____ THE
______ MATHEMATICS ___
_________ EDUCATOR _____
Volume 17 Number 1
Summer 2007
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Kyle T. Schultz
Dear TME Reader,
Associate Editors
Rachael Brown
Kelly Edenfield
Ryan Fox
Na Young Kwon
Eileen Murray
Susan Sexton
Catherine Ulrich
On behalf of our editorial staff and the Mathematics Education Student Association
of the University of Georgia, I am pleased to present to you the first issue of Volume 17
of The Mathematics Educator. We have worked hard to bring to you a collection of
articles that share new and interesting ideas in our field. We hope that you find them
intriguing and that they may provide further fuel for your practice and research.
Advisor
Dorothy Y. White
MESA Officers
2006-2007
President
Rachael Brown
Vice-President
Filyet Asli Ersoz
Secretary
Eileen Murray
Treasurer
Jun-ichi Yamaguchi
NCTM
Representative
Ginger A. Rhodes
Undergraduate
Representative
Laine Bradshaw
Rachel Stokely
In his guest editorial that leads off this issue, Tad Watanabe shares some ideas about
what makes school mathematics curricula focused and coherent. Using his knowledge of
Japanese school curricula and curriculum development, he provides some direction for
improving U.S. curricula. Next, Rick Anderson shares his research of the development
of mathematical identity within secondary-level students and provides practical advice
for educators regarding how to foster positive mathematical identities within students.
Shelly Sheats Harkness and Lisa Portwood, share their experiences in implementing a
lesson with prospective and practicing teachers that focuses upon the mathematics of
quilting. In addition, they provide a critical look at our notion of mathematical activity.
Next, Zelha Tunç-Pekkan shares a unique look into the diverse practices of three
professors who have taught a graduate-level mathematics curriculum course. In the final
article, Jon Warwick reflects on his experiences in teaching undergraduate students
mathematical modeling and provides suggestions for developing strong modeling skills.
Concluding this issue, Ginny Powell reviews Mathematics Education Within the
Postmodern, an “eye-opening and thought-provoking” book.
This is my final issue as editor of TME. I would like to thank my staff of associate
editors, previous editors and the Mathematics Education faculty and staff for their
guidance and support, our reviewers who provide crucial feedback on each submitted
manuscript, and my family and friends for their encouragement and patience. The
journal will be in good hands with Kelly Edenfield and Ryan Fox assuming leadership of
the journal as co-editors. I wish them success in the coming year.
Kyle T. Schultz
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
www.coe.uga.edu/tme
About the Cover
The Exploding Cube. Using Geometer’s Sketchpad, this image shows three stages of the partitioning of the cube (a +
b)3 into components. These components correspond to the algebraic expansion of this expression.
This publication is supported by the College of Education at The University of Georgia
___________ THE ________________
__________ MATHEMATICS ________
_____________ EDUCATOR ____________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Summer 2007
Volume 17 Number 1
Table of Contents
2 Guest Editorial… In Pursuit of a Focused and Coherent School Mathematics
Curriculum
TAD WATANABE
7 Being a Mathematics Learner: Four Faces of Identity
RICK ANDERSON
15 A Quilting Lesson for Early Childhood Preservice and Regular Classroom
Teachers: What Constitutes Mathematical Activity?
SHELLY SHEATS HARKNESS & LISA PORTWOOD
24 Graduate Level Mathematics Curriculum Courses: How Are They Planned?
ZELHA TUNÇ-PEKKAN
32 Some Reflections on the Teaching of Mathematical Modeling
JON WARWICK
42 Book Review… The View from Here: Opening Up Postmodern Vistas
GINNY POWELL
45 Upcoming conferences
46 Submissions information
47 Subscription form
© 2007 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2007, Vol. 17, No. 1, 2–6
Guest Editorial…
In Pursuit of a Focused and Coherent
School Mathematics Curriculum
Tad Watanabe
Most, if not all, readers are familiar with the
criticism of a typical U.S. mathematics curriculum
being “a mile wide and an inch deep” (Schmidt,
McKnight, & Raizen, 1997). A recent analysis by the
Center for the Study of Mathematics Curriculum
(Reys, Dingman, Sutter, & Teuscher, 2005) reaffirms
the crowdedness of most state mathematics standards.
However, criticism of U.S. mathematics curricula is
nothing new.
In April 2006, the National Council of Teachers of
Mathematics (NCTM) released Curriculum Focal
Points for Prekindergarten through Grade 8
Mathematics: A Quest for Coherence. This document
is an attempt by NCTM to initiate a discussion on what
mathematical ideas are important enough to be
considered as “focal points” at a particular grade level.
But why is it so difficult to have a focused and
cohesive school mathematics curriculum? Besides,
what makes a curriculum focused and cohesive? In this
paper, I would like to offer my opinions on what a
focused and cohesive mathematics curriculum may
look like and discuss some obstacles for producing
such a curriculum.
What makes a curriculum focused?
Clearly, a crowded curriculum naturally tends to be
unfocused. A major cause for the crowdedness of many
U.S. textbook series seem to be the amount of
“reviews,” topics that have been discussed at previous
grade levels. Some amount of review is probably
necessary and helpful. However, in many cases, the
topics are redeveloped as if they have not been
previously discussed. For example, in teaching linear
measurement, most of today’s textbooks follow this
general sequence of instruction: (a) direct comparison,
(b) indirect comparison, (c) measuring with arbitrary or
Tad Watanabe is an Associate Professor of Mathematics Education
at Kennesaw State University. He received his PhD in Mathematics
Education from Florida State University in 1991. His research
interests include teaching and learning of multiplicative concepts
and various mathematics education practices in Japan, including
lesson study and curriculum materials.
2
non-standard units, and (d) measuring with standard
units. Often, the discussion of linear measurement in
Grades K, 1, and 2 textbooks involve all four stages of
measurement instruction at each grade level. In
contrast, a Japanese elementary mathematics course of
study (Takahashi, Watanabe, & Yoshida, 2004)
discusses the first 3 stages in Grade 1, and the
discussion in Grade 2 focuses on the introduction of
standard units. Most Grade 2 textbooks, therefore, start
their discussion of linear measurement by establishing
the need for (and usefulness of) standard units through
problem situations in which the use of arbitrary units is
not sufficient.
This redevelopment of the same topic in multiple
grade levels may be both the symptom and the cause of
a misinterpretation of the idea of a “spiral curriculum.”
In the past few years, several elementary mathematics
teachers who are using a “reform” curriculum told me
that it is acceptable for children to not understand some
ideas the first time (or even the second or third time)
since they will see it again later. Such a view does not
describe a spiral. Rather, it seems to be based on the
belief that, by introducing a topic early and discussing
it often, students will come to understand it. This view
is incompatible with a focused curriculum.
However, simply removing some topics from any
given grade level does not necessarily result in a
focused curriculum. If all items on a given grade level
receive equal amount of attention, regardless of
mathematical significance, then the curriculum lacks a
focus. The Focal Points (NCTM, 2006) present three
characteristics for a concept or a topic to be considered
as a focal point:
•
Is it mathematically important, both for further
study in mathematics and for use in
applications in and outside of school?
•
Does it “fit” with what is known about
learning mathematics?
•
Does it connect logically with the mathematics
in earlier and later grade levels? (p. 5)
A Focused and Coherent Curriculum
Whether or not we agree with this particular set of
characteristics, if a curriculum is to be focused, it must
be based on a set of explicitly stated criteria for
organizing its contents.
What makes a curriculum coherent?
It goes without saying that a coherent mathematics
curriculum must have its contents sequenced in such a
way that a new idea is built on previously developed
ideas. Most agree that mathematics learning is like
putting together many building blocks. Of course, there
is typically more than one way to put together ideas.
However, a cohesive curriculum and, ultimately,
teachers must have a vision of how learners can build a
new idea based on what has previously been discussed.
This idea seems to be so obvious, but it is also very
easy to overlook.
Furthermore, I believe that textbook writers have
the responsibility to make clear the potential learning
paths they envision to support teachers who use their
materials. This is where many U.S. mathematics
textbooks seem to fall short. Too often, teachers’
manuals are filled with many suggestions without
explicitly discussing how the target ideas may be
developed from ideas previously discussed. Thus,
teachers are left with an overwhelming amount of
information without any guidance regarding how it can
be organized and put to work.
Another important factor that contributes to the
coherence of a mathematics curriculum is how one part
of a curriculum relates to another. For example, the
Focal Points (NCTM, 2006) states that, in Grade 4,
students are to “develop fluency with efficient
procedures, including the standard algorithm, for
multiplying whole numbers, understand why the
procedures work (on the basis of place value and
properties of operations), and use them to solve
problems” (p. 16). However, in the “Connections to the
Tad Watanabe
Focal Points”, the document also states, “Building on
their work in grade 3, students extend their
understanding of place value and ways of representing
numbers to 100,000 in various contexts” (p. 16).
Therefore, when students are developing fluency with
multiplication procedures, the curriculum writers and
teachers must pay attention to the products of the
assigned problems to insure they will be in the
appropriate range. As not all products of two 3-digit
numbers will be less than 100,000, these two
statements together suggest that the focus of a
curriculum should be on helping students understand
how and why their multiplication procedures work,
rather than focusing solely on students’ proficiency
with multiplying two 3-digit numbers.
The coherence of a mathematics curriculum is also
influenced by its mathematical thoroughness. For
example, in many elementary and middle school
mathematics curricula, students are asked to find the
area of the parallelogram like the one shown in Figure
1. It is expected that most students will cut off a
triangular section from one end and move it to the
other side to form a rectangle, whose area they can
calculate. This idea is discussed in Principles and
Standards for School Mathematics (NCTM, 2000) as
well. Based on this experience, most textbooks will
then conclude that the formula for calculating the area
of a parallelogram is base × height. However, this is an
overgeneralization. For example, if this is the only
experience students have, they will not be able to
determine the area of the parallelogram shown in
Figure 2, unless they already know the Pythagorean
theorem. As a result, students cannot conclude that any
side of a parallelogram may be used as the base to
calculate its area.
3
However, we will then need the Pythagorean theorem
to determine the lengths of the base and the height.
Therefore, for a curriculum to be cohesive,
students should be provided with the opportunity to
determine the area of the parallelogram like the one
shown in Figure 2. Figure 3 shows some of the ways
students may calculate its area. Some of these methods
suggest that we could indeed use the horizontal side as
the base if we consider the height to be the distance
between the parallel lines containing the two horizontal
sides.
In addition to having a thorough sequence of
mathematical ideas, the coherence of a curriculum may
be enhanced by the selection of learning tasks and
representations. For example, in a Japanese textbook
series (Hironaka & Sugiyama, 2006), the following
four problems were used in Grade 6 units on
multiplication and division of fractions:
•
With 1 dl of paint, you can paint
3 2
m of
5
boards. How many m2 can you paint with 2 dl
of paint?
•
•
! paint
With 3 dl of paint, you can
4 2
m of
5
•
With
3
2 2
dl of paint, you can paint
m of
4
5
boards. How many m2 can you paint with 1
dl?
! this particular
By !
selecting the same problem context,
textbook series hopes that students can identify these
problem situations as multiplication or division
situations, even though fractions are involved. We
know from research (e.g., Bell, Fischbein, & Greer,
1984) that this decision is not trivial for students. Once
the operations involved are identified, the series asks
students to investigate how the computation can be
carried out.
A consistent use of the same or similar items
across related mathematical ideas is not limited to the
problem contexts. Another way the coherence may be
enhanced is through the consistent use of
representation. Figure 4 shows how Hironaka and
Sugiyama (2006) use similar representations as they
discuss multiplicative ideas across grade levels. In
early grades, the representations are used primarily to
represent the ways quantities are related to each other
but, later on, students are expected to use the diagrams
as tools to solve problems.
boards. How many m2 can you paint with 1
dl?
Why has it been so difficult to produce a focused
and cohesive curriculum?
4 2
m of
5
2
boards. How many m2 can you paint with dl
3
We can probably list many different reasons to answer
this question. For example, there is a general
With 1 dl of paint, you !
can paint
of paint?
!
!
4
A Focused and Coherent Curriculum
reluctance to remove any topic from an existing
curriculum. Thus, today’s curricula include many ideas
that probably were not included 50 years ago, yet
virtually all topics from 50 years ago are still included
in today’s curricula as well. However, I would like to
discuss another idea that may be undermining our
efforts to create a focused and cohesive curriculum: the
lure of replacement units.
The idea of replacement units, high quality
materials used in place of a unit in a textbook series,
may have started with a good intention. Some reform
curriculum materials appear to be created so that parts
of the curricula may be used as replacement units.
Although many are indeed of very high quality,
replacement units may have encouraged the
compartmentalization and rearrangement of topics
within a curriculum as necessary. Thus, a publisher
may be able to “individualize” their textbook series to
match different state curriculum standards. If
multiplication is introduced in Grade 2 in one state but
in Grade 3 in others, there is no problem. One can
simply package the introduction of multiplication unit
in the appropriate grade level. However, it should be
very clear that a focused and cohesive curriculum is
much more than simply a sequence of mathematics
topics that match the curriculum standards. In addition,
as NCTM (2000) states, a curriculum is more than just
a collection of problems and tasks (p. 14). One must
pay close attention to the internal consistency and
coherence of curriculum materials. A Japanese
textbook series (Hosokawa, Sugioka, & Nohda, 1998)
warned against teachers changing the order of units
presented in the series. This is a stark contrast to a
rather casual approach that some in this country seem
to possess.
Figure 4. Consistent use of similar representations from Hironaka & Sugiyama (2006): (a) multiplication and division
of whole numbers in Grade 3; (b) multiplication of a decimal number by a whole number in Grade 4; (c) multiplying
and dividing by a decimal number in Grade 5; and (d) multiplying and dividing by a fraction.
Tad Watanabe
5
What will it take to produce a focused and coherent
curriculum?
The most obvious response to this question is
closer collaboration among teachers, researchers, and
curricula producers. In Japan, such collaboration is
achieved through lesson study. Although lesson study
(e.g., Lewis, 2002; Stigler & Hiebert, 1999) is often
considered to be a professional development activity, it
also serves a very important role in curriculum
development, implementation, and revision in Japan.
At the beginning of a lesson study cycle, teachers
engage in an intensive study of curriculum materials.
The participating teachers ask questions such as,
•
Why is this topic taught at this particular point
in the curriculum?
•
What previously learned materials are related
to the current topic?
•
How are students expected to use what they
have learned previously to make sense of the
current topic?
Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in
verbal arithmetic problems: The effect of number size,
problem structure and context. Educational Studies in
Mathematics, 15, 129-147.
•
How will the current topic be used in the future
topics?
Hironaka, H. & Sugiyama, Y. (2006). Mathematics for Elementary
School. Tokyo: Tokyo Shoseki. [English translation of New
Mathematics for Elementary School 1, by Hironaka &
Sugiyama, 2000.]
Is the sequence of topics presented in the
textbooks the most optimal one for their
students?
During this process, teachers will read, among
other things, existing research reports and often invite
researchers to participate as consultants. After this
intensive investigation of curriculum materials, the
group develops a public lesson based on their findings.
The public lesson is both their research report and a
test of the hypothesis derived from their investigation.
Through critical reflection on the observation of public
lesson, the group produces their final written report.
Japanese textbook publishers often support local lesson
study groups, and the reports from those groups are
carefully considered in the revision of their textbook
series.
Moreover, teachers examine the new curriculum
ideas carefully through lesson study. Through this
experience, teachers gain a deeper understanding of
these new ideas, and they explore effective ways to
teach them to their students. Because researchers,
university-based mathematics educators, district
•
6
mathematics supervisors, and even the officials from
the Ministry of Education regularly participate in
lesson study open houses, lesson study serves as an
important feedback mechanism for curriculum
development, implementation, and revision.
Lesson study is becoming more and more popular
in the United States; however, the involvement by
mathematics education researchers and curriculum
developers is still rather limited. Moreover, the
examination of curriculum materials is often limited as
well. A closer collaboration between classroom
teachers engaged in lesson study and mathematics
education researchers and other university-based
mathematics educators is critical if U.S. lesson study is
to become a useful feedback mechanism to produce a
more focused and coherent school mathematics
curriculum.
References
Hosokawa, T., Sugioka, T., & Nohda, N. (1998). Shintei Sansuu
(Elementary school mathematics). Osaka: Keirinkan. (In
Japanese).
Lewis, C. (2002). Lesson study: Handbook of teacher-led
instructional change. Philadelphia: Research for Better
Schools.
National Council of Teachers of Mathematics (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (2006). Curriculum
focal points for prekindergarten through grade 8
mathematics: A quest for coherence. Reston, VA: Author.
Reys, B. J., Dingman, S., Sutter, A., & Teuscher, D. (2005).
Development of state-level mathematics curriculum
documents: Report of a survey. Columbia, Mo.: University of
Missouri, Center for the Study of Mathematics Curriculum.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas in
the world’s teachers for improving education in the
classroom. New York: Free Press.
Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A
splintered vision: An investigation of U.S. science and
mathematics education. Dordrecht, The Netherlands: Kluwer.
Takahashi, A., Watanabe, T., & Yoshida, M. (2004). Elementary
school teaching guide for the Japanese Course of Study:
Arithmetic (Grades 1-6). Madison, NJ: Global Education
A Focused and Coherent School Mathematics Curriculum
The Mathematics Educator
2007, Vol. 17, No. 1, 7–14
Being a Mathematics Learner: Four Faces of Identity
Rick Anderson
One dimension of mathematics learning is developing an identity as a mathematics learner. The social learning
theories of Gee (2001) and Wenger (1998) serve as a basis for the discussion four “faces” of identity:
engagement, imagination, alignment, and nature. A study conducted with 54 rural high school students, with
half enrolled in a mathematics course, provides evidence for how these faces highlight different ways students
develop their identity relative to their experiences with classroom mathematics. Using this identity framework
several ways that student identities—relative to mathematics learning—can be developed, supported, and
maintained by teachers are provided.
This paper is based on dissertation research completed at Portland State University under the direction of Dr. Karen Marrongelle. The
author wishes to thank Karen Marrongelle, Joyce Bishop, and the TME editors/reviewers for comments on earlier drafts of this paper.
Learning mathematics is a complex endeavor that
involves developing new ideas while transforming
one’s ways of doing, thinking, and being. Building
skills, using algorithms, and following certain
procedures characterizes one view of mathematics
learning in schools. Another view focuses on students’
construction or acquisition of mathematical concepts.
These views are evident in many state and national
standards for school mathematics (e.g., National
Council of Teachers of Mathematics [NCTM], 2000).
A third view of learning mathematics in schools
involves becoming a “certain type” of person with
respect to the practices of a community. That is,
students become particular types of people—those who
view themselves and are recognized by others as a part
of the community with some being more central to the
practice and others situated on the periphery (Boaler,
2000; Lampert, 2001; Wenger, 1998).
These three views of mathematics learning in
schools, as listed above, correspond to Kirshner’s
(2002) three metaphors of learning: habituation,
conceptual construction, and enculturation. This paper
focuses on the third view of learning mathematics. In
this view, learning occurs through “social
participation” (Wenger, 1998, p. 4). This participation
includes not only thoughts and actions but also
membership within social communities. In this sense,
learning “changes who we are by changing our ability
to participate, to belong, to negotiate meaning”
(Wenger, 1998, p. 226). This article addresses how
students’ practices within a mathematics classroom
Rick Anderson is an assistant professor in the Department of
Mathematics & Computer Science at Eastern Illinois University.
He teaches mathematics content and methods courses for future
elementary and secondary teachers.
Rick Anderson
community shape, and are shaped by, students’ sense
of themselves, their identities.
Learning mathematics involves the development of
each student’s identity as a member of the mathematics
classroom community. Through relationships and
experiences with their peers, teachers, family, and
community, students come to know who they are
relative to mathematics. This article addresses the
notion of identity, drawn from social theories of
learning (e.g., Gee, 2001; Lave & Wenger, 1991;
Wenger, 1998), as a way to view students as they
develop as mathematics learners. Four “faces” of
identity are discussed, illustrated with selected
quotations from students attending a small, rural high
school (approximately 225 students enrolled in grades
9–12) in the U.S. Pacific Northwest.
Method
The students in this study were participants in a
larger study of students’ enrollment in advanced
mathematics classes (Anderson, 2006). All students in
the high school were invited to complete a survey and
questionnaire. Of those invited, 24% responded.
Fourteen students in grades 11 and 12 were selected for
semi-structured interviews so that two groups were
formed: students enrolled in Precalculus or Calculus
(the most advanced elective mathematics courses
offered in the school) and students not taking a
mathematics course that year. These students
represented the student body with respect to postsecondary intentions, as reported on the survey, and
their interest and effort in mathematics classes, as
reported by their teacher. All of the students had taken
the two required and any elective high school
mathematics in the same high school. One teacher
taught most of these courses. When interviewed, this
7
teacher indicated the “traditional” nature of the
curriculum and pedagogy: “We’ve always stayed
pretty traditional. … We haven’t really changed it to
the really ‘out there’ hands-on type of programs.”
Participant observation and interviews with students
corroborated this statement. Calvin, a high school
senior, had enrolled in a mathematics class each year
of high school and planned to study mathematics
education in college. During an interview, he described
a typical day:
Just go in, have your work done. First the teacher
explains how to do it. Like for the Pythagorean
Theorem, for example, she tells you the steps for it.
She shows you the right triangle, the leg, the
hypotenuse, that sort of thing. She makes us write
up notes so we can check back. And then after that
she makes us do a couple [examples] and then if
we all get it right, she shows us. She gives us time
to work. Do it and after that she shows us the
correct way to do it. If we got it right, then we
know. She makes us move on and do an
assignment.
Identity
As used here, identity refers to the way we define
ourselves and how others define us (Sfard & Prusak,
2005; Wenger, 1998). Our identity includes our
perception of our experiences with others as well as
our aspirations. In this way, our identity—who we
are—is formed in relationships with others, extending
from the past and stretching into the future. Identities
are malleable and dynamic, an ongoing construction of
who we are as a result of our participation with others
in the experience of life (Wenger, 1998). As students
move through school, they come to learn who they are
as mathematics learners through their experiences in
mathematics classrooms; in interactions with teachers,
parents, and peers; and in relation to their anticipated
futures.
Mathematics has become a gatekeeper to many
economic, educational, and political opportunities for
adults (D’Ambrosio, 1990; Moses & Cobb, 2001;
NCTM, 2000). Students must become mathematics
learners—members of mathematical communities—if
they are to have access to a full palette of future
opportunities. As learners of mathematics, they will not
only need to develop mathematical concepts and skills,
but also the identity of a mathematics learner. That is,
they must participate within mathematical communities
in such a way as to see themselves and be viewed by
others as valuable members of those communities.
8
Identity as a Mathematics Learner: Four Faces
The four faces of identity of mathematics learning
are engagement, imagination, alignment, and nature.
Gee’s (2001) four perspectives of identity (nature,
discursive, affinity, institutional) and Wenger’s (1998)
discussion of three modes of belonging (engagement,
imagination, alignment) influenced the development of
these faces. Each of the four faces of identity as a
mathematics learner is described below.
Engagement
Engagement refers to our direct experience of the
world and our active involvement with others (Wenger,
1998). Much of what students know about learning
mathematics comes from their engagement in
mathematics classrooms. Through varying degrees of
engagement with the mathematics, their teachers, and
their peers, each student sees her or himself, and is
seen by others, as one who has or has not learned
mathematics.
Engaging in a particular mathematics learning
environment aids students in their development of an
identity as capable mathematics learners. Other
students, however, may not identify with this
environment and may come to see themselves as only
marginally part of the mathematics learning
community. In traditional mathematics classrooms
where students work independently on short, singleanswer exercises and an emphasis is placed on getting
right answers, students not only learn mathematics
concepts and skills, but they also discover something
about themselves as learners (Anderson, 2006; Boaler,
2000; Boaler & Greeno, 2000). Students may learn that
they are capable of learning mathematics if they can fit
together the small pieces of the “mathematics puzzle”
delivered by the teacher. For example, Calvin stated,
“Precalculus is easy. It’s like a jigsaw puzzle waiting
to be solved. I like puzzles.”
Additionally, when correct answers on short
exercises are emphasized more than mathematical
processes or strategies, students come to learn that
doing mathematics competently means getting correct
answers, often quickly. Students who adopt the
practice of quickly getting correct answers may view
themselves as capable mathematics learners. In
contrast, students who may require more time to obtain
correct answers may not see themselves as capable of
doing mathematics, even though they may have
developed effective strategies for solving mathematical
problems.
Four Faces of Identity
One way students come to learn who they are
relative to mathematics is through their engagement in
the activities of the mathematics classroom:
The thing I like about art is being able to be
creative and make whatever I want… But in math
there’s just kind of like procedures that you have to
work through. (Abby, grade 11, Precalculus class,
planning to attend college)
Math is probably my least favorite subject… I just
don’t like the process of it a lot— going through a
lot of problems, going through each step. I just get
dragged down. (Thomas, grade 12, Precalculus
class, planning to attend college)
Students who are asked to follow procedures on
repetitive exercises without being able to make
meaning on their own may not see themselves as
mathematics learners but rather as those who do not
learn mathematics (Boaler & Greeno, 2000). A
substantial portion of students’ direct experience with
mathematics happens within the classroom, so the
types of mathematical tasks and teaching and learning
structures used in the classroom contribute
significantly to the development of students’
mathematical identities. In the quotation above, Abby
expressed her dislike of working through procedures
that she did not find meaningful. In mathematics class,
she was not able to exercise her creativity as she did in
art class. As a result, she may not consider herself to be
a capable mathematics learner.
On one hand, when students are able to develop
their own strategies and meanings for solving
mathematics problems, they learn to view themselves
as capable members of a community engaged in
mathematics learning. When their ideas and
explanations are accepted in a classroom discussion,
others also recognize them as members of the
community. On the other hand, students who do not
have the opportunity to connect with mathematics on a
personal level, or are not recognized as contributors to
the mathematics classroom, may fail to see themselves
as competent at learning mathematics (Boaler &
Greeno, 2000; Wenger, 1998).
Imagination
The activities in which students choose to engage
are often related to the way they envision those
activities fitting into their broader lives. This is
particularly true for high school students as they
become more aware of their place in the world and
begin to make decisions for their future. In addition to
learning mathematical concepts and skills in school,
students also learn how mathematics fits in with their
Rick Anderson
other activities in the present and the future. Students
who engage in a mathematical activity in a similar
manner may have very different meanings for that
activity (Wenger, 1998).
Imagination is the second face of identity: the
images we have of ourselves and of how mathematics
fits into the broader experience of life (Wenger, 1998).
For example, the images a student has of herself in
relation to mathematics in everyday life, the place of
mathematics in post-secondary education, and the use
of mathematics in a future career all influence
imagination. The ways students see mathematics in
relation to the broader context can contribute either
positively or negatively to their identity as mathematics
learners.
When asked to give reasons for their decisions
regarding enrollment in advanced mathematics classes,
students’ responses revealed a few of the ways they
saw themselves in relation mathematics. For example,
students had very different reason for taking advanced
mathematics courses. One survey respondent stated, “I
need math for everyday life,” while another claimed,
“They will help prepare me for college classes.” These
students see themselves as learners of mathematics and
members of the community for mathematics learning
because they need mathematics for their present or
future lives. Others (e.g., Martin, 2000; Mendick,
2003; Sfard & Prusack, 2005) have similarly noted that
students cite future education and careers as reasons
for studying mathematics.
Conversely, students’ images of the way
mathematics fits into broader life can also cause
students to view their learning of further mathematics
as unnecessary. Student responses for why they chose
not to enroll in advanced mathematics classes included
“the career I am hoping for, I know all the math for it”
and “I don’t think I will need to use a pre-cal math in
my life.” Students who do not see themselves as
needing or using mathematics outside of the immediate
context of the mathematics classroom may develop an
identity as one who is not a mathematics learner. If
high school mathematics is promoted as something
useful only as preparation for college, students who do
not intend to enroll in college may come to see
themselves as having no need to learn mathematics,
especially advanced high school mathematics
(Anderson, 2006).
Students may pursue careers that are available in
their geographical locale or similar to those of their
parents or other community members. If these careers
do not require a formal mathematics education beyond
high school mathematics, these students may limit their
9
image of the mathematics needed for work to
arithmetic and counting. In addition, due to the lack of
formal mathematical training, those in the workplace
may not be able to identify the complex mathematical
thinking required for their work. For example, Smith
(1999) noted the mathematical knowledge used by
automobile production workers, knowledge not
identified by the workers but nonetheless embedded in
the tasks of the job. When students are not able to
make connections between the mathematics they learn
in school and its perceived utility in their lives, they
may construct an identity that does not include the
need for advanced mathematics courses in high school.
The students cited in this paper lived in a rural
logging community. Their high school mathematics
teacher formally studied more mathematics than most
in the community. Few students indicated personally
knowing anyone for whom formal mathematics was an
integral part of their work. As a result, careers
requiring advanced mathematics were not part of the
images most students had for themselves and their
futures.
Alignment
A third face of identity is revealed when students
align their energies within institutional boundaries and
requirements. That is, students respond to the
imagination face of identity (Nasir, 2002). For
example, students who consider advanced mathematics
necessary for post-secondary educational or
occupational opportunities direct their energy toward
studying the required high school mathematics. High
school students must meet many requirements set by
others—teachers, school districts, state education
departments,
colleges
and
universities,
and
professional organizations. By simply following
requirements and participating in the required
activities, students come to see themselves as certain
“types of people” (Gee, 2001). For example a “collegeintending” student may take math classes required for
admission to college.
As before, students’ anonymous survey responses
to the question of why they might choose to enroll in
advanced mathematics classes provide a glimpse into
what they have learned about mathematics
requirements and how they respond to these
requirements. Students were asked why they take
advanced mathematics classes in high school. One
student responded, “Colleges look for them on
applications,” and another said, “Math plays a big part
in mechanics.” Likewise, students provided reasons for
why they did not take advanced mathematics courses
10
in high school, including “I have already taken two
[required] math classes,” and “I might not take those
classes if the career I choose doesn’t have the
requirement.” While some students come to see
themselves, and are recognized by others, as
mathematics learners from the requirements they
follow, the opposite is true for others. Students who
follow the minimal mathematics requirements, such as
those for graduation, may be less likely to see
themselves, or be recognized by others, as students
who are mathematics learners.
The three faces of identity discussed to this point
are not mutually exclusive but interact to form and
maintain a student’s identity. When beginning high
school, students are required to enroll in mathematics
courses. This contributes to students’ identity through
alignment. As they participate in mathematics classes,
the activities may appeal to them, and their identity is
further developed through engagement. Similarly,
students—like the one mentioned above who is
interested in mechanics—may envision their
participation in high school mathematics class as
preparation for a career. Mathematics is both a
requirement for entrance into the career and necessary
knowledge to pursue the career. Thus, identity in
mathematics is maintained through both imagination
and alignment.
Nature
Q: Why are some people good at math and some
people aren’t good at math?
A: I think it’s just in your makeup… genetic I
guess. (Barbara, grade 12, Precalculus, planning to
attend vocational training after high school)
The nature face of identity looks at who we are
from what nature gave us at birth, those things over
which we have no control (Gee, 2001). Typically,
characteristics such as gender and skin color are
viewed as part of our nature identity. The meanings we
make of our natural characteristics are not independent
of our relationships with others in personal and broader
social settings. That is, these characteristics comprise
only one part of the way we see ourselves and others
see us. In Gee’s social theory of learning, the nature
aspect of our identity must be maintained and
reinforced through our engagement with others, in the
images we hold, or institutionalized in the
requirements we must follow in the environments
where we interact.
Mathematics teachers are in a unique position to
hear students and parents report that their mathematics
learning has been influenced by the presence or
Four Faces of Identity
absence of a “math gene”, often crediting nature for
not granting them the ability to learn mathematics. The
claim of a lack of a math gene—and, therefore, the
inability to do mathematics—contrasts with Devlin’s
(2000b) belief that “everyone has the math gene” (p. 2)
as well as with NCTM’s (2000) statement that
“mathematics can and must be learned by all students”
(p. 13). In fact, cognitive scientists report,
“Mathematics is a natural part of being human. It arises
from our bodies, our brains, and our everyday
experiences in the world” (Lakoff & Núñez, 2000, p.
377). Mathematics has been created by the human
brain and its capabilities and can be recreated and
learned by other human brains. Yet, the fallacy persists
for some students that learning mathematics requires
special natural talents possessed by only a few:
I’m good at math. (Interview with Barbara, grade
12, Precalculus class)
I’m not a math guy. (Interview with Bill, grade 12,
not enrolled in math, planning to join the military
after high school)
Math just doesn’t work for me. I can’t get it
through my head. (Interview with Jackie, grade 12,
not enrolled in math, planning to enroll in a
vocational program after high school)
Although scientific evidence does not support the
idea that mathematics learning is related to genetics,
some students attribute their mathematics learning to
nature. The high school student who says “I’m not a
math guy” may feel that he is lacking a natural ability
for mathematics. He is likely as capable as any other
student but has come to the above conclusion based on
his experience with mathematics and the way it was
taught in his mathematics classes. Students who are not
the quickest to get the correct answers may learn, albeit
erroneously, that they are not capable of learning
mathematics. They do not engage in practices that are
recognized, in this case, to be the accepted practices of
the community. As a result, they view themselves, and
are viewed by others, to be peripheral members of the
community of mathematics learners.
As shown by the provided responses from students,
each of the four faces of identity exists as a way that
students come to understand their practices and
membership within the community of mathematics
learners. I have chosen to represent these faces of
identity as the four faces of a tetrahedron1 (Figure 1). If
we rotate a particular face to the front, certain features
of identity are highlighted while others are diminished.
Each face suggests different ways to describe how we
see ourselves as mathematics learners although they
are all part of the one whole. This representation of
identity maintains the idea that, as Gee (2001) wrote,
“They are four strands that may very well all be
present and woven together as a given person acts
within a given context” (p. 101). When considering the
four faces of identity as a mathematics learner, this
context is a traditional high school mathematics
classroom.
While all four faces contribute to the formation of
students’ identities as mathematics learners, the nature
face provides the most unsound and unfounded
explanations for students’ participation in the
mathematics community. To allow for the development
of all students to identify as mathematics learners,
students and teachers must discount the nature face and
build on the other three faces of identity.
Developing an Identity as a Mathematics Learner
To conclude this article, recommendations are
offered to teachers for developing and supporting
students’ positive identities as mathematics learners—
members of a community that develops the practices of
→
Figure 1. The four faces of identity
Rick Anderson
11
mathematics learning. The four faces of identity
described here are used to understand how students see
themselves as mathematics learners in relation to their
experiences in the mathematics classroom and through
the ways these experiences fit into broader life
experience. Students’ experiences will not necessarily
reflect just one of the four faces described (Gee, 2001).
In fact, some experiences may be stretched over two or
more faces. For example, learning advanced
mathematics in high school can contribute to a
students’ identity in two ways: (a) through imagination
with the image of math as an important subject for
entrance to higher education and (b) through alignment
since advanced mathematics is required to attend some
colleges. Taken together, however, we can see that a
focus on a particular face of identity suggests particular
experiences that can help to develop strong positive
identities as a mathematics learner in all students. The
engagement face of identity is developed through
students’ experiences with mathematics and, for most
high school students, their mathematics experiences
occur in the mathematics classroom. Therefore, the
most significant potential to influence students’
identities exists in the mathematics classroom. To
develop students’ identities as mathematics learners
through engagement, teachers should consider
mathematical tasks and classroom structures where
students are actively involved in the creation of
mathematics while learning to be “people who study in
school” (Lampert, 2001). That is, students must feel
the mathematics classroom is their scholarly home and
that the ideas they contribute are valued by the class
(Wenger, 1998). As indicated earlier, teacher-led
classrooms with students working independently on
single-answer exercises can cause students to learn that
mathematics is not a vibrant and useful subject to
study. Boaler (2000), for example, identified
monotony, lack of meaning, and isolation as themes
that emerged from a study of students and their
mathematics experiences. As a result many of these
students were alienated from mathematics and learned
that they are not valuable members of the mathematics
community.
Hence, mathematical tasks that engage students in
doing mathematics, making meaning, and generating
their own solutions to complex mathematical problems
can be beneficial in engaging students and supporting
their identity as a mathematics learner (NCTM, 2000).
A good starting point is open-ended mathematical
tasks, questions or projects that have multiple
responses or one response with multiple solution paths
(Kabiri & Smith, 2003). The mathematics classroom
12
can also be organized to encourage discussion, sharing,
and collaboration (Boaler & Greeno, 2000). In this
type of classroom setting, teachers “pull knowledge
out” (Ladson-Billings, 1995, p. 479) of students and
make the construction of knowledge part of the
learning experience.
With respect to imagination, the development of
students’ identities as mathematics learners requires
long-term effort on the part of teachers across
disciplines. The various images students have of
themselves and of mathematics extending outside the
classroom—in the past, present, or future—may be
contradictory and change over time. Teachers and
others in schools can consistently reinforce that
mathematics is an interesting body of knowledge worth
studying, an intellectual tool for other disciplines, and
an admission ticket for colleges and careers.
Since students’ identity development through
imagination extends beyond the classroom, teachers
can provide students with opportunities to see
themselves as mathematics learners away from the
classroom. For example, working professionals from
outside the school can be invited to discuss ways they
use mathematics in their professional lives; many
students may not be aware of the work of engineers,
actuaries, or statisticians. Another suggestion is to
require students to keep a log and record the ways in
which they use mathematics in their daily lives in order
to become aware of the usefulness of mathematics
(Masingila, 2002). This activity could provide an
opportunity for assessing students’ views of
mathematics and discussing the connections between
the mathematics taught in school and that used outside
the classroom.
Although many of students’ mathematical
requirements are beyond the control of teachers and
students, teachers can foster the alignment face of
identity. Teachers can hold their students to high
expectations so that these expectations become as
strong as requirements. Also, knowledge of
mathematics
requirements
for
post-secondary
education and careers can help students decide to
enroll in other mathematics courses. Because students
are known to cite post-secondary education and careers
as reasons for studying mathematics (Anderson, 2006;
Martin, 2000; Sfard & Prusak, 2005), teachers can
facilitate this alignment face by keeping students
abreast of the mathematics requirements for entrance to
college and careers.
Students may commonly reference the nature face
of identity, but this face is the least useful—and
potentially the most detrimental—for supporting
Four Faces of Identity
students as they become mathematics learners. As
mentioned earlier, the ability to learn mathematics is
not determined by genetics or biology (Lakoff &
Núñez, 2000). All students can become mathematics
learners, identifying themselves and being recognized
by others as capable of doing mathematics. Thinking
about the tetrahedron model of identity, if the other
faces are strong and at the fore, the nature face can be
turned to the back As suggested above, the other three
faces of identity can sustain mathematics learners’
identities—through
engaging
students
with
mathematics in the classroom, developing positive
images of students and mathematics, and establishing
high expectations and requirements—regardless of
students’ beliefs in an innate mathematical ability. Gee
(2001) points out that the nature face of identity will
always collapse into other sorts of identities. …
When people (and institutions) focus on them as
“natural” or “biological,” they often do this as a
way to “forget” or “hide” (often for ideological
reasons) the institutional, social-interactional, or
group work that is required to create and sustain
them. (p. 102)
Teachers need to be aware of the four faces of
identity of mathematics learners and of how their
students see themselves as mathematics learners and
doers. Detailed recommendations for developing
students’ identities as mathematics learners are
provided in Figure 2.
The four faces of identity discussed in this article
contribute to our understanding of how students come
to be mathematics learners. Through consistent and
sustained efforts by mathematics teachers to develop
positive identities in their students, more students can
come to study advanced mathematics and improve
their identities as mathematics learners. As I have
pointed out throughout this article, identities are
developed in relationships with others, including their
teachers, parents, and peers. We cannot assume that all
students will develop positive identities if they have
experiences that run to the contrary. We must take
action so each face of identity mutually supports the
others in developing all students’ identities as
mathematics learners.
Developing and Supporting Students’ Identities as Mathematics Learners
Engagement
Use mathematical tasks that allow students to develop strategies for solving problems and meanings for
mathematical tools.
•
Organize mathematics classrooms that allow students to express themselves creatively and communicate their
meanings of mathematical concepts to their peers and teacher.
•
Focus on the process and explanations of problem solving rather than emphasize quick responses to single-answer
exercises.
•
Imagination
Make explicit the ways mathematics is part of students’ daily lives. That is, help students identify ways they create
and use mathematics in their work and play.
•
Have working professionals discuss with high school students ways in which they use mathematics in their
professional lives, emphasizing topics beyond arithmetic.
•
•
Include mathematics topics in classes that relate to occupations, for example, geometric concepts that are part of
factory work or carpentry (e.g., see Smith, 1999; Masingila, 1994).
Alignment
•
•
Maintain expectations that all students will enroll in mathematics courses every year of high school.
Take an active role in keeping students informed of mathematics requirements for careers and college and university
entrance.
Figure 2. Recommendations
Rick Anderson
13
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of the practice of mathematics education in a rural high
school. Unpublished doctoral dissertation, Portland State
University, Oregon.
Boaler, J. (2000). Mathematics from another world: Traditional
communities and the alienation of learners. Journal of
Mathematical Behavior, 18, 379–397.
Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing
in mathematics worlds. In J. Boaler (Ed.), Multiple
perspectives on mathematics teaching and learning (pp. 171–
200). Westport, CT: Ablex.
Boaler, J., & Humphreys, C. (2005). Connecting mathematical
ideas: Middle school video cases to support teaching and
learning. Portsmouth, NH: Heinemann.
D’Ambrosio, U. (1990). The role of mathematics education in
building a democratic and just society. For the Learning of
Mathematics, 10, 20–23.
Devlin, K. (2000a). The four faces of mathematics. In M. J. Burke
& F. R. Curcio (Eds.), Learning mathematics for a new
century (pp. 16–27). Reston, VA: NCTM.
Devlin, K. (2000b). The math gene: How mathematical thinking
evolved and why numbers are like gossip. New York: Basic
Books.
Gee, J. P. (2001). Identity as an analytic lens for research in
education. Review of Research in Education, 25, 99–125.
Kabiri, M. S., & Smith, N. L. (2003). Turning traditional textbook
problems into open-ended problems. Mathematics Teaching in
the Middle School, 9, 186–192.
Kirshner, D. (2002). Untangling teachers’ diverse aspirations for
student learning: A crossdisciplinary strategy for relating
psychological theory to pedagogical practice. Journal for
Research in Mathematics Education, 33, 46–58.
Ladson-Billings, G. (1995). Toward a theory of culturally relevant
pedagogy. American Educational Research Journal, 32, 465–
491.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes
from: How the embodied mind brings mathematics into being.
New York: Basic Books.
Lampert, M. (2001). Teaching problems and the problems of
teaching. New Haven, CT: Yale University Press.
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Lave, J., & Wenger, E. (1991). Situated learning: Legitimate
peripheral participation. Cambridge, UK: Cambridge
University Press.
Martin, D. B. (2000). Mathematics success and failure among
African-American youth: The roles of sociohistorical context,
community forces, school influences, and individual agency.
Mahwah, NJ: Lawrence Erlbaum.
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Anthropology & Education Quarterly, 25, 430–462.
Masingila, J. O. (2002). Examining students’ perceptions of their
everyday mathematics practice. In M. E. Brenner & J. N.
Moschkovich (Eds.), Everyday and academic mathematics in
the classroom (Journal for Research in Mathematics Education
Monograph No. 11, pp. 30–39). Reston, VA: National Council
of Teachers of Mathematics.
Mendick, H. (2003). Choosing maths/doing gender: A look at why
there are more boys than girls in advanced mathematics
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CT: Praeger.
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1
Others have used the idea of “faces” to convey the many
interrelated aspects of a topic. For example, Devlin (2000a)
describes “The Four Faces of Mathematics.”
Four Faces of Identity
The Mathematics Educator
2007, Vol. 17, No. 1, 15–23
A Quilting Lesson for Early Childhood Preservice and Regular
Classroom Teachers: What Constitutes Mathematical Activity?
Shelly Sheats Harkness
Lisa Portwood
In this narrative of teacher educator action research, the idea for and the context of the lesson emerged as a
result of conversations between Shelly, a mathematics teacher educator, and Lisa, a quilter, about real-life
mathematical problems related to Lisa’s work as she created the templates for a reproduction quilt. The lesson
was used with early childhood preservice teachers in a mathematics methods course and with K-2 teachers who
participated in a professional development workshop that focused on geometry and measurement content. The
goal of the lesson was threefold: (a) to help the participants consider a nonstandard real-world contextual
problem as mathematical activity, (b) to create an opportunity for participants to mathematize (Freudenthal,
1968), and (c) to unpack mathematical big ideas related to measurement and similarity. Participants’ strategies
were analyzed, prompting conversations about these big ideas, as well as an unanticipated one.
What would happen if the activities we,
mathematics teacher educators, use to model best
practices and standards-based teaching in mathematics
methods courses and professional development
workshops honored mathematical activity that is
nonstandard in the sense that it is sometimes not even
considered mathematics? Mary Harris (1997) describes
how she uses “nonstandard problems that are easily
solved by any woman brought up to make her family’s
clothes” (p. 215) in mathematics courses for both
preservice and classroom teachers. To make a shirt,
“all you need (apart from the technology and tools) is
an understanding of right angles, parallel lines, the idea
of area, some symmetry, some optimization and the
ability to work from two-dimensional plans to threedimensional forms” (p. 215). Although none of these
considerations are trivial, making a shirt is not
typically considered mathematical activity. Harris
raises questions about why this is true. Is it because the
seamstress is a woman or because only school
mathematics is valued by our society? And, more
broadly, what constitutes mathematical activity?
We contend that mathematical activity is both
physical and mental. It requires the use of tools, such
Shelly Sheats Harkness, Assistant Professor at the University of
Cincinnati in the Secondary Education Program, is a mathematics
educator. Her research interests include Ethnomathematics,
mathematics and social justice issues, and the impact of listening
and believing in mathematics classrooms.
Lisa Portwood is a self-taught quiltmaker and quilt historian. She
has been quilting for the past 21 years and is an active member of
the American Quilt Study Group. She is also a financial secretary
at Miami University in the Teacher Education Department.
Shelly Sheats Harkness & Lisa Portwood
as physical materials and oral and written languages
that are used to think about mathematics (Heibert et al.,
1997). In the process of doing mathematics, one thinks
or reasons in logical, creative, and practical ways.
According to Sternberg (1999), American schools have
a closed system that consistently rewards students who
are skilled in memory and analytical reasoning,
whether in mathematics or other domains. This system,
however, fails to reward, in the sense of grades,
students’ creativity, practical skills, and thinking. In
problem-solving situations, students should be
encouraged to use both physical and mental activity to
do mathematics in order to (a) identify the nature of the
problem; (b) formulate a strategy; (c) mentally
represent the problem; (d) allocate resources such as
time, energy, outside help, and tools; and (e) monitor
and evaluate the solution (Sternberg, 1999).
Researchers who studied the consumer and vendor
sides of mathematical reasoning found skills revealed
by a practical test were not revealed on an abstractanalytical, or school-type, test (Lave, 1998; Nunes,
1994).
Too often, mathematics is viewed as the mastery of
bits and pieces of knowledge rather than as sense
making or as sensible answers to sensible questions
(Schoenfeld, 1994). Sensible questions arise from
many nonstandard contexts. If we design problems that
are based on those questions, model best practices, and
elicit mathematical big ideas, our students might begin
to see mathematics as a human endeavor. They may
use logical, creative, and practical thinking to solve
those problems.
15
As sometimes happens, an idea for a nonstandard
real-world lesson took root in an unexpected place.
During lunchroom conversations, Lisa, a member of
the American Quilt Study Group (AQSG), described
her work on a reproduction quilt with Shelly, a
mathematics teacher educator. AQSG members
participate in efforts to preserve quilt heritage through
various publications, an extensive research library, and
a yearly seminar. At this seminar, the AQSG invites
members to make smaller versions, or reproductions,
of antique quilts from a specified time period so that
many of these can be displayed in one area.
Lisa and Shelly discussed the mathematical
problems Lisa encountered as she designed the
reproduction quilt. The square quilt she was
reproducing, one her neighbor owned, had a side
length of about 88 inches; however, the display quilt
could not be more than 200 inches in perimeter. She
wanted to trace the templates for the design from her
neighbor’s quilt (see Figure 1) and then use a copy
machine to reduce these traced sketches (see Figure 2).
She needed to decide which reduction factor to use and
how much of each color fabric–white and blue–to
purchase. She wanted to buy the least possible amount
of fabric. These reproduction quilt problems became
the context for the lesson that Shelly created and used
with both early childhood education preservice
teachers enrolled in a mathematics methods course and
with K-2 classroom teachers in a professional
development workshop.
Figure 1: Reproduction (left) and original quilt (right)
Figure 2: Templates
16
A Quilting Lesson
Literature Review of Lessons Related to
Mathematics and Quilting
Because of a desire to know more about the
connections between mathematics and quilting, we
began by searching for literature related to lessons for
teachers. We found some rich resources that included a
wide range of mathematical topics embedded in these
lessons.
Transformational geometry was the foundation for
several lessons. Whitman (1991) provided activities for
high school students related to Hawaiian quilting
patterns with a focus on line symmetry and rotational
symmetry. Ernie (1995) showed examples of how
middle school students used modular arithmetic and
transformational geometry to create quilt designs. Most
recently, Anthony and Hackenberg (2005) described an
activity for high school students that made “Southern”
quilts by integrating an understanding of planar
symmetries with wallpaper patterns.
The patterns and sequences found in quilt designs
provided a basis for mathematical topics in other
lessons. Rubenstein (2001) wrote about several
methods that high school students used to solve a
mathematical problem related to quilting: finite
differences, the formula for the sum of consecutive
natural numbers, and a statistical-modeling approach
using a graphing calculator. Westegaard (1998)
described several quilting activities for students in
grades 7-12 that reinforced coordinate geometry skills
and concepts such as identifying coordinates,
determining slope as positive or negative, finding
intercepts, and writing equations for horizontal and
vertical lines. Mann and Hartweg (2004) showcased
third graders’ responses to an activity in which they
covered two different quilt templates with pattern
blocks and then determined which template had the
greatest area.
Reynolds, Cassell, and Lillard (2006) shared
activities based on a book by Betsy Franco, Grandpa’s
Quilt, which they incorporated into lessons for their
second-graders. In these activities, students made
connections to patterns, measurement, geometry, and a
“lead-in” to multiplication. In a lesson for third
graders, Smith (1995) described how she linked the
mathematics of quilting–problem solving, finding
patterns, and making conjectures–with social studies
through the use of a children’s book, Jumping the
Broom. Also with a connection to integration with
social studies, Neumann (2005) focused on the
significance of freedom quilts, the Underground
Railroad, the book Sweet Clara and the Freedom Quilt
Shelly Sheats Harkness & Lisa Portwood
(Hopkinson, 1993), and mathematics–the properties of
squares, rectangles, and right triangles–in her lesson
for upper elementary school children.
In their book Mathematical Quilts: No Sewing
Required!, Venters and Ellison (1999) included 51
activities for giving “pre- and post-geometry students
practice in spatial reasoning” (p. x). These activities
are situated within four chapters: The Golden Ratio
Quilts, The Spiral Quilts, The Right Triangle Quilts,
and The Tiling Quilts. The authors noted that the quilts
that inspired their book were created when they were
teaching mathematics and taking quilting classes in the
mid-1980s:
Because we had no patterns for our [mathematical]
quilts, we had to draft the design and solve the
many problems that arise in this process involving
measurement, color, and the sewing skills needed
for construction ... Taking on a project and working
it through to completion provide invaluable
experiences in problem solving. (p. ix)
These were the same challenges that Lisa faced when
she created her reproduction quilt.
What was missing from this extensive list of
resources was any reference to using mathematics and
quilting in lessons for preservice teachers or
professional development workshops for teachers. We
felt that Lisa’s real-world task would provide the
opportunity for the preservice and classroom teachers
to mathematize,1 recognize big mathematical ideas, and
consider what constitutes mathematical activity. We
thought the big ideas that would emerge from this task
included:
•
Measurement is a way to estimate and compare
attributes.
•
A scale factor can be used to describe how two
figures are similar.
Within this article, we briefly describe the
preservice and classroom teachers who we worked
with, the quilting lesson that we created–based upon
the actual mathematical questions that Lisa faced as
she created her reproduction quilt–and the mathematics
that the preservice and classroom teachers used as they
mathematized. We then summarize our follow-up
conversations about participants’ general reactions to
the reading, An Example of Traditional Women’s Work
as a Mathematics Resource (Harris, 1997), and our
question: What constitutes mathematical activity?
Finally, we describe the emergence an unanticipated
mathematical big idea, based on the ways that the
preservice and classroom teachers approached the
problem.
17
We piloted this lesson during the first semester of
the 2004-2005 school year with a class of preservice
teachers at a large public university in the Midwestern
United States. We then obtained IRB approval to
collect data in the form of work on chart paper that
groups in a second class did the following semester.
We used the lesson again during the summer of 2005
with a group of 20 classroom teachers, grades K-2, in a
professional development workshop.
The K-2 Preservice Teachers and Classroom
Teachers
Teaching Math: Early Childhood (TE300) was a 2credit methods course that preservice teachers enrolled
in prior to student teaching. The course included a twoweek field experience in which the preservice teachers
wrote one standards-based, “best practices” lesson
plan, and then taught the lesson. During each class
session throughout the semester, we focused on a
chapter from Young Mathematicians at Work (Fosnot
& Dolk, 2001) and one of the content or process
standards from Principles and Standards for School
Mathematics (PSSM) (National Council of Teachers of
Mathematics [NCTM], 2000). The reading from PSSM
for the week of the quilting activity focused on
measurement.
All but one of the sixty TE300 preservice teachers
were female. In mathematical autobiographies written
during the first week of the course, about two-thirds of
these preservice teachers said they either disliked or
had mixed feelings about their previous school
mathematical experiences, K-12 and post-secondary.
Some who reported dislike for mathematics described
feeling physically sick before math class, helplessness,
and lack of self-confidence. Those with mixed feelings
wrote about grades of A’s and B’s as “good times” and
grades of D’s and F’s as “bad times.” Many described
board races and timed math tests over basic facts as
dreaded experiences. Generally speaking, they hoped
to help their own students experience the success that
they did not enjoy in math classes. These mathematical
experiences posed a special challenge for us because
we felt that their beliefs about teaching and learning
mathematics had to be addressed. Due to time
constraints, we attempted to address them at the same
time that we talked about best practices and standardsbased methods.
The K-2 classroom teachers participated in a
professional development workshop offered the
summer after we implemented the lesson with the
preservice teachers. Most opted for free tuition to earn
graduate credit; each of them also received a $300
18
stipend to spend on math books, manipulatives, or
other items. All but one of the 20 teachers were female.
The majority of the classroom teachers described their
own mathematical experiences as less than pleasant
and their fear of mathematics was evident from the
beginning. Similar to the preservice teachers, they were
very bold about their dislike of mathematics. They
openly discussed their views of mathematics as a set
rules and procedures to be memorized. Although the
focus of the workshop was on improving the teachers’
content knowledge in geometry and measurement, we
felt that we also needed to address their beliefs about
teaching and learning mathematics within that context.
As a springboard for the semester and the
workshop, we discussed the meaning of mathematics
and mathematizing. To initiate discussion, we posed
the following question: “Is mathematics a noun or a
verb?” Some thought it was a noun because
mathematics is a discipline or subject you study in
school. Others argued that it was a verb because you
“do” it. About half argued that it could be considered
both.
When both the preservice and classroom teachers
read Young Mathematicians at Work (Fosnot & Dolk,
2001), we again negotiated the meaning of
mathematics and mathematizing (Freudenthal, 1968),
reaching a consensus consistent with Fosnot and
Dolk’s interpretation: “When mathematics is
understood as mathematizing one’s world—
interpreting, organizing, inquiring about, and
constructing meaning with a mathematical lens, it
becomes creative and alive” (p. 12-13). These are all
processes that “beg a verb form” (p. 13) because
mathematizing centers around an investigation of a
contextual problem.
The Lesson
According to Fosnot and Dolk (2001), situations
that are likely to be mathematized by learners have at
least three components:
•
The potential to model the situation must be built
in.
•
The situation needs to allow learners to realize
what they are doing. The Dutch used the term
zich realizern, meaning to picture or imagine
something concretely (van den Heuvel-Panhizen,
1996).
•
The situation prompts learners to ask questions,
notice patterns, and wonder why or what if.
Guided by these components, we planned the lesson
within Lisa’s quilting context.
A Quilting Lesson
Throughout the semester and the workshop, Shelly
read part of a picture book, Sweet Clara and the
Freedom Quilt by Deborah Hopkinson (1993), in order
to provide a context for the quilting problem. In this
book, Sweet Clara is a slave on a large plantation. Her
Aunt Rachel teaches her how to sew so that Clara can
work in the Big House. There, she overhears other
slaves’ talk of swamps, the Ohio River, the
Underground Railroad, and Canada. Listening intently
to these conversations, Clara visualizes the path to
freedom and creates a quilt that is a secret map from
the plantation to Canada.
To launch the problem, Lisa told the groups about
her work in the AQSG and explained why she wanted
to produce a replica of a two-color quilt from the
period 1800–1940. As it happened, her neighbor found
a quilt in her basement and showed it to Lisa, who
could hardly believe her luck! Not only did Lisa like
the design, but she liked the two colors, blue and white,
as well. She decided to use her neighbor’s quilt as the
original for her reproduction.
We then shared the parameters for the reproduction
quilt, as given by the AQSG:
•
Display: Each participant is limited to one quilt.
Each quilt must be accompanied by a color
image of the original and the story of why it was
chosen.
•
Size: The maximum perimeter of the replica is
200 inches. This may require reducing the size of
the original quilt. Size is limited to facilitate the
display of many quilts.
•
Color: “Two-color” indicates a quilt with an
overall strong impression of only two colors. A
single color can include prints that contain other
colors but read as a single color.
We also explained how the square-shaped original quilt
had side lengths of 88 inches and showed them a photo
of both the original and reproduction quilts (see Figure
1).
In order to help both the preservice and classroom
teachers immerse themselves in mathematizing and
consider what constitutes mathematical activity related
to measurement and similarity, we posed the following
three questions that were actual questions that Lisa
faced as she prepared to create the study quilt:
1.
By what percent did she need to reduce the
original quilt to fit the 200 inches measured
around all four sides (the perimeter)? The
original was 88 inches on one side. (Lisa wanted
to use the copy machine and a scale factor to
reduce the pattern pieces she traced from the
Shelly Sheats Harkness & Lisa Portwood
original quilt to create the pattern pieces for the
reproduction.)
2.
How much white fabric did she need to buy for
the front and back of the quilt? (Please note:
Fabric from bolts measures 44-45 inches wide.)
3.
How much blue fabric did she need to buy for
the appliqués, borders, and binding around the
edges? (Use the templates from the original quilt
to determine your answer.)
Before they began to work, we also showed them an
actual bolt of fabric and explained how fabric is sold
from the bolt because we were not sure they would
know what this meant (and many did not!). We gave
each group original-sized copies of the templates used
for the appliqué blue pieces on the original quilt (see
figure 2) to use to answer the third question.
We asked them to keep a record on chart paper of both
the mathematics and mathematical thinking or
processes they used to answer the questions so that
they could share the results in a whole group
discussion. Calculators, rulers, meter-yard sticks, tape
measures, string, scissors, and tape were also available.
We walked around, listened, and watched the
groups work. Some had questions we had not expected:
Is there white underneath the blue? Does the back have
to be all one piece of fabric? Can we round our
numbers? Should we allow for extra fabric? The
students’ questions made us realize that, even though
the three questions we posed might seem trivial for
some quilters and mathematicians, they served as a
springboard for the rich mathematical discussion that
followed the small group work.
The Mathematics
We assessed the groups’ strategies while they
worked to answer the three questions by listening to
their discussions and analyzing the chart paper record
of their strategies. We noticed that most groups, both in
the class for preservice teachers and the workshop for
classroom teachers, took the directions quite literally
(i.e. that the reproduction perimeter must be exactly
200 inches) and used similar strategies.
After the groups posted their chart paper on the
walls, we began a whole-group discussion by posing
the question: Are the two quilts, the reproduction and
the original, mathematically similar? All agreed that
they were but when asked why, their responses focused
on the notion that they just looked similar. They knew
the quilts were not congruent because they were
different sizes. We told them that we would return to
this question later so that we could negotiate a
mathematical definition of similarity.
19
Figure 3: A solution to Question 1
Question 1
All but two groups thought of the perimeter
parameter as exact and created scale factors to reduce
the quilt so it would have a perimeter as near to 200
inches as possible. These groups said that the pattern
should be reduced on the copy machine by either 57%
or 43%; this led to an interesting conversation about
how these were related and which one made more
sense. Would we enter 57% or 43% into the copy
machine? Which number makes the most sense based
on what we know about copying machines and how
they reduce images?
The two groups that did not use the method
adopted by the majority used the same scale factor as
Lisa, approximately 67.7%. We asked these groups to
share their thinking (see Figure 3) because it seemed
like their mathematical calculations and reasoning
were also valid. How could there be different answers?
Instead of focusing on perimeter, these groups created
a ratio of the total area of the reproduction quilt to the
total area of the original quilt, 2500/7744 (assuming
the quilt would measure 50 inches by 50 inches); the
ratio was 32.3%. So, the area needed to be reduced by
67.7%. At first, we wondered why the percents differed
when groups compared areas instead of perimeters for
the same geometric figure. This led to the opportunity
to discuss an unexpected big idea, something we had to
think deeply about ourselves before we realized why
the results for the groups were different: When the
perimeter of a rectangle is reduced by a scale factor,
the area is not reduced by the same scale factor. In fact,
the ratio of the areas is the square of the ratio of the
perimeters. In addition, this is true for any size of
rectangular quilt or similar figures.
We had to encourage the preservice and classroom
teachers to think about why the two percents were
different. When pressed, they realized that, because
area is a square measure, taking the ratio of two areas
resulted in a different value than taking the ratio of two
20
perimeters or side lengths. In fact, 32.3% was
approximately the square of the ratio for the perimeter
comparison, (50/88)2.
This led us to rethink our questions about the copy
machine. Does the word “reduction” lead to
mathematical misconceptions? How does the reduction
scale factor change the perimeter and the area? For
instance, is the original image reduced by the selected
percentage or does the machine create an image that is
that percentage of the original? Experimentation with
the copy machine reduction function helped us answer
this question (we leave it to the reader to explore).
Interestingly, even though most groups considered
the perimeter parameter as strict, Lisa knew that the
reproduction quilt could be no larger than 200 inches
so she decided to use a 50% reduction—this made her
study quilt 44 inches per side with a perimeter of 176
inches, which was “close enough.” Like Lisa, two
groups decided that 50% was a reasonable and
“friendly” number to use, making other calculations for
the quilt less cumbersome. This led to a conversation
about when close enough is sufficient for measurement
and other uses of mathematics. We felt this was
especially important because many of the preservice
and classroom teachers experienced mathematics as
problems with one exact answer. The idea that
measurement can be precise but not exact was
something they needed to think about.
Questions 2 and 3
For the second and third questions, the chart paper
revealed that the groups had a wide range of answers
and some mathematical misconceptions. Most had
answers close to 3 yards of white fabric and 1.5 yards
of blue fabric. Some groups drew sketches of the fabric
(see Figure 4). Groups that drew sketches or
representations had the best estimates for conserving
fabric. Even though one might think that determining
A Quilting Lesson
Figure 4: A solution for Question 2
the amount of white material would result in trivial
mathematical conversations, we noticed that most
groups tried various ways to overcome the fact that the
width of the material (44-45”) posed a real contextual
dilemma, as it was shorter than the width of the quilt.
In other words, they had to consider both area and
length in their attempt to minimize the amount of
fabric needed.
For the second question, one group decided that
Lisa needed to buy 47 yards of white fabric. This group
felt the sides of the quilt should measure 41 inches
because the fabric was 44-45 inches wide (see Figure
5). This was similar to Lisa’s thinking and within the
200-inch parameter for total perimeter.
However, we were shocked by their answer of 47
yards! They did not take into account the notion that
you must divide by 144 to convert square inches to
square feet and by 9 to convert square feet to square
yards. This mistake is one that could have been
predicted with out-of-context problems, but Lisa had
shown her reproduction quilt before they began
working in their groups. What was most disturbing
about this answer was the fact that 47 square yards
made no sense given the size of one square yard.
Similarity
Returning to the definition of similarity, we again
posed the question: Why are the two quilts
mathematically similar? The preservice and classroom
teachers negotiated a definition that made sense to
them. They talked about “not the same size but the
same shape” in terms of scale factors and created a
Figure 5: A solution for Question 2
Shelly Sheats Harkness & Lisa Portwood
21
working definition: The scale factors or ratios of the
corresponding sides of the two quilts are equal (or
proportional) and the ratio of the areas is the square of
the ratio of the side lengths.
Generally speaking, we felt as though this task
provided opportunities to talk about many
mathematical notions related to measurement and
similarity including exactness versus precision,
estimation, ratio, proportion, percent, scale factor,
perimeter, and area. We briefly discussed the kinds of
symmetry—reflection (flip), rotation (turn), and
translation (slide)—in the quilt but this was not a
focus. The preservice and classroom teachers also
noted that they used all five NCTM process standards
(problem solving, communication, reasoning and
proof, connections, and representation) as they worked
in their groups and during our class discussion.
What Constitutes Mathematical Activity?
As a way to help the preservice and classroom
teachers consider what constitutes mathematical
activity, we gave them copies of An Example of
Traditional Women’s Work as a Mathematics Resource
(Harris, 1997) to read before our next class or
professional development session. According to Harris,
in mathematical activity, women are disadvantaged in
two ways: (a) until very recently, female
mathematicians were barely mentioned; and (b), in a
world where women's intellectual work is not taken
very seriously, the potential for receiving credit for
thought in their practical work is severely limited. In
her book, Harris showed her students a Turkish flat
woven rug, called a kilim, and her students explored
the mathematics involved in its construction. She also
displayed a right cylindrical pipe created by an
engineer and a sock knitted by a grandmother. She then
posed the following questions: Why is it that the
geometry in the kilim is not usually considered serious
mathematics? Is it because the weaver has had no
schooling, is illiterate, and is a girl? How do we know
that the weaver is not thinking mathematically? Why is
designing the pipe considered mathematical activity
but knitting the heel of the sock is not?
This reading helped create an opportunity for the
K-2 preservice and classroom teachers, groups that are
mostly female, to talk about their own beliefs regarding
what constitutes mathematical activity in the context of
women’s work. Many of them had considered school
mathematics as the only kind of mathematics. After
doing the quilting activity and discussing their reading
22
of Harris (1997), however, they began to talk about
doing mathematics within traditional women’s work
such as measuring and hanging wallpaper, cooking,
creating flower garden blueprints, playing musical
instruments, and determining the number of gallons of
paint needed to paint a room. We did not discuss the
nature of mathematical activity as both physical and
mental activity but, after thinking more deeply about it
ourselves, we now realize that this was a missed
opportunity. It seems that our conversation should also
focus on the logical, creative, and practical ways in
which we think and reason while doing mathematics.
Concluding Remarks
Through our collaborative effort to create a lesson
with real-world applications and a reading related to
what constitutes mathematical activity, the preservice
and classroom teachers saw mathematics as something
you do outside of school. They were mathematizing,
organizing, and interpreting the world through a
mathematical lens as they made conjectures about the
same questions that Lisa faced when she created her
reproduction study quilt.
Analysis of the student strategies revealed
opportunities to discuss big ideas related to
measurement and similarity and what constitutes
mathematical activity. It also prompted Lisa to take
another picture of the two quilts, to illustrate the big
idea that emerged from the groups’ sense-making:
reducing the perimeter by 50% created a reproduction
quilt with one-fourth the area of the original quilt (see
Figure 6).
Within this lesson, we modeled the kind of
teaching we hope these preservice and classroom
teachers will think about and use in their classrooms:
helping students mathematize, make connections to big
ideas and real-world mathematics, and question what
constitutes mathematical activity. As Harris (1997)
noted, the role of mathematics teachers should not be
to teach some theory and then look for applications,
but to analyze and elucidate the mathematics that
grows out of the students' experience and activity.
Using nonstandard contextual problems creates
opportunities to honor school mathematics and
mathematical activity that exists within the real world
of everyday activity. By doing so, we also honor and
respect our students’ logical, creative, and practical
thinking. We give voice to their mathematics.
A Quilting Lesson
Figure 6: The reproduction quilt on top of the original quilt
References
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without sewing: Investigating planar symmetries in Southern
quilts. Mathematics Teacher 99(4), 270–276.
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F. Coxford (Eds.) Connecting mathematics across the
curriculum (pp. 170–181). Reston, VA: National Council of
Teachers of Mathematics.
Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work:
Constructing number sense, addition, and subtraction.
Portsmouth, NH: Heinemann.
Freudenthal, H. (1968). Why to teach mathematics so as to be
useful. Educational Studies in Mathematics 1, 3–8.
Harris, M. (1997). An example of traditional women’s work as a
mathematics resource. In A. B. Powell & M. Frankenstein
(Eds.) Ethnomathematics: Challenging Eurocentrism in
mathematics education (pp. 215–222). Albany, NY: State
University of New York Press.
Heibert, J., Carpenter, T. P., Fennema, E., Fuson, K.C., Wearne, D.,
Murray, H., Olivier, A., & Human, P. (1997). Making sense
teaching and learning mathematics with understanding.
Portsmouth, NH: Heinemann.
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Encyclopedia of human intelligence, Vol. 2 (pp. 1045–1049).
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multiple solutions. Mathematics Teacher 94(3), 176.
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exploration of “Grandpa’s Quilt”. Teaching Children
Mathematics 12(7), 340–345.
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curricula? Journal of Mathematical Behavior 13, 55–80.
Smith, J. (1995). Links to literature: A different angle for
integrating mathematics. Teaching Children Mathematics
1(5), 288–293.
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L. V. Stiff & F.R. Curcio (Eds.), Developing mathematical
reasoning in grades K-12 (pp. 37–44). Reston, VA: National
Council of Teachers of Mathematics.
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mathematics education. Series on Research in Education,
no.19. Utrecht, Netherlands: Utrecht University.
Venters, D., & Ellison, E. K. (1999). Mathematical quilts: No
sewing required! Berkley, CA: Key Curriculum Press.
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Shelly Sheats Harkness & Lisa Portwood
1
The term mathematize was coined by Freudenthal (1968)
to describe the human activity of modeling reality with the
use of mathematical tools.
23
The Mathematics Educator
2007, Vol. 17, No. 1, 24–31
Graduate Level Mathematics Curriculum Courses:
How Are They Planned?
Zelha Tunç-Pekkan
Even though there is much research related to the teaching and learning of K-12 mathematics, there are few
studies in the literature related to university professors’ teaching. In this research report, I investigated how
three professors of mathematics education structure their graduate level curriculum courses. The results show
that three factors influence the ways that the professors design this course: (a) their view of mathematics
curriculum, (b) their view of graduate students’ contributions to classroom atmosphere, and (c) their learning
goals for the graduate students.
One of the goals in writing this article is to make
the practice of teacher educators more public. This goal
was motivated by Shulman’s (1998) exhortation to
teacher educators:
Now, think of your functioning as teachers. How
much of what you do as a teacher—these great acts
of creativeness, these judgments you make all the
time as a teacher, the courses you design, the
internships you tinker with, modify and strike gold
with—how much of that ever becomes public?
How much is susceptible to critical review by your
colleagues or becomes a building block in the work
of other members of the teacher education
community throughout your own institution, much
less the nation or the world? (p.18)
Preparing to teach a mathematics curriculum
course requires more than deciding what mathematics
topics to teach and in which order, as is the case for
many content courses. It requires more than choosing
K–12 mathematics classroom ideas to analyze with
future teachers, as is the case in some methods courses.
Preparing to teach graduate-level mathematics
curriculum courses is a complex endeavor because
these courses involve the integration of various aspects
of mathematics education—curriculum, content and
children’s mathematical learning—with the goals of
educating graduate students. Such complexity seems to
warrant the investigation of how professors design
graduate-level mathematics curriculum courses.
However, there are not many studies investigating how
Zelha Tunç-Pekkan is a doctoral student in the Department of
Mathematics and Science Education at the University of Georgia.
Her research explores how 8th grade students use their fractional
knowledge for the construction of algebraic knowledge including
algebraic written notations. She received her bachelor’s degree
from Middle East Technical University, Turkey and her master’s
degree from Indiana University, Indianapolis.
24
university professors conceptualize the courses they
teach,
especially
graduate-level
mathematics
curriculum courses. This study had two purposes: (a) to
gain insight into university teachers’ decision-making
processes in planning a graduate-level curriculum
course and (b) to make these insights public so that
educators who teach or plan to teach similar courses
will have a stronger base of information to guide their
decision-making processes.
Literature Review
Shuell (1993) stated that teaching and learning at
all grade levels are dynamic and reciprocal processes
and that research should attempt to account for the
complex and simultaneous effects of developmental,
affective, and motivational influences, as well as
cognitive factors. Many investigations of K–12
mathematics teachers’ practices have been conducted
to explore motivational influences and cognitive
factors that affect these complex reflexive processes
(e.g., research on teachers’ knowledge, beliefs, and
motivation). However, there is a paucity of research on
university professors’ motivation or cognitive
processes when they reflect upon their practice as
teachers of graduate level mathematics education
courses.
Looking beyond mathematics, there are a few
general studies of university professors’ beliefs about
students’ learning, and how they perceive their practice
as course instructors. For example, considering
professors’ teaching practices, Kugel (1993) theorized
that professors undergo three stages of development as
teachers: self, subject, and student. During the first
stage of their career, professors primarily focus upon
their own role in the classroom and how they feel
about their own abilities, i.e., self. In the second stage,
professors focus on teaching the subject, which
Graduate Level Mathematics Curriculum Courses
includes the subject matter and the materials they use
when teaching. In the final stage, professors focus on
students and how they learn. This stage is the least
stressful part of the development, according to Kugel,
because at this stage professors have already mastered
the previous two, and are better prepared to focus on
their students. Kugel admitted that not necessarily all
professors experience these stages, but he claimed that
they were a commonality in the experiences of many.
Stark and Lattuca (1997) made an additional
generalization about professors and wrote about their
unsystematic way of thinking and acting in their
teaching practices:
professors’ practices. In addition, Jackson (1994)
claimed that context and values play a fundamental
role in professors’ designing processes at the
undergraduate level. These general studies offer some
insight into investigating how mathematics education
professors conceive their practices. However, there is
still a need for studies that investigate professors’
design processes for graduate-level courses, where
many students are mature adults coming to school to
learn more about their own profession. Thus, this study
focuses on the following research question: How do
university professors decide what to teach in graduate
level mathematics curriculum courses?
Instructional methods are chosen more often
according to personal preferences or trial and error
rather than through systematic attention to [the]
nature of the expected learning, the nature of the
student group or audience, and many varied
practical constraints, such as size of the class (p.
288).
Research Design & Methods
In contrast to these authors, I believe, most
professors use some trial and error in their classroom in
a systematic way. They revise their practice
considering the learning goals for their courses and the
nature of the student group. Therefore, the definition of
trial and error and how it affects teaching needs more
investigation at the graduate level. Hence, there is a
need to understand how professors take into
consideration the nature of the material they want to
teach, the group of students they teach, and their
expected learning goals when making decisions on
how to teach the course differently. With this study, I
hoped to gain insight in these issues.
Jackson (1994) observed: "We do not know what
teachers in higher education think about their teaching
and we do not know the cognitive processes in which
they engage when they develop curriculum" (p. 2). In
her study, she interviewed 11 university professors
from different disciplines to understand their
conception of curriculum design. In particular, her
principal questions were “How do university teachers
see themselves as curriculum makers; how do they
think and make decisions about their teaching; how do
they interpret their experiences and give meaning to
their work?” (p. 2). In her findings, Jackson indicated
that professors’ decisions about the curriculum design
of a course are based upon the context of the course as
well as their individual and disciplinary values.
As a summary, Kugel (1993) indicated what
professors might focus on when they reflect on their
practice, whereas Stark and Lattuca (1997) suggested
that unsystematic trial and error might be part of
Zelha Tunç-Pekkan
The study consisted of three cases. Patton (2002)
said, "Cases are units of analysis. What constitutes a
case, or a unit of analysis, is usually determined during
the design stage and becomes the basis for purposeful
sampling in qualitative inquiry" (p. 447). In this vein, I
selected three professors who approach teaching from
different theoretical frameworks as my units of
analysis. These professors, Martin, Rafaela, and
Adam1, all have taught mathematics education courses
at the undergraduate and graduate levels.
To a certain extent, this study was an intrinsic case
study (Stake, 2000) because I had a personal interest in
trying to better understand the selected cases. As both a
graduate student and a future instructor of curriculum
courses, I was interested in how those particular
professors decided what to teach in their curriculum
courses and how they perceived graduate students. I
used interviews and course artifacts to get detailed
information about the cases and interviewed each
participant for one hour. An overall interview guide
(see Appendix) was developed for this semi-structured
interview. The use of the interview guide followed
Patton‘s (2002) suggestions:
The interview guide provides topics or subject
areas within which the interviewer is free to
explore, probe, and ask questions that will
elucidate and illuminate that particular subject.
Thus, the interviewer remains free to build a
conversation within a particular subject area, to
word questions spontaneously, and to establish a
conversational style but within the focus on a
particular subject that has been predetermined. (p.
343)
The interview guide consisted of ten open-ended
questions. The foci of the guide were the professors’
understanding of curriculum, their goals for the
curriculum course, and the difference between teaching
25
graduate level mathematics education curriculum
courses and teaching undergraduate mathematics
content and method courses. The interviews were
audiotaped, transcribed, and analyzed. Participants
were asked to answer follow-up questions based on the
initial data analysis. I also collected course syllabi,
selected books, organized readers, and other artifacts
used in these professors’ courses to help with the
analysis of the interview data.
Context and Participants
This study focused on a graduate-level curriculum
course taught in a large university in the southeastern
region of the United States. The course was listed as a
three credit hour graduate course in the university
graduate catalog, and had the following description:
“Mathematics curriculum of the secondary schools,
with emphasis on current issues and trends.” It was a
required course for graduate students in the master’s
program. In addition, several doctoral students chose to
take the course as preparation for doctoral-level
advanced curriculum studies if they had not taken a
similar course in their master’s program.
One female and two male professors participated
in this study. One of the participants, Martin, had
taught this course more than 20 times. The last time he
taught the course, Martin used videotapes from the
Third International Mathematics and Science Study
(TIMSS) to discuss curriculum and analyze current
curriculum issues. He also used recent publications
from the National Council of Teachers of Mathematics
(NCTM) and the state’s K–12 mathematics standards.
In addition, Martin drew upon his own articles and
experiences related to curriculum development.
Rafaela had taught this course three times at this
institution. Every semester, she began this course with
a book called The Saber-Tooth Curriculum by
Peddiwell (1939) to create a discussion about the
nature and purpose of curriculum. She also valued
NCTM’s (2000) Principals and Standards for School
Mathematics and thought of NCTM’s standards for the
K–2 grade band as the basis for how she
conceptualized K–12 mathematics education. Rafaela
divided each course session into two parts. In the first
part, she conducted a more theoretical discussion of
selected articles regarding current issues in
mathematics education curriculum (e.g., “math wars”
or equity). In the second part, she incorporated
activities and investigations related to mathematics
education curriculum. For example, she recently used
innovative curriculum materials that were funded by
the National Science Foundation. She asked her
26
students to consider what their classrooms would look
like if they taught with these curricula.
The third participant, Adam, had taught this course
five times. He did not ask students to read articles or
have discussions about curriculum in his course; rather,
he preferred to work on their mathematical knowledge
by engaging them in mathematical investigations.
Adam’s aim was to make mathematics teachers
creators of curriculum by strengthening their
mathematical knowledge.
Data Analysis
After analyzing the cases individually, I searched
for themes that cut across all three cases. Three themes
emerged: professors’ views of curriculum, their views
of graduate students’ contributions to the course, and
their learning goals for the graduate students. In the
remainder of this article, I focus on data related to
these three themes.
During the analysis, I realized that the three
participants did not share similar conceptions of what I
considered to be basic terminology, such as
mathematics curriculum. For example, Martin’s focus
was on curriculum as instantiated by textbooks, state
standards and other documents. Rafaela and Adam, on
the other hand, deeply questioned what curriculum is.
However, Rafaela and Adam’s ways of questioning
curriculum also differed due to their backgrounds and
roots in mathematics education.
Views of Curriculum
What is curriculum? Clements (2002) summarized
a few classic definitions of curriculum in the United
States as follows: the ideal curriculum is what experts
propound; the available curriculum is the textbooks
and teaching materials; the adopted curriculum is the
one that is adopted by authorities; the implemented
curriculum is what teachers teach in the class; the
achieved curriculum is what students have learned; and
“the tested curriculum is determined by the spectrum
of credibility tests” (p.601). The three different
perceptions of curriculum that emerged in this study
are related to different aspects of curriculum as defined
by Clements.
Martin’s view of curriculum was closer to the idea
of the available curriculum. For him, mathematics
curriculum is represented in textbooks; curriculum is
how teaching materials are organized. Because of the
importance he gave to this view of curriculum, Martin
used many different curriculum materials in his course.
He discussed how these materials are organized, and
why certain things are added to or omitted from school
Graduate Level Mathematics Curriculum Courses
mathematics curriculum. In doing so, he followed the
history of the available mathematics curriculum.
Interviewer: What are your goals for your students
in this course?
Martin: Well, I want to get them [the grad students]
to analyze, first of all the structure of the
curriculum. How the curriculum is [organized],
that is really, where we start usually. The structure
of the curriculum, how the U.S. curriculum is
organized, and why it is organized the way it is,
and various attempts to change the organization.
How new topics have come in, how other topics
have gone away, how certain things have been
emphasized in different times. So when we look at
old textbooks, we look at the kinds of problems
that are posed, we look at the organization of
topics, we look at how, and what definitions are
offered for certain things. To compare them with
each other and to see, for example, how certain
definitions have changed, or how the kinds of
activities in the books have been changed.
Curriculum for Rafaela included a wide spectrum
of issues and, when talking about curriculum, she
utilized many of the aspects mentioned by Clements
(2002). To her, curriculum is not just textbooks or that
which is taught by mathematics teachers: curriculum is
a complex political and theoretical concept. She
encouraged her graduate students to discuss what
curriculum is and how it affects students and teachers
in K-12 schools. Even though Rafaela incorporated
some discussion on curriculum theory into her class,
she believed there were underlying expectations at her
university about what to teach in this curriculum
course, e.g., history of curriculum and NCTM
Standards. When asked why she did not include more
theoretical discussions of curriculum in her course, her
answer provided insight into her ideas about
curriculum.
Adam included both teachers’ and students’
actions in his definition. This definition combines
Clements’ notions of implemented and achieved
curriculum. Adam’s view of curriculum was not
related to what is in textbooks or how they are
organized. Thus, he created curricula based upon the
actions of his students as he taught these courses. He
modeled his view of curriculum for his students by
using mathematical investigations. Based on his
knowledge about what is essential middle- and high
school-aged children’s mathematics, Adam formed a
possible curriculum, what it should include implicitly,
and then further developed the investigations for his
students as he interacted with them throughout the
course.
Interviewer: How do you perceive curriculum?
Adam: … You can view curriculum like books on
the shelf: it is already in place. It serves me to
teach and it is objective. That is one view to
curriculum that is a normal view people in
mathematics education take. … It is already there
in place and already there before the teacher. And
the teacher just implements. My view of
curriculum is quite different: my view of
curriculum is, it is done by the teacher and by the
student, it is a dynamic growing, evolving thing,
defined by the participants in the classroom.
Views of Graduate Students
When talking about the graduate students in his
course, Martin mainly focused on their teaching
experiences. He saw that the classroom discussions
changed immensely depending on whether there were
many graduate students with teaching experience.
Hence the teaching experiences of the students enabled
him to conduct the undergraduate- and graduate-level
curriculum courses differently.
Interviewer: What do you think why you don’t talk
about it [the theory]?
Interviewer: So, was summertime different than
how you taught it [the graduate course] in spring or
fall semester?
Rafaela: The way I read what it says in the
description of the course, this is not really a theory
of curriculum course. It is math ed. curriculum in
schools … so I understand it more as a discussion
of curricula that is out there. Because I have taught
curriculum theory classes, I try to include some of
the theory in my class. So, in my class we talk
about things like different types of curriculum. I try
to get students to think about what is curriculum: is
it your textbook, is it the politicians? … We talk
about hidden curriculum: the things you teach but
you don’t even know you are teaching, like values
… so I bring all that in …
Martin: Well, summertime is, of course, shorter. It
makes the course a little bit different, but the
course is basically the same. What changed the
course the most is whether most of the students
have done teaching. I had classes where almost
everybody had teaching experience, if not
everybody, so these were experienced teachers. On
the other hand, I’ve taught [when] almost nobody
had done any teaching except possibly, student
teaching, and that makes the course very different.
That makes more of a difference, I think, than
when the course is given, whether it is given in the
summer …
Zelha Tunç-Pekkan
27
Interviewer: So you think that there are differences
between those graduate students because …
Martin: If they have never taught, then there are
some of these issues [that] don’t occur to them or
are not realistic for them. Or they have trouble
seeing some of the issues. Let me give one
example. I have in recent years—since the TIMSS
video studies came out—I have occasionally used
the TIMMS videos. I have shown it in the course to
discuss curriculum issues. I have shown it in
[another graduate course] also, but occasionally
shown it in [the curriculum graduate course] so we
can talk about some … curriculum questions …
We were concentrating in that course only [on]
American teachers [in the videos] … and I’ve
noticed that they [graduate students with teaching
experience] see different things; they notice
different things about the topics that are being
taught. … They have different reflections on the
video.
Martin could do more in the graduate-level course by
using different curriculum materials and readings about
the history of school mathematics because curriculum
is more real to graduate students who have taught and
experienced it in their professions.
Interviewer: Are there
undergraduate curriculum?
any
courses
for
Martin: Yes. Mostly in that course we remind them
[undergraduate students] what the curriculum looks
like and there is very little history. …There is not
very much analysis of new materials—some
innovative materials. There is a little bit of look at
new textbooks, … but we try to balance that
because we recognize that most of these people
would not be using these new materials right
away... so part of what we try to do is familiarize
them with the most common materials out there.
And again, since none of them have taught … in
that course, it is really a very different orientation
because usually in [the graduate curriculum course]
you can expect some people have done some
teaching, and so they can talk about some of these
curriculum ideas from their own perspective: this is
what we had in our school, these are the materials,
this is what we like, this is what we didn’t like, and
so forth. The undergraduate course doesn’t have
that kind of discussion.
Rafaela also indicated that as a professor she could
try different things with her graduate students
compared to her undergraduate students. On the other
hand, her vision of graduate students not only differed
in their teaching experiences but also in their
willingness to try NCTM materials. Rafaela believed
that most graduate students who live in the academic
28
environment are familiar with NCTM (1989, 2000)
standards and already believe they can teach with these
standards. On the other hand, she observed differences
among the graduate students who are currently
practicing teachers. These graduate students had varied
views of NCTM’s standards and other reform-oriented
curriculum materials. She thought that these practicing
teachers especially needed to be exposed to NCTM
standards in order to analyze their own teaching
practice and observe similarities and differences
between their practice and Standards-based teaching.
Therefore, to accommodate these practicing teachers,
Rafaela planned student demonstrations of teaching a
topic from non-traditional textbooks as an important
part of her curriculum class.
Rafaela: I don’t think it is my goal to convince
them [about NCTM standards]. I think my goal is
to help them analyze what they believe. They can
be critical and write a paper about why they don’t
agree with that [NCTM standards]. So, I think,
especially teachers, they finish the course thinking
that it is a good idea, but you can’t really
implement it. Some of it, it is hard to convince
them that they can do it. And in the methods
course, I am more interested in convincing them
what they can do. I don’t do as much of that in the
curriculum course. But I try to give them a vision
of what it would look like if they were to try it.
And I have changed the materials used in the
course over the years. Last year we had Connected
Mathematics Project materials. … Two or three of
the students would be teachers from that class and
we were the students. Because I started noticing
that some of the students didn’t have a vision:
“What would it look like if I were to do what the
standards say? If I wanted to do that in the
classroom, what does it look like?” … So I started
giving them more of an idea, well, this is a
different thing. Some of them liked it … I am just
talking about the classroom teachers who come
back, not the regular students who are in this
environment that talk about NCTM and change. …
Two years ago, I had one classroom teacher who
came to me and said, “I am very lost. You really
took the carpet from my feet” … for him things
could be different … and I have had other students
saying that “I have been doing that but I did not
know how to call this.” But I also have had
teachers, come and leave thinking that I am a
dreamer. You know, anything I said is not possible.
Similarly to Martin, Adam mainly talked about
how he viewed the graduate students and their
contributions to his teaching by comparing them to his
undergraduate students. Graduate students’ teaching
experience was an important component of how he
Graduate Level Mathematics Curriculum Courses
viewed graduate students and their contributions.
Teaching experience provided the possibility for
graduate students to be involved in secondary school
students’ mathematical thinking. For Adam, having
previously engaged in students’ mathematical thinking
made graduate students more able to appreciate the
importance of the basic mathematical concepts and
operations they investigated in his course. These
graduate students were able to establish meanings of
basic ideas in mathematics from the point of view of a
teacher, not just a student.
Adam: It is very difficult for them [undergraduate
students] because they are struggling. They
struggle for the actual thinking that is involved. …
There is a qualitative distinction between the
natures of the students in the two courses.
Interviewer: Nature of the students?
Adam: The way students view themselves, the way
they view what those courses should be about. …
[Undergraduates] are not as mature as graduate
students in actually working with students. They
just did not have a chance to become involved with
others people’s thinking. So, they don’t appreciate
how important their thinking is in trying to
understand the thinking of other students. So, their
basic orientation in [the undergraduate course] is
not to understand the thinking of the students. It is
more, “what do I have to do when I go teach the
topic that is already given?” That is their
orientation.
Interviewer: But don’t some students have that
kind of orientation in [the graduate] class?
Adam: Oh yes. By all means, they had that
orientation. But I think they are more mature and
probably little bit willing to consider the
possibilities. OK. But [for the] most part few
students that went through the course always knew
what we were doing and quite appreciated what we
are doing. The distinction between [the] two
classes is quite profound in the maturity of the
students and appreciation for investigating basic
mathematical concepts and operations [and the]
meaning of basic ideas in mathematics from the
point of view: How do I make these things? How
do I make meaning for them? How do I formulate a
constructive itinerary of mathematics and the
relationships and the connections to mathematics? I
think [the graduate course] students are much more
able to deal with that than the [undergraduate
course] students.
Goals for Graduate Students
Martin’s overall goal was to make students aware
of current curriculum issues. For this goal, discussing
NCTM’s (2000) new standards was important for him.
Interviewer: What is the purpose when you are
using NCTM standards and why do you want to
use those?
Martin: Well, to acquaint them [the students] with
some of the issues in the field. These current
publications reflect efforts in the profession to
change, in the case of the curriculum standards, to
change the curriculum. So, I think it is important
for them to know what people are advocating. … I
usually add in some critiques of this, or if we don’t
read a critique we actually make a critique
ourselves, … especially if they are experienced
teachers, they don’t necessarily agree with all of
the things that are in these documents so we
discuss them. … So, my purpose is to get them
thinking about current issues… As it says here
[pointing to his course syllabus], I wanted them to
… “gain some skill in analyzing issues and trends.”
Because these people, whatever they end up doing
in [their] profession, they are going to be using, or
at least knowing about, curricula and they are
going to know, I hope, that [there] will be issues
out there.
Martin felt that graduate students needed to look at
mathematical topics locally (for a grade) and globally
(across grades) when discussing curriculum issues. For
example, he discussed the emphasis on proof in
NCTM’s 2000 standards as opposed to the earlier
Curriculum and Evaluation Standards for School
Mathematics (NCTM, 1989) and added that there
would always be debate on certain curriculum issues,
such as the inclusion of real life applications and
technology. Hence, his goal was to make graduate
students aware of those issues and enhance their skills
in critiquing those issues.
Martin: For example, how much emphasis should
be put on proof? … now we have PSSM and a
stronger emphasis on proof, … but … the 89
Standards didn’t emphasize it.…the way that 2000
Principles and Standards is structured, it raises the
question of what is to be done about proof in early
grades and what is to be done about proof at the
later grades. And this raises questions of how the
curriculum is organized across the grades. Even
though the focus is on the secondary curriculum,
there is always a question, “How does it build on
the elementary curriculum?”
Rafaela’s overall goal was related to her
conceptualization of the curriculum. Similarly to
Zelha Tunç-Pekkan
29
Martin, she wanted graduate students to think about
and reflect on curriculum. However, her goal was to
make an implicit change in graduate students’ teaching
practice. She wanted them to think about how
curriculum played a role in their own teaching and the
effects of their use of curricula on their teaching.
Rafaela believed in the existence of a hidden
curriculum that teachers implemented but were not
aware of. Therefore, her goal was to help graduate
students clarify their own teaching goals.
Interviewer: What are your goals? Is it the little
course description?
Rafaela: To think about, “What is curriculum?” is
my goal, probably because…I come from this
curriculum studies perspective. … You have to
decide, what do you want to teach? As a teacher,
what are your goals? And those are things, I can’t
help anyone to decide but I can help them to think
about it. So, my overall goals are to bring the class
… to think … “Yes, there is a hidden curricula that
I teach and never thought about …Why am I
teaching this? What kind of people am I trying to
educate? What [are] my goals as a teacher for my
students?” … That is what I want them to reflect
on. Inside that there is my view that … we want to
create thinkers. … I think the NCTM standards …
are a good venue for helping create thinkers who
reflect mathematically … so I do present it from
that perspective. …Who decides all those things?
Who decides the curriculum? Who decides [the
state standards]? Do we have to follow? What kind
of people are we going to create by following that?
With this course, Adam also wanted to make a
change in his graduate students’ educational
experiences. His main goal was to reorient graduate
students to think about the basics of school
mathematics. In order to understand and value K–12
students’ mathematical activities, he believed that
teachers need to have mathematical experiences such
as understanding and formulating mathematical rules
they use everyday in their teaching. Therefore, he
provided learning opportunities in mathematical
investigations and hoped graduate students would
develop meaningful itineraries for some mathematical
topics.
Adam: How the teacher thinks is totally critical. …
How students think is totally critical. So, my view
of curriculum is manifested in how I acted in the
[graduate] course. I involved … the participants
deeply in doing basic mathematical activities in a
way that they probably haven’t thought about
before. … Investigate the basic ways of reasoning
in mathematics, the basic meaning of … linear
30
functions. … Where they come from, what is the
constructive itinerary for that? So, I want the
participants to become aware how they think
mathematically. OK. I want them to be aware of
what they are doing mathematically … For
example, addition of fractions: half plus a third is
viewed as a procedure, as an algorithm. … I want
them to go back to very basic ways of reasoning …
How would I formulate that for the sum, if I don’t
know already those rules? What do those rules
mean? … I think that attitude is very essential for
teachers because they have to respect … productive
thought and creativity, and potential creativity, of
the students. So, they are not just giving the
mathematics procedurally to students, but the
students are constructing it meaningfully.
Final Comments
Depending on the professor, the learning
experiences graduate students have in this curriculum
course may differ immensely. The professors’ views of
curriculum (e.g., static as in textbooks or already given
as in the school standards versus dynamic views),
views of graduate students, and their goals for the
course influenced what kinds of materials they chose
and how they used these materials.
Martin, Rafaela, and Adam all believed this
curriculum course should make graduate students
better thinkers and better analyzers. Whereas Martin
and Rafaela focused on discussing existing curriculum
materials when talking about their learning goals,
Rafaela was also concerned about changes in her
students’ teaching practice. Adam, on the other hand,
focused on helping graduate students become better at
analyzing their own and their K-12 students’
mathematical activities.
The professors’ learning goals were closely
connected to their views of curriculum and their views
of graduate students. For example, because Martin
regarded school mathematics curriculum as textbooks,
written documents, and the evolution of mathematical
topics in those documents over time, he took these
components into consideration in his planning. Martin
focused on the organization of the materials with his
graduate students and used a variety of current and
historical curriculum materials for that purpose. He
aimed to help his students better analyze current issues.
In addition, he viewed graduate students’ teaching
experiences as the factor that most affected the quality
of discussions.
Rafaela also used reading materials, but she
concentrated on the discussions of how graduate
students conceptualize curriculum, what NCTM
standards mean in terms of teaching and learning, and
Graduate Level Mathematics Curriculum Courses
who is making curriculum. For Rafaela, curriculum
meant a theoretical discussion of teaching practice, so
using NCTM materials as an orientation was a good
venue for that purpose. She believed that some of the
graduate students, mostly the currently practicing
teachers, were hesitant to think about curriculum
differently. Therefore, NCTM and other reform
materials provided a context for this discussion. Using
this context, she could expose teachers to new ideas
that they could try in their practice.
For Adam, curriculum was a dynamic phenomenon
that is formed by teachers and students in the
classroom. He thought graduate students should be
creators of curriculum, like him, with their own
students inside the classroom. In his classes, he tried to
provide a model of this view by dynamically creating a
curriculum with his graduate students. Teachers’
mathematical knowledge, as well as their teaching
experiences, played an important role in that creation.
He interacted with graduate students using a
mathematical domain as the medium. His aim was to
provide opportunities to graduate students to rethink
mathematics curriculum in schools by engaging them
with the basics of mathematics.
This investigation of a graduate-level curriculum
course reveals that various factors affect the ways in
which professors design graduate-level courses.
However, further research is needed to investigate the
learning experiences of graduate students and how
professors’ ideas about teaching curriculum are
compatible with their practices in the classroom.
References
Clements, D. H. (2002). Linking research and curriculum
development. In L. D. English (Ed.), Handbook of
international research in mathematics education (pp. 599–
630). Mahwah, New Jersey: Lawrence Erlbaum.
Jackson, S. (1994, April). Deliberation on teaching and curriculum
in higher education. Paper presented at the Annual Meeting of
American Educational Research Association, New Orleans.
Patton, M. Q. (2002). Qualitative research and evaluation methods
(3rd edition ed.). Thousand Oaks, CA: Sage.
Shulman, L. S. (1998, February). Teaching and teacher education
among the professions. Paper presented at the American
Association of Colleges for Teachers Education 50th Annual
Meeting, New Orleans, Louisiana.
Stake, R. (2000). The case study method in social inquiry. In R.
Gomm, M. Hammersley, & P. Foster (Eds.), Case study
method. London: Sage.
Stark, J. S., & Lattuca, L. R. (1997). Shaping the college
curriculum: Academic plans in action. Boston: Allyn and
Bacon.
1
Appendix: Interview Protocol
1.
How many times have you taught this course? Have
you taught similar courses in different institutions?
2.
How is this course different from any mathematics
education content courses or method courses you taught
before? How does curriculum have a special or different
emphasis in your design of the course?
3.
What are your goals for the course? How do these
goals affect your decisions when you are designing the
course?
4.
Since this course is for graduate level students, how
do you take this audience into consideration (graduate
students might be in-service teachers) when you design
the course?
5.
How do you know your graduate students
understood the curriculum ideas emphasized in the
course? How do you check it?
6.
What components of the K-12 mathematics
curriculum are important in your design of this
curriculum course? How do you know you have
emphasized them enough when teaching this course?
7.
How do you revise the content of the course or the
way you teach the course each time? What factors do
you take into account? (colleagues, recent related
research, students’ success or responses, the
departmental needs, etc.) and How?
8.
What would you like to gain as a teacher when
teaching this course and how does this affect your
design of the course?
9.
How does your research affect your teaching of
graduate level mathematics curriculum courses? Or vice
versa?
10.
In which ways do you think your [graduate
curriculum] class is similar/different from the [graduate
curriculum class] taught by other instructors?
Kugel, P. (1993). How do professors develop as teachers? Studies
in Higher Education, 18, 315–328.
National Council of Teachers of Mathematics. (1989). Curriculum
and Evaluation Standards for School Mathematics. Reston,
VA: Author.
National Council of Teachers of Mathematics. (2000). Principals
and Standards of School Mathematics. Reston, VA: Author.
Peddiwell, J. A. (1939). The saber-tooth curriculum. New York:
McGraw-Hill.
Shuell, T. J. (1993). Toward an integrated theory of teaching and
learning. Educational Psychologist, 28, 291–311.
Zelha Tunç-Pekkan
All names used in this article are pseudonyms.
31
The Mathematics Educator
2007, Vol. 17, No. 1, 32–41
Some Reflections on the Teaching of Mathematical Modeling
Jon Warwick
This paper offers some reflections on the difficulties of teaching mathematical modeling to students taking
higher education courses in which modeling plays a significant role. In the author’s experience, other aspects of
the model development process often cause problems rather than the use of mathematics. Since these other
aspects involve students in learning about and understanding complex problem situations the author conjectures
that problems arise because insufficient time within mathematical modeling modules is spent reflecting on
student work and enabling “learning to learn” about problem situations. Some suggestions for the content and
delivery of mathematical modeling modules are given.
Over the last 20 or so years of teaching in higher
education, I have had the pleasure of teaching
various aspects of the mathematical sciences to
students at levels ranging from pre-degree to master
levels. Although each module1 that one teaches
presents challenges, the one subject that has been the
most challenging to my students and myself has been
that of mathematical modeling.
In this article, I reflect on the mathematical
modeling process and how it has influenced the way
I teach modeling. My own experience of modeling
has been acquired within the management science
domain. This domain is concerned not only with
modeling physical processes but can also include
considerations of systems and organizational culture.
Although this may give my views a different slant
than those of someone working as a modeler in the
pure sciences, the issues discussed apply across many
modeling domains.
By mathematical modeling I mean the “pencil
and paper” type of modeling characterized by written
assumptions, equations, and so on, as opposed to
computer-based simulation models that can be built
using graphical interfaces. Students usually enjoy the
latter since the medium is interesting. These
situations often divert attention from tough modeling
considerations and the need to see the dynamic
equations! This, however, is another story and I wish
Jon Warwick completed his first degree in Mathematics and
Computing at South Bank Polytechnic in 1979 and was awarded
a PhD in Operations Research in 1984. He has many years of
experience in teaching mathematics, mathematical modeling, and
operations research in the Higher Education sector and is
currently Professor of Educational Development in
Mathematical Sciences at London South Bank University. He is
also the Faculty Director of Learning and Teaching.
32
to restrict my discussion to mathematical models
derived without the use of software.
Examples of these pencil-and-paper models are
often presented in management science or
operational research texts and would include some
standard models relating to inventory control,
waiting line models, and mathematical programs.
These models can be written in terms of equations
that give optimal order quantities, average waiting
times, and so on for differing sets of conditions. For
these standard models the underlying assumptions
are well known. Students taking my modules at the
undergraduate level are encouraged to develop their
own models which may be based on a standard form
but must be described using mathematical notation
and with pencil and paper.
In practice I have often used academic library
management as a contextual area where, over the last
forty years, mathematical modeling has been applied
to good effect, producing a wealth of accessible
literature and different types of models (Kraft and
Boyce, 1991). By way of example, I shall describe
some experiences from an introductory modeling
module given to undergraduate students studying
mathematics related to management. Having first
spent some time with the students studying examples
of a number of the standard model forms found in
management science (stochastic, deterministic,
simulation, etc.), the students are given a simple
situation to start the modeling process. Briefly, this
involves the students working in groups to develop a
model that can be used for determining the effect of
changing the loan period of a single title (multiple
copy) text appearing on a class reading list. My
students must develop a model (and if possible solve
it) using pencil and paper only. A crude measure of
The Teaching of Mathematical Modeling
Investigation and Problem
Identification
Mathematical Formulation
of the Model
Collect Data and Obtain a
Mathematical Solution to
the Model
Interpret the Solution
Compare with Reality
Implement the Solution and
Report Writing
Figure 1. Stages in the modeling process.
library user satisfaction is the likelihood of finding a
book on the shelf when desired. Students are asked to
find loan periods that provide certain satisfaction
levels for differing numbers of copies and class size.
The idea is, at this stage, to encourage simplicity in
modeling and highlight the importance of
assumptions. Working in groups is also important as
group discussion facilitates the model development
process.
The Modeling Process
Examination of textbooks dealing with
undergraduate mathematical modeling (or any of the
related fields, such as management science) will
normally yield a description of the modeling process
in general terms incorporating the stages as outlined
in Figure 1. There are many variations on this theme
from both specialist texts on mathematical modeling
(see Edwards & Hamson, 2001) or texts on more
general quantitative analysis (see Lawrence &
Pasternack, 2002), but the basic structure of the
process is usually similar to that shown. There are
two things to notice about the process. First, as
described in Figure 1, it is essentially a looping
process. Second, it is a process that students
Jon Warwick
generally find difficult to undertake, despite the fact
that the process is fairly simple to state, the steps are
logical, and the language fairly non-technical.
The Art of Modeling
As a student of mathematical modeling, I was
introduced many years ago to an article that dealt
with the process of mathematical modeling and
attempted to give some hints and tips as to how the
novice modeler might proceed (Morris, 1967). It is a
paper I often recommend to my students as it
recognizes the difficulties that many of them are
facing. Morris makes the valid point that, when
students read about the development of mathematical
models and look at examples of models that have
been developed by others, the writing is nearly
always in the spirit of justification rather than the
spirit of inquiry. By this we mean the writing
justifies the final product and comments on the
results obtained, the validity of the model, etc.
However, it does not dwell on the frustrations and
problems that may have been encountered on the way
to the final model, the models that were discarded, or
false trails that were followed. Adopting the latter
style of writing, describing the ups and downs of the
33
inquiry process that eventually produced the final
model, would be far more illuminating to students
than just a description of the final model.
In addition, Morris (1967) gives a nice
description of the art of modeling and notes that the
model development process has a looping structure
with two major loops. The first looping process is
developing a working model from a set of
assumptions and continually testing the model
against real data until it may be regarded as
acceptable within the limits set by the realism of its
assumptions. The second involves changing the
assumptions either by relaxing those that seem
unrealistic or by imposing new assumptions if the
model is becoming too complex. These two looping
processes are often in operation at the same time as
the modeler strives to balance model tractability with
performance. Model tractability here means the ease
manipulating and solving the model. Morris refers to
the looping process through which model
assumptions are relaxed and the model enhanced as
enrichment and elaboration.
In addition to these two primary looping
processes, Morris (1967) gives a checklist of hints
and tips that he suggests will help the novice
modeler. These may be summarized as:
34
•
Try to establish the purpose of the model to
give clues to model form and perhaps the
level of detail necessary.
•
Break the problem down into manageable
parts so these smaller pieces can be solved
before being reincorporated into the larger
whole.
•
If possible, use past experiences or other
similar problems already solved to give clues
as to the solution required by the current
model. This is the process of seeking
analogies and is a powerful weapon in the
modeler’s armory (see for example Warwick,
1992).
•
Consider specific numerical examples. This
may give clues as to where assumptions
might be needed or how the problem
situation is structured.
•
Establish some notation as soon as possible
and begin building relationships in the form
of mathematical equations.
•
Write down the obvious!
These hints and tips together with an
appreciation of the general looping processes
involved in model development are the core activities
that students need to master in order to build models.
They are easily learned, or perhaps memorized, and
yet still students find model building difficult. There
are at least two learning processes with which
students are required to engage in order to become
proficient in modeling. Each makes quite distinct
demands of the student.
The first learning process requires the student to
become conversant with the tools of the trade such as
mathematical symbolism, algebraic manipulation, the
stages involved in model building, the looping
processes, and archetypical model forms. These
elements, often as not, form the core content of
mathematical modeling modules. In terms of the type
of learning that is being undertaken, we can refer to
Bloom’s (1956) taxonomy of learning in the
cognitive domain that describes different categories
of learning arranged sequentially. The learning
required to become proficient in the mechanics of
model building is primarily within the three lowest
categories–knowledge,
comprehension,
and
application–and my students seem to have few
problems here. Problems begin to surface when we
consider the second learning process, which is not
explicit in Figure 1.
In this second process the modeler is coming to
terms with the intricacies of the problem being
modeled, the subtleties of the situation being studied,
and the implications these will have for the model
being developed. No model can be developed
successfully unless the modeler has a clear
understanding of what is to be modeled and this
learning will need to take place as the modeling
proceeds. Yet there is nothing in the modeling
process model that helps the modeler with this. In
other words, there is a requirement for the skills of
learning to learn to be appreciated by students as
every modeling situation they meet will be different
and often complex.
Learning to Learn
What do we mean by learning to learn?
Reference to the literature allows three general
observations. First, this idea has been the subject of
research for more than 30 years with researchers
considering learning-to-learn issues at the K-12
(Greany and Rodd, 2003) and university levels
(Wright, 1982), as well as within the work
environment (Ortenblad 2004). Learning-to-learn
The Teaching of Mathematical Modeling
Table 1
Considerations for learning to learn and comparison with Morris (1967).
Achieving Learning to
Learn
Begin with the past;
Some Key Considerations
for the Individual
It is important to look back and
consider what was your previous
experience about how you learn, how
was learning structured before and
what worked well in similar
circumstances.
Considerations from Morris (1967)
Proceed to the present;
There needs to be a clear reason for
doing what you are doing! Which parts
are important? Which should be
tackled first? What is controllable and
what is not and which bits are already
learned to form a basis for further
learning?
Try and establish early on the purpose of
the model so that this will give clues to
model form and perhaps the level of
detail necessary for the model.
Consider the process …
What is the structure of the work to be
learned? Get a feel for the general
theme, the main points, key words. Are
they understood?
Write down the obvious! Establish some
notation as soon as possible and begin
building relationships i.e. writing
equations. Break the problem down into
manageable parts so that these smaller
pieces can be solved before combination
back into the larger whole.
… and the subject matter;
How much of this subject is known
about already? How much is known
about related subjects and what is the
link? What resources are available and
are they accessible now? Decisions
need to be made about how quickly to
proceed through material, when to
attempt questions, when to seek
guidance etc.
Seek analogies and associations with
other, related, modeling problems.
Consider specific numerical examples—
this may give clues as to where
assumptions might be needed or how the
problem situation is structured.
Build in review;
Decide here what went well and what
did not and how this might affect
further learning attempted.
This is a key area that Morris describes
as lacking in modeling articles and
reports. In practice, I have found it
useful for students to keep a log or
workbook that includes reflections on
the various models built during the
course of a taught module.
research also spans academic disciplines with
examples from such diverse subject areas as history
(Knight, 1997), physical education (Howarth, 1997),
and science (Hamming, 1997; Elby, 2001). Little has
been written in the context of mathematical
modeling. Second, the recent interest in learning to
learn has coincided with the development of research
in cognitive and metacognitive strategies (Waeytens,
Lens, and Vandenberghe, 2002) and the expansion of
higher education. As a result, many universities now
recruit students from a variety of backgrounds and
consequently with a range of abilities and previous
Jon Warwick
If possible, try and use past experiences
or other similar problems already solved
to give clues as to the solution required.
educational experiences. Third, there is little
agreement about the definition of learning to learn or
how it should be taught. Some researchers have a
narrow view in which learning to learn involves
essentially study skills, hints, and tips, whereas
others take the broader view that students should be
able to apply skills in critical analysis, goal setting,
personal planning, and so on (Rawson, 2000).
Regarding how learning to learn should be
taught, there has been debate as to whether it is
appropriate to approach it as a separate, isolated
module or whether it should be embedded into other
35
regular study modules. These days, conventional
wisdom suggests it must be taught within regular
modules and not as an isolated subject (Waeytens et
al., 2002). In my view learning to learn incorporates
a broad set of skills including reflective and critical
thinking and it should be approached within the
context of a module. In fact, it is crucial in
developing effective modeling skills.
Because learning to learn is now becoming a key
part of many university learning and teaching
strategies, one way of approaching it is to consider
the key elements as shown in Table 1 (amended from
Landsberger, 2005). These considerations apply as
much to the learning of mathematical modeling as
they do to any other subject. The hints and tips given
by Morris (1967) do, in fact, sit quite well within this
framework, as seen in Table 1. In other words,
Morris seems to be tacitly addressing the learning-tolearn difficulties associated with modeling through
his practical advice.
We can further strengthen this idea that modeling
is as much about learning as it is about applying
mathematics. To accomplish this, we must reconsider
the classic process model of mathematical modeling
(see Figure 1) and re-formulate it to emphasize the
learning processes that are truly going on. True to the
spirit of Morris (1967), we can do this by seeking
analogies with other models of the learning process.
A particularly useful representation has been
developed within the field of organizational learning.
Organizational Learning: Single and Double Loop
Learning
According to Senge (1990), learning enables us
to do things we were never able to do, change our
perception of the world and our relationship to it, and
extend our capacity to create. In this context, learning
organization is an “organization that is continually
expanding its capacity to create its future” (Senge,
1990, p.14). In their classic work on organizational
learning, Argyris and Schon (1978) define learning
as occurring under two conditions: (a) when there is
a match between an expected or desired outcome and
the actuality or real outcome, and (b) when there is a
mismatch between expected or desired outcomes and
reality that is identified and corrected so that the
mismatch becomes a match.
Argyris and Schon (1978) describe two types of
learning response that can occur when a mismatch is
detected, single loop learning and double loop
learning. Single loop learning is described as
focusing on the status quo by narrowing the gap
36
between desired and actual conditions (University of
Luton, 2006). It is a simple feedback loop where the
learner’s actions are changed to accommodate
mismatches between expected or desired and
observed results in the perceived real world. Single
loop learning has also been described as an errorcorrecting or fine-tuning process. There are,
however, a number of limitations to single loop
learning (Peschl, 2005):
•
It is an essentially conservative process that
seeks to retain the existing knowledge
structures rather than exploring new
alternatives.
•
There is very little chance that new insights
will be gained or that anything new or
innovative will be learned.
It is a process that lacks any form of
reflection.
Double loop learning (or reflective learning), on
the other hand, tries to overcome these limitations by
first examining and altering the current mental model
and then the actions. It is single loop learning with an
extension, or second feedback loop, that allows for
the possibility of change in assumptions, premises,
mental models, etc. As Peschi (2005) states:
•
In double loop learning a second feedback loop
introduces a completely new dynamic in the
whole process of learning: each modification in
the set of premises or in the framework of
reference causes a radical change in the structure,
dimensions, dynamics, etc. of the space of
knowledge. By that process, entirely new and
different knowledge, theories, interpretation
patterns, etc. about reality become possible. (p.
92)
This allows us to adapt our mental models in the
light of experience and information. An example of
the structure of single and double loop learning is
shown in Figure 2.
To illustrate the difference between these
models, consider a fall in enrollment numbers on a
previously popular course. In single loop learning
(i.e. identifying a mismatch between desired and
actual outcome), faculty members may respond by
increasing efforts to publicize the course in the
media, with feeder schools, and with colleges, as
well as working more closely with the local
community. Fundamental beliefs are unchallenged
but actions are amended to address the mismatch. An
alternative response characterized by double loop
learning would be to re-examine beliefs about the
The Teaching of Mathematical Modeling
course, such as the suitability of its curriculum, the
attractiveness of the subject area to potential
students, and whether its current state is “fit for
purpose”. This double-loop-learning response may
result in radical change to the course offering.
Organizations as well as individuals derive and
amend their mental models through experience,
observing, and interpreting the outcomes of their
actions and decisions (Argyris and Schon, 1978;
Bartunek, 1984; Levitt and March, 1988). In this
sense, double loop learning requires the generation of
new knowledge, insights, and intuitions by
modifying existing models.
Double Loop Learning and the Modeling Process
We now can see how the mathematical modeling
process can be placed within the framework of
double loop learning. When we develop a
mathematical model, there are two aspects to be
considered. First, when we develop a model based on
a set of assumptions derived from our current
understanding of the problem situation, we
effectively engage in single loop learning. The
assumptions we have made determine the
formulation of the model, the data requirements, and
so on. Once the data has been collected, we solve the
model and interpret this solution within the context
of the problem situation. This leads to model
validation and verification considerations. The
validation and verification process may indicate
problems with the model, a mismatch between our
expectations and real situation dynamics. In this case,
it may be that the model has not been formulated
correctly in terms of the assumptions, that the model
contains errors in its formulation, or that the data
used is unreliable or inappropriate. In any event, the
model needs to be amended. Within the limitations of
our current set of active assumptions about the
problem situation, we seek to find a model that does
not deviate from our expectations. This is a single
loop learning process.
When the model is decided to be valid, then we
can begin the process of enrichment and elaboration,
extending and developing the model by broadening
our understanding of the problem situation, in terms
of both the breadth and sophistication. This produces
an amended set of working assumptions for the
model requiring further development. This second
looping process is double loop learning. It requires
from the student not just the technical mathematical
and statistical skills, but also the learning to learn
skills that were described above. The mathematical
modeling process is outlined in Figure 3.
Information
Feedback
Comparison with
the Real World
Single Loop
Learning
Expectations,
Desires,
Decisions
Double Loop
Learning
Personal
Mental
Models
Worldview,
Strategy
Figure 2. Single and double loop learning – adapted from Sterman (2000).
Jon Warwick
37
Assessment of
verification and
validation results.
Find and interpret
the solution to the
model.
Collect the appropriate
data
Enrichment and Elaboration
By Learning to Learn
Single Loop
Learning
Appreciation
of the problem
situation,
structure and
complexity
Double
Loop
Learning
Derive/amend or
correct the
mathematical
formulation of
the model
Assumptions made
to determine the
model structure,
boundary etc.
Figure 3. The amended modeling process.
To paraphrase Dooley (1999), the single loop
learning phase can be described as “building the
model right” whilst the double loop learning phase
relates to “building the right model” (p. 13). This
mathematical modeling process model (Figure 3) is
richer than the conventional process model used with
students. It allows the discussion of mathematical
modeling to be extended to include elements related
to the double loop learning aspect of the process.
These are the difficult elements of the modeling
process for both teachers and students. Yet these are
just the skills that enable effective modeling and
engage the students in the higher levels of learning as
described by Bloom’s taxonomy (i.e. synthesis and
evaluation).
Now, returning to our example drawn from
library management, my teaching experience
suggests that students will initially adopt a variety of
model forms often using analogy as recommended by
Morris (1967). Common themes here are the
conceptualization as either one of an inventory
control problem or as a waiting line (queuing)
problem. In the case of the inventory model, the
copies of the title on the shelf are the stock being
demanded (borrowed) by students and then
immediately re-ordered. The average inventory level
is a measure of the satisfaction level and lead times
are assumed fixed initially, corresponding to an
assumption that all books are kept for the full loan
period and then returned promptly.
38
For the waiting line model, the service
mechanism represents the copies of the title (one
server for each copy) and average service time
equates to the loan period. Actual borrowing times
are assumed to be random in the basic waiting line
model. The queue itself might represent reservations
having been made for the title if it is not immediately
available. In this case, the satisfaction level is related
to the probability of finding idle servers. Calculations
can also be made of average waiting times to get the
book depending on the number of copies, the loan
period, and the class size.
Students are provided with some sample data,
and then test their models. If necessary, they refine
and correct any faults until they are satisfied with the
results. This is iteration around the single-loop
learning phase. The complexity of the situation is
then increased gradually so that students will, at first,
try to adopt single loop learning in order to
accommodate any new information within their
existing models. Eventually, they must consider
broader and more complex issues that may require
radically changed assumptions, significant new
modeling and understanding, and, in the extreme,
adopting a completely new model formulation.
For example, I might begin by asking students to
relax their assumption about borrowing times by
allowing users to return their copy early or late
according to some probability distribution. This
modification can be built in to both models described
above relatively easily but requires the students to
The Teaching of Mathematical Modeling
research how to do this. As a result, their models
become more complex, moving away from standard
inventory or waiting line models into more
specialized versions.
A higher level of complexity is introduced by
allowing feedback into the system. Students are
asked to consider that demands for the title will not
be regular but depend on the perceived likelihood of
obtaining a copy in reasonable time. If many copies
are available in the library (high satisfaction levels),
then this encourages use of the library, increasing
demand and eventually reducing satisfaction levels.
Otherwise, if copies are never available, potential
borrowers might go elsewhere (or buy it for
themselves), lowering demand.
Dealing with these new complexities requires
students to engage with aspects of double loop
learning. For example, they need to explore their
existing model, ask further questions of the system,
and revisit their assumptions and their understanding
of the situation to incorporate these new factors. At
this stage, students often get stuck dealing with the
additional complexity and need help moving forward
with double loop learning. I have been able to help
students with this by using structured discussion.
The Importance of Advocacy, Inquiry, and
Reflection
In this paper, I have argued that the skills that are
most difficult for students to master are those related
to the double loop-learning cycle in the modeling
process. We have borrowed the notion of double loop
learning from the field of organizational learning
and, in completing this analogy, we can shed some
light on how this sort of learning can be fostered in
students. Senge (1990) argues that, in helping
organizations undertake double loop learning,
members of the organization should be able to
combine advocacy and inquiry. Advocacy refers to
the ability to solve problems by taking a particular
view, making the appropriate decisions, and then
gathering whatever support and resources are
necessary to make things happen. Inquiry, on the
other hand, is being open to questions, asking
questions of others, inquiring into the reasoning of
others, and expressing one’s own reasoning. Senge
states:
When both advocacy and inquiry are high, we
are open to disconfirming data as well as
confirming data–because we are genuinely
interested in finding flaws in our views.
Likewise, we expose our reasoning and look for
Jon Warwick
flaws in it, and we try to understand others
reasoning. (p. 200)
Thus, creative outcomes are far more likely as a
consequence of using advocacy and inquiry.
When working with a group of students
modeling a complex situation, they should be
encouraged to use advocacy and inquiry to challenge
and explore modeling ideas. There are a number of
guidelines proposed by Senge (1990) that, when used
as prompts, can encourage students to explore the
problem situation. For example, when advocating
personal views the guidelines may be summarized as:
•
Make your reasoning explicit.
•
Encourage others to explore your views.
•
Encourage others to provide different views.
Actively inquire into others’ views that differ
from yours.
Or, when inquiring into others’ views, try to:
•
•
State any assumptions you are making about
the views of others.
•
State the data on which your assumptions are
based.
•
Ask what data or logic might change their
view.
If there are disagreements, design an
experiment or collect data that might provide
new information.
Discussion among the groups of students can be
structured using these types of prompts, resulting in
creative thinking about the way modeling should
proceed (I have rarely seen aspects of creative
thinking mentioned as part of mathematical modeling
module descriptions!).
In practice, students find this sort of debate and
discussion difficult. They often need prompting from
the teacher when group discussion has reached a
dead end. Eventually, a greater understanding of the
new problem is achieved. This usually leads to the
amendment of the existing model, incorporating new
assumptions and factors. As a result, students
investigate stock control models with variable
demand patterns or waiting line models with nonindependent arrival patterns. In this way, students
develop further mathematical knowledge and
research skills as well as engage in a cycle of
learning.
In extreme cases of paradigm shift, students will
reject the existing model completely in favor of a
•
39
new formulation. This was the case with one group
who rejected a simple inventory control model as too
restrictive in favor of a model built using simple
differential equations linking the number of copies
available on the shelf with the number of potential
borrowers. If the number of copies available on the
shelf is low, then frustration will reduce the number
of potential borrowers. In time, this causes the
number of copies available to increase (reduced
demand), leading to an increase in potential
borrowers, and so on. These students found stable
solutions to their model and investigated its
sensitivity to changes in the loan period.
Finally, we turn to reflection. Having looked at
one way to encourage double loop learning, we need
to then give students the skills to reflect individually
on their performance, their learning, and how they
can further improve their modeling skills. Thus, it is
important to get students into the habit of reflecting
on their work and the work of others. As King (2002)
states, when undertaking reflection, “a variety of
outcomes can be expected, for example, development
of a theory, the formulation of a plan of action, or a
decision or resolution of some uncertainty” (p. 2).
Furthermore, “reflection might well provide material
for further reflection, and most importantly, lead to
learning and, perhaps, reflection on the process of
learning.” (King, 2002, p. 2)
Morris (1967) pointed out that reflective writing
is sadly lacking in the professional literature.
However, recent educational research has addressed
reflective writing (see for example Moon, 2000) and
how skills in reflection and reflective writing can be
developed. King (2002), for example, suggests a
model of the reflective process as having seven
stages: Purpose, Basic Observation, Additional
Information, Revisiting, Standing Back, Moving On,
and either Resolution or More Reflection. Although
UK Higher Education courses are expected to
promote reflective thinking in many aspects of
student’s work (Southern England Consortium for
Credit Accumulation and Transfer, 2003), I would
argue that it is a particularly crucial aspect of the
mathematical modeler’s toolkit.
Some General Conclusions
Reflecting on my own teaching of mathematical
modeling over the years has led to a number of
changes to the way modules are designed, delivered,
and assessed. When working with students (whether
undergraduate or graduate) the following has been
40
useful in meeting some of the issues referred to in
this article:
•
Ensure that the content of the module
includes some mathematical and statistical
theory (as required by the particular
program) but also sessions on creative
thinking, learning to learn, and reflective
writing.
•
Although some smaller models are used for
the purposes of example, students are
encouraged to work on a progressively more
complex problem during the course of the
module. This gives the opportunity for the
development of successive models through
enrichment and elaboration. Furthermore, it
is helpful if the problem at hand can be
modeled using a variety of approaches. This
enables students to identify alternatives and
to reflect upon the criteria for selection.
•
For longer projects, allow students to work in
groups. Group meetings are held during class
time so that the instructor can observe the
discussion and try to move the students
towards double loop learning as they seek to
enrich their models. Questioning each other
using the prompts discussed earlier can help
here. Students take minutes of their meetings
so that there is a record of the inquiry
process.
Assessment is based upon the models
students produce as a group as well as
students’ individual reflection, both on the
model development process and on their own
learning. Each student each keeps a
reflective log of his or her work during the
module, commenting on the skills and
lessons learned and identifying the skills
needing further development.
It is difficult to say whether students who
complete a mathematical modeling unit with this
type of structure are better modelers at the end. What
I can say, from my experience, is that this structure
engages students more readily than modeling taught
as a more technically-oriented and solitary
experience. The basic skills required of a
mathematical modeler are probably little different
now from when Morris (1967) originally wrote his
guide. Technology, of course, has advanced
enormously, but the individual’s ability to learn
•
The Teaching of Mathematical Modeling
about, understand, and unpack a complex problem
remains at the heart of modeling.
References
Argyris, C., & Schon, D. A. (1978). Organisational learning: a
theory of action perspective. Reading, MA: AddisonWesley.
Levitt, B., & March, J. G. (1988). Organisational learning.
Annual Review of Sociology, 14, 319–340.
Moon, J. A. (2000). Reflection in learning and professional
development. Abingdon, United Kingdom: Routledge
Falmer.
Morris, W. (1967). On the art of modelling. Management
Science, 13(12), 707–717.
Bartunek, J. M. (1984). Changing interpretive schemes and
organisational restructuring. Administrative Science
Quarterly, 29, 355–372.
Ortenblad, A. (2004). The learning organisation: Towards an
integrated model. The Learning Organisation, 11(2), 129–
144.
Bloom, B. S. (1956). Taxonomy of educational objectives,
Handbook I: The cognitive domain. New York: David
McKay.
Peschl, M. F. (2005). Acquiring basic cognitive and intellectual
skills for informatics: Facilitating understanding and
abstraction in a virtual cooperative learning environment. In
P. Micheuz, P. Antonitsch, & R. Mittermeir (Eds.),
Innovative concepts for teaching informatics (pp. 86–101).
Vienna: Carl Ueberreuter.
Dooley, J. (1999). Problem solving as a double loop learning
system. Retrieved on January 12, 2006, from
http://www.well.com/user/dooley/Problem-solving.pdf
Edwards, D., & Hamson, M. (2001). Guide to mathematical
modelling. Basingstoke, United Kingdom: Palgrave
Macmillan.
Elby, A. (2001). Helping physics students learn how to learn.
American Journal of Physics, 69(7), 54–64.
Greany, T., & Rodd, J. (2003). Creating a learning to learn
school. London: Network Educational Press.
Hamming, R. W. (1997). The art of doing science and
engineering: Learning to learn. Amsterdam: Gordon and
Breach Science.
Howarth, K. (1997, March). The teaching of thinking skills in
physical education: Perceptions of three middle school
teachers. Paper presented at the annual meeting of the
American Educational Research Association, Chicago.
King, T. (2002). Development of student skills in reflective
writing. In A. Goody & D. Ingram (Eds.), Spheres of
Influence: Ventures and Visions in Educational
Development. Proceedings of the 4th World Conference of
the International Consortium for Educational Development.
Perth: The University of Western Australia.
Knight, P. T. (1997, May). Learning How to Learn in High
School History. Paper presented at ORD-Congress
(Onderwijsresearchdagen), Leuven, Belgium.
Kraft, D. H., & Boyce, B. R. (1991). Operations research for
libraries and information agencies: Techniques for the
evaluation of management decision alternatives. San Diego:
Academic Press.
Landsberger, J. (2005). Learning to learn. Retrieved on January
12, 2006, from http://www.studygs.net/metacognition.htm
Lawrence, J. A., & Pasternack, B. A. (2002). Applied
management science. New York: John Wiley and Sons.
Jon Warwick
Rawson, M. (2000). Learning to learn: More than a skill set.
Studies in Higher Education, 25(2), 225–238.
Senge, P. M. (1990). The fifth discipline: The art and practice of
the learning organisation. New York: Doubleday.
Southern England Consortium for Credit Accumulation and
Transfer. (2003). Credit level descriptors for further and
higher education. Retrieved on January 17, 2006, from
http://www.seec-office.org.uk/SEEC%20FE-HECLDsmar03def-1.doc
Sterman, J. (2000). Business dynamics: systems thinking and
modelling for a complex world. Boston: McGraw-Hill
Irwin.
University of Luton. (2006). Effecting change in higher
education. Retrieved on January 13, 2006, from
http://www.effectingchange.luton.ac.uk/approaches_to_cha
nge/ index.php?content=ol
Waeytens, K., Lens, W., & Vandenberghe, R. (2002). Learning
to learn: Teachers’ conceptions of their supporting role.
Learning and Instruction, 12, 305–322.
Warwick, J. (1992). Modelling by analogy: An example from
library management. Teaching Mathematics and its
Applications, 11(3), 128–133.
Wright, J. (1982). Learning to learn in higher education.
London: Croom Helm.
1
My intention is to use 'module' to mean part of a
course of study so that a student studies several modules
per year.
41
The Mathematics Educator
2007, Vol. 17, No. 1, 42–44
Book Review…
The View from Here: Opening Up Postmodern Vistas
Ginny Powell
Walshaw, M. (Ed.). (2004). Mathematics education within the postmodern. Greenwich, CT:
Information Age. 254 pp. ISBN 1-59311-130-4 (pb). $34.95
The term “postmodern” has been used in many
different ways by many different people. And that’s
just fine with postmodernists. Those given credit for
the creation of postmodernism cared little for the name,
and today’s evangelizers feel no need to pin it down to
one meaning. That is the point, after all.
Postmodernism was born as a reaction against the
“modernist project” of finding the one final answer to
every question. The real world, postmodernists would
say, is much more complex than that.
But whether you consider yourself a postmodernist
or not, you will find Mathematics Education within the
Postmodern (edited by Margaret Walshaw, $34.95) an
eye-opening and thought-provoking book. As
postmodern pioneer Valerie Walkerdine says in the
preface, the purpose of this volume is to “challenge
accepted wisdoms” (p. vii) about mathematics,
mathematics education, and mathematics educators. Up
to now, “the post” has led to few insights into
mathematics education, even as its contribution to
other disciplines has grown.1 A volume such as this is a
sign that this interesting perspective is growing in
popularity and recognition, pulling up a seat at the
table, and joining the fray.
For those new to postmodernism, editor Walshaw
provides a nice introduction in the first chapter, along
with an explanation of modernism, for those who are
not aware they are embedded in it. Accurately, though
disturbingly, she describes the postmodern approach as
“unsettling” and about “exploring tentativeness” (p. 3).
There is an unapologetic lack of answers here: no “tips
for teachers” (p. 222), as Cotton says in the final
chapter. This book, like this approach, is about opening
up new ways of thinking about matters we did not
realize we needed to think about, things we thought
Ginny Powell is an Instructor of Mathematics at Georgia Perimeter
College, and a doctoral student in Mathematics Education at
Georgia State University. She is interested in the teacher-student
dynamic, especially in cross-cultural and tertiary settings, and in
the fairness, or lack thereof, of standardized assessment.
42
were fixed and decided, the “unthinkable.” I like to
picture postmodernism as pointing out a new path I
never noticed before, which invites endless
exploration, but also carries possible dangers.
This book is the fourth volume in the International
Perspectives in Mathematics Education 2 series, and
international certainly describes it; contributors hail
from, or have worked in, Colombia, Brazil, the United
States, Australia, Kiribati, Denmark, the United
Kingdom, and New Zealand, though only the last two
countries have multiple representatives. As a
mathematics teacher in the United States, I found
nothing that seemed lost in the translation across
cultures. It was all relevant and recognizable,
sometimes troublingly so.
Organization of the Book
After the introductory chapter, the book is divided
into three roughly equal parts. The three chapters in
“Part I: Thinking Otherwise for Mathematics
Education” treat the broad subjects of how
postmodernism might lead to new ways of thinking
about mathematics itself, research in mathematics
education, and the practice of mathematics education.
“Part II: Postmodernism within Classroom Practices”
includes four chapters attempting to show, with
varying success, how a postmodern attitude has
changed or could change the classroom. The final part,
“Part III: Postmodernism within the Structures of
Mathematics Education,” takes on teacher training,
curriculum design, and assessment from the
postmodern perspective.
This structure seems arbitrary. Chapters from the
first and last parts could have easily been put together,
as all treat large issues in mathematics education. Then
again, some of them treat classroom practices, and
might have fit better in the second part. The chosen
organization seems to echo that of the previous volume
in the series rather than any logical arrangement. That
Walshaw’s volume has one less chapter and is one
Opening Up Postmodern Vistas
hundred pages shorter than the previous volume leads
one to wonder if some of the contributions would not
have been accepted if there had been more
submissions. Perhaps in the future there will be more
researchers willing and able to contribute to future
mathematics education publications in the postmodern
vein.
The Big Picture
Paul Ernest starts off the first part by taking
mathematics to task for its failure to respond
meaningfully to fundamental issues, or rather for
responding by gobbling up each new paradox and
going on as though nothing had happened. “Gödel’s
Theorem did not even cause mathematics to break its
stride as it stepped over this and other limitative
results” (p. 17). Some would see that as a strength, but
Ernest, consistent with the postmodern emphasis on
deconstruction, would rather explore where those
issues might take us. Also in this chapter, he discusses
several postmodernists and pre-postmodernists, such as
Lyotard, Foucault, and Lakatos, illuminating the
origins of some of the basic ideas of postmodernism.
Ernest gently points out how we sometimes have to
unlearn what we thought we knew in order to learn
something new, whether it’s “addition makes a bigger
number” or “there is one best way to teach.”
In the next chapter, Valero takes us into a
classroom in Colombia to explore what we think we
know about mathematics students. Adopting a
postmodern attitude, she offers insight into the
unreality of the “laboratory children” (p. 43)
mathematics education researchers claim to have
knowledge about. Real children are much more
complicated, of course, and she discusses how we
might better approach them. Even if some children are
willing to make it easy for us by playing by the rules of
mathematics and the classroom, she asks if that is
really all we want for them.
Finishing up Part I of the book, Neyland tackles
ethics and what postmodern ethics might mean. He
charges current, “modern” ethics with being
“undesirable and illusory” (p. 56), leading to
educational “reforms” like national curricula and
standardized tests that seem to him flawed from the
ground up. His goal is not to replace this state of affairs
with a new set of unquestionable standards—an
“objectively founded and universal ethical code is
impossible to obtain” (p. 60)—but to explore other
possibilities based on the individual self, possibilities
that might lead to a “re-enchantment” (p. 60) with
mathematics.
Ginny Powell
Hands On
The next section, on classroom practices, should be
the most enjoyable for the neophyte postmodernist
reader, if only for its concreteness. Those set afloat by
the endless questioning of the first part will find
something to hold onto here, as we see teachers and
students interacting in recognizable ways. But soon we
will be led to question what we thought we knew about
such a familiar setting, as the authors point out the
obvious-once-you-hear-them, shocking undercurrents
of teacher-student relationships.
Unfortunately, the first chapter in this section
seems completely out of place in this volume. In it,
Macmillan discusses interactions in preschool
mathematics groups in Australia. Her occasional use of
a word from the official postmodern lexicon (e.g.,
agency, discourse) cannot hide her essentially
constructivist approach. She painstakingly systematizes
everything and explicitly and unquestioningly accepts
the conventions of the current modernist classroom.
One wishes she had read and learned from Valero’s
chapter above, or Hardy’s below.
The rest of the second section, however, is filled
with questions and new perspectives. In her piece,
Hardy vividly dissects an “exemplar” teacher training
video. Drawing on Foucault, she discusses the
normalizing effect on students and teachers, asking
how we might “choose to do otherwise” (p. 116).
Speaking as though for the entire book, she hopes that
“by working through alternatives, by exploiting the
lack of stability of many of our professional notions,
we might open up spaces from which we can counter
ill-posed problems and look for sites of resistance” (p.
117).
Editor Walshaw contributes a chapter to this
section as well, bringing Lacanian psychoanalysis into
the fray. Once again she acts as helper to the reader,
defining and explaining constructivist and sociocultural
theories of knowing and outlining Lacan for us, before
bringing him to bear on a single student and that
student’s interactions with her mathematics teacher.
What she has to say about the idea of a “model” pupil
will have you deconstructing all your notions about
your relationship to your students as a teacher and your
relationship with your teachers as a student.
Cabral continues the Lacanian analysis by
describing a classroom where postmodern ideas have
already influenced practice. The result is, as promised,
unsettling. She coins a new phrase, “pedagogical
transference” (p. 142), to explain her ideas about how
learning is affected by the unconscious, by feelings and
moods. But even without adopting her terms, the
43
reader can come to the, by now familiar, space of
constant questioning as they experience this new
vision. The result is not necessarily the urge to run out
and replicate her classroom, but the reader becomes
aware of yet another space for change in her or his own
teaching and learning.
way. His vision is clear, but the example he gives of
his attempt to actually use his new assessment shows
just how difficult change can be, as ten-year-olds
demonstrate how embedded they already are in the
world that standardized testing has wrought.
More Big Issues
This book is not an exhaustive treatise on
mathematics education or on postmodernism. It is only
a beginning, a step towards possible change. While the
authors come at postmodernism from different angles
and through the work of different thinkers (Lacan,
Foucault, Lyotard, Deleuze, etc.), they all share one
goal: to make us think about mathematics education
differently. They remind us that there is no one “right”
way to teach or learn mathematics. Through their own
examples, they inspire us to seek new insights of our
own. As Walshaw says in the introductory chapter,
“Ultimately it is the hope of all the authors that the
ongoing engagement will mark a fruitful and
productive
convergence
between
mathematics
education and postmodernism” (p. 11).
This book would be an interesting and useful read
for anyone involved in the teaching or learning of
mathematics at any level, kindergarten through college,
and for administrators and policymakers who are in a
position to make broad decisions about mathematics
education. It can be read all at once, or the reader can
use the index or Walshaw’s excellent introductions in
Chapter 1 to find something applicable to their own
situation. Whether or not anyone embraces
postmodernism as a result of reading this book is
irrelevant. As long as new questions are asked,
progress has been made toward a more flexible system
of mathematics education.
The last part of the book returns to confrontation
with current large issues in mathematics education. In
their article, Brown, Jones, and Bibby search for
insight into the thinking of elementary school teacher
trainees. They find nearly universal fear and lack of
facility with mathematics. This chapter isn’t “teacher
bashing,” nor is it a panacea, but rather it exposes a
reality that needs to be addressed. Brown, Jones, and
Bibby ask how a teacher’s identity and feelings about
her- or himself as a mathematics learner and teacher
affect future students, and how we might change those
feelings for the betterment of all.
Meaney, in the next chapter, takes us back a step to
look at curriculum design. Specifically, she looks at
how she, as an outside consultant brought in to
facilitate the development of a mathematics curriculum
among the Mäori, negotiated the many power relations
inherent in the situation, and how she might do it
differently next time. There is much here about crosscultural pitfalls, but also about the dangers of top-down
decision making. Once again, she asks only that we
begin to think about how and why things are done, and
what alternatives are possible.
“Do you ever think about what you don’t think
about?” (p. 202) Fleener asks provocatively at the
beginning of her chapter. The sole American
contributor, she draws on Deleuze and Guattari, as well
as popular movies, to question the most basic
structures of mathematics education. “Why do we
teach division after multiplication? …Why is
mathematical aptitude considered evidence of
intelligence? … Why do we teach 400-year-old algebra
and calculus and 2500-year-old geometry?” (p. 202).
Echoing Ernest, she castigates the mathematical
community for ignoring the possibilities that
foundational problems create, expressing a wish that
we “celebrate rather than bemoan the loss of certainty
and structure” (p. 203) and help our students “fall in
love” (p. 205) with mathematics, not just regurgitate it.
In the final chapter, Cotton challenges current
assessment practices and how they can take on a selfperpetuating life of their own. Drawing on Lyotard, he
deconstructs assessment as it is currently practiced in
the UK and then sets forth his own criteria for a better
44
Putting It All Together
References
Linn, R. (1996). A teacher's introduction to postmodernism.
Urbana, IL: National Council of Teachers of English.
Walshaw, M. (Ed.). (2004). Mathematics education within the
postmodern. Greenwich, CT: Information Age.
1
For example, in 1996 the National Council of Teachers
of English (NCTE), in the NCTE Teacher's Introduction
Series, published the book A Teacher's Introduction to
Postmodernism (Linn, 1996).
2
Volume 1 was Multiple Perspectives on Mathematics
Teaching and Learning (2000), edited by Jo Boaler; Volume
2 was Researching Mathematics Classrooms: A Critical
Examination of Methodology (2002), edited by Simon
Goodchild and Lyn English; and Volume 3 was Which Way
Social Justice in Mathematics Education? (2003), edited by
Leone Burton, who is also series editor.
Opening Up Postmodern Vistas
CONFERENCES 2007, 2008…
AMESA
13th Annual National Congress
Mpumalanga, South
Africa
July 2-6, 2007
Seoul, South Korea
July 8-13, 2007
Salt Lake City, UT
July 29-August
2, 2007
First Joint Meeting with the Polish Mathematical Society
http://www.ams.org
Warsaw, Poland
July 31-August
3, 2007
GCTM
Georgia Council of Teachers of Mathematics Annual Conference
Rock Eagle, GA
October 17-19,
2007
Indianapolis, IN
November 1517, 2007
Lake Tahoe, NV
October 25-28,
2007
San Diego, CA
January 6-9,
2008
Tulsa, OK
January 24-26,
2008
Oklahoma City, OK
March 6-8, 2008
Salt Lake City, UT
April 7-9, 2008
Salt Lake City, UT
April 7-12, 2008
New York, NY
March 24-28,
2008
http://www.amesa.org.za/AMESA2007/
PME-31
International Group for the Psychology of Mathematics Education
http://pme31.org
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2007/
http://www.gctm.org/
SSMA
School Science and Mathematics Association
http://www.ssma.org
PME-NA
North American chapter
International Group for the Psychology of Mathematics Education
http://pmena.org
MAA-AMS
Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org
AMTE
Association of Mathematics Teacher Educators
http://amte.net
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML/conference2007.html
NCSM
National Council of Supervisors of Mathematics
http://www.ncsmonline.org/
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
AERA
American Education Research Association
http://www.aera.net
45
The Mathematics Educator (ISSN 1062-9017) is a semiannual publication of the Mathematics Education
Student Association (MESA) at the University of Georgia. The purpose of the journal is to promote the interchange
of ideas among the mathematics education community locally, nationally, and internationally. The Mathematics
Educator presents a variety of viewpoints on a broad spectrum of issues related to mathematics education. The
Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on
Mathematical Education).
The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other
professionals in mathematics education including:
•
•
•
•
•
•
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reports of research (including experiments, case studies, surveys, philosophical studies, and historical studies),
curriculum projects, or classroom experiences;
commentaries on issues pertaining to research, classroom experiences, or public policies in mathematics
education;
literature reviews;
theoretical analyses;
critiques of general articles, research reports, books, or software;
mathematical problems (framed in theories of teaching and learning; classroom activities);
translations of articles previously published in other languages;
abstracts of or entire articles that have been published in journals or proceedings that may not be easily
available.
The Mathematics Educator strives to provide a forum for collaboration of mathematics educators with varying levels
of professional experience. The work presented should be well conceptualized; should be theoretically grounded; and
should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and
researchers.
Guidelines for Manuscripts:
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Manuscripts should be double-spaced with one-inch margins and 12-point font, and be a maximum of 25 pages
(including references and footnotes). An abstract should be included and references should be listed at the end of
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An electronic copy is required. (A hard copy should be available upon request.) The electronic copy may be in
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47
In this Issue,
Guest Editorial… In Pursuit of a Focused and Coherent School Mathematics
Curriculum
TAD WATANABE
Being a Mathematics Learner: Four Faces of Identity
RICK ANDERSON
A Quilting Lesson for Early Childhood Preservice and Regular Classroom Teachers:
What Constitutes Mathematical Activity?
SHELLY SHEATS HARKNESS & LISA PORTWOOD
Graduate Level Mathematics Curriculum Courses: How Are They Taught?
ZELHA TUNÇ-PEKKAN
Some Reflections on the Teaching of Mathematical Modeling
JON WARWICK
Book Review… The View from Here: Opening Up Postmodern Vistas
GINNY POWELL