1 ©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies,...

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©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies, California Preschool Instructional Network (CPIN). (08/2013)

Math Foundations: Geometry©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies, California Preschool Instructional Network (CPIN). (08/2013)

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Agenda• The relationship between the NCTM Standards,

the Preschool Learning Foundations, and the DRDP-R

• The Research

• Exceptional Children--Division of Early Childhood Recommended Practices

• English Learner Strategies

• Taking It Back to the Classroom

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Outcomes• Become familiar with the mathematics

foundations with an emphasis on the Geometry strand.

• Explain how the DRDP-R and the NCTM Standards relate to the mathematics foundations.

• Discover the research behind the practice.

• Explore strategies to make geometry come to life in the classroom.

• Plan for taking it back to your classroom.

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California’s Early Learning and Development System

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Two California Department of Education Resources

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California Preschool Learning Foundations

“Geometry is the study of space and shape (Clements, 1999). Geometry and spatial reasoning offer a way to describe, interpret, and imagine the world.”

PLF, Vol. 1, p. 164

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©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies, California Preschool Instructional Network (CPIN). (08/2013) ©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies, California Preschool Instructional Network (CPIN). (08/2013)

What’s Your What’s Your Shape?Shape?

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sphere coneprism

cylindercube pyramid cuboid

Name Your Shape

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Peeking Inside the Mathematics Peeking Inside the Mathematics FoundationsFoundations


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FoundationsFoundations

With appropriate supportWith appropriate support

After 1After 1stst or 2 or 2ndnd year year of preschoolof preschool

At 48 and 60 monthsAt 48 and 60 months

High-quality programHigh-quality program


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The Sections

• Introduction

• The Foundations

• Bibliographic Notes

• Glossary

• References and Source Material

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StrandStrand

AgeAge

DomainDomainMathematics

SubstrandSubstrand

ExamplesExamples

FoundationFoundation


Map of the Foundations

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DomainDomain

Foundations Organization

Age Age

StrandStrand

Substrand Substrand


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Geometry: It’s more than shapes!

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Geometry: It’s more than shapes!

• Shapes

• Locations, Directions, and Spatial Orientation

• Visualization and Spatial Reasoning

• Transformation and Symmetry

NTCM, Showcasing Mathematics for the Young Child, pp. 56-57.

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Universal Design for LearningProvides for:Provides for:• Multiple Multiple

means of means of representationrepresentation

• Multiple Multiple means of means of engagementengagement

• Multiple Multiple means of means of expressionexpression

PCF, Vol. 1, p. 13PCF, Vol. 1, p. 13


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Multiple means of…

Credit: CAST: http://www.cast.org/udl/

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Today’s Lesson: The Circle

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Circle UDLCircleDraw one… Make one…

Find one… Show us…

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What can we What can we learn from the learn from the

research about research about children and children and

shapes?shapes?


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Circles

• Correctness:– 92% - 4 year olds– 96% - 5 year olds – 99% - 6 year olds

• Few youngest chose ellipse and curved shape• Described as “round”• Thus, easily recognized but difficult to describe• Matched shapes to visual prototype

1

2

4

8

6

5

7

12

3

10

13

9

14

15

11

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Squares

• Only slightly less accurate in identifying squares:82%, 86%, and 91%

• Youngest lower on non-square rhombi, but not on “tilted squares”

• Minority reasoned about properties.

– Was relationship between such responses and correct selections

1

2 3 4 5

6 78

9

12

13

1110

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Triangles Task

• Lower, but not low: about 60%

• Property responses present, but only 18%

• Inverse-U pattern: 5’s more likely than4’s or 6’s accept bothnon-standard trianglesand those withcurved sides

1

23

4

5 6

7

8

9

10

11

12

13

14

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Rectangles

• Slightly more than 50%

• Four-year-olds were more likely to accept the squares

• All accepted “long” quads w/ pair of parallel sides(numbers 3, 6, 10, and 14)

• Properties less frequently

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1

2

3

4 5

68

9

10

11

12

13

14

15

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Compare to Elementary—Rectangles

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Compare to Elementary—Triangles

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What Children See

Ring the triangle

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What Children See

More recent, but not more mathematically precise

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Geometry Must Move Beyond “Basic” Shape

Naming to:• Parts & Properties

– Shape attributes

– Include analysis and description

• Mental images and transformations

• Composing and decomposing

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Learning Trajectory for Shapes

1. Shape Matcher

2. Shape Prototype Recognizer

3. Shape Recognizer

4. Side Recognizer

5. Angle Recognizer

6. Shape Class Identifier

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Curriculum Framework

•Interactions and Strategies

•Teaching Vignettes

•Teachable Moments

•Guiding Principles©2013 California Department of Education (CDE) with the WestEd Center for Child &

Family Studies, California Preschool Instructional Network (CPIN). (08/2013)

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Interaction and StrategyInteraction and Strategy

Talk about shapes and discuss their attributes. PCF, Vol. 1, p. 284


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What it might look like…

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Application for Dual-Application for Dual-Language LearnersLanguage Learners


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Talk About the Here and Now• Gives the child the chance to narrow down Gives the child the chance to narrow down

what the conversation is aboutwhat the conversation is about

• Allows child to focus in on a more restricted Allows child to focus in on a more restricted number of options to respondnumber of options to respond

• Share the same physical space and have Share the same physical space and have reference to the same informationreference to the same information

• Allows the teacher to provide some missing Allows the teacher to provide some missing vocabulary informationvocabulary information

✔ Stage 1: Home Language

✔ Stage 2: Observational/Listening

✔ Stage 3: Telegraphic/Formulaic

✔ Stage 4: Fluid Use


Handout 4

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Helpers/Buddies Helpers/Buddies


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Geometry: It’s More than Space

Geometric measuring concepts can be motivated through motion.

Barrett, J.E., Sarama, J., Clements, D.H., Cullen, C., McCool, J., Witkowski-Rumsey, C., & Klanderman, D. (2012

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Geometry Geometry andandSpatial SenseSpatial Sense

Get ReadyGet Ready

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Geometry SnapshotsReady

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Snapshots with Pattern BlocksReady GO!

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Learning Trajectory for Composing Geometric Shapes

1. Pre-composer

2. Picture maker

3. Shape composer

4. Substitution composer

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Language is a critical element in mathematics. (PCF, Vol. 1, p. 235)

• Shape Flip Book!

• The Greedy Triangle

• Mirror books

• The Shape of Things

• Dot and Line

• What other books do you know about?

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Children Think About Geometry

• Pat H.’s preschool

was studying bees.

• Children noticed the

hexagon.

• Teacher: Why do

you think they chose

this shape?

• Let’s see some of the

others…

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Children Talk About Geometry

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Play It - Chart It - Share It• Find your chart. Play your game. Answer the following

questions as a group on the chart paper.

• What was fun about the game?• What foundations were addressed in the game?• What components of the learning trajectory are

emphasized in the game?• What might be important to consider for dual-language

learners (use the strategies handout)?• If a teacher had observed you playing the activity,

which measure of the DRDP might he or she have collected data about?

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How will it all fit together?


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Teachers Put it all together

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What are you packing for home?

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Thank you for coming; now go

take a rest!

1 ©2013 California Department of Education (CDE) with the WestEd Center for Child & Family Studies,...

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