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Early experience with numbers and operations is fundamental for acquiring more complex math concepts and skills. The practice guide describes how to teach the main aspects of early number knowledge, from basic number skills to operations, according to a developmental progression. Developmental progressions can provide teachers with a road map for delivering developmentally appropriate instruction to students of varying skill levels. With each step in a developmental progression, teachers can first focus on working with small collections of objects (one to three items) and then move to progressively larger collections of objects.
1.
Teaching Math to Young Children
Practice Guide
Educators’ Practice Guide Summary • WHAT WORKS CLEARINGHOUSETM
The five recommendations Introduction
in this WWC practice guide,
Teaching Math to Young Children are interested in math well before they start school.
Children, build on children’s They notice basic shapes, construct and extend simple patterns,
natural interest in math and learn to count. The Teaching Math to Young Children practice
to make preschool and guide presents five recommendations designed to help early
kindergarten more engaging education teachers capitalize on children’s natural interest in
and beneficial. math. The first two recommendations identify early math content
areas that should be included in the preschool, prekindergarten,
and kindergarten curricula. The last three recommendations
focus on strategies and teaching techniques that incorporate
math content into the classroom.
This summary introduces the recommendations and supporting
evidence described in the WWC’s Teaching Math to Young Children
practice guide. Download your free copy of the guide at
http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=18.
2.
Teaching Math to Young Children Practice Guide Summary
Recommendation 1. 2. Next, promote accurate one-to-one counting
as a means of identifying the total number
Teach number and operations
of items in a collection. Once children have
using a developmental progression. started to connect numbers with quantity, they
can then begin to use one-to-one counting to
identify “how many” are in larger collections. In
order to count accurately, one—and only one—
number word should be assigned to each item
in the collection being counted, using the last
number counted to determine the total quantity.
The practice guide also includes a discussion
of the common counting errors children make
when developing counting skills and provides
suggestions teachers can use to correct those
errors when working with children in one-on-one
or small-group situations (see page 19 of the
practice guide).
Modeling one-to-one counting with one
to three items
Early experience with number and operations is
fundamental for acquiring more complex math
concepts and skills. The practice guide describes
how to teach the main aspects of early number While pointing at each object, count:
knowledge, from basic number skills to opera-
tions, according to a developmental progression. (with emphasis)
Developmental progressions can provide teachers “one” “two” “three”
with a road map for delivering developmentally
appropriate instruction to students of varying skill
levels. With each step in a developmental progres-
sion, teachers can first focus on working with small
collections of objects (one to three items) and then “There are three (squares) here.”
move to progressively larger collections of objects.
How to carry out the recommendation
3. Once children can recognize or count collec-
1. First, provide opportunities for children to tions, provide opportunities for children to
practice recognizing the total number of use number words and counting to com-
objects in small collections (one to three pare quantities. To begin magnitude compari-
items) and labeling them with a number word sons, teachers can ask children to compare small
without needing to count them. Teachers can collections of one to three objects visually. Next,
give children experience with immediately recogniz- children can count or match larger collections
ing and labeling quantities of a collection by having of items one-to-one to determine which set has
them answer the question “How many are there?” “more.” Once children can comfortably determine
when looking at collections of one to three objects. “more,” the word “fewer” and the use of writ-
Small-group activities, snack time, and transitions ten number lists can be introduced. Eventually,
between classroom activities can provide quick teachers can encourage children to use their
opportunities for children to practice recogniz- knowledge of which number comes after another
ing small quantities. To help children construct a in the counting sequence to determine the larger
more abstract concept of number, teachers can of two verbal numbers.
emphasize that collections of three similar objects
and three dissimilar objects are both “three.”
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3.
Teaching Math to Young Children Practice Guide Summary
4. Encourage children to label collections with more adept, teachers can progress to more
number words and numerals. After children difficult problems with slightly larger numbers.
practice recognizing, counting, and comparing
quantities, teachers can introduce numerals to Summary of evidence
children as a way to represent a quantity. As
described in the practice guide, teachers can pair The WWC identified 23 studies that examined the
numerals with collections of classroom objects so effects of interventions that included targeted instruc-
that children start to learn, for example, that the tion in number and operations. The research shows
numeral 3, three objects, and the spoken word a strong pattern of positive effects on children’s early
“three” represent the same thing (see page 21 of math skills across a range of curricula with a focus
the practice guide). on number and operations. Eleven studies demon-
strated positive effects for interventions that used a
5. Once children develop these fundamental developmental progression to guide instruction in
number skills, encourage them to solve number and operations. While the panel could not
basic problems. Once children can determine confirm whether a developmental progression guided
the total number of items in a collection by using instruction in 12 other studies that taught number and
small-number recognition or counting and under- operation, the panel felt that the body of evidence as
stand the concepts of “more” and “fewer,” they a whole provided sufficient support to warrant a level
can explore the effects of adding and subtract- of evidence rating of moderate. For more details, see
ing items from a collection. As children become Recommendation 1 in the practice guide, page 12.
Recommendation 2.
Teach geometry, patterns, measure-
ment, and data analysis using a
developmental progression.
The practice guide suggests that children’s exposure
to math should extend beyond number and operations
to include a range of math content areas, including
geometry (shapes and space), patterns, measurement,
and data analysis. These math content areas should
be taught according to developmental progressions.
Learning skills beyond number and operations creates
a foundation for future math instruction, and children and master the concepts of “in,” “on,” “under,”
with strong backgrounds in these areas are more likely “beside,” “above,” or “below.”
to be able to succeed in later grades.
2. Encourage children to look for and identify
patterns, then teach them to extend, cor-
How to carry out the recommendation rect, and create patterns. Pattern instruction
1. Help children to recognize, name, and can begin by encouraging children to notice
compare shapes, and then teach them and experiment with basic repeating patterns
to combine and separate shapes. Using their in the world around them, such as stripes on
surrounding environment, teachers can provide clothing and shapes and designs in rugs. Once
opportunities for children to make comparisons children have become familiar with the nature
and distinctions about the basic features of shapes. of patterns, they should learn to predict what
Once children are comfortable recognizing and will happen next in a pattern, based on what
comparing shapes, teachers should encourage has happened so far. Teachers will find ideas in
children to explore how shapes can be combined the practice guide for using manipulatives, like
and separated to form new shapes. Page 29 of the the one shown in the example on the next page,
practice guide suggests activities to help children to teach children about patterns of increasing
to learn about spatial relationships between shapes complexity (see page 30 of the practice guide).
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4.
Teaching Math to Young Children Practice Guide Summary
3. Promote children’s understanding of mea- total number in each set relative to other sets.
surement by teaching them to make direct Once children are familiar with sorting and orga-
comparisons and to use both informal or nizing the information they have collected, they
nonstandard (e.g., the child’s hand or foot) can learn to summarize their information visually
and formal or standard (e.g., a ruler) units through simple graphs and tallies.
and tools. Teachers can help children learn to use
measurement vocabulary words (long and short,
big and small) to describe similarities and differ-
Summary of evidence
ences while sorting, arranging, and classifying The WWC identified 13 studies that examined the
objects. Once children have become comfortable effects of interventions that provided targeted
making direct comparisons between and among instruction in one or more of the early math content
objects, teachers can provide opportunities to areas (geometry, patterns, measurement, and data
measure objects using nonstandard tools, such as analysis). Positive effects were found for geometry,
children’s own hands and feet, pencils, blocks, or operations, basic number concepts, and general
books. Children can then be introduced to the con- numeracy outcomes. Studies also reported no
cept of standard units and tools of measurement. discernible effects for general numeracy, geometry,
and basic number concept outcomes. Although
4. Help children collect and organize infor- the evidence to support this recommendation is
mation, then teach them to represent that promising, the WWC could not isolate the effects of
information graphically. Teachers can intro- teaching the early math content areas of geometry,
duce children to the concept of organizing and patterns, measurement, and data analysis. As a
displaying information by asking them to count result, the WWC assigned a level of evidence rating
and sort familiar items like toys and blocks. This of minimal. For more details, see Recommendation 2
will help them learn both the characteristics that in the practice guide, page 25.
distinguish the items from one another and the
Complex pattern (boy, boy, girl, girl, boy, boy, girl, girl, boy, boy, girl, girl)
Recommendation 3.
Use progress monitoring to
ensure that math instruction
builds on what each child knows.
Progress monitoring can be a useful way to ensure
that children are participating in targeted, purposeful,
and meaningful math instruction. By continually
monitoring children’s progress, teachers can gather
the information they need to match lessons to chil-
dren’s knowledge levels. Incorporating children’s indi-
vidual differences into lesson planning by monitoring
progress and tailoring instruction can help ensure that
children learn fundamental concepts that are appro-
priately challenging and that will extend their learning.
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5.
Teaching Math to Young Children Practice Guide Summary
Progress monitoring is a systematic approach 3. Assess, record, and monitor each child’s
to assessment with the goal progress so that instructional goals and
of improving skills. methods can be adjusted as needed. The
chart below illustrates the flow of progress moni-
toring. The practice guide describes how teach-
How to carry out the recommendation ers can apply this ongoing process during math
instruction (see page 39 of the practice guide).
1. Use introductory activities, observations,
It is important to continually monitor progress
and assessments to determine each child’s
so that children can be consistently engaged in
existing math knowledge, or the level
activities that are neither too far below their level
of understanding or skill he or she has
(and therefore not interesting) nor too far above
reached on a developmental progression.
it (and therefore frustrating).
When employing progress monitoring, teachers
can first gather specific information about each
child’s skill level in order to determine where to Summary of evidence
focus instruction. The practice guide suggests
using introductory activities, observation, and The WWC identified 12 studies which examined the
formal assessments to determine each child’s effects of progress monitoring on children’s math
level of math understanding. knowledge when used in conjunction with other
practices the WWC recommends for teaching math
2. Tailor instruction to each child’s needs, to young children. For example, four studies exam-
and relate new ideas to his or her existing ining a comprehensive early math curriculum that
knowledge. Once teachers have information included supports for regular assessments found
about a child’s skill level, they can use a devel- that, on average, children participating in the inter-
opmental progression to determine what the vention scored higher on math outcomes than did
child should learn next and design instructional children in the comparison condition. Although the
activities with those needs in mind. Teachers can evidence to support this recommendation is promis-
design small-group activities that are at or slightly ing, the WWC could not isolate the effects of the use
above the children’s level of understanding. As of progress monitoring, since other recommended
detailed in the practice guide, teachers can link practices were also implemented. As a result, the
math activities to children’s existing interests and WWC assigned a level of evidence rating of minimal.
across other content areas to build knowledge, For more details, see Recommendation 3 in the
including music, art, games, and reading. practice guide, page 36.
Ongoing Progress Monitoring
Ongoing Progress Monitoring
Use a developmental progression to choose an activity that targets a math concept.
Assess: Observe and record Plan activities Implement
5
6.
Teaching Math to Young Children Practice Guide Summary
Recommendation 4.
Teach children to view and describe
their world mathematically.
The practice guide outlines the steps teachers can
take to help children view and describe the world
mathematically. Teachers can encourage children
to describe math ideas in the world around them,
gradually moving from informal representations
and language to formal representations and math
vocabulary as their understanding grows. At first,
children can use informal tools such as their fin-
gers, tally marks, or other concrete objects to rep-
resent math ideas. Once children are comfortable
using math informally, teachers can help them link
their informal knowledge to formal math vocabu-
lary and representations, such as math symbols.
Open-ended questions can be used to prompt
children to think about how to describe their ideas
mathematically. Teachers can reinforce these ideas
by encouraging children to look for opportunities
to use their developing math skills throughout the
school day. Lesson ideas for linking familiar concepts
to formal symbols
How to carry out the recommendation Symbol Concept Lesson
1. Encourage children to use informal numerals counting Have children count and record the
methods to represent math concepts, number of children in attendance
processes, and solutions. Initially, teachers each day.
should link math ideas to informal and familiar +,– operations Have children solve problems involv-
experiences, terms, or analogies, resisting the ing adding or subtracting with leaves
urge to use more formal methods until children collected from the playground.
have a conceptual foundation for understanding = equal Show the class four pennies. Next,
them. For example, teachers can first use terms show three pennies, verbally label
them (“I have one, two, three pen-
such as “more” and “all together,” terms that
nies”), and put them in a can. Then,
represent children’s informal understanding of show one more penny, verbally label
addition, before using the more formal, sym- it (“I have one more penny”), and put
bolic representation. it in the can. Ask the class, “Are three
pennies and one more penny the
2. Help children link formal math vocabulary, same number as four pennies?”
symbols, and procedures to their informal <,> unequal Show the class five pennies, verbally
knowledge or experiences. Once children label them, and put them in a can.
develop an understanding of math ideas using Next, show four pennies, verbally label
informal terms, teachers can show them how them, and put them in a different can.
Ask the class, “Which can has more?
their informal knowledge connects to formal Which can has fewer?”
math terms and representations. Teachers can
start with informal vocabulary, like the phrase
“take away,” for example, and then later explain offers teachers ideas for using math vocabulary
that the formal term “subtract” and the “–” sym- with children throughout the school day (see
bol have the same meaning. The practice guide page 44 of the practice guide).
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7.
Teaching Math to Young Children Practice Guide Summary
3. Use open-ended questions to prompt ways to encourage math conversation and prob-
children to apply their math knowledge. lem solving during recess, snack, transition, and
Open-ended questions not only help children small- and large-group time in the practice guide
to develop cognitive and language skills, but (see page 46).
they also can encourage children to use math
vocabulary to explain what they have learned. Summary of evidence
The practice guide provides teachers with ideas
for promoting math-related conversation among The WWC identified 16 studies that examined the
young children using open-ended questions. effects of interventions designed to help children
For example, teachers can use questions that view and describe their world mathematically. Some
begin with “what,” “why,” or “how” to encourage of the interventions provided math vocabulary
children to use math vocabulary to explain what words and suggestions for stories, songs, or ques-
they have learned. tions that supported children in learning to view
and describe their world mathematically. Studies
4. Encourage children to recognize and talk examining these interventions found positive
about math in everyday situations. Teachers effects on children’s math outcomes. Although the
can encourage math thought and conversation evidence to support this recommendation is promis-
by asking children for their help with problems ing, the WWC could not isolate the effects of helping
that arise throughout the day. Once children children view and describe their world mathemati-
solve the problem, teachers can ask a sequence cally, since other recommended practices were also
of questions that prompt the children to share implemented. As a result, the WWC assigned a level
the solution and the strategies used to reach the of evidence rating of minimal. For more details, see
solution. Teachers will find several examples of Recommendation 4 in the practice guide, page 42.
Recommendation 5.
Dedicate time each day to teaching
math, and integrate math instruc-
tion throughout the school day.
Devoting class time to planned, daily math lessons
helps children strengthen math skills. By connect-
ing math to a variety of everyday situations and
routines, teachers can make math meaningful and
provide opportunities for children to practice what
they have learned in a purposeful manner. A math-
rich environment can help generate excitement
among children and encourage them apply their
math knowledge in a meaningful way.
How to carry out the recommendation
1. Plan daily instruction targeting specific
math concepts and skills. In order for young instruction for children at different develop-
children to develop a strong foundation of math mental levels (see page 49 of the practice guide).
skills, teachers can dedicate time each day for
purposeful math instruction. During math lessons, 2. Embed math in classroom routines and
children can learn specific skills and build upon activities. A daily or weekly schedule provides
them throughout the rest of the day. The practice many opportunities to reinforce math concepts
guide describes how large and small groups can outside of the dedicated math instruction
be used during dedicated math time to tailor period. Routines such as taking attendance or
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8.
Teaching Math to Young Children Practice Guide Summary
snack time can serve a math purpose in addition 5. Use games to teach math concepts and
to a practical one, providing opportunities for skills and to give children practice in
counting, comparisons, and graphing. applying them. Games can provide an engag-
ing opportunity for children to practice and
3. Highlight math within topics of study extend skills, and the practice guide supplies
across the curriculum. Teachers can integrate several examples of games teachers can use to
math concepts into non-math lessons by high- encourage children to apply their math knowl-
lighting the aspects of math that are already edge. Teachers can use games that are included
present in the curriculum. This gives children in math curricula, purchase games, make games
opportunities to count and measure objects, themselves, or use games that come up dur-
examine shapes, and analyze data (depending ing natural play, like hopscotch or jump rope,
on the current math objectives). The practice to reinforce math concepts. Teachers can get
guide provides examples of ways to integrate involved with the game-playing in order to
different math content areas into literature, sci- ensure educational play or to challenge children
ence, art, health and safety, and social studies to extend their skills.
lessons (see page 51 of the practice guide).
4. Create a math-rich environment where Summary of evidence
children can recognize and meaningfully The WWC identified 20 studies that examined the
apply math. Teachers can provide opportunities effects of interventions that included dedicated time
for children to see and use math concepts on a for math instruction, integration of math into other
regular basis by creating a math-rich classroom aspects of the school day, and the use of games to
environment. This enrichment can be done by practice math skills. A group of studies found that
making math-related objects and tools readily children who played number-based board games
available, labeling and organizing them so they performed better in the domain of basic number
are easy to find and use, and organizing activities concepts than did children who played color-based
and routines with numeric systems. Teachers can board games or no board games. Although the
also explicitly teach children how to use math evidence to support this recommendation is promis-
tools by modeling their use during small- or ing, the WWC could not isolate the effects of the use
large-group time. of dedicated time for math instruction, integration
of math into other aspects of the school day, and
the use of games to practice math skills, since other
An example of a math-rich environment recommended practices were also implemented. As a
in the classroom. result, the WWC assigned a level of evidence rating of
minimal. For more details, see Recommendation 5 in
the practice guide, page 47.
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