Childhood Mathematics

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Sharp Tutor Tue, Jan 11, 2022 05:20 PM UTC

A positive attitude toward mathematics and a strong foundation for mathematics learning begins in early childhood. These good beginnings reflect all the characteristics of good early childhood education: a deep understanding of children’s development and learning; a strong community of teachers, families, and children; research-based knowledge of early childhood curriculum and teaching practices; continuous assessment in the service of children’s learning; and abiding respect for young children’s families, cultures, and communities.

1. POSITION STATEMENT
Early Childhood Mathematics:
Promoting Good Beginnings
A joint position statement of the National Association for the Education of Young Children (NAEYC)
and the National Council of Teachers of Mathematics (NCTM). Adopted in 2002. Updated in 2010.
Position solid foundation for success in school. In elemen-
tary and middle school, children need mathemat-
The National Council of Teachers of Mathemat-
ical understanding and skills not only in math
ics (NCTM) and the National Association for the
courses but also in science, social studies, and
Education of Young Children (NAEYC) affirm that
other subjects. In high school, students need
high-quality, challenging, and accessible mathe-
mathematical proficiency to succeed in course
matics education for 3- to 6-year-old children is a
work that provides a gateway to technological
vital foundation for future mathematics learning.
literacy and higher education [1–4]. Once out
In every early childhood setting, children should
of school, all adults need a broad range of basic
experience effective, research-based curriculum
mathematical understanding to make informed
and teaching practices. Such high-quality class-
decisions in their jobs, households, communities,
room practice requires policies, organizational
and civic lives.
supports, and adequate resources that enable
Besides ensuring a sound mathematical
teachers to do this challenging and important
foundation for all members of our society, the
nation also needs to prepare increasing numbers
of young people for work that requires a higher
The challenges proficiency level [5, 6]. The National Commission
on Mathematics and Science Teaching for the
Throughout the early years of life, children notice
21st Century (known as the Glenn Commission)
and explore mathematical dimensions of their
asks this question: “As our children move toward
world. They compare quantities, find patterns,
the day when their decisions will be the ones
navigate in space, and grapple with real problems
shaping a new America, will they be equipped
such as balancing a tall block building or sharing
with the mathematical and scientific tools needed
a bowl of crackers fairly with a playmate. Math-
to meet those challenges and capitalize on those
ematics helps children make sense of their world
opportunities?” [7, p. 6]
outside of school and helps them construct a
Copyright © 2002 National Association for the Education of Young Children
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2. Early Childhood Mathematics
Since the 1970s a series of assessments of In 2000, with the growing evidence that the
U.S. students’ performance has revealed an over- early years significantly affect mathematics learn-
all level of mathematical proficiency well below ing and attitudes, NCTM for the first time includ-
what is desired and needed [5, 8, 9]. In recent ed the prekindergarten year in its Principles and
years NCTM and others have addressed these Standards for School Mathematics (PSSM) [19].
challenges with new standards and other re- Guided by six overarching principles—regarding
sources to improve mathematics education, and equity, curriculum, teaching, learning, assess-
progress has been made at the elementary and ment, and technology—PSSM describes for each
middle school levels—especially in schools that mathematics content and process area what chil-
have instituted reforms [e.g., 10–12]. Yet achieve- dren should be able to do from prekindergarten
ment in mathematics and other areas varies through second grade.
widely from state to state [13] and from school
district to school district. There are many en-
couraging indicators of success but also areas of NCTM Principles for School
continuing concern. In mathematics as in Mathematics
literacy, children who live in poverty and who are
members of linguistic and ethnic minority groups Equity: Excellence in mathematics education
demonstrate significantly lower levels of achieve- requires equally high expectations and
ment [14–17]. strong support for all students.
If progress in improving the mathematics Curriculum: A curriculum is more than a col-
proficiency of Americans is to continue, much lection of activities; it must be coherent,
greater attention must be given to early math- focused on important mathematics, and well
ematics experiences. Such increased awareness articulated across the grades.
and effort recently have occurred with respect to Teaching: Effective mathematics teaching re-
early foundations of literacy. Similarly, increased quires understanding of what students know
energy, time, and wide-scale commitment to the and need to learn and then challenging and
early years will generate significant progress in supporting them to learn it well.
mathematics learning.
Learning: Students must learn mathematics
The opportunity is clear: Millions of young
with understanding, actively building new
children are in child care or other early educa-
knowledge from experience and prior knowl-
tion settings where they can have significant
edge.
early mathematical experiences. Accumulating
research on children’s capacities and learning Assessment: Assessment should support the
in the first six years of life confirms that early learning of important mathematics and fur-
experiences have long-lasting outcomes [14, 18]. nish useful information to both teachers and
Although our knowledge is still far from com- students.
plete, we now have a fuller picture of the math- Technology: Technology is essential to teach-
ematics young children are able to acquire and ing and learning mathematics; it influences
the practices to promote their understanding. the mathematics that is taught and enhances
This knowledge, however, is not yet in the hands students’ learning.
of most early childhood teachers in a form to ef-
Note: These principles are relevant across all
fectively guide their teaching. It is not surprising
grade levels, including early childhood.
then that a great many early childhood programs
have a considerable distance to go to achieve
The present statement focuses on children
high-quality mathematics education for children
over 3, in large part because the knowledge
age 3-6.
base on mathematical learning is more robust
for this age group. Available evidence, however,
Copyright © 2002 National Association for the Education of Young Children
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3. NAEYC/NCTM Joint Position Statement
indicates that children under 3 enjoy and benefit Recognition of the importance of good begin-
from various kinds of mathematical explorations nings, shared by NCTM and NAEYC, underlies
and experiences. With respect to mathematics this joint position statement. The statement de-
education beyond age 6, the recommendations scribes what constitutes high-quality mathemat-
on classroom practice presented here remain ics education for children 3–6 and what is nec-
relevant. Further, closely connecting curriculum essary to achieve such quality. To help achieve
and teaching for children age 3–6 with what is this goal, the position statement sets forth 10
done with students over 6 is essential to achieve research-based, essential recommendations to
the seamless mathematics education that chil- guide classroom1 practice, as well as four recom-
dren need. mendations for policies, systems changes, and
other actions needed to support these practices.
In high-quality mathematics education 8. provide ample time, materials, and teacher
for 3- to 6-year-old children, teachers and support for children to engage in play, a
other key professionals should context in which they explore and manipulate
mathematical ideas with keen interest
1. enhance children’s natural interest in math-
ematics and their disposition to use it to make 9. actively introduce mathematical concepts,
sense of their physical and social worlds methods, and language through a range of ap-
propriate experiences and teaching strategies
2. build on children’s experience and knowl-
edge, including their family, linguistic, cultural, 10. support children’s learning by thoughtfully
and community backgrounds; their individual and continually assessing all children’s math-
approaches to learning; and their informal- ematical knowledge, skills, and strategies.
knowledge
To support high quality mathematics edu-
3. base mathematics curriculum and teaching
cation, institutions, program developers,
practices on knowledge of young children’s
and policy makers should
cognitive, linguistic, physical, and social-
emotional development 1. create more effective early childhood teach-
er preparation and continuing professional
4. use curriculum and teaching practices that
development
strengthen children’s problem-solving and
reasoning processes as well as representing, 2. use collaborative processes to develop well
communicating, and connecting mathematical aligned systems of appropriate high-quality
ideas standards, curriculum, and assessment
5. ensure that the curriculum is coherent and 3. design institutional structures and policies
compatible with known relationships and se- that support teachers’ ongoing learning, team-
quences of important mathematical ideas work, and planning
6. provide for children’s deep and sustained 4. provide resources necessary to overcome
interaction with key mathematical ideas the barriers to young children’s mathematical
proficiency at the classroom, community, insti-
7. integrate mathematics with other activities
tutional, and system-wide levels.
and other activities with mathematics
1
Classroom refers to any group setting for 3- to 6-year-olds
(e.g., child care program, family child care, preschool, or
public school classroom).
Copyright © 2002 National Association for the Education of Young Children
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4. Early Childhood Mathematics
Recommendations 2. Build on children’s experience and knowl-
edge, including their family, linguistic,
Within the classroom cultural, and community backgrounds;
To achieve high-quality mathematics edu- their individual approaches to learning;
cation for 3- to 6-year-old children, teach- and their informal knowledge.
ers2 and other key professionals should Recognizing and building on children’s individ-
1. Enhance children’s natural interest in ual experiences and knowledge are central to
mathematics and their disposition to use it effective early childhood mathematics educa-
to make sense of their physical and social tion [e.g., 20, 22, 29, 30]. While striking similari-
worlds. ties are evident in the mathematical issues that
Young children show a natural interest in and interest children of different backgrounds [31],
enjoyment of mathematics. Research evidence it is also true that young children have varying
indicates that long before entering school chil- cultural, linguistic, home, and community expe-
dren spontaneously explore and use mathemat- riences on which to build mathematics learning
ics—at least the intuitive beginnings—and their [16, 32]. For example, number naming is regular
mathematical knowledge can be quite complex in Asian languages such as Korean (the Korean
and sophisticated [20]. In play and daily activi- word for “eleven” is ship ill, or “ten one”), while
ties, children often explore mathematical ideas English uses the irregular word eleven. This
and processes; for example, they sort and clas- difference appears to make it easier for Korean
sify, compare quantities, and notice shapes and children to learn or construct certain numeri-
patterns [21–27]. cal concepts [33, 34]. To achieve equity and
educational effectiveness, teachers must know
Mathematics helps children make sense of the as much as they can about such differences
physical and social worlds around them, and and work to build bridges between children’s
children are naturally inclined to use math- varying experiences and new learning [35–37].
ematics in this way (“He has more than I do!”
“That won’t fit in there—it’s too big”). By capi- In mathematics, as in any knowledge domain,
talizing on such moments and by carefully plan- learners benefit from having a variety of ways
ning a variety of experiences with mathemati- to understand a given concept [5, 14]. Building
cal ideas in mind, teachers cultivate and extend on children’s individual strengths and learn-
children’s mathematical sense and interest. ing styles makes mathematics curriculum and
instruction more effective. For example, some
Because young children’s experiences fun- children learn especially well when instruc-
damentally shape their attitude toward tional materials and strategies use geometry to
mathematics, an engaging and encouraging convey number concepts [38].
climate for children’s early encounters with
mathematics is important [19]. It is vital for Children’s confidence, competence, and inter-
young children to develop confidence in their est in mathematics flourish when new expe-
ability to understand and use mathematics— riences are meaningful and connected with
in other words, to see mathematics as within their prior knowledge and experience [19, 39].
their reach. In addition, positive experiences At first, young children’s understanding of a
with using mathematics to solve problems mathematical concept is only intuitive. Lack of
help children to develop dispositions such as explicit concepts sometimes prevents the child
curiosity, imagination, flexibility, inventiveness, from making full use of prior knowledge and
and persistence that contribute to their future connecting it to school mathematics. There-
success in and out of school [28]. fore, teachers need to find out what young
children already understand and help them
Teachers refers to adults who care for and educate begin to understand these things mathematical-
groups of young children.
Copyright © 2002 National Association for the Education of Young Children
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5. NAEYC/NCTM Joint Position Statement
ly. From ages 3 through 6, children need many opment and her sensitivity to the individual
experiences that call on them to relate their child’s frustration tolerance and persistence
knowledge to the vocabulary and conceptual [45, 46].
frameworks of mathematics—in other words,
For some mathematical topics, researchers have
to “mathematize” what they intuitively grasp.
identified a developmental continuum or learn-
Toward this end, effective early childhood
ing path—a sequence indicating how particular
programs provide many such opportunities
concepts and skills build on others [44, 47, 48].
for children to represent, reinvent, reorganize,
Snapshots taken from a few such sequences are
quantify, abstract, generalize, and refine that
given in the accompanying chart (pp. 19–21).
which they grasp at an experiential or intuitive
level [28]. Research-based generalizations about what
many children in a given grade or age range can
3. Base mathematics curriculum and teaching do or understand are key in shaping curriculum
practices on knowledge of young children’s and instruction, although they are only a start-
cognitive, linguistic, physical, and social- ing point. Even with comparable learning op-
emotional development. portunities, some children will grasp a concept
All decisions regarding mathematics curricu- earlier and others somewhat later. Expecting
lum and teaching practices should be grounded and planning for such individual variation are
in knowledge of children’s development and always important.
learning across all interrelated areas—cogni- With the enormous variability in young chil-
tive, linguistic, physical, and social-emotional. dren’s development, neither policymakers nor
First, teachers need broad knowledge of teachers should set a fixed timeline for children
children’s cognitive development—concept to reach each specific learning objective [49].
development, reasoning, and problem solving, In addition to the risk of misclassifying indi-
for instance—as well as their acquisition of vidual children, highly specific timetables for
particular mathematical skills and concepts. skill acquisition pose another serious threat,
Although children display mathematical ideas especially when accountability pressures are
at early ages [e.g., 40–43] their ideas are often intense. They tend to focus teachers’ attention
very different from those of adults [e.g., 26, 44]. on getting children to perform narrowly defined
For example, young children tend to believe skills by a specified time, rather than on laying
that a long line of pennies has more coins than the conceptual groundwork that will serve
a shorter line with the same number. children well in the long run. Such prescrip-
Beyond cognitive development, teachers need tions often lead to superficial teaching and rote
to be familiar with young children’s social, emo- learning at the expense of real understanding.
tional, and motor development, all of which Under these conditions, children may develop
are relevant to mathematical development. only a shaky foundation for further mathemat-
To determine which puzzles and manipulative ics learning [50–52].
materials are helpful to support mathematical
4. Use curriculum and teaching practices that
learning, for instance, teachers combine their
strengthen children’s problem-solving and
knowledge of children’s cognition with the
reasoning processes as well as represent-
knowledge of fine7 motor development [45].
ing, communicating, and connecting math-
In deciding whether to let a 4-year-old struggle
ematical ideas.
with a particular mathematical problem or to
offer a clue, the teacher draws on more than Problem solving and reasoning are the heart of
an understanding of the cognitive demands in- mathematics. Teaching that promotes profi-
volved. Important too are the teacher’s under- ciency in these and other mathematical pro-
standing of young children’s emotional devel- cesses is consistent with national reports on
Copyright © 2002 National Association for the Education of Young Children
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6. Early Childhood Mathematics
mathematics education [5, 19, 53] and recom- The big ideas or vital understandings in early
mendations for early childhood practice [14, childhood mathematics are those that are
46]. While content represents the what of early mathematically central, accessible to children
childhood mathematics education, the process- at their present level of understanding, and
es—problem solving, reasoning, communica- generative of future learning [28]. Research and
tion, connections, and representation—make it expert practice indicate that certain concepts
possible for children to acquire content know and skills are both challenging and accessible
edge [19]. These processes develop over time to young children [19]. National professional
and when supported by well designed opportu- standards outline core ideas in each of five
nities to learn. major content areas: number and operations,
geometry, measurement, algebra (including pat-
Children’s development and use of these
terns), and data analysis [19]. For example, the
processes are among the most longlasting and
idea that the same pattern can describe differ-
important achievements of mathematics educa-
ent situations is a “big idea” within the content
tion. Experiences and intuitive ideas become
area of algebra and patterning.
truly mathematical as the children reflect on
them, represent them in various ways, and con- These content areas and their related big ideas,
nect them to other ideas [19, 47]. however, are just a starting point. Where does
one begin to build understanding of an idea
The process of making connections deserves
such as “counting” or “symmetry,” and where
special attention. When children connect
does one take this understanding over the
number to geometry (for example, by count-
early years of school? Articulating goals and
ing the sides of shapes, using arrays to under-
standards for young children as a develop-
stand number combinations, or measuring the
mental or learning continuum is a particularly
length of their classroom), they strengthen
useful strategy in ensuring engagement with
concepts from both areas and build knowledge
and mastery of important mathematical ideas
and beliefs about mathematics as a coherent
[49]. In the key areas of mathematics, research-
system [19, 47]. Similarly, helping children con-
ers have at least begun to map out trajectories
nect mathematics to other subjects, such as
or paths of learning—that is, the sequence in
science, develops knowledge of both subjects
which young children develop mathematical
as well as knowledge of the wide applicability
understanding and skills [21, 58, 59]. The ac-
of mathematics. Finally and critically, teaching
companying chart provides brief examples of
concepts and skills in a connected, integrated
learning paths in each content area and a few
fashion tends to be particularly effective not
teaching strategies that promote children’s
only in the early childhood years [14, 23] but
progress along these paths. Information about
also in future learning [5, 54].
such learning paths can support developmen-
5. Ensure that the curriculum is coherent tally appropriate teaching, illuminating various
and compatible with known relationships avenues to understanding and guiding teachers
and sequences of important mathematical in providing activities appropriate for children
ideas. as individuals and as a group.
In developing early mathematics curriculum, 6. Provide for children’s deep and sustained
teachers need to be alert to children’s experi- interaction with key mathematical ideas.
ences, ideas, and creations [55, 56]. To create
In many early childhood programs, mathemat-
coherence and power in the curriculum, how-
ics makes only fleeting, random appearances.
ever, teachers also must stay focused on the
Other programs give mathematics adequate
“big ideas” of mathematics and on the connec-
time in the curriculum but attempt to cover
tions and sequences among those ideas
so many mathematical topics that the result
[23, 57].
Copyright © 2002 National Association for the Education of Young Children
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7. NAEYC/NCTM Joint Position Statement
is superficial and uninteresting to children. 7. Integrate mathematics with other activities
In a more effective third alternative, children and other activities with mathematics.
encounter concepts in depth and in a logical
Young children do not perceive their world as if
sequence. Such depth and coherence allow
it were divided into separate cubbyholes such
children to develop, construct, test, and reflect
as “mathematics” or “literacy” [61]. Likewise,
on their mathematical understandings [10,
effective practice does not limit mathematics
23, 59, 60]. This alternative also enhances
to one specified period or time of day. Rather,
teachers’ opportunities to determine gaps in
early childhood teachers help children develop
children’s understanding and to take time to
mathematical knowledge throughout the day
address these.
and across the curriculum. Children’s everyday
Because curriculum depth and coherence activities and routines can be used to introduce
are important, unplanned experiences with and develop important mathematical ideas [55,
mathematics are clearly not enough. Effective 59, 60, 62–67]. For example, when children are
programs also include intentionally organized lining up, teachers can build in many opportu-
learning experiences that build children’s nities to develop an understanding of mathe-
understanding over time. Thus, early childhood matics. Children wearing something red can be
educators need to plan for children’s in-depth asked to get in line first, those wearing blue to
involvement with mathematical ideas, includ- get in line second, and so on. Or children wear-
ing helping families extend and develop these ing both something red and sneakers can be
ideas outside of school. asked to head up the line. Such opportunities
to build important mathematical vocabulary
Depth is best achieved when the program fo-
and concepts abound in any classroom, and
cuses on a number of key content areas rather
the alert teacher takes full advantage of them.
than trying to cover every topic or skill with
equal weight. As articulated in professional Also important is weaving mathematics into
standards, researchers have identified number children’s experiences with literature, language,
and operations, geometry, and measurement science, social studies, art, movement, music,
as areas particularly important for 3- to 6-year- and all parts of the classroom environment. For
olds [19]. These play an especially significant example, there are books with mathematical
role in building the foundation for mathemat- concepts in the reading corner, and clipboards
ics learning [47]. For this reason, researchers and wall charts are placed where children are
recommend that algebraic thinking and data engaged in science observation and record-
analysis/probability receive somewhat less ing (e.g., measuring and charting the weekly
emphasis in the early years. The beginnings of growth of plants) [65, 66, 68–71]. Projects
ideas in these two areas, however, should be also reach across subject-matter boundaries.
woven into the curriculum where they fit most Extended investigations offer children excel-
naturally and seem most likely to promote lent opportunities to apply mathematics as well
understanding of the other topic areas [19]. as to develop independence, persistence, and
Within this second tier of content areas, pat- flexibility in making sense of real-life problems
terning (a component of algebra) merits special [19]. When children pursue a project or inves-
mention because it is accessible and interesting tigation, they encounter many mathematical
to young children, grows to undergird all alge- problems and questions. With teacher guid-
braic thinking, and supports the development ance, children think about how to gather and
of number, spatial sense, and other conceptual record information and develop representa-
areas. tions to help them in understanding and using
the information and communicating their work
to others [19, 72].
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8. Early Childhood Mathematics
Another rationale for integrating mathematics have emerged in their play. Teachers enhance
throughout the day lies in easing competition children’s mathematics learning when they ask
for time in an increasingly crowded curriculum. questions that provoke clarifications, exten-
Heightened attention to literacy is vital but can sions, and development of new understandings
make it difficult for teachers to give mathemat- [19].
ics and other areas their due. With a strong
Block building offers one example of play’s
interdisciplinary curriculum, teachers can still
value for mathematical learning. As children
focus on one area at times but also find ways
build with blocks, they constantly accumulate
to promote children’s competence in literacy,
experiences with the ways in which objects
mathematics, and other subjects within inte-
can be related, and these experiences become
grated learning experiences [73].
the foundation for a multitude of mathematical
An important final note: As valuable as integra- concepts—far beyond simply sorting and seri-
tion is within early childhood curriculum, it ating. Classic unit blocks and other construc-
is not an end in itself. Teachers should ensure tion materials such as connecting blocks give
that the mathematics experiences woven children entry into a world where objects have
throughout the curriculum follow logical predictable similarities and relationships [66,
sequences, allow depth and focus, and help 76]. With these materials, children reproduce
children move forward in knowledge and skills. objects and structures from their daily lives
The curriculum should not become, in the and create abstract designs by manipulating
name of integration, a grab bag of any mathe- pattern, symmetry, and other elements [77].
matics-related experiences that seem to relate Children perceive geometric notions inherent
to a theme or project. Rather, concepts should in the blocks (such as two square blocks as the
be developed in a coherent, planful manner. equivalent of one rectangular unit block) and
the structures they build with them (such as
8. Provide ample time, materials, and teacher
symmetric buildings with parallel sides). Over
support for children to engage in play, a
time, children can be guided from an intuitive
context in which they explore and manipu-
to a more explicit conceptual understanding of
late mathematical ideas with keen interest.
these ideas [66].
Children become intensely engaged in play.
A similar progression from intuitive to explicit
Pursuing their own purposes, they tend to tack-
knowledge takes place in other kinds of play.
le problems that are challenging enough to be
Accordingly, early childhood programs should
engrossing yet not totally beyond their capaci-
furnish materials and sustained periods of
ties. Sticking with a problem—puzzling over it
time that allow children to learn mathemat-
and approaching it in various ways—can lead
ics through playful activities that encourage
to powerful learning. In addition, when sev-
counting, measuring, constructing with blocks,
eral children grapple with the same problem,
playing board and card games, and engaging in
they often come up with different approaches,
dramatic play, music, and art [19, 64].
discuss, and learn from one another [74, 75].
These aspects of play tend to prompt and pro- Finally, the teacher can observe play to learn
mote thinking and learning in mathematics and more about children’s development and inter-
in other areas. ests and use this knowledge to inform curricu-
lum and instruction. With teacher guidance, an
Play does not guarantee mathematical develop-
individual child’s play interest can develop into
ment, but it offers rich possibilities. Significant
a classroom-wide, extended investigation or
benefits are more likely when teachers fol-
project that includes rich mathematical learn-
low up by engaging children in reflecting on
ing [78–82]. In classrooms in which teachers
and representing the mathematical ideas that
are alert to all these possibilities, children’s
Copyright © 2002 National Association for the Education of Young Children
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9. NAEYC/NCTM Joint Position Statement
play continually stimulates and enriches math- game more mathematically powerful and more
ematical explorations and learning. appropriate for children of differing develop-
mental levels [55, 83].
9. Actively introduce mathematical concepts,
methods, and language through a range Use of materials also requires intentional plan-
of appropriate experiences and teaching ning and involvement on the teacher’s part.
strategies. Computer technology is a good example [84].
Teachers need to intentionally select and use
A central theme of this position statement is
research-based computer tools that comple-
that early childhood curriculum needs to go
ment and expand what can be done with other
beyond sporadic, hit-or-miss mathematics. In
media [59]. As with other instructional materi-
effective programs, teachers make judicious
als, choosing software and determining how
use of a variety of approaches, strategies, and
best to incorporate computer use in the day-to-
materials to support children’s interest and
day curriculum requires thoughtful, informed
ability in mathematics.
decision-making in order for children’s learning
Besides embedding significant mathemat- experiences to be rich and productive.
ics learning in play, classroom routines, and
In short, mathematics is too important to be
learning experiences across the curriculum, an
left to chance, and yet it must also be con-
effective early mathematics program also pro-
nected to children’s lives. In making all of these
vides carefully planned experiences that focus
choices, effective early childhood teachers
children’s attention on a particular mathemati-
build on children’s informal mathematical
cal idea or set of related ideas. Helping children
knowledge and experiences, always taking chil-
name such ideas as horizontal or even and odd
dren’s cultural background and language into
as they find and create many examples of these
consideration [23].
categories provides children with a means to
connect and refer to their just-emerging ideas 10. Support children’s learning by thought-
[35, 37]. Such concepts can be introduced and fully and continually assessing all children’s
explored in large- and small-group activities mathematical knowledge, skills, and strate-
and learning centers. Small groups are particu- gies.
larly well suited to focusing children’s attention
Assessment is crucial to effective teaching
on an idea. Moreover, in this setting the teacher
[85]. Early childhood mathematics assess-
is able to observe what each child does and
ment is most useful when it aims to help young
does not understand and engage each child in
children by identifying their unique strengths
the learning experience at his own level.
and needs so as to inform teacher planning.
In planning for new investigations and activi- Beginning with careful observation, assessment
ties, teachers should think of ways to engage uses multiple sources of information gath-
children in revisiting concepts they have ered systematically over time—for example, a
previously explored. Such experiences enable classroom book documenting the graphs made
children to forge links between previously by children over several weeks. Mathematics
encountered mathematical ideas and new appli- assessment should follow widely accepted prin-
cations [19]. ciples for varied and authentic early childhood
assessment [85]. For instance, the teacher
Even the way that teachers introduce and
needs to use multiple assessment approaches
modify games can promote important mathe-
to find out what each child understands—and
matical concepts and provide opportunities for
may misunderstand. Child observation, docu-
children to practice skills [55, 57]. For example,
mentation of children’s talk, interviews, collec-
teachers can modify any simple board game in
tions of children’s work over time, and the use
which players move along a path to make the
Copyright © 2002 National Association for the Education of Young Children
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10. Early Childhood Mathematics
of open-ended questions and appropriate per- and ongoing professional development is an
formance assessments to illuminate children’s urgent priority. In mathematics, as in literacy
thinking are positive approaches to assessing and other areas, the challenges are formidable,
mathematical strengths and needs [86, 87]. but research-based solutions are available [14,
92–95]. To support children’s mathematical
Careful assessment is especially important
proficiency, every early childhood teacher’s
when planning for ethnically, culturally, and lin-
professional preparation should include these
guistically diverse young children and for chil-
connected components: (1) knowledge of the
dren with special needs or disabilities. Effective
mathematical content and concepts most
teachers use information and insights gathered
relevant for young children—including in-depth
from assessment to plan and adapt teaching
understanding of what children are learning
and curriculum. They recognize that even
now and how today’s learning points toward
young children invent their own mathematical
the horizons of later learning [5]; (2) knowledge
ideas and strategies and that children’s ideas
of young children’s learning and development
can be quite different from those of adults [44].
in all areas—including but not limited to cogni-
They interpret what the child is doing and
tive development—and knowledge of the issues
thinking, and they attempt to see the situation
and topics that may engage children at differ-
from the child’s point of view. With this basis
ent points in their development; (3) knowledge
in thoughtful assessment, teachers are able to
of effective ways of teaching mathematics to
make informed decisions about what the child
all young learners; (4) knowledge and skill in
might be able to learn from new experiences.
observing and documenting young children’s
Reliance on a single group-administered test to mathematical activities and understanding;
document 3- to 6-year-old children’s mathemati- and (5) knowledge of resources and tools that
cal competence is counter to expert recom- promote mathematical competence and enjoy-
mendations on assessment of young children ment [96].
[85, 88–91]. Educators must take care that as-
Essential as this knowledge is, it can be
sessment does not narrow the curriculum and
brought to life only when teachers themselves
inappropriately label children. If assessment
have positive attitudes about mathematics.
results exclude some children from challenging
Lack of appropriate preparation may cause
learning activities, they undercut educational
both preservice and experienced teachers to
equity. Teachers and education policy makers
fail to see mathematics as a priority for young
need to stay in control of the assessment pro-
children and to lack confidence in their ability
cess, ensuring that it helps build mathematical
to teach mathematics effectively [97]. Thus,
competence and confidence. Well conceived,
both preservice education and continuing
well implemented, continuous assessment is an
professional development experiences need to
indispensable tool in facilitating all children’s
place greater emphasis on encouraging teach-
engagement and success in mathematics.
ers’ own enjoyment and confidence, building
positive mathematical attitudes and disposi-
Beyond the classroom tions.
To support excellent early mathematics
Even graduates of four-year early childhood
education, institutions, program develop-
programs with state licensure usually lack
ers, and policy makers should
adequate preparation in mathematics. State
1. Create more effective early childhood legislatures often address their concern over
teacher preparation and continuing profes- teachers’ weak background in mathematics by
sional development. simply increasing the number of required math-
Improving early childhood teacher preparation ematics courses needed for teacher licensure.
Copyright © 2002 National Association for the Education of Young Children
10

11. NAEYC/NCTM Joint Position Statement
This remedy lacks research support [5, 92]. ticipation of staff who work in similar settings;
Credit hours or yearly training requirements do content focused both on what and how to
little or nothing unless the content and delivery teach; active learning techniques; and profes-
of professional development are designed to sional development as part of a coherent pro-
produce desired outcomes for teachers and gram of teacher learning [5, 99]. Innovative and
children [93]. effective professional development models may
use a variety of research-based approaches. In
Teachers of young children should learn the
addition, classroom-based inquiry, team teach-
mathematics content that is directly relevant to
ing by mathematics and early childhood educa-
their professional role. But content alone is not
tion specialists, discussion of case studies, and
enough. Effective professional programs weave
analysis of young children’s work samples tend
together mathematics content, pedagogy, and
to strengthen teachers’ confidence and engage-
knowledge of child development and family
ment in early childhood mathematics [5, 97, 99,
relationships [98]. When high-quality, well
100].
supervised field work is integrated throughout
a training program, early childhood teachers Delivering this kind of ongoing professional
can apply their knowledge in realistic con- development requires a variety of innovative
texts. Courses or inservice training should strategies. For early childhood staff living in
be designed to help teachers develop a deep isolated communities or lacking knowledgeable
understanding of the mathematics they will trainers, distance learning with local facilita-
teach and the habits of mind of a mathematical tors is a promising option. Literacy initiatives
thinker. Courses, practicum experiences, and are increasingly using itinerant or school-wide
other training should strengthen teachers’ abil- specialists; similarly, mathematics education
ity to ask young children the kinds of questions specialists could offer resources to a number
that stimulate mathematical thinking. Effective of early childhood programs. Partnerships
professional development, whether preservice between higher education institutions and local
or inservice, should also model the kind of flex- early childhood programs can help provide this
ible, interactive teaching styles that work well support. Finally, school-district-sponsored pro-
with children [92]. fessional development activities that include
participants from community child care cen-
Preservice and inservice professional develop-
ters, family child care, and Head Start programs
ment present somewhat differing challenges. In
along with public school kindergarten/primary
preservice education, the major challenge is to
teachers would build coherence and continuity
build a sound, well integrated knowledge base
for teachers and for children’s mathematical
about mathematics, young children’s develop-
experiences.
ment and learning, and classroom practices [5].
Inservice training shares this challenge but also 2. Use collaborative processes to develop
carries risks of superficiality and fragmentation. well aligned systems of appropriate
high-quality standards, curriculum, and
To avoid these risks, inservice professional
assessment.
development needs to move beyond the one-
time workshop to deeper exploration of key In mathematics, as in other domains, the task
mathematical topics as they connect with of developing curriculum and related goals and
young children’s thinking and with classroom assessments has become the responsibility not
practices. Inservice professional development only of the classroom teacher but also of other
in mathematics appears to have the greatest educators and policy makers. State agencies,
impact on teacher learning if it incorporates six school districts, and professional organizations
features: teacher networking or study groups; are engaged in standards setting, defining de-
sustained, intensive programs; collective par- sired educational and developmental outcomes
Copyright © 2002 National Association for the Education of Young Children
11

12. Early Childhood Mathematics
for children below kindergarten age [13]. This the principles articulated by national groups
trend represents an opportunity to improve concerned with appropriate assessment for
early childhood mathematics education but young children [88–91].
also presents a challenge. The opportunity is
District- or program-level educators are often
to develop a coherent, developmentally ap-
responsible for selecting or developing cur-
propriate, and well aligned system that offers
riculum. Decision makers can be guided by the
teachers a framework to guide their work. The
general criteria for curriculum adoption articu-
challenge, especially at the preschool and kin-
lated in the position statement jointly adopted
dergarten levels, is to ensure that such a frame-
by NAEYC and the National Association of Early
work does not stifle innovation, put children
Childhood Specialists in State Departments of
into inappropriate categories, ignore important
Education [85]. In addition, decision makers
individual or cultural differences, or result in
should insist that any mathematics curriculum
narrowed and superficial teaching that fails to
considered for adoption has been extensively
give children a solid foundation of understand-
field tested and evaluated with young children.
ing [49].
3. Design institutional structures and policies
To avoid these risks, state agencies and others
that support teachers’ ongoing learning,
must work together to develop more coherent
teamwork, and planning.
systems of standards, curriculum, instruction,
and assessment that support the development National reports stress the need for teacher
of mathematical proficiency. To build coher- planning and collaboration [5, 7, 101, 102], yet
ence between preschool and early elementary few early childhood programs have the struc-
mathematics, the processes of setting stan- tures and supports to enable these processes to
dards and developing early childhood curricu- take place regularly. Teachers of young children
lum and assessment systems must include the face particular challenges in planning mathe-
full range of stakeholders. Participants should matics activities. Early childhood teachers work
include not only public school teachers and in diverse settings, and some of these settings
administrators but also personnel from center- pose additional obstacles to teamwork and col-
based programs and family child care, private laboration. Many early childhood programs, in
and public prekindergarten, and Head Start, as or out of public school settings, have little or no
well as others who serve young children and time available for teacher planning, either indi-
their families. Families too should participate vidually or in groups. Team meetings and staff
as respected partners. Relevant expertise development activities occur infrequently.
should be sought from professional associa- The institutional divide between teachers in
tions and other knowledgeable sources. child care, Head Start, or preschool programs
As in all effective standards-setting efforts, and those in public kindergarten and primary
early childhood mathematics standards should programs presents a barrier to the communi-
be coupled with an emphasis on children’s cation necessary for a coherent mathematics
opportunities to learn, not just on expectations curriculum. Without communication opportu-
for their performance. Standards also should nities, preschool teachers often do not know
be accompanied by descriptions of what young what kindergarten programs expect, and early
children might be expected to accomplish elementary teachers may have little idea of the
along a flexible developmental continuum [49]. content or pedagogy used in prekindergarten
Standards for early childhood mathematics mathematics education. New strategies and
should connect meaningfully but not rigidly structures, such as joint inservice programs
with curriculum. Assessment also should align and classroom visits, could support these
with curriculum and with standards, following linkages.
Copyright © 2002 National Association for the Education of Young Children
12

13. NAEYC/NCTM Joint Position Statement
In addition, many programs have limited ac- To support effective teaching and learning,
cess to specialists who might help teachers mathematics-rich classrooms require a wide
as they try to adopt new approaches to early array of materials for young children to explore
childhood mathematics. Administrators need and manipulate [45, 59, 107]. Equity requires
to reexamine their allocation of resources and that all programs, not just those serving afflu-
their scheduling practices, keeping in mind the ent communities, have these resources.
value of investing in teacher planning time.
Finally, resources are needed to support
4. Provide the resources necessary to over- families as partners in developing their young
come the barriers to young children’s children’s mathematical proficiency. The grow-
mathematical proficiency at the classroom, ing national awareness of families’ central role
community, institutional, and system-wide in literacy development is a good starting point
levels. from which to build awareness of families’
equally important role in mathematical de-
A variety of resources, some financial and some
velopment [108, 109]. Public awareness cam-
less tangible, are needed to support imple-
paigns, distribution of materials in ways similar
mentation of this position statement’s recom-
to the successful Reach Out and Read initia-
mendations. Partnerships among the business,
tive, computer-linked as well as school-based
philanthropic, and government sectors at the
meetings for families, Family Math Nights,
national, state, and local levels will improve
and take-home activities such as mathematics
teaching and learning in all communities,
games and manipulative materials tailored to
including those that lack equitable access to
the ages, interests, languages, and cultures of
mathematics education. Universally available
the children—these are only a few examples of
early childhood mathematics education can
the many ways in which resources can support
occur only in the context of a comprehensive,
families’ engagement in their young children’s
well financed system of high-quality early
mathematical learning [110, see also the online
education, including child care, Head Start, and
“Family Math” materials at www.lhs.berkeley.
prekindergarten programs [103–106]. To sup-
edu/equals/FMnetwork.htm and other resourc-
port universal mathematical proficiency, access
es at www.nctm.org/corners/family/index.htm].
to developmentally and educationally effective
programs of early education, supported by
adequate resources, should be available to all Conclusion
A positive attitude toward mathematics and a
Improvement of early childhood mathematics strong foundation for mathematics learning begin
education also requires substantial investment in early childhood. These good beginnings reflect
in teachers’ professional development. The all the characteristics of good early childhood
mathematics knowledge gap must be bridged education: deep understanding of children’s
with the best tools, including resources for dis- development and learning; a strong community of
seminating models of effective practice, videos teachers, families, and children; research-based
showing excellent mathematics pedagogy in knowledge of early childhood curriculum and
real-life settings, computer-based professional teaching practices; continuous assessment in
development resources, and other materials. the service of children’s learning; and an abiding
In addition, resources are needed to support respect for young children’s families, cultures,
teachers’ involvement in professional confer- and communities.
ences, college courses, summer institutes, and To realize this vision, educators, adminis-
visits to model sites. trators, policy makers, and families must work
together—raising awareness of the importance
of mathematics in early education, informing
Copyright © 2002 National Association for the Education of Young Children
13

14. Early Childhood Mathematics
others about sound approaches to mathematical 13. Education Week. 2002. Quality Counts 2002: Build-
teaching and learning, and developing essential ing blocks for success: State efforts in early-child-
hood education. Education Week (Special issue) 21
resources to support high-quality, equitable
(17).
mathematical experiences for all young children.
14. Bowman, B.T., M.S. Donovan, & M.S. Burns, eds.
2001. Eager to learn: Educating our preschoolers.
Washington, DC: National Academy Press.
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