Early Math PG 111313
Early Math PG 111313
Early Math PG 111313
NCEE 2014-4005
U.S. DEPARTMENT OF EDUCATION
The Institute of Education Sciences (IES) publishes practice guides in education to bring the best
available evidence and expertise to bear on current challenges in education. Authors of practice
guides combine their expertise with the findings of rigorous research, when available, to develop
specific recommendations for addressing these challenges. The authors rate the strength of the
research evidence supporting each of their recommendations. See Appendix A for a full description
of practice guides.
The goal of this practice guide is to offer educators specific, evidence-based recommendations
that address the challenge of teaching early math to children ages 3 to 6. The guide provides
practical, clear information on critical topics related to teaching early math and is based on the
best available evidence as judged by the authors.
Practice guides published by IES are available on our website at http://whatworks.ed.gov.
at
Urbana-Champaign
and
University
Margaret Burchinal
University of North Carolina
Sharon M. Carver
Carnegie Mellon University Childrens School
Nancy C. Jordan
University of Delaware
Judy McDowell
School District of Philadelphia
Staff
M. C. Bradley
Elizabeth Cavadel
Julia Lyskawa
Libby Makowsky
Moira McCullough
Bryce Onaran
Michael Barna
Mathematica Policy Research
Marc Moss
Abt Associates
Project Officers
Joy Lesnick
Diana McCallum
Institute of Education Sciences
NCEE 2014-4005
U.S. DEPARTMENT OF EDUCATION
of
Denver
This report was prepared for the National Center for Education Evaluation and Regional Assistance,
Institute of Education Sciences under Contract ED-IES-13-C-0010 by the What Works Clearinghouse,
which is operated by Mathematica Policy Research.
Disclaimer
The opinions and positions expressed in this practice guide are those of the authors and do not
necessarily represent the opinions and positions of the Institute of Education Sciences or the
U.S. Department of Education. This practice guide should be reviewed and applied according to
the specific needs of the educators and education agency using it, and with full realization that
it represents the judgments of the review panel regarding what constitutes sensible practice,
based on the research that was available at the time of publication. This practice guide should be
used as a tool to assist in decisionmaking rather than as a cookbook. Any references within the
document to specific education products are illustrative and do not imply endorsement of these
products to the exclusion of other products that are not referenced.
U.S. Department of Education
Arne Duncan
Secretary
Institute of Education Sciences
John Q. Easton
Director
National Center for Education Evaluation and Regional Assistance
Ruth Neild
Commissioner
November 2013
This report is in the public domain. Although permission to reprint this publication is not necessary,
the citation should be:
Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math
to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for Education
Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. Retrieved from the NCEE website: http://whatworks.ed.gov
What Works Clearinghouse practice guide citations begin with the panel chair, followed by the
names of the panelists listed in alphabetical order.
This report is available on the IES website at http://whatworks.ed.gov.
Alternate Formats
On request, this publication can be made available in alternate formats, such as Braille, large print, or
CD. For more information, contact the Alternate Format Center at (202) 260-0852 or (202) 260-0818.
Table of Contents
Teaching Math to Young Children
Table of Contents
Overview of Recommendations
. . . . . . . . . . . . . . . . . . . . . . . .1
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Institute of Education Sciences Levels of Evidence for Practice Guides
. . . . . . .4
List of Tables
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Example
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List of Figures
Figure
Figure
Figure
Figure
Figure
Figure
Figure
( iv )
Overview of Recommendations
Recommendation 1.
Teach number and operations using a developmental progression.
First, provide opportunities for children to practice recognizing the total number of objects
in small collections (one to three items) and labeling them with a number word without needing
to count them.
Next, promote accurate one-to-one counting as a means of identifying the total number of items
in a collection.
Once children can recognize or count collections, provide opportunities for children to use number
words and counting to compare quantities.
Encourage children to label collections with number words and numerals.
Once children develop these fundamental number skills, encourage them to solve basic problems.
Recommendation 2.
Teach geometry, patterns, measurement, and data analysis using a developmental progression.
Help children to recognize, name, and compare shapes, and then teach them to combine and
separate shapes.
Encourage children to look for and identify patterns, and then teach them to extend, correct,
and create patterns.
Promote childrens understanding of measurement by teaching them to make direct comparisons
and to use both informal or nonstandard (e.g., the childs hand or foot) and formal or standard
(e.g., a ruler) units and tools.
Help children to collect and organize information, and then teach them to represent that information graphically.
Recommendation 3.
Use progress monitoring to ensure that math instruction builds on what each child knows.
Use introductory activities, observations, and assessments to determine each childs existing
math knowledge, or the level of understanding or skill he or she has reached on a developmental progression.
Tailor instruction to each childs needs, and relate new ideas to his or her existing knowledge.
Assess, record, and monitor each childs progress so that instructional goals and methods can
be adjusted as needed.
(1)
Overview of Recommendations
(continued)
Recommendation 4.
Teach children to view and describe their world mathematically.
Encourage children to use informal methods to represent math concepts, processes,
and solutions.
Help children link formal math vocabulary, symbols, and procedures to their informal
knowledge or experiences.
Use open-ended questions to prompt children to apply their math knowledge.
Encourage children to recognize and talk about math in everyday situations.
Recommendation 5.
Dedicate time each day to teaching math, and integrate math instruction throughout the school day.
Plan daily instruction targeting specific math concepts and skills.
Embed math in classroom routines and activities.
Highlight math within topics of study across the curriculum.
Create a math-rich environment where children can recognize and meaningfully apply math.
Use games to teach math concepts and skills and to give children practice in applying them.
(2)
Acknowledgments
he panel appreciates the efforts of M. C. (Cay) Bradley, Elizabeth Cavadel, Julia Lyskawa,
Libby Makowsky, Moira McCullough, Bryce Onaran, and Michael Barna from Mathematica Policy
Research, and Marc Moss from Abt Associates, who participated in the panel meetings, described
the research findings, and drafted the guide. We also thank Scott Cody, Kristin Hallgren, David Hill,
Shannon Monahan, and Ellen Kisker for helpful feedback and reviews of earlier versions of the guide.
Douglas Frye
Arthur J. Baroody
Margaret Burchinal
Sharon M. Carver
Nancy C. Jordan
Judy McDowell
(3)
his section provides information about the role of evidence in Institute of Education Sciences
(IES) What Works Clearinghouse (WWC) practice guides. It describes how practice guide panels
determine the level of evidence for each recommendation and explains the criteria for each of the
three levels of evidence (strong evidence, moderate evidence, and minimal evidence).
A rating of moderate evidence refers either to
evidence from studies that allow strong causal
conclusions but cannot be generalized with
assurance to the population on which a recommendation is focused (perhaps because the
findings have not been widely replicated) or to
evidence from studies that are generalizable
but have some causal ambiguity. It also might
be that the studies that exist do not specifically examine the outcomes of interest in the
practice guide, although they may be related.
The level of evidence assigned to each recommendation in this practice guide represents the
panels judgment of the quality of the existing
research to support a claim that, when these
practices were implemented in past research,
favorable effects were observed on student
outcomes. After careful review of the studies
supporting each recommendation, panelists
determine the level of evidence for each
recommendation using the criteria in Table 1.
The panel first considers the relevance of
individual studies to the recommendation
and then discusses the entire evidence base,
taking the following into consideration:
(4)
MODERATE
Evidence Base
MINIMAL
Evidence Base
Validity
Effects on
relevant
outcomes
A preponderance of evidence
of positive effects. Contradictory evidence (i.e., statistically significant negative
effects) must be discussed
by the panel and considered
with regard to relevance to
the scope of the guide and
intensity of the recommendation as a component of the
intervention evaluated.
Relevance to
scope
Relevance to scope (ecological validity) may vary, including relevant context (e.g.,
classroom vs. laboratory),
sample (e.g., age and characteristics), and outcomes
evaluated. At least some
research is directly relevant
to scope (but the research
that is relevant to scope does
not qualify as strong with
respect to validity).
Relationship
between
research and
recommendations
Criteria
(continued)
(5)
STRONG
Evidence Base
Panel has a high degree of
confidence that this practice
is effective.
MODERATE
Evidence Base
The panel determines that
the research does not rise
to the level of strong but
is more compelling than a
minimal level of evidence.
Panel may not be confident
about whether the research
has effectively controlled
for other explanations or
whether the practice would
be effective in most or all
contexts.
MINIMAL
Evidence Base
In the panels opinion, the
recommendation must be
addressed as part of the
practice guide; however, the
panel cannot point to a body
of research that rises to the
level of moderate or strong.
Role of expert
opinion
Not applicable
Not applicable
Not applicable
The panel relied on WWC evidence standards to assess the quality of evidence supporting educational programs and practices. The WWC evaluates evidence for the causal validity of instructional
programs and practices according to WWC standards. Information about these standards is available at http://whatworks.ed.gov. Eligible studies that meet WWC evidence standards for group
designs or meet evidence standards with reservations are indicated by bold text in the endnotes
and references pages.
(6)
Introduction
Introduction to the Teaching Math to Young Children Practice Guide
hildren demonstrate an interest in math well before they enter school.6 They notice basic
geometric shapes, construct and extend simple patterns, and learn to count. The Teaching
Math to Young Children practice guide presents five recommendations designed to capitalize on
childrens natural interest in math to make their preschool and school experience more engaging
and beneficial. These recommendations are based on the panel members expertise and experience and on a systematic review of the available literature. The first two recommendations identify
which early math content areas7 (number and operations, geometry, patterns, measurement, and
data analysis)8 should be a part of the preschool, prekindergarten, and kindergarten curricula,
while the last three recommendations discuss strategies for incorporating this math content in
classrooms. The recommendations in this guide can be implemented using a range of resources,
including existing curricula.
In recent years, there has been an increased
emphasis on developing and testing new
early math curricula.9 The development of
these curricula was informed by research
focused on the mechanisms of learning
math,10 and recent studies that test the impact
of early math curricula show that devoting
time to specific math activities as part of the
school curriculum is effective in improving
childrens math learning before and at the
beginning of elementary school.11 Research
evidence also suggests that childrens math
achievement when they enter kindergarten
can predict later reading achievement; foundational skills in number and operations may
set the stage for reading skills.12
This practice guide provides concrete suggestions for how to increase the emphasis on
math instruction. It identifies the early math
content areas that are important for young
childrens math development and suggests
instructional techniques that can be used to
teach them.
The panels recommendations are in alignment
with state and national efforts to identify what
children should know, such as the Common
Core State Standards (CCSS) and the joint
position statement from the National Association for the Education of Young Children
(NAEYC) and National Council of Teachers of
Math (NCTM).17 The early math content areas
described in Recommendations 1 and 2 align
with the content area objectives for kindergartners in the CCSS.18 The panel recommends
teaching these early math content areas using
a developmental progression, which is consistent with the NAEYC/NCTMs recommendation
to use curriculum based on known sequencing
of mathematical ideas. Some states, such as
New York, have adopted the CCSS and developed preschool standards that support the
CCSS. The New York State Foundation to the
Common Core is guided by principles that are
similar to recommendations in this guide.19
Introduction (continued)
effectiveness has not been examined individually. As a result, the body of evidence does not
indicate whether each recommendation would
be effective if implemented alone. However,
the evidence demonstrates that when all of the
recommendations are implemented together,
students math achievement improves.20
Therefore, the panel suggests implementing all
five recommendations in this guide together
to support young children as they learn math.
The first two recommendations identify important content areas. Recommendation 1 identifies number and operations as the primary
early math content area, and Recommendation
2 describes the importance of teaching four
other early math content areas: geometry,
patterns, measurement, and data analysis.
Recommendations 3 and 4 outline how teachers can build on young childrens existing
math knowledge, monitor progress to individualize instruction, and eventually connect
childrens everyday informal math knowledge
to the formal symbols that will be used in later
math instruction. Finally, Recommendation 5
provides suggestions for how teachers can
dedicate time to math each day and link math
to classroom activities throughout the day.
(8)
Introduction (continued)
teaching number and operations based
on their expertise and understanding of
the research on how children learn math
(see Table 3). The panel acknowledges that
different developmental progressions exist;
for example, the Building Blocks curriculum
is based on learning trajectories that are
similar but not identical to the developmental progression presented.22 For a discussion of learning trajectories in mathematics
broadly, as well as the connection between
learning trajectories, instruction, assessment, and standards, see Daro, Mosher,
and Corcoran (2011).
Children should have regular and meaningful opportunities to learn and use
math. The panel believes that math should
be a topic of discussion throughout the
school day and across the curriculum. Early
math instruction should build on childrens
current understanding and lay the foundation for the formal systems of math that will
be taught later in school. These instructional
methods guide Recommendations 4 and 5,
which focus on embedding math instruction
throughout the school day.23
Introduction (continued)
The panel considered two bodies of literature
to develop the recommendations in the
practice guide: (1) theory-driven research,
including developmental research25 and
(2) research on effective practice. The theorydriven research provided a foundation from
which the panel developed recommendations by providing an understanding of how
young children learn math. As this first body
of literature did not examine the effectiveness of interventions, it was not reviewed
under WWC standards, but it did inform the
panels expert opinion on how young children
learn math. The second body of literature
provided evidence of the effectiveness of
practices as incorporated in existing interventions. This body of literature was eligible for
review under WWC standards and, along with
the panels expert opinion, forms the basis
for the levels of evidence assigned to the
recommendations.
Introduction (continued)
members knowledge of and experience
working in preschool, prekindergarten, and
kindergarten classrooms. The panel identified
evidence indicating that student performance
improves when these recommendations are
implemented together.
Recommendation
1. Teach number and operations using a developmental
progression.
Moderate
Evidence
Minimal
Evidence
( 11 )
Recommendation 1
( 12 )
Recommendation 1 (continued)
Table 3. Examples of a specific developmental progression for number knowledge
Subitizing
(small-number
recognition)
Subitizing refers to a childs ability to immediately recognize the total number of items in a collection and label it
with an appropriate number word. When children are presented with many different examples of a quantity (e.g., two
eyes, two hands, two socks, two shoes, two cars) labeled
with the same number word, as well as non-examples labeled
with other number words (e.g., three cars), children construct
precise concepts of one, two, and three.
Developmental Progression
Meaningful object counting is counting in a one-to-one fashion and recognizing that the last word used while counting is
the same as the total (this is called the cardinality principle).
A child is ready for the next step when, for example,
if given five blocks and asked, How many? he or she counts
by pointing and assigning one number to each block: One,
two, three, four, five, and recognizes that the total is five.
Counting-based
comparisons
of collections
larger than three
Number-after
knowledge
Once children recognize that counting can be used to compare collections and have number-after knowledge, they can
efficiently and mentally determine the larger of two adjacent
or close numbers (e.g., that nine is larger than eight).
A child has this knowledge when he or she can answer
questions such as, Which is more, seven or eight? and can
make comparisons of other close numbers.
Number-after
equals one more
( 13 )
Recommendation 1 (continued)
Summary of evidence: Moderate Evidence
( 14 )
Recommendation 1 (continued)
How to carry out the recommendation
1. First, provide opportunities for children to practice recognizing the total number
of objects in small collections (one to three items) and labeling them with a number
word without needing to count them.
Being able to correctly determine the number
of objects in a small collection is a critical
skill that children must develop to help them
learn more complex skills, including counting larger collections and eventually adding
and subtracting. To give children experience
with subitizing40 (also known as small-number
recognition), teachers should ask children to
answer the question How many (name of
object) do you see? when looking at collections of one to three objects.41 As described
in the first step of Table 3, children should
practice stating the total for small collections without necessarily counting. Research
indicates that young children can learn to
use subitizing to successfully determine the
quantity of a collection.42
( 15 )
Recommendation 1 (continued)
Example 1. The Basic Hiding game43
Objective
Practice subitizingimmediately recognizing and labeling small numbers and constructing
a concept of one to threeand the concept of number constancy (rearranging items in a set
does not change its total).
Materials needed:
Objects. Use a small set of identical objects early on and later advance to larger sets or
Directions: With a small group of children, present one to three objects on a mat for a few
seconds. Cover them with a cloth or box and then ask the children, Who can tell me how many
(name of objects) I am hiding? After the children have answered, uncover the objects so that
the objects can be seen. The children can count to check their answer, or the teacher can reinforce the answer by saying, for example, Yes, two. See, there are two (objects) on the mat: one,
two. Continue the game with different numbers of objects arranged in different ways. Teachers
can also tailor the Basic Hiding game for use with the whole class or individual children.
Early math content areas covered
Subitizing
Increasing magnitude up to five items
or she sees. Then, cover the objects and ask again. For larger collections (greater than
three), allow the child to check his or her answer by counting.
Integrating the activity into other parts of the day
Consider playing the game at various points during the day with different sets of objects,
including objects that are a part of childrens everyday experience (e.g., spoons and blocks).
Using the activity to increase math talk in the classroom
Use both informal (more or less) and formal (add and subtract) language to
( 16 )
Recommendation 1 (continued)
2. Next, promote accurate one-to-one counting as a means of identifying the total number
of items in a collection.
Small-number recognition provides a basis for
learning the one-to-one counting principle in
a meaningful manner.47 Often, children begin
learning about number from an early age by
reciting the count sequence (one, two, three,
four). But learning to assign the numbers of
the count sequence to a collection of objects
that are being counted can be a challenging
step. Once children are able to reliably recognize
and label collections of one to three items immediately (without counting), they have started to
connect numbers with quantity. As illustrated
in the second step of Table 3, they should then
begin to use one-to-one counting to identify
how many are in larger collections.48
To count accurately, oneand only onenumber word must be assigned to each item in the
collection being counted. For example, when
counting four pennies, children must point to a
penny and say one, point to a second penny
and say two, point to a third penny and say
three, and point to the final penny and say
four. During this activity the child will need to
keep track of which pennies have been labeled
and which still need to be labeled. The child
can also practice recognition of the cardinality
principle: that the last number word is the total
(cardinal value) of the collection. Although
children can learn to count one-to-one by rote,
they typically do not recognize at the outset
that the goal of this skill is to specify the total
of a collection or how many there are. For
example, when asked how many they just
counted, some children count again or just
guess. By learning one-to-one counting with
small collections that they already recognize,
children can see that the last word used in the
counting process is the same as the total.49
two
three
Recommendation 1 (continued)
Example 2. The Hidden Stars game52
Objective
Practice using one-to-one counting and the final number counted to identify how many objects.
Materials needed:
Star stickers in varying quantities from one to ten, glued to 5-by-8-inch cards
Paper for covering cards
Directions: Teachers can tailor the Hidden Stars game for use with the whole class, a small
group, or individual children. Show children a collection of stars on an index card. Have one
child count the stars. Once the child has counted the stars correctly, cover the stars and
ask, How many stars am I hiding?
Early math content areas covered
Counting
Cardinality (using the last number counted to identify the total in the set)
accuracy and say the amount using the cardinality principle (the last number counted
represents the total).
When children repeat the full count sequence, model the cardinality principle. For
example, for four items, if a child repeats the count sequence, say, One, two, three, four.
So I need to remember four. There are four stars hiding.
Have a child hide the stars while telling him or her how many there are, emphasizing
( 18 )
Recommendation 1 (continued)
Table 4. Common counting errors
Type of Counting Error
Example
Remedy
SEQUENCE ERROR
Saying the number sequence
out of order, skipping numbers, or using the same number more than once.
Skips 15:
113, 14, 16, 17, 18.
10
Practice reciting (or singing) the singledigit sequence, first focusing on one to
ten, then later moving on to numbers
greater than ten.
Highlight and practice exceptions, such
as fif + teen. Fifteen and thirteen are commonly skipped because they are irregular.
Same as above.
5,6
SKIM
No effort at one-to-one counting or keeping track.
NO CARDINALITY RULE
Not recognizing that the last
number word used in the counting process indicates the total.
( 19 )
Recommendation 1 (continued)
3. Once children can recognize or count collections, provide opportunities for children
to use number words and counting to compare quantities.
Once children can reliably determine how
many objects are in a collection, either by
subitizing or counting, teachers can provide
them with opportunities to compare the magnitudes of different collections using number
words (steps 3 through 6 in the developmental progression illustrated in Table 3).
10
Teachers can provide opportunities for practicing the application of the increasing magnitude
principle while playing games that involve
keeping score. A teacher can have two children
( 20 )
Recommendation 1 (continued)
a good opportunity to reinforce these types of
questions. Children can answer a quick Which
is more? question before transitioning to the
next activity.
find that a number list, or a series of numerals in order, can be used to compare numbers (see Figure 3).58 Children can see which
numbers are more or fewer based on the
numbers positions on the list. Number lists
may be particularly helpful for comparing two
collections: by counting with a number list,
children can see that numbers earlier and later
in the list denote lesser and greater cardinalities and, therefore, indicate smaller and larger
quantities. As children practice, these comparisons can be done without the aid of a number
list. Transitioning between activities provides
A number list is a series of numerals beginning with 1 and ordered by magnitude. Number lists are similar to number lines; however,
they do not include 0 and are an easier tool
for young children to use when counting and
learning numerals.
10
Recommendation 1 (continued)
Example 3. The Concentration: Numerals and Dots game
Objective
Match numerals with corresponding quantities.
Materials needed:
One set of twenty cards: ten cards with numerals from 1 to 10 along with the corre-
sponding number of dots, and ten cards with pictures of objects (the numbers of
objects corresponding to a numeral 110).
For even more advanced play, once children are proficient at numerals 110, teachers
the idea that appearance does not matter when it comes to number.
Monitoring childrens progress and tailoring the activity appropriately
Play the game with a small group of children, noting each childs progress in practicing
Use fewer cards, lower numbers, or cards with dots to scaffold. As children gain proficiency with the concepts, increase the number of cards and the size of the numbers.
Using the activity to increase math talk in the classroom
Before asking, How many? ask, How can we find out how many?
5. Once children develop these fundamental number skills, encourage them to solve
basic problems.
recognition or counting and can understand
the concepts of more and fewer, they can
explore the effects of adding and subtracting items from a collection. One way to help
children apply their knowledge is to create
activities that involve manipulating small
Using their number knowledge to solve arithmetic problems can give children a context
to apply and expand this knowledge and
gain confidence in their math ability.60 Once
children can determine the total number of
items in a collection by using small-number
( 22 )
Recommendation 1 (continued)
sets of objects.61 Children can change small
collections of objects by combining or removing objects (e.g., adding two blocks to three
blocks) and then count to determine how
many they have in the new collection. As
children become more adept, teachers should
present more difficult problems with slightly
larger numbers. Problem solving can be useful
even if children have not completely mastered
fundamental number skills, as problem solving
may serve as a vehicle for childrens learning.
Problem solving challenges children to use
their math knowledge to answer and explain
math-related questions, providing them with
an opportunity to strengthen their math skills.
of objects (e.g., pennies) when the final outcome is hidden from view.63 This arrangement
can be in a hiding game that is an extension
of the Basic Hiding game (see Example 1) or
Hidden Stars (see Example 2). Teachers can
place three or four objects in a line while the
children watch. Teachers can then cover the
objects (with a cloth or with a box that has an
opening on the side) and, while the objects are
covered, take one or two additional objects
and add them to the objects under the cover.
(Alternatively, they can reach beneath the cover
to take one or two objects away.) The children
see the initial group of objects and the objects
being added or taken away, but they do not
see the final set of objects. The children must
then determine, without looking at the final set
of objects, how many are hiding. Children may
solve this problem by counting on their fingers
or in their heads. After the children give their
answer, the teacher can take the cover away,
and the children can count to check the answer.
Snack time is also a great opportunity to provide children with authentic comparisons of
adding and subtracting or more and fewer.
As children receive or eat their snacks, they
can count how many items they have. Teachers can also adapt this activity for children of
varying skill levels by asking each child different questions, such as How many will you
have after you eat one? or How many will
you have after your friend gives you one?
Because the number will change, this activity
provides good practice for understanding
comparisons of more and fewer and combining or removing objects.
Once children have experience with combining or separating objects in a collection they
can see, they can do the same with collections
Recommendation 1 (continued)
service projects, such as canned-food drives,
which can provide opportunities for children
to count, sort, label, and organize donations.
Sports can also provide children with chances
to practice mathfor example, measuring
the distance for a race on the playground,
recording times, and making a chart to
display results. Teachers can also consider
sharing their own interests with children and
highlighting whatever math is involved, such
as the measurement involved in cooking or
sewing, the geometry involved in woodworking, and so on.
( 24 )
Recommendation 2
( 25 )
Recommendation 2 (continued)
Seven interventions taught measurement.78
These interventions were examined in nine
studies. Positive effects were found in the
domains of general numeracy, geometry,
and basic number concepts.79
( 26 )
Recommendation 2 (continued)
Together, these three limitations resulted in
the panel not being able to claim with certainty that the effects seen were due solely to
targeted instruction in the early math content
areas of geometry, patterns, measurement,
and data analysis. Nevertheless, the panel
believes the positive effects found for interventions based on a developmental progression when compared to instruction that does
not appear to be based on a developmental
progression support their recommendation
Teachers should provide examples and nonexamples of shapes so children can learn
to categorize them.85 A non-example of a
shape lacks one or more critical attributes
that define the shape. For instance, a long,
thin rectangle is a non-example of a square
because all the sides are not equal; a diamond
(rhombus) is a non-example of a triangle
because it has four sides instead of three.
These and other examples and non-examples
allow children to make distinctions about the
basic features of shapes, paving the way for
learning about relationships among shapes.
Once children are comfortable recognizing and comparing shapes, teachers should
encourage children to explore how shapes
can be combined and separated to form new
shapes.86 For example, two identical squares
can be combined to form a rectangle, and a
square can be cut along the diagonal to form
two triangles or across the middle to form
two rectangles, as shown in Figure 4.
( 27 )
Recommendation 2 (continued)
Figure 4. Combining and separating shapes
Two identical squares can be combined to form a rectangle.
=
A square can be cut across the middle to form two rectangles.
=
the critical attributes of the shape.87 They can
also learn about spatial relationships between
objects, such as in, on, under, beside,
above, or below.
( 28 )
Recommendation 2 (continued)
Example 4. The Shapes game
Objective
Identify and discuss attributes of various shapes and how to manipulate shapes to fit
inside a larger field.
Materials needed:
A large piece of poster board with a large shape drawn on it
Various (precut) foam or plastic geometric shapes
Directions: Children draw from a basket or bag containing a variety of small shapes to put
on the large shape drawn on a piece of poster board. The children take turns choosing a small
shape from the basket and then identifying it, describing it, and placing it on top of the large
shape. The group works together to fit as many small shapes as possible within the borders of
the large shape without overlapping any of the shapes. When children have finished filling the
large shape, they can count how many of each small shape they used and how many shapes
were used in total. For subsequent games, the children can try to choose and place shapes
strategically so the group can fit more small shapes inside the large shape. Teachers can tailor
the Shapes game for use with the whole class, a small group, or individual children.
Early math content areas covered
Geometry (shapes and attributes of shapes)
As children become more proficient with the activity, increase the complexity of the shapes.
Integrating the activity into other parts of the day
Blocks offer an opportunity for children to strategically manipulate and combine
( 29 )
Recommendation 2 (continued)
2. Encourage children to look for and identify patterns, then teach them to extend,
correct, and create patterns.
Pattern instruction should begin by encouraging children to experiment with basic repeating patterns. For example, teachers can select
a child to establish the pattern in which the
rest of the class will line up for an activity
Winter
January
February
March
Sunday
Monday
Summer
Spring
Fall
April
July
October
May
August
November
June
September
December
Tuesday
Wednesday
Thursday
Friday
Saturday
Recommendation 2 (continued)
and ask children to correct the errors. As
childrens understanding grows, teachers can
provide opportunities for children to create
patterns based on a set of instructions. For
example, teachers could present the beads
and strings to children and ask them to make
a pattern in which two red beads follow every
Directions: Distribute short strings and handfuls of colored beads to the children. Create
an example of a pattern, such as a red bead followed by a blue bead followed by another
red bead. First, ask the children to recreate the existing pattern. Next, ask the children to
predict which color will come next in the pattern. As the childrens understanding grows,
create patterns with deliberate errors (for example, following the blue bead with a second
blue bead in the exercise above) and then ask the children to identify incorrect sequences.
Finally, instruct the children to create patterns on their own. Teachers can tailor this activity for use with the whole class, a small group, or individual children.
Early math content areas covered
Patterns
(e.g., red, blue, red, blue, red, blue). If the child grasps the exercise quickly, use more
complicated patterns (e.g., red, blue, red, blue, blue, red, blue, blue, blue).
Integrating the activity into other parts of the day
Adapt the exercise to include patterns children find in the world around them. For
example, encourage children to look for patterns in the tiles on the classroom floor
(square tiles and rectangular tiles), the bricks on the outside of the school (big bricks
and small bricks), the clothing they wear (stripes, plaids, and other designs), or music
they hear (verses and choruses).
Using the activity to increase math talk in the classroom
Ask children to create patterns using themselves when lining up, and emphasize that
Recommendation 2 (continued)
3. Promote childrens understanding of measurement by teaching them to make direct
comparisons and to use both informal or nonstandard (e.g., the childs hand or foot)
and formal or standard (e.g., a ruler) units and tools.
Teachers should show children how to
compare objects for the purpose of sorting,
arranging, and classifying them.90 Teachers
can help children understand what it means
to compare the characteristics of two objects
and identify similarities and differences.
For example, as childrens understanding of
comparisons develops, children can begin to
compare the lengths of two pieces of string to
determine which is shorter or longer. Teachers
can expand on this concept by demonstrating
how to arrange a collection of pieces of
string from shortest to longest. When making comparisons, teachers should reinforce
measurement vocabulary words that describe
the characteristics of the objects and the
differences between them. Table 5 provides
examples of vocabulary words associated
with different types of measurement.
Length
Size
Temperature
Time
Weight
( 32 )
Recommendation 2 (continued)
to the restroom, by counting the number of
steps between the two locations. Teachers
could emphasize that childrens measurements may vary depending on the size of the
steps they take. Once children have learned
to assign numerical values and use measurement vocabulary and tools, they can measure
the distance in standard feet and inches using
rulers and yardsticks.
warmer than yesterday) through different seasons, and differences in time (We eat breakfast
in the morning, and we eat dinner at night).
Children will learn that thermometers, scales,
and rulers produce more consistent measurements than nonstandard tools. Understanding
the numerical values associated with measurement will then help children make comparisons
between objects. Children can utilize their existing knowledge of number to determine that an
object with a length of 10 inches is longer than
an object with a length of 5 inches because ten
is more than five.
4. Help children collect and organize information, and then teach them to represent
that information graphically.
Teachers should provide children with opportunities to count and sort familiar items to
introduce them to the concept of organizing
and displaying information.92 This information
can take the form of tangible objects, such
as toys or blocks, or abstract concepts, such
as characteristics (e.g., which children are 4
years old and which children are 5 years old)
or preferences (e.g., favorite snacks, colors,
or animals). The goal of such exercises is to
demonstrate both the characteristics that
distinguish the items and the total number in
each set relative to other sets. For example,
teachers could introduce sorting exercises
when children are cleaning up and putting
away toys. For children interested in building, teachers could encourage recording
the number of different types of blocks. For
children interested in drawing, teachers could
encourage sorting, counting, and recording
( 33 )
Recommendation 2 (continued)
Example 6. The Favorites game
Objective
Have children practice sorting and grouping.
Materials needed:
Signs for each sorting category, located in different areas of the classroom. In this
( 34 )
Recommendation 2 (continued)
Potential roadblocks and solutions
Roadblock 2.2. Some children are struggling with basic vocabulary skills or are being
exposed to English for the first time.
Suggested Approach. Teachers can link
visual representations of the most important
vocabulary and concepts for geometry, patterns, measurement, and data analysis with
terms in the childs home language, as well as
in English, particularly when multiple children
in the classroom speak the same language.95
Teachers can help English-speaking children
learn to count in their classmates native
languages to learn about each other. Songs
and fingerplays are helpful tools for learning
new words and math concepts. Using math
manipulatives and inviting children to arrange
materials or draw to show their answers can
also help bridge the language gap.
( 35 )
Recommendation 3
Recommendation 3 (continued)
The 12 studies examined curricula that
included regular, short assessments during
lessons. These assessments may have been
informal, computer-based, or supported by
rubrics to be used by the teacher during
small-group instruction. Two interventions that
included regular, short assessments were examined in six studies. Four of the six studies examined an intervention that included supports for
assessments. On average, children who participated in the intervention scored higher on math
outcomes than did children in the comparison
condition.102 Two of the six studies examined a
number sense curriculum that included regular
informal assessments to support the tailoring
of review sessions. Once again, children who
participated in the intervention tended to score
higher on math outcomes than children in the
comparison condition.103
The panel concluded that the body of evidence assessed in relation to Recommendation 3 was promising. However, it was not
sufficient to warrant a moderate evidence
rating as the panel was unable to definitively
attribute the effects in the studies to the
strategies included in Recommendation 3 due
to two characteristics of the studies. First, the
interventions examined in the studies were
multi-component interventions that included
strategies related to Recommendation 3 and
other recommendations in the guide.105 As
such, it was difficult to determine whether
the use of progress monitoring alone, or in
combination with other program components,
was responsible for the effects seen in math
achievement. It is also possible that progress
monitoring had no effect, and other components (or practices) were responsible for
effects observed. Second, in most studies, the
difference in the amount and type of progress
monitoring the intervention and comparison
groups received was not always specified,106
and thus was not considered a direct test of
a key component of the recommendation.
Based on its expertise and the effects of
interventions that include progress monitoring, the panel believes the studies generally
support this recommendation despite the
limitations to the body of evidence.
Recommendation 3 (continued)
these shapes should differ in size and
color for each child. After presenting a
lesson on the different shapes, the teacher
could ask younger children to name and
compare the shapes in their bags, inquiring whether there are fewer blue circles
or green triangles in the bag, which
rectangle is the longest, or which circle
is the smallest. Teachers could challenge
older children to remove a shape from the
baga rectangle, for exampleand to tell
the group how they know it is a rectangle.
This kind of introductory activity can
provide an opportunity for the teacher to
assess a childs ability to sort shapes with
similar features and classify them using
math vocabulary.
2. Tailor instruction to each childs needs, and relate new ideas to his or her
existing knowledge.
Teachers should continually monitor a childs
learning by employing a combination of
strategies from the first step in this recommendation and should then use that information to design instructional activities.107 Once
teachers have information about a childs skill
level, they can use a developmental progression to determine what the child should learn
next and then can choose activities that are at
or slightly above the childrens level of understanding. For example, once a child can use
small-number recognition to compare small
collections, he or she can use meaningful
object counting to determine the larger of two
collections (for more details on a developmental progression for number and operations,
see Table 3). Activities that are only slightly
Recommendation 3 (continued)
in music, teachers can design math activities
that involve musical instruments. Children
can determine how many instruments they
need for everyone to play together or how
many sticks are needed to play all the drums;
they can count, sort, and compare different
sets of instruments (how many drums, how
many wind instruments, etc.); they can count
along with musical beats, claps, or marching;
and they can create musical patterns (e.g.,
one drum beat, two claps, one drum beat,
two claps). By engaging children in activities
that are interesting and applicable to their
daily lives, children can connect skills across
different activities and content areas.
3. Assess, record, and monitor each childs progress so that instructional goals and
methods can be adjusted as needed.
It is important to continually monitor progress so that children can be consistently
engaged in activities that are neither too far
below their level (and therefore not interesting) nor too far above it (and therefore
frustrating). Progress monitoring also allows
teachers to plan what children should learn
next. Example 7 contains a model of the
flow of progress monitoring. In this model,
a teacher focuses small-group instruction
on counting small collections. The teacher
Assess:
Observe and record
Plan activities
( 39 )
Implement
Recommendation 3 (continued)
child can do particularly well. For example, a
teacher can observe a child counting objects
to assess whether the child can successfully
count with one-to-one correspondence. If the
teacher notices a child making a coordination
or sequencing error, the teacher can note the
type of error to help determine which activities
the child should work on next to practice this
skill. (See Table 4 for common counting errors.)
Child
Largest Set
Counted
Successfully
Types of
Errors Made
Date
Activity
Sarah
September
counting
stars
skips six
when
counting
Bill
September
counting
stars
10
sometimes
double-counts
a star
Recommendation 3 (continued)
Roadblock 3.3. What if I do not have
required assessments, or the assessments
do not fit well with the skills that are targeted
in the developmental progression?
( 41 )
Recommendation 4
Teach children to view
and describe their world
mathematically.
Teachers can encourage children to look
for opportunities to describe math ideas
in the world around them, gradually
moving from informal representations
and language to formal representations
and math vocabulary as childrens
understanding grows.109 By exploring
their environment and interacting with
manipulatives, children can begin to
apply their math knowledge.110 At first,
children should use informal tools such
as their fingers, tally marks, or other
concrete objects to represent math ideas.
For example, children can be encouraged
to use blocks to model and solve simple
addition problems (e.g., If I have two
blocks, and I add three more, how many
Representations are objects, actions, words,
blocks do I have?). Once children are
pictures, or symbols that stand for ideas.
comfortable using math informally,
teachers can help them link their
informal knowledge to more abstract math concepts, formal math vocabulary, and formal
representations such as math symbols.111
If children hear math vocabulary in context and then practice using it, they may be better able
to understand the underlying math concepts.112 The panel believes there is evidence of a positive
relationship between math-related talk and childrens math knowledge.113 As one part of mathrelated talk, teachers can use open-ended questions to prompt children to think about how to
describe their ideas mathematically and to increase the amount of math-related dialog in the
classroom. If a child can describe his or her method for solving a problem to someone else and
then hear other children describe their approach to a problem, all the children may learn to
apply their math knowledge in new ways.114 Teachers can reinforce this idea by encouraging
children to look for opportunities to use their developing math skills throughout the school day.
Recommendation 4 (continued)
and geometry domains.120 In two studies, math
conversation, whether with a peer or an adult,
resulted in higher math achievement.121
Informal Representation
whole
number
three
equal
unequal
more than
or fewer than
addition
and or more Start with a collection and add more items to make it larger. For example, start with three crayons and add one more. Then ask, How many?
Start with a collection and take away some items to make it smaller.
For example, start with three crayons and take away one. Then ask,
How many?
( 43 )
Recommendation 4 (continued)
2. Help children link formal math vocabulary, symbols, and procedures to their informal
knowledge or experiences.
other math concepts. For example, teachers
can make a comment about which child is
standing first in line or which child has
more or fewer objects than another child.
As another example, while the child is drawing a picture of his or her family, a discussion
could focus on the number of family members and who is older or younger.
Concept
Lesson
numerals
counting
Have children count and record the number of children in attendance each day.
+,
operations
equal
Show the class four pennies. Next, show three pennies, verbally
label them (I have one, two, three pennies), and put them in a
can. Then, show one more penny, verbally label it (I have one more
penny), and put it in the can. Ask the class, Are three pennies and
one more penny the same number as four pennies?
<,>
unequal
Show the class five pennies, verbally label them, and put them in
a can. Next, show four pennies, verbally label them, and put them
in a different can. Ask the class, Which can has more? Which can
has fewer?
( 44 )
Recommendation 4 (continued)
3. Use open-ended questions to prompt children to apply their math knowledge.
Table 8. Examples of open-ended questions
( 45 )
Recommendation 4 (continued)
4. Encourage children to recognize and talk about math in everyday situations.
Teachers can encourage math thought and
conversation by asking children for their
help with problems that arise throughout
the day.129 For example, a teacher might say,
I have to figure out how many cups we are
going to need for the birthday party. Can you
help me? How should we do that?
answer, teachers might ask, How did you figure that out? or Show me how you did that.
If children share a strategy, teachers might also
ask, Is there another way to solve that problem? or What would happen if I changed?
Asking children to compare and contrast also
helps them clarify their ideas (How are these
[shapes, numbers, patterns, measuring] tools
alike or different?). These questions are appropriate for any math content area.
( 46 )
Recommendation 5
Dedicate time each day
to teaching math, and
integrate math instruction
throughout the school day.
Dedicated time that is devoted to planned,
daily math lessons can allow children
to develop important skills in number
and operations, geometry, patterns,
measurement, and data analysis. By
connecting 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.131 If
teachers coordinate their current math
objectives with activities in the classroom
and lessons in other subject areas, children
can master skills and extend the concepts
to higher levels or broader contexts.132
A classroom environment that contains mathrelated objects can help children recognize and apply math knowledge. For example, games
can provide an enjoyable and meaningful way to learn a range of math ideas and practice a
wide variety of basic skills.133 Games can build on childrens math knowledge, provide a reason
for learning skills and concepts, supply repeated practice that is not boring, give children and
teachers an opportunity to discuss strategies and ideas, and generate excitement.134
concepts.137 These activities can be implemented during various times of the day,
such as circle time, transitions, or mealtimes.
Children in classrooms using Math Is Everywhere scored higher in the general numeracy
domain than children in classrooms where the
teachers continued their regular classroom
instruction. These higher scores could be due
to teachers providing daily math lessons and
incorporating math into various times of the
day; however, the scores could also be due to
aspects of other recommendations present in
the intervention.
One of the studies examined Math Is Everywhere, a collection of 85 suggested activities (e.g., books, music, games, discussions,
and group projects) that reinforce math
Recommendation 5 (continued)
played color-based board games or no board
games.138 However, the effects of numberbased board games on measures of number
recognition and operations were mixed.139 The
interventions in which playing a board game
was part of a larger curriculum included not
only elements of this recommendation but
also other recommendations in the guide.140
Recommendation 5 (continued)
Example 9. Linking large groups to small groups
Objective
Understand the differences and similarities between triangles, rectangles, and squares.
Materials needed:
Book: Bear in a Square, by Stella Blackstone
A variety of other objects (based on availability, but could include the following)
Large pieces of paper cut into varied shapes for painting
Lunch trays and a small amount of sand
Geoboards with rubber bands
Directions, large group: Read the book in a large group, highlighting the names of all the
shapes but focusing specifically on the difference between the number and length of sides
and types of angles in triangles, rectangles, and squares.
Directions, small group: Once children are divided into small groups, highlight the number
and length of sides and types of angles in each of the shapes the children create in the activities below. Children should be encouraged to use informal terms to describe the shapes, such
as long and short sides and big and little angles for triangles. These activities will vary
based on the types of materials available, but they could include the following:
Provide paint, chalk, or other art materials so that children can add a stripe around the edge
of a large paper cutout of a triangle or rectangle. Then, have the children continue to add
more of the same shapes inside the original shape to create a design with concentric shapes.
Lead children to use their fingers to draw shapes in sand on a tray or in a sandbox.
They might draw shapes within shapes or combine shapes to make other figures.
Encourage children to experiment with placing rubber bands on a geoboard to make
triangles, rectangles, and squares of different sizes and orientations.
Early math content areas covered
Geometry (shapes and attributes of shapes)
children may notice the number of sides on other shapes, such as a pentagon, or may
ask about the number of sides in a circle.
Integrating the activity into other parts of the day
Take a group walk outside to collect sticks of different sizes, and then use them to
room or on their clothing, to identify examples of triangles, rectangles, and squares. When
children locate a shape, ask them to explain it to the group: How can you tell that shape is
a
? Prompt the children to identify the number and length of sides and type of angles.
( 49 )
Recommendation 5 (continued)
by saying, for example, We have 8 girls and we
have 10 boys. We have 18 children all together:
8 plus 10 equals 18. The class could then display the results of attendance for several days
using a chart that has columns or rows titled
with the days of the week or a pie chart with
the number of slices in the pie matching the
total number of children in the class on a particular day. Teachers can also engage children
( 50 )
Recommendation 5 (continued)
3. Highlight math within topics of study across the curriculum.
Teachers can integrate math concepts into
non-math lessons by highlighting the aspects
of math that are already present in the curriculum.145 Teachers can point out opportunities to
count objects, examine shapes, analyze data, or
measure objects (depending on the current math
objectives and where children are in the developmental progressions for these content areas).
During literacy time, for example, when reading a story, the teacher can ask questions that
encourage children to solve a math problem
based on the story. If the class is reading a
Science
Literacy
Geometry
Patterns
Measurement
Data Analysis
We All Went on
Safari, Krebs
Bear in a Square,
Blackstone
A Pair of Socks,
Murphy
How Big Is a
Foot?, Myller
Its Probably
Penny, Leedy
Mouse Count,
Walsh
Mouse Shapes,
Walsh
Pattern Bugs,
Harris
Spence Is Small,
Chevalier
7 Little Rabbits,
Becker and
Cooney
Shapes,
Silverstein
Pattern Fish,
Harris
Tall, Alborough
Tiger Math,
Nagda and Bickel
Describe objects
from nature (e.g.,
rocks, leaves,
and insects) in
geometric terms.
Graph the
amount the
classroom plant
grows each day.
Use precut
shapes to make
animals.
Design a model
of an insect
using a pattern
design.
Measure the
growth of a plant
in the classroom each day
and predict how
much time it will
take before flowers will be visible
on the plant.
Identify shapes
in artwork.
Use patterns to
make pictures
or frames for
pictures.
Make a graph of
the childrens
favorite colors.
Art
The Grouchy
Ladybug, Carle
Graph animals
with two legs,
four legs, and
more than four
legs.
Tally childrens
opinions about
artwork. For
example, ask,
Which painting do you like
better?
(continued)
( 51 )
Recommendation 5 (continued)
Table 9. Integrating math across the curriculum (continued)
Math Content Area
Social Studies
Number and
Operations
Count the length
of time it takes
to wash your
hands.
List rules for
washing hands
or playing safely
outside.
In a unit about
families, order
people by size
or from youngest
to oldest.
Geometry
Patterns
Measurement
Data Analysis
Jump rope or
play hopscotch
with an alternating pattern.
Measure your
bodys growth
over time.
Graph your
height or foot
size.
During a unit on
recycling, children can count
how many of a
certain object
they have collected to recycle.
Study patterns
in clothes from
different parts of
the world.
Make a map of
the neighborhood using
measuring,
Look for patterns geometry, spatial
thinking, and
in flags from
positioning
other countries.
words.
4. Create a math-rich environment where children can recognize and meaningfully apply math.
Teachers can provide opportunities for children to see and use math concepts regularly
by creating a math-rich classroom environment. This enrichment can be done by making
math-related objects and tools readily available, labeling and organizing math-related
objects and tools so they are easy to find and
Table 10. Examples of tools that can be useful in each math content area
Number and
Operations
Geometry
Patterns
Measurement
blocks
geoshapes
beads
rulers
abacuses
precut foam
shapes
different-colored
cubes
tape measures
art materials,
such as stamps
and markers
scales
number lists
number puzzles
clocks
measuring
spoons and cups
Data Analysis
clipboard and
paper for tallying the question
of the day
hula hoops or
small hoops that
bend for Venn
diagrams
sorting bins
( 52 )
Recommendation 5 (continued)
Teachers can explicitly teach children how
to use tools by modeling their use during
small- or large-group time.147 For example,
the teacher can use shapes or blocks to demonstrate how a rectangle and a triangle can
be combined to make a house. As another
example, the teacher can bring different
types of measuring tools to circle time to
demonstrate how to use tools to measure
objects of varying sizes (e.g., placing the
ruler next to the object to be measured, with
the end of the ruler at one end of the object,
then reading the number closest to the opposite edge of the object).
10
11 12
9
8
Art Station 1
3
7
Art Station 2
( 53 )
Recommendation 5 (continued)
5. Use games to teach math concepts and skills and to give children practice in
applying them.
correspondence and cardinality. Games that
target different math content areas are often
included in math curricula. Games can also
be purchased separately or be made by the
teacher. Some math concepts may also be
highlighted in games that come up during
natural play, such as hopscotch or jump rope.
( 54 )
Recommendation 5 (continued)
Teachers can get involved with the game-playing to ensure educational play. For example,
if children are playing a game to learn oneto-one correspondence and cardinality, the
teacher can emphasize moving one space at
a time and then reinforce the total number of
When purchasing materials, strategic planning can help save resources. Teachers can
choose games that teach the math content
areas children are most interested in. They
can also choose games that are accessible
to a range of skill levels to avoid having to
purchase more than one game. For example,
if the teacher is playing a memory game with
younger or less advanced children, the group
can play with all the cards face-up, or they
can play with fewer cards than the whole set.
The teacher can play the same game with
older or more advanced children by flipping
the cards over and using the whole set.
Teachers can also turn to existing community resources. For example, they can take
advantage of the local public library to find
math-related books for their classroom. Many
librarians can help teachers by selecting
( 55 )
Recommendation 5 (continued)
Roadblock 5.4. Parents may wonder why
their children are playing games in school.
( 56 )
Glossary
A
An assessment provides information on how much a child knows about a particular topic or the skills
a child has in a particular area. Assessments may include an adults observation of a child in classroom
activities, an adults rating of the child, or an adult's scoring of a child's accuracy on a particular task (e.g.,
test or worksheet). Assessments may be formal, such as standardized tests, standardized rating scales,
teacher-developed tests, or worksheets. Teachers may also conduct informal assessments to check to
see what a child knows or can do. Assessments can be formative, with the results used to determine
the extent to which the child learned the intended skills from instruction as part of progress monitoring.
Finally, assessments may be summative, with the result documenting a childs performance, for example,
on an end-of-chapter test or state developed test. The particular type of assessment (formal or informal,
formative or summative) should be chosen based on how the results will be utilized.
C
Cardinality is the total number of items in a collection. The cardinality principle is the understanding
that when counting, the number word assigned to the last item of a collection represents the total quantity.
A collection is a group of discrete objects or things.
D
A developmental progression refers to a sequence of skills and concepts that children acquire as
they build math knowledge. It effectively defines the developmental prerequisites for a skill or concept.
For grouping outcomes within WWC reviews for this practice guide, the panel defines a domain as a
group of outcomes related to a childs math achievement. For this practice guide, the panel has identified
six domains into which all outcomes are grouped: general numeracy, basic number concepts, number
recognition, operations, geometry, and patterns and classification. The domains are not meant to be
synonymous with any early math content area (see early math content areas).
E
Early math content areas are the specific math topics the panel believes should become the foundation of preschool, prekindergarten, and kindergarten curricula. The panel has identified number and
operations, geometry, patterns, measurement, and data analysis as critical to childrens math learning.
Outcome domains defined for grouping outcomes in WWC reviews cover the range of skills within the
early math content areas, but in some cases, the skills are grouped slightly differently (see domain).
F
Formal representations are the typically school-taught standard mathematical terms and symbols
that represent mathematical ideas. Informal representations are familiar everyday objects, pictures,
or words that stand for those ideas. Informal units, a type of informal representation, are non-standard
forms of measurement, such as blocks or childrens hands and feet. By contrast, examples of formal or
standard measurement tools include rulers and scales. Informal methods are childrens self-invented
strategies to solve mathematical problems, and these may be supported and encouraged by teachers.
I
The increasing magnitude principle is the idea that a number word later in the counting sequence
represents a larger quantity than a number word earlier in the counting sequence.
( 57 )
Glossary (continued)
M
Math knowledge is a childs understanding of math concepts and skills. Math achievement refers
to a childs performance on a variety of math tasks, including assessments.
A multi-component intervention is a set of instructional practices that are implemented together
and evaluated as a set.
N
A non-example illustrates what a concept is not. For example, whereas five and six come after four
and are examples of numbers larger than four, two and three come before four and are not larger.
Non-examples are teaching tools designed to illustrate the difference between two things, and thus
to help children learn the boundaries of a concept.
Number refers to a system for representing quantity. Number knowledge consists of an understanding of numbers and the relations among them. It includes the ability to recognize quantity, count,
identify numerals (written numbers), and perform number operations.
Number-after knowledge is a counting skill that comes from experience with the number sequence.
Children with number-after knowledge are able to identify the next number in the counting sequence
without starting the count from one.
A number list is a series of numerals beginning with 1 and ordered by magnitude.
Number sense refers to a persons general understanding of number and operations along with
the ability to use this understanding in flexible ways to make math judgments and to develop useful
strategies for solving complex problems.150
Numerals, or written numbers, are symbols that represent numbers. For example, the numeral 8 is
the symbol that represents the number eight.
O
The one-to-one counting principle refers to understanding one-to-one correspondence; that is,
when counting, each item in a collection must be assigned one and only one number word.
The panel uses the term operations to refer to arithmetic. Addition and subtraction are examples
of operations.
P
Prekindergarten (Pre-K) refers to the year before children enter kindergarten, usually when children
are 4 years old. Preschool refers to the year before the prekindergarten year, when most children
are 3 years old.
Progress monitoring is a systematic approach to assessment with the goal of improving skills.
Progress monitoring begins with an evaluation of the childs current level of knowledge. Changes in
the childs skills are then tracked through regular assessment, and goals and teaching strategies are
adjusted based on the childs progress.
S
Subitizing refers to a childs ability to immediately recognize the total number of items in a collection and label it with an appropriate number word. For example, subitizing enables a child to see a
collection of three toy animals and immediately know, without counting, that there are three.151 This
ability is also known as small-number recognition.
( 58 )
Appendix A
Postscript from the Institute of Education Sciences
What is a practice guide?
The Institute of Education Sciences (IES) publishes practice guides to share evidence and expert guidance on addressing education-related challenges not readily solved with a single program, policy, or
practice. Each practice guides panel of experts develops recommendations for a coherent approach
to a multifaceted problem. Each recommendation is explicitly connected to supporting evidence.
Using common standards, the supporting evidence is rated to reflect how well the research demonstrates the effectiveness of the recommended practices. Strong evidence means positive findings
are demonstrated in multiple well-designed, well-executed studies, leaving little or no doubt that the
positive effects are caused by the recommended practice. Moderate evidence means well-designed
studies show positive impacts, but there are questions about whether the findings can be generalized
beyond the study samples or whether the studies definitively show evidence that the practice is effective. Minimal evidence means that there is not definitive evidence that the recommended practice is
effective in improving the outcome of interest, although there may be data to suggest a correlation
between the practice and the outcome of interest. (See Table 1 for more details on levels of evidence.)
IES practice guides are then subjected to
external peer review. This review is done
independently of the IES staff that supported
the development of the guide. A critical task
of the peer reviewers of a practice guide is
to determine whether the evidence cited in
support of particular recommendations is
up-to-date and that studies of similar or better quality that point in a different direction
have not been overlooked. Peer reviewers
also evaluate whether the level of evidence
category assigned to each recommendation is
appropriate. After the review, a practice guide
is revised to meet any concerns of the reviewers and to gain the approval of the standards
and review staff at IES.
Appendix A (continued)
the recommendations. As a result, it is possible that two teams of recognized experts
working independently to produce a practice
guide on the same topic would come to
very different conclusions. Those who use
the guides should recognize that the recommendations represent, in effect, the advice
of consultants. However, the advice might
be better than what a school or district could
obtain on its own. Practice guide authors
( 60 )
Appendix B
About the Authors
Panel
Douglas Frye, Ph.D., is an associate professor at the University of Pennsylvanias Graduate
School of Education and is the director of the
Interdisciplinary Studies in Human Development program. Dr. Fryes research efforts
are concentrated on two topics in cognitive
development: childrens early math development and theories of mind. His math research
focuses on the developmental sequence of
early math reasoning skills and activities to
support young childrens development of those
skills. Dr. Frye has been involved in designing
and evaluating several strategies-based
emergent numeracy interventions, including
a computer-based program (Kids Count I at
Yale University) as well as classroom-based
interventions implemented in urban Head Start
classrooms (e.g., Kids Count II and the Evidencebased Program for Integrated Curricula [EPIC] at
the University of Pennsylvania). Dr. Frye also
investigates how young childrens theories of
mind relate to their understanding of teaching
and learning. He has been an associate editor
of Child Development and the Journal of Cognition and Development.
Appendix B (continued)
as teacher of the year for all of Philadelphia
Head Start in 2002, and was twice nominated
as teacher of the year for the School District of
Philadelphia (2001 and 2004). Throughout her
career, Ms. McDowell has welcomed research
teams into her classroom and developed collaborative relationships with researchers. As a
consultant with the Evidence-based Program for
Integrated Curricula (EPIC) project, she translated research-based developmental principles
into early childhood classroomfriendly math,
literacy, and socio-emotionally focused activities. She has mentored other teachers implementing this curriculum and leads professional
development meetings. Ms. McDowell is also a
COR assessment trainer/mentor for Philadelphia
Head Start. She has co-authored a book chapter
with Ageliki Nicoloupolo and Carolyn Brockmeyer on how play motivates and enhances
childrens developmentbased on research
conducted in her classroom. She has presented
at professional conferences including National
Head Start Association conferences and the
International Conference on Imagination and
Education. Ms. McDowell is certified in early
childhood education, elementary education,
and special education.
Staff
M. C. (Cay) Bradley, Ph.D., is a researcher
at Mathematica Policy Research. She has both
delivered and evaluated education and social
work programs. Dr. Bradley supported the
panel in the review and documentation of evidence. She has reviewed evidence for previous
What Works Clearinghouse practice guides and
topic areas. Dr. Bradley has also conducted
or participated in other meta-analyses and
syntheses focused on paraprofessional homevisiting programs and interventions for oppositional defiant disorder.
Elizabeth W. Cavadel, Ph.D., is a researcher
at Mathematica Policy Research. She has a background in child development and psychology.
Dr. Cavadel assisted the panel in the writing of
this practice guide, drawing on her experience
in early numeracy and in translating research to
practice. Dr. Cavadel has worked on large-scale
Appendix B (continued)
psychological and educational interventions
and evaluations in Head Start and private
preschool settings. Her current work focuses on
child outcomes across a range of topics including early child-care quality, childrens school
readiness, child behavior, social-emotional
development, and early childhood assessment.
Dr. Cavadel is a certified What Works Clearinghouse reviewer and has also reviewed evidence
and synthesized reports focusing on homevisiting programs.
appendices for this practice guide. She is a certified What Works Clearinghouse reviewer and
is a deputy principal investigator for the Early
Childhood Education topic area. She has experience assisting with evidence documentation
and writing for numerous practice guides on
topics such as effective fractions instruction.
Marc Moss, Ed.D., is a researcher at Abt
Associates. Dr. Moss has directed numerous
large-scale, national evaluations that examined
the implementation and impact of various
reforms in the field of education. He has
reviewed evidence for and participated in the
writing of this practice guide. Dr. Moss has
also reviewed evidence for other What Works
Clearinghouse practice guides and topic areas.
( 63 )
Appendix C
Disclosure of Potential Conflicts of Interest
Practice guide panels are composed of individuals who are nationally recognized experts on the topics
about which they are making recommendations. IES expects the experts to be involved professionally
in a variety of matters that relate to their work as a panel. Panel members are asked to disclose these
professional activities and institute deliberative processes that encourage critical examination of their
views as they relate to the content of the practice guide. The potential influence of the panel members
professional activities is further muted by the requirement that they ground their recommendations
in evidence that is documented in the practice guide. In addition, before all practice guides are published, they undergo an independent external peer review focusing on whether the evidence related
to the recommendations in the guide has been presented appropriately.
Douglas Frye and Elizabeth Cavadel
collaborated on developing the initial mathematics portion of Evidence-based Program for
Integrated Curricula (EPIC). Judy McDowell
was involved in the development of EPIC as
a classroom teacher. As EPIC is not currently
available for purchase, none of these authors
receive royalties for the curriculum.
Nancy Jordan was involved in the development of a number sense curriculum that is
reviewed in the guide. As the curriculum is
not currently available for purchase, Dr. Jordan
does not receive royalties for the curriculum.
Dr. Jordan is also the co-developer of the
Number Sense Screener assessment tool. She
receives royalties on the sale of Number Sense
Screener from Brookes Publishing.
( 64 )
Appendix D
Rationale for Evidence Ratings152
This appendix discusses studies that examined the effectiveness of recommended practices using
strong designs for addressing questions of causal inference including randomized controlled trials
(RCTs) and quasi-experimental designs (QEDs) that met What Works Clearinghouse (WWC) standards
and were used to determine the level of evidence rating. The studies were identified through an
initial search for research on practices for improving young childrens early math achievement. The
search focused on studies published between 1989 and 2011 that examined practices for teaching
number, operations, and other early math content areas to children in preschool, prekindergarten,
and kindergarten. Studies examined children in both the United States and other countries. Interventions could target children who were typically developing, at risk of facing challenges in math,
or exhibiting challenges with math or school in general. The search was supplemented with studies
recommended by the panel based on its expertise in the area of early math.
The panel identified more than 2,300 studies
through this search, including 78 studies with
rigorous designs reviewed according to WWC
standards. Twenty-eight of these studies met
evidence standards with or without reservations and tested interventions related to one
or more recommendations. Study effects were
calculated and classified as having a positive
or negative effect when the result was either:
statistically significant153 or
substantively important as defined
by the WWC.154
When a result met none of these criteria,
it was classified as having no discernible
effects. A study was described as having
mixed effects if it had any combination of
positive, negative, and no discernible effects.
Appendix D (continued)
The patterns and classification domain
includes measures that assess a childs
ability to identify, replicate, and extend
patterns. Also included are assessments
of a childs ability to sortfor example,
placing all red blocks on one shelf or all
triangle blocks on another shelf.
Appendix D (continued)
Table D.1. Summary of studies contributing to the body of evidence, by recommendation
Contributes to the body of evidence for
Citation
Rec. 1
Rec. 2
Rec. 3
Rec. 4
Rec. 5
X
X
X
X
Monahan (2007)
X
X
Siegler (1995)
Siegler and Ramani (2008)
X
X
Sood (2009)
Sophian (2004)
Weaver (1991)
X The comparison was included in the body of evidence for this recommendation.
Appendix D (continued)
Pre-K Mathematics is a supplemental curriculum designed to develop informal
math knowledge and skills in preschool
children.167 Math content is organized into
seven units. Specific math concepts and
skills from each unit are taught in the classroom through teacher-guided, small-group
activities with concrete manipulatives.
Take-home activities with materials that
parallel the small-group classroom activities
are designed to help parents support their
childrens math development at home.168
Appendix D (continued)
group). The effectiveness of an intervention
must be assessed in the context of a specific
comparison. For example, a finding based
on an intervention group that received math
instruction and a comparison group that
received reading instruction concerns the
effect of both the math content provided and
how it was taught. A finding based on a comparison between intervention children taught
math using manipulatives and comparison
children taught math without manipulatives
concerns the effect of manipulatives.
Appendix D (continued)
included core components of Recommendations 3, 4, and 5. As a result, the panel was
unable to isolate the effects of instruction
in number and operations. Without studies
providing an isolated (or direct) test of this
recommendation, it is impossible to say conclusively that the causes of the effects seen are
the result of practices aligned with the panels
suggestions of how to implement this recommendation. However, in the panels estimation,
teaching number and operations was a primary component of many of the interventions
that showed positive effects.
Appendix D (continued)
children also received targeted instruction in
number and operations; however, these children may not have received the same amount
of targeted instruction in number and operations or may not have received instruction in
which a developmental progression shaped
the sequence in which number and operations
topics were introduced. Findings in these five
studies were positive in the domains of basic
number concepts, geometry, and general
numeracy; mixed findings were reported in
the domain of operations.187
Appendix D (continued)
researchers found a positive effect on childrens performance in general numeracy when
compared to children receiving regular classroom instruction.202
Table D.2. Studies of early math curricula that taught number and operations and contributed
to the level of evidence rating
Population
Characteristics2
RCT
Meets evidence
standards without
reservations
Findings (Domain:
Assessment (Effect Size,
Significance))4
Teach Number
and
Operations
Skills
Citation, Design,
and WWC Rating1
Use a
Developmental
Progression
Recommendation
Components Tested
Study Characteristics
Math Is Everywhere
vs. regular classroom instruction
X7
X7
Geometry: BB Assessment
Geometry Scale
X7
X7
Comparison3
Clements and
Sarama (2007b)5,6
RCT
Meets evidence
standards with
reservations
Preschool classrooms in
state-funded or Head Start
programs
Children: 68 total (30 intervention; 38 comparison)
Age range: 2.9 to 4.8 years
Positive (1.40*)
( 72 )
(continue d)
Appendix D (continued)
Table D.2. Studies of early math curricula that taught number and operations and contributed
to the level of evidence rating (continued)
Recommendation
Components Tested
Population
Characteristics2
Clements and
Sarama (2008)5,8
RCT
Meets evidence
standards without
reservations
Findings (Domain:
Assessment (Effect Size,
Significance))4
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Study Characteristics
Building Blocks
vs. regular classroom instruction
(Where Bright
Futures Begin;
Opening the
World of Learning; Investigations
in Number, Data,
and Space; DLM
Early Childhood
Express)
X11
X11
X11
X11
X11
X11
Rightstart vs.
regular classroom
instruction
X13
X13
Comparison3
Positive (1.07*)
RCT
Meets evidence
standards without
reservations
Clements et al.
(2011)5,9,10
Kindergarten students in
public schools in inner-city
areas in Massachusetts
Positive (0.48*)
Basic number concepts:
REMANumbers Total Score
Positive (0.39*)
Geometry: REMAGeometry
Total Score
Positive (0.64*)
Positive (1.79*)
QED
Meets evidence
standards with
reservations
Klein et al. (2008)5
RCT
Meets evidence
standards without
reservations
Pre-K Mathematics
combined with
DLM Early Childhood Express vs.
regular classroom
instruction (Creative Curriculum,
High Scope, Montessori, locally
developed)
( 73 )
(continued)
Appendix D (continued)
Table D.2. Studies of early math curricula that taught number and operations and contributed
to the level of evidence rating (continued)
Recommendation
Components Tested
Population
Characteristics2
Sarama et al.
(2008)14
RCT
Meets evidence
standards without
reservations
Comparison3
Building Blocks
combined with
Pre-K Mathematics vs. regular
classroom
instruction
Findings (Domain:
Assessment (Effect Size,
Significance))4
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Study Characteristics
Positive (0.62*)
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the
comparison group may have participated in instruction that taught number and operations and that may have used a developmental
progression to guide that instruction.
X The intervention included this component.
BB Assessment = Building Blocks Assessment of Early Mathematics206
REMA = Research-Based Early Math Assessment207
CMA = Child Math Assessment208
TEMA-2 = Test of Early Mathematics Ability, second edition209
NKT = Number Knowledge Test210
1
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random procedure.
2
SD = Standard deviation. The information presented includes the following: (a) the type of program and unit of assignment, if the study
is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) age of children in the sample.
3
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group. If details
were available on the curriculum or curricula the comparison teachers used, it is noted parenthetically.
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic may
not have been math. If details were available on what was provided, it is noted parenthetically.
4
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ from
author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes that were
significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked with an asterisk (*);
ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed here. Positive findings favor
the intervention group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger). Negative findings
favor the comparison group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger).
No discernible refers to findings that are neither significant nor substantively important.
5
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook,
Version 2.1 (http://whatworks.ed.gov).
6
Clements and Sarama (2007b) also reported scores for the subscales of the Numbers and Geometry scales; positive effects were
seen for each subscale. Findings from Clements and Sarama (2007b) were previously reported in the WWC intervention report on
SRA Real Math Building Blocks PreK. The panel reports the same findings as presented in the intervention report.
7
In Clements and Sarama (2007b), the difference between the intervention and comparison groups included any aspect of instruction that differed between Building Blocks and the curricula used in the comparison classrooms, including the branded comprehensive
early childhood curriculum Creative Curriculum. The intervention group participated in Building Blocks, a math curriculum that included
instruction in number and operations guided by a developmental progression. The comparison group participated in a variety of curricula, including Creative Curriculum, which also included instruction in number and operations guided by a developmental progression.
8
For Clements and Sarama (2008), the WWC is reporting author-reported effect sizes consistent with prior reporting of findings
from this study in the WWC intervention report on SRA Real Math Building Blocks PreK.
9
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
10
Clements et al. (2011) also reported the subscale scores from the REMA. Findings for the subscale scores were consistent with
the total score findings and generally positive (9 of 13 scores). No discernible effects were seen for 4 of the 13 subscale scores (two in
the geometry domain: transformations/turns and comparing shapes; one in the operations domain: arithmetic; and one in the basic
number concepts domain: composition of number).
11
In Clements et al. (2011), the difference between the intervention and comparison groups included any aspect of instruction that
differed between Building Blocks and the various branded curricula used in the comparison classrooms, including DLM Early Childhood Express, a comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a math curriculum
that included instruction in number and operations guided by a developmental progression. The comparison group participated in a
( 74 )
Appendix D (continued)
number of branded curricula, including DLM Early Childhood Express, an early childhood curriculum that included instruction in number
and operations but was not guided by a developmental progression in the same manner as Building Blocks instruction.
12
Griffin, Case, and Capodilupo (1995) and related publication Griffin, Case, and Siegler (1994) reported other outcomes for
which no pretest data were provided. The WWC was unable to conduct a review that included these outcomes, as baseline equivalence
could not be established.
13
In Klein et al. (2008), the difference between the intervention and comparison groups included any aspect of instruction that differed between the combined Pre-K Mathematics curriculum and DLM Early Childhood Express intervention and the curricula used in
the comparison classrooms, including the branded comprehensive early childhood curriculum Creative Curriculum. The intervention
group, which participated in a combination of Pre-K Mathematics curriculum and DLM Early Childhood Express, included instruction
in number and operations using a developmental progression. The comparison group participated in a number of branded curricula,
including Creative Curriculum, a comprehensive early childhood curriculum that included instruction in number and operations guided
by a developmental progression.
14
Sarama et al. (2008) reported subscale scores as well; however, only the means were provided, so the WWC was unable to calculate
effect sizes for the subscales.
Appendix D (continued)
Table D.3. Studies of comprehensive curricula with an explicit math component that
taught number and operations and contributed to the level of evidence rating
Population
Characteristics2
Aunio,
Hautamaki, and
Van Luit (2005)5,6
RCT
Meets evidence
standards without
reservations
Comparison3
Let's Think! combined with Maths!
vs. regular classroom instruction
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
ENTRelational Scale,
Posttest
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
RCT
Meets evidence
standards with
reservations
Fantuzzo,
Gadsden, and
McDermott (2011)8
RCT
Meets evidence
standards without
reservations
Operations: WJ-Revised
Applied Math Problems
Subtest
Evidence-based
Program for
Integrated
Curricula (EPIC)
vs. regular classroom instruction
(DLM Early Childhood Express)
General numeracy:
LEMathematics, Wave 4
X7
X7
X9
X9
Positive (0.18*)
( 76 )
(continued)
Appendix D (continued)
Table D.3. Studies of comprehensive curricula with an explicit math component that
taught number and operations and contributed to the level of evidence rating (continued)
Recommendation
Components Tested
Population
Characteristics2
PCER Consortium
(2008, Chapter 2)5,10
Prekindergarten teachers
working in public programs
were randomly assigned the
year before the study began.
RCT
Meets evidence
standards with
reservations
Comparison3
Creative Curriculum vs. regular
classroom instruction (teacherdeveloped nonspecific curricula)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Study Characteristics
Operations: WJ-IIIApplied
Problems, Posttest
PCER Consortium
(2008, Chapter 2)5,10
RCT
Meets evidence
standards with
reservations
Prekindergarten teachers
working in public programs
were randomly assigned the
year before the study began.
Children: 198 total (98 intervention; 100 comparison)
Bright Beginnings
vs. regular classroom instruction
(teacher-developed nonspecific
curricula)
Operations: WJ-IIIApplied
Problems, Posttest
Preschoolers attending
Head Start centers
RCT
Meets evidence
standards with
reservations
Operations: WJ-IIIApplied
Problems, Posttest
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the
comparison group may have participated in instruction that taught number and operations and that may have used a developmental
progression to guide that instruction.
X The intervention included this component.
ENT = Early Numeracy Test219
SRT = Spatial Relationships Test220
( 77 )
Appendix D (continued)
CMA-A = Child Math AssessmentAbbreviated221
WJ-III = Woodcock-Johnson, third edition222
LE = Learning Express223
1
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random
procedure.
2
SD = Standard deviation. The information presented includes the following: (a) the type of program and unit of assignment, if the
study is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) the age of children
in the sample.
3
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group.
If details were available on the curriculum or curricula the comparison teachers used, it is noted parenthetically.
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic may
not have been math. If details were available on what was provided, it is noted parenthetically.
4
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ from
author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes that were
significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked with an asterisk (*);
ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed here. Positive findings favor
the intervention group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger). Negative findings
favor the comparison group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger).
No discernible refers to findings that are neither significant nor substantively important.
5
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook,
Version 2.1 (http://whatworks.ed.gov).
6
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
7
In Barnett et al. (2008), the difference between the intervention and comparison groups with respect to math instruction is not
known. The intervention group participated in Tools of the Mind, a comprehensive early childhood curriculum with a math component
that supports incorporating math into other parts of the school day. The comparison group participated in a district-created balanced
literacy curriculum. From the information provided, it is not clear how the intervention and comparison groups differed with respect to
instruction in number and operations or the use of a developmental progression to guide instruction in number and operations.
8
Fantuzzo, Gadsden, and McDermott (2011) reported on four waves of data collection. The panel decided to use Wave 1 as pretest
data, because it was collected prior to the delivery of math content. Wave 4 was used as the posttest, as it was collected at the end
of the school year and delivery of the intervention. Waves 2 and 3 could be viewed as intermediary outcomes, but the panel chose to
focus on posttests when determining levels of evidence.
9
In Fantuzzo, Gadsden, and McDermott (2011), the difference between the intervention and comparison groups included any
aspect of instruction that differed between EPIC and DLM Early Childhood Express, a branded comprehensive early childhood curriculum. The intervention group participated in EPIC, a comprehensive early childhood curriculum that included instruction in number and
operations guided by a developmental progression. The comparison group participated in another branded comprehensive early childhood curriculum, DLM Early Childhood Express, which included number and operations content but was not guided by a developmental
progression in the same manner as instruction using EPIC.
10
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel rated the study differently but reports the same findings as presented in the intervention report. The difference in study
rating is due to the use of WWC Version 2.1 standards as opposed to WWC Version 1.0 standards. Findings from this study of Bright
Beginnings were previously reported in the WWC intervention report on Bright Beginnings. The panel reports the same findings as
reported in the intervention report. For both Creative Curriculum and Bright Beginnings, the authors report on additional outcomes that
were assessed in the spring of kindergarten.
11
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel reports the same findings as presented in the intervention report.
Appendix D (continued)
such as number recognition, oral counting,
number sequencing, verbal subitizing, and
counting on fingers. Results indicated that the
supplemental curriculum produced a positive
effect on childrens general numeracy and
operations skills, whether the comparison was
with regular classroom instruction or with a
treated comparison group that participated in
a supplemental language intervention. Some of
these effects were maintained at a follow-up.227
Appendix D (continued)
and classification domains. Children in the
numeracy instruction group scored, on average, lower on outcomes in the basic number
concepts, operations, and patterns and classification domains than their classmates who
participated in the cognitive instruction.
( 80 )
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating
Population
Characteristics2
Baroody, Eiland,
and Thompson
(2009)6
RCT
Meets evidence
standards without
reservations
Comparison3
Semi-structured
discovery learning
vs. treated comparison (haphazard
practice)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Operations: Percentage of
children scoring at least
85% accurate (E-3 Scale)
for n+0/0+n items
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
X7
X7
RCT
Meets evidence
standards without
reservations
Operations: Percentage of
children scoring at least
85% accurate (E-3 Scale)
for n+0/0+n items
X8
X8
RCT
Meets evidence
standards without
reservations
Operations: Percentage of
children scoring at least
85% accurate (E-3 Scale)
for n+0/0+n items
X9
X9
Positive (1.20*)
( 81 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
Preschoolers attending a
university-based program
in California
RCT
Meets evidence
standards without
reservations
Comparison3
Adult support
(adults counted
weights or pegs
and repeated final
number to indicate
cardinal value of
set) vs. no adult
support
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
Balance Beam Scores
Large Difference in
Weights
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
X12
X12
X12
X12
Adult support
(adults counted
weights or pegs
and repeated final
number to indicate
cardinal value of
set) vs. no adult
support
X12
X12
X12
X12
( 82 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
RCT
Meets evidence
standards without
reservations
Comparison3
Supplemental researcher-developed
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers)
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy: NSB
Total Score, Posttest
X15
X15
X15
X15
X15
X15
X15
X15
X17
X17
X17
X17
X17
X17
X17
X17
X18
X18
X18
X18
X18
X18
X18
X18
Positive (0.64*)
Operations: WJ-IIITotal
Score, Posttest
Positive (0.29, ns)
General numeracy: NSB
Total Score, Maintenance
(6 weeks)
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
Positive (0.65*)
Operations: WJ-IIITotal
Score, Maintenance
(6 weeks)
No discernible (0.18, ns)
Jordan et al.
(2012)10,16
RCT
Meets evidence
standards without
reservations
Supplemental researcher-developed
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers or
Math Connects)
Jordan et al.
(2012)10,16
RCT
Meets evidence
standards without
reservations
Supplemental researcher-developed
number sense
curriculum vs.
treated comparison
(supplemental language intervention
with Math Trailblazers or Math
Connects)
( 83 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
Comparison3
Numeracy vs.
treated comparison
(art)
RCT
Meets evidence
standards without
reservations
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
Conservation Test
X19
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
X19
X19
X19
X19
10
RCT
Meets evidence
standards without
reservations
Numeracy vs.
treated comparison
(cognitive
instruction in
oddity principle,
inserting objects
into series, and
conservation)
X20
Negative (0.68*)
Operations: WJ-III
Applied Problems
X20
X20
Negative (0.68*)
Patterns and classification: OLSAT
Classification Scale
X20
X20
Operations: Gain in
Performance on Inversion
Trials
X21
Positive (0.54*)
( 84 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
Comparison3
Monahan (2007)8,22
Children attending
Head Start centers in
Philadelphia, Pennsylvania
Number-based
board games vs.
treated comparison
(color-based board
games)
RCT
Meets evidence
standards without
reservations
Use a
Developmental
Progression
Findings (Domain:
Assessment (Effect Size,
Significance))4
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
X23
Monahan (2007)8,22
RCT
Meets evidence
standards without
reservations
Children attending
Head Start centers in
Philadelphia, Pennsylvania
Children: 81 total (42 intervention; 39 comparison)
X24
Monahan (2007)8,22
RCT
Meets evidence
standards without
reservations
Children attending
Head Start centers in
Philadelphia, Pennsylvania
Children: 78 total (39 intervention; 39 comparison)
X25
Preschoolers attending
Head Start programs
RCT
Meets evidence
standards without
reservations
X27
Positive (0.74*)
Basic number concepts:
Numerical Magnitude
Comparison, Posttest
X27
Positive (0.99*)
Number recognition:
Number Identification,
Posttest
X27
Positive (0.69*)
Basic number concepts:
Counting, Maintenance
(9 weeks)
X27
Positive (0.66*)
Basic number concepts:
Numerical Magnitude
Comparison, Maintenance
(9 weeks)
X27
Positive (0.77*)
Number recognition:
Number Identification,
Maintenance
(9 weeks)
X27
Positive (0.80*)
( 85 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
Preschool-aged children
attending Head Start or one
of three childcare centers
RCT
Meets evidence
standards without
reservations
Sood (2009)5,10
RCT
Meets evidence
standards with
reservations
Comparison3
Linear, numberbased board
games vs. treated
comparison (colorbased board
games)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
Number Line Estimation
Percent Absolute Error
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
X28
Positive (0.86*)25
Basic number concepts:
Percent of Correctly
Ordered Numbers
X28
Positive (1.17*)
Researcherdeveloped number
sense curriculum
vs. regular classroom instruction
(district-mandated
curriculum)
Positive (1.23*)
Basic number concepts:
Five and Ten Frame Identification and Representation, Posttest
( 86 )
(continued)
Appendix D (continued)
Table D.4. Studies of targeted interventions that taught number and operations and
contributed to the level of evidence rating (continued)
Population
Characteristics2
Sood (2009)5,10
Kindergarten classrooms
in an urban elementary
school in Pennsylvania
RCT
Meets evidence
standards with
reservations
(continued)
Comparison3
Researcherdeveloped number
sense curriculum
vs. regular classroom instruction
(district-mandated
curriculum)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
Counting From,
Maintenance (3 weeks)
Use a
Developmental
Progression
Citation, Design,
and WWC Rating1
Teach Number
and
Operations
Skills
Recommendation
Components Tested
Study Characteristics
Positive (1.09*)
Basic number concepts:
Number Relationships,
Maintenance (3 weeks)
Positive (1.19*)
Operations: Five and Ten
Frame Calculations,
Maintenance (3 weeks)
QED
Meets evidence
standards with
reservations
Researcherdeveloped measurement-focused
curriculum vs.
treated comparison (literacy
instruction)
X29
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the
comparison group may have participated in instruction that taught number and operations and that may have used a developmental
progression to guide that instruction.
X The intervention included this component.
ENCO = Emergent Numeracy and Cultural Orientations Assessment240
NSB = Number Sense Brief241
OLSAT = Otis-Lennon School Ability Test242
WJ-III = Woodcock-Johnson, third edition243
DSC = Developing Skills Checklist244
( 87 )
Appendix D (continued)
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random
procedure.
2
SD = Standard deviation. The information presented includes the following: (a) the type of program and unit of assignment, if the study
is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) the age of children in the sample.
3
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group.
If details were available on the curriculum or curricula the comparison teachers used, it is noted parenthetically.
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic may
not have been math. If details were available on what was provided, it is noted parenthetically.
4
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ from
author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes that were
significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked with an asterisk (*);
ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed here. Positive findings favor
the intervention group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger). Negative findings
favor the comparison group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger).
No discernible refers to findings that are neither significant nor substantively important.
5
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook,
Version 2.1 (http://whatworks.ed.gov).
6
Baroody, Eiland, and Thompson (2009) reported six different scoring methods as well as the TEMA. The panel selected the E-3
scale, which excluded answers the child determined by counting (verbal, finger, or object counting) and excluded response biases (i.e.,
nonselective application of a strategy that was used on more than half the items and that did not make sense for at least one of the
items). The TEMA was not reported for a comparison of interest to the panel and thus was not included in the review.
7
In Baroody, Eiland, and Thompson (2009), there were three intervention groups that could be compared with a single comparison
group. In this contrast, the difference between the intervention and comparison groups was in the manner of presentation of the same number and operations material (number-after, n+0/0+n facts, n+1/1+n items, and other combinations). Both the intervention and comparison
groups participated in a core manipulativebased and game-based curriculum that developed the prerequisites for mental addition. During
the second phase, all groups used a computer-supported curriculum to promote mastery of addition and estimation skills, although the
nature of the curriculum differed. The intervention group participated in a semi-structured, computer-supported discovery-learning condition. Children practiced number-after, n+0/0+n facts, n+1/1+n items, and other combinations in four blocks of five items. The comparison
group had haphazard practice of the same four types of items (number-after, n+0/0+n facts, n+1/1+n items, and other combinations).
8
In Baroody, Eiland, and Thompson (2009), there were three intervention groups that could be compared with a single comparison group.
In this contrast, the difference between the intervention and comparison groups was in the manner of presentation of the same number and
operations material (number-after, n+0/0+n facts, n+1/1+n items, and other combinations). Both the intervention and comparison groups participated in a core manipulativebased and game-based curriculum that developed the prerequisites for mental addition. During the second phase,
all groups used a computer-supported curriculum to promote mastery of addition and estimation skills, although the nature of the curriculum
differed. The intervention group practiced in an implicitly structured discovery-learning manner. Children practiced three items consecutively
to highlight relations between number-after, related n+1/1+n combinations, and related n+0/0+n facts. The comparison group had haphazard
practice of the same four types of items (number-after, n+0/0+n facts, n+1/1+n items, and other combinations).
9
In Baroody, Eiland, and Thompson (2009), there were three intervention groups that could be compared with a single comparison
group. In this contrast, the difference between the intervention and comparison groups was in the manner of presentation of the same
number and operations material (number-after, n+0/0+n facts, n+1/1+n items, and other combinations). Both the intervention and comparison groups participated in a core manipulativebased and game-based curriculum that developed the prerequisites for mental addition. During the second phase, all groups used a computer-supported curriculum to promote mastery of addition and estimation skills,
although the nature of the curriculum differed. The intervention group practiced in an explicitly structured discovery-learning manner.
Adults provided explicit instruction (i.e., When we add one, its just the number after the other number), while children practiced three
items consecutively to highlight relations between number-after, related n+1/1+n combinations, and related n+0/0+n facts. The comparison group had haphazard practice of the same four types of items (number-after, n+0/0+n facts, n+1/1+n items, and other combinations).
10
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
11
For Experiment 1 in Curtis, Okamoto, and Weckbacher (2009), the panel decided to use the counting outcome as the pretest
for the post-hoc difference-in-difference adjustments. There was no pretest for the specific outcomes, but the counting measure was
deemed an acceptable substitute by the panel.
12
In both experiments in Curtis, Okamoto, and Weckbacher (2009), the difference between the intervention and comparison groups
in teaching number and operations was whether the children received adult support in counting items. Children in the intervention group
completed math tasks with an adult pointing to and counting aloud the number of items with repetition of the final number to reinforce the
cardinality of the set. The comparison group did not receive assistance from the adult in counting or determining cardinality of the set.
13
Experiment 2 in Curtis, Okamoto, and Weckbacher (2009) did not report pretest data for the outcomes. The panel decided to use
the quantity estimation pretest in the post-hoc difference-in-difference adjustments.
14
Dyson, Jordan, and Glutting (2013) reported total and subscale scores for the NSB, as well as the WJ-IIIApplied Problems and
WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation
Problems subscales. Positive effects were found for all subscales at posttest and maintenance, except for the WJ-IIIApplied Problems
subscale, for which no discernible effects were seen at posttest or maintenance.
15
In Dyson, Jordan, and Glutting (2013), the difference between the intervention and comparison groups was the additional 12 hours
of math instruction the intervention group received. The intervention group participated in 30-minute sessions, generally 3 a week, for
a total of 24 sessions (or 12 hours). The sessions included instruction in number and operations that was based on a developmental progression. The comparison group did not receive this additional instruction; rather, they received only regular classroom math instruction.
1
( 88 )
Appendix D (continued)
The regular classroom math instruction, for both the intervention and comparison children, was Math Trailblazers, a branded math curriculum used to teach number and operations but not guided by a developmental progression.
16
Jordan et al. (2012) reported posttest and maintenance effects for total and subscale scores for the NSB, as well as the WJ-IIIApplied
Problems and WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation Problems subscales. Positive effects were found for all but seven of the NSB outcomes, which were reported as no discernible effects.
17
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The sessions included instruction in number and operations that
was based on a developmental progression. The comparison group did not receive this additional instruction in math; rather, they
received only regular classroom instruction. The regular classroom instruction, for both the intervention and comparison children, was
Math Trailblazers or Math Connects. Both of these are commercially available curricula that teach number and operations; the panel
determined that Math Trailblazers does not use a developmental progression to guide instruction.
18
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute
sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The sessions included instruction in number and operations that was
based on a developmental progression. The comparison group did not receive this additional instruction in math; rather, they only
received regular classroom instruction and additional literacy instruction. The regular classroom instruction, for both the intervention
and comparison children, was Math Trailblazers or Math Connects. Both of these are commercially available curricula that teach number
and operations; the panel determined that Math Trailblazers does not use a developmental progression to guide instruction.
19
There were two comparisons in Kidd et al. (2008). In this comparison, the difference between the intervention and comparison
groups was the nature of the supplemental instruction each group received. Both groups received weekly 10- to 15-minute sessions of
supplemental small-group instruction during circle time. The intervention group received supplemental instruction in numeracy: games
they played with adults taught them to recognize numbers and count. The children first learned the numbers 110 and then focused on
numbers 1030. The comparison group participated in supplemental art activities during their sessions.
20
There were two comparisons in Kidd et al. (2008). In this comparison, the difference between the intervention and comparison
groups was the nature of the supplemental instruction each group received. Both groups received weekly 10- to 15-minute sessions of
supplemental small-group instruction during circle time. The intervention group received supplemental instruction in numeracy: games
they played with adults taught them to recognize numbers and count. The children first learned the numbers 110 and then focused on
numbers 1030. The comparison condition participated in supplemental cognitive instruction: they played games to learn the oddity,
seriation, and conservation.
21
In Lai, Baroody, and Johnson (2008), the difference between the intervention and comparison groups was the type of instruction
in number and operations each group received. The intervention group participated in training sessions based on an adaptation of Gelmans magic task. Children practiced tasks that were addition only, subtraction only, and a mixture of addition and subtraction, over
the course of the two-week training phase. The tasks helped to concretely teach reversibility (undoing operations). The comparison
group played decomposition/composition games to help them estimate a large collection of 5 to 11 items.
22
For Recommendation 1, the panel was not interested in comparisons between the three intervention conditions, although those findings are of interest in other recommendations.
23
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups in teaching number and operations was whether they were taught number sense in a pull-out activity. The intervention group
participated in a pull-out number sense curriculum using activities adapted from Big Math for Little Kids. The comparison group participated
in pull-out sessions of the same length in which they read stories about shapes and patterns, rather than the number sense curriculum.
24
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups in teaching number and operations was whether they were taught number sense in a pull-out activity. The intervention group
participated in pull-out activities that involved reading stories to teach a number sense curriculum. The comparison group participated in
pull-out sessions of the same length in which they read stories about shapes and patterns, rather than the number sense curriculum.
25
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups in teaching number and operations was whether they were taught number sense in a pull-out activity. The intervention group
participated in pull-out activities that involved movement to teach a number sense curriculum. The comparison group participated in
pull-out sessions of the same length in which they read stories about shapes and patterns, rather than the number sense curriculum.
26
Findings from these studies were previously reported in the WWC practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. The panel reports the same findings as discussed in that practice guide.
27
The effect is in the desired direction with the intervention making fewer errors than the comparison group, resulting in a negative
effect size. However, to present the findings in a consistent manner, the effect size is reported as positive.
28
In both Ramani and Siegler (2008) and Siegler and Ramani (2008), the difference between the intervention and comparison
groups in teaching number and operations was the nature of the board games played. The intervention group played a number-based
version of The Great Race, with each space on the board having a number and children stating the number as they moved their token
thus practicing number and operations. The comparison group played The Great Race but with spaces that were colored, rather than
numbered, and children stating the colors as they moved their token.
29
In Sophian (2004), the difference between the intervention and comparison groups was whether the children received math instruction using a researcher-developed, measurement-focused curriculum. The intervention group participated in a researcher-developed,
measurement-focused curriculum that emphasized the concept of unit and taught number and operations. The comparison group participated in a literacy curriculum. There is no description of the math instruction children in the comparison group may have received
as part of their regular classroom instruction.
( 89 )
Appendix D (continued)
Despite the presence of largely positive effects,
when assigning the level of evidence for this
recommendation, the panel identified three
concerns regarding how well the evidence supported this recommendation: (1) interventions
were multi-component, leading to concerns
about how much of the demonstrated effect
was the result of targeted teaching of geometry,
patterns, measurement, and data analysis;
(2) the degree to which the intervention and
comparison groups received different amounts
of targeted instruction in geometry, patterns,
measurement, and data analysis could not be
determined in all studies; and (3) many studies
only reported on outcomes that were not
aligned with the early math content areas
included in this recommendation. As such, it
was difficult for the panel to determine the
extent to which targeted instruction in geometry, patterns, measurement, and data analysis
according to a developmental progression
was responsible for the effects seen in math
achievement. Based on their expertise and the
effects of interventions that include targeted
instruction in geometry, patterns, measurement, and data analysis, the panel believes the
studies generally support this recommendation,
despite the limitations to the body of evidence.
Appendix D (continued)
The panel determined that, due to the multicomponent nature of many of the interventions comprising the body of evidence for
Recommendation 2, it was not possible to attribute the demonstrated effects to the teaching
of geometry, patterns, measurement, and data
analysis.254 In addition to teaching number and
operations (Recommendation 1), many of the
interventions examined also included aspects
of Recommendations 3, 4, and 5. For example,
Building Blocks, EPIC, and the Pre-K Mathematics curriculum include progress monitoring
(the focus of Recommendation 3) as a core
component of the intervention and involve the
targeted teaching of number and operations
(Recommendation 1), in addition to geometry,
patterns, measurement, and data analysis. The
panel cautions that the effects in these studies
of comprehensive curricula may not be replicated when only the elements relating to Recommendation 2 are implemented. In fact, the
panel believes that all the recommendations
in this guide should be implemented together,
as was the case in many of the interventions
demonstrating positive effects.
When reviewing the evidence for Recommendation 2, the panel considered the degree
to which the groups being compared differed (i.e., strength of the contrast) related
to targeted instruction in geometry, patterns, measurement, and data analysis. The
panel identified three studies in which the
intervention group received targeted instruction, while the comparison group did not.255
Although the specific nature of comparison
group instruction was not clear in 6 of the 13
studies, the panel determined that the comparison group may have received targeted
instruction in the same early math content
areas.256 This group of six studies found both
positive257 and no discernible258 effects in the
domains of operations, geometry, and general
numeracy. Both the intervention and comparison groups received targeted instruction in
the specific early math content areas in 4 of
the 12 studies;259 positive effects were found
in these studies in the outcome domains of
geometry,260 general numeracy,261 and basic
number concepts.262
Appendix D (continued)
teachers to use to teach geometry with no
clear tie to a developmental progression.267
Appendix D (continued)
using the DLM Early Childhood Express curriculum.281 Building Blocks or DLM Early Childhood
Express, either with or without the Pre-K Mathematics curriculum, was the tested intervention in
five of the studies.282 Seven of the interventions
discussed in these studies283 were also examined
as evidence for the effects of teaching geometry
to children, as previously discussed.
( 93 )
Appendix D (continued)
Table D.5. Studies of interventions that taught geometry, patterns, measurement,
or data analysis and contributed to the level of evidence rating
RCT
Meets evidence
standards with
reservations
Casey et al.
(2008)6,7
RCT
Meets evidence
standards without
reservations
Geometry: Building
Block Test
X5
X5
Use a Developmental
Progression
Barnett et al.
(2008)
Comparison3
Findings (Domain:
Assessment (Effect
Size, Significance))4
Teach Measurement
Population
Characteristics2
Teach Patterns
Citation, Design,
and WWC Rating1
Teach Geometry
Recommendation
Components Tested
Study Characteristics
X5
X8
X8
X8
X8
X8
X8
X9
X8
X9
X8
X9
X8
Casey et al.
(2008)6,7
RCT
Meets evidence
standards without
reservations
Geometry: Building
Block Test
No discernible (0.06, ns)
Geometry: WISC-IV
Block Design Subtest
Positive (0.35, ns)
Geometry: Mental
Rotation
No discernible (0.16, ns)
Clements and
Sarama (2007b)6,10
RCT
Meets evidence
standards with
reservations
Preschool classrooms in
state-funded or Head Start
programs
Children: 68 total (30 intervention; 38 comparison)
Age range: 2.9 to 4.8 years
Building Blocks
vs. regular classroom instruction
(Creative Curriculum or locally
developed)
X11
X11
X11
X11
X11
X11
X11
X11
X11
X11
Positive (0.75*)
Geometry: BB AssessmentGeometry Scale
Positive (1.40*)
( 94 )
(continued)
Appendix D (continued)
Table D.5. Studies of interventions that taught geometry, patterns, measurement,
or data analysis and contributed to the level of evidence rating (continued)
RCT
Meets evidence
standards without
reservations
General numeracy:
REMA
General numeracy:
REMATotal
Evidence-based
Program for
Integrated
Curricula (EPIC)
vs. regular classroom instruction
(DLM Early Childhood Express)
General numeracy:
LEMathematics, Wave 4
Use a Developmental
Progression
Clements and
Sarama (2008)6,12
Comparison3
Findings (Domain:
Assessment (Effect
Size, Significance))4
Teach Measurement
Population
Characteristics2
Teach Patterns
Citation, Design,
and WWC Rating1
Teach Geometry
Recommendation
Components Tested
Study Characteristics
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X16
X16
X16
Positive (1.07*)
Meets evidence
standards without
reservations
Fantuzzo,
Gadsden, and
McDermott (2011)15
RCT
Meets evidence
standards without
reservations
Positive (0.48*)
Basic number concepts:
REMANumbers Total
Positive (0.39*)
Geometry: REMA
Geometry Total
Positive (0.64*)
X16
Positive (0.18*)
( 95 )
(continued)
Appendix D (continued)
Table D.5. Studies of interventions that taught geometry, patterns, measurement,
or data analysis and contributed to the level of evidence rating (continued)
RCT
Meets evidence
standards without
reservations
Use a Developmental
Progression
Comparison3
Findings (Domain:
Assessment (Effect
Size, Significance))4
Teach Measurement
Population
Characteristics2
Teach Patterns
Citation, Design,
and WWC Rating1
Teach Geometry
Recommendation
Components Tested
Study Characteristics
X19
X19
X19
X17
X17
Positive (0.71*)
Patterns and classification: Oddity Test
X17
Positive (0.87*)
Patterns and classification: OLSAT Classification Scale
X17
X17
Positive (0.62*)
Kidd et al. (2008)7
RCT
Meets evidence
standards without
reservations
X18
Positive (0.68*)
Operations: WJ-III
Applied Problems
X18
X18
Positive (0.68*)
Patterns and classification: OLSAT Classification Scale
X18
X18
Pre-K Mathematics
combined with
DLM Early Childhood Express vs.
regular classroom
instruction (Creative Curriculum,
High Scope,
Montessori,
locally developed)
( 96 )
General numeracy:
CMA
X19
X19
Positive (0.57*)
(continued)
Appendix D (continued)
Table D.5. Studies of interventions that taught geometry, patterns, measurement,
or data analysis and contributed to the level of evidence rating (continued)
RCT
Meets evidence
standards with
reservations
Operations: WJ-III
Applied Problems,
Posttest
Use a Developmental
Progression
Prekindergarten teachers
working in public programs
the year before the study
began
PCER Consortium
(2008, Chapter 2)6,20
Comparison3
Findings (Domain:
Assessment (Effect
Size, Significance))4
Teach Measurement
Population
Characteristics2
Teach Patterns
Citation, Design,
and WWC Rating1
Teach Geometry
Recommendation
Components Tested
Study Characteristics
Geometry: Shape
Composition, Posttest
No discernible (0.12, ns)
PCER Consortium
(2008, Chapter 2)6,20
RCT
Meets evidence
standards with
reservations
Prekindergarten teachers
working in public programs
the year before the study
began
Children: 198 total
(98 intervention;
100 comparison)
Bright Beginnings
vs. regular classroom instruction
(teacher-developed nonspecific
curricula)
Operations: WJ-III
Applied Problems,
Posttest
No discernible (0.16, ns)
General numeracy:
CMA-A, Posttest
No discernible (0.14, ns)
Geometry: Shape
Composition, Posttest
No discernible (0.03, ns)
PCER Consortium
(2008, Chapter 3)21
Preschoolers attending
Head Start centers
RCT
Meets evidence
standards with
reservations
Operations: WJ-III
Applied Problems,
Posttest
No discernible (0.20, ns)
General numeracy:
CMA-AMathematics
Composite, Posttest
No discernible (0.10, ns)
Geometry: Shape
Composition, Posttest
No discernible (0.19, ns)
Operations: WJ-III
Applied Problems,
Maintenance (spring
of kindergarten year)
No discernible (0.09, ns)
( 97 )
(continued)
Appendix D (continued)
Table D.5. Studies of interventions that taught geometry, patterns, measurement,
or data analysis and contributed to the level of evidence rating (continued)
RCT
Meets evidence
standards with
reservations
(continued)
General numeracy:
CMA-AMathematics
Composite, Maintenance
(spring of kindergarten
year)
Use a Developmental
Progression
Preschoolers attending
Head Start centers
PCER Consortium
(2008, Chapter 3)21
Comparison3
Findings (Domain:
Assessment (Effect
Size, Significance))4
Teach Measurement
Population
Characteristics2
Teach Patterns
Citation, Design,
and WWC Rating1
Teach Geometry
Recommendation
Components Tested
Study Characteristics
RCT
Meets evidence
standards without
reservations
Sarama et al.
(2008)22
Building Blocks
combined with
Pre-K Mathematics
vs. regular classroom instruction
General numeracy:
REMA
Researcherdeveloped
measurementfocused curriculum
vs. treated comparison (literacy
instruction)
General numeracy:
DSCMathematics
Subscale
Geometry: Geometry
Score
Positive (0.62*)
QED
Meets evidence
standards with
reservations
Weaver (1991)24
RCT
Meets evidence
standards without
reservations
X23
X23
X25
X25
X25
Positive (0.86*)
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the
comparison group may have participated in instruction that taught geometry, patterns, measurement and data analysis, and
that may have used a developmental progression to guide that instruction.
X The intervention included this component.
BB Assessment = Building Blocks Assessment of Early Mathematics297
CMA = Child Math Assessment298
CMA-A = Child Math AssessmentAbbreviated299
DSC = Developing Skills Checklist300
LE = Learning Express301
OLSAT = Otis-Lennon School Ability Test302
REMA = Research-Based Early Math Assessment303
WISC-IV = Wechsler Intelligence Scale for Children, fourth edition304
WJ-Revised = Woodcock-Johnson, revised edition305
WJ-III = Woodcock-Johnson, third edition306
( 98 )
Appendix D (continued)
1
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random
procedure.
2
SD = Standard deviation. The information presented includes the following: (a) the type of program and unit of assignment, if
the study is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) the age of
children in the sample.
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group.
If details were available on the curriculum the comparison teachers used, it is noted parenthetically.
3
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic
may not have been math. If details were available on what was provided, it is noted parenthetically.
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ
from author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes
that were significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked
with an asterisk (*); ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed
here. Positive findings favor the intervention group and are either significant or substantively important (i.e., the effect size is
0.25 SD or larger). Negative findings favor the comparison group and are either significant or substantively important (i.e., the
effect size is 0.25 SD or larger). No discernible refers to findings that are neither significant nor substantively important.
4
5
In Barnett et al. (2008), the difference between the intervention and comparison groups with respect to math instruction is
not known. The intervention group participated in Tools of the Mind, a comprehensive early childhood curriculum with a math
component that supported incorporating math into other parts of the school day. The comparison group participated in a district-created balanced literacy curriculum. From the information provided, it was not clear how the intervention and comparison
groups differed with respect to instruction in the early math content areas of geometry and patterns or the use of a developmental progression to guide instruction in these early math content areas.
6
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct
for clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards
Handbook, Version 2.1 (http://whatworks.ed.gov).
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
7
8
In Casey (2008), the difference between the intervention and comparison groups in teaching measurement was in the instructional use of building blocks. The intervention group engaged in sequenced block-building activities in a story-telling context.
The comparison group had the opportunity to engage in unstructured play with blocks. There is not sufficient information to
know what other activities were conducted in to teach geometry.
9
In Casey (2008), the difference between the intervention and comparison groups in teaching measurement was in the instructional use of building blocks. The intervention group engaged in sequenced block-building activities. The comparison group had
regular math instruction. There is not sufficient information to know what other activities were conducted in the comparison
group to teach geometry.
Clements and Sarama (2007b) also reported scores for the subscales of the Number and Geometry scales; positive effects
were seen for each subscale. Findings from Clements and Sarama (2007b) were previously reported in the WWC intervention
report on SRA Real Math Building Blocks PreK. The panel reports the same findings as presented in the intervention report.
10
In Clements and Sarama (2007b), the difference between the intervention and comparison groups encompassed any aspect
of instruction that differed between Building Blocks and the curricula used in the comparison classrooms, including the branded
comprehensive early childhood curriculum Creative Curriculum. The intervention group participated in Building Blocks, a math
curriculum that included instruction in geometry, patterns, measurement, and data analysis guided by a developmental progression. The comparison group participated in a variety of curricula, including Creative Curriculum, which also included instruction
in geometry, patterns, measurement, and data analysis guided by a developmental progression.
11
For Clements and Sarama (2008), the WWC is reporting author-reported effect sizes consistent with prior reporting of findings from this study in the WWC intervention report on SRA Real Math Building Blocks PreK.
12
Clements et al. (2011) also reported the subscale scores from the REMA. Findings for the subscale scores were consistent
with the total score findings and generally positive (9 of 13 scores). No discernible effects were seen for 4 of the 13 subscale
scores (two in the geometry domain: transformations/turns and comparing shapes; one in the operations domain: arithmetic;
and one in the basic number concepts domain: composition of number).
13
In Clements et al. (2011), the difference between the intervention and comparison groups encompassed any aspect of instruction that differed between Building Blocks and the various branded curricula used in the comparison classrooms, including DLM
Early Childhood Express, a comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a
math curriculum that included instruction in geometry, patterns, measurement, and data analysis guided by a developmental
progression. The comparison group participated in a number of branded curricula, including DLM Early Childhood Express, an
early childhood curriculum that included instruction in geometry, patterns, measurement, and data analysis but was not guided
by a developmental progression in the same manner as Building Blocks instruction.
14
15
Fantuzzo, Gadsden, and McDermott (2011) reported on four waves of data collection. The panel decided to use Wave 1 as
pretest data, because it was collected prior to the delivery of math content. Wave 4 was used as the posttest, as it was collected
( 99 )
Appendix D (continued)
at the end of the school year and delivery of the intervention. Waves 2 and 3 could be viewed as intermediary outcomes, but the
panel chose to focus on posttests when determining levels of evidence.
In Fantuzzo, Gadsden, and McDermott (2011), the difference between the intervention and comparison groups included
any aspect of instruction that differed between EPIC and DLM Early Childhood Express, a branded comprehensive early childhood
curriculum. The intervention group participated in EPIC, a comprehensive early childhood curriculum that included instruction
in geometry, measurement, and data analysis guided by a developmental progression. The comparison group participated in
another branded comprehensive early childhood curriculum, DLM Early Childhood Express, which included math content in the
early math content areas of geometry, patterns, measurement, and data analysis but was not guided by a developmental progression in the same manner as instruction using EPIC.
16
17
There were two comparisons in Kidd et al. (2008). In this comparison, the difference between the intervention and comparison groups was the nature of the supplemental instruction each group received. Both groups received weekly 10- to 15-minute
sessions of supplemental small-group instruction during circle time. The intervention group received supplemental cognitive
instruction in oddity, seriation, and conservation: games they played with adults enabled them to practice these concepts. The
comparison group participated in supplemental art activities during their sessions.
There were two comparisons in Kidd et al. (2008). In this comparison, the difference between the intervention and comparison groups was the nature of the supplemental instruction each group received. Both groups received weekly 10- to 15-minute
sessions of supplemental small-group instruction during circle time. The intervention group received supplemental cognitive
instruction in oddity, seriation, and conservation; games they played with adults enabled them to practice these concepts. The
comparison group participated in supplemental instruction in numeracythey played games with adults that taught them to recognize numbers and count. The children first learned the numbers 110 and then focused on numbers 1030.
18
In Klein et al. (2008), the difference between the intervention and comparison groups encompassed any aspect of instruction that differed between the combined Pre-K Mathematics curriculum and DLM Early Childhood Express intervention and the
curricula used in the comparison classrooms, including Creative Curriculum. The intervention group, which participated in a
combination of Pre-K Mathematics and DLM Early Childhood Express, received instruction in geometry, patterns, measurement,
and data analysis using a developmental progression. The comparison group participated in a number of branded curricula,
including Creative Curriculum, a comprehensive early childhood curriculum that included instruction in geometry, patterns,
measurement, and data analysis guided by a developmental progression.
19
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel rated the study differently but reports the same findings as presented in the intervention report. The difference in
study rating is due to the use of WWC Version 2.1 standards as opposed to WWC Version 1.0 standards. Findings from this study
of Bright Beginnings were previously reported in the WWC intervention report on Bright Beginnings. The panel reports the same
findings as presented in the intervention report. For both Creative Curriculum and Bright Beginnings, the authors report on additional outcomes that were assessed in the spring of kindergarten.
20
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel reports the same findings as presented in the intervention report.
21
22
Sarama et al. (2008) reported subscale scores as well; however, only the means were provided, so the WWC was unable to
calculate effect sizes for the subscales.
23
In Sophian (2004), the difference between the intervention and comparison groups was whether children received math instruction using a researcher-developed, measurement-focused curriculum. The intervention group participated in a researcherdeveloped, measurement-focused curriculum that emphasized the concept of unit and taught geometry and measurement. The
comparison group participated in a literacy curriculum. There was no description of the math instruction children in the comparison group may have received as part of their regular classroom instruction.
Weaver (1991) also assessed the impact of computer-based LOGO compared with floor-based LOGO for preschool children.
This contrast is not evidence for this recommendation, as both groups of children used LOGO; there were no discernible effects
found for computer-based LOGO.
24
In Weaver (1991), the difference between the intervention and comparison groups was the use of a LOGO turtle (floor- or
computer-based) to practice geometry concepts including paths, knowledge of right and left, and perspective-taking abilities.
The intervention group and a portion of the comparison group (the path portion) participated in three non-computer lessons
on the properties of paths. The other portion of the comparison group, the control group, did not receive any portion of the
path curriculum. The intervention group then worked with either a floor- or computer-based turtle to practice concepts related
to paths. LOGO is a programming language that results in a turtle drawing a path while following commands entered by the
child to indicate both a graphic command (right, left, forward, or backward) and a distance to move or the degrees to turn.
25
( 100 )
Appendix D (continued)
monitoring, the panel believes the studies generally support this recommendation despite
the limitations to the body of evidence.
Appendix D (continued)
monitoring and emphasis on starting with
a childs informal knowledge; however, the
comparison group was identified as participating in an intervention that included at
least one of these two components as well.319
Positive effects were reported in outcome
domains of general numeracy,320 basic number concepts,321 and geometry.322
The panel did not view the evidence as sufficient to warrant a moderate evidence rating
for two key reasons. First, the studies incorporated practices associated with Recommendation 3 and practices associated with the other
recommendations in the guide (i.e., they were
multi-component interventions).323 For example, all studies included targeted instruction in
number and operations (see Recommendation
1), and 8 of the 12 included targeted instruction in at least one of the early math content
areas addressed in Recommendation 2.324 As
such, it was difficult for the panel to determine
the extent to which the use of progress monitoring was responsible for the effects seen
in math achievement. Second, the difference
in the amount and type of progress monitoring the intervention and comparison groups
received was not distinct in most studies325 and
thus was not a direct test of a key component
of the recommendation. The panel believes
progress monitoring should be a deliberate
process that identifies a childs knowledge as a
starting point and regularly assesses progress
in developing connections between new math
knowledge and what the child already knows.
The panel further believes that when progress monitoring is implemented with other
recommendations in this guide, it will lead to
improved math achievement for children.
The fourth study332 of Building Blocks implemented an intervention in New York and
California Head Start programs and statefunded prekindergarten classrooms. The
intervention in this study combined the
software component of the curriculum and
correlated non-computer activities with twiceweekly small-group sessions from the Pre-K
Mathematics curriculum. The combined program was designed to follow research-based
learning trajectories specifying developmental
progressions of levels of competence. The
children receiving the intervention curriculum
performed significantly better, on average,
than their counterparts (who received the
regular classroom math curriculum) on the
REMA, a measure of general numeracy.
Appendix D (continued)
tested the effectiveness of a math intervention that combined elements from the Pre-K
Mathematics curriculum and DLM Early Childhood Express. The curriculum provided teachers with suggestions for scaffolding activities
and downward and upward extensions of
the activities. Assessment record sheets for
the small-group activities allowed teachers
to track the progress of individual children,
and time was built into the curriculum for
reviewing activities with children experiencing difficulty. Children in the comparison
group continued to receive regular classroom
instruction (the preschool curricula used in
their programs). The study found that, on
average, the intervention had a positive effect
on childrens general numeracy as measured
by the Child Math Assessment (CMA).334
The third study focused on the Evidencebased Program for Integrated Curricula (EPIC),
a comprehensive, stand-alone curriculum that
was not specific to math.339 Teachers used
curriculum-based assessments, called EPIC
Integrated Check-Ins (ICIs), three times yearly
to identify competencies of individual children, monitor progress, and inform instruction. Researchers randomly assigned 80 Head
Start classrooms to implement either EPIC or
Appendix D (continued)
DLM Early Childhood Express, another standalone curriculum. Teachers implementing the
comparison curriculum, DLM Early Childhood
Express, used the High/Scope Educational
Research Foundations Preschool Child Observation Record to conduct individual assessments of children and monitor their progress.
The researchers noted that teachers in both
the intervention and comparison conditions
conducted a comparable number of assessments. On an assessment of general numeracy, children taught using EPIC performed
better, on average, than students whose
teachers used DLM Early Childhood Express.
Population
Characteristics2
Arnold et al.
(2002)5
RCT
Meets evidence
standards without
reservations
Clements and
Sarama (2007b)5,6
RCT
Meets evidence
standards with
reservations
Comparison3
Math Is Everywhere vs.
regular classroom
instruction
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy: TEMA-2
Start with a
Childs
Informal Math
Knowledge
Citation, Design,
and WWC Rating1
Use Progress
Monitoring
to Tailor
Instruction
Recommendation
Components Tested
Study Characteristics
X7
X7
X7
X7
Positive (1.40*)
( 104 )
(continued)
Appendix D (continued)
Table D.6. Studies of interventions that used a deliberate progress-monitoring process
and contributed to the level of evidence rating (continued)
Population
Characteristics2
Clements and
Sarama (2008)5,8
RCT
Meets evidence
standards without
reservations
Comparison3
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy:
REMATotal
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers)
Start with a
Childs
Informal Math
Knowledge
Citation, Design,
and WWC Rating1
Use Progress
Monitoring
to Tailor
Instruction
Recommendation
Components Tested
Study Characteristics
X11
X11
X11
X11
X11
X11
Positive (1.07*)
RCT
Prekindergarten classrooms
in two urban public school
districts
Meets evidence
standards without
reservations
Clements et al.
(2011)5,9,10
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 121 total (56 intervention; 65 comparison)
Mean age: 5.5 years
(SD 4.0 months)
Positive (0.48*)
Basic number concepts:
REMANumbers Total
Positive (0.39*)
Geometry: REMAGeometry
Total
Positive (0.64*)
X13
Positive (0.64*)
Operations: WJ-IIITotal
Score, Posttest
X13
X13
Positive (0.65*)
Operations: WJ-IIITotal Score,
Maintenance (6 weeks)
X13
RCT
Meets evidence
standards without
reservations
Evidence-based
Program for
Integrated Curricula (EPIC) vs.
regular classroom
instruction (DLM
Early Childhood
Express)
( 105 )
General numeracy:
LEMathematics, Wave 4
X15
X15
Positive (0.18*)
(continued)
Appendix D (continued)
Table D.6. Studies of interventions that used a deliberate progress-monitoring process
and contributed to the level of evidence rating (continued)
Population
Characteristics2
Comparison3
Kindergarten students in
public schools in inner-city
areas in Massachusetts
Rightstart vs.
regular classroom
instruction
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers or
Math Connects)
Start with a
Childs
Informal Math
Knowledge
Findings (Domain:
Assessment (Effect Size,
Significance))4
Citation, Design,
and WWC Rating1
Use Progress
Monitoring
to Tailor
Instruction
Recommendation
Components Tested
Study Characteristics
Positive (1.79*)
QED
Meets evidence
standards with
reservations
Jordan et al.
(2012)17
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 86 total (42 intervention; 44 comparison)
Mean age: 5.5 years
(SD 4.38 months)
X18
Positive (1.10*)
Operations: WJ-IIITotal,
Posttest
X18
Positive (0.91*)
General numeracy: NSB
Total, Maintenance (8 weeks)
X18
Positive (0.77*)
Operations: WJ-IIITotal,
Maintenance (8 weeks)
X18
Positive (0.56*)
Jordan et al.
(2012)17
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools in
one district in the MidAtlantic region of the
United States
Children: 84 total (42 intervention; 42 comparison)
Mean age: 5.5 years
(SD 4.38 months)
Supplemental
researcherdeveloped
number sense
curriculum vs.
treated comparison (supplemental language
intervention with
Math Trailblazers
or Math Connects)
X19
Positive (0.91*)
Operations: WJ-III, Total,
Posttest
X19
Positive (0.84*)
General numeracy: NSB,
Total, Maintenance (8 weeks)
X19
Positive (0.62*)
Operations: WJ-III, Total,
Maintenance (8 weeks)
X19
Positive (0.75*)
Klein et al. (2008)
RCT
Meets evidence
standards without
reservations
Pre-K Mathematics
combined with
DLM Early Childhood Express vs.
regular classroom
instruction (Creative Curriculum,
High Scope, Montessori, locally
developed)
( 106 )
X20
X20
Positive (0.57*)
(continued)
Appendix D (continued)
Table D.6. Studies of interventions that used a deliberate progress-monitoring process
and contributed to the level of evidence rating (continued)
Population
Characteristics2
PCER Consortium
(2008, Chapter 2)5,21
Prekindergarten teachers
working in public programs
the year before the study
began
RCT
Meets evidence
standards with
reservations
Comparison3
Creative Curriculum vs. regular
classroom instruction (teacherdeveloped nonspecific curricula)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Operations: WJ-IIIApplied
Problems, Posttest
Start with a
Childs
Informal Math
Knowledge
Citation, Design,
and WWC Rating1
Use Progress
Monitoring
to Tailor
Instruction
Recommendation
Components Tested
Study Characteristics
Prekindergarten teachers
working in public programs
the year before the study
began
Children: 198 total (98 intervention; 100 comparison)
Bright Beginnings
vs. regular classroom instruction
(teacher-developed nonspecific
curricula)
Operations: WJ-IIIApplied
Problems, Posttest
Preschoolers attending
Head Start centers
RCT
Meets evidence
standards with
reservations
( 107 )
(continued)
Appendix D (continued)
Table D.6. Studies of interventions that used a deliberate progress-monitoring process
and contributed to the level of evidence rating (continued)
Population
Characteristics2
Sarama et al.
(2008)23
RCT
Meets evidence
standards without
reservations
Comparison3
Building Blocks
combined with
Pre-K Mathematics
vs. regular classroom instruction
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy: REMA
Start with a
Childs
Informal Math
Knowledge
Citation, Design,
and WWC Rating1
Use Progress
Monitoring
to Tailor
Instruction
Recommendation
Components Tested
Study Characteristics
Positive (0.62*)
( 108 )
Appendix D (continued)
10
Clements et al. (2011) also reported the subscale scores from the Early Mathematics Assessment. Findings for the subscale scores
are consistent with the total score findings and are generally positive (9 of 13 scores). No discernible effects are seen for 4 of the 13
subscale scores (transformations/turns, comparing shapes, arithmetic, and composition of number).
11
In Clements et al. (2011), the difference between the intervention and comparison groups encompassed any aspect of instruction
that differed between Building Blocks and the various branded curricula used in the comparison classrooms, including DLM Early Childhood Express, a comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a math curriculum
that incorporated regular progress monitoring and encouraged using childrens existing knowledge as a starting point. The comparison
group participated in a number of branded curricula, including DLM Early Childhood Express, which included progress monitoring but
did not appear to emphasize starting with a childs informal math knowledge to the same extent as Building Blocks.
12
Dyson, Jordan, and Glutting (2013) reported total and subscale scores for the NSB, as well as the WJ-IIIApplied Problems and
WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation
Problems subscales. Positive effects were found for all subscales at posttest and maintenance, except for the WJ-IIIApplied Problems
subscale, for which no discernible effects were seen at posttest or maintenance.
13
In Dyson, Jordan, and Glutting (2013), the difference between the intervention and comparison groups was the additional 12 hours
of math instruction the intervention group received. The intervention group participated in 30-minute sessions, generally 3 a week, for a
total of 24 sessions (or 12 hours). The sessions included using deliberate progress monitoring to tailor instruction. The comparison group
did not receive this additional instruction; rather, they received only the regular classroom math instruction. The regular classroom math
instruction, for both the intervention and comparison children, was Math Trailblazers, a branded math curriculum that uses deliberate
progress monitoring
14
Fantuzzo, Gadsden, and McDermott (2011) reported on four waves of data collection. The panel decided to use Wave 1 as pretest
data, because it was collected prior to the delivery of math content. Wave 4 was used as the posttest, as it was collected at the end
of the school year and delivery of the intervention. Waves 2 and 3 could be viewed as intermediary outcomes, but the panel chose to
focus on posttests when determining levels of evidence.
15
In Fantuzzo, Gadsden, and McDermott (2011), the difference between the intervention and comparison groups included any
aspect of instruction that differed between EPIC and DLM Early Childhood Express, a branded comprehensive early childhood curriculum.
Both curricula used progress monitoring and encouraged starting with a childs informal math knowledge.
16
Griffin, Case, and Capodilupo (1995) and related publication Griffin, Case, and Siegler (1994) reported other outcomes for
which no pretest data were provided. The WWC was unable to conduct a review that included these outcomes, as baseline equivalence
could not be established.
17
Jordan et al. (2012) reported posttest and maintenance effects for total and subscale scores for the NSB, as well as the WJ-III
Applied Problems and WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and
WJ-IIICalculation Problems subscales. Positive effects were found for all but seven of the NSB outcomes that were reported as no
discernible effects.
18
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute
sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The sessions included instruction that used deliberate progress
monitoring to tailor instruction. The comparison group did not receive this additional instruction in math; rather, they received only the
regular classroom instruction. The regular classroom instruction, for both the intervention and comparison children, was Math Trailblazers or Math Connects. Both of these are commercially available curricula. The panel confirmed that Math Trailblazers uses progress
monitoring but could not confirm whether Math Connects includes deliberate progress monitoring.
19
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute
sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The sessions included instruction that used deliberate progress
monitoring to tailor instruction. The comparison group did not receive this additional instruction in math; rather, they received only
the regular classroom instruction and additional literacy instruction. The regular classroom instruction, for both the intervention and
comparison children, was Math Trailblazers or Math Connects. Both of these are commercially available curricula. The panel confirmed
that Math Trailblazers uses progress monitoring but could not confirm whether Math Connects includes deliberate progress monitoring.
20
In Klein et al. (2008), the difference between the intervention and comparison groups encompassed any aspect of instruction that
differed between the combined Pre-K Mathematics and DLM Early Childhood Express intervention and the curricula used in the comparison classrooms, including Creative Curriculum. The intervention group, which participated in a combination of Pre-K Mathematics
and DLM Early Childhood Express, incorporated regular progress monitoring and emphasized using childrens existing knowledge as
a starting point for instruction. The comparison group participated in a number of branded curricula, including Creative Curriculum, a
comprehensive early childhood curriculum that included progress monitoring but did not appear to emphasize starting with a childs
informal knowledge in the same manner as the intervention group curricula.
21
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel rated the study differently but reports the same findings as presented in the intervention report. The difference in study
rating is due to the use of WWC Version 2.1 standards as opposed to WWC Version 1.0 standards. Findings from this study of Bright
Beginnings were previously reported in the WWC intervention report on Bright Beginnings. The panel reports the same findings as
reported in the intervention report. For both Creative Curriculum and Bright Beginnings, the authors report on additional outcomes
that were assessed in the spring of kindergarten.
22
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel reports the same findings as presented in the intervention report.
23
Sarama et al. (2008) reported subscale scores as well; however, only the means were provided, so the WWC was unable to calculate effect sizes for the subscales.
( 109 )
Appendix D (continued)
The interventions examined in each of the
16 studies included guidance for teachers
and/or activities that, if implemented, would
support children in learning how to view
and describe their world mathematically.
However, the intervention groups also participated in instructional activities that were
good examples of the practices addressed in
other recommendations in the practice guide.
For example, to teach math vocabulary and
encourage math conversation, teachers need
to teach the early math content areas that
are the focus of Recommendations 1 and 2.
Twelve of the 16 studies included key components of Recommendation 3.357 Fourteen of
the 16 studies also included key components
of Recommendation 5.358 Finding positive
effects in interventions with co-occurrence
of key components of multiple recommendations supports the panels belief that childrens
math achievement will improve when they
are exposed to instruction that includes
most, or all, of the core elements for all
five recommendations.
Further, it is difficult to directly test the implementation of specific vocabulary or communication activities, because teaching academic
vocabulary and encouraging communication
are core activities for preschool, prekindergarten, and kindergarten classrooms, regardless
of the subject matter.359 The panel identified
four studies in which the intervention group
appears to have received additional instruction
that encouraged the use of math vocabulary
or math conversations.360 Positive effects were
reported in the domains of general numeracy
and operations in two of the four studies.361
Both positive and no discernible effects in
general numeracy and operations were found
in a third study.362 The final study found both
positive and negative effects in operations,
depending upon the particular type of feedback the intervention and comparison groups
received.363 The amount of math vocabulary
and math conversation, as well as the degree
to which instruction deliberately linked informal
math knowledge to formal math representations, was not clear for the comparison group
in 9 of the 16 studies.364 This group of studies
( 110 )
Appendix D (continued)
reported mixed effects in the outcome domains
of general numeracy365 and geometry366 and
only positive effects in the outcome domain
of basic number concepts.367 Three studies
reported no discernible effects in the operations
outcome domain.368 The panel determined that
the comparison group had participated in an
intervention with core elements of Recommendation 4 in three studies369 that found positive
effects in general numeracy,370 basic number
concepts,371 and geometry.372
Despite the limitations of the body of evidence for this recommendation, the panel
believesbased on its own expertise and the
presence of these practices in multiple studies
with positive effects on math outcomesthat
teaching math vocabulary and providing
children with opportunities to talk about math
are important for the development of childrens early math skills.
Appendix D (continued)
Together, these activities prepare children
to be introduced to the formal concepts of
addition and subtraction, including the use of
formal symbols to represent the operations.
One study found that children participating
Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and
linking informal knowledge to formal knowledge and contributed to the level of evidence rating
Arnold et al.
(2002)5
RCT
Meets evidence
standards without
reservations
Barnett et al.
(2008)
RCT
Meets evidence
standards with
reservations
Clements and
Sarama (2007b)5,7
RCT
Meets evidence
standards with
reservations
Comparison3
Math Is Everywhere vs.
regular classroom
instruction
Operations: WJ-Revised
Applied Math Problems
Subtest
RCT
Meets evidence
standards without
reservations
X6
X6
X8
X8
X8
X8
X8
X8
Findings (Domain:
Assessment (Effect Size,
Significance))4
Link Informal
Knowledge to
Formal Representations
Population
Characteristics2
Use Math
Vocabulary
Citation, Design,
and WWC Rating1
Encourage
Math
Communication
Recommendation
Components Tested
Study Characteristics
Positive (1.40*)
Building Blocks vs.
regular classroom
instruction (locally
developed)
( 112 )
(continued)
Appendix D (continued)
Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and
linking informal knowledge to formal knowledge and contributed to the level of evidence rating
(continued)
Clements et al.
(2011)5,10,11
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 121 total (56 intervention; 65 comparison)
Comparison3
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy:
REMATotal
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction
(Math Trailblazers)
Link Informal
Knowledge to
Formal Representations
Population
Characteristics2
Use Math
Vocabulary
Citation, Design,
and WWC Rating1
Encourage
Math
Communication
Recommendation
Components Tested
Study Characteristics
X12
X12
X12
X12
X12
X12
X12
X12
X12
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X14
X16
X16
X16
Positive (0.48*)
Basic number concepts:
REMANumbers Total
Positive (0.39*)
Geometry: REMAGeometry Total
Positive (0.64*)
Positive (0.64*)
Operations: WJ-IIITotal
Score, Posttest
Positive (0.29, ns)
General numeracy: NSB
Total Score, Maintenance
(6 weeks)
Positive (0.65*)
Operations: WJ-IIITotal
Score, Maintenance
(6 weeks)
No discernible (0.18, ns)
Fantuzzo,
Gadsden, and
McDermott (2011)15
RCT
Meets evidence
standards without
reservations
Fuchs, L. S.,
Fuchs, D., and
Karns (2001)5,17
RCT
Meets evidence
standards without
reservations
Evidence-based
Program for
Integrated Curricula (EPIC) vs.
regular classroom
instruction (DLM
Early Childhood
Express)
General numeracy:
LEMathematics, Wave 4
Peer-assisted
learning strategies
(PALS ) vs. regular
classroom instruction (same curriculum as PALS, a
district curriculum
including Math
Advantage Grade
K Basal)
( 113 )
Positive (0.18*)
X18
X18
(continued)
Appendix D (continued)
Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and
linking informal knowledge to formal knowledge and contributed to the level of evidence rating
(continued)
Comparison3
Kindergarten students in
public schools in inner-city
areas in Massachusetts
Rightstart vs.
regular classroom
instruction
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers or
Math Connects)
Link Informal
Knowledge to
Formal Representations
Population
Characteristics2
Use Math
Vocabulary
Findings (Domain:
Assessment (Effect Size,
Significance))4
Citation, Design,
and WWC Rating1
Encourage
Math
Communication
Recommendation
Components Tested
Study Characteristics
X21
X21
X21
X21
X21
X21
X21
X21
X21
X21
X21
X21
X22
X22
X22
X22
X22
X22
X22
X22
X22
X22
X22
X22
X23
X23
X23
Positive (1.79*)
QED
Meets evidence
standards with
reservations
Jordan et al.
(2012)20
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 86 total (42 intervention; 44 comparison)
Mean age: 5.5 years
(SD 4.38 months)
Positive (1.10*)
Operations: WJ-IIITotal,
Posttest
Positive (0.91*)
General numeracy: NSB
Total, Maintenance (8 weeks)
Positive (0.77*)
Operations: WJ-IIITotal,
Maintenance (8 weeks)
Positive (0.56*)
Jordan et al.
(2012)20
RCT
Meets evidence
standards without
reservations
Supplemental
researcherdeveloped
number sense
curriculum vs.
treated comparison (supplemental language
intervention with
Math Trailblazers
or Math Connects)
RCT
Meets evidence
standards without
reservations
( 114 )
(continued)
Appendix D (continued)
Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and
linking informal knowledge to formal knowledge and contributed to the level of evidence rating
(continued)
PCER Consortium
(2008, Chapter 2)5,24
Prekindergarten teachers
working in public programs
the year before the study
began
RCT
Meets evidence
standards with
reservations
Comparison3
Creative Curriculum vs. regular
classroom instruction (teacherdeveloped nonspecific curricula)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Operations: WJ-IIIApplied
Problems, Posttest
Link Informal
Knowledge to
Formal Representations
Population
Characteristics2
Use Math
Vocabulary
Citation, Design,
and WWC Rating1
Encourage
Math
Communication
Recommendation
Components Tested
Study Characteristics
PCER Consortium
(2008, Chapter 2)5,24
RCT
Meets evidence
standards with
reservations
Prekindergarten teachers
working in public programs
the year before the study
began
Children: 198 total (98 intervention; 100 comparison)
Bright Beginnings
vs. regular classroom instruction
(teacher-developed nonspecific
curricula)
Operations: WJ-IIIApplied
Problems, Posttest
No discernible (0.16, ns)
General numeracy: CMA-A,
Posttest
No discernible (0.14, ns)
PCER Consortium
(2008, Chapter 3)25
Preschoolers attending
Head Start centers
RCT
Meets evidence
standards with
reservations
Operations: WJ-IIIApplied
Problems, Posttest
No discernible (0.20, ns)
General numeracy:
CMA-AMathematics
Composite, Posttest
No discernible (0.10, ns)
Geometry: Shape Composition, Posttest
No discernible (0.19, ns)
Operations: WJ-IIIApplied
Problems, Maintenance
(spring of kindergarten year)
No discernible (0.09, ns)
General numeracy:
CMA-AMathematics Composite, Maintenance (spring
of kindergarten year)
No discernible (0.14, ns)
Geometry: Shape Composition, Maintenance (spring
of kindergarten year)
No discernible (0.01, ns)
( 115 )
(continued)
Appendix D (continued)
Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and
linking informal knowledge to formal knowledge and contributed to the level of evidence rating
(continued)
Recommendation
Components Tested
Sarama et al.
(2008)26
RCT
Meets evidence
standards without
reservations
Findings (Domain:
Assessment (Effect Size,
Significance))4
Link Informal
Knowledge to
Formal Representations
Population
Characteristics2
Use Math
Vocabulary
Citation, Design,
and WWC Rating1
Encourage
Math
Communication
Study Characteristics
Building Blocks
combined with
Pre-K Mathematics vs. regular classroom
instruction
Feedback with
explanation of
own reasoning
vs. treated
comparison
(feedback only)
X27
X27
Feedback with
explanation of
raters reasoning
vs. treated
comparison
(feedback only)
X28
X28
Feedback with
explanation of
own reasoning
vs. treated
comparison
(feedback with
explanation of
raters reasoning)
X29
X29
X30
X30
Comparison3
Positive (0.62*)
University-based preschool,
university-based day care
center, or day care center in
a middle-class community
Children: 30 total (15 intervention; 15 comparison)
University-based preschool,
university-based day care
center, or day care center in
a middle-class community
Children: 30 total (15 intervention; 15 comparison)
University-based preschool,
university-based day care
center, or day care center in
a middle-class community
Children: 30 total (15 intervention; 15 comparison)
Age range: 4.5 to 6.1 years
Negative (0.88*)
QED
Meets evidence
standards with
reservations
X30
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the comparison group may have participated in instruction that taught math vocabulary, encouraged communication about math, and supported
children in linking informal and formal math knowledge.
X The intervention included this component.
BB Assessment = Building Blocks Assessment of Early Mathematics382
CMA = Child Math Assessment383
CMA-A = Child Math AssessmentAbbreviated384
SESAT = Stanford 7 Plus385
SAT-P = Stanford Achievement TestPrimary 1386
NSB = Number Sense Brief387
( 116 )
Appendix D (continued)
REMA = Research-Based Early Math Assessment388
WJ-Revised = Woodcock-Johnson, revised edition389
WJ-III = Woodcock-Johnson, third edition390
DSC = Developing Skills Checklist391
1
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random
procedure.
2
SD = Standard deviation. The information presented includes the following: (a) the type of program and unit of assignment, if the
study is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) the age of children
in the sample.
3
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group.
If details were available on the curriculum the comparison teachers used, it is noted parenthetically.
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic may
not have been math. If details were available on what was provided, it is noted parenthetically.
4
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ from
author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes that were
significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked with an asterisk (*);
ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed here. Positive findings favor
the intervention group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger). Negative findings
favor the comparison group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger).
No discernible effects are findings that are neither significant nor substantively important.
5
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook,
Version 2.1 (http://whatworks.ed.gov).
6
In Barnett et al. (2008), the difference between the intervention and comparison groups with respect to math instruction is not
known. The intervention group participated in Tools of the Mind, a comprehensive early childhood curriculum with a math component
that supported incorporating math into other parts of the school day. The comparison group participated in a district-created balanced
literacy curriculum. From the information provided, it was not clear how the intervention and comparison groups differed with respect
to teaching children to use math vocabulary or encouraging them to communicate about math.
7
Clements and Sarama (2007b) also reported scores for the subscales of the Number and Geometry scales; positive effects were
seen for each subscale. Findings from Clements and Sarama (2007b) were previously reported in the WWC intervention report on
SRA Real Math Building Blocks PreK. The panel reports the same findings as presented in the intervention report.
8
In Clements and Sarama (2007b), the difference between the intervention and comparison groups encompassed any aspect of
instruction that differed between Building Blocks and the curricula used in the comparison classrooms, including Creative Curriculum,
a branded comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a math curriculum that
taught children to view and describe their world mathematically. Building Blocks included teaching math vocabulary, encouraging communication about math, and supporting children in linking informal and formal math knowledge. The comparison group participated
in a variety of curricula, including Creative Curriculum, which taught math vocabulary and encouraged communication about math but
did not appear to support linking informal and formal knowledge.
9
For Clements and Sarama (2008), the WWC is reporting author-reported effect sizes consistent with prior reporting of findings
from this study in the WWC intervention report on SRA Real Math Building Blocks PreK.
10
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
11
Clements et al. (2011) also reported the subscale scores from the REMA. Findings for the subscale scores were consistent with the
total score findings and were generally positive (9 of 13 scores). No discernible effects were seen for 4 of the 13 subscale scores (two
in the geometry domain: transformations/turns and comparing shapes; one in the operations domain: arithmetic; and one in the basic
number concepts domain: composition of number).
12
In Clements et al. (2011), the difference between the intervention and comparison groups encompassed any aspect of instruction
that differed between Building Blocks and the various branded curricula used in the comparison classrooms, including DLM Early Childhood Express, a comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a math curriculum
that taught children to view and describe their world mathematically. Building Blocks included teaching math vocabulary, encouraging
communication about math, and supporting children in linking informal and formal math knowledge. The comparison group participated in a number of branded curricula, including DLM Early Childhood Express, which taught math vocabulary, encouraged communication about math, and supported children in linking informal and formal knowledge.
13
Dyson, Jordan, and Glutting (2013) reported total and subscale scores for the NSB, as well as the WJ-IIIApplied Problems and WJIIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation Problems
subscales. Positive effects were found for all subscales at posttest and maintenance, except for the WJ-IIIApplied Problems subscale,
for which no discernible effects were seen at posttest or maintenance.
14
In Dyson, Jordan, and Glutting (2013), the difference between the intervention and comparison groups was the additional 12
hours of math instruction the intervention group received. The intervention group participated in 30-minute sessions, generally 3 a
week, for a total of 24 sessions (or 12 hours). The intervention group participated in additional number sense instruction that included
teaching math vocabulary, encouraging communication about math, and supporting children in linking informal and formal knowledge. The comparison group did not receive this additional instruction; rather, they received only the regular classroom math instruction. The regular classroom math instruction, for both the intervention and comparison children, was Math Trailblazers, a branded
math curriculum used to teach number and operations but not guided by a developmental progression.
15
Fantuzzo, Gadsden, and McDermott (2011) reported on four waves of data collection. The panel decided to use Wave 1 as pretest
data, because it was collected prior to the delivery of math content. Wave 4 was used as the posttest, as it was collected at the end
( 117 )
Appendix D (continued)
of the school year and delivery of the intervention. Waves 2 and 3 could be viewed as intermediary outcomes, but the panel chose to
focus on posttests when determining levels of evidence.
16
In Fantuzzo, Gadsden, and McDermott (2011), the difference between the intervention and comparison groups included any aspect
of instruction that differed between EPIC and DLM Early Childhood Express, a branded comprehensive early childhood curriculum. Both
curricula taught math vocabulary, encouraged communication about math, and supported linking informal and formal math knowledge.
17
Fuchs, L. S., Fuchs, D., and Karns (2001) did not provide a pretest for the SAT-P. The panel decided to use the SESAT pretest for
the post-hoc difference-in-difference adjustment.
18
In Fuchs, L. S., Fuchs, D., and Karns (2001), the difference between the intervention and comparison groups was the use of
peer-assisted learning strategies to practice math problem solving. Both the intervention and comparison groups participated in similar
math instruction using the district curriculum, which included the Math Advantage Grade K Basal. The intervention group also took
turns working in pairs, with both children serving as coach while solving math problems together; this provided an opportunity for
children to practice communicating about math with peers. The comparison group did not participate in any peer-assisted learning
strategies to practice math skills.
19
Griffin, Case, and Capodilupo (1995) and related publication Griffin, Case, and Siegler (1994) reported other outcomes for
which no pretest data were provided. The WWC was unable to conduct a review that included these outcomes, as baseline equivalence
could not be established.
20
Jordan et al. (2012) reported posttest and maintenance effects for total and subscale scores for the NSB, as well as the WJ-III
Applied Problems and WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and
WJ-IIICalculation Problems subscales. Positive effects were found for all but seven of the NSB outcomes that were reported as no
discernible effects.
21
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The intervention group participated in additional number sense
instruction that included teaching math vocabulary, encouraging communication about math, and supporting children in linking informal and formal knowledge. The comparison group did not receive this additional instruction in math; rather, they only had the regular
classroom instruction. The regular classroom instruction, for both the intervention and comparison children, was Math Trailblazers or
Math Connects, both of which are commercially available curricula.
22
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in
30-minute sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The intervention group participated in additional number
sense instruction that included teaching math vocabulary, encouraging communication about math, and supporting children in linking
informal and formal knowledge. The comparison group did not receive this additional instruction in math; rather, they received only
the regular classroom instruction and additional literacy instruction. The regular classroom instruction, for both the intervention and
comparison children, was Math Trailblazers or Math Connects, both of which are commercially available curricula.
23
In Klein et al. (2011), the difference between the intervention and comparison groups encompassed any aspect of instruction that
differed between the combined Pre-K Mathematics and DLM Early Childhood Express intervention and the curricula used in the comparison
classrooms, including Creative Curriculum, a branded comprehensive early childhood curriculum. The intervention group participated in
a combination of Pre-K Mathematics and DLM Early Childhood Express, which taught math vocabulary, encouraged communication about
math, and supported children in linking informal and formal knowledge. The comparison group participated in a number of branded curricula, including Creative Curriculum, a comprehensive early childhood curriculum that included regular math lessons.
24
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel rated the study differently but reports the same findings as presented in the intervention report. The difference in study
rating is due to the use of WWC Version 2.1 standards as opposed to WWC Version 1.0 standards. Findings from this study of Bright
Beginnings were previously reported in the WWC intervention report on Bright Beginnings. The panel reports the same findings as
reported in the intervention report. For both Creative Curriculum and Bright Beginnings, the authors reported on additional outcomes
that were assessed in the spring of kindergarten.
25
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel reports the same findings as presented in the intervention report.
26
Sarama et al. (2008) reported subscale scores as well; however, only the means were provided, so the WWC was unable to calculate
effect sizes for the subscales.
27
There are three comparisons in Siegler (1995). In this comparison, the difference between the intervention and comparison groups
was the explanation of the solution provided in conjunction with feedback. Both groups received feedback on their response. The
children in the intervention group provided an explanation of their own reasoning. Children in the comparison group did not provide
or receive any explanation.
28
There are three comparisons in Siegler (1995). In this comparison, the difference between the intervention and comparison groups
was the explanation of the solution provided in conjunction with feedback. Both groups received feedback on their response. Children
in the intervention group listened to the raters explanation of their response. Children in the comparison group did not provide or
receive any explanation.
29
There are three comparisons in Siegler (1995). In this comparison, the difference between the intervention and comparison groups
was the explanation of the solution provided in conjunction with feedback. Both groups received feedback on their response. Children
in the intervention group provided an explanation of their own reasoning. Children in the comparison group listened to the raters
explanation of their reasoning.
30
In Sophian (2004), the difference between the intervention and comparison groups was whether children received math instruction
using a researcher-developed, measurement-focused curriculum. The intervention group participated in a researcher-developed, measurement-focused curriculum that emphasized the concept of unit, provided math vocabulary, encouraged communication about math,
and supported children in linking informal and formal knowledge. The comparison group participated in a literacy curriculum. There is no
description of the math instruction children in the comparison group may have received as part of their regular classroom instruction.
( 118 )
Appendix D (continued)
classrooms are teaching math either as a
particular subject or in conjunction with other
content areas,403 the panel determined that
it was highly unlikely that the comparison
groups were receiving no math instruction.
For this reason, the panel did not consider the
studies to be direct tests of Recommendation
5. Based on their expertise and the effects of
interventions that include dedicated time each
day to teach math and/or efforts to integrate
math instruction throughout the school day,
the panel believes the studies generally support this recommendation despite the limitations of the body of evidence.
The panel believes that teachers should dedicate time to math instruction daily as well as
take advantage of opportunities to integrate
math into other classroom activities, including
games and instruction in other content areas.
Math instruction was a regular, if not daily,
activity for the intervention groups in 14 of
the 20 studies.399 Integration of math into
other content areas was a focus in the interventions examined in 11 of the 20 studies.400
The panel was able to determine that 6 of
the 20 studies deliberately played games to
reinforce math skills.401
The panel identified two areas of concern
regarding the evidence associated with this
recommendation. First, the interventions
examined always included key elements
of other recommendations (i.e., they were
multi-component interventions).402 Thus, the
panel was unable to attribute the effects seen
to the instruction of math both at specific
points during the day and during instruction
in other content areas. Second, since many
preschool, prekindergarten, and kindergarten
Appendix D (continued)
and the differences between the intervention
and comparison groups. Every intervention
examined for this recommendation included
components of other recommendationsfor
example, targeted instruction in number and
operations, geometry, patterns, measurement, and data analysiswhich may have
contributed to the overall effects seen. Furthermore, based on their own experiences,
the members of panel recognize that it is rare
to find a preschool, prekindergarten, or kindergarten classroom that is not doing some
sort of math activity. However, the panel
believes that children in classrooms that both
provide regular math time and integrate
math into other content areas will learn more
math than children in classrooms that do not
include these experiences.
Appendix D (continued)
participating in interventions that supported
reinforcing and extending math concepts in
the classroom environment, routines, and
other activities scored higher, on average,
than children in the comparison group, in the
domains of general numeracy,420 basic number concepts,421 and geometry.422
Several other studies investigated interventions in which childrens math concepts were
reinforced by playing board games, an activity
specified in the panels recommendation.423 In
these studies, children played The Great Race,
a numerical board game, one-on-one with
the experimenter over the course of three to
four 15- to 20-minute sessions.424 The studies generally found that children who played
number-based board games performed better
in the domain of basic number concepts than
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating
Arnold et al.
(2002)5
RCT
Meets evidence
standards without
reservations
Aunio, Hautamaki,
and Van Luit
(2005)5,7
RCT
Meets evidence
standards without
reservations
Findings (Domain:
Assessment (Effect Size,
Significance))4
Incorporate
Math into Other
Parts of the Day
Population
Characteristics2
Include Regular
Math Lessons
Citation, Design,
and WWC Rating1
X6
Comparison3
Use Games to
Reinforce Math
Skills
Recommendation
Components Tested
Study Characteristics
Positive (0.87*)
Geometry: Geometrical
Analogies, Posttest
( 121 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Aunio, Hautamaki,
and Van Luit
(2005)5,7
RCT
Meets evidence
standards without
reservations
(continued)
Comparison3
Lets Think! combined with Maths!
vs. regular classroom instruction
Findings (Domain:
Assessment (Effect Size,
Significance))4
Geometry: SRT, Posttest
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
Meets evidence
standards with
reservations
Clements and
Sarama (2007b)5,9
Preschool classrooms in
state-funded or Head Start
programs
RCT
Meets evidence
standards with
reservations
Operations: WJ-Revised
Applied Math Problems
Subtest
X8
X10
X10
X10
X10
X10
X10
Positive (0.75*)
Geometry: BB Assessment
Geometry Scale
Positive (1.40*)
( 122 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Clements et al.
(2011)5,7,12
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 121 total (56 intervention; 65 comparison)
Comparison3
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy:
REMATotal
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers)
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
X13
X13
X13
X13
X13
X13
X13
X13
X13
Positive (0.48*)
Basic number concepts:
REMANumbers Total
Positive (0.39*)
Geometry: REMAGeometry
Total
Positive (0.64*)
X15
X15
X15
X15
X15
X15
X15
X15
Positive (0.64*)
Operations: WJ-IIITotal
Score, Posttest
Positive (0.29, ns)
General numeracy: NSBTotal
Score, Maintenance (6 weeks)
Positive (0.65*)
Operations: WJ-IIITotal
Score, Maintenance (6 weeks)
No discernible (0.18, ns)
Fantuzzo,
Gadsden, and
McDermott (2011)16
RCT
Meets evidence
standards without
reservations
Evidence-based
Program for
Integrated Curricula (EPIC) vs.
regular classroom
instruction (DLM
Early Childhood
Express)
General numeracy:
LEMathematics, Wave 4
Rightstart vs.
regular classroom
instruction
X17
X17
Positive (0.18*)
Kindergarten students in
public schools in inner-city
areas in Massachusetts
Positive (1.79*)
QED
Meets evidence
standards with
reservations
( 123 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
19
RCT
Meets evidence
standards without
reservations
Comparison3
Supplemental
researcherdeveloped
number sense
curriculum vs.
regular classroom
instruction (Math
Trailblazers or
Math Connects)
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy: NSB
Total, Posttest
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
X20
X20
X20
X20
X20
X20
X20
X20
X21
X21
X21
X21
X21
X21
X21
X21
Positive (1.10*)
Operations: WJ-IIITotal,
Posttest
Positive (0.91*)
General numeracy: NSB
Total, Maintenance (8 weeks)
Positive (0.77*)
Operations: WJ-III, Total,
Maintenance (8 weeks)
Positive (0.56*)
Jordan et al.
(2012)19
RCT
Meets evidence
standards without
reservations
Kindergarten students
attending full-day kindergarten in one of five schools
in one district in the MidAtlantic region of the
United States
Children: 84 total (42 intervention; 42 comparison)
Mean age: 5.5 years
(SD 4.38 months)
Supplemental
researcherdeveloped
number sense
curriculum vs.
treated comparison (supplemental language
intervention with
Math Trailblazers
or Math Connects)
Monahan (2007)23
RCT
Meets evidence
standards without
reservations
X22
X22
X24
X24
X22
Positive (0.57*)
( 124 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Monahan (2007)
23
RCT
Meets evidence
standards without
reservations
Comparison3
Findings (Domain:
Assessment (Effect Size,
Significance))4
Operations: WJ-IIIApplied
Problems, Posttest
X25
X25
X26
X26
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
23
RCT
Meets evidence
standards without
reservations
Prekindergarten teachers
working in public programs
the year before the study
began
Children: 193 total (93 intervention; 100 comparison)
Prekindergarten teachers
working in public programs
the year before the study
began
Children: 198 total (98 intervention; 100 comparison)
Bright Beginnings
vs. classroom instruction (teacherdeveloped nonspecific curricula)
Operations: WJ-IIIApplied
Problems, Posttest
( 125 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
PCER Consortium
(2008, Chapter 3)28
Preschoolers attending
Head Start centers
RCT
Meets evidence
standards with
reservations
Comparison3
Creative Curriculum vs. regular
classroom instruction (teacherdeveloped nonspecific curricula)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Operations: WJ-IIIApplied
Problems, Posttest
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
Preschoolers attending
Head Start programs
RCT
Meets evidence
standards without
reservations
Number-based
board games vs.
treated comparison (color-based
board games)
X30
Positive (0.74*)
Basic number concepts: Numerical Magnitude Comparison, Posttest
X30
Positive (0.99*)
X30
Positive (0.69*)
Basic number concepts:
Counting, Maintenance
(9 weeks)
X30
Positive (0.66*)
Basic number concepts: Numerical Magnitude Comparison, Maintenance (9 weeks)
X30
Positive (0.77*)
Number recognition:
Number Identification,
Maintenance (9 weeks)
X30
Positive (0.80*)
( 126 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
RCT
Meets evidence
standards without
reservations
Comparison3
Linear, numberbased board
games vs. treated
comparison (number string counting, numeral
identification, and
object counting)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Number recognition:
Numeral Identification
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
X32
X32
No discernible (ns)24
Operations: Arithmetic
Absolute Error
X32
X33
X33
X33
Meets evidence
standards without
reservations
Preschool-aged children
attending Head Start or one
of three childcare centers
RCT
Meets evidence
standards without
reservations
Building Blocks
combined with
Pre-K Mathematics vs. regular classroom
instruction
Positive (0.62*)
X30
Positive (0.86*)31
Basic number concepts:
Percent of Correctly
Ordered Numbers
X30
Positive (1.17*)
( 127 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Preschoolers attending
Head Start programs
or one of two childcare
centers
RCT
Meets evidence
standards without
reservations
Comparison3
Linear numberbased board
games vs. treated
comparison
(number string
counting, numeral
identification, and
object counting)
Findings (Domain:
Assessment (Effect Size,
Significance))4
Basic number concepts:
Number Line Estimation
Percent Absolute Error
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
X36
Positive (0.63*)31
Basic number concepts:
Numerical Magnitude
Comparison
X36
No discernible (ns)34
X36
No discernible (ns)34
Number recognition: Number
Identification
X36
No discernible (ns)34
Operations: Arithmetic
Percentage Answered
Correctly
X36
No discernible (ns)34
Operations: Arithmetic
Percent Absolute Error
X36
No discernible (ns)34
Siegler and Ramani
(2009)7,29
RCT
Meets evidence
standards without
reservations
X37
No discernible (ns)31,34
Basic number concepts:
Numerical Magnitude
Comparison
X37
No discernible (ns)34
X37
No discernible (ns)34
Number recognition: Number
Identification
X37
No discernible (ns)34
Operations: Arithmetic
Percentage Answered
Correctly
X37
No discernible (ns)34
Operations: Arithmetic
Percent Absolute Error
X37
No discernible (ns)34
( 128 )
(continued)
Appendix D (continued)
Table D.8. Studies of interventions that included regular math time, incorporated math into
other aspects of the school day, and used games to reinforce math skills and contributed to
the level of evidence rating (continued)
Sophian (2004)
Preschoolers attending
Head Start programs
5,8
QED
Meets evidence
standards with
reservations
Comparison3
Researcher-developed, measurement-focused
curriculum vs.
treated comparison (literacy
instruction)
Findings (Domain:
Assessment (Effect Size,
Significance))4
General numeracy: DSC
Mathematics Subscale
Use Games to
Reinforce Math
Skills
Population
Characteristics2
Incorporate
Math into Other
Parts of the Day
Citation, Design,
and WWC Rating1
Include Regular
Math Lessons
Recommendation
Components Tested
Study Characteristics
X38
? There was not sufficient description of the type and nature of the instruction the comparison group received. Children in the comparison group may have participated in instruction that included regular math lessons, incorporated math into other parts of the day,
or used games to reinforce math skills.
X The intervention included this component.
BB Assessment = Building Blocks Assessment of Early Mathematics426
REMA = Research-Based Early Math Assessment427
NKT = Number Knowledge Test428
DSC = Developing Skills Checklist429
LE = Learning Express430
WJ-Revised = Woodcock-Johnson, revised edition431
WJ-III = Woodcock-Johnson, third edition432
CMA = Child Math Assessment433
TEMA-2 = Test of Early Mathematics Ability, second edition434
ENCO = Emergent Numeracy and Cultural Orientations Assessment435
NSB = Number Sense Brief436
1
RCT = Randomized controlled trial. Children, classrooms, or schools were randomly assigned to intervention conditions.
QED = Quasi-experimental design. Children, classrooms, or schools were assigned to intervention conditions by a non-random
procedure.
2
SD = Standard deviation. The information presented includes the following: (a) type of program and unit of assignment, if the study
is an RCT and it differs from the unit of analysis; (b) the number of children by intervention status; and (c) the age of children in the sample.
3
Regular classroom instruction: The researchers did not provide any additional instructional material to the comparison group. If
details were available on the curriculum the comparison teachers used, it is noted parenthetically.
Treated comparison: The comparison group received additional instruction or materials from the researchers, although the topic may
not have been math. If details were available on what was provided, it is noted parenthetically.
4
All effect sizes and significance levels are calculated by the WWC unless otherwise noted. WWC calculations sometimes differ from
author-reported results, due to WWC adjustments for baseline differences, clustering, or multiple comparisons. Effect sizes that were
significant (p 0.05) by WWC calculations or author calculations where no WWC adjustment was required are marked with an asterisk (*);
ns refers to effects that were not significant. Only outcomes that met WWC evidence standards are listed here. Positive findings favor
the intervention group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger). Negative findings
favor the comparison group and are either significant or substantively important (i.e., the effect size is 0.25 SD or larger).
No discernible effects are findings that are neither significant nor substantively important.
5
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
clustering within classrooms or schools. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook,
Version 2.1 (http://whatworks.ed.gov).
6
The difference between the intervention and comparison groups was the use of the Math Is Everywhere activities to help teachers
incorporate math in other parts of the school day, such as circle time, transitions from one activity to another, or meals.
7
The level of statistical significance was reported by the study authors or, where necessary, calculated by the WWC to correct for
multiple comparisons. For an explanation of these adjustments, see the WWC Procedures and Standards Handbook, Version 2.1
(http://whatworks.ed.gov).
8
In Barnett et al. (2008), the difference between the intervention and comparison groups with respect to math instruction is not
known. The intervention group participated in Tools of the Mind, a comprehensive early childhood curriculum with a math component
that supported incorporating math into other parts of the school day. The comparison group participated in a district-created balanced
literacy curriculum. From the information provided, it was not clear how the intervention and comparison groups differed with respect
to incorporating math into other aspects of the school day.
( 129 )
Appendix D (continued)
9
Clements and Sarama (2007b) also reported scores for the subscales of the Number and Geometry scales; positive effects were
seen for each subscale. Findings from Clements and Sarama (2007b) were previously reported in the WWC intervention report on
SRA Real Math Building Blocks PreK. The panel reports the same findings as presented in the intervention report.
10
In Clements and Sarama (2007b), the difference between the intervention and comparison groups encompassed any aspect of
instruction that differed between Building Blocks and the curricula used in the comparison classrooms, including Creative Curriculum,
a branded comprehensive early childhood curriculum. The intervention group participated in Building Blocks, a math curriculum that
included regular math lessons, incorporated math into other aspects of the school day, and used games to reinforce math skills. The
comparison group participated in a variety of curricula, including Creative Curriculum, which included regular math lessons.
11
For Clements and Sarama (2008), the WWC is reporting author-reported effect sizes consistent with prior reporting of findings
from this study in the WWC intervention report on SRA Real Math Building Blocks PreK.
12
Clements et al. (2011) also reported the subscale scores from the REMA. Findings for the subscale scores were consistent with the
total score findings and were generally positive (9 of 13 scores). No discernible effects were seen for 4 of the 13 subscale scores (two
in the geometry domain: transformations/turns and comparing shapes; one in the operations domain: arithmetic, and one in the basic
number concepts domain: composition of number).
13 In Clements et al. (2011), the difference between the intervention and comparison groups encompassed any aspect of instruction
that differed between Building Blocks and the various branded curricula used in the comparison classrooms, including DLM Early Childhood Express. The intervention group participated in Building Blocks, a math curriculum that included regular math lessons, incorporated math into other aspects of the school day, and used games to reinforce math skills. The comparison group participated in a
number of branded curricula, including DLM Early Childhood Express, an early childhood curriculum that included regular math lessons,
incorporated math into other aspects of the school day, and used games to reinforce math skills.
14
Dyson, Jordan, and Glutting (2013) reported total and subscale scores for the NSB, as well as the WJ-IIIApplied Problems and
WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation Problems
subscales. Positive effects were found for all subscales at posttest and maintenance, except for the WJ-IIIApplied Problems subscale,
for which no discernible effects were seen at posttest or maintenance.
15
In Dyson, Jordan, and Glutting (2013), the difference between the intervention and comparison groups as the additional 12 hours
of math instruction the intervention group received. The intervention group participated in 30-minute sessions, generally 3 a week, for a
total of 24 sessions (or 12 hours). The sessions included instruction in number and operations in regular supplemental lessons and used
games to reinforce skills, including The Great Race. The comparison group did not receive this additional instruction; rather, they received
only the regular classroom math instruction. The regular classroom math instruction, for both the intervention and comparison children,
was Math Trailblazers, a branded math curriculum used to teach number and operations but not guided by a developmental progression.
16
Fantuzzo, Gadsden, and McDermott (2011) reported on four waves of data collection. The panel decided to use Wave 1 as pretest
data, because it was collected prior to the delivery of math content. Wave 4 was used as the posttest, as it was collected at the end
of the school year and delivery of the intervention. Waves 2 and 3 could be viewed as intermediary outcomes, but the panel chose to
focus on posttests when determining levels of evidence.
17
In Fantuzzo, Gadsden, and McDermott (2011), the difference between the intervention and comparison groups included any
aspect of instruction that differed between EPIC and DLM Early Childhood Express, a branded comprehensive early childhood curriculum. Both curricula provided regular math lessons and incorporated math into other aspects of the school day.
18
Griffin, Case, and Capodilupo (1995) and related publication Griffin, Case, and Siegler (1994) reported other outcomes for
which no pretest data were provided. The WWC was unable to conduct a review that included these outcomes, as baseline equivalence
could not be established.
19
Jordan et al. (2012) reported posttest and maintenance effects for total and subscale scores for the NSB, as well as the WJ-IIIApplied
Problems and WJ-IIICalculation Problems subscales and a WJ-III Total, which is the sum of the WJ-IIIApplied Problems and WJ-IIICalculation Problems subscales. Positive effects were found for all but seven of the NSB outcomes that were reported as no discernible effects.
20
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The intervention group participated in additional number sense
instruction that included regular math lessons and used games to reinforce math skills, including The Great Race. The comparison
group did not receive this additional instruction in math; rather, they only had the regular classroom instruction. The regular classroom
instruction, for both the intervention and comparison children, was Math Trailblazers or Math Connects, both of which are commercially available curricula.
21
There were two comparisons in Jordan et al. (2012). In this comparison, the difference between the intervention and comparison
groups was the additional 12 hours of math instruction the intervention group received. The intervention group participated in 30-minute
sessions, 3 times a week, for a total of 24 sessions (or 12 hours). The intervention group participated in additional number sense instruction that included regular math lessons and used games to reinforce math skills, including The Great Race. The comparison group did not
receive this additional instruction in math; rather, they only had the regular classroom instruction and additional literacy instruction. The
regular classroom instruction, for both the intervention and comparison children, was Math Trailblazers or Math Connects, both of which
are commercially available curricula.
22
In Klein et al. (2008), the difference between the intervention and comparison groups encompassed any aspect of instruction that
differed between the combined Pre-K Mathematics and DLM Early Childhood Express intervention and the curricula used in the comparison classrooms, including Creative Curriculum. The intervention group, which participated in a combination of Pre-K Mathematics and
DLM Early Childhood Express, included regular math lessons, incorporated math into other aspects of the school day, and used games
to reinforce math skills. The comparison group participated in a number of branded curricula, including Creative Curriculum,
a comprehensive early childhood curriculum that included regular math lessons.
23
The panel focused on the comparisons between the three intervention groups for this recommendation.
24
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups was the manner in which number sense instruction was delivered. The intervention group participated in number sense
instruction using stories to reinforce concepts and skills. The comparison group participated in the same number sense curriculum
delivered in small groups, without the use of stories.
( 130 )
Appendix D (continued)
25
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups was the manner in which number sense instruction was delivered. The intervention group participated in number sense
instruction using movement to reinforce concepts and skills. The comparison group participated in the same number sense curriculum
delivered in small groups, without the use of movement.
26
There were three possible comparisons in Monahan (2007). In this comparison, the difference between the intervention and comparison groups was the manner in which number sense instruction was delivered. The intervention group participated in number sense
instruction using movement to reinforce concepts and skills. The comparison group participated in the same number sense instruction
using stories without movement to reinforce concepts and skills.
27
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel rated the study differently but reports the same findings as presented in the intervention report. The difference in study
rating is due to the use of WWC Version 2.1 standards as opposed to WWC Version 1.0 standards. Findings from this study of Bright
Beginnings were previously reported in the WWC intervention report on Bright Beginnings. The panel reports the same findings as
reported in the intervention report. For both Creative Curriculum and Bright Beginnings, the authors report on additional outcomes
that were assessed in the spring of kindergarten.
28
Findings from this study of Creative Curriculum were previously reported in the WWC intervention report on Creative Curriculum.
The panel reports the same findings as presented in the intervention report.
29
Findings from these studies (Ramani & Siegler, 2008; Siegler & Ramani, 2008; Siegler & Ramani, 2009) were previously
reported in the WWC practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. The panel reports
the findings as discussed in that practice guide.
30
In both Ramani and Siegler (2008) and Siegler and Ramani (2008), the difference between the intervention and comparison
groups was the nature of the board games played. The intervention group played a number-based version of The Great Race with each
space on the board having a number and children stating the number as they moved their token. The comparison group also played
The Great Race, but with spaces that were colored and children stating the color as they moved their token.
31
The effect is in the desired direction, with the intervention making fewer errors than the comparison group, which results in a negative
effect size. However, to present the findings in a consistent manner, the effect size is reported as positive.
32
There are two comparisons in Ramani and Siegler (2011). In this comparison, the difference between the intervention and comparison groups was whether they played linear, number-based board games to reinforce math concepts and skills. The intervention
group played a linear version of The Great Race with each space on the board having a number and children stating the number as
they moved their token. The comparison group practiced counting number strings and objects and identifying numerals.
33
There are two comparisons in Ramani and Siegler (2011). In this comparison, the difference between the intervention and comparison groups was whether they played circular number-based board games to reinforce math concepts and skills. The intervention
group played a circular version of The Great Race with each space on the board having a number and children stating the number as
they moved their token. The comparison group practiced counting number strings and objects and identifying numerals.
34
Sarama et al. (2008) reported subscale scores as well; however, only the means were provided, so the WWC was unable to calculate effect sizes for the subscales.
35
The authors reported non-significant findings for these outcomes and comparisons but did not report effect sizes or provide sufficient information for the WWC to calculate effect sizes. The panel reports on these outcomes and comparisons in a manner similar
to the WWC practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade.
36
There are two comparisons in Siegler and Ramani (2009). In this comparison, the difference between the intervention and
comparison groups was whether they played linear number-based board games to reinforce math concepts and skills. The intervention group played The Great Race with each space on the board having a number and children stating the number as they moved their
token. The comparison group practiced counting number strings and objects and identifying numerals.
37
There are two comparisons in Siegler and Ramani (2009). In this comparison, the difference between the intervention and
comparison groups was whether they played circular number-based board games to reinforce math concepts and skills. The intervention group played The Great Race with each space on the board having a number and children stating the number as they moved their
token. The comparison group practiced counting number strings and objects and identifying numerals.
38
In Sophian (2004), the difference between the intervention and comparison groups was whether children received math instruction using a researcher-developed, measurement-focused curriculum. The intervention group participated in a researcher-developed,
measurement-focused curriculum that emphasized the concept of unit and incorporated math into other aspects of the school day.
The comparison group participated in a literacy curriculum. There is no description of the math instruction children in the comparison
group may have received as part of their regular classroom instruction.
( 131 )
Endnotes437
Please note that there will still be some footnotes in the guidethese will be attached to titles of the sections
specifically to state that, Eligible studies that meet WWC evidence standards or meet evidence standards with
reservations are indicated by bold text in the endnotes and references pages.
1. Following WWC guidelines, improved outcomes are indicated by either a positive
statistically significant effect or a positive, substantively important effect size.
The WWC defines substantively important,
or large, effects on outcomes to be those
with effect sizes greater than or equal to
0.25 standard deviations. See the WWC
guidelines at http://whatworks.ed.gov.
2. For more information, see the WWC Frequently Asked Questions page for practice guides, http://whatworks.ed.gov.
3. This includes randomized controlled trials (RCTs) and quasi-experimental design
studies (QEDs). Studies not contributing
to levels of evidence include single-case
designs (SCDs) evaluated with WWC pilot
SCD standards and regression discontinuity designs (RDDs) evaluated with pilot
RDD standards.
4. The research may include studies generally meeting WWC standards and supporting the effectiveness of a program, practice, or approach with small sample sizes
and/or other conditions of implementation or analysis that limit generalizability.
The research may include studies that
support the generality of a relation but do
not meet WWC standards; however, they
have no major flaws related to internal
validity other than lack of demonstrated
equivalence at pretest for QEDs. QEDs
without equivalence must include a pretest covariate as a statistical control for
selection bias. These studies must be
accompanied by at least one relevant
study meeting WWC standards.
5. American Educational Research Association, American Psychological Association,
and National Council on Measurement in
Education (1999).
6. Ginsburg, Klein, and Starkey (1998).
7. Underlined terms in this practice guide
are defined in the Glossary.
8. Early math content areas are the specific
math topics the panel believes should
become the foundation of preschool, prekindergarten, and kindergarten curricula.
The panel has identified number and operations, geometry, patterns, measurement,
9.
10.
11.
12.
13.
14.
15.
16.
17.
( 132 )
Endnotes (continued)
18. National Governors Association Center
for Best Practices, Council of Chief State
School Officers (2010).
19. New York State Department of Education
(2011).
20. For example, positive effects in math
achievement are seen in Clements
and Sarama (2007b); Clements and
Sarama (2008); Clements et al. (2011);
Fantuzzo, Gadsden, and McDermott
(2011); Klein et al. (2008); Sarama et
al. (2008); Arnold et al. (2002); Barnett et al. (2008); Dyson, Jordan, and
Glutting (2013); Griffin, Case, and
Capodilupo (1995) and related publication Griffin, Case, and Siegler (1994);
Jordan et al. (2012).
21. Although there is little direct evidence that
identifies which specific developmental
progression is most effective in teaching
math to young children, the panel believes
there is indirect evidence from Clements
et al. (2011); Dyson, Jordan, and Glutting (2013); and Fantuzzo, Gadsden,
and McDermott (2011) demonstrating
developmental progressions are necessary. The panel also considered research
that supports the use of the specific steps
outlined in the sequence of a developmental progression, even if the developmental
progression as a whole was not directly
tested, for example, Purpura, Baroody, and
Lonigan (in press).
22. Sarama and Clements (2009a).
23. Although unstructured or unguided learning
opportunities certainly play a role in young
childrens mathematical learning, structured
or guided opportunities also have an important role. For instance, Mix, Moore, and
Holcomb (2011) found that 3-year-olds provided with toys and a matching container
(e.g., wiffle balls and a muffin tin) and asked
to complete a challenging equivalence
(number-matching) task outperformed children who were provided with the same toys
but not given a container.
24. Although instructional methods may vary
across countries, the content of early math
is quite similar internationally; therefore,
the panel did not consider a geographic or
language restriction in the review.
25. The guide focuses on teaching math to
children attending preschool, prekindergarten, or kindergarten. The panel considered
research examining the math competen-
26.
27.
28.
29.
30.
( 133 )
Endnotes (continued)
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
( 134 )
Endnotes (continued)
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
Endnotes (continued)
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
( 136 )
Endnotes (continued)
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
( 137 )
Endnotes (continued)
118. Fantuzzo, Gadsden, and McDermott
(2011).
119. Clements and Sarama (2007b); Clements and Sarama (2008); Clements
et al. (2011); Sarama et al. (2008).
120. Studies that found positive effects in general numeracy: Clements and Sarama
(2008); Clements et al. (2011); Fantuzzo, Gadsden, and McDermott
(2011); Klein et al. (2008). Studies
that found positive effects in basic number concepts: Clements and Sarama
(2007b); Clements et al. (2011). Studies that found positive effects in geometry: Clements and Sarama (2007b);
Clements et al. (2011).
121. Fuchs, L. S., Fuchs, D., and Karns
(2011); Siegler (1995).
122. Table D.1 summarizes which studies are
linked to which recommendations.
123. Six of the 16 studies contributing to the
body of evidence for Recommendation 4
did not provide sufficient information for
the panel to determine whether the comparison group participated in instruction
that included elements of Recommendation 4. Appendix D includes additional
information on the studies contributing to
the body of evidence, including descriptions of the intervention and comparison
group conditions.
124. For examples of evaluations of curricula
that begin with a childs informal and familiar math knowledge, see Arnold et al.
(2002); Clements and Sarama (2007b);
Clements and Sarama (2008); Clements et al. (2011); Fantuzzo, Gadsden,
and McDermott (2011); Griffin, Case,
and Capodilupo (1995) and related
publication Griffin, Case, and Siegler
(1994); Klein et al. (2008); Sarama et
al. (2008); Sophian (2004).
125. NAEYC and NCTM (2010); NAEYC (2009).
126. For examples of evaluations of curricula
that teach children to use math vocabulary,
see Barnett et al. (2008); Clements and
Sarama (2007b); Clements and Sarama
(2008); Clements et al. (2011); Dyson,
Jordan, and Glutting (2013); Jordan
et al. (2012); Klein et al. (2008); PCER
Consortium (2008, Chapter 2); PCER
Consortium (2008, Chapter 3); Sarama
et al. (2008).
127. For examples of evaluations of curricula
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
( 138 )
Endnotes (continued)
139. Dyson, Jordan, and Glutting (2013);
Jordan et al. (2012); Ramani and Siegler
(2008); Ramani and Siegler (2011);
Siegler and Ramani (2009).
140. Table D.1 summarizes which studies are
linked to which recommendations.
141. Table D.1 summarizes which studies are
linked to which recommendations.
142. Six of the 20 studies contributing to the
body of evidence for Recommendation 5
did not provide sufficient information for
the panel to determine whether the comparison condition included dedicated time
for math or integration of math instruction
throughout the day. Appendix D includes
additional information on the studies contributing to the body of evidence, including descriptions of the intervention and
comparison group conditions.
143. Clements and Sarama (2008).
144. For examples of evaluations of curricula
that incorporate math concepts throughout the day, see Arnold et al. (2002);
Barnett et al. (2008); Clements and
Sarama
(2007b); Clements
and
Sarama (2008); Clements et al. (2011);
Fantuzzo, Gadsden, and McDermott
(2011); Griffin, Case, and Capodilupo
(1995) and related publication Griffin,
Case, and Siegler (1994); Klein et al.
(2008); Monahan (2007); Sarama et al.
(2008); Sophian (2004).
145. For examples of evaluations of curricula
that incorporate math concepts into
other parts of the day, see Arnold et al.
(2002); Barnett et al. (2008); Clements
and Sarama (2007b); Clements and
Sarama (2008); Clements et al. (2011);
Fantuzzo, Gadsden, and McDermott
(2011); Griffin, Case, and Capodilupo
(1995) and related publication Griffin,
Case, and Siegler (1994); Klein et al.
(2008); Monahan (2007); Sarama et al.
(2008); Sophian (2004).
146. Boggan, Harper, and Whitmire (2010).
147. Baroody, Purpura, and Reid (2012); Baroody, Tiilikainen, and Tai (2006); Chi
(2009); Clements and Sarama (2012); Uttal, Scudder, and DeLoache (1997).
148. For examples of evaluations of curricula
that use board games to practice math
skills, see Dyson, Jordan, and Glutting
(2013); Jordan et al. (2012); Ramani
149.
150.
151.
152.
153.
154.
155.
156.
157.
158.
159.
( 139 )
Endnotes (continued)
160.
161.
162.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173.
174.
( 140 )
Endnotes (continued)
and Johnson (2008) was in Taiwan.
175. Positive effects were seen in general
numeracy in Arnold et al. (2002); Clements and Sarama (2008); Clements
et al. (2011); Dyson, Jordan, and Glutting (2013); Fantuzzo, Gadsden, and
McDermott (2011); Jordan et al. (2012);
Klein et al. (2008); Monahan (2007);
Sarama et al. (2008); Sophian (2004).
Positive effects were seen in basic number concepts in Aunio, Hautamaki, and
Van Luit (2005); Clements and Sarama
(2007b); Clements et al. (2011); Curtis, Okamoto, and Weckbacher (2009);
Griffin, Case, and Capodilupo (1995)
and related publication Griffin, Case, and
Siegler (1994); Ramani and Siegler
(2008); Siegler and Ramani (2008);
Sood (2009). Positive effects were seen
in operations in Baroody, Eiland, and
Thompson (2009); Dyson, Jordan, and
Glutting (2013); Jordan et al. (2012);
Kidd et al. (2008); Lai, Baroody, and
Johnson (2008); Sood (2009). Positive
effects were seen in number recognition
in Sood (2009). Positive effects were
seen in patterns and classification in Sood
(2009). Positive effects were seen in geometry in Aunio, Hautamaki, and Van Luit
(2005); Clements and Sarama (2007b);
Clements et al. (2011).
176. In Curtis, Okamoto, and Weckbacher
(2009), a negative effect was found in
the basic number concepts domain: children who did not receive adult support
in counting scored higher than children
who did receive adult support in counting
during the balance-beam task with large
differences in weights. The authors note
that they did not expect the intervention
to favor either group for this outcome;
they expected all children to pass the
items, given their age and the targeted
age of the items. In Kidd et al. (2008),
negative effects were seen in basic number concepts, operations, and patterns
and classification when comparing the
numeracy instruction condition with
the cognitive instruction condition. The
authors expected negative effects in the
patterns and classification outcomes; the
outcomes are more closely aligned with
the cognitive instruction condition. The
authors suggest children in the cognitive
instruction condition also scored higher
on basic number concepts and operation
177.
178.
179.
180.
181.
182.
183.
( 141 )
Endnotes (continued)
184.
185.
186.
187.
188.
189.
190.
191.
192.
( 142 )
Endnotes (continued)
193.
194.
195.
196.
197.
198.
199.
200.
201.
202.
203.
204.
205.
206.
207.
208.
209.
210.
211.
212.
213.
214.
215.
216.
217.
218.
219.
220.
221.
222.
223.
( 143 )
Endnotes (continued)
224. Baroody, Eiland, and Thompson
(2009); Curtis, Okamoto, and Weckbacher (2009); Dyson, Jordan, and
Glutting (2013); Lai, Baroody, and
Johnson (2008); Jordan et al. (2012);
Kidd et al. (2008); Monahan (2007);
Ramani and Siegler (2008); Siegler
and Ramani (2008); Sood (2009);
Sophian (2004).
225. Dyson, Jordan, and Glutting (2013);
Jordan et al. (2012); Kidd et al.
(2008); Monahan (2007); Sood (2009);
Sophian (2004).
226. Dyson, Jordan, and Glutting (2013);
Jordan et al. (2012).
227. At maintenance (six weeks after the intervention), counting skills of the children
who participated in the intervention were
no longer significantly different from the
control group, as measured by the Number Sense Brief (NSB; Jordan et al., 2010)
and the Woodcock-Johnson, third edition
(WJ-III; Woodcock, McGrew, & Mather,
2007). All other effects were maintained.
228. Sood (2009).
229. Sood (2009). At the three-week follow-up,
effects were maintained in number relationships, five-and-ten-frame identification and
representation, five-and-ten-frame calculation, and nonverbal calculations. Effects
were not maintained for counting from
and number identification.
230. Monahan (2007).
231. Sophian (2004).
232. Kidd et al (2008).
233. Curtis, Okamoto, and Weckbacher
(2009).
234. Curtis, Okamoto, and Weckbacher
(2009) report a negative effect in the basic
number concepts domain, with children
who did not receive adult support in counting scoring higher than children who did
receive adult support in counting during the
balance-beam task with large differences
in weights. The authors note that they did
not expect the intervention to favor either
group for this outcome; they expected all
children to pass the items, given their age
and the targeted age of the items.
235. Baroody, Eiland, and Thompson (2009).
236. Lai, Baroody, and Johnson (2008).
237. Ramani and Siegler (2008); Siegler
and Ramani (2008).
238. Ramani and Siegler (2008).
239.
240.
241.
242.
243.
244.
245.
246.
247.
248.
249.
250.
( 144 )
Endnotes (continued)
251. Tools of the Mind, Building Blocks, LOGO,
Pre-K Mathematics Curriculum with Building Blocks, Pre-K Mathematics Curriculum
with DLM Early Childhood Express, Bright
Beginnings, and Creative Curriculum.
252. Building Blocks, EPIC, LOGO, Pre-K Mathematics Curriculum with Building Blocks,
Pre-K Mathematics Curriculum with
DLM Early Childhood Express, and two
researcher-developed curricula.
253. Building Blocks, EPIC, Bright Beginnings,
Creative Curriculum, Pre-K Mathematics Curriculum with Building Blocks, and
Pre-K Mathematics Curriculum with DLM
Early Childhood Express.
254. Table D.1 summarizes which studies are
linked to which recommendations.
255. Weaver (1991) offered supplemental
instruction in geometry, patterns, and
measurement and found positive effects
in the geometry domain. Sophian (2004)
offered targeted instruction in geometry
and measurement, while the comparison
group received a literacy intervention;
positive effects were found in the domain
of general numeracy. Kidd et al. (2008)
offered targeted instruction in oddity,
seriation and conservation, while the
comparison group received either an art
intervention or a numeracy intervention;
positive effects were found in the domains
of basic number concepts, operations,
and patterns and classification.
256. Barnett et al. (2008); Casey et al.
(2008); Clements and Sarama (2008);
PCER Consortium (2008, Chapter 2);
PCER Consortium (2008, Chapter 3);
Sarama et al. (2008).
257. Positive effects were found in the domains
of geometry (Casey et al., 2008) and
general numeracy (Clements & Sarama,
2008; Sarama et al., 2008).
258. No discernible effects were found in the
domains of geometry (Casey et al.,
2008; PCER Consortium, 2008, Chapter 2; PCER Consortium, 2008, Chapter 3), operations, (Barnett et al., 2008;
PCER Consortium, 2008, Chapter 2;
PCER Consortium, 2008, Chapter 3),
and general numeracy (PCER Consortium, 2008, Chapter 2; PCER Consortium, 2008, Chapter 3).
259. Clements and Sarama (2007b); Clements et al. (2011); Fantuzzo, Gadsden, and McDermott (2011); Klein et
260.
261.
262.
263.
264.
265.
266.
267.
268.
269.
270.
( 145 )
al. (2008).
Clements and Sarama (2007b); Clements et al. (2011).
Clements et al. (2011); Fantuzzo,
Gadsden, and McDermott (2011);
Klein et al. (2008).
Clements and Sarama (2007b); Clements et al. (2011).
Clements and Sarama (2008); Fantuzzo, Gadsden, and McDermott
(2011); Klein et al. (2008); Sarama et
al. (2008); Sophian (2004).
Both positive effects (Casey et al., 2008;
Clements & Sarama, 2007b; Clements
& Sarama, 2008; Weaver, 1991) and no
discernible effects (Casey et al., 2008;
PCER Consortium, 2008, Chapter 2;
PCER Consortium, 2008, Chapter 3)
were found in geometry. The panel reports
scale scores from Clements et al. (2011);
however, subscales were reported and
include positive effects for three of the five
geometry subscales, no discernible effects
for two of the five geometry subscales, and
positive effects for outcomes assessing
measurement and patterns and classification. Positive effects were found in patterns
and classification (Kidd et al., 2008).
Barnett et al. (2008); Casey et al.
(2008); Clements and Sarama (2007b);
Clements and Sarama (2008); Clements et al. (2011); Fantuzzo, Gadsden, and McDermott (2011); Klein et
al. (2008); PCER Consortium (2008,
Chapter 2); PCER Consortium (2008,
Chapter 3); Sarama et al. (2008);
Sophian (2004); Weaver (1991).
Barnett et al. (2008); Clements and
Sarama (2007b); Clements and Sarama
(2008); Clements et al. (2011); Fantuzzo, Gadsden, and McDermott (2011);
Klein et al. (2008); PCER Consortium
(2008, Chapter 2); PCER Consortium
(2008, Chapter 3); Sarama et al. (2008).
Casey et al. (2008); PCER Consortium
(2008, Chapter 2); Sophian (2004);
Weaver (1991).
Clements and Sarama (2007b); Clements and Sarama (2008).
Clements and Sarama (2007b); Clements and Sarama (2008); Clements
et al. (2011).
Sarama et al. (2008).
Endnotes (continued)
271. Klein et al. (2008).
272. The geometry scale of the Building Blocks
Assessment (Clements & Sarama, 2007c).
273. The total score on the Research-Based
Early Math Assessment (REMA; Clements,
Sarama, & Liu, 2008) or the Child Math
Assessment (CMA; Klein, Starkey, & Wakeley, 2000).
274. Weaver (1991). The publication also
assessed the impact of computer-based
LOGO compared with floor-based LOGO
for preschool children. This contrast is
not evidence for this recommendation,
as both groups of children used LOGO;
the study found no discernible effects for
computer-based LOGO.
275. Fantuzzo, Gadsden, and McDermott
(2011).
276. Sophian (2004).
277. Casey et al. (2008).
278. Casey et al. (2008) also reported no discernible effects and a single negative effect
in the study. The negative finding was
for a new assessment of mental rotation
ability. The regular classroom instruction
group scored higher on the assessment
at pretest and posttest. The authors suggested that the intervention experience,
which involved mentally rotating individual blocks, may not have transferred
to the mental rotation of more complex
figuresthe focus of the assessment.
279. Barnett et al. (2008); Clements and
Sarama (2007b); Clements and Sarama
(2008); Clements et al. (2011); Klein
et al. (2008); PCER Consortium (2008,
Chapter 2); PCER Consortium (2008,
Chapter 3); Sarama et al. (2008);
Weaver (1991).
280. Clements and Sarama (2007b); Clements and Sarama (2008); Clements
et al. (2011); Klein et al. (2008);
Sarama et al. (2008); Weaver (1991)
found positive effects in the domains of
general numeracy and geometry. Kidd et
al. (2008) found positive effects in basic
number concepts, operations, and patterns and classification. Barnett et al.
(2008) found no discernible effects in the
operations domain. No discernible effects
were reported in general numeracy, operations, and geometry for PCER Consortium (2008, Chapter 2); PCER Consortium (2008, Chapter 3).
281. Clements et al. (2011). The development of the Research-Based Early Math
Assessment (REMA) is discussed in Clements, Sarama, and Liu (2008).
282. Building Blocks was the focal intervention
in Clements and Sarama (2007b); Clements and Sarama (2008); Clements
et al. (2011). DLM Early Childhood
Express was combined with the Pre-K
Mathematics Curriculum in Klein et al.
(2008). Building Blocks was combined
with the Pre-K Mathematics Curriculum
in Sarama et al. (2008).
283. Barnett et al. (2008); Clements and
Sarama (2007b); Clements and Sarama
(2008); Clements et al. (2011); Klein
et al. (2008); PCER Consortium (2008,
Chapter 2); PCER Consortium (2008,
Chapter 3); Sarama et al. (2008);
Weaver (1991).
284. Building Blocks was examined in Clements
and Sarama (2007b); Clements and
Sarama (2008); Clements et al. (2011);
Sarama et al. (2008). Bright Beginnings
and Creative Curriculum were assessed in
PCER Consortium (2008, Chapter 2);
PCER Consortium (2008, Chapter 3).
EPIC was examined in Fantuzzo, Gadsden, and McDermott (2011). Pre-K
Mathematics Curriculum was studied
in Klein et al. (2011). Two researcherdeveloped curricula were examined in
Sophian (2004); Weaver (1991).
285. Building Blocks, Bright Beginnings, Creative Curriculum, EPIC, and the Pre-K
Mathematics Curriculum.
286. Building Blocks, Creative Curriculum, and
EPIC.
287. Positive effects were seen in general numeracy by Clements and Sarama (2008);
Clements et al. (2011); Fantuzzo,
Gadsden, and McDermott (2011); Klein
et al. (2011); Sarama et al. (2008). Positive effects were seen in geometry by Clements and Sarama (2007b); Clements
et al. (2011); Weaver (1991). Positive
effects were seen in basic number concepts
by Clements and Sarama (2007b); Clements et al. (2011). No discernible effects
were seen in general numeracy, operations,
and geometry by PCER Consortium
(2008, Chapter 2); PCER Consortium
(2008, Chapter 3).
288. Clements and Sarama (2007b); Clements and Sarama (2008); Clements et
( 146 )
Endnotes (continued)
289.
290.
291.
292.
293.
294.
295.
296.
297.
298.
299.
300.
301.
302.
303.
304.
305.
306.
307.
308.
309.
310.
311.
312.
313.
314.
315.
316.
317.
318.
319.
320.
321.
322.
323.
324.
( 147 )
Endnotes (continued)
325. For example, in Fantuzzo, Gadsden, and
McDermott (2011), the teachers in the
comparison condition used the High/Scope
Educational Research Foundations Preschool Child Observation Record, which is
a progress-monitoring tool. In other studies,
there was limited information on the comparison condition; thus, the panel is unsure
whether progress monitoring occurred (e.g.,
PCER Consortium, 2008, Chapter 3).
326. Clements and Sarama (2007b); Clements and Sarama (2008); Clements
et al. (2011); Sarama et al. (2008).
327. Clements and Sarama (2007b); Clements et al. (2011).
328. Clements and Sarama (2007b).
329. Clements et al. (2011).
330. Clements and Sarama (2008).
331. Clements, Sarama, and Liu (2008).
332. Sarama et al. (2008).
333. Klein et al. (2008).
334. The CMA was developed as described in
Klein, Starkey, and Wakeley (2000).
335. PCER Consortium (2008, Chapter 2).
336. PCER Consortium (2008, Chapter 3).
337. Dyson, Jordan, and Glutting (2013);
Fantuzzo, Gadsden, and McDermott
(2011); Jordan et al. (2012).
338. Dyson, Jordan, and Glutting (2013);
Jordan et al. (2012).
339. Fantuzzo, Gadsden, and McDermott
(2011).
340. Arnold et al. (2002); Griffin, Case,
and Capodilupo (1995) and related
publication Griffin, Case, and Siegler
(1994).
341. Arnold et al. (2002).
342. Griffin, Case, and Capodilupo (1995)
and related publication Griffin, Case,
and Siegler (1994).
343. Clements and Sarama (2007c).
344. Clements, Sarama, and Liu (2008).
345. Klein, Starkey, and Wakeley (2000).
346. Klein and Starkey (2002).
347. Jordan et al. (2010).
348. McDermott et al. (2009).
349. Woodcock, McGrew, and Mather (2007).
350. Arnold et al. (2002); Barnett et al.
(2008); Clements and Sarama (2007b);
Clements and Sarama (2008); Clements et al. (2011); Dyson, Jordan,
and Glutting (2013); Fantuzzo, Gads-
351.
352.
353.
354.
355.
356.
357.
( 148 )
Endnotes (continued)
358.
359.
360.
361.
362.
363.
364.
365.
366.
367.
368.
369.
370.
371.
372.
373.
374.
375.
376.
377.
378.
379.
380.
381.
( 149 )
Endnotes (continued)
382.
383.
384.
385.
386.
387.
388.
389.
390.
391.
392.
393.
394.
395.
396.
397.
(2011).
Clements and Sarama (2007c).
Klein, Starkey, and Wakeley (2000).
Klein and Starkey (2002).
Madden, Gardner, and Collins (1987).
Gardner et al. (1987).
Jordan et al. (2010).
Clements, Sarama, and Liu (2008).
Woodcock and Johnson (1990).
Woodcock, McGrew, and Mather (2007).
CTB/McGraw Hill (1990).
Arnold et al. (2002); Aunio, Hautamaki, and Van Luit (2005); Barnett
et al. (2008); Clements and Sarama
(2007b); Clements and Sarama (2008);
Clements et al. (2011); Dyson, Jordan,
and Glutting (2013); Fantuzzo, Gadsden, and McDermott (2011); Jordan et
al. (2012); Klein et al. (2008); Monahan (2007); PCER Consortium (2008,
Chapter 2); PCER Consortium (2008,
Chapter 3); Ramani and Siegler (2008);
Ramani and Siegler (2011); Sarama et
al. (2008); Siegler and Ramani (2008);
Siegler and Ramani (2009).
Griffin, Case, and Capodilupo (1995)
and related publication Griffin, Case,
and Siegler (1994); Sophian (2004).
Aunio, Hautamaki, and Van Luit
(2005); Clements and Sarama (2007b);
Clements et al. (2011); Griffin, Case,
and Capodilupo (1995) and related
publication Griffin, Case, and Siegler
(1994); Ramani and Siegler (2008);
Siegler and Ramani (2008); Siegler
and Ramani (2009).
Positive effects: Arnold et al. (2002); Clements and Sarama (2008); Clements et
al. (2011); Dyson, Jordan, and Glutting
(2013); Fantuzzo, Gadsden, and McDermott (2011); Jordan et al. (2012); Klein
et al. (2008); Monahan (2007); Sarama
et al. (2008); Sophian (2004). No discernible effects: Monahan (2007); PCER
Consortium (2008, Chapter 2); PCER
Consortium (2008, Chapter 3).
Positive effects: Ramani and Siegler
(2008). No discernible effects: Ramani
and Siegler (2011); Siegler and Ramani
(2009).
Positive effects: Dyson, Jordan, and
Glutting (2013); Jordan et al. (2012);
Ramani and Siegler (2011). No dis-
398.
399.
400.
401.
402.
403.
404.
405.
( 150 )
Endnotes (continued)
406.
407.
408.
409.
410.
411.
412.
413.
414.
415.
416.
417.
418.
419.
420.
421.
422.
423.
424.
425.
426.
427.
428.
429.
430.
431.
432.
433.
434.
435.
436.
437.
( 151 )
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