What does the transformations- based focus of the Common Core State Standards mean? Nanette Seago WestEd

2 Presentation Overview I.Geometric Transformations and Common Core State Standards II.Students’ Struggles with Transformations and Similarity III.Static and Transformations-based approaches IV.Rectangle Problem & Video Clip Discussion V.Learning and Teaching Geometry Project’s Research on Teacher and Student Learning VI.Resources for Resources for PD

Geometric Transformations and the Common Core Standards

Geometric Transformations and the 8 th Grade Common Core Standards Understand congruence and similarity using physical models, transparencies, or geometry software. 1.Verify experimentally the properties of rotations, reflections, and translations 2.Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 3.Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 4.Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.

5 Challenges In 2011, a national task force of mathematicians, mathematic educators, state leaders, and teacher leaders made recommendations about which CCSS mathematics domains should be targeted as priority areas for professional development and resources in grades K-8. They targeted Grade 8 geometry as one of the five recommended priority areas, stating: Geometry in the Common Core State Standards is based on transformations, an approach that is significantly different from previous state standards. This is a change for students, teachers, and teachers of teachers. Challenges include attention to precision and language about transformations…. The transformational approach to congruence and similarity is likely unfamiliar to many middle grades teachers (McCallum, 2011).

Students’ Struggles with Transformations and Similarity

7 Eighth Grade NAEP Item This 2007 NAEP item was classified as “Use similarity of right triangles to solve the problem”. 1% of eighth- grade students answered correctly

8 Twelfth- Grade NAEP Item This 1992 NAEP item was classified as “Find the side length given similar triangle”. 24% of high school seniors answered correctly

Static and Transformations-based Approaches to Similarity

Static Conceptions of Similarity Focus on comparison of numerical relationships between corresponding parts of similar figures Focus on setting up and solving proportions that are not connected to geometric meaning Similarity is conceptualized in discrete terms as a numeric relationship between two figures

11 A Transformations-based Conception of Similarity Similarity is conceptualized as enlarging or reducing figures proportionally to create a class of similar figures. Focus is on geometric transformations that result in similar figures Attention is on all possible figures in a similarity class enabled by visual representations of dilating figures The ratio of lengths of corresponding sides of similar figures is the scale factor of the dilation Ratio of lengths within a single figure is invariant across the similarity class

A Geometric Transformations Approach to the 8 th Grade NAEP problem Center of Dilation

A Geometric Transformations Approach to the 12 th Grade NAEP problem Center of Rotation

A Geometric Transformations Approach to the 12 th Grade NAEP problem Center of Dilation

A two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations and dilations Transformations-Based Definitions of Congruence & Similarity A two dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. 15

Rectangle Problem & Randy

17 Rectangle Problem Which rectangles are similar to rectangle a?

How is Randy solving the problem? What relationships is he attending to?

Unpacking Randy’s Method What did Randy do? (What was his method?) Why might we argue that Randy’s conception of similarity is more transformations-based than static- based? 19

Representing Similar Rectangles as Dilation Images 20

Learning and Teaching Geometry Project’s Research on Teacher and Student Learning

22 Learning and Teaching Geometry Project Overview Funded by the U.S. National Science Foundation Developing video case-based professional development materials Targeted for mathematics teachers of grades 5-10 The materials include: – 1 Foundation Module (10, 3-hour sessions) – 4 Extension Modules (2, 3-hour sessions)

23 Learning and Teaching Geometry Project Staff Staff: Nanette Seago (PI), Mark Driscoll (Co-PI), Jennifer Jacobs, Michael Matassa, 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

Design Built around authentic video clips – 3-6 minutes in length – Grades 5-9 classrooms Video clips intended as objects of inquiry, not exemplars Well-specified facilitator support materials: – Detailed agendas and resources – Make explicit the goals and underlying core principles Focus on geometric similarity

LTG Foundation Module Ten 3-hr Sessions Session 1 A dynamic, transformatio nal view of congruence Session 2 A dynamic, transformati onal view of Similarity Session 3 Relationship Between Dilation and Similarity Session 4 Properties of Dilation Session 5 Preservation of Angles & Proportional Lengths through Dilation Session 6 Ratios Within and Between Similar Figures Session 7 Ratios Within and Between Similar Figures. Part 2 Session 8 Connections between Similarity, Slope & Graphs of Linear Functions Session 9 Area of Similar Figures Session 10 Closure and Re-capping of Big Ideas Defining Congruence and Similarity Relationships and Attributes of Similar Figures ConnectionsClosure Teacher Learning Goals: To examine a transformations-based view of similarity, and geometry in general To enhance teachers’ mathematical knowledge for teaching similarity To gain insight into students’ developing conceptions of similarity

In each 3-hour PD session, teachers: Grapple with the same mathematical task(s) the videotaped students tackled View, analyze & discuss the video clip(s) Consider issues around content, student thinking, and pedagogy

Research Questions (Horizon Research, Inc.) What is the impact of participation in the LTG professional development program on teachers’ mathematical knowledge for teaching? What is the impact of teachers’ participation in the LTG program on their students’ performance in geometry?

Field Test Sites

Field Test Participants 126 Participants o 87 treatment o 39 comparison o Mix of K-12 in-service teachers, teacher leaders/coaches and pre-service teachers

Field Test Data Collection Activities Facilitator & Teacher Questionnaire Teacher Content and Embedded Assessments Facilitator Session Logs Professional Development Observation Facilitator Interview Student Content Assessment ( only)

Teacher Content and Embedded Assessments Teacher Content Assessment – Geometry Assessment – 25 multiple-choice items (pre/post) Congruence Transformations Dilation Properties of Similarity Ratios and Proportions Scaling Embedded Assessments – Video Analysis [Randy] (pre/post) – Sorting Rectangles Math Task (pre/post)

LTG Teacher Content Assessment Field Test Percent Correct Scores PrePostGain Scores Effect Size Mean Treatment (N=83) † Comparison (N=38) † On average, teachers in the treatment condition demonstrated significant improvements in percent correct scores while comparison teachers did not (repeated-measures ANOVA; p < 0.05).

Embedded Teacher Assessments: Video Analysis & Sorting Rectangles Task Treatment teachers significantly improved on 2 of the 3 math task questions, and on 3 of 3 video analysis questions. Comparison teachers didn’t demonstrate significant improvement on any of the 6 questions. Session 1 A dynamic, transformatio nal view of congruence Session 2 A dynamic, transformat ional view of Similarity Session 3 Relationship Between Dilation and Similarity Session 4 Properties of Dilation Session 5 Preservation of Angles & Proportional Lengths through Dilation Session 6 Ratios Within and Between Similar Figures Session 7 Ratios Within and Between Similar Figures, Part 2 Session 8 Connections between Similarity, Slope & Graphs of Linear Functions Session 9 Area of Similar Figures Session 10 Closure and Re-capping of Big Ideas Defining Congruence and Similarity Relationships and Attributes of Similar Figures ConnectionsClosure Randy Math task SORTING RECTANGLES TASK

LTG Student Geometry Assessment Field Test Percent Correct Scores PrePostGain Scores Effect Size Mean Treatment (N=162) † Comparison (N=104) † On average, students of teachers in the treatment condition demonstrated significantly larger gains in percent correct scores than students of comparison teachers (repeated-measures ANOVA; p < 0.05).

Tentative Conclusions The LTG PD program appears to improve teachers’ mathematical knowledge for teaching in the area of transformations-based geometry. Students of the participating teachers appear to show improved knowledge of transformations-based geometry.

Resources for PD

Some Potential Resources for Leaders LTG Resources:  Learning and Teaching Geometry Video Case Materials, WestEd. Expected publication date: Spring 2014  Field Guide to Geometric Transformations, Congruence & Similarity available at  NCSM: Illustrating the Standards for Mathematical Practice Other Resources:  Illustrative Mathematics Project: Tools for the Common Core  Transformational Geometry, Richard Brown (1989)

38 Transformations-based Definitions

NCSM Modules

Congruence & Similarity Module Participants examine the meaning of defining congruence and similarity through transformations as articulated in the Common Core State Standards. Participants are asked to compare and contrast static definitions of congruence and similarity with dynamic definitions of congruence and similarity. Participants consider implications for instruction that the dynamic definitions have on teaching and learning mathematics.

Similarity, Slope, Lines Module Participants unpack the connection between similarity, slope, and the graphs of linear functions by doing mathematical tasks, analyzing student thinking, and exploring a computer-based applet. Participants consider strategies to support students' to reason mathematically and learn to use precise language in their explanations.

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