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Preschool and
Kindergarten
Mathematics: A
National Conference
NCTM’s Principles and Standards for
School Mathematics (2000) addresses
preschool mathematics education, which
is a first for the NCTM’s Standards documents. We
celebrate this initial coverage but wonder whether
Principles and Standards has enough detail for
early childhood teachers and caregivers. We are
concerned that although the document offers a good
start, it might not provide sufficient guidelines.
Without these guidelines, we face the danger that a
wide variety of incoherent standards will be pro-
duced, some of which may be developmentally
inappropriate. A lack of consistency across various
standards and guidelines will continue to
result in “mile wide and inch deep” curricula
(National Center for Education Statistics
1996) as publishers struggle to meet a variety
of different content standards and guidelines.
Because we believe in the importance of
supporting early communication and coordi-
nating efforts among educational leaders and agen-
cies, we held a national Conference on Standards
for Preschool and Kindergarten Mathematics Edu-
cation in May 2000, in Arlington, Virginia. This
conference was funded by grants from the Exxon-
Mobil Foundation and the National Science Foun-
dation (NSF) to the State University of New York
at Buffalo. The conference was a historic event, the
first to bring together a comprehensive range of
experts in the diverse fields concerned with creat-
ing educational standards. The participants includ-
ed representatives from state departments of edu-
cation and the U.S. Department of Education;
mathematicians; early childhood teachers and
mathematics teachers; early childhood policymak-
ers and researchers; mathematics education
researchers; curriculum developers; representatives
from NCTM, including the writing group for
grades pre-K–2 of Principles and Standards; and
representatives from the funding foundations, NSF
and ExxonMobil.
The conference was, according to the partici-
pants, a resounding success. The presentations and
panels were lively and informative. The discus-
sions, in which ideas were shared, were also pro-
ductive and enjoyable. Information from the con-
ference, including transcripts of every discussion,
will soon be published in a book. This article
touches the surface of the extensive information
from the conference and in the book.
Conference Highlights
With so many different perspectives represented in
the discussions, we would expect to find many dis-
agreements. Surprisingly, the most passionate
debate centered on a single question: Should we
establish standards for young children at all?
Those connected with NCTM’s Principles and
Standards were in support of standards and wanted
Early Childhood
Corner
510 TEACHING CHILDREN MATHEMATICS
Douglas Clements, clements@buffalo.edu, and Julie Sarama, jsarama@buffalo.edu, teach at
the State University of New York at Buffalo, Buffalo, NY 14260. Ann-Marie DiBiase, dibiase
@iaw.on.ca, is a doctoral candidate in elementary education at the same university and is
also a certified elementary school teacher.
Edited by Julie Sarama and Douglas Clements. This department addresses the early child-
hood teacher’s need to support young children’s emerging mathematics understandings and
skills in a context that conforms with current knowledge about the way that children in
prekindergarten and kindergarten learn mathematics. Readers are encouraged to send manu-
scripts for this section to “Early Childhood Corner,” NCTM, 1906 Association Drive, Reston,
VA 20191-9988.
Douglas H. Clements,
Julie Sarama, and
Ann-Marie DiBiase
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
511
MAY 2002
FIGURE 1
Big ideas in mathematics for early childhood learning
to elaborate on those in the organization’s docu-
ment. Several other people, especially teachers,
were concerned about, and even vehemently
argued against, the very notion of standards for
early childhood learning. The discussion revealed
that these two groups were talking about two dif-
ferent types of standards, thereby leading to the
following conclusion from the conference:
There is a substantial and critical difference between stan-
dards as a vision of excellence and standards as narrow and
rigid requirements for mastery. Only the former, including
flexible guidelines and ways to achieve learning goals, is
appropriate for early childhood mathematics education at
the national level.
The conference participants also addressed the
question of whether standards should be estab-
lished for children or for programs or for both. Sue
Bredekamp, one of the creators of the ideas of
“developmentally appropriate practice,” answered
the question by saying, “We need both.” Most par-
ticipants agreed that although standards for chil-
dren can certainly be abused, we can avoid abuses
and realize four advantages in setting standards.
First, standards for children can demystify beliefs
about children’s abilities. Teachers usually wel-
come more specific guidance on learning goals that
are linked with age and grade levels, such as those
published in a recent joint position statement on
developmentally appropriate practices in early lit-
eracy (NAEYC and International Reading Associ-
ation 1998). Second, standards can give teachers of
young children needed guidance about appropriate
expectations for children’s learning. Moreover,
they can focus that learning on important knowl-
edge and skills, including critical thinking skills.
Third, standards can help parents better understand
their children’s development and learning and pro-
vide appropriate experiences for them. Finally,
standards can help achieve equity by ensuring that
the mathematical potential of all young children is
developed throughout their lives.
Having agreed on the advantages of setting stan-
dards, the conference participants made two
• Numbers can be used to tell us how many, describe
order, and measure; they involve numerous relations, and
can be represented in various ways.
• Operations with numbers can be used to model a variety
of real-world situations and to solve problems; they can
be carried out in various ways.
Number and Operations • Patterns can be used to recognize
relationships and can be extended
to make generalizations.
Algebra
• Geometry can be used to understand
and to represent the objects, direc-
tions, locations in our world, and the
relationships between them.
• Geometric shapes can be described,
analyzed, transformed, and com-
posed and decomposed into other
shapes.
Geometry
• Comparing and measuring can be used to
specify “how much” of an attribute (e.g., length)
objects possess.
• Measures can be determined by repeating a
unit or using a tool.
Measurement
• Data analysis can be
used to classify, repre-
sent, and use informa-
tion to ask and answer
questions.
Data Analysis
Problem Solving
Communication
Representation Reasoning
Connections
FIGURE 2
Developmental guidelines for number and operations—a sample continuum
PreK* Kindergarten
Topic Ages 2–4 Ages 4–5 Ages 5–6
a. A key element of object-
counting readiness is non-
verbally representing and
gauging the equivalence of
small collections.
Counting
Counting can be used to find out how many in a collection.
Make and imagine small col-
lections of one to four items
nonverbally, such as seeing
••, which is covered, then
putting out ••.
Find a match equal to a collection of one to four items, such as
matching :: or four drum beats to collections of four with different
arrangements, dissimilar items, or mixed items (e.g., ✯• ∆).
b. Another key element of
object-counting readiness is
learning standard sequences
of number words, learning that
is facilitated by discovering
patterns.
Verbally count by ones from . . .—————————————————————————————————
1 to 10 1 to 30 (and more), with
emphasis on counting pat-
terns (e.g., knowing that
twenty-one, twenty-two . .. is
parallel to one, two . . .).
1 to 100, with emphasis on pat-
terns (e.g., the decades, such as
sixty, seventy, parallel the corre-
sponding ones, six, seven; also,
the teens, such as fourteen to
*Ages reflect those typically found in classes or groups of children; for example, in the first category, ages 2–4 years, a typical classroom of three-year-olds may begin the
year with some two-year-olds and end the year with some children who are just turning four.
Flexibly start verbal count-by-one sequence from any point; that
is, start a count from a number other than 1 (ends early in first
grade for some).
Flexibly state the next number word . .. ————————————————————————–>
after 2 to 9 with a running start. after 2 to 9 without a running start to 9; also, state the word
before any number from 2 to 9.
Verbally count backward . . .———————————————————
from 5.
by tens.
from 10.
Skip count . .. ——————————
Grades and Ages
c. Object counting involves
creating a one-to-one corre-
spondence between a number
word in a verbal counting
sequence and each item of a
collection, using some action
to indicate each item as a
number word is spoken.
Count the items in a collection and know that the last counting word tells how many. ——————————
1 to 4 items 1 to 10 items 1 to 20 items
1 to 4 items 1 to 10 items 1 to 20 items
Count out (produce) a collection of a specified size; this ability lags a bit behind counting items in a
collection. ————————––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
d. Number patterns can facili-
tate determining the number
of items in a collection or rep-
resenting it.
Verbally subitize (quickly “see” and label with a number) . .. —————————————————————
Represent collections with a finger pattern . .. ———————————————————————————
collections of 1 to 3.
for 1 and 2. up to 5. up to 10.
collections of 1 to 5. collections of 1 to 6; patterns
up to 10.
assumptions about developing standards for early
childhood learning. First, emphasis should be
placed on standards for programs and for teaching.
Given the wide range of development in the early
years, rigid standards for young children are not as
useful as standards for high-quality programs and
teaching. However, curricula and teaching must be
built on extensive knowledge of young children’s
mathematical acting, thinking, and learning.
The second assumption was that knowledge of
what young children can do and learn, as well as
specific learning goals, is necessary for teachers to
realize a vision of high-quality early childhood
education. How is this knowledge communicated
if not as rigid standards? The decision at the con-
ference was to structure this knowledge as curricu-
lum standards—descriptions of the ideas and skills
that programs should enable children to under-
stand and perform. We believe that mathematics
curriculum standards for early childhood educa-
tion should be written as flexible guidelines along
learning paths for young children’s mathematical
understanding. These guidelines should meet the
following criteria:
1. Guidelines should be based on available
research and expert practice.
2. Guidelines should focus on the “big ideas” in
mathematics for children.
3. Guidelines should represent a range of expecta-
tions for outcomes that are developmentally
appropriate.
We specified big ideas for each of the five main
topics of Principles and Standards (see fig. 1). We
knew that teachers, curriculum developers, and
others also needed far more specificity. For the
finest level of detail, we designed a developmental
or learning continuum for each idea in each topic.
The goal of these representations is to illuminate
potential developmental paths and to encourage
teachers to present their students with activities
that are appropriate to their abilities, that is, activ-
ities that can be mastered but are challenging for
each child. Figure 2 shows a small portion of the
continuum for number, which is one of the topics.
We caution readers that the competencies in
these paths are developmental guidelines, not
detailed directions for curriculum, teaching, or
assessment. The activities in which children
engage to acquire these competencies should pro-
vide rich, integrated experiences that enable them
to develop several competencies simultaneously,
including ones that go beyond those in the contin-
uum. For example, consider the activities from the
Building Blocks project, which is an NSF-funded
project to develop and evaluate an innovative
513
MAY 2002
1 to 1000, with emphasis on
patterns (e.g. the hundreds,
one hundred, two hundred,
parallel one, two).
by fives, twos. by threes, fours.
from 20.
1 to 100 items
1 to 100 items
using groups of 10
Use skip counting to determine how many. –——>
Switch among counts (e.g.,
“100, 200, 300, 310, 320,
321, 322, 323”).
2, 5, or 10 at a time
for teens as 10 and more;
uses tens flexibly to count
on, etc.
Grade 1 Grade 2
Ages 6–7 Ages 7–8
——————————————————————––——>
nineteen, parallel
the corresponding
ones, four through
nine).
—————————————>
————————————————————>
———————————>
–––––––––––––––––—>
————————>
————————————>
514 TEACHING CHILDREN MATHEMATICS
preschool–grade-2 curriculum. The Building
Blocks program incorporates both old and new
technologies, from blocks and puzzles to multimedia
computer programs. Preliminary evaluations show
that the program’s approach of finding the mathe-
matics in, and developing mathematics from, chil-
dren’s everyday activities allows children to learn
and do more mathematics than was previously
thought possible.
In the Double Trouble activity from Building
Blocks, the teacher tells a story of Mrs. Double, who
is throwing a birthday party for her twins. The twins
like their cookies to have the same number of choco-
late chips. Pretending that rugs are cookies and chil-
dren are chips, the teacher has children act out situa-
tions from the story. For example, one group of four
students might sit on a rug cookie and pretend to be
chips, and another group makes a cookie with the
same number of chips. Several follow-up activities
can be conducted. The Double Trouble computer
activity has numerous levels of difficulty. In one, the
on-screen character Mrs. Double asks children to
make a “twin” cookie with the same number of chips
as a cookie that Mrs. Double has made. A later activ-
ity is similar, but Mrs. Double makes a cookie, then
covers it with a napkin.
Another group of follow-up activities consists of
cookie games that children play on a mat that con-
tains a picture of a dinner plate at the top and cook-
ies with no chips below. Player 1 rolls a die and puts
that many chips, say, six, on her “plate.” Player 2
must agree that player 1 is correct. If so, player 1
puts the chips on her cookies, trying to get four (or
whatever number the children are working on that
day) on each. For example, if player 1 rolled six,
she could put four chips on one cookie and start
another cookie with two chips. Players take turns.
The winner is the first player to get four chips on
each cookie. Teachers should usually try to have all
children be “winners,” one after another.
These simple activities are appropriate for
preschool, and they meet multiple goals. For exam-
ple, the cookie game addresses almost all the goals
in figure 3. Children make small collections, non-
verbally if they prefer; that is, a child sees two on a
die and puts two chips on the plate. Children also
subitize, or quickly see a group and tell how many.
They count by ones and learn that the last counting
word tells “how many.” Children count out, or pro-
duce, a collection of a specified size. To check one
another, children identify whether collections are the
“same number.” As do all good activities, whether
planned, such as these, or the equally important inci-
dental, informal activities of the day, the cookie
game helps children develop several skills and con-
cepts at once.
Further Information
Please visit our Web site, www.gse.buffalo.edu/org/
conference/index.htm, for more information and
news about the upcoming publication of the book.
We will also keep site visitors informed about one
final, exciting outcome of the conference: the NCTM
and National Association for the Education of Young
Children have formed a committee to issue a joint
position statement on early childhood mathematics.
References
National Association for the Education of Young Children
(NAEYC) and International Reading Association. “Learning
to Read and Write: Developmentally Appropriate Practices for
Young Children.” Young Children (July 1998): 30–46.
National Center for Education Statistics (NCES). Pursuing Excel-
lence. NCES 97-198. Washington, D.C.: U.S. Government
Printing Office, 1996. www.ed.gov/NCES/timss.
National Council of Teachers of Mathematics (NCTM). Princi-
ples and Standards for School Mathematics. Reston, Va.:
NCTM, 2000.
Time to prepare this material was partially provided
by three grants, two from the National Science Foun-
dation (NSF): ESI-98-17540, “Conference on Stan-
dards for Preschool and Kindergarten Mathematics
Education” (www.gse.buffalo.edu/org/conference/
index.htm), and ESI-9730804, “Building Blocks—
Foundations for Mathematical Thinking, Pre-
Kindergarten to Grade 2: Research-Based Materials
Development” (www.gse.buffalo.edu/org/building
blocks/), and one from the ExxonMobil Foundation,
also titled “Conference on Standards for Preschool
and Kindergarten Mathematics Education.” Any
opinions, findings, and conclusions or recommenda-
tions expressed in this publication are those of the
authors and do not necessarily reflect the views of
either foundation. ▲
FIGURE 3
Goals from figure 2 that are addressed by the cookie game
1. Make and imagine small collections of items nonverbally
2. Count by ones to 10
3. Know that the last counting word tells “how many”
4. Count out (produce) a collection of a specified size
5. Subitize (quickly “see” and label with a number)
6. Identify whether collections are the “same” number or which is “more” visually
