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From Tinkering to Practice—The Role of Teachers in the Application of Realistic Mathematics Education Principles in the United States



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Chapter 2
From Tinkering to Practice—The Role
of Teachers in the Application of Realistic
Mathematics Education Principles
in the United States
David C. Webb and Frederick A. Peck
Abstract The history of Realistic Mathematics Education (RME) in the United
States has positioned teachers at the centre of innovation from its early years to
present day. From the first proof-of-concept study at a high school in Milwaukee to
localised professional development opportunities, the application and spread of RME
is best characterised as a teacher-centred approach to principled reconsideration of
how students learn mathematics. Such reconsideration of beliefs and conceptions
is often motivated when teachers re-experience mathematics through the lens of
progressive formalisation and related didactic approaches. Through a series of cases
that articulate teacher interpretation and application of RME in U.S. classrooms, we
highlight how teacher participation has led to greater exploration of student-centred
practices. These efforts, while inspired and supported by professional development
and curricula, have been inspired and sustained by teachers who provide colleagues
a proof-of-concept in local contexts.
Keywords Design principles ·Teacher beliefs ·Professional development ·
Mathematics instruction ·Progressive formalisation
2.1 Introduction
Mathematics education in the United States is not typically perceived as a field
that has demonstrated significant innovation in teaching. Many who experienced
mathematics as students speak of the predictability in how it was presented, the
boredom, and the limited relevance. “When are we ever going to use this?” is an
all-too-common refrain shared by students to their teachers. With great consistency
D. C. Webb (B)
University of Colorado Boulder, Boulder, USA
F. A. Peck
University of Montana, Missoula, USA
© The Author(s) 2020
M. van den Heuvel-Panhuizen (ed.), International Reflections
on the Netherlands Didactics of Mathematics, ICME-13 Monographs, 030-20223- 1_2
22 D. C. Webb and F. A. Peck
students’ mathematical experience is one that has been described as hard and boring,
an unfortunate combination that rarely leads to future pursuits in mathematics. It is in
this context that Realistic Mathematics Education (RME), and its design principles
for curriculum, instruction and assessment, recast the mathematical experience as
one that should be meaningful, relevant, and accessible.
In this chapter, we describe the case of how some teachers in the United States
have been influenced by and have benefitted from contemporary Dutch principles of
mathematics education, specifically RME. Our collective experience includes pro-
fessional development, curriculum development, educational research, and the role
of the teacher. Even though there are many others who could articulate similar cases
about their experience with RME, our stance is focused primarily on the character-
istic teacher-centred approach that we have observed and experienced over the past
twenty years through which RME was piloted, disseminated, and integrated into
various mathematics resources in the United States.
2.1.1 The Role of Teachers in Advancing RME in the United
There are several challenges against advancing mathematics teaching beyond the pre-
vailing transmission model of instruction towards more student-centred approaches
that are called for in RME. The challenges that are relevant to this chapter include
the apprenticeship of observation, the inherent complexity of ambitious teaching,
and the system nature of teaching. While these three constructs are not specific to
mathematics education nor education in the United States, these aspects of schooling
all involve teachers and the ways in which their classroom practices are envisioned
and enacted.
In a classic study of teaching in K–12 schools, Lortie (1975) proposed the hypoth-
esis that teachers emulate practices that they experienced as students. During this
‘apprenticeship of observation’, which Lortie (1975) estimated at 13,000 hours of
observed practice, the role of the teacher and the norms and routines of the class-
room are interpreted by future teachers and later imitated. Such recollection of teacher
practice, modelled repeatedly, serves as a basis for recreating the surface features of
classrooms that the novice teacher has come to value as productive. Continuing the
argument, if one’s experiences as a student supported his or her success with math-
ematics (successful enough to pursue a career path into teaching), then those same
practices should benefit future generations of students. Historically, with respect to
K–12 mathematics teaching, this often led to a mathematics that was predominantly
procedural, mechanistic, and predictable. From a case study of secondary schools, a
typical day in a mathematics classroom was described as follows.
First, answers were given for the previous day’s assignment. The more difficult problems
were worked by the teacher or a student at the chalkboard. A brief explanation, sometimes
none at all, was given of the new material, and problems were assigned for the next day. The
remainder of the class was devoted to working on the homework while the teacher moved
2 From Tinkering to Practice—The Role of Teachers … 23
about the room answering questions. The most noticeable thing about math classes was the
repetition of this routine. (Welsh, 1978, p. 391)
What is remarkable is that even though this case study was published almost 40 years
ago, the persistence of this routine of review, lesson and practice can be found when
observing U.S. mathematics classrooms today despite major policy initiatives and
significant resources invested in curricula and professional development. The appren-
ticeship of observation hypotheses provides one possible (if controversial; see Mew-
born & Tyminski, 2006) reason for this.
From an RME perspective, the prevailing routine of school mathematics reflects
what Hans Freudenthal (1983) critiqued as the ‘anti-didactical inversion’ of teaching
the results of mathematical activity, rather than engaging students in the activity itself.
Why should problem solving in realistic contexts be presented as an afterthought,
deferred until the end of skill and concept development as applications when, histor-
ically, authentic problem solving was a motivation for developing new mathematics?
From this, the central tenet of RME was born: Mathematics should be thought of,
first and foremost, as the human activity of mathematising the world.
Supporting student engagement in authentic problem solving requires new mod-
els of teaching. One such model is ambitious teaching (Lampert & Graziani, 2009),
which “requires that teachers teach in response to what students do as they engage
in problem solving performances, all while holding students accountable to learn-
ing goals that include procedural fluency, strategic competence, adaptive reasoning,
and productive dispositions” (Kazemi, Franke, & Lampert, 2009, p. 11). Ambitious
teaching is improvisational, student-centred, and focussed on the development of the
full range of learning goals for mathematical reasoning. This resonates with another
tenet of RME, in that mathematising the world requires authentic problem solving
for students and student-centred instruction by teachers. Ambitious teaching is com-
plex and requires nothing less than a complete overhaul of the prevailing routine in
school mathematics.
Changing this routine is challenged by the system nature of teaching (Hiebert
& Grouws, 2007). Teachers’ decisions and actions are influenced by a milieu of
personal and contextual factors that include teachers’ prior experiences (including
the apprenticeship of observation), teachers’ beliefs about mathematics and about
teaching and learning, local curricular policies, available resources, the expectations
of the community, and other factors. The opportunity for innovation lies at the nexus
of these teacher and context variables. Thus, policies and administrative directives,
on their own, are ineffective approaches to motivate changes in practice. Similarly,
professional development and other opportunities for teacher learning, on their own,
are also insufficient. Sustained innovation in teaching requires systemic changes that
align policies, resources, and activities towards common goals.
In the case of RME, the vision for teaching and learning mathematics that was
articulated by Hans Freudenthal found political support in the United States in 1989,
when the National Council for Teachers of Mathematics (NCTM) published Curricu-
lum and Evaluation Standards for School Mathematics (NCTM, 1989), and related
state and national policies were disseminated throughout the United States. The
24 D. C. Webb and F. A. Peck
vision found further support in curricular resources that were developed using RME
for use in the United States (described in greater detail below). Finally, through
activities that integrated professional development, classroom practice, and academic
research, teachers played a central role in the dissemination and integration of RME
in the United States.
2.1.2 Attractive Features of RME to U.S. Teachers
From years of observing the use and application of RME at all levels of mathemat-
ics, and participating in its implementation, we have developed several hypotheses
regarding its uptake by teachers in their classrooms.
First, with respect to curriculum, RME offers a different approach to engage stu-
dents in new mathematics content. RME’s unique context-first approach frequently
places students’ mathematical engagement on a somewhat level playing field for
students from a wide variety of experiences. The use of problem contexts to learn
new mathematics provides meaningful anchors for student discussions and mathe-
matical activity. Even though this design principle contradicts many U.S. teachers’
prior experiences with mathematics when they were students, the accessibility of
mathematical principles when they are situated within carefully selected contexts
invites more students to participate and contribute to the mathematical discourse.
Mathematically engaged students are a powerful motivator for teachers.
In addition, teachers are attracted to the wide variety of ‘pre-formal’ models and
tools—such as double number lines, percentage bars, and combination charts to
support simultaneous calculations with two variables—that are explained as ways
to promote progressive formalisation from an RME perspective. In the classroom,
these models emerge from realistic activity and are made general though subsequent
activity. As such, they serve as powerful resources for students to do mathematics
and they invite students to make sense of mathematics (Peck & Matassa, 2016;
Webb, Boswinkel, & Dekker, 2008). To teachers, these pre-formal models and tools
often demonstrate ways in which curricular design can support improved student
learning. In professional development, when these models and tools are first used
with teachers, we often hear excitement, followed by puzzlement about why this was
the first time they were seeing such powerful didactical devices (Webb, 2017).
Finally, RME has been attractive to teachers in the United States due to its robust
approach to assessment. Most would recognise that mathematising involves more
than working with procedures and algorithms with precision. Mathematising includes
several characteristic features that involve modelling, problem solving, inductive and
deductive reasoning, developing logical arguments from a set of assumptions, and
so forth. To support teachers in achieving these broader goals, RME offers a com-
prehensive assessment framework (Dekker, 2007; Verhage & De Lange, 1997). This
framework, usually illustrated as an assessment pyramid with three dimensions, has
been used by teachers to support students’ mathematical reasoning in their class-
2 From Tinkering to Practice—The Role of Teachers … 25
2.2 Introduction of RME in the United States: Late
1980s—Mid 1990s
During the 1980s, RME was being articulated in primary and secondary school
reforms in the Netherlands. During the latter half of that decade, Thomas A. Romberg,
a professor from the University of Wisconsin who was deeply interested in curricu-
lum and policy in mathematics education, was chairing a committee that was putting
the final touches on Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989). In the spring of 1988 Jan de Lange, the director of the Freudenthal
Institute was invited to meet Romberg at the National Center for Research in Mathe-
matical Sciences Education (NCRMSE) at the University of Wisconsin-Madison. It
was a beneficial development that these two mathematics educators with a passion
for reforming mathematics teaching and learning, on opposite sides of the Atlantic,
would become colleagues and partners. One might observe that such international
partnerships are somewhat rare in mathematics education, with few publications
co-authored by colleagues from different countries.
The aforementioned NCTM standards articulated a student-centred model of
mathematics education oriented around problem solving. In recounting the story
as told by Romberg, it was understood that the release of the ‘Standards’ would be
followed soon after with significant support from the National Science Foundation
(NSF) for the development of instructional materials, professional development, and
multiple systemic initiatives to support the vision for school mathematics. It is worth
noting that decades before, Romberg was a graduate student at Stanford working
with Ed Begle and the School Mathematics Study Group in documenting how the
post-Sputnik New Math materials impacted teaching and learning (an effort, much
of which, was also funded by the NSF). So Romberg was no stranger to the need
for exemplar instructional materials that could support teacher practice and student
learning at scale. In Romberg’s (1997, p. 139) opinion, one of the exemplar cases
might be found in the work of the Freudenthal Institute, based on the “international
reputation arising from the work of Hans Freudenthal and his colleagues…and the
fact that the performance of Dutch students ranked very high on all international
comparative studies.” This observation led to the first pilot study of RME in the
United States—the Whitnall Study.
2.2.1 The Whitnall Study
As an outcome of a meeting of various scholars and curriculum developers hosted
by Romberg, De Lange proposed a pilot study of RME in a U.S. school. The content
focus would be statistics. The school would be Whitnall High School, located in
a suburb of Milwaukee. Six teachers and their classrooms would be involved in
the study, including Gail and Jack Burrill. Jan de Lange rallied several Freudenthal
Institute faculty who moved to the Milwaukee area for four weeks, to work closely
26 D. C. Webb and F. A. Peck
with the Whitnall High School mathematics department and develop instructional
materials based on observed classroom activities from the previous day.
Even though a blueprint for the instructional unit was well-established, regular
adaptations were made to the daily activities exemplifying the student-centredness
of the approach. As Jack Burrill described the process: “Sometimes we would get
copies of that day’s lesson the night before. Sometimes the same morning!” (personal
communication). Much of the Whitnall Study has been recounted elsewhere (Van
Reeuwijk, 1992; De Lange, Burrill, Romberg, & Van Reeuwijk, 1993). The more
important point to make here is that the initial entrée of RME into the United States
was through dedicated teachers who were willing to face the unknown, take risks in
front of their students and colleagues, and perhaps be humbled in the process. Both
Jack and Gail Burrill had vivid recollections of the experience—in fact, one might
say the experience was transformative. Gail Burrill recounted her experience in this
[T]here was still no real anticipation of the radical changes we would be called on to make in
our classrooms. We knew about the NCTM ‘Curriculum and evaluation standards for school
mathematics’. We were prepared for something new but not so different. As we worked
throughout the project, however, the ‘Standards’ came to life. We began to recognize that
we not only needed new ways of teaching but a new way of thinking about the mathematics
we should teach. (De Lange, Burrill, Romberg, & Van Reeuwijk, 1993, p. 154)
One of the main challenges for the teachers was in shifting from a teacher-centred to
a student-centred classroom. The materials were designed to support student inquiry;
they were not designed for a teacher to show the students how to do the problems.
This transition to ‘letting the students do the mathematics’ was not easy, as well-
established instructional routines by experienced teachers were found to be difficult
habits to break. Eventually, the Whitnall teachers began to internalise the approach
used by RME and even began to self-correct their practice. As Jack Burrill recalled,
later in the study when he finished teaching a lesson he would meet with the Freuden-
thal group in the back of the class, and before anything else was said, he would ask,
“I blew it again, didn’t I?” after recognising that he was doing the mathematics for
the students, rather than the having the students to the work.
From a researcher/curriculum developer point-of-view, the experience must
have been equally exhilarating. Martin van Reeuwijk was one of the Freudenthal
researchers who co-designed materials to be used by the Whitnall teachers. As Van
Reeuwijk (1992, p. 516) wrote later in an article summarising the experience:
After the first week of the project, problems with the new mathematics decreased drastically.
Students were interested in the class and commented that they liked mathematics now more
than before, that it was not so boring, and that they had discovered that mathematics can be
used in real-life situations. When questions arose about homework, they came after school
to discuss them. Even the low-level and least motivated students got involved in the data-
visualisation unit and liked it.
The shift in student engagement and participation observed by Van Reeuwijk did not
go unnoticed by the teachers. The students’ response to the RME experiment moti-
vated the teachers to emphasise practices that supported student inquiry and problem
2 From Tinkering to Practice—The Role of Teachers … 27
solving. One of the key findings from the Whitnall Study was the importance of
teacher professional development and support if RME was going to be implemented
at scale in the United States. But this is not a challenge unique to the United States.
As described further by Van Reeuwijk (1992, p. 517), “[t]he difficulties that students
and teachers had in reaction to a new approach to mathematics were the same as those
experienced in the Netherlands when the mathematics curriculum was changed.” The
difference between the Netherlands and the United States is a student population of
over 40 million students.
2.2.2 Going to Scale with Mathematics in Context
The Whitnall Study provided a proof-of-concept that RME could work in U.S. class-
rooms so much so that it motivated Romberg and De Lange to apply for a curriculum
development grant at a much larger scale. In the autumn of 1991 the NSF funded the
project Mathematics in Context: A Connected Curriculum for Grades 5–8 (MiC), one
of thirteen mathematics instructional material development projects funded by NSF
in the early 1990s. This project involved a five-year collaboration between research
and development teams at the Freudenthal Institute and the University of Wisconsin
and scores of elementary and middle school teachers. Focussed on middle grades
mathematics, forty units were developed for Grades 5 through 8, which reflected
the middle grade band described in the NCTM Standards. Freudenthal researchers
were responsible for initial drafts of the units and then these drafts were modified by
University of Wisconsin faculty, staff and doctoral students before they were piloted
in U.S. schools by teachers and students. To support this work, several Freudenthal
researchers moved to Madison, Wisconsin, to work directly with the University of
Wisconsin team, and local teachers, as early drafts of the materials were piloted.
Given that this project launched before the advent of public email or broadband
internet, most communication occurred either in person in Madison, or using trans-
Atlantic mail and conference calls.
As the MiC units moved from piloting to field testing, there was a need to recruit
many participating teachers across the United States who worked in a diverse set of
school contexts. In addition to a significant number of teachers across Wisconsin,
field testing of MiC included teachers in California, Florida, Iowa, Massachusetts,
Missouri, Puerto Rico, Tennessee and Virginia. Local site coordinators were also
recruited to support ongoing communication between research team and teachers, and
coordinate classroom level data collection that could be used to inform subsequent
revisions of the student books and teacher guides. Encyclopaedia Britannica agreed
to publish the materials, and also supported efforts to market the materials even before
they were available in their final printed form. Teachers’ response to the field testing of
MiC was generally positive; however, the challenges observed in the Whitnall Study
suggesting a need to support teachers as they transitioned to student-centred practices
were magnified further since there were not as many project personnel who worked
locally with teachers on a regular basis. Nevertheless, teachers provided copious
28 D. C. Webb and F. A. Peck
input from the field, leading to improvements to the student activities and teacher
support materials. As the MiC units moved from field testing to the publication
of the textbook series Mathematics in Context (National Center for Research in
Mathematical Sciences Education & Freudenthal Institute, 19971998), many of the
teachers in the original field test sites decided to adopt MiC after they observed its
impact on student engagement and achievement (e.g., Webb et al., 2001; Webb &
Meyer, 2002).
Recognition of the need for teacher support led to a commitment on the part
of Encyclopaedia Britannica to provide professional development to schools that
adopted MiC, which also required the recruitment of lead teachers (many of who
piloted and field tested MiC) to facilitate workshops across the United States. It was
through this rapidly expanding professional development network that early adopters
of MiC were put in the position of communicating RME principles to their colleagues,
school administrators, parents and a multitude of teachers who attended MiC work-
shops. To frame the goals and purpose of MiC, RME was explicitly discussed in
these workshops with ample reference to Hans Freudenthal and the historical work
of the Freudenthal Institute. Teachers and the co-designers of MiC communicated
what RME was, and how it related to the vision of the NCTM Standards. MiC
became a U.S. exemplification of RME that demonstrated how formal mathematics
could emerge from students’ activity in realistic contexts. The careful development
of concepts and skills in algebra, number and geometry in MiC became early instan-
tiations of progressive formalisation, and led others to reference these examples in
mathematics education research (Driscoll, 1999; Gutstein, 2003). Towards the end
of the 1990s, MiC was referenced in the National Academies Press publication How
People Learn (Bransford, Brown, & Cocking, 1999, p. 137), where it was described
as an innovative approach “to the development of curricula that support learning
with understanding and encourage sense making”. In this widely disseminated book
several key principles of RME have been described in lay terms:
The idea of progressive formalization is exemplified by the algebra strand for middle school
students using Mathematics in Context (National Center for Research in Mathematical Sci-
ences Education & Freudenthal Institute, 19971998). It begins by having students use their
own words, pictures, or diagrams to describe mathematical situations to organize their own
knowledge and work and to explain their strategies. In later units, students gradually begin
to use symbols to describe situations, organize their mathematical work, or express their
strategies. At this level, students devise their own symbols or learn some nonconventional
notation. Their representations of problem situations and explanations of their work are a
mixture of words and symbols. Later, students learn and use standard conventional alge-
braic notation for writing expressions and equations, for manipulating algebraic expressions
and solving equations, and for graphing equations. Movement along this continuum is not
necessarily smooth, nor all in one direction. (Bransford et al., 1999, p. 137)
With respect to contributions to mathematics education research, this period also saw
the publication of RME related studies in practitioner journals and highly regarded
research journals, which offered many cases of the theory and application of RME
in U.S. classrooms. During this time there were also a multitude of classroom-based
research studies that used RME related materials. These studies were completed as
2 From Tinkering to Practice—The Role of Teachers … 29
dissertations and focussed on a range of research topics such as curriculum imple-
mentation (Brinker, 1996;Clarke,1995), teacher change (Clarke, 1997), teacher con-
tent knowledge (Hutchinson, 1996), student learning (Hung, 1995; Spence, 1997)
and classroom assessment (Shafer, 1996; Van den Heuvel-Panhuizen, 1996; Webb,
2.2.3 Assessing RME
Even though MiC was published and competing for adoption in school districts
across the United States, requests for additional support came in from school admin-
istrators and teachers regarding assessment. Several assessment initiatives emerged
during this time, some funded by the publisher to work directly with teacher in New
York City and others funded by the U.S. Department of Education, for example,
the RAP (Research in Assessment Practices) project and the CATCH (Classroom
Assessment as a Basis of Teacher Change) project. These projects involved a team
of Freudenthal researchers, including Jan de Lange, Els Feijs, Truus Dekker, Nanda
Querelle, Mieke Abels, Martin van Reeuwijk and Monica Wijers. Working together
with several researchers from the University of Wisconsin, and teachers in Philadel-
phia, Providence (RI), and South Milwaukee, this research project studied ways to
support teachers’ assessment practices. These projects provided an opportunity to
articulate the research domain of classroom assessment as it relates to not only RME,
but other scholarly literature regarding mathematical literacy, the use of context in
task design, non-routine problem solving and formative assessment. All three of the
districts had adopted MiC to some extent, but the research also included teachers
who were using other NSF-funded curricula or traditional textbooks. The research
team worked closely with teachers as they developed their own classroom assessment
experiments, which were opportunities to try new and innovative assessment prac-
tices. In many cases this involved using assessment tasks that asked for more than
recall of procedures, which revealed other forms of students’ mathematical reasoning
that had previously been under-addressed in quizzes and tests, or classroom instruc-
tion. These classroom assessment experiments were transformative experiences for
many of the participating teachers, who emerged as leaders in their district and later
shared their findings with other mathematics teachers and school administrators at
national conferences. Towards the end of the project, greater attention was given to
the ways teachers could support student communication, problem solving, and use
of representations through formative assessment.
As we entered the new millennium, mathematics education in the United States
was amid a public debate over school mathematics and the way it should be taught
(Schoenfeld, 2004). A significant outcome of these so-called ‘Math Wars’ was a call
to draw together mathematics educators, research mathematicians and education psy-
chologists to prepare a revision to the 1989 NCTM Standards. The publication Prin-
ciples and Standards for School Mathematics (NCTM, 2000) subsequently sparked
a new wave of revision of NSF-funded instructional materials, and led to a new group
30 D. C. Webb and F. A. Peck
of lead teachers and schools being engaged in RME through their involvement in the
revision of MiC.
2.2.4 Two Other Collaborations
Two other productive collaborations are worth mentioning here. The first, ‘Math in
the City’, began as a collaboration between Cathy Fosnot from the City College of
New York and Maarten Dolk and Willem Uittenbogaard from the Freudenthal Insti-
tute. The project had two goals: to learn more about student learning, and to reform
both mathematics teaching and the mathematics curriculum. Teacher participation
was integral in achieving both goals. The project was centred on teachers, and over
450 teachers participated in courses and summer institutes designed to allow them
to re-experience mathematics as mathematising, and to focus on how children learn
mathematics. As well, teachers worked with instructional coaches in their classrooms
to develop, test, and tinker with instructional activities. These classroom sessions
were recorded, and the videos became data that Fosnot and Dolk used to learn more
about student learning. Ultimately, this led to innovative developmental progres-
sions that inscribe student learning as movement within metaphorical ‘landscapes’
of mathematical strategies, big ideas, and models. The collaboration produced a book
series written for teachers that shares the activities and the landscapes of learning
produced over the five-year project (Fosnot & Dolk, 2001a,2001b,2002). The books
prominently feature vignettes of teachers engaging their students in RME activities.
Moreover, the collected activities that emerged from the collaboration were published
as Contexts for Learning Mathematics (Fosnot, 2007).
The second collaboration involved Paul Cobb and colleagues in the United States,
and Koeno Gravemeijer from the Freudenthal Institute. In the United States, Cobb and
colleagues were researching student learning in mathematics classrooms. In looking
for heuristics to guide instructional design to promote student learning, they learned
about RME and began a collaboration with Gravemeijer to develop, implement, and
revise RME-based instructional sequences to promote student learning. In the course
of this collaboration, the research team produced instructional sequences for early
number (Cobb, Gravemeijer, Yackel, McClain, & Whitenack, 1997; Gravemeijer,
1999) and statistics (Cobb, McClain, & Gravemeijer, 2003; McClain & Cobb, 2001;
McClain, Cobb, & Gravemeijer, 2000). In addition, the team made two conceptual
shifts in the ways that they viewed student learning in classrooms and in so doing they
provided numerous contributions to research on mathematics teaching and learning
(e.g., Cobb, Stephan, McClain, & Gravemeijer, 2001). Teachers played a large role
in these shifts.
The first conceptual shift occurred when the research team began to view class-
rooms as activity systems, composed of interdependent means of support, including
norms, tools, discourse, and activities. This shift was precipitated by a teacher’s
question, and the research team’s realisation that what counted as an ‘answer’ was
an interactional achievement and not an a priori given nor solely a product of an indi-
2 From Tinkering to Practice—The Role of Teachers … 31
vidual student’s personal knowledge (Yackel & Cobb, 1996). In light of this, they
shifted their design focus from designing for individual student learning to design-
ing for the mathematical development of classrooms. Because of the central medi-
ating role of teachers in classrooms, this shift entailed a new focus: “[D]evelop[ing]
instructional activities that would result in a range of solutions on which the teacher
could capitalise as she planned whole class discussions” (Cobb, Zhao, & Visnovska,
2008, p. 117). Hence, teachers assumed a central design role in the interactive con-
stitution of classroom activity systems. In addition, the research team came to view
a teacher’s enaction of instructional sequences as a fundamentally creative activ-
ity, arguing “although designed curricula and textbooks are important instructional
resources, teachers are the designers of the curricula that are actually enacted in their
classrooms” (Visnovska, Cobb, & Dean, 2012, p. 323, emphasis in original). As they
came to recognise the creative role of the teacher, the research team made a second
conceptual shift: from designing instructional sequences for teachers to implement,
to designing supports for teacher learning.
In light of these conceptual shifts, the research team developed three adaptations
to RME design theory: (1) a shift in focus, from designing instructional activities
and sequences, to designing entire activity systems—including activity sequences
but also social norms and classroom discourse; (2) a shift from designing activities
to achieve student learning directly, to designing activities that a teacher can use
to achieve a class-wide instructional outcome; and (3) incorporating teacher profes-
sional development to support teachers’ productive adaptations of designed resources
(Cobb et al., 2008).
2.2.5 FIUS: Developing RME Networks in the United States
The increasing interest in ways to improve the teaching and learning of mathematics
using principles of RME motivated the establishment of the Freudenthal Institute
United States (FIUS) at the University of Wisconsin-Madison in 2003. During the
early years of FIUS, research proposals were submitted to extend the application
of RME into special education and courses typically taught in high schools and
community colleges. In 2005, FIUS hosted the first ‘Realistic Mathematics Edu-
cation Conference’, which included presentations by Dutch and U.S. researchers
and educators describing past, current and emerging use of RME in K–12 curricula,
professional development and assessment.
In the autumn of 2005, FIUS relocated to the University of Colorado Boulder. Over
the next 10 years, RME was integrated into a number of pre-service and graduate
level courses focussed on mathematics and science education, with several of these
courses being jointly taught by instructors from the Freudenthal Institute and the
University of Colorado Boulder.
In addition, FIUS helped to facilitate several cross-national collaborations involv-
ing personnel from the Freudenthal Institute in the Netherlands, FIUS, and U.S.
teachers. These collaborations resulted in several classroom studies in middle, high,
32 D. C. Webb and F. A. Peck
and post-secondary classrooms that were similar in approach to the Whitnall Study.
In post-secondary, Monica Geist and other mathematics faculty at Front Range Com-
munity College collaborated with Henk van der Kooij to develop a unit that would
deepen students’ understanding of exponential and logarithmic functions. The imple-
mentation of the unit resulted in a dramatic shift in student engagement and math-
ematical reasoning in ways that were unexpected for a relatively brief two-week
unit (Webb, Van der Kooij, & Geist, 2011). In middle school, a number of produc-
tive collaborations were realised. Peter Boon and Mieke Abels worked with middle
school teachers in Denver to pilot sequences of applet-based activities organised in
the Digital Mathematics Environment. One of the findings from their study was the
influence of new contexts and models in the applet sequences; teacher observation
of students’ productive use of representations resulted in teacher uptake of the same
representations during non-tech portions of the unit. A second collaboration involved
David Webb from FIUS, Truus Dekker and Mieke Abels from the Freudenthal Insti-
tute, mathematics faculty at University of Colorado Boulder, and over thirty teachers
in the Boulder Valley (Colorado, U.S.) School District. In this three-year collabora-
tion, teachers designed and redesigned assessments and activity sequences according
to RME design principles (Webb, 2009,2012; Webb et al., 2008). In high school,
Fred Peck participated in a series of collaborations with members of the Freudenthal
Institute and FIUS as a teacher and researcher.
To give a sense of what these collaborations were like ‘from the inside’, and the
powerful effect that they have for teachers, we now turn to a first-person account of
the high school collaborations.
2.3 Guided Reinvention of High School Mathematics: Fred
Peck’s Personal Account
I was introduced to RME during my second year as a high school mathematics teacher,
in a school in the Boulder Valley School District. David Webb had just brought FIUS
to the University of Colorado. He came to our school for an afternoon, and introduced
the mathematics teachers to Peter Boon and Henk van der Kooij, from the Freudenthal
Institute. Peter and Henk were interested in collaborating with teachers in the United
States. I was interested in reform mathematics education, including active learning
and sense-making, but I had very little design experience. After some brief personal
introductions, David passed out the ‘Hot dogs and lemonade’ task shown in Fig. 2.1,
and we all got to work.
What was immediately clear to me as I worked on the problem was the principled
use of context. Many of my colleagues set up a system of simultaneous linear equa-
tions. This was my first instinct, too. But rather than join my colleagues in formal
algebra, I found myself drawn to the context. I combined the orders in various ways
to make new combinations, eventually eliminating the hot dogs. By that time in my
life, I had used Gaussian elimination to solve systems of equations hundreds of times.
2 From Tinkering to Practice—The Role of Teachers … 33
Fig. 2.1 ‘Hot dogs and lemonade’ task (Webb et al., 2001,p.5)
But I never understood why it worked. Why can one just combine rows of a matrix
to make a new row (or combine two equations to make a third equation)? I knew that
elimination worked, but had no idea why. And, of course, someone had to teach the
method to me. Nothing about a formal matrix or formal system of equations invited
exploration or sense-making.
The ‘Hot dogs and lemonade’ task was different. The context was not just a
‘wrapper’ for formal mathematics—something to peel away in order to find the
system of linear equations hidden within. Of course, the problem could be interpreted
that way, but the context invited mathematical exploration. It was begging to be
mathematised. As I engaged in realistic activity in the context, making combinations
of hot dogs and lemonade, I finally understood elimination! I was hooked. It was
clear to me that RME was a powerful tool for didactical design.
The principled use of contexts—that emerged from Freudenthal’s (1983) didac-
tical phenomenology—initially drew me to RME. Soon I learned about guided rein-
vention and emergent modelling/progressive formalisation (e.g., Freudenthal, 1991;
Gravemeijer, 1999; Webb et al., 2008), and I became even more excited about RME.
Together, didactical phenomenology, emergent modelling, and guided reinvention
offered a set of powerful heuristics to design activity sequences such that formal
mathematics can emerge from realistic activity.
For the next six years, I endeavoured to apply these design principles to all my
classes. Slowly but surely, I developed a repertoire of activity sequences. In Calcu-
34 D. C. Webb and F. A. Peck
lus, I developed an activity sequence involving see-saws and chains of see-saws to
guide students to reinvent the chain rule for derivatives, and another involving the
path of a ‘vomit comet’ as it climbs and free-falls to guide students to reinvent the
second derivative as a point of inflection (vomit comets are airplanes that engage
in a sequence of free falls followed by steep climbs, and are used to simulate zero-
gravity). In Probability, I developed activity sequences involving overlapping dart
boards and branching rivers to guide students to reinvent joint probability.
I also taught Algebra I with a colleague, Jen Moeller. Jen and I collaborated with
David Webb, Peter Boon, and Henk van der Kooij to develop an entire curriculum for
Algebra I using RME design principles. We developed a sequence for single-variable
equations that guided students to reinvent balance strategies and backtracking strate-
gies, balance models and arrow chain models, and formal expressions as objects to
be manipulated and processes to be undone. We developed a sequence for quadratic
functions that guided students to reinvent two powerful models for polynomials—
an area model and a Cartesian model—and from there to reinvent the fundamental
theorem of algebra: that ‘line times line equals parabola’ and more generally that
polynomials are composed of linear factors. Jen and I presented these sequences at
local and national conferences (Peck & Moeller, 2010,2011).
Another colleague, Michael Matassa, joined our school as a mathematics coach,
and he and I started to conduct design research in my classroom. We designed, tested,
and refined two local instructional theories using RME: one for fractions as they are
used in algebra (Peck & Matassa, 2016), and one for slope and linear functions
(Peck, 2014). We made theoretical contributions to RME, including a deep analysis
of the ways that models transform students’ mathematical activity and mathematical
understandings (Peck & Matassa, 2016), and a new way of thinking about emergent
modelling as a ‘cascade of artifacts’ (Peck, 2015).
I went to graduate school and wrote my dissertation on RME. I became involved
in professional development and conducted workshops for teachers on RME, includ-
ing emergent modelling and how models transform students’ mathematical activity
and understanding. Now, I am exploring how cultural theories of learning can con-
tribute to RME, and I am teaching pre-service teachers about RME. I just heard from
some former students—now teachers—that they are working together to develop
mathematics games using RME design principles.
I still use the ‘Hot dogs and lemonade’ task.
2.4 Summary Remarks
The influence of RME on mathematics education in the United States has been
significant. Its approach to the use of context and models has influenced state and
national curricula. Models that were used extensively in early RME resources—such
as the empty number line, percentage bar and ratio table—were introduced to many
teachers in the United States through RME related instructional materials. RME
2 From Tinkering to Practice—The Role of Teachers … 35
is also well represented in mathematics education research published in U.S. and
international journals.
There are many other stories that could be shared that describe RME’s more
subtle influence in the non-public space, such as conversations among teachers,
school administrators, and university faculty who are seeking ways to improve stu-
dent engagement with mathematics or impart a more meaningful mathematical expe-
rience to the next generation of students. As mentioned previously, at FIUS and the
University of Colorado Boulder, principles of RME have influenced undergradu-
ate mathematics and science education and the design of instructional materials to
support active learning. RME has been applied elsewhere in the United States by
mathematics faculty who are interested in studying and improving student learning
in abstract algebra (Larsen, Johnson, & Bartlo, 2013), differential equations (Ras-
mussen & Kwon, 2007), and other advanced mathematics topics.
In our opinion, what is remarkable about many of these individual stories is the
involvement of teachers. From the first pilot study of RME in the United States at
Whitnall High to the development of comprehensive curricula, teachers have been
central to the dissemination, use, and development of RME in the United States.
Teachers have collaborated with researchers to develop and improve RME sequences
and curricula, they have become instructional leaders who facilitate professional
development on RME, and many continue to participate in the RME community—
for example, by sharing their experiences in using RME at the biennial international
conference on RME.
Publications such as Mathematics in Context and Contexts for Learning Math-
ematics represent the most durable reifications of teachers’ participation in RME
in the United States. Perhaps even more important, however, are the hidden ways
that teachers continue to incorporate RME instructional principles into their class-
rooms, striving to find meaningful ways to engage students in the human activity of
This has been, and continues to be, challenging work. While artefacts of RME have
gained wide acceptance in the United States, RME itself is not widely known. Even
though Mathematics in Context was adopted by several major school districts in New
York City, Philadelphia, and Washington DC, its presense as instructional materials
has since waned. The extent to which Contexts for Learning Mathematics is used
presently in U.S. schools is also unclear. Thus, while certain models are widely used,
the design principles that give the models such power—didactical phenomenology,
emergent modelling/progressive formalisation, and guided reinvention—are often
unknown to teachers and thus are incorporated only sparingly. In the U.S. academy,
RME remains a niche topic of research and development. Mathematics education
scholarship, meanwhile, has taken a sociocultural turn, in which learning is under-
stood as an ontological enterprise and not just an epistemic one. There is a need to
continue the theoretical development of RME in light of these advances in learning
As we look towards the future, we are hopeful that these challenges will be
recognised as opportunities rather than barriers. As they always have, teachers will
play a key role in making that vision a reality.
36 D. C. Webb and F. A. Peck
Acknowledgements We would like to acknowledge the many researchers and teachers who have
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... Horizontal mathematization is done through informal steps before students enter more formal mathematics. The process of horizontal-vertical mathematization is expected to give students more opportunities to understand more abstract mathematics easily [23,27,28,29,30]. ...
... By considering the average value of N-Gain algebraic thinking of the two groups of learning, it can be concluded that the overall enhancement in the algebraic thinking ability of students who received learning of RME is better than students who received expository learning. RME as a learning innovation that begins with a realistic situation has helped students to master the concept of algebra gradually, by formalizing progressively [27]. Students are allowed to reinvent the concept of algebra, so that algebraic mastery can be better, more meaningful, and can master formal algebra material [32], as a representation of the increased ability of students to think algebra. ...
... RME theory has helped mathematics teachers to renovate their teaching process and teaching effciency, and improve students' interest in learning. RME is also used to develop mathematics education programs and textbooks (Dickinson & Hough, 2012;Dossey et al., 2016;Gravemeijer et al., 2016;Venkat et al., 2009) The mathematization of the world requires authentic problem solving, with student-centered problems and teacher instruction (Webb & Peck, 2020). Therefore, teachers have an important role in the light of RME theory. ...
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The launch ofa new book series is always a challenging eventn ot only for the Editorial Board and the Publisher, but also, and more particularly, for the first author. Both the Editorial Board and the Publisher are delightedt hat the first author in this series isw ell able to meet the challenge. Professor Freudenthal needs no introduction toanyone in the Mathematics Education field and it is particularly fitting that his book should be the first in this new series because it was in 1968 that he, and Reidel, produced the first issue oft he journal Edu cational Studies in Mathematics. Breakingfresh ground is therefore nothing new to Professor Freudenthal and this book illustrates well his pleasure at such a task. To be strictly correct the ‘ground’ which he has broken here is not new, but aswith Mathematics as an Educational Task and Weeding and Sowing, it is rather the novelty oft he manner in which he has carried out his analysis which provides us with so many fresh perspectives. It is our intention that this new book series should provide those who work int he emerging discipline of mathematicseducation with an essential resource, and at a time of considerable concern about the whole mathematics cu rriculum this book represents just such resource. ALAN J. BISHOP Managing Editor vii A LOOK BACKWARD AND A LOOK FORWARD Men die, systems last.