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### Introduction to GTWs Understanding of Mathematics

### Terri Video Clips Understanding of Mathematics

As you arrive, please work on the Geometric Transformation Workout (copies are on your table along with tracing paper if you wish to use it)

Warm Up Problem

2.

Using VideoCases to Develop Teachers’ Conceptual Understanding of MathematicsNanette Seago, WestEdPatrick Callahan, Affiliation?Mark Driscoll, EDCJohannah Nikula, EDCJennifer Jacobs, University of Colorado

Session Agenda Understanding of Mathematics

Overview of Project

Geometric Transformation Workouts

View and Discuss potential video clips for Definition Module

Using videocases in PD

Q & A

L Understanding of Mathematicsearningand Teaching GeometryProject Overview

In year four of a 5-year National Science Foundation project

Developing videocase-based, PD materials

1 Foundation Module

4 Extension Modules

Staff: Nanette Seago (PI), Mark Driscoll (Co-PI), Jennifer Jacobs, Johannah Nikula, Patrick Callahan, Hilda Borko

Advisory Board:Harold Asturias, Tom Banchoff, Phil Daro, Megan Franke, Karen Koellner, Glenda Lappan, Hung-Hsi Wu

Evaluation Team: [Horizon Research, Inc.] Dan Heck, Kristen Malzahn, Courtney Nelson

LTG Materials Understanding of Mathematics

Built around authentic video clips from grades 6-8 classrooms

Focus on similarity and its mathematical use in teaching

Modular in design--coherent, sequenced set of videocase professional development sessions

Well-specified facilitator support materials:

Explicitly communicates the underlying core principles

Clearly laid out rationale for principles

Detailed sample agendas and mathematical notes designed with an eye toward making the design and values explicit

Foundation module: ten, 3 hour sessions

LTG Foundation Module Map Understanding of Mathematics

A sequence of learning experiences

LTG Foundation Module Goals Understanding of Mathematics

Help teachers develop a deep, flexible understanding of similarity

Promote a dynamic, transformational view of similarity, and geometry in general.

Provide insight into students developing conceptions of similarity

Equip teachers with specialized content knowledgein the area of similarity

Specialized Understanding of MathematicsContent Knowledge (SCK)

“The mathematical knowledge and skill unique to teaching- not typically needed for purposes other than teaching.

This work involves an uncanny kind of unpacking of mathematics that is not needed--or even desirable--in settings other than teaching. Many of the everyday tasks of teaching are distinctive to this special work.”

For example:

Evaluating the plausibility of students’ claims (often quickly)

Giving or evaluating mathematical explanations

Choosing and developing useable definitions

Recognizing what is involved in using a particular representation

Linking representations to underlying ideas and to other representations

(Ball, Thames, & Phelps, 2008)

The 2007 8th grade NAEP item below was classified as “Use similarity of right triangles to solve the problem.”

Only 1% of students answered this item correctly.

The 1992 12 Understanding of Mathematicsth grade NAEP item below was classified as “Find the side length given similar triangles.”

Only 24% of high school students answered this item correctly.

Common Core Standards Understanding of Mathematics

- Need to list the ones that fit

GTW Discussion Understanding of Mathematics

2.

Task 3 Understanding of Mathematics

13

Intro to Use of VideoCases in LTG Understanding of Mathematics

- Not intended as models
- Used to analyze the dynamic nature of classroom interactions
- Used to generate discussion, conjectures and evidence
- Intended to respect teachers, students and teaching
- Use of lesson graphs to situate the video clips within the context of a whole lesson

Extension Modules Understanding of Mathematics

I. Mathematical Definition Through Similarity

II. Communication & Language in Similarity

III. Using Representations and Tools in Similarity

IV. Justifying and Proving Geometric Similarity

Definition Module Understanding of Mathematics

- Draft Ideas
- Goals
- Determine the components and implications of mathematical definitions through the example of similarity
- Examine how mathematical definitions are introduced, developed, and included in classroom lessons through the example of similarity.

- A few mathematical punch lines:
- There is not one correct definition of similarity, but there are incorrect definitions
- Definitions differ in their application and context
- A part of practice is being able to analyze, compare, and work with the different definitions floating around in the classroom
- Geometry is dense with vocabulary, so the idea of definition is highlighted

- Goals

Definition of Similarity Used in Understanding of Mathematics

the Foundation Module

Two figures are similar if one is the same as an enlargement or reduction of the other.

“the same” ~ “congruent” or “can be matched by rotating, translating, and/or reflecting.”

“an enlargement or reduction”~ “stretching or shrinking” or “dilating a figure where the distance from the center of dilation can be multiplied by a fixed scale factor.”

Potential Video Clips Understanding of Mathematics

- Series of three video clips from an 8th grade classroom in Arkansas that the staff thinks have potential to fit within the Definition Module.
- Examine these clips with an eye towards their use in furthering teachers understanding of the specialized knowledge in defining similarity

Video Clips Discussion Understanding of Mathematics

- What opportunities to learn do the use of these three clips offer teachers around mathematical knowledge specific to teaching?
- Do you see other learning opportunities within these three clips?

Q & A Understanding of Mathematics

Stepping Back Understanding of Mathematics

- A central goal of the Foundation Module is to support teachers to help their students progress from using informal definitions of similarity to using increasingly precise definitions. Currently, we are planning to show the Melanie clips early on in the Module. Given the goal of the module, do you think this is a wise decision? Why or why not?
- Pilot/Field-test sites needed!

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