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Linking research and curriculum development

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23 Linking research and
curriculum development
Douglas H. Clements
University of Buffalo, State University of New York
Commercially published, traditional textbooks predominate mathematics curriculum materi-
als in U.S. classrooms and, to a great extent, determine teaching practices (Goodlad, 1984),
even in the context of reform efforts (Grant, Peterson, & Shojgreen-Downer, 1996). Various
standards (NCTM, 1989, 2000) and state and local curriculum frameworks are designed to
govern or at least guide these materials. However, publishers attempt to meet the criteria of
all such frameworks, including scope and sequence requirements, and thus the educational
vision of any one is, at best, diluted. This results in “mile wide, inch deep” (NCES, 1996) cur-
ricula that are a primary cause of the poor performance of U.S. students in mathematics (A.
Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005; Kouba et al., 1988; McKnight, Trav-
ers, Crosswhite, & Swafford, 1985; Mullis et al., 1997). Problems exist both in the quantity
of topics that are treated and how they are treated (Clements & Battista, 1992; A. Ginsburg,
Leinwand, Anstrom, & Pollock, 2005; Porter, 1989). In one main focus of study, geometry,
for example, textbooks are not only ineffective in promoting higher levels of geometric think-
ing (Fuys, Geddes, & Tischler, 1988), they often hinder students’ development of this think-
ing (Jaime, Chapa, & Gutiérrez, 1992; Mansfi eld & Happs, 1992).
Why do curricula in the United States not improve? One reason is that the vast majority
of curriculum development efforts do not follow systematic, much less scientifi c, research
procedures (Battista & Clements, 2000; Clements & Battista, 2000). In this chapter, I dis-
cuss the nature and relationship of science, research, and curricula; how curricula are usually
developed; and recent alternative research and development models. I describe one model in
depth.
SCIENCE, RESEARCH, AND CURRICULUM
Science includes the observation, description, analysis, experimental investigation, and the-
oretical explanation of phenomena. Scienti c knowledge is accepted as more reliable than
commonsense knowledge because the way in which it is developed is explicit and repeatable.
“Our faith [in it] rests entirely on the certainty of reproducing or seeing again a certain
phenomenon by means of certain well de ned acts” (Valéry, 1957, p. 1253, as quoted in
Glasersfeld, 1995). These acts are the method of science. Scientifi c method, or research, is
disciplined inquiry (Cronbach & Suppes, 1969). “Inquiry” suggests that the investigation’s
goal is answering a specifi c question. “Disciplined” suggests that the investigation should
be guided by concepts and methods from disciplines and connected to relevant theory in
those disciplines, and also that it should be in the public view so that the inquiry can be
inspected and criticized. The use of research methods, and the conscious documentation and
full reporting of these processes—data collection, argumentation, reasoning, and checking
590 Douglas H. Clements
for counterhypotheses—distinguishes disciplined inquiry from other sources of opinion and
belief (Cronbach & Suppes, 1969; NRC, 2002; Shulman, 1997).
The research process usually involves recursive phases. Summarizing Maturana’s formula-
tion, Glasersfeld (1995, p. 117) describes these as follows.
1. The conditions (constraints) under which the phenomenon is observed are made explicit,
so that the observation can be repeated.
2. A hypothetical mechanism is proposed that could serve as explanation of how the inter-
esting or surprising aspects of the observed phenomenon may arise.
3. From the hypothetical mechanism, a prediction is deduced, concerning an event that has
not yet been observed.
4. The conditions under which the mechanism should lead to the observation of the pre-
dicted event are generated; and these conditions are again made explicit.
One implication is that science is not conceived as producing the “truth” or a single correct
view. It provides reliable ways of dealing with experiences and pursuing and achieving goals
(Glasersfeld, 1995). Science involves the process of progressive problem solving and advance-
ment beyond present limits of competence (Scardamalia & Bereiter, 1994). Problem redefi ni-
tion at increasingly high levels is the goal.
A limitation of this description is that it may promote the misconception that scientists
are solitary explorers. Science exists and develops in communities. As one example, scholarly
journals, with their editors, editorial boards, reviewers, and contributors, are a signi cant
force in the progressive development of knowledge (Latour, 1987). In contrast to articles
in other periodicals, in which the authors’ goal is frequently to present the same informa-
tion in a superfi cially “new” way, each article in a scholarly journal must advance scientifi c
knowledge.
Thus, scientifi c knowledge is valued because it offers reliable, self-correcting, documented,
shared knowledge based on research methodology (Mayer, 2000; NRC, 2002). Curricu-
lum development is a design science (Brown, 1992; H. A. Simon, 1969; Wittmann, 1995)
and knowledge created during curriculum development should be both generated and placed
within a scientifi c research corpus, peer reviewed, and published. However, I do not promote
a simple, causal, deterministic view of science. Nor, as the remainder of this article should
make clear, do I privilege narrow views of scientifi c research, such as quantitative experiments
(Dewey, 1929). Consider the polar positions in the following passage.
For some, only when educational researchers agree about how one goes about creating
knowledge in educational research will education start producing “reliable knowledge”
(Zinman, 1978) and become a “proper science.” For others, however, the very nature of
the educational activity—the complexity of the objects of study—means that educational
research can never become a “science” in the traditional (and narrow) sense. (Lester &
Wiliam, 2002, p. 490)
Although I agree that the nature of educational research is complex and interpretive, I do not
accept the “traditional” and “narrow” sense of the term “science” or believe that research can
produce “truth” or a single correct view. I take the perspective that scientifi c research, as dis-
ciplined inquiry, provides reliable ways of dealing with experiences and pursuing and achiev-
ing goals, as described previously (Glasersfeld, 1987). In curriculum development research,
the mechanism centers on a curriculum and its implementation.
Further, scientifi c research involves co-mutual infl uences within social and political con-
texts (Cobb, 2001) and is itself social and political (Latour, 1987), with researchers garner-
ing support for their global perspectives, research issues, individual studies, and even results.
That is, scientifi c research is not free from social-historical movements, values, controversies,
Linking research and curriculum development 591
politics, competition, status hierarchies, and egotism. Scienti c knowledge advances are ulti-
mately achieved by theself-regulating norms of a scienti c community over time. One
implication is that the goal cannot be to develop a single “ideal” curriculum, but rather
dynamic problem solving, progress, and advancement beyond present limits of competence
(Dewey, 1929; Scardamalia & Bereiter, 1994; Tyler, 1949). Ironically, another implication is
that curricula should be based on research—as de ned here. Given that all such factors affect
curriculum as well—probably to a much greater degree, particularly in the realm of fi nancial
gain—the checks and balances of scientifi c research are essential to support full disclosure as
well as progress.
Finally, scientifi c research is necessary but not suf cient, for the continued development of
high-quality curricula.
You make a great, a very great mistake, if you think that psychology, being the science
of the mind’s laws, is something from which you can deduce defi nite programmes and
schemes and methods of instruction for immediate classroom use. Psychology is a sci-
ence, and teaching is an art; and sciences never generate arts directly out of themselves.
An intermediary inventive mind must make the application, by using its originality.
(James, 1958, pp. 23–24)
James argues that scienti c knowledge is applied artfully to create teaching materials. Such
research-to-practice methods are included in the framework. However, because it constitutes
one-way translations of research results, a strict research-to-practice model is fl awed in its
presumptions, insensitive to changing goals in the content area, and unable to contribute to a
revision of the theory and knowledge on which it is built—the second critical goal of a scien-
tifi c curriculum research program. Thus, developers must draw from existing research so that
what is already known can be applied to the anticipated curriculum; structure and revise the
nature and content of curricular components in accordance with models of students’ think-
ing and learning in a domain; and conduct formative and summative evaluations in a series of
progressively expanding social contexts (Clements, 2006). That is, research should be present
in all phases of the curriculum development process, from James’ initial scientifi c base to for-
mative and summative evaluation (Brown, 1992), and thus be integrated even into the most
creative phases (Dewey, 1929), to achieve the documentation of decisions and the ultimate
checking of hunches and full reporting of all procedures (Cronbach & Suppes, 1969). Such
documentation requires a common language for connections between curriculum develop-
ment and research. Toward this goal, I next defi ne “curriculum” and how traditional curricula
are often developed in the United States and then discuss alternative relationships between
research and curriculum development, including an explication of categories and phases of
curriculum research.
CURRICULUM DEFINITIONS AND HOW
TRADITIONAL CURRICULA ARE DEVELOPED
There are many defi nitions of “curriculum,” including the following (Burkhardt, Fraser, &
Ridgway, 1989, pp. 5–6).
The ideal curriculum is what experts propound; because it is not fi rmly grounded in rel-
evant experience, it is fundamentally speculative but important in defi ning directions for
change that should be pursued.
The available curriculum is the one for which teaching materials exist, though these will
not always be matched to the capabilities of all teachers.
592 Douglas H. Clements
The adopted curriculum is the one which some state or local authority says must be
taught.
The implemented curriculum is what teachers actually teach in the classroom; because
teachers vary enormously in their capabilities, there is a wide distribution of implemented
curricula.
The achieved curriculum is what the students actually learn; its distribution is even wider
across many variables.
The tested curriculum is determined by the spectrum of tests which carry public credibility,
and through that, infl uence what happens in classrooms.
In this chapter, I use the single word “curriculum” to mean theavailable curriculum”
(both traditional and innovative). In this meaning, curriculum is an instructional blueprint
and set of materials for guiding students’ acquisition of certain culturally-valued concepts,
procedures, intellectual dispositions, and ways of reasoning (Battista & Clements, 2000). I
will use adjectives to discriminate other uses of the term (e.g., some use the Standards devel-
oped by the NCTM, 1989, 2000, to defi ne an “ideal curriculum”).
As previously stated, traditional commercial textbooks dominate mathematics curriculum
materials and thus have a large infl uence on teaching practices in the United States (Goodlad,
1984). Textbooks are widely used in classroom instruction and structure 75–90% of class-
room instruction (Grouws & Cebulla, 2000; Woodward & Elliot, 1990). About two-thirds
of teachers report they use textbooks almost every day (Grouws & Cebulla, 2000). According
to Ginsburg, K lein, and Starkey (1998), the most in uential U.S. publishers are a few large
conglomerates that usually have profi t, rather than the mathematics learning of students,
as their main goal. This leads them to painstakingly follow state curriculum frameworks,
attempting to meet every objective of every state—especially those that mandate adherence
to their framework (permitting the curriculum to be listed as acceptable for purchase by state
schools). The publishers’ marketing departments determine content and approach as much
as the educators do. The writing team is comprised of an editorial staff, writing staff, and
the of cial authors, who increasingly play merely a consultant role, helping to frame philoso-
phy and approach, but have minimal infl uence on the precise form and content of the fi nal
product.
Publishers are also more concerned that their materials appear to meet national standards
and state frameworks than that they actually do (H. P. Ginsburg et al., 1998). Teachers
appeals for textbooks that are easy to use, along with conservative political forces, usually
contradict ideal curriculum guidelines. Focus groups of teachers, for example, frequently
emphasize that reform movements are not based in the real world, that drill and practice
should predominate curricula, and that “good textbooks” are those that get one through
mathematics as quickly and effortlessly (for both student and teacher) as possible by supplying
simple activities and familiar routines (H. P. Ginsburg et al., 1998). Thus, publishers give the
appearance of meeting standards and frameworks but actually provide traditional lessons.
The result is that publishers produce an incoherent mix of traditional didactic-presentation-
plus-drill pages, pages that are designed to give the appearance of higher-order thinking but
often do not. They provide a false sense of innovation. This reveals “the skill of publishers in
including materials which appear to support the new aspects of the curriculum that are needed
for adoption, presented in such a way as not to embarrass those who wish to continue teaching
mathematics the way they have always done it” (Burkhardt et al., 1989, p. 16).
This is an unfortunate situation. The following factors and problems are possible
contributors.
Social and political forces
Already mentioned, but beyond the scope of this chapter, are the diverse forces that—often
Linking research and curriculum development 593
misinterpreting visions of reform and lacking knowledge of mathematics education—work to
block reform and maintain traditional conservative practices. Social support for this position
is widely available in the U.S. culture’s instrumentalist views of mathematics and knowledge
acquisition as simple transmission (Thompson, 1992). Without scienti c research as a guide-
line and a constraint, available and implemented curricula move toward these conservative,
instrumentalist, and transmissive views. In a similar vein, the previously-described lack of
consistency across various standards and guidelines in the U.S. continues to produce “mile
wide and inch deep” (NCES, 1996) curricula as publishers struggle to meet a variety of dif-
ferent content standards and guidelines (Clements, Sarama, & DiBiase, 2004; Tyson-Bern-
stein, 1988). (Note that the recent “Curriculum Focal Points” work of the U.S. NCTM is an
attempt to address this problem directly.)
Further, the process of curriculum adoption in the United States is a “conspiracy of good
intentions”—resulting from “destructive and interactive effects of various state and local text-
book adoption policies and selection practices” (Tyson-Bernstein, 1988, p. 10) that jointly
militate against the use of research-based curricula. For example, gatherings of experts agree
that standards and curricula lack focus, but when they list a topic for deletion, at least one per-
son argues forcefully to retain it. Tyson-Bernstein argues that textbooks should be selected on
the basis of qualities known to bene t students; however, without criteria for such knowledge,
we will not escape our present cycle of battling opinions and biases.
Social and political forces at universities also retard progress. Design has less status than
other avenues to scholarship and is thus not valued (Wittmann, 1995).
Rejection of all published curricula
From the other end of the spectrum, some educators, discouraged with the lack of depth of
traditional materials, often disparage textbooks, leading to the view that good teachers do not
use textbooks. Such an overgeneralization limits the role that high-quality curricula may play
(Ball & Cohen, 1996).
Lack of standards for curriculum development
There are no established standards for development, peer-review, communication, or profes-
sional training for curriculum development. While good curriculum development can benefi t
from a variety of perspectives and expertise, the lack of standards has had a deleterious effect
on the fi eld. Such standards are not developed, discussed, and applied in part because cur-
riculum development is viewed as not requiring a substantial research component.
In addition, there is little refl ection on or documentation of the process of curriculum
development. We have worked for a variety of curriculum development projects as authors or
consultants. We have not seen any development team record the reasons for their decisions or
otherwise document their process and progress. When we have suggested that they should,
about half, usually for-profi t organizations, state that this is not their mission. The other
half, usually funded by outside agencies, admit that it would be wise, but claim that there is
insuffi cient time or funding to do so. As a community, then, we are left with no structure to
support the development and sharing of knowledge.
Limited involvement of, and communication between, relevant parties
Due to this lack of structure and the diverse nature of the people and organizations that
develop curricula, often designers and teachers have few conversations with one another (Ben-
Peretz, 1990; Dow, 1991). This has a negative effect on all aspects of education. The point for
this chapter is that curriculum developers do not have standards that require them to interact
with teachers. This is unfortunate, given that individual teachers shape the curriculum in fun-
594 Douglas H. Clements
damental ways. The result is that the relationship between textbooks and teachers has rarely
been taken up with much care or creativity (Ball & Cohen, 1996). Instead, developers too
often assume that curriculum materials can be used almost independently (Dow, 1991).
This also limits the contribution of curriculum materials to professional practice. Such a
contribution would be enhanced if the materials were created with attention to processes of
curriculum enactment (Ball & Cohen, 1996)
RESEARCH-BASED CURRICULUM DEVELOPMENT
Relationships between research and research-based curriculum development
My position, then, is that the isolation of curriculum development, classroom teaching, and
mathematics educational research deleteriously affects each of these three areas of mathe-
matics education. This is not the same as saying that curriculum development should be
research. The goal of scienti c research is the creation of knowledge, both theoretical and
empirical. The goal of curriculum development is the production of instructional materi-
als. Although knowledge is usually created during curriculum development, it is usually not
explicated (Gravemeijer, 1994b), placed in the context of scienti c theory or an empirical
research corpus, reviewed, and shared. Thus, I do not propose that curriculum development
become research. I propose fusing the two. Research-based curriculum development efforts
can contribute to (a) more effective curriculum materials because the research reveals critical
issues for instruction, (b) better understanding of students’ mathematical thinking, and (c)
research-based change in mathematics curriculum (Clements, Battista, Sarama, & Swamina-
than, 1997; Schoenfeld, 1999). Many curricula claim to be based on research; it is therefore
necessary to clarify what I mean by this phrase.
Curriculum development might be “based” on research in a variety of ways. Consider the
following possibilities in mathematics education.
1. Broad philosophies, theories, and empirical results on learning and teaching are consid-
ered when creating curriculum.
2. Empirical fi ndings on making activities educationally effective—motivating and effi ca-
cious—serve as general guidelines for the generation of activities.
3. Research is used to identify mathematics that is developmentally appropriate and interest-
ing to students in the target population.
4. Activities are structured to be consistent with empirically-based learning models of stu-
dents’ thinking and learning.
5. Sets of activities are sequenced according to research-based sequences through the con-
cepts and skills that constitute a domain of mathematics.
6. Activities or activity sets are eld-tested from their fi rst inception and early intensive
interpretive work, to classroom-based studies, and are revised substantially after each
test.
7. Summative evaluation studies are conducted, including issues of scalability.
8. Following the creation of a curriculum, research results that are ostensibly consistent
with it are cited post hoc.
9. Each stage of the development process is documented, refl ected upon, analyzed, and
reported in the scientifi c literature.
So my position is not misconstrued, I concur that research, especially psychological
research, has played a substantial role in education. It has been used not so much to produce
practical materials for teaching, but to interpret the phenomena of mathematics education (H.
P. Ginsburg et al., 1998). This role is indirect, but important. Here, in contrast, I examine
how research as been directly applied to curriculum development. We believe that most of
Linking research and curriculum development 595
the ways that curriculum development might be “based” on research should be employed,
strategically. We next consider a small number of early attempts to base curriculum develop-
ment on research.
Early attempts to base curriculum development on
research: The research-to-practice model
Early efforts to write research-based mathematics curricula often were grounded in the broad
philosophies, theories, and empirical results on learning and teaching of general theories. For
example, in early childhood, early applications of Piaget’s theories often trained students on
Piagetian clinical tasks or incorporated materials directly adapted from those tasks (Forman
& Fosnot, 1982). These were not particularly successful. Even detailed analyses of Piagetian
research failed to guide curricula development in directly useful ways (Duckworth, 1979).
Others have based their educational programs on Piaget’s constructivist foundation. For
example, Duckworth encouraged students to “have wonderful ideas” (Duckworth, 1973).
Such programs have been arguably more successful, although the interpretations varied widely
(Forman, 1993). Indeed, the curricula per se were very different. The broad philosophy and
theory, unsurprisingly, leaves much room for interpretation and provides little guidance for
curriculum development.
Even theories that are born in instruction, when used as a general framework, may not be
successful. For example, in one study, a curriculum based on the van Hielian theory of levels
of geometric thinking, featuring informal experiences before formal arguments, was not bet-
ter than a traditional approach (Han, 1986).
In summary, the research-to-practice model has a less than successful history (Clements &
Battista, 2000; Cobb, 2001; Gravemeijer, 1994b). Based on the notion of a one-way transla-
tion of research results to principles to instructional designs, it is fl awed in its presumptions,
insensitive to changing goals in the subject matter fi eld, unable to contribute to a revision of
the theory and knowledge on which it is built, and thus limited in its contribution to either
theory or practice.
More comprehensive research and curriculum development efforts
Other recent curriculum development efforts incorporate more of the aforementioned ways of
basing curriculum development on research. In this section, I briefl y describe several of these
efforts in PreK–12 education. (This is a select list prepared to refl ect the international picture;
many laudable efforts have not been included specifi cally, even if they have contributed to the
ideas included here, such as didactical engineering (Artigue, 1994), Hoyles and Noss in the
U.K., Griffi n and Case and Confrey in the U.S., and others.)
One of the longest standing, comprehensive, and innovative projects is in the Netherlands
under the name of Realistic Mathematics Education (RME). The curriculum design pro-
cess is part of a research approach the authors term “developmental research” (Gravemeijer,
1994b). Developmental research is best described as an integration of design and research.
The design of instructional sequences serves as research on an instruction theory. Curriculum
development is conceived as purposeful and sensible tinkering, guided by theory and produc-
ing theory (Gravemeijer, 1994b).
A team begins the process by conducting an anticipatory thought experiment. They for-
mulate a “hypothetical learning trajectory” that involves conjectures about both a possible
learning route that aims at signifi cant mathematical ideas, and a specifi c means that might
be used to support and organize learning along this route (I will return to the learning tra-
jectory construct, including its defi nition, in a following section). These means of support
are construed broadly in three categories: (1) resources, including instructional activities,
notational schemes, and the physical and computer-based tools that students might use; (2)
596 Douglas H. Clements
the classroom social context, including the general structure of classroom participation and
the nature of the specifi c mathematical discourse; and (3) the teacher’s role in supporting the
emergence of increasingly sophisticated mathematical reasoning.
The learning trajectory is conceived of through a thought experiment in which the his-
torical development of mathematics is used as a heuristic; more recently, students’ informal
solution strategies also have been used as a source of inspiration. The original design is a set of
instructional activities with guidelines suggesting an order for the activities and the learning
trajectory, or the mental activities in which the students are to engage as they work through
the instructional activities. This original design is often not worked out in detail because
activities are revised extensively during fi eld testing. That is, the activities that are actually
used in the classroom are determined on a day-to-day basis considering what was learned from
implementing the preceding activities in the classroom.
Second, the educational experiment, this preliminary design is elaborated, refi ned, and
adjusted in a series of intense cyclic processes of deliberations on and trials of instructional
activities (Gravemeijer, 1999). Third, the knowledge gained is used to construct an optimal
instructional sequence. The goal is to develop and describe the local instruction theory (a
more general description of the learning trajectories that emerged in specifi c classrooms) that
underlies this entire instructional sequence and to justify it with both theoretical delibera-
tions and empirical data (Gravemeijer, 1994a, 1994b, 1999). The ideal is that such a local
instruction theory will provide a framework that teachers can use to construe hypothetical
learning trajectories that fi t their own classroom situations.
Recent collaborators with the Netherlands developers (McClain, Cobb, Gravemeijer, &
Estes, 1999), Cobb and his colleagues have similar philosophical and curriculum develop-
ment perspectives (Cobb & McClain, 2002). Theirs is likewise a methodological approach in
which instructional design serves as a primary setting for the development of theory (Cobb,
2001). Like that of the Netherlands, their work posits learning trajectories and frequently
conducts classroom tests. The learning trajectories are hypothetical, and are revised as needed
with each test. The goal is not to “prove” that the initial trajectory is correct or that the origi-
nal instruction plan is effective, but to improve both by modifying them as required by the
daily analyses of students’ thinking and the classroom environment (Cobb, 2001).
These authors argue that their model’s daily cycle of planning, instruction, and analysis
is consistent with the practices of teachers who are skilled in nurturing students’ develop-
ment of deep mathematical understandings (cf. Lampert, 1988; M. A. Simon, 1995; Stigler
& Hiebert, 1999). Therefore, the fi ndings and products of such research and development
efforts are immediately applicable to other classrooms.
Results are also applicable to knowledge development in larger domains. This is because,
although concerns that arise during an experiment relate directly to the goal of supporting
the participating students’ learning, the retrospective analysis of an experiment contributes to
the development of instructional theory. This theory, emerging from analyses of the several
cycles of teaching and learning, explains the relationships between the two and thus generates
grounded generalizations.
Some of their recent changes are especially noteworthy. More than a decade ago, the team
followed a standard psychological approach that focused on individual students’ internal men-
tal reasoning. The demands of working in the classroom led them to adopt a perspective more
consistent with cultural-historical activity theory (Cobb & McClain, 2002). This includes
considering the overall goal or motive of students’ activity. For example, in building the
structure for activities involving data analysis, a goal was for students to participate in discus-
sions of the data creation process. This gave the data a history for the students, refl ecting the
purposes for which it was initially created. In addition, the team has developed interpretive
frameworks that enables the analysis of students’ learning as it occurs in the social context
of the classroom, documenting both the developing reasoning of individual students as they
Linking research and curriculum development 597
participate in classroom practices and the collective learning of the classroom community over
extended periods (Cobb, 2001).
A second recent emphasis is on tools, including computer tools. Their basic design prin-
ciple is to eschew attempts to “build mathematics into” the tools and instead, to focus on
how students use the tools and what they might learn in such activity. Thus, the focus is not
on the tool as “carrying” meaning, but refl ects an increased emphasis on the use of tools
(internal, not external, to students’ activity) as compared to earlier work (Cobb, 1995). In
this experiment, the authors claim that the idea of data sets as distributions would not have
become a signi cant part of the classroom discourse if the design of the computer tools had
been different.
Instructional planning at this level of detail is unusual in the United States. It is more
typical in Japan, where members of professional teaching communities often spend several
years teaching and revising the hypothesized learning trajectories that underpin a sequence of
mathematics lessons (Stigler & Hiebert, 1999).
Japanese educators call these “research lessons”—actual classroom lessons taught to one’s
own students, with a set of unique characteristics (Lewis & Tsuchida, 1998). First, these les-
sons are observed by other teachers and often outside educators as well. Second, they are care-
fully planned, usually in collaboration with one or more colleagues. Third, they are designed to
implement a certain educational vision, similar to the NCTM Standards in the United States,
and simultaneously to illustrate a successful approach to teaching a certain topic. Fourth, they
are recorded and discussed. Videotapes, audiotapes, narrative and checklist observations, and
copies of student work document the lesson. This documentation helps later refl ection and
discussion, including the faculty that developed the lesson, and, frequently, outside educators
and researchers. They are a strong part of professional image and development (Lewis reports
that one teacher said that “if we didn’t do research lessons, we wouldn’t be teachers”).
Many research lessons follow general steps, often with a group of a half-dozen teachers
working through the process together (this account is from Stigler & Hiebert, 1999). First,
the instructional problem is defi ned. This may come from the teachers’ practices, or may be
posed by the National Ministry of Education and addressed by many groups throughout
Japan. It may be a general problem, such as motivating students’ interest in mathematics, or
a specifi c one, such as understanding subtraction with regrouping. In either case, the group
focuses the lesson on the problem until it can be addressed by one lesson.
Second, the group plans the lesson. They look at books and articles by other teachers to
form a hypothesis and a goal that are used to create an effective lesson and to understand why
it was effective. They engage in numerous detailed discussions of the problem with which the
lesson would begin, including: (a) the exact numbers and wording; (b) the materials students
would use; (c) the anticipated solutions and thoughts students might develop; (d) the ques-
tions that could promote student thinking; (e) how chalkboard space would be used; (f) how
to handle individual differences; and (g) how to end the lesson to advance student understand-
ing. The initial plan is presented at a school meeting to solicit criticism. The critical responses
are used to revise the lesson. After several months, the lesson is ready to be implemented.
Third, the lesson is taught. One member of the group teaches it, but everyone in the group
helps in its preparation, including gathering materials and role playing the lesson the night
before. The group observes the lesson as it is taught the next day. Fourth, the group evaluates
the lesson, criticizing weak parts (of the lesson, not the teacher who taught it). Fifth, they
revise the lesson, often based on specifi c student misunderstandings.
Sixth, one member of the group teaches the revised lesson. The audience now includes all
the members of the school faculty. Seventh, the entire faculty, and sometimes outside experts,
evaluate and refl ect on the lesson. The original hypothesis is discussed, as are general issues
of teaching and learning that were illuminated by the lesson and its implementation. Eighth,
the results are shared. A report is published in book form, for the school and sometimes, com-
598 Douglas H. Clements
mercially, for the nation. Teachers from other schools are invited to observe the teaching of
the fi nal version of the lesson.
Yerushalmy (1997) proposed thinking about technology and functions as the founda-
tion of post-arithmetic curriculum. She suggested that the major agenda of algebra teaching
should be equipping learners with tools for mathematizing the perception of the situation
context and that placing function as a central object of the learning could support this evolu-
tion of mathematization. To research such an approach demanded new actions, print, and
computer materials; a reformulation of classroom structures and discourse including new
roles for both student and teacher; and thus a professional willing to make a long-term com-
mitment to the project. Yerushalmy prepared an experimental curriculum, including a full
sequence of algebra activities, innovative software tools, specially designed materials for stu-
dents and teachers, and workshops and materials for professional development. She found
that it is possible to begin with any of the external representations of the function—symbolic
language, numerical language, graphical language, and natural language—and then proceed
to any other representation.
Across several projects, Yerushalmy became convinced of the need to document the reform
processes intensely. The changes that emerged during a relatively short period of time are
relevant to educational reforms; however, the complexity of the learning environment makes
it diffi cult for someone from the outset to understand and replicate the full process. They
closely documented the implementation process along three themes:
1. Following a single classroom with the same teacher and same students for 3 years and
documenting the learning mainly by writing protocols of weekly observations, conversa-
tions with the teacher and the students, and collecting portfolios of students.
2. Conducting a longitudinal study, interviewing 12 pairs of students (of various ability lev-
els) from 4 different classrooms (two different middle schools of different socioeconomic
background and two different teachers in each) twice a year during 3 years on mathemati-
cal problem solving tasks that were not directly addressed in the curriculum but instead
involved conceptual mathematical thinking.
3. Videotaping and analyzing teaching and learning episodes, mainly those that involve
nontraditional classroom discourse.
These data are used to refl ect on and improve the curriculum materials, which are still
undergoing development and revision.
We have been involved intensely in the development of the Investigations in Number,
Data, and Space curriculum, a K–5 reform-based mathematics program. Various units of this
curriculum illustrate a variety of ways curriculum can be based on research. Some of the units
consciously use scienti c models (Battista & Clements, 2000; Clements & Battista, 2000),
resulting not only in research-based curriculum units (Akers, Battista, Goodrow, Clements,
& Sarama, 1997; Battista & Clements, 1995a, 1995b; Clements et al., 1995a; Clements et al.,
1995b; Clements, Russell, Tierney, Battista, & Meredith, 1995) but also in various research
publications reporting the results of these efforts (Battista & Clements, 1996, 1998; Battista,
Clements, Arnoff, Battista, & Borrow, 1998; Clements, Battista, Sarama, & Swaminathan,
1996; Clements et al., 1997; Clements, Sarama, & Battista, 1996, 1998; Clements, Sarama,
Battista, & Swaminathan, 1996). In contrast, most of the other units were built upon knowl-
edge of research on the part of the developers and informal research in classrooms involving
eld testing the materials. They may be as or more effective, but there is less documentation
of their effectiveness. Even worse, there is little or no record of the curriculum development
process from which others might learn and upon which others might build.
Many of these curricula have also been used widely, but specifi c reporting of results of
larger evaluations have only begun to appear (e.g., Mokros, 2003; Stree and, 1991, and
Cobb’s group is planning on working with 10 classrooms). However, such evaluations are
becoming increasing common, such as those by Fuson and colleagues of their own curricula
Linking research and curriculum development 599
and of Everyday Math (Fraivillig, Murphy, & Fuson, 1999; Fuson, Carroll, & Drueck, 2000;
Fuson, Smith, & Lo Cicero, 1997) and the studies of Connected Mathematics 2.1
PRINCIPLES FOR COMPREHENSIVE RESEARCH-
BASED CURRICULUM DEVELOPMENT
From projects such as these (I provided no description of Sarama and my own work, as it is
used to elaborate and illustrate the model I describe in the following section), I abstract sev-
eral principles for comprehensive research-based curriculum development.
Create and maintain connections between research and curriculum
development as integrated, interactive, processes
A synthesis of curriculum development, classroom teaching, and research in mathematics
educational is necessary to contribute both to a better understanding of mathematical think-
ing, learning, and teaching and to progressive change in mathematics curricula. Without
curriculum development projects, rich tasks and authentic settings would be unavailable to
researchers. Such projects serve as sources of, and testing sites of, important research ideas.
Without concurrent research, the curriculum developers and teachers will miss opportunities
to learn about the importance of critical aspects of students’ thinking, and the particular fea-
tures of software, curricula, and teaching actions that engender mathematical development.
I believe that development of research-based curriculum such as that presented here will help
ameliorate this critical problem (Clements et al., 1997; Schoenfeld, 1999).
Th is does not mean that all the va rieties of ways to “base” curriculum on research, as previ-
ously enumerated, must be employed in every project. It does mean that extant research (and
curricula) should be studied and used as a foundation on which to build and that curriculum
development needs to proceed linked with its own dynamic research.
What is the nature of this research? Schoenfeld (1999) places research in a two-by-two
matrix, asking whether the researcher seeks fundamental understanding on one dimension
and whether the researcher considers the application of the fi ndings on the other. The yes/no
cell is pure basic research, the yes/yes is use-inspired basic research, and no/yes is pure applied
research.2 Research such as that I espouse here is placed solidly in the yes/yes cell, seeking
fundamental understanding and direct application of the fi ndings.
Use a broad range of scientifi c methodologies
Scientifi c research in mathematics education curriculum development is variegated. Some
take the stance of traditional aims of science: explanation, prediction, and control. Others
take interpretative and other qualitative perspectives, such as those based on anthropologi-
cal research (Erickson, 1986; Strauss & Corbin, 1990), and seek to understand the mean-
ings that curriculum would have for teachers and students. Taking the perspective of action
research, others examine how to help teachers and students gain autonomy and effective-
ness in their teaching and learning endeavors. None of these perspectives is irrelevant to
research in the service of curriculum development (Mayer, 2000). They underlie, to different
degrees, the enterprise of curriculum development integrated with research, the topic of this
chapter.
Use hypothetical learning trajectories
Many of the successful approaches use learning trajectories (or conceptually similar con-
structs). Such trajectories are descriptions of students’ thinking and learning of a specifi c
600 Douglas H. Clements
mathematical domain and a conjectured route for that learning to follow through a set of
instructional activities (Gravemeijer, 1999; M. A. Simon, 1995). The route and activities
specify the mental actions in which it is hypothesized students engage as they participate in
the instructional activities. Signifi cant also is that there is evidence that superior teachers use
learning trajectories. In one study of a reform-based curriculum, the few teachers that had
worthwhile, in-depth discussions, saw themselves not as moving through a curriculum, but
as helping students move through a progression or range of solution methods—a learning
trajectory (Fuson et al., 2000).
Develop or use models of cognition and models of mathematics
Learning trajectories are often based on specifi c models of students’ thinking and learning.
In addition, the instructional activities frequently use a different type of model—a model of
mathematics used to support that thinking and learning. Gravemeijer (1999) describes how
these models in RME undergo a transition in which such a model initially emerges as a model
of informal mathematical activity (“model of”) and then gradually develops into a model for
more formal mathematical reasoning (“model for”). Both types of models are important; in
our approach, the two are coordinated and synthesized, which we believe provides additional
explanatory and instructional power (Clements & Battista, 2000).
Use stages and cycles of revisions
Most of the successful approaches have well-conceptualized stages. Of ten, a preliminary design
is created (including a learning trajectory and correlated set of instructional activities), then
elaborated and revised through a series of cyclic empirical fi eld tests, leading to fi nal products
that include both an effective curriculum and theoretical and empirical research reports (Cle-
ments & Battista, 2000; Gravemeijer, 1999; M. A. Simon, 1995). Each stage involves the
creation or revision of both instructional activities and psychological and instructional theo-
ries. The cyclic alternations of curriculum development and research is often considered more
effi cient and effective when they are as short as possible (Burkhardt et al., 1989; Clements &
Battista, 2000; Cobb, 2001; Cobb & McClain, 2002; Gravemeijer, 1994b).
Maintain close connections between activities and students’ mathematical thinking
Throughout the stages, it is critical to maintain direct linkages between the instructional
activities and students’ mathematical thinking. In the end, if the project does not help other
people understand students’ thinking and design better activities to promote learning, the
project has failed a major research goal.
Ensure the curriculum is informed by ecological perspectives, including
research on teachers and the social and cultural context
Curriculum does not stand apart from teachers. Teachers’ knowledge, theories, and belief
systems in uence their instructional plans, decisions, and actions, including their implemen-
tation of curricula. Developers must consider these factors, as well as the classroom social
context, including the nature of classroom interactions and roles. Several relevant questions
must be addressed. How do the developers conceive the teacher’s role in supporting the cur-
riculum as it is realized in the classroom? What are the patterns of participation in which
teachers and students engage? What supports do teachers need to realize the vision the cur-
riculum embodies?
Linking research and curriculum development 601
Document and describe the development, implementation,
and evaluation procedures in detail for each stage
Any scientifi c research carefully documents the procedures used. This requirement is espe-
cially intense for research-based curriculum development, when myriad decisions of many
types are made on a variety of bases. This documentation is required to maintain the connec-
tions between instruction and learning and thus to generate grounded generalizations. To
accomplish this and all the previous principles, it is important to have the senior researchers
directly involved in all aspects of the research and development (Clements & Battista, 2000;
Cobb, 2001).
The literature and projects discussed to this point have led us to create two tightly inter-
related structures for curriculum development based on research. The next section briefl y
describes my taxonomy of research categories and phases for research in the context of cur-
riculum development. The following section describes Julie Sarama and my model for cur-
riculum development that includes those categories and phases.
These categories and phases involve a combination of research methods; no single method
would be adequate. For example, design experiments (Brown, 1992; Cobb, Confrey, diSessa,
Lehrer, & Schauble, 2003; The Design-Based Research Collective, 2003), developed as a
way to conduct formative research to test and refi ne educational designs (Collins, Joseph, &
Bielaczyc, 2004) are central, but are usually limited to pilot testing (Fishman, Marx, Blu-
menfeld, Krajcik, & Soloway, 2004; NRC Committee for a Review of the Evaluation Data
on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Cur-
riculum Materials, 2004, p. 75), put too little focus on the development of curricula, and
do not address the full range of questions. The Curriculum Research Framework (CRF) is
based on the assumption that all appropriate methods should be synthesized into a coherent,
complete curriculum framework, as described in Table 23.1 (see Clements, 2006, for a full
description).
A MODEL FOR RESEARCH-BASED CURRICULUM
AND SOFTWARE DEVELOPMENT
As an example of an approach that embodies these principles, I describe Julie Sarama and my
model for integrated research and curriculum development, emphasizing the development of
software. The model is a modifi cation of our previously-presented work (Clements & Battista,
2000; Sarama & Clements, 2002), taking into consideration our own recent experience and
what we have learned from analyzing similar efforts, as described previously. The description
that follows omits elaborated reports of the work, both our own (Battista & Clements, 1991;
Clements & Battista, 1991) and that of others (Biddlecomb, 1994; Olive, 1996; Steffe &
Wiegel, 1994), from which the original model was abstracted; see the original (Clements &
Battista, 2000) for more extensive, concrete, stage-by-stage illustrations from these works.
This design model posits that we now have models of teaching and learning mathematics
with suffi cient explanatory power to permit design to grow concurrently with the refi nement
of these models. Thus, curriculum and software design can and should interact with the
ongoing development of theory and research—reaching toward the ideal of testing a theory
by testing the software. Capitalizing fully on both research and curriculum development
opportunities requires the maintenance of explicit connections between these two domains
and formative research with users throughout the development process (c.f. Laurillard &
Taylor, 1994). Thus, we include all the research phases from our Curriculum Research Frame-
work (CRF) as described in Table 23.1. Ideal, but dif cult, is the inclusion of a third-party
researcher to study the team as it works and to serve as an external auditor for the team’s
research (Lincoln & Guba, 1985). At the very least, the design team needs to document all
their decisions and their reasons for these decisions.
602 Douglas H. Clements
Table 23.1 Categories and phases of the Curriculum Research Framework (CRF), adapted from
Clements (2006)
Categories Questions asked Phases
A Priori Foundations.
In variants of the research-to-
practice model, extant
research is reviewed and
implications for the nascent
curriculum development
effort drawn.
What is already known that
can be applied to the
anticipated curriculum?
Established review procedures (e.g., Light &
Pillemer, 1984) and content analyses (NRC
Committee for a Review of the Evaluation
Data on the Effectiveness of NSF-Supported
and Commercially Generated Mathematics
Curriculum Materials, 2004) are employed to
garner knowledge concerning the specifi c
subject matter content, including the role it
would play in students’ development (phase
1); general issues concerning psychology,
education, and systemic change (phase 2); and
pedagogy, including the effectiveness of
certain types of activities (phase 3).
Learning Model.
Activities are structured in
accordance with empirically-
based models of students’
thinking and learning in the
targeted subject-matter
domain.
How might the curriculum
be developed to be
consistent with models of
students’ thinking and
learning (which are posited
to have characteristics and
developmental courses that
are not arbitrary and
therefore not equally
amenable to various
instructional approaches or
curricular routes)?
In phase 4, the nature and content of activities
is based on models of students’ mathematical
thinking and learning (cf. James, 1958; Tyler,
1949). In addition, a set of activities (the
hypothetical mechanism of the research) may
be sequenced according to specifi c learning
trajectories (D. H. Clements & Sarama,
2004). What distinguishes phase 4 from phase
3, which concerns pedagogical a prior
foundations, is not only the focus on the
student’s learning, rather than teaching
strategies alone, but also the iterative nature
of its application. That is, in practice, such
models are usually applied and revised (or, not
infrequently, created anew) dynamically,
simultaneously with the development of
instructional tasks, using grounded theory
methods, clinical interviews, teaching
experiments, and design experiments.
Evaluation.
In these phases, empirical
evidence is collected to
evaluate the curriculum,
realized in some form. The
goal is to evaluate the appeal,
usability, and effectiveness of
an instantiation of the
curriculum.
How can market share for
the curriculum be
maximized?
Is the curriculum usable by,
and effective with, various
student groups and
teachers? How can it be
improved in these areas or
adapted to serve diverse
situations and needs?
Phase 5 focuses on marketability, using
strategies such as gathering information about
mandated educational objectives and surveys
of consumers.
Formative phases 6 to 8 seek to understand the
meaning that students and teachers give to
the curriculum objects and activities in
progressively expanding social contexts; for
example, the usability and effectiveness of
specifi c components and characteristics of the
curriculum as implemented by a teacher who
is familiar with the materials in small groups
(phase 6) and whole classes (phase 7) and,
later, by a diverse group of teachers (phase 8).
Methods include interpretive work using a
mix of model testing and model generation
strategies, including design experiments,
microgenetic, microethnographic, and
phenomenological approaches (phase 6),
classroom-based teaching experiments and
ethnographic participant observation (phase
7), and these plus content analyses (phase 8).
The curriculum is altered based on empirical
results, with the focus expanding to include
aspects of support for teachers.
Linking research and curriculum development 603
Our design model is speci c for the instructional use of research-based microworlds. How-
ever, with minor revisions, it would be applicable to most software based on similar cognitive
perspectives. Although the model contains a linear sequence, in practice development can
“fold back” to previous stages.
Stage I: Draft initial goals
The fi rst stage begins with the identifi cation of domains of mathematics that are deemed
signifi cant in at least two ways. First, the learning of the domain would make a substantive
contribution to students’ mathematical development. Second, learning about students’ math-
ematical activity in the domain would make a similar contribution to research and theory.
The fi rst way implies both that the domain should play a central role in mathematics per se
and that the concepts and procedures of the domain are generative in students’ development
of mathematical understanding. From the beginning, then, there is involvement of a diverse
set of experts, including mathematicians, mathematics educators (teachers and researchers),
and cognitive psychologists.
Establishing mathematical learning goals might begin with a critical examination of cur-
riculum recommendations (NCTM, 2000, as well as NCTM’s Curriculum Focal Points, and
similar documents or curricula from other countries). These documents were created by a dia-
lectical process among many legitimate stakeholders, and thus serve as a valuable starting point,
as are comparisons to other successful curricula. These are scientifi c research-oriented strate-
gies that constitute part of comprehensive content analyses (cf. NRC Committee for a Review
of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated
Mathematics Curriculum Materials, 2004). A specifi c content analysis should be conducted on
the fi rst draft of the content specifi cations for the curricu lum produced in t his stage. Ascerta in-
ing the role specifi c subject matter content plays in students’ mathematical development also
requires a thorough review of research in the domain (phase 1 in Table 23.1).
The second way to identify domains of mathematics that are deemed signifi cant appears
straightforward, involving the identifi cation of lacuna in research and theory. However, after
identifi cation of a mathematical domain such as addition, mathematical activity has to be
understood from the perspective of the students, which may be distinctly different. Thus,
drafting the initial goals requires analyzing adults’ conceptions of the domain and also devel-
Categories Questions asked Phases
What is the effectiveness
(e.g., in affecting teaching
practices and ultimately
student learning) of the
curriculum, now in its
complete form, as it is
implemented in realistic
contexts?
Summative phases 9 and 10 both use
randomized fi eld trials and differ from each
other most markedly on the characteristic of
scale. That is, phase 10 examines the fi delity or
enactment, and sustainability, of the
curriculum when implemented on a large
scale, and the critical contextual and
implementation variables that infl uence its
effectiveness. Experimental or carefully
planned quasi-experimental designs,
incorporating observational measures and
surveys, are useful for generating political and
public support, as well as for their research
advantages. In addition, qualitative
approaches continue to be useful for dealing
with the complexity and indeterminateness of
educational activity (Lester & Wiliam, 2002).
604 Douglas H. Clements
oping a model of the concepts and strategies of students as they engage in activities that could
be called mathematical.
I take an example from our NSF-funded project, Building Blocks—Foundations for Math-
ematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development
(Clements & Sarama, 2007). One of the domains from the geometry and spatial sense line is
composing geometric forms. The basic competence is combining shapes to produce compos-
ite shapes. This is one section of the larger composing/decomposing trajectory in geometry.
We determined this domain to be signifi cant in that the concepts and actions of creating and
then iterating units and higher-order units in the context of constructing patterns, measur-
ing, and computing are established bases for mathematical understanding and analysis (Cle-
ments et al., 1997; Reynolds & Wheatley, 1996; Steffe & Cobb, 1988). However, there was a
lack of research on the trajectories students might follow in this geometric domain.
The product of this fi rst stage is a description of an important, and often problematic,
aspect of mathematics. This description should be as detailed as possible.
Stage II: Build an explicit model of students’ knowledge
including hypothesized learning trajectories
Developers build a cognitive model of students’ learning that is suf ciently explicit to describe
the processes involved in learning the goal mathematics (phase 4 in Table 23.1). Extant mod-
els may be available, although they vary in degree of specifi city. Especially when details are
lacking, developers use clinical interviews and observations to examine students’ knowledge
of the content domain, including conceptions, strategies, intuitive ideas, and informal strate-
gies used to solve problems. In these experiments, the teacher tries to set up a situation or
task that will elicit pertinent concepts and processes. Once a (static) model has been partially
developed, it is tested and extended with exploratory teaching (Steffe, Thompson, & Glaser-
sfeld, 2000).
These cognitive models are synthesized or extended to form the basis for hypothesized
learning trajectories (Cobb & McClain, 2002; Gravemeijer, 1999; M. A. Simon, 1995).
These trajectories ultimately include “the learning goal, the learning activities, and the think-
ing and learning in which the students might engage” (M. A. Simon, 1995, p. 133). In con-
trast to other approaches (Gravemeijer, 1994b), we believe that existing research should be
the primary means of writing the fi rst draft of these learning trajectories (which may, in turn,
ameliorate the diffi culty many development teams appear to have incorporating the research
of others). The defi nition of learning trajectories for our work is more speci c: “descriptions
of children’s thinking and learning in a specifi c mathematical domain, and a related, conjec-
tured route through a set of instructional tasks designed to engender those mental processes
or actions hypothesized to move children through a developmental progression of levels of
thinking, created with the intent of supporting children’s achievement of specifi c goals in that
mathematical domain” (Clements & Sarama, 2004, p. 83).
As an example, our synthesis of research for the Building Blocks project posits the following
developmental sequence in levels of thinking. The basic structure of this sequence was deter-
mined by observations made in the context of early research (Sarama, Clements, & Vukelic,
1996) and were later refi ned through a research review and a series of clinical interviews and
focused observations by research staff and teachers (see Clements, Wilson, & Sarama, 2004,
for a complete description and empirical support).
1. Precomposer. Manipulates shapes as individuals, but is unable to combine them to com-
pose a larger shape.
2. Piece Assembler. Similar to Step 1, but can concatenate shapes to form pictures. Each
shape represents a unique role, or function in the picture. Can fi ll simple frames using
trial and error. Uses turns or fl ips to do so, but again by trial and error; cannot use
Linking research and curriculum development 605
motions to see shapes from different perspectives. Thus, students at Steps 1 and 2 view
shapes only as wholes and see no geometric relationship between shapes or between parts
of shapes (i.e., a property of the shape).
3. Picture maker. Matches shapes using Gestalt confi guration or one component such as side
length. If several sides of the existing arrangement form a partial boundary of a shape
(instantiating a schema for it), the student can fi nd and places that shape. If such cues
are not present, the student matches by a side length. The student may attempt to match
corners, but does not possess angle as a quantitative entity, so will try to match shapes
into corners of existing arrangements in which their angles do not fi t. Rotating and fl ip-
ping are used, usually by trial-and-error, to try different arrangements (a “picking and
discarding” strategy). Thus, there is intentionality and anticipation (“I know what will
t”), based on shapes’ components.
4. Shape Composer. Matches shapes using angles as well as side lengths. Eventually considers
several alternative shapes with angles equal to the existing arrangement. Rotation and
ipping are used intentionally (and mentally, i.e., with anticipation) to select and place
shapes. Can fi ll complex frames or cover regions. Is beginning to form substitution rela-
tionships among shapes (e.g., two pattern block trapezoids make a hexagon).
5. Substitution Composer. Forms composite units of shapes by trial-and-error. May combine
these composite units by simple duplication.
6. Shape Composite Iterater. Builds and operates on composite units intentionally (i.e., stu-
dents conceptualize each unit as being constituted of multiple singletons and as being
one higher-order unit). Can continue a pattern of shapes that leads to a “good covering,”
but without coordinating units of units.
7. Shape Composer with Superordinate Units. Builds and applies units of units (superordinate
units). For example, in constructing spatial patterns, students extend their patterning
activity to create a tiling with a new unit shape—a (higher-order) unit of unit shapes that
they recognize and consciously construct.
There may be a misunderstanding of the role of such developmental progressions, similar
to a misinterpretation of the results of teaching experiments. That is, some believe that experi-
ments that involve a small number of students are not applicable to classrooms. However,
from such work we can take the cognitive models, developmental progressions, and potential
activities (components of a learning trajectory)—these can be realized within curricula (from
early drafts to fi nal product) and again within various classrooms. The general framework
guides such realization, but multiple developmental progressions and activity sequences can
be tailored to each situation. This is important to the design of curricula both for students
and for teachers (Ball & Cohen, 1996).
The end result of this stage is an explicit cognitive model of students’ learning of math-
ematics in the target domain. Ideally, such models specify knowledge structures, the devel-
opment of these structures, mechanisms or processes of development, and developmental
progressions of nascent learning trajectories that specify hypothetical routes that students
might take in learning the mathematics.
Stage III: Create an initial design for software and activities
Throughout stages 1–3, reviews of pertinent literature in psychology and education (phase 2
and phase 3 in Table 23.1) are studied or performed as needed. Especially in this stage, reviews
of specifi c pedagogical issues, such as the effectiveness of certain types of activities, are con-
ducted to guide the design of software and activities. For examples, refl ecting on the actions
and activities that are enabled by a new technology can catalyze a reconceptualization of the
nature and the content of the mathematics that could and should be learned. The fl exibility
of computer technologies can generate visions less hampered by the limitations of traditional
materials and pedagogical approaches (Confrey, 1996; Lukens, 1984; Papert, 1998). For
606 Douglas H. Clements
example, computer-based communication can extend the model for science and mathematical
learning beyond the classroom, and computers can allow external representations and actions
not possible with other media.
With those general guidelines in mind, the main work at this stage is to create a basic
design to describe the objects that will constitute the software environment and the actions
that may be performed on these objects, based on the model of students’ learning generated
in Stage II. These actions-on-objects should mirror the hypothesized mathematical activity
of students.
Continuing the Building Blocks example, we wanted students to work with shapes and
composite shapes as objects. We wanted them to act on these objects—to create, duplicate,
position (with geometric motions), combine, and break apart both individual shapes (units)
and composite shapes (units). Offering students such objects and actions on these objects is
consistent with the Vygotskian theory that med iation by tools and signs is critical in t he devel-
opment of human cognition (Steffe & Tzur, 1994). Further, designs based on objects and
actions force the developer to focus on explicit actions or processes and what they will mean
to the students. These characteristics mirror the benefi t attributed to cognitive science models
of human thinking; they do not allow “black boxes” to hide weaknesses in the theory.
Designs are not determined fully by this line of reasoning. Intuition and the art of teaching
(Confrey, 1996; Hiebert, 1999; James, 1958) play critical roles in the design of the objects
and actions, as well as the activities.
The developers next create a sequence of instructional activities that use objects and actions
to move students through the hypothesized developmental progressions—the activities, of
course, constitute the third component of the learning trajectory. They consult the profes-
sional literature and published curricula and draw from their own experiences and creativity,
as they write these activities. They consider the unique potential of technology for providing
cognitive tools, “concrete mathematics,” and “situated abstractions” (Clements, 1994, 2000;
Hoyles, 1993). They also seek extensive advice from teachers.
Returning to the Building Blocks example, we initially created a sequence of activities
aligned with the learning trajectory. An essential task is combining shapes to produce com-
posite shapes (e.g., to fi ll a frame or create an imagined design or picture). Research shows this
type of activity to be motivating for young children (Sales, 1994; Sarama et al., 1996). The
setting for such tasks will be constantly changing (making pictures, fi xing “broken” objects
which “work” or are animated when fi xed, completing jigsaw-like puzzles with pictures, com-
pleting wallpaper patterns or “fl oor tilings,” etc.). For the purposes of brief illustration of the
essential features, only the mathematically signifi cant basic elements are described in Figure
23.1 (further, most activities allow for open-ended projects using the objects and actions).
Given the importance yet paucity of student-designed projects in mathematics education
and the support that the computer can offer such projects (Clements, 2000), provision for
such self-motivated, self-maintained work should be considered. Open-ended activities using
the objects and actions should therefore be a part of the design so that the software envi-
ronment can be a setting in which students think creatively. Design activity on the part of
students is frequently the best way for that to happen. In this, as well as the other activities,
developers, teachers, and students should not be constrained by the scienti cally-based trajec-
tory (see Figure 23.1).
Specifi c assessment and teaching strategies should be included as part of the plan (c.f.
Hoyles & Noss, 1992). Teachers should be encouraged to go beyond the activities and help
students to use the environment not just as a “model of” informal mathematical activity but
eventually as a “model for” investigating other situations, and “esoteric” mathematical prob-
lems and relationships.
Several additional points should be made regarding technology. Thanks to recent develop-
Linking research and curriculum development 607
1. Piece Assembler. In the fi rst level of the “Piece Puzzler” activity sequence,
students complete a picture given a frame that suggests the placement of the
shapes, each of which plays a separate semantic role in the picture and that
requires no fl ips or turns.
2. Picture Maker. Students complete a picture given a frame that suggests the
placement of the individual shapes but in which several shapes together may
play a single semantic role in the picture. As the student succeeds, she is given
pictures that include such combinations more frequently and that require
applying (small) turn actions to the shapes (note: the computer environment
helps bring this action to an explicit level of awareness because the student
must consciously choose the turn tool and because sound effects and speech
are used to explicate the turning action). The student is challenged to fi ll an
open region and is provided shapes in which matching side lengths is a useful
strategy.
3. Shape Composer. The student must use given shapes to completely fi ll a
region that consists of multiple corners, requiring selecting and placing shapes
to match angles. Later tasks challenge students to fi ll complex frames or
regions in which shape placement is ill defi ned, allowing for multiple solutions.
These tasks require use of turning and fl ipping and eventually the
discrimination of these.
4. Substitution Composer. The student is challenged to fi nd as
many different ways as possible to fi ll in a frame or region,
emphasizing substitution relationships (as the student is doing to
the hexagons; the tangram puzzle must be solved at least two
different ways).
5. Shape Composite Iterater. The student works in a toy factory, learning to use
the glue and duplicate tools to make several copies of the same (composite)
toy. The student then completes a toy puzzle (made completely from multiple
copies of a tetromino) using the glue, duplicate, and “do it again” tools to
make and iterate composite units in fi lling space.
6. Shape Composer with Superordinate Units. The student covers
regions by building superordinate units of tetrominoes with the
glue tool that are then duplicated, slid, turned, and fl ipped, and
iterated systematically to tile the plane. For example, she might fi ll
the rectangle at the right with a strategy that combines four “T”
tetrominoes into a superordinate square.
Figure 23.1 An illustrative learning trajectory for shape composition.
As an example, after they have successfully completed one of the series of tasks
described previously, students are invited to create their own designs in the
“Piece Puzzler Free Explore” activity. They choose either pattern blocks or
tangram shapes, then design their own picture, using duplicate, slide, fl ip,
turn, and scissors tools. For example, this student cut rhombuses in half to
create the arms and legs of “Jack jumping over the candlestick.” (The teacher
employed the research-based suggestion of challenging students with specifi c
tasks, in this case, creating a character from a story or nursery rhyme.)
Once they are satisfi ed with their picture, they click the “Play” button and
their picture is converted into a puzzle. Given this student used all the tools,
this puzzle can actually be more challenging than those in the “Piece Puzzler”
sequence.
608 Douglas H. Clements
ments, even students with physical and emotional disabilities can use a computer with ease—if
the designers plan for it. Developers should plan for the adaptations the software will need for
people with disabilities (e.g., hearing—adjustable volume and register for all speech, simpli-
ed captions and visual animation by the agent; visual—high contrast versions of all screens;
physical—key press and single switch access); here I wish to emphasize that the environment
and activities be designed based on research on specifi c effective interventions for learning
disabled (LD) and retarded students (e.g., Baroody, 1996; Kameenui & Carnine, 1998; Mas-
tropieri, Scruggs, & Shiah, 1991; Swanson & Hoskyn, 1998).
In addition, computers can be programming to assess students’ progress, and move them,
along learning trajectories as appropriate. This can be a substantial benefi t for individualiza-
tion, “catching up” students who have been absent, and assessment.
As a fi nal note for this stage, at appropriate points in the design process, evidence should
be collected regarding the marketability of the materials (phase 5 in Table 23.1). In t he United
States, those who ignore concerns of publishers, teachers, and marketability in general often
do not achieve wide adoption (Tushnet et al., 2000). Therefore, such information is collected
as needed throughout the curriculum development process.
The main product of the work at this stage is an initial design for the software and activi-
ties. Aspects of these will be tested in the following stage.
Stage IV: Investigate the components
This stage is especially interwoven with the previous one. For example, in this stage, compo-
nents of the software are tested using clinical interviews and observations of a small number
of students. A critical issue concerns how students interpret and understand the screen design,
objects, and actions. A mix of model (or hypothesis) testing and model generation (e.g., a
microethnographic approach, see Spradley, 1979) are used to understand the meaning that
students give to the objects and actions (phase 6 in Table 23.1). To accomplish this, developers
may use paper or physical material mock-ups of the software or early prototype versions.
In this and the next stage, communication between the developer and programmer is
essential. In much of our work, the same people conduct the design, programming, and
research. If programming is carried out separately, full communication about all of the aspects
(e.g., goals, actions, objects, aesthetics, etc.) should be ensured.
Equity must be addressed throughout the stages. As an example relevant to this and sub-
sequent stages, thought should be given to the students who are envisioned as users and who
participate in fi eld tests; a convenience sample is usually inappropriate. Systemic classroom
and home participation patterns and sociocultural issues should be considered as well (Cobb,
2001).
A small example from the Building Blocks project is our research on students’ initial inter-
pretation of the actions that each icon might engender. For the decomposition of units, we
had created a hammer icon. Students did not interpret this tool as breaking things apart, even
with minor prompts, but as “nailing down” items (“It will hammer the shapes down harder”)
or “hammering it off” the paper or screen. We therefore tested new icons, determining that a
“split” button, with an icon of a broken shape, was the most communicative.
The product of this stage is similar to that of Stage III, producing a design for the software
and activities. This stage has produced a piloted version of the critical components. These
components will be tested and combined to create a curriculum in the following stages.
Stage V: Assess prototypes and curriculum
The developers continue to evaluate the prototype (phase 6), rendered in a more complete
form. A major goal is to test hypotheses concerning features of the computer environment
that are designed to correspond to students’ thinking. Do their actions on the objects sub-
Linking research and curriculum development 609
stantiate the actions of the researcher’s mental model of students’ mathematical activity? If
not, should the mental model be changed, or the way in which this model is instantiated in
the software? Do students use the tools to perform the actions, either spontaneously or with
prompting? If the latter, what type is successful? In all cases, are students’ actions-on-objects
enactments of their cognitive operations (Steffe & Wiegel, 1994), and as models of informal
mathematical activity (c.f. Gravemeijer, 1999), in the way the model posits, or merely trial-
and-error or random manipulation?
Similarly, the developers test the learning trajectories and adjust them as needed. Teach-
ing experiments are used initially. Often, a free exploration stage precedes the introduction
of activities. In addition, the developer interprets the students’ contributions, and poses new
tasks or questions. Students’ responses may indicate a need—or, more positively stated, an
opportunity—to change the cognitive model, software environment, trajectories, and activi-
ties. Some activities and teaching strategies emerge from, and are mutually constituted by, the
developer-teacher and the student in the software context. Thus, empirical data may be gar-
nered from the interactions of the students with the software, the activities (writ large), their
peers, the teacher-developer, and combinations of these. In addition, responses and advice of
teachers playing the role of students are sought.
Throughout this testing period, the ironic goal is to “fail often”; that is, to fi nd gaps or
inaccuracies in the cognitive model, the learning trajectories, and the activities, and adjust
them through intensive and extensive cycles of testing and refl ection. Indeed, this is the
most iterative research-design stage; sometimes evaluation and redesign may cycle in quick
succession, often as much as every twenty-four hours (Burkhardt et al., 1989; Char, 1990;
Clements & Sarama, 1995; Cobb, 2001). In this way, the computer environment is modifi ed
in ways not originally anticipated to fi ne tune, correct problems, check speed, and add func-
tions whose need was not foreseen. Similarly, the cognitive model, learning trajectories, and
activities are revised. Activities may be completed reconstituted, with edited or newly-created
activities tried the next day.
Finally, using the cognitive model and learning trajectories as guides, and the software and
activities as catalysts, the developer creates more refi ned models of particular students. Simul-
taneously, the developer describes what elements of the teaching and learning environment
were observed as having contributed to student learning. The theoretical model may involve
disequilibrium, modeling, internalization of social processes, practice, and combinations of
these and other processes. The connection of these processes with specifi c environmental
characteristics and teaching strategies and student learning is critical.
This brings us to another point. Many pedagogical issues can and should be addressed
in this and the following stages. Space limitations prevent us from listing them all here, but
developers ca nnot ignore them. Perhaps most important, my focus here is on the development
of software programs, which should not be misinterpreted as designating less importance to
the social ecology. Further, the computer tools themselves can contribute to discourse and
communication in several ways. For example, students might produce new information in the
form of “notes” and enter them into a database that the whole class shares (Scardamalia &
Bereiter, 1992). Web-based communication is another avenue with myriad possibilities. In
this way, classroom discourse and classroom activity structures (Cobb & McClain, 2002) are
considered when planning and assessing the prototype; however, they are developed more
completely in the following stages.3
With so many research and development processes occurring, and so many possibilities,
extensive documentation is vital. Videotapes (for later microgenetic analysis), audiotapes,
and fi eld notes are collected. Computers might store data documenting students’ ongoing
activity. Solution-path recording is a particularly useful technique (Gerber, Semmel, & Sem-
mel, 1994; Lesh, 1990). Solution paths can be re-executed and examined by the teacher,
student, or researcher (and analyzed in many ways); they also can be modi ed. Issues such as
the ef ciency, simplicity, and elegance of particular solutions—even those that result in the
610 Douglas H. Clements
same answer—can be assessed (Lesh, 1990). Techniques such as videorecording a mix of two
inputs, traditional camera video, and computer screen output serve similar purposes. This
documentation also should be used also to evaluate and refl ect on those components of the
design that were based on intuition, aesthetics, and subconscious beliefs.
In the Building Blocks project, we tested the tools for composition and decomposition.
We had found that students using the computer tools develop compositional imagery. Mitch-
ell started making a hexagon out of triangles. After placing two, he counted with his fi nger
on the screen around the center of the incomplete hexagon, imaging the other triangles. He
announced that he will need four more. After placing the next one, he said, “Whoa! Now,
three more!” Whereas off-computer, Mitchell had to check each placement with a physical
hexagon, the intentional and deliberate actions on the computer lead him to form images
(decomposing the hexagon mentally) and predict each succeeding placement.
As a second example, consider Alyssa, whose work is illustrated in the fi rst picture of
Step 4 of Stage III. As Alyssa fi lls the hexagons, she evinces understanding of both anticipa-
tory use of geometric motions and substitution relationships and therefore notions of area,
equivalence, and congruence. The second activity challenged Alyssa to fi nish covering a wall
with wallpaper. She was partially successful, but the developer recorded that she sometimes
ounders. When she slid a square near a 60° corner, the developer suggested, “Look at the
corners.” By the end of this activity, Alyssa showed that she could completely fi ll a region and
therefore understood covering a plane and did so by matching angles (Alyssa slid a square and
later turned and slid the large angle of the rhombus into matching angles). Based on these
assessments, the developer decided to move to activity Level 5 for Alyssa’s next session.
The product of this stage is the fi rst full realization of the curriculum. In addition, research
data continues to be collected and analyzed, as it is in every stage.
Stage VI: Conduct pilot tests in a classroom
Teachers are involved in all stages of the design model. Starting with this stage, a special
emphasis is placed on the process of curricular enactment (Ball & Cohen, 1996). Curricu-
lum materials should help teachers interpret students’ thinking about the activities and the
mathematics content they are designed to teach; support teachers’ learning of that content,
especially which is probably new to teachers; and provide guidance regarding the external
representations of content that the materials use (Ball & Cohen, 1996).
There are two research thrusts (see phase 7 in Table 23.1). First, teaching experiments
continue, but in a different form. Classroom-based teaching experiments are conducted with
one or two students. The goal is making sense of the curricular activities as they were expe-
rienced by individual students (Gravemeijer, 1994a). Such classroom-based teaching experi-
ments serve similar research purposes as traditional teaching experiments but are conducted
in a naturalistic classroom setting. Videotapes and extensive fi eld notes are required so that
students’ performance can be examined repeatedly for evidence of their interpretations and
learning. Developers evaluate whether the objects-and-actions serve not just as a “model of
informa l mathematical activity, but also develop into a “model for” more formal mat hematical
reasoning (Gravemeijer, 1999).
Second and simultaneously, the entire class is observed for information concerning the
usability and effectiveness of the software and curriculum. Ethnographic participant observa-
tion is used heavily to study the teacher and students as they establish new types of classroom
cultures and interactions together (Spradley, 1980). Thus, the focus is on how the materials
are used and how the teacher guides students through the activities (for our preschool mate-
rials, child care providers and parents are also involved; class dynamics cannot be taken as a
given). Attention is given to how software experiences reinforce, complement, and extend
learning experiences with manipulatives or print (Char, 1989) as well as the diversity in the
practices of students’ homes.
Linking research and curriculum development 611
This pilot test stage involves teachers working closely with the developers. The class is
taught either by a team including one of the developers and the teacher, or by a teacher
familiar with and intensively involved in curricula development. The Building Blocks project
included several tests of learning trajectories (e.g., the entire shape composition, or counting
strand) in this stage.
The product of this stage includes a more polished, usable, version of the software and the
curriculum. Although only teachers experienced with the curriculum teach at this stage, sup-
ports for teachers are addressed in the revisions.
Stage VII: Conduct fi eld tests in multiple classrooms
The size and scope of evaluation studies is gradually expanded (Burkhardt et al., 1989), from
studies of students’ learning in the previous stages (1 to 10 students) to studies of different
kinds of teaching and their effects on student learning (10 to 100 students, phase 8 in Table
23.1) to studies of what can actually be achieved with typical teachers under realistic circum-
stances (100 to 1000 students, phases 9 to 10). These fi eld tests are conducted with teachers
not initially connected intimately with development.
In several classrooms, the entire class is observed for information about the effectiveness
and usability of the software and the curriculum, but more emphasis is placed on the usability
by such teachers. There is too little research done at this level. Innovative materials too often
provide less support than the textbooks with which teachers are accustomed, even when they
are teaching familiar material (Burkhardt et al., 1989). We need to understand what the cur-
riculum should include to fully support teachers of all levels of experience and enthusiasm for
adopting the new curriculum. Thus, in this stage developers evaluate whether the software
and its supporting materials are fl exible enough to support multiple situations (e.g., variation
in the number of computers available), various modes of instruction (e.g., demonstration to a
class, class discussion, small group work), and different modes and styles of management (e.g.,
how teachers track students’ progress while using the materials, monitor students’ problem
solving with the materials, and assess students’ learning). Another question is whether the
materials support teachers if they desire to delve more deeply into their students’ thinking and
teach differently, such as consistent with the vision of the NCTM Standards. Can teachers
adapt the materials to their own vision? Again, ethnographic research (Spradley, 1979, 1980)
is important, especially because teachers may agree with the curriculum’s goals and approach
but their implementation of these may not be veridical to the developers’ vision (Sarama, Cle-
ments, & Henry, 1998), and in this stage, developers need to determine the meaning that the
various curricular materials have for both teachers and students. In addition, of course, the
nal fi eld tests may include summative evaluations.
To supplement these data, two additional types of data are collected from numerous other
classrooms, many of which are located some distance from the developers. First, surveys of
all teacher participants can be used to compare data collected before and after they have used
the software. Simultaneously with these surveys, developers analyze data collected by the
computer and from paper-and-pencil assessments produced by all participating students. Such
data are important in generating political and public support for any innovative materials.
Such research is more similar to, but still differs from, traditional summative evaluation. In
this design model, a theoretical framework is essential; comparison of scores outside of such a
framework, permitted in traditional curriculum evaluation, is inadequate.
The combined interpretive and survey data address whether such supports are viewed
as helpful by teachers and other caretakers and whether their teaching practices have been
infl uenced. Do before-and-after comparisons indicate that they have learned about students’
thinking in specifi c mathematical domains and adopted new teaching practices? Have they
changed previous approaches to teaching and assessment of mathematics?
My hypothesis is that such software developments will have signifi cant positive infl uence
612 Douglas H. Clements
on teachers, thus addressing numerous issues such as equity (adults in diverse settings and
possessing various levels of educational background receive research-based, and thus quality,
prompts that will help them guide students’ learning), assessment (teachers will have authen-
tic, observational assessment information), and thus scalability (reaches a diverse population
and also permits data collection on a wide scale). Moreover, one ideal is that teachers build
new activities from the software environments, take control of their curriculum, and develop
“research lessons” of their own. Certainly, creating, or at least documenting specifi c concerns
for, materials for professional development should be well underway by this stage.
The fi rst summative (phase 9) evaluation of the Building Blocks PreK curriculum resulted
in signifi cant differences, with effect sizes of 1.71 for number and 2.12 for geometry (Clements
& Sarama, in press). Effect sizes of the fi rst of two large scale evaluations (phase 10) ranged
from .46 (compared to another research-based curriculum) to 1.11 (compared to a “home
grown” control curriculum). Achievement gains of the experimental group were thus compa-
rable to the sought-after 2-sigma effect of individual tutoring (Bloom, 1984).
The product of this stage is a complete curriculum, with appropriate supports for a variety
of teachers. The fi rst seven stages provide a comprehensive approach to obtaining both advice
from users and signifi cant research data. Not every project can or should employ each stage;
however, the reasons for omitting any stage and the coherence of the stages that are to be
included must be considered and documented.
Stage VIII: Recurse as necessary
Not really a distinct process so much as a reminder, this stage involves iterative and recursive
actions within and between stages. The intensive and extensive cycles of design and analysis
and evaluation are critical to the success of both curriculum and research. There are three
types of cycles: daily revisions of software environment and activities, longer cycles encom-
passing an entire learning trajectory or curriculum, and cycles that operate across projects.
Substantive progress is often made when a complete project (e.g., Battista & Clements, 1991;
Clements & Battista, 1991) is revisited, refi ned, reconceptualized, and reborn in similar (Cle-
ments & Meredith, 1994; Clements & Sarama, 1995) or different (Clements & Sarama,
1998) forms.
As an example, the Building Blocks curriculum progressed through printed versions for
formative evaluations before being published in each of its two published versions. Here we
wish to emphasize substantial changes that occurred in the second version. Some teachers,
and most of the program directors from the National Science Foundation, were not pleased
that the learning trajectories, although embedded in the activities of the fi rst published ver-
sion (Clements & Sarama, 2003; Schiller, Clements, Sarama, & Lara-Alecio, 2003), were not
explicitly stated in the teacher materials. We collaborated with the publisher to eld test and
publish another version in which, for all parties concerned, especially the teacher, the learning
trajectories stood at the core (Clements & Sarama, 2007). Each week begins with a discus-
sion of the learning goals, developmental progressions, and activities linked to these progres-
sions—the three components of a learning trajectory. Correlational charts show teachers the
connections between every activity and the levels of thinking each was designed to engender.
Full developmental progressions for each learning trajectory are printed in the appendix. The
software includes the same information, with reports available for individual students and for
the class by learning trajectory. And so forth. We have found this to make a substantial differ-
ence in the effect of the materials on teachers and, in turn, on students (Clements & Sarama,
2006).
Stage IX: Publish
The software and curricula may be disseminated through a variety of channels, from com-
Linking research and curriculum development 613
mercial publishers to the Internet. As simple as this seems, this stage is not unproblematic for
either curriculum development or research.
On the curriculum side, negotiations and cooperation with a commercial publisher can
have a substantive in uence on the fi nal software and print materials. The demands on, and
of, publishers, were detailed in previous sections. Suffi ce it to say that these same pressures
are exerted on any curriculum that is commercially published. In addition, multimedia based
materials often require even more support and cooperation from publishers. Therefore, there
may be less freedom for developers to publish their own version of their materials. These pres-
sures often are exerted regardless of the research base for the materials, resulting in software,
originally designed to support in-depth problem solving and student evaluation of math-
ematical strategies and products, to shift towards activities characterized by simpler problems
and feedback. To publish the Building Blocks materials in the form in which they were written,
we talked with several publishers, and started to work with two, before fi nding one that was
committed to research-based curricula.
On the research side, there also are constraints to publication. Many interesting pieces
of software have been created; however, the expertise developed during the production of
that software has not been disseminated. Whether this is because resources are exhausted
(fi nances, time, and emotional energy) or because there is no interest, nonpublication has a
strong deleterious effect on the fi eld of curricula development and research.
Stage X: Continue to conduct large-scale evaluations and revise
Evaluations must be con rmed by researchers unrelated to the developers of the curriculum
(Darling-Hammond & Snyder, 1992), with attention given to issues of adoption and diffu-
sion of the curriculum (Fishman et al., 2004; Rogers, 2003; Zaritsky, Kelly, Flowers, Rogers,
& O’Neil, 2003). Information collected here may be used to develop or refi ne supplementary
materials (e.g., professional development materials) as well as to guide the next edition of the
curriculum.
CONCLUSIONS
Commercially published textbooks strongly infl uence teaching practices in traditional and
reform classrooms (Goodlad, 1984; Grant et al., 1996). They constitute an essential resource
for many teachers. Although their in uence is often conservative, it is not reasonable to expect
teachers to teach well without mathematics curriculum materials. Furthermore, such materi-
als can play positive roles in teaching and reform (Ball & Cohen, 1996; Sosniak & Stodolsky,
1993). This chapter is based on the view that these materials should be centered on scienti c
research. To that end, I have presented the nature and relationship of science, research, and
curriculum and described several models for linking research and curriculum development.
Those implementing such a model assume a responsibility to describe the details of their
theoretical and empirical foundations and their design and to conduct the research deemed
necessary not only to see if the design is successful, but also to trace whether that success
can be attributed to the posited, theory-design connections. Realizing the full potential of
both the research and the curricula development opportunities requires consistent, coherent,
formative research using multiple methodologies. Some have been discussed, among them
clinical interviews, protocol analyses of students’ problem solving, classroom observations,
and interviews with teachers, students, and administrators. Others, such as paired teach-
ers’ observations, students’ immediate retrospective reports of their strategies, performance
assessments, portfolio development, and content analyses of students’ work, may be more
suitable in certain situations. In any case, repeated intensive investigations are required.
I offer several caveats and suggestions.
614 Douglas H. Clements
1. Deve loper s must consi der t ha t crit er ia fo r suc cess s hould al so inclu de t he vi sion a nd t heor y
that underlie the curriculum. For example, the following might be evaluated: students’
mathematical power and their opportunities to develop it through creation of their own
strategies; students’ expression and communication of their mathematical thinking; and
students’ beliefs and attitudes toward mathematics and the tools and resources provided
them to do mathematics. Similarly, the ways the curriculum is understood, adopted, and
adapted by teachers, as well as the way it affects their beliefs, attitudes, knowledge, and
future practice, are relevant. All of these fi ndings inform the curriculum development
process and the research base.
2. Technology and its use in our culture are changing rapidly. Designs, research questions,
and methodologies should remain sensitive to new possibilities. Research indicates that
technological bells and whistles should not become a central concern, however, although
they can affect motivation, they rarely emerge as critical to students’ learning. Instead,
the critical feature is the degree to which the computer environment successfully imple-
mented education principles born from speci c research on the teaching and learning of
specifi c mathematical topics (Sarama, 2000).
3. While this model offers comprehensive, rich data collection, the diversity of the method-
ologies employed could lead to incoherence and confusion between theoretical assump-
tions. Constant refl ection and checks are ever more important in models such as this
one. We must consistently ask questions such as the following. What are we attempting
to learn? What evidence would convince us? What types of circularity in our design and
research work might lead to spurious conclusions?
4. Along a similar vein, our theoretical models and software—to an extent, instantiations of
these models—may funnel our perceptions and conceptions. Testing or refi ning our the-
ories by testing or refi ning our software has signifi cant advantages: We make our theories
more explicit and we extend our visions of what students can do mathematically. Given
the emotional investment in such a complex process, however, we must take precautions
that our work does not contain self-gratifying, self-fulfi lling circularities.
5. The model and examples described here emphasize one class of effective software. The
developmental model would need to be modifi ed for other classes, such as intelligent
tutorials with microadaptation assessment. However, the basic goals and procedures
could be quite similar.
6. Subtle differences in activities can enhance or sabotage the principles (Sarama, 2000). The
basic research principles must be refi ned and especially elaborated by ongoing research
and development work that tracks the effectiveness of every specifi c implementation. This
means that research cannot be considered only something upon which curriculum and
software development are a priori based. Research must also be conducted throughout
the development process.
IMPLICATIONS
In this fi nal section, I describe some of the ramifi cations of this chapter’s arguments.
Curriculum developers must accept new responsibilities
The most direct implication is that curriculum developers must accept new responsibilities.
The models described herein make daunting demands. Curriculum developers must expand
their knowledge to include scientifi c research procedures and ideas—and a wide range at that.
They must consider issues of mathematics, psychology, instruction, and implementation in
turn (Gravemeijer, 1994b). In our vision, curriculum development is painted as an extremely
creative, complex enterprise in which multiple demands must be met and multiple resources
used.
Linking research and curriculum development 615
Developers should study all research
The position taken here is that theoretical purity is less important than a consideration of
all relevant theories and empirical work. The complexity of the fi eld often creates a Babel of
disciplines (Latour, 1987) in which the lack of communication prevents progress. This is one
conceit curriculum developers can ill afford. Instead, they must meld academic issues and
practical teaching demands no less than a serious consideration of what researchers and teach-
ers from other philosophical positions experience and report. This does not imply inconsistent
positions. It does imply that overzealous applications (often misinterpretations and overgen-
eralizations) can limit practical effectiveness. As merely one illustration, constructivism does
not imply that practice is not necessary and does not dictate specifi c pedagogical practices
(Clements, 1997; M. A. Simon, 1995).
Along a related vein, this chapter presented our own design model, based on principles
abstracted from our own and others’ work. However, these other models, and still more not
described here, have unique features and advantages that any curriculum developer should
also investigate.
Developers should remain receptive to the successes of varied approaches
Given its scientifi c basis, can research-based curricula be outperformed? Of course. I dis-
cussed previously that curriculum development is an art as well as a science. James has more
to say on this subject.
The science of logic never made a man reason rightly, and the science of ethics (if there be
such a thing) never made a man behave rightly. The most such sciences can do is to help
us catch ourselves up and check ourselves, if we start to reason or to behavior wrongly;
and to criticise ourselves more articulately after we have made mistakes. A science only
lays down lines within which the rules of the art must fall, laws which the follower of
the art must not transgress; but what particular thing he shall positively do within those
lines is left exclusively to his own genius. One genius will do his work well and succeed
in one way, while another succeeds as well quite differently; yet neither will transgress
the lines…. And so everywhere the teacher must agree with the psychology, but need not
necessarily be the only kind of teaching that would so agree; for many diverse methods of
teaching may equally well agree with psychological laws. (James, 1958, p. 24)
Thus, there are many approaches, but each should be consistent with what is known about
teaching and learning. Researcher-developers should be amenable to the lessons learned by
any curriculum that leads to desirable outcomes. If such approaches are not based on research,
they should use research methodologies to document these outcomes and investigate why the
approach is successful. Without such research, the curricula will be limited in their contribu-
tion to all succeeding curriculum development projects.
Developer/researchers should use the multiple phases in
the curriculum research framework (CRF)
Particular research designs and methods are suited for specifi c kinds of investigations and
questions, but can rarely illuminate all the questions and issues in a line of inquiry (cf. NRC,
2002, p. 4; NRC Committee for a Review of the Evaluation Data on the Effectiveness of
NSF-Supported and Commercially Generated Mathematics Curriculum Materials, 2004).
This is why different methods are used in various phases of the CRF (Clements, 2006).
For example, although iterating through one or two of the phases might lead to an effective
curriculum, this would not meet all the goals of an integrated research and development
program. As a simple example, the curriculum might be effective in some settings, but not
616 Douglas H. Clements
others, or it might be too dif cult to scale up. Moreover, we would not know why the cur-
riculum is effective.
Some might argue that using multiple stages and phases are logistically or practi-
cally infeasible. Consider, with the hundreds of millions of dollars undoubtedly spent on
developing and testing mathematics curricula without producing satisfactory evaluation
data (NRC Committee, 2004), is it impracticable to use the proposed framework? I argue it
is impractical to spend such sums without using it.
Developers should support professional development and systemic change
Curriculum has a large effect on teaching and learning in the United States. This does not
mean, however, that this “intended curriculum” determines classroom practice (Sosniak &
Stodolsky, 1993). Beliefs and former experiences in uence how teachers interpret an innova-
tion (Haimes, 1996; Sarama et al., 1998). If research-based curricula are developed, teachers
will not necessarily adopt their philosophy, especially if it con icts with their traditional beliefs
and practices. Teachers may instead give priority to curriculum content coverage, emphasize
methods and procedures, and adopt teacher-focused pedagogical practices (Haimes, 1996).
Changing teacher beliefs is incredibly dif cult, but necessary (Prawat, 1992). Essential, then,
is the provision of meaningful and accessible support materials and pre- and in-service training
(Haimes, 1996; Sarama et al., 1998). These efforts must acknowledge that teachers face many
competing requests for reforms in many different content areas (Grant et al., 1996; Sarama
et al., 1998), that they are not adequately knowledgeable about teaching practices consistent
with reform standards (Gravemeijer, 1994b; Kemis & Lively, 1997), and that “teachers who
take this path must work harder, concentrate more, and embrace larger pedagogical responsi-
bilities than if they only assigned text chapters and seatwork” (Cohen, 1988, p. 255, as cited
in Prawat, 1992). Also important are issues of systemic change, and thus studies and curricu-
lum change efforts a much larger levels than the curriculum development process described
here (Burkhardt et al., 1989).
The education community should support and heed the
results of research-based curriculum development
Given the grounding in both comprehensive research and classroom experience, the curricu-
lar products and empirical fi ndings of such integrated research and development programs
should be implemented in classrooms. Curriculum developers should follow models and base
their development on the fi ndings and lessons learned from these projects. Administrators
and policy makers should accept and promote curricula based upon similar research-based
models. Educators at all levels should eschew software that is not developed consonant with
research on students’ learning of mathematics and that does not have the support of empirical
evaluation. This would eliminate much of what is presently used in classrooms. This is a strong
position, but one that may avoid a backlash against the use of computers in education, and the
use of innovative curricula in general, and that will, I believe, ultimately benefi t students.
Fortunately, the design models discussed here, with their tight cycles of planning, instruc-
tion, and analysis, are consistent with the practices of teachers who develop broad conceptual
and procedural knowledge in their students (Cobb, 2001; Lampert, 1988; M. A. Simon,
1995; Stigler & Hiebert, 1999). Therefore, the curriculum and fi ndings are not only appli-
cable to other classrooms but also support exactly those practices.
Universities should legitimize research-based curriculum development
There is a long history of bias against design sciences.
Linking research and curriculum development 617
As professional schools, including the independent engineering schools, are more and
more absorbed into the general culture of the university, they hanker after academic
respectability. In terms of the prevailing norms, academic respectability calls for sub-
ject matter that is intellectually tough, analytic, formalizable, and teachable. In the past,
much, if not most, of what we knew about design and about the arti cial sciences was
intellectually soft, intuitive, informal, and cookbooky. Why would anyone in a university
stoop to teach or learn about designing machines or planning market strategies when
he could concern himself with solid-state physics? The answer has been clear: he usually
wouldn’t. (H. A. Simon, 1969, pp. 56–57)
In particular, the more that schools of education in prestigious research universities “have
rowed toward the shores of scholarly research the more distant they have become form the
public schools they are bound to serve” (Clifford & Guthrie, 1988, p. 3, as cited in Wittman,
1995). This is a dangerous prejudice, and one we should resist. Mathematics education might
be seen largely as a design science, with a unique status and autonomy (Wittmann, 1995).
“Attempts to organize mathematics education by using related disciplines as models miss the
point because they overlook the overriding importance of creative design for conceptual and practi-
cal innovations” (Wittmann, 1995, p. 363). The converse of this argument is that universities
benefi t because the approaches described here will prove practically useful, they will legitimize
academic research per se.
Some argue that curriculum should be carried out only by experts (Battista & Clements,
2000). “…A teacher can be compared more to a conductor than to a composer or perhaps
better to a director…than to a writer of a play” (Wittmann, 1995, p. 365). In this chapter, I
mitigate this argument to welcome creative efforts, but argue forcibly that research method-
ologies be used to evaluate every curriculum offered to others and that specifi c curriculum
development projects follow research-based models. Further, I argue that although the publi-
cation of a research-based curriculum should yield the authors full scholarly credit, such credit
depends on the parallel publication of the full body research that the curriculum development
process necessarily involved.
Funding agencies should reconsider time frames and funding
requirements for curriculum development
Until a much larger body of research-based development is created, greater funding oppor-
tunities for research-based curriculum development are needed. Software is even more costly.
Multimedia components, speech production and recognition, well-designed tools, interactive
diagrams, and the like are expensive. They greatly increase the cost of software, with which,
even in its traditional forms, is dif cult to make a profi t.
Such funding should also consider the time period such development requires. In the
development of traditional curricula, there are deadlines, but any extra time that might exist
is used to improve the product, rather than for refl ection and research (Gravemeijer, 1994b).
Curriculum projects that are funded usually are given implausible time frames that make such
refl ection and research nearly impossible, such as 5 years to develop 5 years of curriculum
(Schoenfeld, 1999).
Policy makers should support and insist on research-based curricula
To garner this type of support, curriculum developers need to be proactive, particularly in the
political arena and especially when they are reform-oriented. “Decisions about educational
reform are driven far more by political considerations, such as the prevailing public mood,
than they are by any systematic effort to improve instruction” (Dow, 1991, p. 5). The pro-
portion of funds presently allocated to research in education is abominably inconsistent with
618 Douglas H. Clements
virtually any other enterprise (Dow, 1991; President’s Committee of Advisors on Science and
Technology—Panel on Educational Technology, 1997; Schoenfeld, 1999).
State policy makers, especially those with strict criteria for getting on the “list” of approved
curriculum materials, should change their criteria to require a research basis for curricula and
for the criteria themselves (and not artifi cially and unjustifi ably limited subsets of research).
Adopting this policy, and the defi nition of research-based curriculum described here, should
help all stakeholders avoid the political (in the pejorative sense of the term) swings typifi ed by
California’s recent transition from one end of the pedagogical spectrum to the other. Done
well, it can serve as a partial antidote to the pervasive anti-intellectualism and fundamental-
ism of American politics that eschews honest refl ection and research (H. P. Ginsburg et al.,
1998). Partial is the best that can be expected, however, as values, more than science, fuel
such debates (H. P. Ginsburg et al., 1998; Hiebert, 1999).
All groups should collaboratively address U.S. implementation barriers.
Software development requires cooperation from publishers who are more connected to
research and development than is the present norm. All curriculum development, however,
benefi ts from informed publishers who put the needs of students higher, relative to profi t
considerations, than is presently done.
Systemic issues must be addressed. As just one example, while the Japanese research les-
sons do not have extensive connections to theoretical and empirical research, they have several
unique advantages that should be considered by countries whose integration of research has
been problematic, such as in the United States. They create demand. According to Lewis
and Tsuchida (1998), the United States suffers not from a low supply of good educational
programs, but from a low demand for those programs. Demand occurs when teachers want
to improve their practice—and when they can see the possibility of doing so. Principals say
that research lessons build momentum for improvement more effectively than direct leader-
ship by the principal. There are also lessons for U.S. policy makers, curriculum leaders and
developers. Supporting conditions for research lessons include a shared, frugal curriculum,
collaboration among teachers, critical self-refl ection, and stability of educational policy. A
common, coherent vision such as that of the new NCTM Standards could provide a useful
framework for such work in the United States. We all need to cooperate to change the system
(Stigler & Hiebert, 1999).
Traditional research is conservative; it studies “what is” rather than “what could be.
When research is an integral component of the design process, when it helps uncover and
invent models of students’ thinking and builds these into a creative curriculum, then research
moves to the vanguard in innovation and reform of education.
ACKNOWLEDGMENT
This chapter was supported in part by the National Science Foundation under Grant No.
ESI-9730804, “Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergar-
ten to Grade 2: Research-based Materials Development” and in small part by the Institute
of Educational Sciences (U.S. Department of Education, under the Interagency Educational
Research Initiative, or IERI, a collaboration of the IES, NSF, and NICHHD) under Grant
No. R305K05157 to Clements, J. Sarama, and J. Lee, “Scaling Up TRIAD: Teaching Early
Mathematics for Understanding with Trajectories and Technologies.” Any opinions, fi ndings,
and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily refl ect the views of the funding agencies.
Linking research and curriculum development 619
NOTES
1. Space constraints prohibit describing the many relevant research-based projects from the fi elds of
mathematics education (e.g., Clements, Sarama, 2002; Confrey, Castro-Filho, & Wilhelm, 2000;
Confrey & Lachance, 2000; Hoyles & Noss, 1992; Hoyles, Noss, & Sutherland, 1989; Lehrer &
Chazan, 1998; Lewis & Tsuchida, 1998; Stigler & Hiebert, 1999; Yerushalmy, 1997) and cogni-
tive science (e.g., Anderson, Corbett, Koedinger, & Pelletier, 1995; Brown, 1992; Griffi n & Case,
1997; Lehrer, Jacobson et al., 1998; Lehrer, Jenkins, & Osana, 1998), as well as different concep-
tions such as didactical engineering (Artigue, 1994).
2. Schoenfeld leaves the no/no cell empty; perhaps this is fi nally a resting place for research conducted
only to complete a dissertation or acquire tenure.
3. We use the term “classroom” but it should be noted that there are many types of situations. For
example, especially for preschoolers, our software and curriculum are used in day care and home
settings, among others. Often there is little (directly educational) social setting available to the stu-
dent; while not ideal, considering such situations in the design of educational materials is both more
accurate and equitable than the traditional approach of positing and relying on specifi c classroom
social interactions exclusively.
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... The question, therefore, refers to micro-processes within the curriculum and how this curriculum is implemented (Thijs & Van Den Akker, 2009). Such empirical findings on developing activities that are educationally effective can be considered research-driven curriculum development (Clements, 2010). Hattie and Timperley (2007) distinguish four levels of feedback messages that address different aspects of students' learning: task, process, self-regulation and self. ...
... The results of our study can help teachers design their lessons in a more optimal way, by tailoring the type of feedback more specifically to the student's learning process and mathematical reasoning (Clements, 2010). In our exploratory study, we intended to examine feedback given by teachers during mathematical reasoning lessons from two different viewpoints. ...
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In our multi‐method study, feedback levels derived from the well‐known feedback model of Hattie and Timperley were used in conjunction with feedback that was related to subject‐specific content; here, mathematical reasoning tasks in primary school. Feedback needs to be aligned with the learning process; in the beginning, more task feedback is valuable. Based on the analyses of videos and questionnaires of 44 teachers of 5th‐ and 6th‐grade primary school classes (N = 804), we demonstrated that feedback for finding an approach and operationalisation were related to feedback on the task. We further showed that feedback at the task level predicted students' achievement in mathematical reasoning via students' interest in mathematics. It might be concluded that the four levels of feedback should be applied by teachers in such a way that they focus on the current problem that is occurring while the student is solving a task.</