Interventions in Early Mathematics: Avoiding Pollution and Dilution

Abstract

Although specific interventions in early mathematics have been successful, few have been brought to scale successfully, especially across the challenging diversity of populations and contexts in the early childhood system in the United States. In this chapter, we analyze a theoretically based scale-up model for early mathematics that was designed to avoid the pollution and dilution that often plagues efforts to achieve broad success. We elaborate the theoretical framework by noting the junctures that are susceptible to dilution or pollution. Then we expatiate the model's guidelines to describe specifically how they were designed and implemented to mitigate pollution and dilution. Finally, we provide evidence regarding the success of these efforts.

Full text

CHAPTER THREE
Interventions in Early
Mathematics: Avoiding Pollution
and Dilution
Julie Sarama
1
, Douglas H. Clements
University of Denver, Denver, CO, United States
1
Corresponding author e-mail address: Julie.Sarama@du.edu
Contents
1. Background 96
2. The TRIAD Model 97
2.1 Theoretical Framework 97
2.2 The TRIAD Model’s 10 Guidelines 102
2.3 How the TRIAD Implementation Was Designed to Militate Against Pollution
and Dilution 105
3. Research Evaluations: Did the TRIAD Design Mitigate Dilution and Pollution? 110
3.1 Initial Instantiation and Evaluations of the TRIAD Model 111
3.2 Full-Scale Implementation and Evaluation of TRIAD 115
3.3 Fighting Dilution Over Time: TRIAD and Sustainability 118
4. Final Words 120
Acknowledgments 121
References 121
Abstract
Although specific interventions in early mathematics have been successful, few have
been brought to scale successfully, especially across the challenging diversity of
populations and contexts in the early childhood system in the United States. In this
chapter, we analyze a theoretically based scale-up model for early mathematics that
was designed to avoid the pollution and dilution that often plagues efforts to achieve
broad success. We elaborate the theoretical framework by noting the junctures that are
susceptible to dilution or pollution. Then we expatiate the model’s guidelines to
describe specifically how they were designed and implemented to mitigate pollution
and dilution. Finally, we provide evidence regarding the success of these efforts.
Although specific interventions in early mathematics have been successful
(Clements & Sarama, 2011), few have been brought to scale successfully,
especially across the challenging diversity of populations and contexts in
Advances in Child Development and Behavior, Volume 53 #2017 Elsevier Inc.
ISSN 0065-2407 All rights reserved.
http://dx.doi.org/10.1016/bs.acdb.2017.03.003
95

the early childhood system in the United States. In this chapter, we describe
a theoretically based scale-up model for early mathematics that has avoided
the pollution and dilution that often plagues efforts to achieve broad success
(Clements & Sarama, 2011).
1. BACKGROUND
Taking promising interventions to scale means implementing them
successfully in larger settings. Moreover, though, such scale-up goes beyond
larger numbers to confront the complexity of such an enterprise. For exam-
ple, there is an increase in not just the number of students and teachers, but
also in both the number of categories of stakeholders and the number of dif-
ferent and often conflicting perspectives they hold. Therefore, we define
scale-up as instantiation of an intervention in varied settings with diverse
populations, addressing the needs of multiple sociopolitical stakeholders,
to achieve satisfactory fidelity of implementation and, thus, intervention
goals. Although this already is a substantial challenge, consider the complex-
ity of addressing the perspectives, needs, and desires of the different catego-
ries of stakeholders, including parents, various community groups, the
professional teaching community, educational leadership in early childhood
and in mathematics (often these are distinct groups), and higher-level
administration (Sarama & Clements, 2013). Achieving an adequate fidelity
of implementation during implementation presents other challenges, such as
sufficient materials, technology, professional development, and in-class
support.
Also, interventions are not useful if they are not maintained after the ini-
tial thrust for the implementation. Thus, attention must be given to includ-
ing transfer to local ownership and sustainability. We define sustainability as
the length of time an innovation continues to be implemented with fidelity
(cf. Baker, 2007).
Implementation, fidelity, and sustainability are threatened by both pol-
lution and dilution. Pollution is the introduction of elements into the inter-
vention environment or teaching practices that vitiate or are inimical to the
intervention. Dilution is the gradual replacement of components of the inter-
vention with other aspects that do not fulfill the same role as the replaced
components.
The next section describes the theoretical framework we developed for
our research and development work on implementation and wider scale-up,
and the model we derived from it. After we define each of the 10 guidelines
96 Julie Sarama and Douglas H. Clements

that constitute the TRIAD model, we describe how they were designed to
mitigate the usual forces of pollution and dilution.
2. THE TRIAD MODEL
2.1 Theoretical Framework
Our theoretical framework (Sarama, Clements, Starkey, Klein, & Wakeley,
2008) is an elaboration of the Network of Influences framework (Sarama,
Clements, & Henry, 1998). We consider successful implementation of an
intervention at scale to involve multiple coordinated efforts to maintain
the integrity of the vision and practices of an innovation through increas-
ingly numerous and complex socially mediated filters.
2.1.1 Interactions
The TRIAD model goes beyond solely adopting new curricula. Instead, it
scales up the support of “interactions among teachers and children around
educational material” (Ball & Cohen, 1999, p. 3). This strategy creates
extensive opportunities for teachers to focus on math, goals, and children’s
thinking and learning, which improves teachers’ knowledge of subject mat-
ter, teaching, and learning, and increases child achievement (Ball & Cohen,
1999; Cohen, 1996, p. 98; Schoen, Cebulla, Finn, & Fi, 2003; Sowder,
2007). The depiction of the Network of Influences framework in Fig. 1
illustrates the hypothesized influences of context and implementation vari-
ables on outcomes such as teacher knowledge, child achievement, and
sustainability.
2.1.2 Administrators and Other School Leaders (Fig. 1, Factors K and I)
Principal leadership is strongly correlated with levels of implementation
and effectiveness (Berends, Kirby, Naftel, & McKelvey, 2001; Bodilly,
1998; Bryk, Sebring, Allensworth, Suppescu, & Easton, 2010; Fullan,
1992; Heck, Weiss, Boyd, & Howard, 2002; Kaser, Bourexis, Loucks-
Horsley, & Raizen, 1999; Klingner, Ahwee, Pilonieta, & Menendez,
2003; Teddlie & Stringfield, 1993). Effective administrators provide
the time for teachers to experiment, discuss, and, in general, construct
their own meanings of the innovation. They communicate continuing
commitment, not just in verbal form, but also in other ways, such as
resource allocation (Bodilly, 1998). School mathematics and early child-
hood leaders can serve as essential bridges between administrators and
teachers (Cobb, McClain, de Silva, & Dean, 2003). District level leaders
97Implementation of TRIAD

and their decisions also impact implementation and ultimately child
achievement (Bodilly, 1998; Klingner et al., 2003; Snipes, Doolittle, &
Herlihy, 2002; Spillane, 2000). Communication between principals
and all other groups is particularly essential. Principals forgetting the study
and their involvement in it and similar communication lapses can be a
huge challenge, which increases with the size of the district (Foorman,
Santi, & Berger, 2007).
2.1.3 Communication
Communication, collaboration, and agreement among all groups are essential.
Our own previous research revealed multiple missed opportunities for facili-
tation of innovations due to the divergent beliefs of social groups, even about
ostensibly observable “facts,” such as “there is adequate technology available”
(Sarama et al., 1998).
Fig. 1 Revised Network of Influences theoretical framework elaborated with junctures
susceptible to dilution or pollution. The shaded text such as “D1”or “P2”refer to hypoth-
esized threats (D ¼dilution, P ¼pollution) to implementation addressed by guidelines
(#1, #2, etc.) of the TRIAD model. The “Follow Though”model at the bottom right is sim-
ple a microcosm of the framework. Contextual variables in dotted ovals include the
school (A–D), teacher (E), and child (FdH) factors from the revised Network of Influ-
ences framework. For example, child socioeconomic status, or SES (G), impacts chil-
dren’s initial mathematics knowledge (H), which influences children’s achievement
(R)—an outcome variable indicated by the solid rectangle. Implementation variables
in solid ovals are features that the project can encourage and support, but cannot con-
trol absolutely. For example, heavy arrows from professional development (J), to teacher
knowledge (N), to implementation fidelity (O), to child achievement (R), indicate the
strong effects in that path. Support from coaches (L) also has a strong effect on imple-
mentation fidelity, while other factors (J, K, and M) are influential, but to a moderate
degree (not all small effects are depicted).
98 Julie Sarama and Douglas H. Clements

2.1.4 Teachers and Professional Development (Fig. 1, Factors E,
N, and Q)
Research suggests that the most critical feature of a high-quality educational
environment is a knowledgeable and responsive adult and that high-quality
professional development is essential to innovation (Fig. 1J and N; Darling-
Hammond, 1997; Ferguson, 1991; National Research Council, 2007;
Sarama & DiBiase, 2004; Schoen et al., 2003; Sowder, 2007). Scaling up
such professional development has special challenges and opportunities
given the early childhood setting, as noted previously. Our model specifies
only a weak effect of initial teacher expertise (Fig. 1, dotted oval E) because
of the low level of mathematics content and pedagogical content knowledge
of most pre-K teachers, regardless of background (Copley, 2004; Sarama,
2002; Sarama & DiBiase, 2004), which is consistent with previous research
(e.g., Bryk et al., 2010). Changes in beliefs follow changes in practice (boxes
N, Q, Showers, Joyce, & Bennett, 1987); moreover, we believe that changes
in beliefs help sustain teacher practices (Fig. 1, boxes Q and S).
Research-based solutions to these challenges can be found (Corcoran,
2007; Klingner et al., 2003; NAEYC, 2002; National Research Council,
2007; Peisner-Feinberg et al., 1999; Sarama, 2002; US Department of
Education, 1999). Use of theory, demonstrations, practice, and feedback,
especially from coaches, quadruples the positive effects of information-only
training (e.g., strong effects from J and L to N and O in Fig. 1,Foorman
et al., 2007;Pellegrino, 2007;Showers et al., 1987). Effective professional
development eschews “one-shot” interventions, begins with a specific cur-
riculum (Fig. 1M), embodies the kind of flexible, interactive teaching styles
that work well with children, and weaves together math content, pedagogy,
and knowledge of child development and family relationships (Baroody &
Coslick, 1998; Schoen et al., 2003; Sowder, 2007). The professional devel-
opment in TRIAD provides a promising path for developing teachers’
understanding of learning, teaching, curriculum, and assessment by focusing
on research-based models of children’s thinking and learning. Research
indicates the efficacy of such cognitive models (Bredekamp, 2004;
Carpenter & Franke, 2004; Hiebert, 1999; Klingner et al., 2003), especially
compared with other approaches such as process–product models (Lawless &
Pellegrino, 2007). Research-based learning trajectories are TRIAD’s core
(Clements, Sarama, & DiBiase, 2003). We defined learning trajectories as
“descriptions of children’s thinking and learning…and a related, conjectured
route through a set of instructional tasks” (Clements & Sarama, 2004, p. 83).
Thus, learning trajectories have three components: a goal (that is, an aspect
99Implementation of TRIAD

of a mathematical domain children should learn), a developmental progres-
sion, or learning path through which children move through levels of think-
ing, and instruction that helps them move along that path (Fig. 2 provides an
example). Learning trajectories are at the heart of both TRIAD’s math cur-
riculum and its professional development. Learning trajectories help teachers
Age (years)
e n
n
cs
s
s
s
g
n
sn
Developmental progression Sample instructional tasks
Fig. 2 A learning trajectory for recognition of number and subitizing.
100 Julie Sarama and Douglas H. Clements

s
sfa s
fa s
d
s
s
m
pv
sc
wp v
2nd
Fig. 2—Cont’d
101Implementation of TRIAD

focus on the “conceptual storyline” of the curriculum, a critical element that
is often missed (Heck et al., 2002; Weiss, 2002). They facilitate teachers’
learning about mathematics, how children think about and learn this math,
and how such learning is supported by the curriculum and its teaching strat-
egies. They address domain-specific components of learning and teaching
that have the strongest impact on cognitive outcomes (Lawless &
Pellegrino, 2007). By illuminating potential developmental progressions,
they bring coherence and consistency to goals, curricula, and assessments
(for a comprehensive description and review of research on these learning
trajectories, see Sarama and Clements (2009a), and for the instructional
component).
2.1.5 Children and Their Families (Fig. 1, Factors F, G, and P)
Interventions are more effective if they involve parents, especially by pro-
viding activities to do with their children (Bryk et al., 2010; Galindo &
Sonnenschein, 2015; Halpern, 2004; Ramey & Ramey, 1998). As with
pre-K teachers, parents have a limited view of the breadth of math appro-
priate for young children (Sarama, 2002). Low-income parents, compared
to middle-income parents, believe that math education is the responsibility
of the preschool and that children cannot learn aspects of math that research
indicates they can learn (Starkey et al., 1999).
2.1.6 Resources, Curriculum, and Technology
The print curriculum, research-based learning trajectories that link develop-
mental progressions to connected instruction, manipulatives, supporting
manuals, the Building Blocks software activities and management system,
and the professional development website, all components of the mathemat-
ics intervention, provided a central focus around which the reform revolved.
Scale-up is more likely to be successful if resources are targeted to
implementing a single curriculum well (Snipes et al., 2002).
2.2 The TRIAD Model’s 10 Guidelines
Building on our theoretical framework and review of research, we created a
scale-up model for scale-up called TRIAD (technology-enhanced, research-
based, instruction, assessment, and professional development, Sarama et al.,
2008). The TRIAD model scales up the support of “interactions among
teachers and children around educational material” (Ball & Cohen, 1999,
p. 3). This strategy creates extensive opportunities for teachers to focus on
mathematics, goals, and students’ thinking and learning, which improves
102 Julie Sarama and Douglas H. Clements

teachers’ knowledge of subject matter, teaching, and learning, and increases
student achievement (Ball & Cohen, 1999; Cohen, 1996, p. 98; Schoen et al.,
2003; Sowder, 2007). This section summarizes 10 guidelines we followed
in creating the TRIAD intervention (Sarama et al., 2008) and the following
section describes how each was expected to minimize pollution and dilution.
1. Involve, and promote communication among, key groups around a shared vision
of the innovation (Hall & Hord, 2001). Emphasize connections between
the project’s goals, national and state standards, and greater societal
need. Promote clarity of these goals and of all participants’ responsibil-
ities. School and project staff must share goals and a vision of the inter-
vention (Bryk et al., 2010; Cobb et al., 2003).
2. Promote equity through equitable recruitment and selection of partici-
pants, allocation of resources, and use of curriculum and instructional
strategies that have demonstrated success with underrepresented
populations (Kaser et al., 1999).
3. Plan for the long term.Recognizing that scale-up is not just an increase in
number, but also of complexity, provide continuous, adaptive support
over an extended time. Communicate clearly that change is not an
event, but a process (Hall & Hord, 2001).
4. Focus on instructional change that promotes depth of children’s thinking, placing
learning trajectories at the core of the teacher/child/curriculum triad to
ensure that curriculum, materials, instructional strategies, and assessments
are aligned with national and state standards and a vision of high-quality
math education, each other, (c) “bestpractice”as determined by research,
especially formative assessment (Ball & Cohen, 1999; Bodilly, 1998;
Bryk et al., 2010; Fullan, 2000; Kaser et al., 1999; National
Mathematics Advisory Panel, 2008; Raudenbush, 2008; Sowder, 2007).
5. Build expectations and camaraderie to support a consensus around adaptation.
Promote “buy-in” in multiple ways, such as dealing with all participants
as equal partners and distributing resources to support the project. Estab-
lish and maintain cohort groups. Facilitate teachers visiting successful
implementation sites. Build local leadership by involving principals
and encouraging teachers to become teacher leaders (Berends et al.,
2001; Borman, Hewes, Overman, & Brown, 2003; Elmore, 1996b;
Fullan, 2000; Glennan, Bodilly, Galegher, & Kerr, 2004; Hall &
Hord, 2001.)
6. Provide professional development that is ongoing, intentional, reflective, focused
on mathematics content knowledge and children’s thinking, grounded in partic-
ular curriculum materials, situated in the classroom and the school. Encourage
103Implementation of TRIAD

sharing, risk taking, and learning from and with peers. Aim at preparing
to teach a specific curriculum and develop teachers’ knowledge and
beliefs that the curriculum is appropriate and its goals are valued and
attainable. Situate work in the classroom, formatively evaluating
teachers’ fidelity of implementation and providing feedback and sup-
port from coaches in real time (Bodilly, 1998; Borman et al., 2003;
Bryk et al., 2010; Cohen, 1996; Elmore, 1996b; Garet, Porter,
Desimone, Birman, & Yoon, 2001; Guskey, 2000; Hall & Hord,
2001; Kaser et al., 1999; Klingner et al., 2003; Pellegrino, 2007;
Schoen et al., 2003; Showers et al., 1987; Sowder, 2007). This also pro-
vides a common language for teachers in working with each other and
other groups (Bryk et al., 2010).
7. Ensure school leaders are a central force supporting the innovation and provide
teachers continuous feedback that children are learning what they are taught and
that these learnings are valued. Leaders, especially principals, must show
that the innovation is a high priority, through statements, resources,
and continued commitment to permanency of the effort. An innova-
tion champion leads the effort within each organization (Bodilly, 1998;
Bryk et al., 2010; Glennan et al., 2004; Hall & Hord, 2001; Rogers,
2003, p. 434; Sarama et al., 1998).
8. Give latitude for adaptation to teachers and schools, but maintain integrity.
Emphasize the similarities of the curriculum with sound practice and
what teachers already are doing. Discourage uncoordinated innovations
(Fullan, 2000; Huberman, 1992; Sarama et al., 1998; Snipes et al., 2002).
9. Provide incentives for all participants, including intrinsic and extrinsic motivators
linked to project work, such as external expectations—from standards to
validation from administrators. Show how the innovation is advanta-
geous to and compatible with teachers’ experiences and needs
(Berends et al., 2001; Borman et al., 2003; Cohen, 1996; Darling-
Hammond, 1996; Elmore, 1996a; Mohrman & Lawler III, 1996;
Rogers, 2003).
10. Maintain frequent, repeated communication, assessment (checking up), and
follow-through efforts emphasizing the purpose, expectations, and visions
of the project, and involve key groups in continual improvement
through cycles of data collection and problem solving (Fullan, 1992;
Hall & Hord, 2001; Huberman, 1992; Kaser et al., 1999; Snipes
et al., 2002). Throughout, connections with parents and community
groups are especially important, to meet immediate and long-range
(sustainability) goals.
104 Julie Sarama and Douglas H. Clements

2.3 How the TRIAD Implementation Was Designed to Militate
Against Pollution and Dilution
This section addresses the TRIAD model regarding pollution and dilu-
tion. That is, for each of these 10 guidelines constituting the model,
we describe the sources of possible pollution or dilution and then describe
how the guideline was expected to minimize the threat to high-quality
implementation.
1. Involve, and promote communication among, key groups around a
shared vision of the innovation. This guideline jumpstarts and later
institutionalizes the intervention; for example, by establishing a com-
mon understanding of the goals and how they are consistent with state
or district standards and by planning for the ongoing socialization and
training of new teachers (Elmore, 1996b; Fullan, 2000; Huberman,
1992; Kaser et al., 1999; Klingner et al., 2003; Sarama et al., 1998).
To begin, the TRIAD model includes initial meetings with adminis-
tration as well as teachers, including a presentation with a brochure
and a question and answer session. Planning for this meeting includes
preliminary discussions with key staff to ensure that the relationships
between the district’s goals and other instructional guidelines and those
of the intervention can be explicated and defended. After the meeting,
presenters emphasize that to move forward, all levels of administration
must agree to participate. (Ideally, at least 80% of teachers, as well as
parents, would agree to participate. However, for the purposes of gen-
eralizability, we did not ask teachers to sign consent forms in the largest
study described here. This way we could generalize our findings to dis-
tricts that used that model but had reluctant or hostile teachers.)
Once agreements are obtained and the project started, ongoing
planning with all key groups that minimizes dilution (see D1’s in
Fig. 1) includes specifics around the intervention, including clear, writ-
ten specifications concerning the time and supports needed to imple-
ment (e.g., all teachers and paraprofessionals are scheduled to attend
all trainings; sufficient resources are available in every classroom, from
blocks to computers; admin). Furthermore, dilution may occur due to
rapid turnover of preschool teachers, introducing an increasing propor-
tion of teachers who are not involved in the initial implementation.
Also, pollution (see P1’s in Fig. 1) can occur if new (and also old)
teachers introduce inconsistent elements into their practice. Therefore,
plans are made for training and socializing new teachers, including a
mentor system and including district supervisors and coaches in the
105Implementation of TRIAD

project’s training sessions so they would be well versed in introducing
the intervention to any new teachers and to monitoring and supporting
all teachers past the end of the project.
2. Promote equity. This ensures that every teacher has the support, both
physical and personnel, necessary to teach and continue to teach the
intervention properly. Without this, both pollution and dilution are
likely (see D2’s and P2 in Fig. 1), especially given the challenges
present-day teachers face in implementing mathematics interventions.
That is, without comprehensive curricular resources, training, and
coaching, fidelity is unlikely to be achieved and, maintained; unfortu-
nately, inequitable distribution of resources frequently plagues pre-
schools, especially those serving children from lower SES
communities. We worked with administrators to ensure all resources
were provided to each teacher and coaches filled in for any missing
materials throughout the implementation period.
3. Plan for the long term. The TRIAD professional development model
includes regular professional development meetings, bi-weekly
coaching visits, and peer coaching. Year 1 professional development
is a “gentle introduction,” in that child data are not collected and
coaches have time to slowly get teachers comfortable with all aspects
of the intervention. “One-shot” interventions may have a temporary
effect, but frequently are diluted or even abandoned soon thereafter
(Bodilly, 1998; Cuban & Usdan, 2003). TRIAD uses a dynamic, mul-
tilevel, feedback, and self-corrective strategy (Bryk et al., 2010;
Coburn, 2003; Guskey, 2000). That is, all work with teachers is
designed so that data about teachers’ understandings and engagement
is collected in real time and used to alter plans for the following sessions.
Interviews with principals occur throughout the project, to ensure they
not only remember the project (Foorman et al., 2007) but provide
feedback on what is progressing successfully and what is not in their
building (see guideline #1). In addition, administrators are provided
with tools to offer additional support to teachers in their building
and fulfill district requirements (to monitor instructional time and qual-
ity of instruction).
4. Focus on instructional change that promotes depth of children’s think-
ing, placing learning trajectories at the core of the teacher/child/cur-
riculum. This is the core of the TRIAD approach so ensuring this
guideline is critical. Implemented well, this focus helps avoid pollution,
mainly, by ensuring fidelity by helping teachers not only use the
106 Julie Sarama and Douglas H. Clements

effective instructional strategy of formative assessment (Penuel &
Shepard, 2016) but also by helping teacher understand not only how
to teach a given activity, but why—in other words, connecting their
growing knowledge of how children think about and learning mathe-
matics to the characteristics of a given activity that promote this learn-
ing. With an integrated understanding of the mathematics, children’s
developmental progressions, and instruction, teachers are less likely
to introduce lethal mutations, changes to the core math content, or
pedagogy that weakens the activity. This focus also prevents dilution,
however, because teachers who observe and understand the depth of
children’s thinking and learning given these activities are more likely
to continue to use the program with fidelity over years (Clements,
Sarama, Wolfe, & Spitler, 2015).
5. Build expectations and camaraderie to support a consensus around
adaptation. Dilution is less likely when teachers are part of a
learning–teaching community. In TRIAD, teachers are intermingled
(across schools) in small group work, but they also work in school clus-
ters for other small groups. In this way, teachers are exposed to ideas
from peers at other sites but had adequate time to address school specific
concerns and address implementation as a team.
6. Provide professional development that is ongoing, intentional, reflec-
tive, focused on mathematics content knowledge and children’s think-
ing, grounded in particular curriculum materials, situated in the
classroom and the school. As guideline #4 is a main focus, this guideline
is TRIAD’s main leverage point—sine qua non of a high-quality
implementation. That is, without well-trained and committed teachers,
dilution (to the degree of no implementation) and/or pollution
(low fidelity) is highly likely. TRAID’s professional-development
model includes both repeated (e.g., >10) full-day sessions of training
in regular meetings and frequent coaching. Training includes all three
components of each learning trajectory, the goal, the developmental pro-
gression, and the instructional activities and strategies (Fig. 2). All three
are critical. To understand the goal, teachers study the mathematical con-
tent. Without knowledge of the mathematical content, pollution is
almost assured. For example, teachers may believe that any series of colors
in fabric is “a pattern” (Fox, 2005) or that “two triangles put together
always make a square” (Thomas, 1982). In TRIAD, teachers learn core
mathematics concepts and procedures for each topic. For example, they
study the system of verbal counting based on cycling through 10 digits
107Implementation of TRIAD

and the concept of place value (i.e., content like that presented in
National Research Council, 2009; Wu, 2011).
A key instructional use of learning trajectories is in formative ass-
essment, an empirically supported pedagogical strategy (National
Mathematics Advisory Panel, 2008; Penuel & Shepard, 2016). How-
ever, dilution is probable if teachers cannot accurately assess children’s
developmental level or select appropriate tasks and instructional strat-
egies based on children’s levels—the second and third components of a
learning trajectory. To learn about the developmental progression,
teachers analyze multiple video segments illustrating each level and dis-
cuss the mental “actions on objects” that constitute the defining cog-
nitive components of each level; order tasks corresponding to those
levels; and practice diagnosis in teams, with a couple of teachers exem-
plifying behaviors of children at different levels, and one teacher iden-
tifying the level of each. Furthermore, teachers need training in
understanding, administering, and especially using data from new
assessment strategies (Foorman et al., 2007). TRIAD training focuses
mainly on the curriculum-embedded assessment of Building Blocks’
“Small Group Record Sheets,” but also provides teachers with tasks
appropriate for clinical interviews.
Formative assessment requires more than identifying children’s
levels of thinking. Teachers must select and modify instructional activ-
ities and strategies that are appropriate and effective for each level. To
learn about instructional tasks and strategies, teachers practice the cur-
riculum’s activities, but also analyze them to establish and justify their
connection to a particular level of the developmental progression. They
extend the team practice sessions to use their diagnoses to select and to
modify activities to match instructional tasks to developmental levels of
individuals.
Across all forms of professional development, the focus is on chil-
dren’s thinking and learning. Conversations about children learning
serve as way to address implementation issues. Although early mathe-
matics is often an uncomfortable topic for early childhood educators,
the newness of the learning trajectories for all participants helps establish
a sense of shared learning and community. Each session in the last third
of professional development includes scheduled time to discuss
“learning stories” (Perry, Perry, Dockett, & Harley, 2007). Teachers
show their record keeping on small group record sheets, and sometimes
108 Julie Sarama and Douglas H. Clements

videos, and discuss their use of learning trajectories in teaching children,
including challenges, questions, and successes. These discussions pro-
mote peer learning and collaboration and also motivate peers to solve
implementation difficulties.
Finally, we do not train the teachers and immediately begin evalu-
ating the project. Instead, an entire year is spent in professional devel-
opment and implementation without evaluations as a “gentle
introduction” without the pressure of assessments.
7. Ensure school leaders are a central force supporting the innovation and
provide teachers continuous feedback that children are learning what
they are taught and that these learnings are valued. School leaders deal
with myriad issues and problems and a single intervention can easily
lose their attention (Foorman et al., 2007), vitiating the support
teachers need from their leaders. In TRIAD, communication is initially
established (see guideline #1) and is not maintained to ensure that
school leaders provide support, encouragement, and feedback to
teachers. Principals receive, and are offered training on, a “walk-
through” fidelity instrument, which when used, heightened teachers’
perception of the importance the leader assigned to the intervention.
Leaders also include supervisors of early childhood education and
mathematics education. Coaches from both these fields are invited to
all trainings. All leaders are taught about the structure and research
of the intervention, to avoid dilution and pollution that can occur
when leaders tell teachers they are “professionals who can choose to
include any activity from any source.”
8. Give latitude for adaptation to teachers and schools, but maintain integrity.
Adaptation to local contexts can be important, as can adaptation to chil-
dren’s interests and backgrounds. However, it is easy for teachers to mis-
takenly dilute or pollute instructional activities in these attempts. TRIAD
training addresses this issue by explicitly comparing productive adaptations
to lethal mutations (Brown & Campione, 1996). For example, teachers
learn to change surface features (e.g., tofitaclass“theme”)withoutaltering
the core mathematical or pedagogical components of the instructional
activities (Winter & Szulanski, 2001).Theylearntomodifyanactivity
to fit children’s interests and needs without (permanently) reducing the
cognitive demand of instructional tasks after their initial introduction
(Stein, Brover, & Hennigsen, 1996). The best productive adaptations
are then shared among the sites and added to future training sessions.
109Implementation of TRIAD

From a school perspective, our fidelity form, including the “walk-
through” version for principals, includes pacing guidelines. However,
these forms are not based on compliance alone. That is, the emphasis is
on integrity and pacing, but with latitude to make changes. For example,
higher, not lower, scores are given to teachers who modify the schedule
following children’s development and learning. Similarly, higher scores
are given for modifying activities, but only for productive adaptations.
9. Provide incentives for all participants, including intrinsic and extrinsic
motivators linked to project work. TRIAD implementations connect
the intervention to the standards and school reports teachers use. Early
successes, even anecdotal, with teachers and children are shared with
principals so they remain committed and pass on the complements
to their teachers. Discussing the small group record sheets and sharing
the learning stories not only motivate those implementing, but also
subtly pressure nonimplementers to “up their game.” Training sessions
are designed to be happy and productive, providing opportunities to
work with peers, to discuss education over good food, and generally
to be treated as professionals. For example, they are asked to share their
ideas for productive adaptations and trainers are serious about using
feedback from them to trainings.
10. Maintain frequent, repeated communication, assessment (checking
up), and follow-through efforts. Of course, assuming that simple intro-
duction and training will lead to complete and continued implemen-
tation is invalid. Throughout the TRIAD implementation, project
staff maintained connections with not only teachers and leaders, as
described previously, but also with parents and local media.
3. RESEARCH EVALUATIONS: DID THE TRIAD DESIGN
MITIGATE DILUTION AND POLLUTION?
Our colleagues and we have conducted several studies to evaluate
instantiations of the TRIAD model and its components (Clements &
Sarama, 2008a, 2008b; Clements, Sarama, Spitler, Lange, & Wolfe, 2011;
Clements, Sarama, Wolfe, & Spitler, 2013; Sarama & Clements, 2009b;
Sarama et al., 2008). Most of these studies used cluster randomized trials
designs. Although financial and logistical constraints did not allow data col-
lection that would directly address every source of dilution and pollution
identified in Fig. 1 and in the previous section, successful outcomes do
provide evidence that these challenges were addressed successfully.
110 Julie Sarama and Douglas H. Clements

3.1 Initial Instantiation and Evaluations of the
TRIAD Model
The first evaluations were limited, designed to complete procedures and
assessments and serve as a proof of concept that the model could be effec-
tive. In the first, we randomly assigned 25 preschool classrooms serving
children at-risk for later school failure to either TRIAD or control groups
(Sarama et al., 2008), including about half Head Start and half public
preschool programs. This was successful so we implemented it more
completely with 36 teachers, which we describe here (Clements &
Sarama, 2008a).
3.1.1 Implementation
We attempted to implement most components of the TRIAD model (some,
such as planning for the long term, guideline #3, and full realization of #10,
were not possible in this limited scale-up). For example, we began by meet-
ing with administrators and teachers to encourage them to have a shared
understanding of the goals of the intervention (guideline #1). We pro-
vided incentives in the form of rationales for the intervention and the pro-
vision of curriculum materials and especially professional development
(guideline #9).
To promote equity (guideline #2), the TRIAD intervention used the
Building Blocks curriculum, which had been empirically validated in class-
rooms serving underrepresented populations (Clements & Sarama, 2007,
2008a). Every classroom received all components and, thanks to the support
of administration, (guidelines #1 and #7) adequate auxiliary materials
such as computers. All teachers were provided with the same professional
development (see guideline #6).
We describe this professional development in detail because, like the
measures, it shared most characteristics with similar work in all future
studies of TRIAD (guidelines #4 and #6). The first year was a pilot/train-
ing year, because our previous experience and others’ research suggested
that teachers often need at least a year of experience before completely
and effectively implementing a curriculum (Berends et al., 2001;
Clements & Sarama, 2014; Cobb et al., 2003). Thus, teachers participated
in 8 days of professional development during the school day in the
first year (a “no stress, gentle introduction” to the curriculum with no
data collection by researchers). Introductory discussions emphasized
the “developmental appropriateness” of the intervention’s mathematics
111Implementation of TRIAD

education and its importance to the teachers and children (guideline #5),
especially in promoting equity (guideline #2). This work focused on the
learning trajectories for each mathematical topic, usually as woven into
the Building Blocks curriculum (guideline #4). Training addressed each of
the three components of the learning trajectories. To understand the goals,
teachers learned core mathematics concepts and procedures for each topic.
For example, they studied the system of verbal counting based on cycling
through 10 digits and the concept of place value (based on content like that
in National Research Council, 2009).
To understand the developmental progressions of levels of thinking,
teachers studied multiple video segments illustrating each level and discussed
the mental “actions on objects” that constitute the defining cognitive com-
ponents of each level. To understand the instructional tasks, teachers studied
the tasks, and they viewed, analyzed, and discussed video of the enactments
of these tasks in classrooms. A central tool to study and to connect all three
components was the internet-based software application, Building Blocks
Learning Trajectories (BBLT, see Fig. 3). Throughout the 2 years, teachers
study the BBLT and engaged in discussions of these videos and correlated
text from the Building Blocks curriculum. BBLT provided scalable access
to the learning trajectories via descriptions, videos, and commentaries.
Two sequential aspects of the learningtrajectories—thedevelopmental
progressions of children’s thinking and connected instruction—are linked
to the others (see Fig. 2). As an example, using these approaches, teachers
studied the learning trajectory for counting. Initial presentations including
viewing the levels of thinking that constitute the developmental progres-
sion, using the BBLT, and discussing the attributes, both mathematical
and psychological, of each level. As an illustration, these included the
cardinal principle (the last counting word “tells how many”), which is
often underemphasized in curricula and teaching; and, at higher levels,
competencies in counting on, and other counting strategies. Teachers
used the “Test Yourself” feature of the BBLT to evaluate their abilities
to diagnose children’s level of counting (by identifying the level evinced
by children in randomly selected videos). They also used the BBLT’s links
to view research-based instructional strategies to promote children’s
progress to the next level. Teachers worked in small groups to plan
how activities from their curriculum might promote, or be modified to
promote, learning for the relevant levels. The coaches joined the teachers
in the participated in professional development, as well as several days
of training on coaching, most of which focused on the unique aspects
112 Julie Sarama and Douglas H. Clements

of coaching early mathematics education. Coaches worked with teachers
during the year to avoid dilution of the intervention and to provide
teachers with continual feedback and support.
Trainers and coaches also promoted productive adaptations and avoidance
of lethal mutations (guideline #8). As an example, for board games, produc-
tive adaptations could include changing the die (from 1–6dotsto1–3dotsfor
children who are at a lower level, or 5–10 dots or even two cubes for children
at higher levels). Such adaptations could simply be changing the theme of the
board and tokens to match a class theme (e.g., a trip through a farm and farm
Fig. 3 Building Blocks Learning Trajectory (BBLT) web application.
113Implementation of TRIAD

animals). In contrast, lethal mutations include changing the game to Candy
Land or other games that use a spinner with colors where children just
advance to the next instance of that color, eliminating number from the activ-
ity. In subsequent sessions, teachers shared and kept lists of their own produc-
tive adaptations and discussed how they avoided lethal mutations.
In year 2, teachers and coaches participated in an additional 5 days of pro-
fessional development (guidelines #6 and #10). They continued to study the
learning trajectories, including discussions of how they conducted various
curricular activities the previous year. As part of this work, teachers brought
case studies of particular situations that occurred in their classrooms to the
group to facilitate these discussions; thus, this work included elements of les-
son study.
3.1.2 Findings
TRIAD children made significantly and substantially greater gains in math-
ematics achievement than the control children (effect size ¼1.07). There
was no evidence that contextual variables influenced these positive effects,
such as the Head Start or public programs. Similarly, there was no evidence
that effects differed by children’s ethnicity or gender. This provides evidence
that TRIAD’s guidelines worked as a whole, and those addressing equity
were successful in particular.
The two observational measures provide additional evidence that greater
scores in achievement by the TRIAD group resulted from the implemen-
tation of the guidelines. Findings on the fidelity instrument indicate that the
teachers implemented the curriculum with “good” fidelity (that is, on a
5-point Likert scales from “strongly disagree” to from “strongly agree”
for each item evaluating the quality of teaching, the average was
“agree”). This provides support for guidelines especially #4, #5, #6, and
#8 as teachers as a group implemented the intervention with high-quality
fidelity. In addition, the higher the teacher’s fidelity score, the higher the
average child gain in achievement in her classroom (although small variance
resulted in lack of statistical significance). Furthermore, the TRIAD inter-
vention resulted in consistently greater scores on the quality and quantity of
the mathematics environments and teaching in the TRIAD, compared to
control, classes. TRIAD classrooms had higher average scores on general
classroom behaviors such as teaching more mathematics, showing knowl-
edge of, enjoyment in, and enthusiasm for mathematics, and using effective
management and instructional strategies. In summary, the intervention
increased the quantity and quality of the mathematics environment and
114 Julie Sarama and Douglas H. Clements

teaching in preschool classrooms, providing empirical support for the guide-
lines, especially #4, #5, #6, and #9.
3.2 Full-Scale Implementation and Evaluation of TRIAD
Our largest implementation and evaluation to date involved 1375 pre-
schoolers in 106 classrooms serving low-resource communities in two states.
Schools in two large urban cities in two states were randomly assigned to
three treatment groups: TRIAD, TRIAD with Follow Through
(TRIAD-FT), and control (business as usual—in both cities, this involved
standards, curriculum, and expectations in preschool mathematics). The
TRIAD-FT differed from the TRIAD implementation only after the pre-
school year, so they are combined for analyses reported here.
3.2.1 Implementation
This implementation was also the most valid and therefore generalizable
because the administration of both school systems agreed to the implemen-
tation across their districts. Therefore, we could both randomly select and
randomly assign schools (with small exceptions—schools that had worked
with us in previous studies were not included in the selection pool). Further,
teachers were not volunteers—a common limitation in many such studies,
especially as math is often not the favorite subject of preschool teachers
(Copley, 2004; Sarama & DiBiase, 2004).
We attempted a complete implementation of the TRIAD-based inter-
vention in two school districts. We again began with extensive explanations
of the intervention, including the rationale for it and the particulars of it,
to broad audiences of administrators, teachers, and parents (guideline #1).
In this effort, we include planning for the long term, (guideline #3); for
example, by planning for training both teachers who were randomly
assigned to the control group and those who are new to the district. We also
more fully realized guideline #10. For example, we scheduled meetings
with district and school leaders more frequently and, when possible, school
boards to update them on our results and progress. As another example, par-
ents received letters explaining what their children were doing that week
and how they could support their children’s learning at home. Beyond this,
they were asked to give the project staff advice on the usefulness of these
letters and on the way the intervention was or was not helping their children.
Such efforts are important to avoid dilution of implementation over years.
Leaders did respond well, buying supplemental materials at one site, and
actively working on integrating the new curriculum with other curricula
115Implementation of TRIAD

at the other. They released and encouraged early childhood coaches to
attend the trainings. Several early childhood center leaders asked for addi-
tional support after the trainings proper.
The professional development followed the same model as previously
described; here we note only the differences. We added a distinction
between “mentors” (coaches who were trained by and worked closely with
project staff ) and in-school teacher coaches within each building. Assigning
schools instead of classrooms allowed us to build consensus, support, and
camaraderie within each building. Both changes were hypothesized to bet-
ter implement guideline #5. As much as possible, we included each districts’
early childhood and mathematics specialists in these meetings and our pro-
fessional development to plan for long-term implementation. We encour-
aged them to rely on teachers in our sessions as teacher leaders to support
wider implementation after the conclusion of the data collection (guidelines
#3, #7, and #10).
3.2.2 Findings
Teachers implemented the intervention with adequate fidelity (Clements
et al., 2011). Again, the modal category for mentor-rated Likert items was
“agree.” That is, most teachers implemented all aspects of the Building
Blocks curriculum. Along with previous studies, this evidence indicates
that interventions in preschool mathematics education, such as this one,
can be successfully implemented on a large scale. Similarly, the TRIAD
intervention enabled teachers to develop richer classroom environments
for mathematics than those of the control classrooms as measured by
the COEMET (the Classroom Observation of Early Mathematics Envi-
ronment and Teaching measures the quality of the mathematics environ-
ment and activities with a half-day observation and can be used with
different curricula, see Clements et al., 2011). The TRIAD classes scored
significantly higher than the control classes on four components of this
measure: the classroom culture subscore, the specific math activities
(SMAs) subscore, the number of SMAs, and the number of computers
on and working for students to use. The classroom culture subscore
assesses teachers’ general approach to mathematics education, indicated
by “environment and interaction” variables such as responsiveness to chil-
dren, use of “teachable moments” as well as “personal attributes of the
teacher” variables, including appearing knowledgeable and confident
about mathematics as well as showing enjoyment in, curiosity about,
and enthusiasm for, teaching mathematics. These suggest that the TRIAD
116 Julie Sarama and Douglas H. Clements

intervention successfully altered teachers’ beliefs and dispositions, beyond
specific curricular practices (guideline #6). Such practices were also pos-
itively affected, given that teachers used the computer component of the
TRIAD curriculum and engaged their children in a greater number of
explicit, targeted, and mathematics activities. The higher scores for the
SMA subscale suggest that the SMAs that teachers in TRIAD classrooms
conducted were of a higher quality than those in the control classrooms.
We will return to the mediational role of these practices after examining
the effects on mathematics achievement. Again, these data support the
efficacy of implementing guidelines #4, #5, #6, and #8 to avoid dilution
and pollution and instead implement with high fidelity, achieving high-
quality mathematics teaching and environments.
Thus, children in the Building Blocks group learned more mathematics
than the children in the control group (effect size, g¼0.72). This is signif-
icant, because, unlike most previous research, the counterfactual condition
was not the “practice-as-usual” control condition involving no published
mathematics curriculum and little district-wide emphasis on mathematics.
Both districts had placed new emphasis on pre-K mathematics and adopted
new literacy curricula that included specific mathematics components. In
addition, the TRIAD intervention itself, especially the first year’s “gentle
introduction,” generated considerable spillover of early mathematics peda-
gogical practices into control classrooms.
The use of curriculum demonstrated as effective with underrepresented
populations, with adequate support for teachers, is supported (guideline #2).
Consistent with other research, analyses support the use of curricula that
have been empirically supported. Analyses also indicate that the computer
software component of the Building Blocks curriculum makes a unique con-
tribution to children’s learning.
Evidence suggested that the Building Blocks curriculum was equally effec-
tive in classes serving low- or mixed-income families or with schools that
have a higher or lower percentage of children with limited English profi-
ciency. Also, the intervention was equally effective for girls and boys, and
for children with or without IEPs. There was evidence that the intervention
was differentially effective for only a single comparison: African-American
children learned less than other children in the same control classrooms and
African-American children learned more than other children in the same
Building Blocks classrooms. It may be that this TRIAD intervention is par-
ticularly effective in ameliorating the negative effects of low expectations for
learning of mathematics (National Mathematics Advisory Panel, 2008).
117Implementation of TRIAD

Thus, there is some support for the implementation avoiding dilution and
pollutions from inequitable support and thereby promoting equity (guide-
line #2, although future research may wish to alter the intervention to close
or narrow other gaps as it did for African-American children here).
3.3 Fighting Dilution Over Time: TRIAD and Sustainability
Avoiding dilution during the intervention is necessary, but it may be more
important to investigate the dilution that most interventions suffer once the
“push” for the intervention has passed—especially for those like TRIAD,
which is introduced from outside the school district. A full concept of suc-
cessful scale-up requires not only consequential implementation and diffu-
sion but also endurance over long periods of time and a transfer of
responsibility from any external organization to the internal resources of a
school district (Coburn, 2003; Dearing & Meyer, 2006). Does implemen-
tation become diluted or polluted over time? Or is it sustainable—continued
professional practice over time, with a focus on the maintenance of core
beliefs and values, and the use of these core beliefs to guide adaptations
(Century, Rudnick, & Freeman, 2012; Scheirer, 2005; Scheirer &
Dearing, 2011; Timperley, 2011). (Persistence of effects, continuation of
the effects of an intervention on individual children’s trajectories of learning,
is a distinct issue, discussed in other publications, Clements et al., 2017;
Clements et al., 2013; Sarama, Clements, Wolfe, & Spitler, 2012; Sarama,
Lange, Clements, & Wolfe, 2012.)
We evaluated the fidelity of implementation and the sustainability of
TRIAD years after any support from the funded project ended in two stud-
ies. The first measured teachers’ implementation 2 years later (Clements
et al., 2015). Although a logical expectation would be that, after the cessa-
tion of external support and professional development provided by the inter-
vention, teachers would show a pattern of decreasing levels of fidelity, these
teachers demonstrated increasing levels of fidelity, continuing to demon-
strate high levels of sustained fidelity in their implementation of the curric-
ulum and underlying pedagogy. Different profiles of variables predicted
separate aspects of sustainability, but by far the most consistent factor was
teachers’ recognition of children’s learning. Teachers who noticed chil-
dren’s substantial development increased their efforts to implement all the
components of the intervention with greater fidelity (Clements et al., 2015).
In our latest study, teachers again demonstrated sustained or increasing
levels of fidelity as long as 6 years after the end of the intervention without
118 Julie Sarama and Douglas H. Clements

support from the project (Sarama, Clements, Wolfe, & Spitler, 2016). Nota-
ble is these teachers’ encouragement and support for discussions of mathe-
matics and their use of formative assessment.
We believe there is considerable evidence that dilution especially was
mitigated by implementing guidelines #1, #7, and #10: Communication
among key groups around a shared vision of the innovation, ensuring school
leaders are a central force, and maintaining communication, assessment, and
follow-through efforts. Most project activities would not be achieved con-
sistently throughout the districts without such communication and contin-
ued collaboration. We found it necessary to repeatedly provide higher-level
administrators with updates and reminders of the projects’ goals and activ-
ities. Similarly, every change in administration had to be monitored and new
people introduced to the project and its successes quickly. The early intro-
duction of the project was facilitated by prior awareness of the researchers’
work and a strong commitment to raising mathematics achievement.
At the classroom level, both dilution and pollution were minimized
because of a strong and consistent implementation of guidelines #6 and
#7, addressing professional development and support from leaders, espe-
cially coaches. Effects of TRIAD depend on the development of teachers’
skills and knowledge. The classroom culture so heavily dependent on the
teacher is a consistent mediator of those positive effects. Note that we pro-
vided approximately 75 h of out-of-class teacher training as well as hours of
mentoring in the classroom, which is substantially more than offered to most
teachers, only 6% of whom participate in mathematics professional develop-
ment for more than 24 h over a year (Birman et al., 2007). A total of 50–70 h
of professional development is consistent with previous research docu-
menting what is necessary to achieve measurable effectiveness (Yoon,
Duncan, Lee, Scarloss, & Shapley, 2007). Similarly, findings indicate that
learning trajectories played a critical role in both teachers’ and children’s
learning, supporting guideline #4: Focus on instructional change that pro-
motes depth of children’s thinking, placing learning trajectories at the core.
Finally, teachers repeatedly commented that the project staff respected
and collaborated with them as professionals, important for generating a
self-sustaining learning community (avoiding dilution via camaraderie sup-
port adaptation, guideline #5). After some initial trepidation, teachers came
to view peer coaching within schools as a positive characteristic of the imple-
mentation. In-class support from mentors and technology support staff
(guideline #6) were also appreciated and viewed as important by the
teachers after they were sure they were there to help and not “police”
119Implementation of TRIAD

implementation. Mentors needed to have the knowledge and skill to dem-
onstrate the implementation of all components of the curriculum, doing
activities right in teachers’ classroom on many occasions and inviting discus-
sion centered on students’ learning.
4. FINAL WORDS
The goal of the TRIAD project was to increase knowledge of scaling
up by conducting research that investigates the effectiveness of a research-
based mathematics education intervention implemented in varied pre-K set-
tings with diverse student populations. Perhaps most unique is TRIAD’s
consistent emphasis on teaching for understanding following developmental
guidelines, or learning trajectories, as well as its use of technology at multiple
levels. TRIAD had a moderate to strong positive effect on children’s math-
ematics achievement and did not harm literacy skills while increasing most
language competencies measured. There was no evidence the approach is
differentially effective for participants in different states or types of programs,
or for children of different SES, ethnic group, or gender. The moderate the
large effect sizes we obtained is notable considering that other comprehen-
sive reform programs, including multitiered teacher support, sustained pro-
fessional development, and in-class coaching, achieve effect sizes such as
0.24 only with great effort (Balfanz, Mac Iver, & Byrnes, 2006).
Most people who discuss TRIAD focus on these findings regarding chil-
dren’s learning, and of course without this evidence, little else matters.
However, to evaluate the ultimate effects of the project, these longitudinal
results are arguably more important—if the intervention is diluted and pol-
luted over time, is it worth the cost and effort? On the other hand, if it grows,
the number of children positively affected is potentially an order of magnitude greater.
Teachers seemed to have internalized the program (Timperley, Wilson,
Barrar, & Fung, 2007). Through the professional development and support,
and then, becoming empowered by their own knowledge of the trajectories
and the practices to support learners through the trajectories, they became
progressively more faithful to the intended program, instead of drifting from
it as time elapsed and support disappeared, dilution found in other studies
(Datnow, 2005; Hargreaves, 2002).
The TRIAD model is not simply about a new curriculum or training
teachers to use it. Success required complex changes, including a change
in instructional structures, pedagogical strategies, and classroom communi-
cation and culture (Grubb, 2008). Given the importance of early
120 Julie Sarama and Douglas H. Clements

competence in mathematics (e.g., Duncan et al., 2007; Paris, Morrison, &
Miller, 2006), the TRIAD implementation described here has implications
for practice and policy, as well as research. TRIAD’s guidelines should be
considered when planning to increase the quality and quantity of not only
early mathematics but also to mitigate dilution and pollution threatening any
intervention.
ACKNOWLEDGMENTS
This research was supported by the Institute of Education Sciences, US Department of
Education through Grants R305A120813, R305K05157, and R305A110188. The
opinions expressed are those of the authors and do not represent views of the US
Department of Education. Although the research is concerned with theoretical issues, not
particular curricula, a small component of the intervention used in this research have been
published by the authors and their collaborators on the project, who thus could have a
vested interest in the results. Researchers from an independent institution oversaw the
research design, data collection, and analysis and confirmed findings and procedures. The
authors wish to express appreciation to the school districts, teachers, and students who
participated in this research.
REFERENCES
Baker, E. L. (2007). Principles for scaling up: Choosing, measuring effects, and promoting the
widespread use of educational innovation. In B. Schneider & S.-K. McDonald (Eds.),
Scale up in education: Ideas in principle: Vol. 1. (pp. 37–54). Lanham, MD: Rowman &
Littlefield.
Balfanz, R., Mac Iver, D. J., & Byrnes, V. (2006). The implementation and impact of
evidence-based mathematics reforms in high-poverty middle schools: A multi-site,
multi-year study. Journal for Research in Mathematics Education,37,33–64.
Ball, D. L., & Cohen, D. K. (1999). Instruction, capacity, and improvement. Philadelphia, PA:
Consortium for Policy Research in Education, University of Pennsylvania.
Baroody, A. J., & Coslick, R. T. (1998). Fostering children’s mathematical power: An investigative
approach to K-8 mathematics instruction. Mahwah, NJ: Erlbaum.
Berends, M., Kirby, S. N., Naftel, S., & McKelvey, C. (2001). Implementation and performance
in new American schools: Three years into scale-up. Santa Monica, CA: Rand Education.
Birman, B. F., LeFloch, K. C., Klekotka, A., Ludwig, M., Taylor, J., Walters, K., …
Yoon, K. S. (2007). State and local implementation of the No Child Left Behind Act, Volume
II—Teacher quality under NCLB: Interim report. Washington, DC: U.S. Department of
Education Office of Planning Evaluation and Policy Development, Policy and
Program Studies Service.
Bodilly, S. J. (1998). Lessons from new American Schools’ scale-up phase. Santa Monica, CA:
RAND Education.
Borman, G. D., Hewes, G. M., Overman, L. T., & Brown, S. (2003). Comprehensive school
reform and achievement: A meta-analysis. Review of Educational Research,73, 125–230.
http://dx.doi.org/10.3102/00346543073002125.
Bredekamp, S. (2004). Standards for preschool and kindergarten mathematics education.
In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young children in math-
ematics: Standards for early childhood mathematics education (pp. 77–82). Mahwah, NJ:
Erlbaum.
121Implementation of TRIAD

Brown, A. L., & Campione, J. C. (1996). Psychological theory and the design of innovative
learning environments: On procedures, principles, and systems. In R. Glaser (Ed.), Inno-
vations in learning: New environments for education (pp. 289–325). Mahwah, NJ: Erlbaum.
Bryk, A. S., Sebring, P. B., Allensworth, E., Suppescu, S., & Easton, J. Q. (2010). Organizing
schools for improvement: Lessons from Chicago. Chicago, IL: University of Chicago Press.
Carpenter, T. P., & Franke, M. L. (2004). Cognitively guided instruction: Challenging the
core of educational practice. In J. Glennan, K. Thomas, S. J. Bodilly, J. R. Galegher, &
K. A. Kerr (Eds.), Expanding the reach of education reforms: Perspectives from leaders in the scale-
up of educational interventions (pp. 41–80). Santa Monica, CA: RAND Corporation.
Century, J., Rudnick, M., & Freeman, C. (2012). In Defining and measuring sustainability of
reform: Factors that affect our abilities to generate enduring change.Paper presented at the annual
meeting of the American Educational Research Association, Vancouver, British, Columbia,
Canada.
Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Math-
ematical Thinking and Learning,6,81–89. http://dx.doi.org/10.1207/s15327833mtl0602_1.
Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Sum-
mative research on the Building Blocks project. Journal for Research in Mathematics Education,
38(2), 136–163.
Clements, D. H., & Sarama, J. (2008a). Experimental evaluation of the effects of a research-
based preschool mathematics curriculum. American Educational Research Journal,45,
443–494. http://dx.doi.org/10.3102/0002831207312908.
Clements, D. H., & Sarama, J. (2008b). In Scaling-up interventions: The case of mathematics.Paper
presented at the American Educational Research Association, New York, NY.
Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science,
333(6045), 968–970. http://dx.doi.org/10.1126/science.1204537.
Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories
approach. New York, NY: Routledge.
Clements, D. H., Sarama, J., & DiBiase, A.-M. (2003). Preschool and kindergarten mathe-
matics: A national conference. Teaching Children Mathematics,8, 510–514.
Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathe-
matics learned by young children in an intervention based on learning trajectories:
A large-scale cluster randomized trial. Journal for Research in Mathematics Education,
42(2), 127–166.
Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2013). Longitudinal evaluation
of a scale-up model for teaching mathematics with trajectories and technologies: Persis-
tence of effects in the third year. American Educational Research Journal,50(4), 812–850.
http://dx.doi.org/10.3102/0002831212469270.
Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2015). Sustainability of a scale-
up intervention in early mathematics: Longitudinal evaluation of implementa-
tion fidelity. EarlyEducationandDevelopment,26(3), 427–449. http://dx.doi.org/
10.1080/10409289.2015.968242.
Clements, D. H., Sarama, J., Layzer, C., Unlu, F., Wolfe, C. B., Fesler, L., …Spitler, M. E.
(2017). Effects of TRIAD on mathematics achievement: Long-term impacts. Submitted for
publication.
Cobb, P., McClain, K., de Silva, T., & Dean, C. (2003). Situating teachers’ instructional
practices in the institutional setting of the school and district. Educational Researcher,
32(6), 13–24.
Coburn, C. E. (2003). Rethinking scale: Moving beyond numbers to deep and lasting
change. Educational Researcher,32(6), 3–12.
Cohen, D. K. (1996). Rewarding teachers for student performance. In S. H. Fuhrman &
J. A. O’Day (Eds.), Rewards and reforms: Creating educational incentives that work
(pp. 61–112). San Francisco, CA: Jossey Bass.
122 Julie Sarama and Douglas H. Clements

Copley, J. V. (2004). The early childhood collaborative: A professional development
model to communicate and implement the standards. In D. H. Clements, J. Sarama,
&A.-M.DiBiase(Eds.),Engaging young children in mathematics: Standards for early child-
hood mathematics education (pp. 401–414). Mahwah, NJ: Erlbaum.
Corcoran, T. B. (2007). Teaching matters: How state and local policymakers can improve the quality
of teachers and teaching. CPRE Policy Briefs (pp. 1–15).
Cuban, L., & Usdan, M. (Eds.), (2003). Powerful reforms with shallow roots: Improving America’s
urban schools. New York, NY: Teachers College.
Darling-Hammond, L. (1996). Restructuring schools for high performance. In S. H. Fuhrman
& J. A. O’Day (Eds.), Rewards and reform: Creating educational incentives that work
(pp. 144–192). San Francisco: Jossey-Bass.
Darling-Hammond, L. (1997). The right to learn: A blueprint for creating schools that work. San
Francisco: Jossey-Bass.
Datnow, A. (2005). The sustainability of comprehensive school reform models in changing
district and state contexts. Educational Administrative Quarterly,41(1), 121–153.
Dearing, J. W., & Meyer, G. (2006). Revisiting diffusion theory. In A. Singhal &
J. W. Dearing (Eds.), Communication of innovations: A journey with Ev Rogers
(pp. 29–60). New Delhi: Sage.
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P.,
…Japel, C. (2007). School readiness and later achievement. Developmental Psychology,
43(6), 1428–1446. http://dx.doi.org/10.1037/0012-1649.43.6.1428.
Elmore, R. F. (1996a). Getting to scale with good educational practices. In S. H. Fuhrman &
J. A. O’Day (Eds.), Rewards and reform: Creating educational incentives that work
(pp. 294–329). San Francisco: Jossey-Bass.
Elmore, R. F. (1996b). Getting to scale with good educational practices. Harvard Educational
Review,66,1–25.
Ferguson, R. F. (1991). Paying for public education: New evidence on how and why money
matters. Harvard Journal on Legislation,28(2), 465–498.
Fox, J. (2005). Child-initiated mathematical patterning in the pre-compulsory years.
In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the interna-
tional group for the psychology in mathematics education (Vol. 2, pp. 313–320). Melbourne,
AU: PME.
Foorman, B. R., Santi, K. L., & Berger, L. (2007). Scaling assessment-driven instruction using
the internet and handheld computers, Volume 2. In B. Schneider & S.-K. McDonald
(Eds.), Scale up in practice (pp. 69–89). Lanham, MD: Rowman & Littlefield.
Fullan, M. G. (1992). Successful school improvement. Philadelphia, PA: Open University
Press.
Fullan, M. G. (2000). The return of large-scale reform. Journal of Educational Change,1,
5–28.
Galindo, C., & Sonnenschein, S. (2015). Decreasing the SES math achievement gap: Initial
math proficiency and home learning environments. Contemporary Educational Psychology,
43,25–38. http://dx.doi.org/10.1016/j.cedpsych.2015.08.003.
Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes
professional development effective? Results from a national sample of teachers. American
Educational Research Journal,38, 915–945.
Glennan, T. K., Jr. (2004). Expanding the reach of education reforms. In S. J. Bodilly,
J. R. Galegher, & K. A. Kerr (Eds.), Perspectives from leaders in the scale-up of educational
interventions. Santa Monica, CA: RAND Corporation.
Grubb, W. N. (2008). Multiple resource, multiple outcomes: Testing the “improved” school
finance with NELS88. American Educational Research Journal,45, 104–144.
Guskey, T. R. (Ed.), (2000). Evaluating professional development. Thousand Oaks, CA: Corwin
Press.
123Implementation of TRIAD

Hall, G. E., & Hord, S. M. (2001). Implementing change: Patterns, principles, and potholes.
Boston, MA: Allyn and Bacon.
Halpern, R. (2004). Parent support and education: Past history, future prospects. Applied
Research in Child Development,6(1), 4–11.
Hargreaves, A. (2002). Sustainability of educational change: The role of social geographies.
Journal of Educational Change,3(3–4), 189–214.
Heck, D. J., Weiss, I. R., Boyd, S., & Howard, M. (2002). Lessons learned about planning and
implementing statewide systemic initiatives in mathematics and science education. In Paper pres-
ented at the annual meeting of the American Educational Research Association. New Orleans,
LA. http://www.horizon-research.com/public.htm.
Hiebert, J. C. (1999). Relationships between research and the NCTM standards. Journal for
Research in Mathematics Education,30,3–19.
Huberman, M. (1992). Critical introduction. In M. G. Fullan (Ed.), Successful school improve-
ment (pp. 1–20). Philadelphia, PA: Open University Press.
Kaser, J. S., Bourexis, P. S., Loucks-Horsley, S., & Raizen, S. A. (1999). Enhancing program
quality in science and mathematics. Thousand Oaks, CA: Corwin Press.
Klingner, J. K., Ahwee, S., Pilonieta, P., & Menendez, R. (2003). Barriers and facilitators in
scaling up research-based practices. Exceptional Children,69, 411–429.
Lawless, K. A., & Pellegrino, J. W. (2007). Professional development in integrating technol-
ogy into teaching and learning knowns, unknowns, and ways to pursue better questions
and answers. Review of Educational Research,77(4), 575–614.
Mohrman, S. A., & Lawler, E. E., III. (1996). Motivation for school reform. In S. H. Fuhrman
& J. A. O’Day (Eds.), Rewards and reform: Creating educational incentives that work
(pp. 115–143). San Francisco: Jossey-Bass.
NAEYC. (2002). NAEYC standards for early childhood professional preparation. Washington,
DC: National Association for the Education of Young Children.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the
National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education,
Office of Planning, Evaluation and Policy Development.
National Research Council. (2007). Taking science to school: Learning and teaching sciences in
grades K-8. Washington, DC: National Academies Press.
National Research Council. (2009). Mathematics learning in early childhood: Paths toward excel-
lence and equity. Washington, DC: National Academy Press.
Paris, S. G., Morrison, F. J., & Miller, K. F. (2006). Academic pathways from preschool
through elementary school. In P. Alexander & P. Winne (Eds.), Handbook of research
in educational psychology (pp. 61–85). Mahwah, NJ: Erlbaum.
Peisner-Feinberg, E. S., Burchinal, M. R., Clifford, R. M., Culkin, M. L., Howes, C.,
Kagan, S. L., …Zelazo, J. (1999). The children of the cost, quality, and outcomes study go
to school. Chapel Hill, NC: Frank Porter Graham Child Development Center,
University of North Carolina at Chapel Hill.
Pellegrino, J. W. (2007). From early reading to high school mathematics: Matching case stud-
ies of four educational innovations against principles for effective scale up. In B. Schneider
& S.-K. McDonald (Eds.), Scale up in practice (pp. 131–139). Lanham, MD: Rowman and
Littlefield.
Penuel, W. R., & Shepard, L. A. (2016). Assessment and teaching. In D. H. Gitomer &
C. A. Bell (Eds.), Handbook of research on teaching (5th ed., pp. 787–850). Washington,
DC: American Educational Research Association.
Perry, B., Perry, B., Dockett, S., & Harley, E. (2007). Learning stories and children’s pow-
erful mathematics. Early Childhood Research & Practice,9(2), 117–134.
Ramey, C. T., & Ramey, S. L. (1998). Early intervention and early experience. American
Psychologist,53, 109–120.
124 Julie Sarama and Douglas H. Clements

Raudenbush, S. W. (2008). Advancing educational policy by advancing research on instruc-
tion. American Educational Research Journal,45, 206–230.
Rogers, E. M. (2003). Diffusion of innovations (5th ed.). New York, NY: The Free Press.
Sarama, J. (2002). Listening to teachers: Planning for professional development. Teaching
Children Mathematics,9,36–39.
Sarama, J., & Clements, D. H. (2009a). Early childhood mathematics education research: Learning
trajectories for young children. New York, NY: Routledge.
Sarama, J., & Clements, D. H. (2009b). In Scaling up successful interventions: Multidisciplinary
perspectives.Paper presented at the American Educational Research Association, San Diego, CA.
Sarama, J., & Clements, D. H. (2013). Lessons learned in the implementation of the TRIAD
scale-up model: Teaching early mathematics with trajectories and technologies.
In T. G. Halle, A. J. Metz, & I. Martinez-Beck (Eds.), Applying implementation science
in early childhood programs and systems (pp. 173–191). Baltimore, MD: Brookes.
Sarama, J., & DiBiase, A.-M. (2004). The professional development challenge in preschool
mathematics. In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young chil-
dren in mathematics: Standards for early childhood mathematics education (pp. 415–446).
Mahwah, NJ: Erlbaum.
Sarama, J., Clements, D. H., & Henry, J. J. (1998). Network of influences in an implemen-
tation of a mathematics curriculum innovation. International Journal of Computers for Math-
ematical Learning,3, 113–148.
Sarama, J., Clements, D. H., Starkey, P., Klein, A., & Wakeley, A. (2008). Scaling up the
implementation of a pre-kindergarten mathematics curriculum: Teaching for under-
standing with trajectories and technologies. Journal of Research on Educational Effectiveness,
1,89–119. http://dx.doi.org/10.1080/19345740801941332.
Sarama, J., Clements, D. H., Wolfe, C. B., & Spitler, M. E. (2012a). Longitudinal eval-
uation of a scale-up model for teaching mathematics with trajectories and technolo-
gies. Journal of Research on Educational Effectiveness,5(2), 105–135. http://dx.doi.org/
10.1080/119345747.2011, 627980.
Sarama, J., Lange, A., Clements, D. H., & Wolfe, C. B. (2012b). The impacts of an early
mathematics curriculum on emerging literacy and language. Early Childhood Research
Quarterly,27, 489–502. http://dx.doi.org/10.1016/j.ecresq.2011.12.002.
Sarama, J., Clements, D. H., Wolfe, C. B., & Spitler, M. E. (2016). Professional development
in early mathematics: Effects of an intervention based on learning trajectories on teachers’
practices. Nordic Studies in Mathematics Education,21(4), 29–55.
Scheirer, M. A. (2005). Is sustainability possible? A review and commentary on empirical
studies of program sustainability. The American Journal of Evaluation,26(3), 320–347.
Scheirer, M. A., & Dearing, J. W. (2011). An agenda for research on the sustainability of
public health programs. American Journal of Public Health,101(11), 2059–2067.
Schoen, H. L., Cebulla, K. J., Finn, K. F., & Fi, C. (2003). Teacher variables that relate to
student achievement when using a standards-based curriculum. Journal for Research in
Mathematics Education,34(3), 228–259.
Showers, B., Joyce, B., & Bennett, B. (1987). Synthesis of research on staff development:
A framework for future study and a state-of-the-art analysis. Educational Leadership,
45(3), 77–87.
Snipes, J., Doolittle, F., & Herlihy, C. (2002). Foundations for success: Case studies of how urban school
systems improve student achievement. Washington, DC: Council of the Great City Schools.
Sowder, J. T. (2007). The mathematical education and development of teachers.
In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and
learning: Vol. 1 (pp. 157–223). New York, NY: Information Age Publishing.
Spillane, J. P. (2000). Cognition and policy implementation: District policy-makers and the
reform of mathematics education. Cognition and Instruction,18, 141–179.
125Implementation of TRIAD

Starkey, P., Klein, A., Chang, I., Qi, D., Lijuan, P., & Yang, Z. (1999, April).
In Environmental supports for young children’s mathematical development in China and the
United States.Paper presented at the society for research in child development, Albuquerque, NM.
Stein, M. K., Brover, B. W., & Hennigsen, M. (1996). Building student capacity for math-
ematical thinking and reasoning: An analysis of mathematical tasks used in reform class-
room. American Educational Research Journal,33, 455–488.
Teddlie, C., & Stringfield, S. (1993). Schools make a difference: Lessons learned from a 10-year
study of school effects. New York: Teachers College Press.
Thomas, B. (1982). An abstract of kindergarten teachers’ elicitation and utilization of children’s prior
knowledge in the teaching of shape concepts. Unpublished manuscript. New York, NY:
School of Education, Health, Nursing, and Arts Professions, New York University.
Timperley, H. (2011). Realizing the power of professional learning. London: Open University
Press.
Timperley, H., Wilson, A., Barrar, H., & Fung, I. (2007). Teacher professional development: Best
evidence synthesis iteration. Wellington, New Zealand: Ministry of Education.
U. S. Department of Education. (1999). New teachers for a new century: The future of early child-
hood professional preparation. Washington, DC: Author.
Weiss, I. R. (2002). Systemic reform in mathematics education: What have we learned?
In Research presession of the 80th annual meeting of the National Council of Teachers of
Mathematics. Las Vegas, NV.
Winter, S. G., & Szulanski, G. (2001). Replication as strategy. Organization Science,12,
730–743.
Wu, H.-H. (2011). Understanding numbers in elementary school mathematics. Providence, RI:
American Mathematical Society.
Yoon, K. S., Duncan, T., Lee, S. W.-Y., Scarloss, B., & Shapley, K. L. (2007). Reviewing the
evidence on how teacher professional development affects student achievement (Issues & Answers
Report, REL 2007–No. 033). Retrieved from Washington, DC. http://ies.ed.gov/
ncee/edlabs.
126 Julie Sarama and Douglas H. Clements