Contributed by:
The outcomes of this course are:
• Become familiar with the research in early childhood and mathematics
• Describe what the implications of this research and instruction are for young children and teachers
• Become familiar with the preschool learning foundations in mathematics with an emphasis on number sense
• Understand the links between the mathematics research in California’s Child Development Division’s initiatives (Preschool Learning Foundations-Mathematics, PreK Guidelines & Desired Results)
• Identify classroom practices that support all children’s mathematical growth and development
1.
1
Mathematics Foundations
Number Sense
A California Department of Education
Child Development Division Presentation
2.
2
Icebreaker
Use an icebreaker of
your choice. There are
two icebreaker slides
at the end of the
presentation for
optional use.
3.
3
Take a Stand
Did you like math when you were in school?
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4
Outcomes
• Become familiar with the research in early childhood
and mathematics
• Describe what the implications of this research and
instruction are for young children and teachers
• Become familiar with the preschool learning
foundations in mathematics with an emphasis on
number sense
• Understand the links between the mathematics
research in California’s Child Development Division’s
initiatives (Preschool Learning Foundations-
Mathematics, PreK Guidelines & Desired Results)
• Identify classroom practices that support all children’s
mathematical growth and development
5.
5
Norms
• Start on time and end on time.
• Turn off cell phones or to vibrate.
• Help the group stay on task.
• Listen to thoughts and ideas of others.
• Contribute your thoughts and ideas.
6.
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Parking Lot
Please write questions on post-its and
place them on chart paper titled
“Parking Lot.”
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Agenda
• What research tells us about the importance of early
mathematics
• National Council of Teachers of Mathematics - 5 Big Ideas
California’s Preschool Initiatives
• The Prekindergarten Learning and Development Guidelines
• The Early Childhood Environment Rating Scale
• The Desired Results Developmental Profile-Revised
• Preschool Learning Foundations
Research to Practice
• Exploring Number Sense in the classroom
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Lessons Learned from Research
• Gaps are striking
• Less is more
• Connect informal and school
math
• Meet the needs of all
learners
• Include geometry
• Use learning trajectories
Doug Clements, 2007 CPIN presentation
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Gaps are Striking
School Mathematics is improving but Not Working
Well Enough for Enough Students
1994-95 2007
Internationally our students
are not mathematically HIGHER HIGHER
competitive 20 countries 9 countries
United States SAME SAME
14 countries 2 countries
LOWER LOWER
7 countries 37 countries
Third International Mathematics and Science Study:Grade 8, 1994–1995 and 2007
9
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Comparisons: SES & Ethnicity
Low SES have gaps - Level 2:
79% of children with mothers with bachelor’s degree passed
vs.
32% of those whose mothers with less than a high school degree
Ethnic groups
70% of Asian and 66% of non-Hispanic white children passed
vs.
42% of African American, 40% of Hispanic, 48% of Hawaiian
Native or Pacific Islander, and 34% of American Indian or
Alaska Native
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International Comparisons
There are cultural differences: High
income and low income Chinese children
performed higher than high income U.S.
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Using Research
Early
mathematics
research helps
teachers be more
effective and
have more fun
teaching math.
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Core Mathematic Content Areas
PreK - 8th grade
Number and Operations
Geometry
Measurement
Algebra
Data Analysis & Probability
National Council Teachers of Mathematics
(NCTM)
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NCTM Focal Points
• Number and Operations
• Geometry
• Measurement
National Council Teachers of Mathematics
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NCTM Focal Points for
Prekindergarten Handout 1
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Research Bites
• In the table groups, open the research
bite.
• Share information and discuss how this
informs participant’s work.
• There will be an opportunity to share
insights with the large group.
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Research Bite #1
Infants are born with the ability to
understand numerical ideas
“Children possess and build mathematical
competencies from their first year and
keep on learning mathematical ideas
throughout their preschool years.”
Clements, D.H. & Sarama, J. “Creative Pathways to Math,” Scholastic Early
Childhood Today Journal, 2003.
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Research Bite # 2
“High quality teaching in mathematics is about
challenge and joy, not imposition and pressure.
Math instruction includes providing a lot of unit
blocks, along with loads of time to use them. It
invites children to experience mathematics as
they play in, describe, and think about their
Clements, D.H., “Mathematics in the Preschool,” Teaching Children, 2001.
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Research Bite #3
“Teachers should provide time, materials,
and support for children to engage in play,
to nourish their interest in exploration and
manipulation of mathematical ideas.”
National Association for the Education of Young Children (NAEYC) & National
Council of Teachers of Mathematics (NCTM) 2002. Early Childhood Mathematics:
Promoting Good Beginnings, A joint position statement of NAEYC & NCTM,
Washington D.C.
21.
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Research Bite #4
“Preschool children explore a variety
of mathematical ideas during play
including comparison, estimation,
patterns, symmetry, and spatial
K.-H., Seo. “What Children’s Play Tells Us about Teaching Mathematics,” Young
Children, January, 2003.
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Research Bite #5
• Find the mathematics in, and develop mathematics
from, children’s activity.
• Help children extend and “mathematize” every day
activities from building blocks, art, songs, and
puzzles.
• Create invitations to mathematical activities based on
children’s experiences and interests.
Clements & Sarama, Building Blocks Real Math Pre K Math Program, 2002.
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California DRDP-R
PrekGuidelines
Learning Preschool
Foundations, Framework,
Volume 1 Volume 1
ECERS-R
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California Preschool Learning
Foundations Handout 2A
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California’s Preschool Learning
Foundations - Mathematics
• Number Sense
• Algebra and Functions
(Classification and Patterning)
• Measurement
• Geometry
• Mathematical
Reasoning
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Map of the Foundations Handout 2B
Mathematics
Strand
Domain
Foundations
Includes notes for
children with
disabilities
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Prekindergarten Learning &
Development Guidelines
Chapter 6 - Curriculum:
Mathematics Learning
and Development
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Prekindergarten Learning & Development
Guidelines for Mathematics
Program develops and builds on children’s
existing informal mathematical knowledge,
recognizing that children enter preschool
with different experiences in mathematics.
Teacher-guided and child-initiated
activities are integrated in a
mathematically rich learning environment,
using multiple instructional approaches.
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Prekindergarten Learning & Development
Guidelines for Mathematics
The program implements a mathematics
curriculum that lays the foundation for
children’s success in mathematics in
elementary school.
The program identifies clear, age appropriate
goals for mathematics learning and
development.
The program establishes a partnership with
parents and other caregivers in preparing
children for mathematics learning.
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Prekindergarten Learning &
Development Guidelines
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Handout 3
Early Childhood Environmental
Rating Scales Revised (ECERS-R)
Item 26 - Math/Number:
Different types of materials
for math/number help
children to experience
counting, measuring,
comparing quantities,
recognizing shapes, and
become familiar with the
written number.
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Desired Results Developmental
Profile-Revised (DRDP-R)
Measures 22 & 23 DRDP-R
• An observational
assessment instrument
• Developmental
continuum
• For all children,
including English-
language learners and
children with disabilities
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Desired Result 2:
Children are effective learners
Indicator: MATH-Preschoolers demonstrate competence
in real-life mathematical concepts
Measure 22: Number sense: Understands quantity and
counting
Measure 23: Number sense: Math operations
Measure 24: Shapes
Measure 25: Time
Measure 26: Classification
Measure 27: Measurement
Measure 28: Patterning
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Number and Operations
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Number and Operations
Handout 4
Clements, D.H., Engaging Young Children In Mathematics, Lawrence Erlbaum Associates, 2004.
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Number and Operations in Play
• Recognizing quantities by sight without actually
counting objects
• Reading numbers
• Putting objects together or taking objects apart
• Adding or taking away numbers
Seo, K.-H. “What Children’s Play Tells Us about Teaching Mathematics,” Young
Children, January, 2003.
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Activities for Multiple Goals
• Make and imagine small collections of items nonverbally
• Count by ones to ten
• Know the last counting words tells “how many”
• Count out (produce a collection)
• Subitize - quickly “see” and label with a number
• Identify whether collections are the “same” number or
which is “more” visually
Clements, D. H., & Sarama, J., Building Blocks Real Math PreK Curriculum, 2007.
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Recognition of Small Number
• Early number is not counting, but
recognition of number
• Recognition of number is not numerals
but the understanding of discreet
quantities
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Clements, D.H., Engaging Young Children In Mathematics, 2004, pg. 17.
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Clements, D.H., Engaging Young Children in Mathematics, 2004, pg. 17.
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Recognition of Small Number
• Simple, but continuous, intervention
makes a big difference - Hannula’s 3-
year-olds
• Large differences in spontaneous
recognition of number
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Recognition of Small Number
Teachers can make huge differences by
providing thousands of spontaneous
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How many triangles did you see?
That’s Subitizing!
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Learning Trajectory for
Subitizing
• Small collection namer
• Maker of small collections
• Perceptual subitizer to 4
• Perceptual subitizer to 5
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Simple, but Continuous,
Intervention Makes a Difference!
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Developmental Sequence of
Counting
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Group Activity
Please Close Folders and Notes
• Take cards from envelope
• As a group, place cards in a
developmental sequence
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Developmental Sequence of
Counting
• Saying number words in sequence. May omit some
numbers when reciting the number words
• Counts a small set of objects (five or six) but may not
have one-to-one correspondence
• May count correctly a larger set of objects (about ten)
by keeping track of counted and uncounted objects
• Understands that the number name of the last
objects counted represents the total number of
objects in the group
Resource: California DRAFT Preschool Mathematics Framework
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Developmental Sequence of
Counting
• Knows to say the number words one to ten in
the correct order, but is still learning the
sequence between ten and twenty
• Creates set with a certain number of objects
• Knows to say the number words up to twenty
correctly
Resource: California DRAFT Preschool Mathematics Framework
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Levels of Math Operations
• Comparing & Ordering - Can tell which group has
more or less
• Adding to/Taking away - Solve problems with three
to five objects
• Composing and Decomposing - Can develop the
ability to recognize that three and two are “hiding
inside” five
• Grouping - Can count groups of objects
Clements, D.H., Engaging Young Children in Mathematics, 2004, pg 20-23.
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Handouts 5 & Measures 22 & 23
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Let’s Play
• Follow the directions to the game.
• After playing the game, talk about the foundations,
developmental sequence, DRDP-R measures involved in
the game. Use the handout to structure the discussion.
• Write a goal for a child in your class. For example, “Maria
will count to five using one-to-one correspondence.”
• Later participants will be asked to present the activity to the
group. Be ready to share the math concepts in the activity.
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Teachers are the key!
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The Importance of Questions
Mathematics is about thinking, not just
doing something with manipulatives.
Handout 6
• How did you know?
• Why did you do it that
way?
• How did you figure
that out?
• What else would
work like this?
• What would happen
if?
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The Power of Asking “Why?” and
“How did you know?”
“You present problems, and they figure out what to
do. Then you ask what process they used. I’m
amazed they learned to! They’ll use this
knowledge to answer science questions. They
really do critical thinking. Asking ‘How do you
know?’ Starting at Prek is very powerful!”
- Anne, preschool teacher in the TRIAD research project
Clements & Sarama - U.S. Dept. of Education's IES-funded TRIAD research project
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In Classrooms that Promote
Number Sense, Teachers Will…
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What Teachers Do to Support
Number Sense in the Classroom…
1. Move to the assigned chart.
2. Take five minutes to record ideas.
3. Move to next chart when signaled.
4. Add ideas to next chart and continue on
at the signal.
5. At the last chart, circle the top three ideas.
6. Share top three ideas with group.
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Activities for Multiple Goals
• Ability to recognize and duplicate a small
collection nonverbally
• Count by ones to ten
• Know the last counting words tells “how many”
• Count out (produce a collection)
• Subitize - quickly “see” and label with a number
• Identify whether collections are the “same”
number or which is “more” visually
Clements, D. H., & Sarama, J., Building Blocks Real Math PreK Curriculum, 2003.
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Tips for Supporting ALL Learners
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Handout 7
Tips for Supporting ALL Learners
• Assess what the child knows – scaffold.
• Slow down! Emphasize accuracy with
counting.
• Guide the child’s hand while counting, if they
are working on one-to-one correspondence.
• Repeated practice.
• Provide “wait” time.
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Tips for Supporting ALL Learners
• Make it concrete! Count “real” objects.
• Ask the child to make a verbal plan.
“Let’s count them by starting at the top.”
• Move items as they are counted.
• Involve the child’s whole body as much
as possible.
• Simplify vocabulary and directions
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Tips for Supporting English
Learners
• Use concise language - speak clearly
• Use oral descriptions when talking
about concepts
• Model and act out - Total Physical
Response (TPR)
• Consider stages of language
development
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Let’s Play - Part 2
Now revisit your game and consider the following:
1. How might you document children’s growth and development?
2. How might you adapt this game for children with disabilities?
3. What are some strategies for English Learners?
Refer to the back side of your Let’s Play worksheet to
facilitate your discussion and record your ideas.
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In Classrooms That Promote
Number Sense, You Will See
Children…
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DRDP-R Measures 22 and 23
1. Review the measures at the
Handout 8
table.
2. Brainstorm what children might
do at each developmental level.
3. Record ideas on the measure
pages provided.
4. How might you document
children’s learning?
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Taking It Back to the Classroom
Consider: Handout 9
• The research
• The activities
• The games played and
displays
• Conversations
• Handouts
• Ideas that were new or
important
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Five Little Seashells Handout 10
Five little seashells sleeping on the shore.
(Hold up 5 fingers, bending down one for each verse)
Swish! went a big wave, and then there were four.
(Move arms, palms up, to make the waves)
Four little seashells quiet as can be.
Swish! went a big wave, and then there were three.
Three little seashells pearly and new.
Swish! went a big wave, and then there were two.
Two little seashells having great fun.
Swish! went a big wave, and then there was one.
One little seashell lying in the sun.
Swish! went a big wave, and then there were none.
Five little seashells gone out to sea.
(Point out to sea)
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CDE Web Site
• At the Web address, the underlined Preschool
Learning Foundations link leads to the 192 page
publication providing easy access to the chapters and
sections.
• The Appendix, on pages 173-192, provides a
summary list of the foundations.
• Frequently Asked Questions (FAQ) are posted on the
Web site. Questions can be sent to
psfoundations@cde.ca.gov.
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To Purchase the Preschool
Learning Foundations Book
• The Preschool Learning Foundations
publication is available for purchase from the
CDE Press for $19.95 plus shipping and
handling.
• Ordering information can be found at the CDE
Web site www.cde.ca.gov/re/pn/rc or by
calling 1- 800-995-4099
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To Purchase PEL Guide
NOW in Spanish!
• Preschool English Learners:
Principles and Practices to
Promote Language, Literacy
and Learning publication is
available for purchase from the
CDE Press for $15.95.
• Ordering information can be
found at the CDE Web site
www.cde.ca.gov/re/pn/rc or by
calling 1-800-995-4099.
• Appendix A is translated into
Chinese, Hmong, Korean,
Spanish, Tagalog, and
Vietnamese, and is available on
the CDE Web site.
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Take a Stand
Did you like math when you were in school?
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CPIN - Region
Contact info added
You can announce
your next event or ???
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Thank You for Coming!
Please take a minute and complete
the evaluation before leaving.
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Optional Icebreakers
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Grade Yourself in Math
• On a scale of “1” (A+) to “5” (F), how did
you perform in math in school?
• In your math group, talk about
experiences with Math in school.
• Be ready to share with the whole group.
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The following slides are
for the debrief of
activities and the Make
and Take directions.
Place them where you
think they fit best.
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How Did the Activity Support
Mathematical Thinking and the
Foundations?
• Share your experience with the activity
& materials used.
• What foundations did the activity
address?
• Are there any changes or additions to
make?
83.
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How to Play “Peace”
• Each player has 5 shuffled cards.
• Place cards face down in a pile in front of
you.
• Turn one card over and partner turns one
over.
• Whoever has the card with the most dots
takes the pair.
• Continue playing until someone has all the
cards.
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