What does the transformationsbased focus of the Common Core
State Standards mean?
Nanette Seago
WestEd
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
2
Geometric Transformations and the
Common Core Standards
Geometric Transformations and the 8th 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 twodimensional figures, describe a sequence that exhibits the similarity
between them.
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).
5
Students’ Struggles with
Transformations and Similarity
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
7
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
8
Static and Transformations-based
Approaches to Similarity
Static Conceptions of Similarity
Similarity is conceptualized in discrete terms as a numeric relationship
between two figures
2
1
3
6
1 3
=
2 6
Focus on comparison of numerical
relationships
between corresponding parts
!
of similar figures
2
1
3
6
1 2
=
3 6
Focus on setting up and solving
!
proportions
that are not connected to
geometric meaning
A Transformations-based Conception of
Similarity
Similarity is conceptualized as enlarging or reducing figures proportionally to
create a class of similar 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
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 11
A Geometric Transformations Approach to
the 8th Grade NAEP problem
Center of
Dilation
A Geometric Transformations Approach to
the 12th Grade NAEP problem
Center of
Rotation
A Geometric Transformations Approach to
the 12th Grade NAEP problem
Center of
Dilation
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.
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
15
Rectangle Problem & Randy
Rectangle Problem
Which rectangles are similar to rectangle a?
17
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 staticbased?
19
Representing Similar Rectangles as
Dilation Images
20
Learning and Teaching Geometry
Project’s Research on Teacher
and Student Learning
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)
22
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
23
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
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
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
Defining
Congruence and Similarity
Session 4
Properties of
Dilation
Session 5
Preservation
of Angles &
Proportional
Lengths
through
Dilation
Session 6
Ratios
Within and
Between
Similar
Figures
Relationships and Attributes
of 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
Connections
Session 10
Closure and
Re-capping
of Big Ideas
Closure
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
o
o
87 treatment
39 comparison
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 (2010-11 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
Pre
Treatment
(N=83)
Post
Mean
Mean
63.66
72.39†
Gain Scores
Effect
Size
+8.73
0.39
Comparison
(N=38)
†On
65.79
67.47
+1.68
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
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
Defining
Congruence 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
Relationships and Attributes
of Similar Figures
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 8
Connections
between
Similarity,
Slope &
Graphs of
Linear
Functions
Session 9
Area of
Similar
Figures
Connections
Session 10
Closure and
Re-capping
of Big Ideas
Closure
SORTING RECTANGLES TASK
LTG Student Geometry Assessment Field Test
Percent Correct Scores
Treatment
(N=162)
Pre
Post
Mean
Mean
36.42
45.28†
Gain Scores
Effect Size
+8.86
0.49
Comparison
(N=104)
†
42.69
45.00
+2.31
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 http://www.wested.org/cs/we/view/rs/1246

NCSM: Illustrating the Standards for Mathematical Practice
http://www.mathedleadership.org/ccss/itp/index.html
Other Resources:

Illustrative Mathematics Project: Tools for the Common Core
http://commoncoretools.me/2011/01/16/the-illustrative-mathematics-project/

Transformational Geometry, Richard Brown (1989)
Transformations-based Definitions
38
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.
THANK YOU!
nseago@wested.org