmarson

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 Context in Elementary Mathematics Teaching

- What is the work of constructing a representational context in elementary mathematics teaching?
- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 1989-90 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 Context in Elementary Mathematics Teaching

- Mathematical Knowledge for Teaching
- Common content knowledge
- Specialized content knowledge
- Knowledge of content and students
- Knowledge of content and teaching

Representations Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

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

Ideas for Teacher Education Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

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

Questions I have Context in Elementary Mathematics Teaching

- 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 Context in Elementary Mathematics Teaching

- 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

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