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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

e-mail: dcwebb@colorado.edu

F. A. Peck

University of Montana, Missoula, USA

e-mail: frederick.peck@umontana.edu

© The Author(s) 2020

M. van den Heuvel-Panhuizen (ed.), International Reﬂections

on the Netherlands Didactics of Mathematics, ICME-13 Monographs,

https://doi.org/10.1007/978-3- 030-20223- 1_2

21

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

States

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-

rooms.

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

way:

[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, 1997–1998), 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, 1997–1998). 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,

2001).

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

mathematising.

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

theory.

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

contributed to the theory and application of RME in the United States, and the vision of Thomas A.

Romberg and Jan de Lange to initiate an international partnership that has contributed to mathematics

education in numerous ways.

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