____ THE
_____ MATHEMATICS ___
________ EDUCATOR _____
Volume 14 Number 1
Spring 2004
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Holly Garrett Anthony
Dear TME readers,
Along with the editorial team, I present the first of two issues to be produced during my brief
tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both
veteran and budding scholars in mathematics education. The articles range in topic and thus invite all
those vested in mathematics education to read on.
Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in
mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical
perspectives he believes can change this, and motivates mathematics educators at all levels to rethink
their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social
Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across
chapters and applauds the picture of social justice painted by the diversity of voices therein.
Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics
educators to step outside themselves and reexamine features of PhD programs and elementary
textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom
teaching, how we engage in and write about research on or with teachers, and what features of a PhD
program can inform teacher education. Beckmann looks abroad to highlight simple diagrams used in
Singapore elementary texts—that facilitate the development of students’ algebraic reasoning and
problem solving skills—and suggests that such representations are worthy of attention in the U. S.
Finally, Bharath Sriraman and Melissa Freiberg offer insights into the creativity of
mathematicians and the organization of rich experiences for preservice elementary teachers,
respectively. Sriraman builds on creativity theory in his research to characterize the creative practices
of five well-published mathematicians in the production of mathematics. Freiberg reminds us of the
daily challenge of mathematics educators—providing preservice teachers rich classroom
experiences—and details the organization, coordination, and evaluation of Family Math Fun Nights in
elementary schools.
It has been my goal thus far to entice you to read what follows, but I now want to focus your
attention on the work of TME. I invite and encourage TME readers to support our journal by getting
involved. Please consider submitting manuscripts, reviewing articles, and writing abstracts for
previously published articles. It is through the efforts put forth by us all that TME continues to thrive.
Last I would like to comment that publication of Volume 14 Number 1 has been a rewarding
process—at times challenging—but always worthwhile. I have grown as an editor, writer, and scholar.
I appreciate the opportunity to work with authors and editors and look forward to continued work this
Fall. I extend my thanks to all of the people who make TME possible: reviewers, authors, peers,
faculty, and especially, the editors.
Associate Editors
Ginger Rhodes
Margaret Sloan
Erik Tillema
Publication
Stephen Bismarck
Laurel Bleich
Dennis Hembree
Advisors
Denise S. Mewborn
Nicholas Oppong
James W. Wilson
MESA Officers
2004-2005
President
Zelha Tunç-Pekkan
Vice-President
Natasha Brewley
Secretary
Amy J. Hackenberg
Treasurer
Ginger Rhodes
NCTM
Representative
Angel Abney
Holly Garrett Anthony
Undergraduate
Representatives
Erin Bernstein
Erin Cain
Jessica Ivey
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@coe.uga.edu
www.ugamesa.org
About the cover
Cover artwork by Thomas E. Ricks. Fractal Worms I of the Seahorse Valley in the Mandelbrot Set, 2004.
For questions or comments, contact: tomricks@uga.edu
Benoit Mandelbrot was the pioneer of fractal mathematics, and the famous Mandelbrot set is his namesake. Based on a simple iterative equation applied to the
complex number plane, the Mandelbrot set provides an infinitely intricate and varied landscape for exploration. Visual images of the set and surrounding points are
made by assigning a color to each point in the complex plane based on how fast the iterative equation’s value “escapes” toward infinity. The points that constitute
the actual Mandelbrot set, customarily colored black, are points producing a finite value. The Mandelbrot set is a fractal structure, and one can see self-similar
forms within the larger set.
Using computing software, anyone can delve within this intricate world and discover views never seen before. Modern computing power acts as a microscope
allowing extraordinary magnification of the set’s detail.
The fanciful drawing Fractal Worms I is based on the structure of spirals residing in the commonly called “Seahorse Valley” of the Mandelbrot Set. Using a
lightboard, Thomas Ricks drew the fractal worms on a sheet of art paper laid over a computer printout of the Seahorse spirals. With the light shining through both
sheets of paper, he drew the various fractal worms following the general curve of the spirals. The printout was produced by a Mandelbrot set explorer software
package called “Xaos”, developed by Jan Hubicka and Thomas Marsh and available at: http://xaos.theory.org/
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
Spring 2004
Volume 14 Number 1
Table of Contents
2
Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways
Respected by the Mathematics Education Community?
CHANDRA HAWLEY ORRILL
8
Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim
Toward Empowering All Children With a Key to the Gate
DAVID W. STINSON
19 The Characteristics of Mathematical Creativity
BHARATH SRIRAMAN
35 Getting Everyone Involved in Family Math
MELISSA R. FREIBERG
42 In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a
Method Demonstrated in Grade 4–6 Texts Used in Singapore
SYBILLA BECKMANN
47 Book Review… Diverse Voices Call for Rethinking and Refining Notions of Equity
AMY J. HACKENBERG
52 Upcoming Conferences
53 Subscription Form
54 Submissions Information
© 2004 Mathematics Education Student Association.
All Rights Reserved
The Mathematics Educator
2004, Vol. 14, No. 1, 2–7
Guest Editorial…
Do You Need a PhD to Teach K–8 Mathematics in Ways
Respected by the Mathematics Education Community?
Chandra Hawley Orrill
The genesis of this editorial was a conversation
about an article in which Ball (1991) provided
descriptions of three teachers’ approaches to working
with their students. In Ball’s article, teachers without
PhDs in mathematics or mathematics education
struggled to engage their students in developing
meaningful concepts of mathematics. They could not
provide
multiple
interpretations
of
concepts—particularly representations that provided
concrete explanations or tie-ins to the real world. They
demonstrated only stepwise approaches to doing
mathematics, clinging tightly to procedures and
algorithms, and provided no evidence that they had a
deeper understanding of the mathematics. In stark
contrast, the same Ball article offered a vignette of
Lampert’s teaching that illustrated a rich mathematical
experience for students. Lampert provided multiple
perspectives, introduced multiple representations, and
demonstrated a deep understanding of both
mathematics and student learning throughout the
episode described.
Given the number of articles in the literature
painting the ‘typical’ mathematical experience as one
that is impoverished, and the growing body of
literature written by PhD researcher-teachers, I
wondered, “Do you need a PhD to teach elementary
Chandra Hawley Orrill is a Research Scientist in the Learning
and Performance Support Laboratory at the University of
Georgia. Her research interest is teacher professional
development with an emphasis on teaching in the midst of
change. She is also interested in how professional development
impacts the opportunities teachers create for student learning.
Acknowledgements
The research reported here came from a variety of projects
spanning six years. These projects were supported by grants from
the Andrew W. Mellon Foundation and the Russell Sage
Foundation, the National Science Foundation, Georgia’s
Teacher Quality Program (formerly Georgia’s Eisenhower
Higher Education Program), and the Office of the Vice President
for Research at the University of Georgia. Opinions expressed
here are my own and do not necessarily reflect those of the
granting agencies. My thanks to Holly Anthony, Ernise Singleton,
Peter Rich, Craig Shepherd, Laurel Bleich, and Drew Polly for
their ongoing discussions with me about whether a teacher needs
a PhD to teach K-8 mathematics.
2
and middle school mathematics in ways that
mathematics educators would value?” After all, the
Balls, Lamperts, and McClains1 in the literature offer
high-quality mathematics instruction, attend to student
thinking, provide opportunities for knowledge
construction, and introduce students to a variety of
tools they can use later (e.g., visual representations and
problem solving strategies). Further, these researcherteachers seem to have a gift for promoting student
thinking and moving an entire class forward by
scaffolding lessons, questioning students, and creating
a classroom community where learners consider each
other’s work critically and interact meaningfully. The
reality, however, is that not all mathematics teachers
have PhDs and it is unlikely that most ever will.2
In working through this question both with the
graduate students with whom I work and in preparation
for this editorial, I have developed some ideas both
about researcher-teachers as a “special” group and
about why having a PhD might matter. Based on my
thoughts I would like to propose two conjectures about
researcher-teacher efforts. First, I conjecture that we
should consider the way we think about researcherteachers versus research on/with teachers. Second, I
propose that certain features of PhD programs can be
applied to teacher professional development and/or
undergraduate education to support all teachers in
creating richer mathematics learning experiences for
their students. This editorial explores these two
conjectures in more detail.
Researcher-Teachers as a Special Group
In order to understand some of the unique qualities
of the teaching exemplified by researcher-teachers, it is
worthwhile to consider why they do what they do so
well. There are a variety of factors that impact both the
way these people teach and the way we, as consumers
of research, read about their teaching. First, researcherteachers teach well because they have significant
knowledge of mathematics and how children learn
mathematics. There is no doubt that teachers, with or
without PhDs, who have strong pedagogical
knowledge and strong content knowledge, create richer
Do You Need a PhD?
learning experiences for their students (e.g., Ball,
Lubienski, & Mewborn, 2001).
Further, in the process of earning a PhD,
researcher-teachers presumably develop reflective
dispositions, grapple with their own epistemological
beliefs, and define their visions of learning and
teaching. This produces teachers who critically
examine the world around them and who are
introspective in ways that are productive for achieving
the classroom environment valued by mathematics
education researchers and described in the NCTM
Standards (NCTM, 2000). By developing this
disposition, researcher-teachers are in a unique position
to make critical changes to the classroom environment
as needs are identified. Too often, regular classroom
teachers do not have the time or skills to analyze
formal or informal data about their students and their
teaching. In fact, many classroom teachers have only
been exposed to the most basic concepts of student
learning theory and research. As a result, even if they
tried to make sense of the data presented in their
classroom, they would be ill-equipped to make
important changes based on those data.
In addition, researcher-teachers have some
pragmatic luxuries that typical teachers do not have.
For example, they usually only teach one subject to
one class per day, while a typical elementary teacher
might teach four subjects to one class, and a middle
school teacher might teach one or two subjects to four
or five classes each day. This provides the researcherteacher with more time for reflection and refinement.
To be fair, researcher-teachers typically do have other
work responsibilities – they do not simply teach for 50
minutes and “call it a day.” However, their situation is
very different from that of a typical classroom teacher.
Researcher-teachers have support with the reflection
process from others studying the classroom, and often
have no additional responsibilities such as conducting
parent conferences, developing individualized plans for
certain students, and attending the team meetings
common in many teachers’ daily experience. While
this difference should not be viewed or used as an
excuse for classroom teachers to avoid improving their
practice, it is undeniable that a researcher-teacher’s job
is fundamentally different from that of the typical
classroom teacher.
In addition to teaching expertise and workload,
researcher-teachers have some advantages over
teachers when participating in others’ studies. Unlike
most “typical” teachers, researcher-teachers are, by
definition, philosophically aligned with and invested in
the goals of the research. They already have agreement
Chandra Hawley Orrill
with the researcher about what good teaching and
learning look like – after all, they are typically either
the researcher (e.g., Ball, 1990a and Lampert, 2001) or
they are a full member of the research team (e.g.,
McClain in Bowers, Cobb, & McClain, 1999). The
importance of this is profound. A researcher-teacher
wants the s a m e (not negotiated or compromised)
outcomes as the researcher, because she either is the
researcher or is a member of the research team. The
researcher-teacher, therefore, attends to those issues
and aspects of the classrooms and student learning that
are the focus of the research. Further, the researcherteacher provides unlimited, or nearly unlimited, rich
access to her thinking for the research effort because,
again, she has a vested interest in capturing that
thinking. Thus, teacher and researcher alignment in
terms of goals, values, and expectations is important.
One potential disadvantage for researcher-teachers
worth noting is the potential for bias to confound the
research. After all, the researcher-teacher has a biased
view of the teaching being studied because it is her
own. Further, because she is invested in the research
and because she is a member of the research team, it is
possible that her teaching is biased to make the
research work. That is, if the researcher is looking for
particular aspects of teaching, such as student-teacher
interactions, the researcher-teacher may attend to those
interactions more in the course of instruction than she
would under other circumstances. Clearly, the impact
of this on the research is determined by both the
research questions and the data collection and analysis
techniques used.
Research On/With Teachers
In order to understand the differences between
researcher-teacher research and research on or with
full-time teachers, it is necessary to explore some of
the issues involved with doing research on/with
teachers. Research in regular classrooms differs in
some significant ways from the researcher-teacher
work alluded to in this editorial. To highlight some of
these differences, I offer examples from my own
experience in working with middle grades mathematics
teachers.
One major difference I alluded to is the values a
teacher holds. In the course of my career, I have been
fortunate to work with several “good” teachers.
However, the ways in which they were “good” were
direct reflections of their own values and the values of
the system within which they were working.
Sometimes, they were good in the eyes of the
administrators with whom they worked because they
3
kept their students under control. Sometimes they were
good for my research in that their practice had the
elements I was interested in, thus making it easier for
me to find the kinds of interactions I was looking for in
their classrooms. Sometimes they were good in that
they were predisposed to reflective practice allowing
me, as a researcher, easier access to their ideas through
observation and interviews. The quality of the teachers,
though, depended on what measure they were held up
against and what measures they, personally, felt they
were trying to align with.
Another important aspect of working with teachers
is a lack of access to certain aspects of their thinking.
For example, I have never been able to analyze a data
set without thinking, at some point, “I wonder what she
was thinking when she did that?” or “Did she not
understand what that student was asking?”
Acknowledging this lack of access to a teacher’s
thinking requires researchers to be careful in their
analysis of the teacher’s actions and beliefs and to
explain how thinking and actions are interpreted.
Further, at times, such limitations require researchers
to analyze situations from their own perspectives as
well as from the teacher’s perspective to understand a
situation.
As a practical example of the influence of
researcher and teacher alignment issues, I offer two
situations from my own work: one addressing the
“good” teacher issue and the other addressing the need
to understand the situation from the teacher’s
perspective. My goal in presenting these two examples
is to highlight issues that arise in research with teachers
who are not members of the research team. In one
study (Orrill, 2001), I worked with two middle school
teachers (one mathematics and one science) in New
York City to understand how to structure professional
development to support uses of computer-based
simulations. My goal for the professional development
was to enhance teachers’ attention to student problemsolving skills in the context of computer-based,
workplace simulations. The mathematics teacher was
considered to be “good” by her principal and other
teachers. In my observations of her classroom, I found
that she taught mathematics in much the same way as
the “typical” teachers we read about in case study after
case study. She offered many procedures but provided
inadequate opportunities for students to interact with
the content in ways that would allow them to develop
deep understanding of the mathematical concepts
underlying those procedures. However, this teacher
had remarkable skill in classroom management, which
was highly valued in her school. Further, she had
4
developed techniques that supported her students in
achieving acceptable scores on the New York
standardized tests. By these standards, she was
considered “good.” When she used the simulations I
was researching, she maintained the same kinds of
approaches, particularly early in the study. She kept
students on task and directed them to work more
efficiently. Given my goal of understanding how to
promote problem solving, her interactions with the
students were inadequate and impoverished. She
typically did not ask the students questions that
provided insight into their thinking and she did not
allow them to struggle with a problem. Instead, she
directed them to an efficient approach for solving the
problem they were working on, which effectively kept
them on task and motivated them to move forward.
While this presented a challenge to me as the
researcher, it would not be fair for me to “accuse” her
of being less than a good teacher when she was clearly
meeting the expectations of the system within which
she worked. This is clearly a case in which there was a
mismatch between what I, the researcher, valued and
what the teacher and system valued. Had I been
researching my own practice or the practice of a
research team member, this tension would have been
removed.
As a second example, a teacher I have worked with
more recently proved a perplexing puzzle for my team
as we considered her teaching. A point of particular
interest was the teacher’s frustration with poor student
performance on tests – regardless of what students did
in class, a significant number failed her tests. In my
analysis of this case, I recognized that this teacher’s
beliefs about teaching and learning significantly
differed from my own. Until I realized this, I was
unable to understand the magnitude of the barrier the
teacher felt she was facing. At the simplest level, she
believed that her role as a good teacher was to present
new material and provide an opportunity for students
to practice that material. The students’ job, in her view,
was to engage in that practice and develop an
understanding from it. Therefore, when students were
not succeeding, she became extremely frustrated since
she had presented information and provided
opportunities for practice. In her worldview, student
success was out of her hands – she had already done
what she could to support them. As the researcher in
that setting, it was difficult to understand her
frustration because I was working from a constructivist
perspective. Specifically, I was looking for an
environment in which the teacher provided students
opportunities to develop their own thinking via an
Do You Need a PhD?
assortment of models, experiences, and collaborative
exchanges. Student test failure, for me, was an
indicator that learning was not complete and that
students needed different opportunities to build and
connect knowledge. It took considerable analysis for
me, as a researcher with a different perspective and
different goals, to understand how the teacher
understood her role and how she enacted her beliefs
about her role in the classroom.
My point in these two examples is that in much
research there are significant and important differences
in the worldviews of the participants and the
researcher. These differences can lead to frustrations in
data collection, hurdles in data analysis, and, in the
worst cases, assessments of the teachers that are simply
not fair. For example, in the early 1990’s there were
many articles written about the implementation of the
standards in California (e.g., Ball, 1990b; Cohen, 1990;
Wilson, 1990). In many of these cases, the teachers
struggled to implement a set of standards that were
written from a particular perspective that they did not
fully understand. This led to implementations that were
far from ideal in the eyes of the researchers who
understood the initial intent of the standards. Too
often, teachers were presented by researchers as
hopeless or inadequate—in contrast, the teachers
reportedly perceived themselves as adhering to these
new standards. Likely, if the researchers and teachers
had philosophical alignment afforded by the
researcher-teacher approach the findings would have
been tremendously different. After all, had these
studies focused on researcher-teachers, the teacher and
the researcher would have had a shared understanding
of the intent of the standards and had a shared vision of
what their implementation should look like.
PhD Program Features That Could Be Useful In
Teacher Development
While not all people who hold PhDs are good
teachers, certain habits of mind are developed as part
of the process of earning a PhD that can significantly
impact the learning environment a teacher designs.
Given the high-quality of teaching exhibited by the
researcher-teachers referred to in this article, it seems
likely that there are aspects of the PhD program that
could be adapted for teacher professional development.
First, the researcher-teacher typically has
developed solid pedagogical knowledge, content
knowledge, and pedagogical content knowledge. This
comes from having time and encouragement to read
about different practices in a focused way,
participating in shared discourse with colleagues,
Chandra Hawley Orrill
conducting research in others’ classrooms, and having
other similar experiences. This is in stark contrast to
the elementary or middle grades teacher who has
typically had four years of college—with courses
spread across the curriculum—and only limited “life
experience” to relate to in the courses that help develop
these knowledge areas. Second, one of the most
powerful outcomes of earning a PhD is the
development of a concrete picture of a desired learning
environment that looks beyond issues of classroom
management and logistics to focus on the kinds of
learning and teaching that will take place. Third, PhDs
develop a rich, precise vocabulary aligned with that of
the standards-writers and the researchers. In becoming
a researcher, the holder of the PhD becomes active in
the conversation of the field—meaning that person has
developed a refined vocabulary and vision that is
shared, in some way, by the field. This is not to say
that there is a definitive definition of K-8 mathematics
education that is shared across the field of mathematics
education, rather that there is a shared way of
discussing and thinking about mathematics education
that allows a more consistent enactment of standards
and practices.
Finally, many researcher-teachers implement or
develop a “special” curriculum. In the case of Lampert
(2001), the teacher was creating open-ended problems
each day to support mathematical topics. In other
cases, the research team has developed materials for
the researcher-teachers to implement. Often, these
materials are far richer than traditional mathematics
textbooks. While there may not be a single disposition
that could be pulled from the process of earning a PhD
that allows researcher-teachers to be successful
implementers of non-traditional materials, it is clear
that there is something different between PhD-holding
researcher-teachers and other teachers. Likely, part of
this ability is related to the knowledge constructs the
researcher-teachers have that allow them to implement
those materials. In my own work, I have found that
teachers who are not well-versed in the curricula, who
lack conceptual knowledge, or who lack the
pedagogical content knowledge to see connections
between various mathematical ideas do not know how
to utilize these kinds of materials to make the
experiences mathematically rich for their students.
Clearly, some attention to the aspects of earning a PhD
that relate to these dispositions would benefit
preservice and inservice teachers.
5
Teacher Development
While it may not be feasible, or even reasonable, to
expect teachers to pursue doctoral degrees, there may
be some characteristics of doctoral education that are
worthwhile for consideration as components or foci of
professional development and undergraduate programs.
To frame this section, I want to draw on the work of
Cohen and Ball (1999) who have argued that the
learning environment is shaped by the interactions of
three critical elements: teachers, students, and
materials/content. This model assumes that for each
element a variety of beliefs, values, and backgrounds
work together to create each unique learning
environment. Considering the classroom from this
perspective is critical to understanding why the
solutions to the problems highlighted in research on
and with teachers are complex.
What We See Now
A quick overview of my definition of the “typical”
classroom may be warranted at this point. Based on the
classrooms described in the literature and those I work
in, the typical mathematics classroom remains focused
on teachers’ delivering information to the students,
typically by working sample problems on the board.
Students are responsible for using this information to
work problems on worksheets or in their books.
Students are asked to do things like name the fractional
portion of a circle that is colored in or to work 20
addition or multiplication problems. Many teachers use
manipulatives or drawn representations to introduce
new ideas to their students. However, their intent is to
provide a concrete example and move the students to
the abstract activities of arithmetic as quickly as
possible or to use the manipulative to motivate the
students to want to do the arithmetic. Mathematics
learning in these classrooms is more about developing
efficient means for working problems than developing
rich understandings of why those methods work.
Referring back to the Cohen and Ball triangle of
interactions, the interactions in these classrooms could
best be characterized by what follows. The teacher
interprets the materials/content and delivers that
interpretation to the students. The students look to
teachers as holders of all information. Teachers are to
provide guidance when students are unable to solve a
problem, to provide feedback about the “rightness” of
student work, and to find the errors students have made
in their work. The students interact with the materials
by working problems. The students may or may not
interact with the concepts at a meaningful level – that
depends on the teacher and the activity. In these
6
classrooms, success is measured in the number of
problems students can answer correctly, often within a
specific amount of time.
How Features of PhD Programs May Change This
To enhance the interactions among teachers,
students, and materials/content there are a number of
elements from doctoral training that may be worth
pursuing. First, teachers can use guided reflection as a
means to step out of the teaching moment to consider
critical aspects of the teaching and learning
environment. Through reflection, teachers have the
opportunity to align their beliefs and practices (e.g.
Wedman, Espinosa, & Laffey, 1998) and to make their
intent more explicit rather than relying on tacit “gut
instinct” (e.g., Richardson, 1990). The reflective
practitioner can learn to look at a learning environment
as a whole by considering how students and materials
are interacting, looking for evidence of conceptual
development, and thinking about ways to improve their
own role in the classroom. The researcher-teachers
(Ball, Lampert, and McClain) cited in this article all
reported using reflection regularly as part of their
practice.
Another element of the PhD experience worth
consideration is the development of solid content and
pedagogical knowledge. Teachers who do not
understand mathematics cannot be as effective as those
who do. For example, teachers who do not know how
to use representations to model multiplication of
fractions cannot use that pedagogical strategy in their
classrooms. Teachers who lack adequate content or
pedagogical knowledge cannot know what to do when
a student suggests an approach to solving a problem
that does not work—too often the only approach the
teacher has is to point out errors to the student and
demonstrate “one more time” the “right” way to work
the problem. I assert that combining teacher
development of content knowledge and pedagogical
knowledge with the development of a reflective
disposition will lead to the emergence of pedagogical
content knowledge. By pedagogical content
knowledge, I refer to knowledge that is a combination
of knowing what content can be learned/taught with
which pedagogies and knowing when to use each of
these approaches to teach students.
Some of the habits of mind developed in a doctoral
program in education translate directly into practice
without focusing on the entire teacher-studentmaterials interaction triad. For example, one
potentially powerful factor to address is the teacherstudent interaction. PhD programs in education offer
Do You Need a PhD?
tremendous opportunities for thinking about this
relationship in meaningful ways, and in the researcherteacher work, attention to this interaction is ubiquitous.
It is absolutely critical to support teachers in learning
to listen to students and respond to them in meaningful
ways. Further, given the poor grounding most teachers
have in learning theory, it may be that developing a
theoretical understanding of how people learn should
be a part of this (this is supported in recent research
such as Philipp, Clement, Thanheiser, Schappelle, &
Sowder, 2003). Finally, focusing professional
development on techniques for questioning that allow
the teacher to access student understanding will
provide teachers with ways to access student thinking.
Conclusion
While it is not realistic to expect that all classroom
teachers will earn doctoral degrees, there are elements
that go into the attainment of a PhD that can lead to
improved classroom teaching. Therefore, it seems
reasonable to capitalize on what we know about the
process of getting and having a doctorate versus more
traditional routes to becoming a teacher.
Granted, there are aspects of researcher-teachers'
activities that are not addressed simply by considering
their educational background or their role in the
research team. For example, high quality materials are
extremely important. Further, it is vital that teachers
are supported in learning how to interact with those
materials (and the content they are trying to convey) if
we want to raise the bar on teaching and learning. No
one can create rich learning experiences around
materials they do not understand. On the other hand,
researcher-teachers have been able to find ways to
capitalize on even the weakest of materials. For
example, Lampert (2001) discusses how she was able
to use the topic ideas from the traditional textbook her
school used to develop rich problems that allowed
students prolonged and repeated exposure to critical
mathematics content—it is clear that the typical teacher
is unable to capitalize on materials in these ways.
Certainly, there is an appropriate place in professional
development efforts to support teachers’ use of
materials.
While this article has only begun to explore the
differences between a typical classroom teacher’s
environment and that of a researcher-teacher, it appears
that researcher-teachers have some advantages over
other teachers. They are better able to understand and
address what is going on in the classroom, as well as
the material they are expected to work with.
Researcher-teachers are also better able to
Chandra Hawley Orrill
communicate with others in the field and to understand
input from the research. Unfortunately, it is not
practical to expect most teachers to earn a doctoral
degree. The question then becomes, “What elements
can we take from earning an advanced degree that will
help teachers in the classroom?” By incorporating
these elements into teacher education and professional
development programs, we can greatly improve
classroom instruction.
REFERENCES
Ball, D. L. (1990a). Halves, pieces, and twoths: Constructing
representational concepts in teaching fractions. East Lansing, MI:
National Center for Research on Teacher Education.
Ball, D. L. (1990b). Reflection and deflections of policy: The case of Carol
Turner. Educational Evaluation and Policy Analysis, 12(3), 247–259.
Ball, D. L. (1991). Research on teaching mathematics: Making subjectmatter knowledge part of the equation. In J. Brophy (Ed.), Advances
in research on teaching (Vol. 3, pp. 1–48). Greenwich, CT: JAI Press.
Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on
teaching mathematics: The unsolved problem of teachers'
mathematical knowledge. In V. Richardson (Ed.), Handbook of
research on teaching (4th ed.). Washington, DC: American
Educational Research Association.
Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical
practices: A case study. Cognition and Instruction, 17(1), 25–64.
Cohen, D. (1990). A revolution in one classroom: The case of Mrs. Oublier.
Educational Evaluation and Policy Analysis, 12(3), 327–345.
Cohen, D., & Ball, D. B. (1999). Instruction, capacity, & improvement (No.
CPRE-RR-43). Philadelphia, PA: Consortium for Policy Research in
Education.
Lampert, M. (2001). Teaching problems and the problems of teaching. New
Have, CT: Yale University Press.
National Council of Teachers of Mathematics. (2000). Principles and
standards for school mathematics. Reston, VA: Author.
Orrill, C. H. (2001). Building technology-based learning-centered
classrooms: The evolution of a professional development framework.
Educational Technology Research and Development, 49(1), 15–34.
Philipp, R. A., Clement, L., Thanheiser, E., Schappelle, B., & Sowder, J. T.
(2003). Integrating mathematics and pedagogy: An investigation of
the effects on elementary preservice teachers' beliefs and learning of
mathematics. Paper presented at the Research Presession of the 81st
Annual Meeting of the National Council of Teachers of Mathematics,
San Antonio, TX. Available online:
http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html.
Richardson, V. (1990). Significant and worthwhile change in teaching
practice. Educational Researcher, 19(7), 10–18.
Wedman, J. M., Espinosa, L. M., & Laffey, J. M. (1998). A process for
understanding how a field-based course influences teachers' beliefs
and practices, Paper presented at the Annual Meeting of the American
Educational Research Association, San Diego, CA.
Wilson, S. M. (1990). A conflict of interests: The case of Mark Black.
Educational Evaluation and Policy Analysis, 12(3), 309–326.
1
I cite examples of each of these researcher-teachers’ work throughout this
editorial. This list is not exhaustive.
2
Reasons why I believe this is true range from the lack of incentives relative
to the effort required to earn a PhD to the mismatch between the intent of
PhD programs and what teachers do in their everyday lives. This is not to
assert that earning a PhD is not helpful for a teacher, rather that it is not
likely in the current educational system.
7
The Mathematics Educator
2004, Vol. 14, No. 1, 8–18
Mathematics as “Gate-Keeper” (?): Three Theoretical
Perspectives that Aim Toward Empowering All Children With
a Key to the Gate
David W. Stinson
In this article, the author’s intent is to begin a conversation centered on the question: How might mathematics
educators ensure that gatekeeping mathematics becomes an inclusive instrument for empowerment rather than
an exclusive instrument for stratification? In the first part of the discussion, the author provides a historical
perspective of the concept of “gatekeeper” in mathematics education. After substantiating mathematics as a
gatekeeper, the author proceeds to provide a definition of empowering mathematics within a Freirian frame, and
describes three theoretical perspectives of mathematics education that aim toward empowering all children with
a key to the gate: the situated perspective, the culturally relevant perspective, and the critical perspective. Last,
within a Foucauldian frame, the author concludes the article by asking the reader to think differently.
My graduate assistantship in The Department of
Mathematics Education at The University of Georgia
for the 2002–2003 academic year was to assist with a
four-year Spencer-funded qualitative research project
entitled “Learning to Teach Elementary Mathematics.”
This assistantship presented the opportunity to conduct
research at elementary schools in two suburban
counties—a new experience for me since my prior
professional experience in education had been within
the context of secondary mathematics education. My
research duties consisted of organizing, coding,
analyzing, and writing-up existing data, as well as
collecting new data. This new data included
transcribed interviews of preservice and novice
elementary school teachers and fieldnotes from
classroom observations.
By January 2003, I had conducted five
observations in 1st, 2nd, and 3rd grade classrooms at
two elementary schools with diverse populations. I was
impressed with the preservice and novice elementary
teachers’ mathematics pedagogy and ability to interact
with their students. Given that my research interest is
equity and social justice in education, I was mindful of
the “racial,” ethnic, gender, and class make-up of the
classroom and how these attributes might help me
explain the teacher-student interactions I observed. My
David W. Stinson is a doctoral candidate in The Department of
Mathematics Education at The University of Georgia. In the fall
of 2004, he will join the faculty of the Middle-Secondary
Education and Instructional Technology Department at Georgia
State University. His research interests are the sociopolitical and
cultural aspects of mathematics and mathematics teaching and
learning with an emphasis on equity and social justice in
mathematics education and education in general.
8
experiences as a secondary mathematics teacher,
preservice-teacher supervisor, and researcher supported
Oakes’s (1985) assertions that often students are
distributed into “ability” groups based on their race,
gender, and class. Nonetheless, my perception after
five observations was that ability grouping according
to these attributes was diminishing—at least in these
elementary schools. In other words, the student makeup of each mathematics lesson that I observed
appeared to be representative of the demographics of
the school.
However, on my sixth observation, at an
elementary school with 35.8 % Black, 12.8 % Asian,
5.3 % Hispanic, 3.5 % Multi-racial, and 0.5 %
American Indian1 children, I observed a 3rd grade
mathematics lesson that was 94.4% White (at least it
was 50% female). The make-up of the classroom was
not initially unrepresentative of the school’s
racial/ethnic demographics, but became so shortly
before the start of the mathematics lesson as some
students left the classroom while others entered. When
I questioned why the students were exchanged between
classrooms, I was informed that the mathematics
lesson was for the “advanced” third graders. Because
of my experience in secondary mathematics education,
I am aware that academic tracking is a nationally
practiced education policy, and that it even occurs in
many districts and schools as early as 5th grade—but
these were eight-year-old children! Has the structure of
public education begun to decide who is and who is not
“capable” mathematically in the 3rd grade? Has the
structure of public education begun to decide who will
be proletariat and who will be bourgeoisie in the 3rd
grade—with eight-year-old children? How did school
Mathematics as “Gate-Keeper” (?)
mathematics begin to (re)produce and regulate racial,
ethnic, gender, and class divisions, becoming a
“gatekeeper”? And (if) school mathematics is a
gatekeeper, how might mathematics educators ensure
that gatekeeping mathematics becomes an inclusive
instrument for empowerment rather than an exclusive
instrument for stratification?
This article provides a two-part discussion centered
on the last question. The first part of the discussion
provides a historical perspective of the concept of
gatekeeper in mathematics education, verifying that
mathematics is an exclusive instrument for
stratification, effectively nullifying the if. The intent of
this historical perspective is not to debate whether
mathematics should be a gatekeeper but to provide a
perspective that reveals existence of mathematics as a
gatekeeper (and instrument for stratification) in the
current education structure of the United States. In the
discussion, I state why I believe all students are not
provided with a key to the gate.
After arguing that mathematics is a gatekeeper and
inequities are present in the structure of education, I
proceed to the second part of the discussion: how
might mathematics educators ensure that gatekeeping
mathematics becomes an inclusive instrument for
empowerment? In this discussion, I first define
empowerment and empowering mathematics. Then, I
make note of the “social turn” in mathematics
education research, which provides a framework for
the situated, culturally relevant, and critical
perspectives of mathematics education that are
presented. Finally, I argue that these theoretical
perspectives replace characteristics of exclusion and
stratification (of gatekeeping mathematics) with
characteristics of inclusion and empowerment. I
conclude the article by challenging the reader to think
differently.
Mathematics a Gatekeeper: A Historical
Perspective
Discourse regarding the “gatekeeper” concept in
mathematics can be traced back over 2300 years ago to
Plato’s (trans. 1996) dialogue, The Republic. In the
fictitious dialogue between Socrates and Glaucon
regarding education, Plato argued that mathematics
was “virtually the first thing everyone has to
learn…common to all arts, science, and forms of
thought” (p. 216). Although Plato believed that all
students needed to learn arithmetic—”the trivial
business of being able to identify one, two, and three”
(p. 216)—he reserved advanced mathematics for those
that would serve as philosopher guardians2 of the city.
David W. Stinson
He wrote:
We shall persuade those who are to perform high
functions in the city to undertake calculation, but
not as amateurs. They should persist in their studies
until they reach the level of pure thought, where
they will be able to contemplate the very nature of
number. The objects of study ought not to be
buying and selling, as if they were preparing to be
merchants or brokers. Instead, it should serve the
purposes of war and lead the soul away from the
world of appearances toward essence and reality.
(p. 219)
Although Plato believed that mathematics was of
value for all people in everyday transactions, the study
of mathematics that would lead some men from
“Hades to the halls of the gods” (p. 215) should be
reserved for those that were “naturally skilled in
calculation” (p. 220); hence, the birth of mathematics
as the privileged discipline or gatekeeper.
This view of mathematics as a gatekeeper has
persisted through time and manifested itself in early
research in the field of mathematics education in the
United States. In Stanic’s (1986) review of
mathematics education of the late 19th and early 20th
centuries, he identified the 1890s as establishing
“mathematics education as a separate and distinct
professional area” (p. 190), and the 1930s as
developing the “crisis” (p. 191) in mathematics
education. This crisis—a crisis for mathematics
educators—was the projected extinction of
mathematics as a required subject in the secondary
school curriculum. Drawing on the work of Kliebard
(c.f., Kliebard, 1995), Stanic provided a summary of
curriculum interest groups that influenced the position
of mathematics in the school curriculum: (a) the
humanists, who emphasized the traditional disciplines
of study found in Western philosophy; (b) the
developmentalists, who emphasized the “natural”
development of the child; (c) the social efficiency
educators, who emphasized a “scientific” approach that
led to the natural development of social stratification;
and (d) the social meliorists, who emphasized
education as a means of working toward social justice.
Stanic (1986) noted that mathematics educators, in
general, sided with the humanists, claiming:
“mathematics should be an important part of the school
curriculum” (p. 193). He also argued that the
development of the National Council of Teachers of
Mathematics (NCTM) in 1920 was partly in response
to the debate that surrounded the position of
mathematics within the school curriculum.
9
The founders of the Council wrote:
Mathematics courses have been assailed on every
hand. So-called educational reformers have
tinkered with the courses, and they, not knowing
the subject and its values, in many cases have
thrown out mathematics altogether or made it
entirely elective. …To help remedy the existing
situation the National Council of Teachers of
Mathematics was organized. (C. M. Austin as
quoted in Stanic, 1986, p. 198)
The backdrop to the mathematics education crisis
was the tremendous growth in school population that
occurred between 1890 and 1940—a growth of nearly
20 times (Stanic, 1986). This dramatic increase in the
student population yielded the belief that the overall
intellectual capabilities of students had decreased;
consequently, students became characterized as the
“army of incapables” (G. S. Hall as quoted in Stanic,
1986, p. 194). Stanic presented the results of this
prevailing belief by citing the 1933 National Survey of
Secondary Education, which concluded that less than
half of the secondary schools required algebra and
plane geometry. And, he illustrated mathematics
teachers’ perspectives by providing George Counts’
1926 survey of 416 secondary school teachers.
Eighteen of the 48 mathematics teachers thought that
fewer pupils should take mathematics, providing a
contrast to teachers of other academic disciplines who
believed that “their own subjects should be more
largely patronized” (G. S. Counts as quoted in Stanic,
p. 196). Even so, the issues of how mathematics should
be positioned in the school curriculum and who should
take advanced mathematics courses was not a major
national concern until the 1950s.
During the 1950s, mathematics education in U.S.
schools began to be attacked from many segments of
society: the business sector and military for graduating
students who lacked computational skills, colleges for
failing to prepare entering students with mathematics
knowledge adequate for college work, and the public
for having “watered down” the mathematics
curriculum as a response to progressivism (Kilpatrick,
1992). The launching of Sputnik in 1957 further
exacerbated these attacks leading to a national demand
for rigorous mathematics in secondary schools. This
demand spurred a variety of attempts to reform
mathematics education: “the ‘new’ math of the 1960s,
the ‘back-to-basic’ programs of the 1970s, and the
‘problem-solving’ focus of the 1980s” (Johnston,
1997). Within these programs of reform, the questions
were not only what mathematics should be taught and
10
how, but more importantly, who should be taught
mathematics.
The question of who should be taught mathematics
initially appeared in the debates of the 1920s and
centered on “ascertaining who was prepared for the
study of algebra” (Kilpatrick, 1992, p. 21). These
debates led to an increase in grouping students
according to their presumed mathematics ability. This
“ability” grouping often resulted in excluding female
students, poor students, and students of color from the
opportunity to enroll in advanced mathematics courses
(Oakes, 1985; Oakes, Ormseth, Bell, & Camp, 1990).
Sixty years after the beginning of the debates, the
recognition of this unjust exclusion from advanced
mathematics courses spurred the NCTM to publish the
Curriculum and Evaluation Standards for School
M a t h e m a t i c s (Standards, 1989) that included
statements similar to the following:
The social injustices of past schooling practices can
no longer be tolerated. Current statistics indicate
that those who study advanced mathematics are
most often white males. …Creating a just society
in which women and various ethnic groups enjoy
equal opportunities and equitable treatment is no
longer an issue. Mathematics has become a critical
filter for employment and full participation in our
society. We cannot afford to have the majority of
our population mathematically illiterate: Equity has
become an economic necessity. (p. 4)
In the Standards the NCTM contrasted societal
needs of the industrial age with those of the
information age, concluding that the educational goals
of the industrial age no longer met the needs of the
information age. They characterized the information
age as a dramatic shift in the use of technology which
had “changed the nature of the physical, life, and social
sciences; business; industry; and government” (p. 3).
The Council contended, “The impact of this
technological shift is no longer an intellectual
abstraction. It has become an economic reality” (p. 3).
The NCTM (1989) believed this shift demanded
new societal goals for mathematics education: (a)
mathematically literate workers, (b) lifelong learning,
(c) opportunity for all, and (d) an informed electorate.
They argued, “Implicit in these goals is a school
system organized to serve as an important resource for
all citizens throughout their lives” (p. 3). These goals
required those responsible for mathematics education
to strip mathematics from its traditional notions of
exclusion and basic computation and develop it into a
dynamic form of an inclusive literacy, particularly
given that mathematics had become a critical filter for
Mathematics as “Gate-Keeper” (?)
full employment and participation within a democratic
society. Countless other education scholars
(Frankenstein, 1995; Moses & Cobb, 2001; Secada,
1995; Skovsmose, 1994; Tate, 1995) have made
similar arguments as they recognize the need for all
students to be provided the opportunity to enroll in
advanced mathematics courses, arguing that a dynamic
mathematics literacy is a gatekeeper for economic
access, full citizenship, and higher education. In the
paragraphs that follow, I highlight quantitative and
qualitative studies that substantiate mathematics as a
gatekeeper.
The claims that mathematics is a “critical filter” or
gatekeeper to economic access, full citizenship, and
higher education are quantitatively substantiated by
two reports by the U. S. government: the 1997 White
Paper entitled Mathematics Equals Opportunity and
the 1999 follow-up summary of the 1988 National
Education Longitudinal Study (NELS: 88) entitled Do
Gatekeeper Courses Expand Education Options? The
U. S. Department of Education prepared both reports
based on data from the NELS: 88 samples of 24,599
eighth graders from 1,052 schools, and the 1992
follow-up study of 12,053 students.
In Mathematics Equals Opportunity, the following
statements were made:
In the United States today, mastering mathematics
has become more important than ever. Students
with a strong grasp of mathematics have an
advantage in academics and in the job market. The
8th grade is a critical point in mathematics
education. Achievement at that stage clears the
way for students to take rigorous high school
mathematics and science courses—keys to college
entrance and success in the labor force.
Students who take rigorous mathematics and
science courses are much more likely to go to
college than those who do not.
Algebra is the “gateway” to advanced mathematics
and science in high school, yet most students do
not take it in middle school.
Taking rigorous mathematics and science courses
in high school appears to be especially important
for low-income students.
Despite the importance of low-income students
taking rigorous mathematics and science courses,
these students are less likely to take them. (U. S.
Department of Education, 1997, pp. 5–6)
This report, based on statistical analyses, explicitly
stated that algebra was the “gateway” or gatekeeper to
advanced (i.e., rigorous) mathematics courses and that
David W. Stinson
advanced mathematics provided an advantage in
academics and in the job market—the same argument
provided by the NCTM and education scholars.
The statistical analyses in the report entitled, D o
Gatekeeper Courses Expand Educational Options? (U.
S. Department of Education, 1999) presented the
following findings:
Students who enrolled in algebra as eighth-graders
were more likely to reach advanced math courses
(e.g., algebra 3, trigonometry, or calculus, etc.) in
high school than students who did not enroll in
algebra as eighth-graders.
Students who enrolled in algebra as eighth-graders,
and completed an advanced math course during
high school, were more likely to apply to a fouryear college than those eighth-grade students who
did not enroll in algebra as eighth-graders, but who
also completed an advanced math course during
high school. (pp. 1–2)
The summary concluded that not all students who
took advanced mathematics courses in high school
enrolled in a four-year postsecondary school, although
they were more likely to do so—again confirming
mathematics as a gatekeeper.
Nicholas Lemann’s (1999) book The Big Test: The
Secret History of the American Meritocracy provides a
qualitative substantiation that mathematics is a
gatekeeper to economic access, full citizenship, and
higher education. In Parts I and II of his book, Lemann
presented a detailed historical narrative of the merger
between the Educational Testing Service with the
College Board. Leman argued this merger established
how mathematics would directly and indirectly
categorize Americans—becoming a gatekeeper—for
the remainder of the 20th and beginning of the 21st
centuries. During World War I, the United States War
Department (currently known as the Department of
Defense) categorized people using an adapted version
of Binet’s Intelligence Quotient test to determine the
entering rank and duties of servicemen. This
categorization evolved into ranking people by
“aptitude” through administering standardized tests in
contemporary U. S. education.
In Part III of his book, Lemann presented a casestudy characterization of contemporary Platonic
guardians, individuals who unjustly (or not) benefited
from the concept of aptitude testing and the ideal of
American meritocracy. Lemann argued that because of
their ability to demonstrate mathematics proficiency
(among other disciplines) on standardized tests, these
individuals found themselves passing through the gates
11
to economic access, full citizenship, and higher
education.
The concept of mathematics as providing the key
for passing through the gates to economic access, full
citizenship, and higher education is located in the core
of Western philosophy. In the United States, school
mathematics evolved from a discipline in “crisis” into
one that would provide the means of “sorting”
students. As student enrollment in public schools
increased, the opportunity to enroll in advanced
mathematics courses (the key) was limited because
some students were characterized as “incapable.”
Female students, poor students, and students of color
were offered a limited access to quality advanced
mathematics education. This limited access was a
motivating factor behind the Standards, and the
subsequent NCTM documents.3
NCTM and education scholars’ argument that
mathematics had and continues to have a gatekeeping
status has been confirmed both quantitatively and
qualitatively. Given this status, I pose two questions:
(a) Why does U.S. education not provide all students
access to a quality, advanced (mathematics) education
that would empower them with economic access and
full citizenship? and (b) How can we as mathematics
educators transform the status quo in the mathematics
classroom?
To fully engage in the first question demands a
deconstruction of the concepts of democratic public
schooling and American meritocracy and an analysis of
the morals and ethics of capitalism. To provide such a
deconstruction and analysis is beyond the scope of this
article. Nonetheless, I believe that Bowles’s
(1971/1977) argument provides a comprehensive, yet
condensed response to the question of why U. S.
education remains unequal without oversimplifying the
complexities of the question. Through a historical
analysis of schooling he revealed four components of
U. S. education: (a) schools evolved not in pursuit of
equality, but in response to the developing needs of
capitalism (e.g., a skilled and educated work force); (b)
as the importance of a skilled and educated work force
grew within capitalism so did the importance of
maintaining educational inequality in order to
reproduce the class structure; (c) from the 1920s to
1970s the class structure in schools showed no signs of
diminishment (the same argument can be made for the
1970s to 2000s); and (d) the inequality in education
had “its root in the very class structures which it serves
to legitimize and reproduce” (p. 137). He concluded by
writing: “Inequalities in education are thus seen as part
12
of the web of capitalist society, and likely to persist as
long as capitalism survives” (p. 137).
Although Bowles’s statements imply that only the
overthrow of capitalism will emancipate education
from its inequalities, I believe that developing
mathematics classrooms that are empowering to all
students might contribute to educational experiences
that are more equitable and just. This development may
also assist in the deconstruction of capitalism so that it
might be reconstructed to be more equitable and just.
The following discussion presents three theoretical
perspectives that I have identified as empowering
students. These perspectives aim to assist in more
equitable and just educative experiences for all
students: the situated perspective, the culturally
relevant perspective, and the critical perspective. I
believe these perspectives provide a plausible answer
to the second question asked above: How do we as
mathematics educators transform the status quo in the
mathematics classroom?
An Inclusive Empowering Mathematics Education
To frame the discussion that follows, I provide a
definition of e m p o w e r m e n t and empowering
mathematics. Freire (1970/2000) framed the notion of
empowerment within the concept of conscientização,
defined as “learning to perceive social, political and
economic contradictions, and to take action against the
oppressive elements of reality” (p. 35). He argued that
conscientização leads people not to “destructive
fanaticism” but makes it possible “for people to enter
the historical process as responsible Subjects4” (p. 36),
enrolling them in a search for self-affirmation.
Similarly, Lather (1991) defined empowerment as the
ability to perform a critical analysis regarding the
causes of powerlessness, the ability to identify the
structures of oppression, and the ability to act as a
single subject, group, or both to effect change toward
social justice. She claimed that empowerment is a
learning process one undertakes for oneself; “it is not
something done ‘to’ or ‘for’ someone” (Lather, 1991,
p. 4). In effect, empowerment provides the subject with
the skills and knowledge to make sociopolitical
critiques about her or his surroundings and to take
action (or not) against the oppressive elements of those
surroundings. The emphasis in both definitions is selfempowerment with an aim toward sociopolitical
critique. With this emphasis in mind, I next define
empowering mathematics.
Ernest (2002) provided three domains of
empowering mathematics—mathematical, social, and
epistemological—that assist in organizing how I define
Mathematics as “Gate-Keeper” (?)
empowering
mathematics.
Mathematical
empowerment relates to “gaining the power over the
language, skills and practices of using mathematics”
(section 1, ¶ 3) (e.g., school mathematics). Social
empowerment involves using mathematics as a tool for
sociopolitical critique, gaining power over the social
domains—“the worlds of work, life and social affairs”
(section 1, ¶ 4). And, epistemological empowerment
concerns the “individual’s growth of confidence in not
only using mathematics, but also a personal sense of
power over the creation and validation of
knowledge”(section 1, ¶ 5). Ernest argued, and I agree,
that all students gain confidence in their mathematics
skills and abilities through the use of mathematics in
routine and nonroutine ways and that this confidence
will logically lead to higher levels of mathematics
attainment. All students achieving higher levels of
attainment will assist in leveling the racial, gender, and
class imbalances that currently persist in advanced
mathematics courses. Effectively, Ernest’s definition of
empowering mathematics echoes the definition of
empowerment stated earlier.
Using Ernest’s three domains of empowering
mathematics as a starting point, I selected three
empowering mathematics perspectives. In doing so, I
kept in mind Stanic’s (1989) challenge to mathematics
educators: “If mathematics educators take seriously the
goal of equity, they must question not just the common
view of school mathematics but also their own takenfor-granted assumptions about its nature and worth” (p.
58). I believe that the situated perspective, culturally
relevant perspective, and critical perspective, in
varying degrees, motivate such questioning and
resonate with the definition I have given of
empowering mathematics. These configurations are
complex theoretical perspectives derived from multiple
scholars who sometimes have conflicting working
definitions. These perspectives, located in the “social
turn” (Lerman, 2000, p. 23) of mathematics education
research, originate outside the realm of “traditional”
mathematics education theory, in that they are rooted
in anthropology, cultural psychology, sociology, and
sociopolitical critique. In the discussion that follows, I
provide sketches of each theoretical perspective by
briefly summarizing the viewpoints of key scholars
working within the perspective. I then explain how
each perspective holds possibilities in transforming
gatekeeping mathematics from an exclusive instrument
for stratification into an inclusive instrument for
empowerment.
David W. Stinson
The Situated Perspective
The situated perspective is the coupling of
scholarship from cultural anthropology and cultural
psychology. In the situated perspective, learning
becomes a process of changing participation in
changing communities of practice in which an
individual’s resulting knowledge becomes a function
of the environment in which she or he operates.
Consequently, in the situated perspective, the dualisms
of mind and world are viewed as artificial constructs
(Boaler, 2000b). Moreover, the situated perspective, in
contrast to constructivist perspectives, emphasizes
interactive systems that are larger in scope than the
behavioral and cognitive processes of the individual
student.
Mathematics knowledge in the situated perspective
is understood as being co-constituted in a community
within a context. It is the community and context in
which the student learns the mathematics that
significantly impacts how the student uses and
understands the mathematics (Boaler, 2000b). Boaler
(1993) suggested that learning mathematics in contexts
assists in providing student motivation and interest and
enhances transference of skills by linking classroom
mathematics with real-world mathematics. She argued,
however, that learning mathematics in contexts does
not mean learning mathematics ideas and procedures
by inserting them into “real-world” textbook problems
or by extending mathematics to larger real-world class
projects. Rather, she suggested that the classroom itself
becomes the context in which mathematics is learned
and understood: “If the students’ social and cultural
values are encouraged and supported in the
mathematics classroom, through the use of contexts or
through an acknowledgement of personal routes and
direction, then their learning will have more meaning”
(p. 17).
The situated perspective offers different notions of
what it means to have mathematics ability, changing
the concept from “either one has mathematics ability or
not” to an analysis of how the environment coconstitutes the mathematics knowledge that is learned
(Boaler, 2000a). Boaler argued that this change in how
mathematics ability is assessed in the situated
perspective could “move mathematics education away
from the discriminatory practices that produce more
failures than successes toward something considerably
more equitable and supportive of social justice” (p.
118).
13
The Culturally Relevant Perspective
Working toward social justice is also a component
of the culturally relevant perspective. Ladson-Billings
(1994) developed the “culturally relevant” (p. 17)
perspective as she studied teachers who were
successful with African-American children. This
perspective is derived from the work of cultural
anthropologists who studied the cultural disconnects
between (White) teachers and students of color and
made suggestions about how teachers could “match
their teaching styles to the culture and home
backgrounds of their students” (Ladson-Billings, 2001,
p. 75). Ladson-Billings defined the culturally relevant
perspective as promoting student achievement and
success through cultural competence (teachers assist
students in developing a positive identification with
their home culture) and through sociopolitical
consciousness (teachers help students develop a civic
and social awareness in order to work toward equity
and social justice).
Teachers working from a culturally relevant
perspective (a) demonstrate a belief that children can
be competent regardless of race or social class, (b)
provide students with scaffolding between what they
know and what they do not know, (c) focus on
instruction during class rather than busy-work or
behavior management, (d) extend students’ thinking
beyond what they already know, and (e) exhibit indepth knowledge of students as well as subject matter
(Ladson-Billings, 1995). Ladson-Billings argued that
all children “can be successful in mathematics when
their understanding of it is linked to meaningful
cultural referents, and when the instruction assumes
that all students are capable of mastering the subject
matter” (p. 141).
Mathematics knowledge in the culturally relevant
perspective is viewed as a version of
ethnomathematics— ethno defined as all culturally
identifiable groups with their jargons, codes, symbols,
myths, and even specific ways of reasoning and
inferring; mathema defined as categories of analysis;
and tics defined as methods or techniques (D’
Ambrosio, 1985/1997, 1997). In the culturally relevant
mathematics classroom, the teacher builds from the
students’ ethno or informal mathematics and orients
the lesson toward their culture and experiences, while
developing the students’ critical thinking skills
(Gutstein, Lipman, Hernandez, & de los Reyes, 1997).
The positive results of teaching from a culturally
relevant perspective are realized when students
develop mathematics empowerment: deducing
mathematical generalizations and constructing creative
14
solution methods to nonroutine problems, and
perceiving mathematics as a tool for sociopolitical
critique (Gutstein, 2003).
The Critical Perspective
Perceiving mathematics as a tool for sociopolitical
critique is also a feature of the critical perspective. This
perspective is rooted in the social and political critique
of the Frankfurt School (circa 1920) whose
membership included but was not limited to Max
Horkheimer, Theodor Adorno, Leo Lowenthal, and
Franz Neumann. The critical perspective is
characterized as (a) providing an investigation into the
sources of knowledge, (b) identifying social problems
and plausible solutions, and (c) reacting to social
injustices. In providing these most general and
unifying characteristics of a critical education,
Skovsmose (1994) noted, “A critical education cannot
be a simple prolongation of existing social
relationships. It cannot be an apparatus for prevailing
inequalities in society. To be critical, education must
react to social contradictions” (p. 38).
Skovsmose (1994), drawing from Freire’s
(1970/2000) popularization of the concept
c o n s c i e n t i z a ç ã o and his work in literacy
empowerment, derived the term “mathemacy” (p. 48).
Skovsmose claimed that since modern society is highly
technological and the core of all modern-day
technology is mathematics that mathemacy is a means
of empowerment. He stated, “If mathemacy has a role
to play in education, similar to but not identical to the
role of literacy, then mathemacy must be seen as
composed of different competences: a mathematical, a
technological, and a reflective” (p. 48).
In the critical perspective, mathematics knowledge
is seen as demonstrating these three competencies
(Skovsmose, 1994). Mathematical competence is
demonstrating proficiency in the normally understood
skills of school mathematics, reproducing and
mastering various theorems, proofs, and algorithms.
Technological competence demonstrates proficiency in
applying mathematics in model building, using
mathematics in pursuit of different technological aims.
And, reflective competence achieves mathematics’
critical dimension, reflecting upon and evaluating the
just and unjust uses of mathematics. Skovsmose
contended that mathemacy is a necessary condition for
a politically informed citizenry and efficient labor
force, claiming that mathemacy provides a means for
empowerment in organizing and reorganizing social
and political institutions and their accompanying
traditions.
Mathematics as “Gate-Keeper” (?)
Transforming Gatekeeping Mathematics
The preceding sketches demonstrate that these
three theoretical perspectives approach mathematics
and mathematics teaching and learning differently than
traditional perspectives. All three perspectives, in
varying degrees, question the taken-for-granted
assumptions about mathematics and its nature and
worth, locate the formation of mathematics knowledge
within the social community, and argue that
mathematics is an indispensable instrument used in
sociopolitical critique. In the following paragraphs I
explicate the degrees to which the three perspectives
address these issues.
The situated perspective negates the assumption
that mathematics is a contextually free discipline,
contending that it is the context in which mathematics
is learned that determines how it will be used and
understood. The culturally relevant perspective negates
the assumption that mathematics is a culturally free
discipline, recognizing mathematics is not separate
from culture but is a product of culture. The critical
perspective redefines the worth of mathematics
through an acknowledgment and critical examination
of the just and, often overlooked, unjust uses of
mathematics.
The situated perspective locates mathematics
knowledge in the social community. In this
perspective, mathematics is not learned from a
mathematics textbook and then applied to real-world
contexts, but is negotiated in communities that exist in
real-world contexts. The culturally relevant perspective
also locates mathematics knowledge in the social
community. This perspective argues teachers should
begin to build on the collective mathematics
knowledge present in the classroom communities,
moving toward mathematics found in textbooks. The
critical perspective does not locate mathematics
knowledge in the social community but is oriented
towards using mathematics to critique and transform
the social and political communities in which
mathematics exists and has its origins.
The situated perspective posits that students will
begin to understand mathematics as a discipline that is
learned in the context of communities. It is in this way
that students may learn how mathematics can be
applied in uncovering the inequities and injustices
present in communities or can be used for
sociopolitical critique. Similarly, one of the two tenets
of the culturally relevant perspective is for the teacher
to assist students in developing a sociopolitical
consciousness. Finally, using mathematics as a means
for sociopolitical critique is essential to the critical
David W. Stinson
perspective, as mathematics is understood as a tool that
can be used for critique.
How do the three aspects of mathematics and
mathematics teaching and learning relate to each other
in these perspectives and how does this relationship
address the three domains of empowering
mathematics? First, mathematics empowerment is
achieved because each perspective questions the
assumptions that are often taken-for-granted about the
nature and worth of mathematics. Although all three
perspectives see value in the study of mathematics,
including “academic”5 mathematics, they differ from
traditional perspectives in that academic mathematics
itself is troubled6 with regards to its contextual
existence, its cultural connectedness, and its critical
utility. Second, students achieve social empowerment
because all three perspectives argue that students
should engage in mathematics contextually and
culturally; and, therefore students have the opportunity
to gain confidence in using mathematics in routine and
nonroutine problems. The advocates for these three
perspectives argue that as students expand the use of
mathematics into nonroutine problems, they become
cognizant of how mathematics can be used as a tool for
sociopolitical critique. Finally students achieve
epistemological empowerment because all three
perspectives trouble academic mathematics that in turn
may lead students to understand that the concept of a
“true” or “politically-free” mathematics is a fiction.
Students will hopefully understand that mathematics
knowledge is (and always has been) a contextually and
culturally (and politically) constructed human
endeavor. If students achieve this perspective of
mathematics, they will better understand their role as
producers of mathematics knowledge, not just
consumers. Hence, the three domains of empowering
mathematics—mathematical,
social,
and
epistemological—are achieved in each perspective or
through various combinations of the three perspectives.
The chief aim of an empowering mathematics is to
transform gatekeeping mathematics from a discipline
of oppressive exclusion into a discipline of
empowering inclusion. (This aim is inclusive of
mathematics educators and education researchers.)
Empowering inclusion is achieved when students (and
teachers of mathematics) are presented with the
opportunity to learn that the foundations of
mathematics can be troubled. This troubling of
mathematics’ foundations transforms the discourse in
the mathematics classroom from a discourse of
transmitting mathematics to a “chosen” few students,
into a discourse of exploring mathematics with all
15
students. Empowering inclusion is achieved when
students (and teachers of mathematics) are presented
with the opportunity to learn that, similar to literacy,
mathemacy is a tool that can be used to reword worlds.
This rewording of worlds (Freire, 1970/2000) with
mathematics transforms mathematics from a tool used
by a few students in “mathematical” pursuits, into a
tool used by all students in sociopolitical pursuits.
Finally, empowering inclusion is achieved when
students (and teachers of mathematics) are presented
with the opportunity to learn that mathematics
knowledge is constructed human knowledge. This
returning to the origins of mathematics knowledge
transforms mathematics from an Ideal of the gods
reproduced by a few students, into a human endeavor
produced by all students.
Concluding Thoughts
The concept of mathematics as gatekeeper has a
very long and disturbing history. There have been
educators satisfied with the gatekeeping status of
mathematics and those that have questioned not only
its gatekeeping status but also its nature and worth. In
my thinking about mathematics as a gatekeeper and the
possibility of transforming mathematics education, I
often reflect on Foucault’s challenge. He challenged us
to think the un-thought, to think: “how is it that one
particular statement appeared rather than another?”
(Foucault, 1969/1972, p. 27). With Foucault’s
challenge in mind, I often think what if Plato had said,
We shall persuade those who are to perform high
functions in the city to undertake ________, but
not as amateurs. They should persist in their studies
until they reach the level of pure thought, where
they will be able to contemplate the very nature of
________…. it should serve the purposes of war
and lead the soul away from the world of
appearances toward essence and reality. (trans.
1996, p. 219)
In the preceding blanks, I insert different human
pursuits, such as writing, speaking, painting, sculpting,
dancing, and so on, asking: does mathematics really
lead the soul away from the world of appearances
toward essence and reality?7 Or could dancing, for
example, achieve the same result? While rethinking
Plato’s centuries old comment, I rethink the privileged
status of mathematics as a gatekeeper (and as an
instrument of stratification). But rather than asking
what is school mathematics as gatekeeper or what does
it mean, I ask different questions: How does school
mathematics as gatekeeper function? Where is school
mathematics as gatekeeper to be found? How does
16
school mathematics as gatekeeper get produced and
regulated? How does school mathematics as
gatekeeper exist? (Bové, 1995). These questions
transform the discussions around gatekeeper
mathematics from discussions that attempt to find
meaning in gatekeeper mathematics to discussions that
examine the ethics of gatekeeper mathematics. Implicit
in this examination is an analysis of how the structure
of schools and those responsible for that structure are
implicated (or not) in reproducing the unethical effects
of gatekeeping mathematics.
Will asking the questions noted above transform
gatekeeping mathematics from an exclusive instrument
for stratification into an inclusive instrument for
empowerment? Will asking these questions stop the
“ability” sorting of eight-year-old children? Will
asking these questions encourage mathematics teachers
(and educators) to adopt the situated, culturally
relevant, or critical perspectives, perspectives that aim
toward empowering all children with a key? Although
I believe that there are no definitive answers to these
questions, I do believe that critically examining (and
implementing) the different possibilities for
mathematics teaching and learning found in the
theoretical perspectives explained in this article
provides a sensible beginning to transforming
mathematics education. In closing, I fervently proclaim
the way we use mathematics today in our nation’s
schools must stop! Mathematics should not be used as
an instrument for stratification but rather an instrument
for empowerment!
REFERENCES
Boaler, J. (2000a). Exploring situated insights into research and
learning. Journal for Research in Mathematics Education,
31(1), 113–119.
Boaler, J. (2000b). Mathematics from another world: Traditional
communities and the alienation of learners. Journal of
Mathematical Behavior, 18(4), 379–397.
Boaler, J. (1993). The role of context in the mathematics
classroom: Do they make mathematics more “real”? For the
Learning of Mathematics, 13(2), 12–17.
Bové, P. A. (1995). Discourse. In F. Lentricchia & T. McLaughlin
(Eds.), Critical terms for literary study (pp. 50–65). Chicago:
University of Chicago Press.
Bowles, S. (1977). Unequal education and the reproduction of the
social division of labor. In J. Karabel & A. H. Halsey (Eds.),
Power and ideology in education (pp. 137–153). New York:
Oxford University Press. (Original work published in 1971)
D’Ambrosio, U. (1997). Ethnomathematics and its place in the
history and pedagogy of mathematics. In A. B. Powell & M.
Frankenstein (Eds.), Ethnomathematics: Challenging
Eurocentrism in mathematics education (pp. 13–24). Albany:
State University of New York Press. (Original work published
in 1985)
Mathematics as “Gate-Keeper” (?)
D’Ambrosio, U. (1997). Foreword. In A. B. Powell & M.
Frankenstein (Eds.), Ethnomathematics: Challenging
Eurocentrism in mathematics education (pp. xv–xxi). Albany:
State University of New York Press.
Moses, R. P., & Cobb, C. E. (2001). Radical equations: Math
literacy and civil rights. Boston: Beacon Press.
Derrida, J. (1997). Of grammatology (Corrected ed.). Baltimore:
Johns Hopkins University Press. (Original work published
1974)
National Council of Teachers of Mathematics. (1995). Assessment
standards for school mathematics. Reston, VA: Author.
Ernest, P. (2002). Empowerment in mathematics education.
Philosophy of Mathematics Education, 15. Retrieved January
26, 2004, from
http://www.ex.ac.uk/~PErnest/pome15/empowerment.htm
Foucault, M. (1972). The archaeology of knowledge (1st American
ed.). New York: Pantheon Books. (Original work published
1969)
Frankenstein, M. (1995). Equity in mathematics education: Class in
the world outside the class. In W. G. Secada, E. Fennema, &
L. Byrd (Eds.), New directions for equity in mathematics
education (pp. 165–190). Cambridge: Cambridge University
Press.
Freire, P. (2000). Pedagogy of the oppressed (30th anniversary
ed.). New York: Continuum. (Original work published 1970)
Gutstein, E. (2003). Teaching and learning mathematics for social
justice in an urban, Latino school. Journal for Research in
Mathematics Education, 34(1), 37–73.
Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997).
Culturally relevant mathematics teaching in a Mexican
American context. Journal for Research in Mathematics
Education, 28(6), 709–737.
Johnston, H. (1997). Teaching mathematics for understanding:
Strategies and activities (Unpublished manuscript). Atlanta:
Georgia State University.
Kilpatrick, J. (1992). A history of research in mathematics
education. In D. A. Grouws (Ed.), Handbook of research on
mathematics teaching and learning (pp. 3–38). New York:
Macmillan.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional
standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1989). Curriculum
and evaluation standards for school mathematics. Reston,
VA: Author.
Oakes, J. (1985). Keeping track: How schools structure inequality.
New Haven: Yale University Press.
Oakes, J., Ormseth, T., Bell, R., & Camp, P. (1990). Multiplying
inequalities: The effects of race, social class, and tracking on
opportunities to learn mathematics and science. Santa
Monica, CA: Rand Corporation.
Plato. (trans. 1996). The republic (R. W. Sterling & W. C. Scott,
Trans.) (Paperback ed.). New York: Norton.
Secada, W. G. (1995). Social and critical dimensions for equity in
mathematics education. In W. G. Secada, E. Fennema, & L. B.
Adajian (Eds.), New directions for equity in mathematics
education (pp. 146–164). Cambridge: Cambridge University
Press.
Skovsmose, O. (1994). Towards a critical mathematics education.
Educational Studies in Mathematics, 27, 35–57.
Spivak, G. C. (1997). Translator’s preface. In J. Derrida (Ed.), Of
grammatology (Corrected ed., pp. ix - lxxxvii). Baltimore:
Johns Hopkins University Press. (Original work published
1974)
Stanic, G. M. A. (1989). Social inequality, cultural discontinuity,
and equity in school mathematics. Peabody Journal of
Education, 66(2), 57–71.
Kliebard, H. M. (1995). The struggle for the American curriculum.
New York: Routledge.
Stanic, G. M. A. (1986). The growing crisis in mathematics
education in the early twentieth century. Journal for Research
in Mathematics Education, 17(3), 190–205.
Ladson-Billings, G. (2001). The power of pedagogy: Does teaching
matter? In W. H. Watkins, J. H. Lewis, & V. Chou (Eds.),
Race and education: The roles of history and society in
educating African American students (pp. 73–88). Boston:
Allyn & Bacon.
Tate, W. F. (1995). Economics, equity, and the national
mathematics assessment: Are we creating a national toll road?
In W. G. Secada, E. Fennema, & L. Byrd (Eds.), New
directions for equity in mathematics education (pp. 191–206).
Cambridge: Cambridge University Press.
Ladson-Billings, G. (1995). Making mathematics meaningful in a
multicultural context. In W. G. Secada, E. Fennema, & L.
Byrd (Eds.), New directions for equity in mathematics
education (pp. 126–145). Cambridge: Cambridge University
Press.
Ladson-Billings, G. (1994). The Dreamkeepers: Successful
teachers of African American children. San Francisco: JosseyBass.
Lather, P. (1991). Getting smart: Feminist research and pedagogy
with/in the postmodern. New York: Routledge.
Lemann, N. (1999). The big test: The secret history of the
American meritocracy (1st ed.). New York: Farrar Straus and
Giroux.
Lerman, S. (2000). The social turn in mathematics education
research. In J. Boaler (Ed.), International perspectives on
mathematics education, (pp. 19–44). Westport, CT: Ablex.
David W. Stinson
U.S. Department of Education. (1997). Mathematics equals
opportunity. White Paper prepared for U.S. Secretary of
Education Richard W. Riley. Retrieved January 26, 2004,
from http://www.ed.gov/pubs/math/mathemat.pdf
U.S. Department of Education. (1999). Do gatekeeper courses
expand education options? National Center for Education
Statistics. Retrieved January 26, 2004, from
http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=1999303
1
The student racial/ethnic data were based on the 2001-2002
Georgia Public Education Report Card; the racial/ethnic
classifications were the State of Georgia’s not this author’s. For
details of racial/ethnic data on all schools in the State of Georgia
see: http://techservices.doe.k12.ga.us/reportcard/
2
Plato (trans. 1996) in establishing his utopian Republic imagined
that the philosopher guardians of the city, identified as the
17
aristocracy, would be children taken from their parents at an early
age and educated at the academy until of age when they would
dutifully rule as public servants and not for personal gain. Plato
believed that these children would be from all classes: “it may
sometimes happen that a silver child will be born of a golden
parent, a golden child from a silver parent and so on” (p. 113); and
from both sexes: “we must conclude that sex cannot be the criterion
in appointments to government positions…there should be no
differentiation” (pp. 146-147). However, Plato’s concept of
aristocracy has been greatly misinterpreted within Western
ideology. The concept has historically and consistently favored the
social positionality of the White heterosexual Christian male of
bourgeois privilege.
3
Throughout the remainder of this article the term NCTM
documents designates the Professional Standards for Teaching
M a t h e m a t i c s (1991), Assessment Standards for School
Mathematics (1995), Principles and Standards for School
Mathematics (2000), and the Curriculum and Evaluation Standards
for School Mathematics (1989).
4
Freire (1970/2000) defined the term Subjects, with a capital S, as
“those who know and act, in contrast to objects, which are known
and acted upon” (p. 36).
5
I define the term “academic” mathematics as D`Ambrosio
(1997) defined the term: mathematics that is taught and learned in
schools, differentiated from ethnomathematics.
6
In this context, I use the term trouble to place academic
mathematics under erasure. Spivak (1974/1997) explained
Derrida’s (1974/1997) sous rature, that is, under erasure, as
learning “to use and erase our language at the same time” (p. xviii).
She claimed that Derrida is “acutely aware… [of] the strategy of
using the only available language while not subscribing to its
premises, or ‘operat[ing] according to the vocabulary of the very
thing that one delimits’ (MP 18, SP 147)” (p. xviii). In other words,
I argue that these three perspectives, while purporting the teaching
of the procedures and concepts of academic mathematics (i.e., the
language of mathematics), also place it sous rature so as not to
limit the mathematics creativity and engagement of all students.
7
Even though I trouble Plato’s remark regarding “essence and
reality,” the purpose of this article is not to engage in that
argument, an argument that I believe will be my life’s work.
18
Mathematics as “Gate-Keeper” (?)
The Mathematics Educator
2004, Vol. 14, No. 1, 19–34
The Characteristics of Mathematical Creativity
Bharath Sriraman
Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the
creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In
order to investigate how mathematicians create mathematics, a qualitative study involving five creative
mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes
involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview
transcripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians’
creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification. It
was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of
mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and
used to interpret the characteristics of mathematical creativity
.
Mathematical creativity ensures the growth of the
field of mathematics as a whole. The constant increase
in the number of journals devoted to mathematics
research bears evidence to the growth of mathematics.
Yet what lies at the essence of this growth, the
creativity of the mathematician, has not been the
subject of much research. It is usually the case that
most mathematicians are uninterested in analyzing the
thought processes that result in mathematical creation
(Ervynck, 1991). The earliest known attempt to study
mathematical creativity was an extensive questionnaire
published in the French periodical L'Enseigement
Mathematique (1902). This questionnaire and a lecture
on creativity given by the renowned 20th century
mathematician Henri Poincaré to the Societé de
Psychologie inspired his colleague Jacques Hadamard,
another prominent 20th century mathematician, to
investigate the psychology of mathematical creativity
(Hadamard, 1945). Hadamard (1945) undertook an
informal inquiry among prominent mathematicians and
scientists in America, including George Birkhoff,
George Polya, and Albert Einstein, about the mental
images used in doing mathematics. Hadamard (1945),
influenced by the Gestalt psychology of his time,
theorized that mathematicians’ creative processes
followed the four-stage Gestalt model (Wallas, 1926)
of preparation-incubation-illumination-verification.
As we will see, the four-stage Gestalt model is a
characterization of the mathematician's creative
process, but it does not define creativity per se. How
Bharath Sriraman is an assistant professor of mathematics and
mathematics education at the University of Montana. His
publications and research interests are in the areas of cognition,
foundational issues, mathematical creativity, problem-solving,
proof, and gifted education.
Bharath Sriraman
does one define creativity? In particular what exactly is
mathematical creativity? Is it the discovery of a new
theorem by a research mathematician? Does student
discovery of a hitherto known result also constitute
creativity? These are among the areas of exploration in
this paper.
The Problem Of Defining Creativity
Mathematical creativity has been simply described
as discernment, or choice (Poincaré, 1948). According
to Poincaré (1948), to create consists precisely in not
making useless combinations and in making those
which are useful and which are only a small minority.
Poincaré is referring to the fact that the “proper”
combination of only a small minority of ideas results in
a creative insight whereas a majority of such
combinations does not result in a creative outcome.
This may seem like a vague characterization of
mathematical creativity. One can interpret Poincaré's
"choice" metaphor to mean the ability of the
mathematician to choose carefully between questions
(or problems) that bear fruition, as opposed to those
that lead to nothing new. But this interpretation does
not resolve the fact that Poincaré’s definition of
creativity overlooks the problem of novelty. In other
words, characterizing mathematical creativity as the
ability to choose between useful and useless
combinations is akin to characterizing the art of
sculpting as a process of cutting away the unnecessary!
Poincaré's (1948) definition of creativity was a
result of the circumstances under which he stumbled
upon deep results in Fuchsian functions. The first stage
in creativity consists of working hard to get an insight
into the problem at hand. Poincaré (1948) called this
the preliminary period of conscious work. This period
is also referred to as the preparatory stage (Hadamard,
19
1945). In the second, or incubatory, stage (Hadamard,
1945), the problem is put aside for a period of time and
the mind is occupied with other problems. In the third
stage the solution suddenly appears while the
mathematician is perhaps engaged in other unrelated
activities. "This appearance of sudden illumination is a
manifest sign of long, unconscious prior work"
(Poincaré, 1948). Hadamard (1945) referred to this as
the illuminatory stage. However, the creative process
does not end here. There is a fourth and final stage,
which consists of expressing the results in language or
writing. At this stage, one verifies the result, makes it
precise, and looks for possible extensions through
utilization of the result. The “Gestalt model” has some
shortcomings. First, the model mainly applies to
problems that have been posed a priori by
mathematicians, thereby ignoring the fascinating
process by which the actual questions evolve.
Additionally, the model attributes a large portion of
what “happens” in the incubatory and illuminatory
phases to subconscious drives. The first of these
shortcomings, the problem of how questions are
developed, is partially addressed by Ervynck (1991) in
his three-stage model.
Ervynck (1991) described mathematical creativity
in terms of three stages. The first stage (Stage 0) is
referred to as the preliminary technical stage, which
consists of "some kind of technical or practical
application of mathematical rules and procedures,
without the user having any awareness of the
theoretical foundation" (p. 42). The second stage
(Stage 1) is that of algorithmic activity, which consists
primarily of performing mathematical techniques, such
as explicitly applying an algorithm repeatedly. The
third stage (Stage 2) is referred to as c r e a t i v e
(conceptual, constructive) activity. This is the stage in
which true mathematical creativity occurs and consists
of non-algorithmic decision making. "The decisions
that have to be taken may be of a widely divergent
nature and always involve a choice" (p. 43). Although
Ervynck (1991) tries to describe the process by which a
mathematician arrives at the questions through his
characterizations of Stage 0 and Stage 1, his
description of mathematical creativity is very similar to
those of Poincaré and Hadamard. In particular his use
of the term “non-algorithmic decision making” is
analogous to Poincaré’s use of the “choice” metaphor.
The mathematics education literature indicates that
very few attempts have been made to explicitly define
mathematical creativity. There are references made to
creativity by the Soviet researcher Krutetskii (1976) in
the context of students’ abilities to abstract and
20
generalize mathematical content. There is also an
outstanding example of a mathematician (George
Polya) attempting to give heuristics to tackle problems
in a manner akin to the methods used by trained
mathematicians. Polya (1954) observed that in "trying
to solve a problem, we consider different aspects of it
in turn, we roll it over and over in our minds; variation
of the problem is essential to our work." Polya (1954)
emphasized the use of a variety of heuristics for
solving mathematical problems of varying complexity.
In examining the plausibility of a mathematical
conjecture, mathematicians use a variety of strategies.
In looking for conspicuous patterns, mathematicians
use such heuristics as (1) verifying consequences, (2)
successively verifying several consequences, (3)
verifying an improbable consequence, (4) inferring
from analogy, and (5) deepening the analogy.
As is evident in the preceding paragraphs, the
problem of defining creativity is by no means an easy
one. However, psychologists’ renewed interest in the
phenomenon of creativity has resulted in literature that
attempts to define and operationalize the word
“creativity.” Recently psychologists have attempted to
link creativity to measures of intelligence (Sternberg,
1985) and to the ability to abstract, generalize
(Sternberg, 1985), and solve complex problems
(Frensch & Sternberg, 1992). Sternberg and Lubart
(2000) define creativity as the ability to produce
unexpected original work that is useful and adaptive.
Mathematicians would raise several arguments with
this definition, simply because the results of creative
work may not always have implications that are
“useful” in terms of applicability in the contemporary
world. A recent example that comes to mind is Andrew
Wiles’ proof of Fermat’s Last Theorem. The
mathematical community views his work as creative. It
was unexpected and original but had no applicability in
the sense Sternberg and Lubart (2000) suggest. Hence,
I think it is sufficient to define creativity as the ability
to produce novel or original work, which is compatible
with my personal definition of mathematical creativity
as the process that results in unusual and insightful
solutions to a given problem, irrespective of the level
of complexity. In the context of this study involving
professional mathematicians, mathematical creativity is
defined as the publication of original results in
prominent mathematics research journals.
The Motivation For Studying Creativity
The lack of recent mathematics education literature
on the subject of creativity was one of the motivations
for conducting this study. Fifteen years ago Muir
Mathematical Creativity
(1988) invited mathematicians to complete a modified
and updated version of the survey that appeared in
L'Enseigement Mathematique (1902) but the results of
this endeavor are as yet unknown. The purpose of this
study was to gain insight into the nature of
mathematical creativity. I was interested in distilling
common attributes of the creative process to see if
there were any underlying themes that characterized
mathematical creativity. The specific questions of
exploration in this study were:
Is the Gestalt model of mathematical creativity still
applicable today?
What are the characteristics of the creative process
in mathematics?
Does the study of mathematical creativity have any
implications for the classroom?
Literature Review
Any study on the nature of mathematical creativity
begs the question as to whether the mathematician
discovers or invents mathematics. Therefore, this
review begins with a brief description of the four most
popular viewpoints on the nature of mathematics. This
is followed by a comprehensive review of
contemporary models of creativity from psychology.
The Nature of Mathematics
Mathematicians actively involved in research have
certain beliefs about the ontological nature of
mathematics that influence their approach to research
(Davis & Hersh, 1981; Sriraman, 2004a). The Platonist
viewpoint is that mathematical objects exist prior to
their discovery and that “any meaningful question
about a mathematical object has a definite answer,
whether we are able to determine it or not” (Davis &
Hersh, 1981). According to this view, mathematicians
do not invent or create mathematics - they discover
mathematics. Logicists hold that “all concepts of
mathematics can ultimately be reduced to logical
concepts” which implies that “all mathematical truths
can be proved from the axioms and rules of inference
and logic alone” (Ernest, 1991). Formalists do not
believe that mathematics is discovered; they believe
mathematics is simply a game, created by
mathematicians, based on strings of symbols that have
no meaning (Davis & Hersh, 1981).
Constructivism (incorporating Intuitionism) is one
of the major schools of thought (besides Platonism,
Logicism and Formalism) that arose due to the
contradictions that emerged in the development of the
theory of sets and the theory of functions during the
early part of the 20th century. The constructivist
Bharath Sriraman
(intuitionist) viewpoint is that “human mathematical
activity is fundamental in the creation of new
knowledge and that both mathematical truths and the
existence of mathematical objects must be established
by constructive methods" (Ernest, 1991, p. 29).
Contradictions like Russell’s Paradox were a major
blow to the absolutist view of mathematical
knowledge, for if mathematics is certain and all its
theorems are certain, how can there be contradictions
among its theorems? The early constructivists in
mathematics were the intuitionists Brouwer and
Heyting. Constructivists claim that both mathematical
truths and the existence of mathematical objects must
be established by constructivist methods.
The question then is how does a mathematician go
about conducting mathematics research? Do the
questions appear out of the blue, or is there a mode of
thinking or inquiry that leads to meaningful questions
and to the methodology for tackling these questions? I
contend that the types of questions asked are
determined to a large extent by the culture in which the
mathematician lives and works. Simply put, it is
impossible for an individual to acquire knowledge of
the external world without social interaction.
According to Ernest (1994) there is no underlying
metaphor for the wholly isolated individual mind.
Instead, the underlying metaphor is that of persons in
conversation, persons who participate in meaningful
linguistic interaction and dialogue (Ernest, 1994).
Language is the shaper, as well as being the
“summative” product, of individual minds
(Wittgenstein, 1978). The recent literature in
psychology acknowledges these social dimensions of
human activity as being instrumental in the creative
process.
The Notion of Creativity in Psychology
As stated earlier, research on creativity has been on
the fringes of psychology, educational psychology, and
mathematics education. It is only in the last twenty-five
years that there has been a renewed interest in the
phenomenon of creativity in the psychology
community. The Handbook of Creativity (Sternberg,
2000), which contains a comprehensive review of all
research then available in the field of creativity,
suggests that most of the approaches used in the study
of creativity can be subsumed under six categories:
mystical, pragmatic, psychodynamic, psychometric,
cognitive, and social-personality. Each of these
approaches is briefly reviewed.
21
The mystical approach
The mystical approach to studying creativity
suggests that creativity is the result of divine
inspiration or is a spiritual process. In the history of
mathematics, Blaise Pascal claimed that many of his
mathematical insights came directly from God. The
renowned 19th century algebraist Leopold Kronecker
said that “God made the integers, all the rest is the
work of man” (Gallian, 1994). Kronecker believed that
all other numbers, being the work of man, were to be
avoided; and although his radical beliefs did not attract
many supporters, the intuitionists advocated his beliefs
about constructive proofs many years after his death.
There have been attempts to explore possible
relationships between mathematicians’ beliefs about
the nature of mathematics and their creativity (Davis
and Hersh, 1981; Hadamard, 1945; Poincaré, 1948;
Sriraman, 2004a). These studies indicate that such a
relationship does exist. It is commonly believed that
the neo-Platonist view is helpful to the research
mathematician because of the innate belief that the
sought after result/relationship already exists.
The pragmatic approach
The pragmatic approach entails “being concerned
primarily with developing creativity” (Sternberg, 2000,
p. 5), as opposed to understanding it. Polya’s (1954)
emphasis on the use of a variety of heuristics for
solving mathematical problems of varying complexity
is an example of a pragmatic approach. Thus,
heuristics can be viewed as a decision-making
mechanism which leads the mathematician down a
certain path, the outcome of which may or may not be
fruitful. The popular technique of brainstorming, often
used in corporate or other business settings, is another
example of inducing creativity by seeking as many
ideas or solutions as possible in a non-critical setting.
The psychodynamic approach
The psychodynamic approach to studying
creativity is based on the idea that creativity arises
from the tension between conscious reality and
unconscious drives (Hadamard, 1945; Poincaré, 1948,
Sternberg, 2000, Wallas, 1926; Wertheimer, 1945).
The four-step Gestalt model (preparation-incubationillumination-verification) is an example of the use of a
psychodynamic approach to studying creativity. It
should be noted that the gestalt model has served as
kindling for many contemporary problem-solving
models (Polya, 1945; Schoenfeld, 1985; Lester, 1985).
Early psychodynamic approaches to creativity were
used to construct case studies of eminent creators such
22
as Albert Einstein, but the behaviorists criticized this
approach because of the difficulty in measuring
proposed theoretical constructs.
The psychometric approach
The psychometric approach to studying creativity
entails quantifying the notion of creativity with the aid
of paper and pencil tasks. An example of this would be
the Torrance Tests of Creative Thinking, developed by
Torrance (1974), that are used by many gifted
programs in middle and high schools to identify
students that are gifted/creative. These tests consist of
several verbal and figural tasks that call for problemsolving skills and divergent thinking. The test is scored
for fluency, flexibility, originality (the statistical rarity
of a response), and elaboration (Sternberg, 2000).
Sternberg (2000) states that there are positive and
negative sides to the psychometric approach. On the
positive side, these tests allow for research with noneminent people, are easy to administer, and objectively
scored. The negative side is that numerical scores fail
to capture the concept of creativity because they are
based on brief paper and pencil tests. Researchers call
for using more significant productions such as writing
samples, drawings, etc., subjectively evaluated by a
panel of experts, instead of simply relying on a
numerical measure.
The cognitive approach
The cognitive approach to the study of creativity
focuses on understanding the “mental representations
and processes underlying human thought” (Sternberg,
2000, p. 7). Weisberg (1993) suggests that creativity
entails the use of ordinary cognitive processes and
results in original and extraordinary products. These
products are the result of cognitive processes acting on
the knowledge already stored in the memory of the
individual. There is a significant amount of literature in
the area of information processing (Birkhoff, 1969;
Minsky, 1985) that attempts to isolate and explain
cognitive processes in terms of machine metaphors.
The social-personality approach
The social-personality approach to studying
creativity focuses on personality and motivational
variables as well as the socio-cultural environment as
sources of creativity. Sternberg (2000) states that
numerous studies conducted at the societal level
indicate that “eminent levels of creativity over large
spans of time are statistically linked to variables such
as cultural diversity, war, availability of role models,
availability of financial support, and competitors in a
domain” (p. 9).
Mathematical Creativity
Most of the recent literature on creativity
(Csikszentmihalyi, 1988, 2000; Gruber & Wallace,
2000; Sternberg & Lubart, 1996) suggests that
creativity is the result of a confluence of one or more
of the factors from these six aforementioned
categories. The “confluence” approach to the study of
creativity has gained credibility, and the research
literature has numerous confluence theories for better
understanding the process of creativity. A review of the
most commonly cited confluence theories of creativity
and a description of the methodology employed for
data collection and data analysis in this study follow.
Confluence Theories of Creativity
The three most commonly cited “confluence”
approaches to the study of creativity are the “systems
approach” (Csikszentmihalyi, 1988, 2000); “the case
study as evolving systems approach” (Gruber &
Wallace, 2000), and the “investment theory approach”
(Sternberg & Lubart, 1996). The case study as an
evolving system has the following components. First, it
views creative work as multi-faceted. So, in
constructing a case study of a creative work, one must
distill the facets that are relevant and construct the case
study based on the chosen facets. Some facets that can
be used to construct an evolving system case study are:
(1) uniqueness of the work; (2) a narrative of what the
creator achieved; (3) systems of belief; (4) multiple
time-scales (construct the time-scales involved in the
production of the creative work); (5) problem solving;
and (6) contextual frame such as family, schooling, and
teacher’s influences (Gruber & Wallace, 2000). In
summary, constructing a case study of a creative work
as an evolving system entails incorporating the many
facets suggested by Gruber & Wallace (2000). One
could also evaluate a case study involving creative
work by looking for the above mentioned facets.
The systems approach
The systems approach takes into account the social
and cultural dimensions of creativity instead of simply
viewing creativity as an individualistic psychological
process. The systems approach studies the interaction
between the individual, domain, and field. The field
consists of people who have influence over a domain.
For example, editors of mathematics research journals
have influence over the domain of mathematics. The
domain is in a sense a cultural organism that preserves
and transmits creative products to individuals in the
field. The systems model suggests that creativity is a
process that is observable at the “intersection where
individuals, domains and fields interact”
Bharath Sriraman
(Csikzentmihalyi, 2000). These three components individual, domain, and field - are necessary because
the individual operates from a cultural or symbolic
(domain) aspect as well as a social (field) aspect.
“The domain is a necessary component of
creativity because it is impossible to introduce a
variation without reference to an existing pattern. New
is meaningful only in reference to the old”
(Csikzentmihalyi, 2000). Thus, creativity occurs when
an individual proposes a change in a given domain,
which is then transmitted by the field through time.
The personal background of an individual and his
position in a domain naturally influence the likelihood
of his making a contribution. For example, a
mathematician working at a research university is more
likely to produce research papers because of the time
available for “thinking” as well as the creative
influence of being immersed in a culture where ideas
flourish. It is no coincidence that in the history of
science, there are significant contributions from
clergymen such as Pascal and Mendel because they
had the means and the leisure to “think.”
Csikszentmihalyi (2000) argues that novel ideas, which
could result in significant changes, are unlikely to be
adopted unless they are sanctioned by the experts.
These “gatekeepers” (experts) constitute the field. For
example, in mathematics, the opinion of a very small
number of leading researchers was enough to certify
the validity of Andrew Wiles’ proof of Fermat’s Last
Theorem.
There are numerous examples in the history of
mathematics that fall within the systems model. For
instance, the Bourbaki, a group of mostly French
mathematicians who began meeting in the 1930s,
aimed to write a thorough unified account of all
mathematics. The Bourbaki were essentially a group of
expert mathematicians that tried to unify all of
mathematics and become the gatekeepers of the field,
so to speak, by setting the standard for rigor. Although
the Bourbakists failed in their attempt, students of the
Bourbakists, who are editors of certain prominent
journals, to this day demand a very high degree of rigor
in submitted articles, thereby serving as gatekeepers of
the field.
A different example is that of the role of proof.
Proof is the social process through which the
mathematical community validates the mathematician's
creative work (Hanna, 1991). The Russian logician
Manin (1977) said "A proof becomes a proof after the
social act of accepting it as a proof. This is true of
mathematics as it is of physics, linguistics, and
biology."
23
In summary, the systems model of creativity
suggests that for creativity to occur, a set of rules and
practices must be transmitted from the domain to the
individual. The individual then must produce a novel
variation in the content of the domain, and this
variation must be selected by the field for inclusion in
the domain.
Gruber and Wallace’s case study as evolving
systems approach
In contrast to Csikszentmihalyi’s (2000) argument
calling for a focus on communities in which creativity
manifests itself, Gruber and Wallace (2000) propose a
model that treats each individual as a unique evolving
system of creativity and ideas; and, therefore, each
individual’s creative work must be studied on its own.
This viewpoint of Gruber and Wallace (2000) is a
belated victory of sorts for the Gestaltists, who
essentially proclaimed the same thing almost a century
ago. Gruber and Wallace’s (2000) use of terminology
that jibes with current trends in psychology seems to
make their ideas more acceptable. They propose a
model that calls for “detailed analytic and sometimes
narrative descriptions of each case and efforts to
understand each case as a unique functioning system
(Gruber & Wallace, 2000, p. 93). It is important to note
that the emphasis of this model is not to explain the
origins of creativity, nor is it the personality of the
creative individual, but on “how creative work works”
(p. 94). The questions of concern to Gruber and
Wallace are: (1) What do creative people do when they
are being creative? and (2) How do creative people
deploy available resources to accomplish something
unique? In this model creative work is defined as that
which is novel and has value. This definition is
consistent with that used by current researchers in
creativity (Csikszentmihalyi, 2000; Sternberg &
Lubart, 2000). Gruber and Wallace (2000) also claim
that creative work is always the result of purposeful
behavior and that creative work is usually a long
undertaking “reckoned in months, years and decades”
(p. 94).
I do not agree with the claim that creative work is
always the result of purposeful behavior. One
counterexample that comes to mind is the discovery of
penicillin. The discovery of penicillin could be
attributed purely to chance. On the other hand, there
are numerous examples that support the claim that
creative work sometimes entails work that spans years,
and in mathematical folklore there are numerous
examples of such creative work. For example, Kepler’s
laws of planetary motion were the result of twenty
24
years of numerical calculations. Andrew Wiles’ proof
of Fermat’s Last Theorem was a seven-year
undertaking. The Riemann hypothesis states that the
roots of the zeta function (complex numbers z, at
which the zeta function equals zero) lie on the line
parallel to the imaginary axis and half a unit to the
right of it. This is perhaps the most outstanding
unproved conjecture in mathematics with numerous
implications. The analyst Levinson undertook a
determined calculation on his deathbed that increased
the credibility of the Riemann-hypothesis. This is
another example of creative work that falls within
Gruber and Wallace's (2000) model.
The investment theory approach
According to the investment theory model, creative
people are like good investors; that is, they buy low
and sell high (Sternberg & Lubart, 1996). The context
here is naturally in the realm of ideas. Creative people
conjure up ideas that are either unpopular or
disrespected and invest considerable time convincing
other people about the intrinsic worth of these ideas
(Sternberg & Lubart, 1996). They sell high in the sense
that they let other people pursue their ideas while they
move on to the next idea. Investment theory claims that
the convergence of six elements constitutes creativity.
The six elements are intelligence, knowledge, thinking
styles, personality, motivation, and environment. It is
important that the reader not mistake the word
intelligence for an IQ score. On the contrary, Sternberg
(1985) suggests a triarchic theory of intelligence that
consists of synthetic (ability to generate novel, task
appropriate ideas), analytic, and practical abilities.
Knowledge is defined as knowing enough about a
particular field to move it forward. Thinking styles are
defined as a preference for thinking in original ways of
one’s choosing, the ability to think globally as well as
locally, and the ability to distinguish questions of
importance from those that are not important.
Personality attributes that foster creative functioning
are the willingness to take risks, overcome obstacles,
and tolerate ambiguity. Finally, motivation and an
environment that is supportive and rewarding are
essential elements of creativity (Sternberg, 1985).
In investment theory, creativity involves the
interaction between a person, task, and environment.
This is, in a sense, a particular case of the systems
model (Csikszentmihalyi, 2000). The implication of
viewing creativity as the interaction between person,
task, and environment is that what is considered novel
or original may vary from one person, task, and
environment to another. The investment theory model
Mathematical Creativity
suggests that creativity is more than a simple sum of
the attained level of functioning in each of the six
elements. Regardless of the functioning levels in other
elements, a certain level or threshold of knowledge is
required without which creativity is impossible. High
levels of intelligence and motivation can positively
enhance creativity, and compensations can occur to
counteract weaknesses in other elements. For example,
one could be in an environment that is non-supportive
of creative efforts, but a high level of motivation could
possibly overcome this and encourage the pursuit of
creative endeavors.
This concludes the review of three commonly cited
prototypical confluence theories of creativity, namely
the systems approach (Csikszentmihalyi, 2000), which
suggests that creativity is a sociocultural process
involving the interaction between the individual,
domain, and field; Gruber & Wallace’s (2000) model
that treats each individual case study as a unique
evolving system of creativity; and investment theory
(Sternberg & Lubart, 1996), which suggests that
creativity is the result of the convergence of six
elements (intelligence, knowledge, thinking styles,
personality, motivation, and environment).
Having reviewed the research literature on
creativity, the focus is shifted to the methodology
employed for studying mathematical creativity.
Methodology
The Interview Instrument
The purpose of this study was to gain an insight
into the nature of mathematical creativity. In an effort
to determine some of the characteristics of the creative
process, I was interested in distilling common
attributes in the ways mathematicians create
mathematics. Additionally, I was interested in testing
the applicability of the Gestalt model. Because the
main focus of the study was to ascertain qualitative
aspects of creativity, a formal interview methodology
was selected as the primary method of data collection.
The interview instrument (Appendix A) was developed
by modifying questions from questionnaires in
L’Enseigement Mathematique (1902) and Muir (1988).
The rationale behind using this modified questionnaire
was to allow the mathematicians to express themselves
freely while responding to questions of a general
nature and to enable me to test the applicability of the
four-stage Gestalt model of creativity. Therefore, the
existing instruments were modified to operationalize
the Gestalt theory and to encourage the natural flow of
ideas, thereby forming the basis of a thesis that would
emerge from this exploration.
Bharath Sriraman
Background of the Subjects
Five mathematicians from the mathematical
sciences faculty at a large Ph.D. granting mid-western
university were selected. These mathematicians were
chosen based on their accomplishments and the
diversity of the mathematical areas in which they
worked, measured by counting the number of
published papers in prominent journals, as well as
noting the variety of mathematical domains in which
they conducted research. Four of the mathematicians
were tenured full professors, each of whom had been
professional mathematicians for more than 30 years.
One of the mathematicians was considerably younger
but was a tenured associate professor. All interviews
were conducted formally, in a closed door setting, in
each mathematician’s office. The interviews were
audiotaped and transcribed verbatim.
Data Analysis
Since creativity is an extremely complex construct
involving a wide range of interacting behaviors, I
believe it should be studied holistically. The principle
of analytic induction (Patton, 2002) was applied to the
interview transcripts to discover dominant themes that
described the behavior under study. According to
Patton (2002), "analytic induction, in contrast to
grounded theory, begins with an analyst's deduced
propositions or theory-derived hypotheses and is a
procedure for verifying theories and propositions based
on qualitative data” (Taylor and Bogdan, 1984, p. 127).
Following the principles of analytic induction, the data
was carefully analyzed in order to extract common
strands. These strands were then compared to
theoretical constructs in the existing literature with the
explicit purpose of verifying whether the Gestalt model
was applicable to this qualitative data as well as to
extract themes that characterized the mathematician’s
creative process. If an emerging theme could not be
classified or named because I was unable to grasp its
properties or significance, then theoretical comparisons
were made. Corbin and Strauss (1998) state that “using
comparisons brings out properties, which in turn can be
used to examine the incident or object in the data. The
specific incidents, objects, or actions that we use when
making theoretical comparisons can be derived from
the literature and experience. It is not that we use
experience or literature as data “but rather that we use
the properties and dimensions derived from the
comparative incidents to examine the data in front of
us” (p. 80). Themes that emerged were social
interaction, preparation, use of heuristics, imagery,
incubation, illumination, verification, intuition, and
25
proof. Excerpts from interviews that highlight these
characteristics are reconstructed in the next section
along with commentaries that incorporate the wider
conversation, and a continuous discussion of
connections to the existing literature.
Results, Commentaries & Discussion
The mathematicians in this study worked in
academic environments and regularly fulfilled teaching
and committee duties. The mathematicians were free to
choose their areas of research and the problems on
which they focused. Four of the five mathematicians
had worked and published as individuals and as
members of occasional joint ventures with
mathematicians from other universities. Only one of
the mathematicians had done extensive collaborative
work. All but one of the mathematicians were unable
to formally structure their time for research, primarily
due to family commitments and teaching
responsibilities during the regular school year. All the
mathematicians found it easier to concentrate on
research in the summers because of lighter or nonexistent teaching responsibilities during that time. Two
of the mathematicians showed a pre-disposition
towards mathematics at the early secondary school
level. The others became interested in mathematics
later, during their university education. The
mathematicians who participated in this study did not
report any immediate family influence that was of
primary importance in their mathematical
development. Four of the mathematicians recalled
being influenced by particular teachers, and one
reported being influenced by a textbook. The three
mathematicians who worked primarily in analysis
made a conscious effort to obtain a broad overview of
mathematics not necessarily of immediate relevance to
their main interests. The two algebraists expressed
interest in other areas of mathematics but were
primarily active in their chosen field.
Supervision Of Research & Social Interaction
As noted earlier, all the mathematicians in this
study were tenured professors in a research university.
In addition to teaching, conducting research, and
fulfilling committee obligations, many mathematicians
play a big role in mentoring graduate students
interested in their areas of research. Research
supervision is an aspect of creativity because any
interaction between human beings is an ideal setting
for the exchange of ideas. During this interaction the
mathematician is exposed to different perspectives on
the subject, and all of the mathematicians in this study
26
valued the interaction they had with their graduate
students. Excerpts of individual responses follow.1
Excerpt 1
A. I've had only one graduate student per semester
and she is just finishing up her PhD right now,
and I'd say it has been a very good interaction
to see somebody else get interested in the
subject and come up with new ideas, and
exploring those ideas with her.
B. I have had a couple of students who have sort
of started but who haven't continued on to a
PhD, so I really can't speak to that. But the
interaction was positive.
C. Of course, I have a lot of collaborators, these
are my former students you know…I am
always all the time working with students, this
is normal situation.
D. That is difficult to answer (silence)…it is
positive because it is good to interact with
other people. It is negative because it can take
a lot of time. As you get older your brain
doesn't work as well as it used to
and…younger people by and large their minds
are more open, there is less garbage in there
already. So, it is exciting to work with younger
people who are in their most creative time.
When you are older, you have more
experience, when you are younger your mind
works faster …not as fettered.
E. Oh…it is a positive factor I think, because it
continues to stimulate ideas …talking about
things and it also reviews things for you in the
process, puts things in perspective, and keep
the big picture. It is helpful really in your own
research to supervise students.
Commentary on Excerpt 1
The responses of the mathematicians in the
preceding excerpt are focused on research supervision;
however, all of the mathematicians acknowledged the
role of social interaction in general as an important
aspect that stimulated creative work. Many of the
mathematicians mentioned the advantages of being
able to e-mail colleagues and going to research
conferences and other professional meetings. This is
further explored in the following section, which
focuses on preparation.
Mathematical Creativity
Preparation and the Use of Heuristics
When mathematicians are about to investigate a
new topic, there is usually a body of existing research
in the area of the new topic. One of goals of this study
was to find out how creative mathematicians
approached a new topic or a problem. Did they try their
own approach, or did they first attempt to assimilate
what was already known about that topic? Did the
mathematicians make use of computers to gain insight
into the problem? What were the various modes of
approaching a new topic or problem? The responses
indicate that a variety of approaches were used.
Excerpt 2
A. Talk to people who have been doing this topic.
Learn the types of questions that come up.
Then I do basic research on the main ideas. I
find that talking to people helps a lot more than
reading because you get more of a feel for
what the motivation is beneath everything.
B. What might happen for me, is that I may start
reading something, and, if feel I can do a better
job, then I would strike off on my own. But for
the most part I would like to not have to
reinvent a lot that is already there. So, a lot of
what has motivated my research has been the
desire to understand an area. So, if somebody
has already laid the groundwork then it's
helpful. Still I think a large part of doing
research is to read the work that other people
have done.
C. It is connected with one thing that simply…my
style was that I worked very much and I even
work when I could not work. Simply the
problems that I solve attract me so much, that
the question was who will die
first…mathematics or me? It was never clear
who would die.
D. Try and find out what is known. I won't say
assimilate…try and find out what's known and
get an overview, and try and let the problem
speak…mostly by reading because you don't
have that much immediate contact with other
people in the field. But I find that I get more
from listening to talks that other people are
giving than reading.
E. Well! I have been taught to be a good scholar.
A good scholar attempts to find out what is
first known about something or other before
they spend their time simply going it on their
Bharath Sriraman
own. That doesn't mean that I don't
simultaneously try to work on something.
Commentary on Excerpt 2
These responses indicate that the mathematician
spends a considerable amount of time researching the
context of the problem. This is primarily done by
reading the existing literature and by talking to other
mathematicians in the new area. This finding is
consistent with the systems model, which suggests that
creativity is a dynamic process involving the
interaction between the individual, domain, and field
(Csikzentmihalyi, 2000). At this stage, it is reasonable
to ask whether a mathematician works on a single
problem until a breakthrough occurs or does a
mathematician work on several problems concurrently?
It was found that each of the mathematicians worked
on several problems concurrently, using a back and
forth approach.
Excerpt 3
A. I work on several different problems for a
protracted period of time… there have been
times when I have felt, yes, I should be able to
prove this result, then I would concentrate on
that thing for a while but they tend to be
several different things that I was thinking
about a particular stage.
B. I probably tend to work on several problems at
the same time. There are several different
questions
that
I
am
working
on…mm…probably the real question is how
often do you change the focus? Do I work on
two different problems on the same day? And
that is probably up to whatever comes to mind
in that particular time frame. I might start
working on one rather than the other. But I
would tend to focus on one particular problem
for a period of weeks, then you switch to
something else. Probably what happens is that
I work on something and I reach a dead end
then I may shift gears and work on a different
problem for a while, reach a dead end there
and come back to the original problem, so it’s
back and forth.
C . I must simply think on one thing and not
switch so much.
D. I find that I probably work on one. There
might be a couple of things floating around but
I am working on one and if I am not getting
27
anywhere, then I might work on the other and
then go back.
work. I have a very geometrically based intuition and
uhh…so very definitely I do a lot of manipulations.
E. I usually have couple of things going. When I
get stale on one, then I will pick up the other,
and bounce back and forth. Usually I have one
that is primarily my focus at a given time, and
I will spend time on it over another; but it is
not uncommon for me to have a couple of
problems going at a given time. Sometimes
when I am looking for an example that is not
coming, instead of spending my time beating
my head against the wall, looking for that
example is not a very good use of time.
Working on another helps to generate ideas
that I can bring back to the other problem.
A. That is a problem because of the particular
area I am in. I can't draw any diagrams, things
are infinite, so I would love to be able to get
some kind of a computer diagram to show the
complexity for a particular ring… to have
something
like
the
Julia
sets
or…mmm…fractal images, things which are
infinite but you can focus in closer and closer
to see possible relationships. I have thought
about that with possibilities on the computer.
To think about the most basic ring, you would
have to think of the ring of integers and all of
the relationships for divisibility, so how do you
somehow describe this tree of divisibility for
integers…it is infinite.
Commentary on Excerpt 3
The preceding excerpt indicates that
mathematicians tend to work on more than one
problem at a given time. Do mathematicians switch
back and forth between problems in a completely
random manner, or do they employ and exhaust a
systematic train of thought about a problem before
switching to a different problem? Many of the
mathematicians reported using heuristic reasoning,
trying to prove something one day and disprove it the
next day, looking for both examples and
counterexamples, the use of "manipulations" (Polya,
1954) to gain an insight into the problem. This
indicates that mathematicians do employ some of the
heuristics made explicit by Polya. It was unclear
whether the mathematicians made use of computers to
gain an experimental or computational insight into the
problem. I was also interested in knowing the types of
imagery used by mathematicians in their work. The
mathematicians in this study were queried about this,
and the following excerpt gives us an insight into that
aspect of mathematical creativity.
Imagery
The mathematicians in this study were asked about
the kinds of imagery they used to think about
mathematical objects. Their responses are reported
here to give the reader a glimpse of the ways
mathematicians think of mathematical objects. Their
responses also highlight the difficulty of explicitly
describing imagery.
Excerpt 4
Yes I do, yes I do, I tend to draw a lot of pictures
when I am doing research, I tend to manipulate things
in the air, you know to try to figure out how things
28
B . Science is language, you think through
language. But it is language simply; you put
together theorems by logic. You first see the
theorem in nature…you must see that
somewhat is reasonable and then you go and
begin and then of course there is big, big, big
work to just come to some theorem in nonlinear elliptic equations…
C. A lot of mathematics, whether we are teaching
or doing, is attaching meaning to what we are
doing and this is going back to the earlier
question when you talked about how do you do
it, what kind of heuristics do you use? What
kind of images do you have that you are using?
A lot of doing mathematics is creating these
abstract images that connect things and then
making sense of them but that doesn't appear
in proofs either.
D. Pictorial, linguistic, kinesthetic...any of them is
the point right! Sometimes you think of one,
sometimes another. It really depends on the
problem you are looking at, they are very
much…often I think of functions as very
kinesthetic, moving things from here to there.
Other approaches you are talking about is
going to vary from problem to problem, or
even day to day. Sometimes when I am
working on research, I try to view things in as
many different ways as possible, to see what is
really happening. So there are a variety of
approaches.
Mathematical Creativity
Commentary on Excerpt 4
Besides revealing the difficulty of describing
mental imagery, all the mathematicians reported that
they did not use computers in their work. This
characteristic of the pure mathematician's work is
echoed in Poincaré's (1948) use of the “choice”
metaphor and Ervynck's (1991) use of the term “nonalgorithmic decision making.” The doubts expressed
by the mathematicians about the incapability of
machines to do their work brings to mind the reported
words of Garrett Birkhoff, one of the great applied
mathematicians of our time. In his retirement
presidential address to the Society for Industrial and
Applied Mathematics, Birkhoff (1969) addressed the
role of machines in human creative endeavors. In
particular, part of this address was devoted to
discussing the psychology of the mathematicians (and
hence of mathematics). Birkhoff (1969) said:
The remarkable recent achievements of computers
have partially fulfilled an old dream. These
achievements have led some people to speculate
that tomorrow's computers will be even more
"intelligent" than humans, especially in their
powers of mathematical reasoning...the ability of
good mathematicians to sense the significant and to
avoid undue repetition seems, however, hard to
computerize; without it, the computer has to pursue
millions of fruitless paths avoided by experienced
human mathematicians. (pp. 430-438)
Incubation and Illumination
Having reported on the role of research supervision
and social interaction, the use of heuristics and
imagery, all of which can be viewed as aspects of the
preparatory stage of mathematical creativity, it is
natural to ask what occurs next. As the literature
suggests, after the mathematician works hard to gain
insight into a problem, there is usually a transition
period (conscious work on the problem ceases and
unconscious work begins), during which the problem is
put aside before the breakthrough occurs. The
mathematicians in this study reported experiences that
are consistent with the existing literature (Hadamard,
1945; Poincaré, 1948).
Excerpt 5
B. One of the problems is first one does some
preparatory work, that has to be the left side
[of the brain], and then you let it sit. I don't
think you get ideas out of nowhere, you have
to do the groundwork first, okay. This is why
people will say, now we have worked on this
problem, so let us sleep on it. So you do the
Bharath Sriraman
preparation, so that the sub-conscious or
intuitive side may work on it and the answer
comes back but you can't really tell when. You
have to be open to this, lay the groundwork,
think about it and then these flashes of
intuition come and they represent the other
side of the brain communicating with you at
whatever odd time.
D. I am not sure you can really separate them
because they are somewhat connected. You
spend a lot of time working on something and
you are not getting anywhere with it…with the
deliberate effort, then I think your mind
continues to work and organize. And maybe
when the pressure is off the idea comes…but
the idea comes because of the hard work.
E. Usually they come after I have worked very
hard on something or another, but they may
come at an odd moment. They may come into
my head before I go to bed …What do I do at
that point? Yes I write it down (laughing).
Sometimes when I am walking somewhere, the
mind flows back to it (the problem) and says
what about that, why don't you try that. That
sort of thing happens. One of the best ideas I
had was when I was working on my thesis
…Saturday night, having worked on it quite a
bit, sitting back and saying why don't I think
about it again…and ping! There it was…I
knew what it was, I could do that. Often ideas
are handed to you from the outside, but they
don't come until you have worked on it long
enough.
Commentary on Excerpt 5
As is evident in the preceding excerpt, three out of
the five mathematicians reported experiences
consistent with the Gestalt model. Mathematician C
attributed his breakthroughs on problems to his
unflinching will to never give up and to divine
inspiration, echoing the voice of Pascal in a sense.
However, Mathematician A attributed breakthroughs to
chance. In other words, making the appropriate
(psychological) connections by pure chance which
eventually result in the sought after result.
I think it is necessary to comment about the
unusual view of mathematician A. Chance plays an
important role in mathematical creativity. Great ideas
and insights may be the result of chance such as the
discovery of penicillin. Ulam (1976) estimated that
there is a yearly output of 200,000 theorems in
29
mathematics. Chance plays a role in what is considered
important in mathematical research since only a
handful of results and techniques survive out of the
volumes of published research. I wish to draw a
distinction between chance in the "Darwinian" sense
(as to what survives), and chance in the psychological
sense (which results in discovery/invention). The role
of chance is addressed by Muir (1988) as follows.
The act of creation of new entities has two aspects:
the generation of new possibilities, for which we
might attempt a stochastic description, and the
selection of what is valuable from among them.
However the importation of biological metaphors
to explain cultural evolution is dubious…both
creation and selection are acts of design within a
social context. (p. 33)
Thus, Muir (1988) rejects the Darwinian
explanation. On the other hand, Nicolle (1932) in
Biologie de L'Invention does not acknowledge the role
of unconsciously present prior work in the creative
process. He attributes breakthroughs to pure chance.
By a streak of lightning, the hitherto obscure
problem, which no ordinary feeble lamp would
have revealed, is at once flooded in light. It is like a
creation. Contrary to progressive acquirements,
such an act owes nothing to logic or to reason. The
act of discovery is an accident. (Hadamard, 1945)
Nicolle's Darwinian explanation was rejected by
Hadamard on the grounds that to claim creation occurs
by pure chance is equivalent to asserting that there are
effects without causes. Hadamard further argued that
although Poincaré attributed his particular
breakthrough in Fuchsian functions to chance, Poincaré
did acknowledge that there was a considerable amount
of previous conscious effort, followed by a period of
unconscious work. Hadamard (1945) further argued
that even if Poincaré's breakthrough was the result of
chance alone, chance alone was insufficient to explain
the considerable body of creative work credited to
Poincaré in almost every area of mathematics. The
question then is how does (psychological) chance
work?
It is my conjecture that the mind throws out
fragments (ideas) that are products of past experience.
Some of these fragments can be juxtaposed and
combined in a meaningful way. For example, if one
reads a complicated proof consisting of a thousand
steps, a thousand random fragments may not be enough
to construct a meaningful proof. However the mind
chooses relevant fragments from these random
fragments and links them into something meaningful.
Wedderburn's Theorem, that a finite division ring is a
30
field, is one instance of a unification of apparently
random fragments because the proof involves algebra,
complex analysis, and number theory.
Polya (1954) addresses the role of chance in a
probabilistic sense. It often occurs in mathematics that
a series of mathematical trials (involving computation)
generate numbers that are close to a Platonic ideal. The
classic example is Euler's investigation of the infinite
series 1 + 1/4 + 1/9 + 1/16 +…+ 1/n2 +…. Euler
obtained an approximate numerical value for the sum
of the series using various transformations of the
series. The numerical approximation was 1.644934.
Euler confidently guessed the sum of the series to be
π2/6. Although the numerical value obtained by Euler
and the value of π 2/6 coincided up to seven decimal
places, such a coincidence could be attributed to
chance. However, a simple calculation shows that the
probability of seven digits coinciding is one in ten
million! Hence, Euler did not attribute this coincidence
to chance but boldly conjectured that the sum of this
series was indeed π2/6 and later proved his conjecture
to be true (Polya, 1954, pp. 95-96).
Intuition, Verification and Proof
Once illumination has occurred, whether through
sheer chance, incubation, or divine intervention,
mathematicians usually try to verify that their
intuitions were correct with the construction of a proof.
The following section discusses how these
mathematicians went about the business of verifying
their intuitions and the role of formal proof in the
creative process. They were asked whether they relied
on repeatedly checking a formal proof, used multiple
converging partial proofs, looked first for coherence
with other results in the area, or looked at applications.
Most of the mathematicians in this study mentioned
that the last thing they looked at was a formal proof.
This is consistent with the literature on the role of
formal proof in mathematics (Polya, 1954; Usiskin,
1987). Most of the mathematicians mentioned the need
for coherence with other results in the area. The
mathematician’s responses to the posed question
follow.
Excerpt 6
B. I think I would go for repeated checking of the
formal proof…but I don't think that that is
really enough. All of the others have to also be
taken into account. I mean, you can believe
that something is true although you may not
fully understand it. This is the point that was
made in the lecture by … of … University on
Mathematical Creativity
Dirichlet series. He was saying that we have
had a formal proof for some time, but that is
not to say that it is really understood, and what
did he mean by that? Not that the proof wasn't
understood, but it was the implications of the
result that are not understood, their
connections with other results, applications
and why things really work. But probably the
first thing that I would really want to do is
check the formal proof to my satisfaction, so
that I believe that it is correct although at that
point I really do not understand its
implications… it is safe to say that it is my
surest guide.
C. First you must see it in the nature, something,
first you must see that this theorem
corresponds to something in nature, then if you
have this impression, it is something relatively
reasonable, then you go to proofs…and of
course I have also several theorems and proofs
that are wrong, but the major amount of proofs
and theorems are right.
D. The last thing that comes is the formal proof. I
look for analogies with other things… How
your results that you think might be true would
illuminate other things and would fit in the
general structure.
E. Since I work in an area of basic research, it is
usually coherence with other things, that is
probably more than anything else. Yes, one
could go back and check the proof and that sort
of thing but usually the applications are yet to
come, they aren't there already. Usually what
guides the choice of the problem is the
potential for application, part of what
represents good problems is their potential for
use. So, you certainly look to see if it makes
sense in the big picture…that is a coherence
phenomenon. Among those you've given me,
that’s probably the most that fits.
Commentary on Excerpt 6
This excerpt indicates that for mathematicians,
valid proofs have varied degrees of rigor. “Among
mathematicians, rigor varies depending on time and
circumstance, and few proofs in mathematics journals
meet the criteria used by secondary school geometry
teachers (each statement of proof is backed by
reasons). Generally one increases rigor only when the
result does not seem to be correct” (Usiskin, 1987).
Proofs are in most cases the final step in this testing
Bharath Sriraman
process. “Mathematics in the making resembles any
other human knowledge in the making. The result of
the mathematician’s creative work is demonstrative
reasoning, a proof; but the proof is discovered by
plausible reasoning, by guessing” (Polya, 1954). How
mathematicians approached proof in this study was
very different from the logical approach found in proof
in most textbooks. The logical approach is an artificial
reconstruction of discoveries that are being forced into
a deductive system, and in this process the intuition
that guided the discovery process gets lost.
Conclusions
The goal of this study was to gain an insight into
mathematical creativity. As suggested by the literature
review, the existing literature on mathematical
creativity is relatively sparse. In trying to better
understand the process of creativity, I find that the
Gestalt model proposed by Hadamard (1945) is still
applicable today. This study has attempted to add some
detail to the preparation-incubation-illuminationverification model of Gestalt by taking into account the
role of imagery, the role of intuition, the role of social
interaction, the use of heuristics, and the necessity of
proof in the creative process.
The mathematicians worked in a setting that was
conducive to prolonged research. There was a
convergence of intelligence, knowledge, thinking
styles, personality, motivation and environment that
enabled them to work creatively (Sternberg, 2000;
Sternberg & Lubart, 1996, 2000). The preparatory
stage of mathematical creativity consists of various
approaches used by the mathematician to lay the
groundwork. These include reading the existing
literature, talking to other mathematicians in the
particular mathematical domain (Csikzentmihalyi,
1988; 2000), trying a variety of heuristics (Polya,
1954), and using a back-and-forth approach of
plausible guessing. One of the mathematicians said that
he first looked to see if the sought after relationships
corresponded to natural phenomenon.
All of the mathematicians in this study worked on
more than one problem at a given moment. This is
consistent with the investment theory view of creativity
(Sternberg & Lubart, 1996). The mathematicians
invested an optimal amount of time on a given
problem, but switched to a different problem if no
breakthrough was forthcoming. All the mathematicians
in this study considered this as the most important and
difficult stage of creativity. The prolonged hard work
was followed by a period of incubation where the
problem was put aside, often while the preparatory
31
stage is repeated for a different problem; and thus,
there is a transition in the mind from conscious to
unconscious work on the problem. One mathematician
cited this as the stage at which the "problem begins to
talk to you." Another offered that the intuitive side of
the brain begins communicating with the logical side at
this stage and conjectured that this communication was
not possible at a conscious level.
The transition from incubation to illumination
often occurred when least expected. Many reported the
breakthrough occurring as they were going to bed, or
walking, or sometimes as a result of speaking to
someone else about the problem. One mathematician
illustrated this transition with the following: "You talk
to somebody and they say just something that might
have been very ordinary a month before but if they say
it when you are ready for it, and Oh yeah, I can do it
that way, can’t I! But you have to be ready for it.
Opportunity knocks but you have to be able to answer
the door."
Illumination is followed by the mathematician’s
verifying the result. In this study, most of the
mathematicians looked for coherence of the result with
other existing results in the area of research. If the
result cohered with other results and fit the general
structure of the area, only then did the mathematician
try to construct a formal proof. In terms of the
mathematician’s beliefs about the nature of
mathematics and its influence on their research, the
study revealed that four of the mathematicians leaned
towards Platonism, in contrast to the popular notion
that Platonism is an exception today. A detailed
discussion of this aspect of the research is beyond the
scope of this paper; however, I have found that beliefs
regarding the nature of mathematics not only
influenced how these mathematicians conducted
research but also were deeply connected to their
theological beliefs (Sriraman, 2004a).
The mathematicians hoped that the results of their
creative work would be sanctioned by a group of
experts in order to get the work included in the domain
(Csikzentmihalyi, 1988, 2000), primarily in the form of
publication in a prominent journal. However, the
acceptance of a mathematical result, the end product of
creation, does not ensure its survival in the Darwinian
sense (Muir, 1988). The mathematical result may or
may not be picked up by other mathematicians. If the
mathematical community picks it up as a viable result,
then it is likely to undergo mutations and lead to new
mathematics. This, however, is determined by chance!
32
Implications
It is in the best interest of the field of mathematics
education that we identify and nurture creative talent in
the mathematics classroom. "Between the work of a
student who tries to solve a difficult problem in
mathematics and a work of invention (creation)…there
is only a difference of degree" (Polya, 1954).
Creativity as a feature of mathematical thinking is not a
patent of the mathematician! (Krutetskii, 1976); and
although most studies on creativity have focused on
eminent individuals (Arnheim, 1962; Gardner, 1993,
1997; Gruber, 1981), I suggest that contemporary
models from creativity research can be adapted for
studying samples of creativity such as are produced by
high school students. Such studies would reveal more
about creativity in the classroom to the mathematics
education research community. Educators could
consider how often mathematical creativity is
manifested in the school classroom and how teachers
might identify creative work. One plausible way to
approach these concerns is to reconstruct and evaluate
student work as a unique evolving system of creativity
(Gruber & Wallace, 2000) or to incorporate some of
the facets suggested by Gruber & Wallace (2000). This
necessitates the need to find suitable problems at the
appropriate levels to stimulate student creativity.
A common trait among mathematicians is the
reliance on particular cases, isomorphic reformulations,
or analogous problems that simulate the original
problem situations in their search for a solution (Polya,
1954; Skemp, 1986). Creating original mathematics
requires a very high level of motivation, persistence,
and reflection, all of which are considered indicators of
creativity (Amabile, 1983; Policastro & Gardner, 2000;
Gardner, 1993). The literature suggests that most
creative individuals tend to be attracted to complexity,
of which most school mathematics curricula has very
little to offer. Classroom practices and math curricula
rarely use problems with the sort of underlying
mathematical structure that would necessitate students’
having a prolonged period of engagement and the
independence to formulate solutions. It is my
conjecture that in order for mathematical creativity to
manifest itself in the classroom, students should be
given the opportunity to tackle non-routine problems
with complexity and structure - problems which
require not only motivation and persistence but also
considerable reflection. This implies that educators
should recognize the value of allowing students to
reflect on previously solved problems to draw
comparisons between various isomorphic problems
(English, 1991, 1993; Hung, 2000; Maher & Kiczek,
Mathematical Creativity
2000; Maher & Martino, 1997; Maher & Speiser,
1996; Sriraman, 2003; Sriraman, 2004b). In addition,
encouraging students to look for similarities in a class
of problems fosters "mathematical" behavior (Polya,
1954), leading some students to discover sophisticated
mathematical structures and principles in a manner
akin to the creative processes of professional
mathematicians.
REFERENCES
Amabile, T. M. (1983). Social psychology of creativity: A
componential conceptualization. Journal of Personality and
Social Psychology, 45, 357−376.
Arnheim, R. (1962). Picasso’s guernica. Berkeley: University of
California Press.
Birkhoff, G. (1969). Mathematics and psychology. SIAM Review,
11, 429−469.
Corbin, J., & Strauss, A. (1998). Basics of qualitative research.
Thousand Oaks, CA: Sage.
Csikszentmihalyi, M. (1988). Society, culture, and person: A
systems view of creativity. In R. J. Sternberg (Ed.), The nature
of creativity: Contemporary psychological perspectives (pp.
325−339). Cambridge UK: Cambridge University Press.
Csikszentmihalyi, M. (2000). Implications of a systems perspective
for the study of creativity. In R. J. Sternberg (Ed.), Handbook
of creativity (pp. 313−338). Cambridge UK: Cambridge
University Press.
Hadamard, J. (1945). Essay on the psychology of invention in the
mathematical field. Princeton, NJ: Princeton University Press.
Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.). Advanced
mathematical thinking (pp. 54−60). Dordrecht: Kluwer.
Hung, D. (2000). Some insights into the generalizations of
mathematical meanings. Journal of Mathematical Behavior,
19, 63–82.
Krutetskii, V. A. (1976). The psychology of mathematical abilities
in school children. (J. Kilpatrick & I. Wirszup, Eds.; J. Teller,
Trans.). Chicago: University of Chicago Press. (Original work
published 1968)
L'Enseigement Mathematique. (1902), 4, 208–211.
L'Enseigement Mathematique. (1904), 6, 376.
Lester, F. K. (1985). Methodological considerations in research on
mathematical problem solving. In E. A. Silver (Ed.), Teaching
and learning mathematical problem solving: Multiple
research perspectives (pp. 41–70). Hillsdale, NJ: Erlbaum.
Maher, C. A., & Kiczek R. D. (2000). Long term building of
mathematical ideas related to proof making. Contributions to
Paolo Boero, G. Harel, C. Maher, M. Miyasaki. (organizers)
Proof and Proving in Mathematics Education. Paper
distributed at ICME9 -TSG 12. Tokyo/Makuhari, Japan.
Maher, C. A., & Speiser M. (1997). How far can you go with block
towers? Stephanie's intellectual development. Journal of
Mathematical Behavior, 16(2), 125−132.
Maher, C. A., & Martino A. M. (1996). The development of the
idea of mathematical proof: A 5-year case study. Journal for
Research in Mathematics Education, 27(2), 194−214.
Davis, P. J., & Hersh, R. (1981). The mathematical experience.
New York: Houghton Mifflin.
Manin, Y. I. (1977). A course in mathematical logic. New York:
Springer-Verlag.
English, L. D. (1991). Young children's combinatoric strategies.
Educational Studies in Mathematics, 22, 451−474.
Minsky, M. (1985). The society of mind. New York: Simon &
Schuster.
English, L. D. (1993). Children's strategies in solving two- and
three-dimensional combinatorial problems. Journal for
Research in Mathematics Education, 24(3), 255−273.
Muir, A. (1988). The psychology of mathematical creativity.
Mathematical Intelligencer, 10(1), 33−37.
Ernest, P. (1991). The philosophy of mathematics education,
Briston, PA: Falmer.
Patton, M. Q. (2002). Qualitative research and evaluation methods.
Thousand Oaks, CA: Sage.
Ernest, P. (1994). Conversation as a metaphor for mathematics and
learning. Proceedings of the British Society for Research into
Learning Mathematics Day Conference, Manchester
Metropolitan University (pp. 58−63). Nottingham: BSRLM.
Policastro, E., & Gardner, H. (2000). From case studies to robust
generalizations: An approach to the study of creativity. In R. J.
Sternberg (Ed.), Handbook of creativity (pp. 213−225).
Cambridge, UK: Cambridge University Press.
Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),
Advanced mathematical thinking (pp. 42−53). Dordrecht:
Kluwer.
Poincaré, H. (1948). Science and method. New York: Dover.
Frensch, P., & Sternberg, R. (1992). Complex problem solving:
Principles and mechanisms. New Jersey: Erlbaum.
Gallian, J. A. (1994). Contemporary abstract algebra. Lexington,
MA: Heath.
Gardner, H. (1997). Extraordinary minds. New York: Basic Books.
Gardner, H. (1993). Frames of mind. New York: Basic Books.
Gruber, H. E. (1981). Darwin on man. Chicago: University of
Chicago Press.
Gruber, H. E., & Wallace, D. B. (2000). The case study method and
evolving systems approach for understanding unique creative
people at work. In R. J. Sternberg (Ed.), Handbook of
creativity (pp. 93-115). Cambridge UK: Cambridge University
Press.
Bharath Sriraman
Nicolle, C. (1932). Biologie de l'invention, Paris: Alcan.
Polya, G. (1945). How to solve it. Princeton, NJ: Princeton
University Press.
Polya, G. (1954). Mathematics and plausible reasoning: Induction
and analogy in mathematics (Vol. II). Princeton, NJ: Princeton
University Press.
Schoenfeld, A. H. (1985). Mathematical problem solving. New
York: Academic Press.
Skemp, R. (1986). The psychology of learning mathematics.
Middlesex, UK: Penguin Books.
Sriraman, B. (2003). Mathematical giftedness, problem solving,
and the ability to formulate generalizations. The Journal of
Secondary Gifted Education. XIV(3), 151−165.
Sriraman, B. (2004a). The influence of Platonism on mathematics
research and theological beliefs. Theology and Science, 2(1),
131−147.
33
Sriraman, B. (2004b). Discovering a mathematical principle: The
case of Matt. Mathematics in School (UK), 3(2), 25−31.
Sternberg, R. J. (1985). Human abilities: An information
processing approach. New York: W. H. Freeman.
Sternberg, R. J. (2000). Handbook of creativity. Cambridge, UK:
Cambridge University Press.
Sternberg, R. J., & Lubart, T. I. (1996). Investing in creativity.
American Psychologist, 51, 677−688.
Sternberg, R. J., & Lubart, T. I. (2000). The concept of creativity:
Prospects and paradigms. In R. J. Sternberg (Ed.), Handbook
of creativity (pp. 93−115). Cambridge, UK: Cambridge
University Press.
Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitative
research methods: The search for meanings. New York: John
Wiley & Sons.
Torrance, E. P. (1974). Torrance tests of creative thinking: Normstechnical manual. Lexington, MA: Ginn.
Ulam, S. (1976). Adventures of a mathematician. New York:
Scribners.
Usiskin, Z. P. (1987). Resolving the continuing dilemmas in school
geometry. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning
and teaching geometry, K-12: 1987 yearbook (pp. 17−31).
Reston, VA: National Council of Teachers of Mathematics.
Wallas, G. (1926). The art of thought. New York: Harcourt, Brace
& Jovanovich.
Weisberg, R. W. (1993). Creativity: Beyond the myth of genius.
New York: Freeman.
Wertheimer, M. (1945). Productive thinking. New York: Harper.
Wittgenstein, L. (1978). Remarks on the foundations of
mathematics (Rev. Ed.).Cambridge: Massachusetts Institute of
Technology Press.
APPENDIX A: Interview Protocol
The interview instrument was developed by modifying questions
from questionnaires in L’Enseigement Mathematique (1902) and
Muir (1988).
1. Describe your place of work and your role within it.
2. Are you free to choose the mathematical problems you tackle
or are they determined by your work place?
3. Do you work and publish mainly as an individual or as part of
a group?
4. Is supervision of research a positive or negative factor in your
work?
5. Do you structure your time for mathematics?
6. What are your favorite leisure activities apart from
mathematics?
7. Do you recall any immediate family influences, teachers,
colleagues or texts, of primary importance in your
mathematical development?
8. In which areas were you initially self-educated? In which
areas do you work now? If different, what have been the
reasons for changing?
9. Do you strive to obtain a broad overview of mathematics not
of immediate relevance to your area of research?
10. Do you make a distinction between thought processes in
learning and research?
11. When you are about to begin a new topic, do you prefer to
assimilate what is known first or do you try your own
approach?
12. Do you concentrate on one problem for a protracted period of
time or on several problems at the same time?
13. Have your best ideas been the result of prolonged deliberate
effort or have they occurred when you were engaged in other
unrelated tasks?
14. How do you form an intuition about the truth of a proposition?
15. Do computers play a role in your creative work (mathematical
thinking)?
16. What types of mental imagery do you use when thinking about
mathematical objects?
Note: Questions regarding foundational and theological
issues have been omitted in this protocol. The discussion
resulting from these questions are reported in Sriraman
(2004a).
34
Mathematical Creativity
The Mathematics Educator
2004, Vol. 14, No. 1, 35–41
Getting Everyone Involved in Family Math
Melissa R. Freiberg
Teachers from the departments of Mathematics and Computer Science, and Curriculum and Instruction at the
University of Wisconsin-Whitewater collaboratively developed and implemented an evening math event,
Family Math Fun Night, at local elementary schools. As an assignment, preservice elementary teachers
developed hands-on mathematical activities, adaptable for different ages and abilities, to engage children and
parents. The pre-service elementary teachers presented a variety of activities at each school site to small groups
of families and school personnel. This paper outlines the purpose, structure, and benefits of the project for all its
participants.
In an age when we continually hear about the
necessity of parent awareness and involvement in
schools, there are still limited connections among
schools, parents, and higher education institutions. It is
especially important for parents and teachers to be
aware of the premises and types of activities that
support effective mathematics learning as advocated by
the National Council of Teachers of Mathematics
(NCTM, 2000). However, many parents did not grow
up learning in ways the NCTM advocates; they see
hands-on activities as a fun “waste of time” rather than
an avenue for providing conceptual underpinnings for
mathematics. Teachers must realize that fun hands-on
activities, though motivating for students, must also
have mathematical integrity in order to be included in
the curriculum. To facilitate both parents and teachers
reaching these goals, our university presents what we
call Family Math Fun Night (FMFN) at area
elementary schools.
Numerous schools and districts report using some
variation of Family Math to help parents understand
their children’s mathematics curricula better (Wood,
1991, 1992; Carlson, 1991; Pagni, 2002; Kyle,
McIntyre, & Moore, 2001). Our program is a variation
of Stenmark, Thompson, and Cossey’s (1986) Family
Math. In contrast to their Family Math, we choose to
have our preservice teachers present activities at
elementary schools. This provides our preservice
teachers with an opportunity to have a positive, early
experience in schools and allows them to test ideas
about mathematics education they have learned in their
university classes. Also, FMFNs provide an
Melissa Freiberg is an associate professor in the Department of
Curriculum and Instruction at the University of WhitewaterWisconsin. She has a PhD in Urban Education with an emphasis
in teacher education. Her research interests are teacher
induction and hands-on learning.
Melissa R. Freiberg
entertaining family experience centered on academics
with very little expense to or preparation by the school.
Finally, FMFNs offer a unique opportunity for
professional interaction among university and school
faculties and staff.
At the University of Wisconsin-Whitewater, we
require the FMFN project for students enrolled in the
Math for Elementary Teachers content courses and
provide it as an optional project for students enrolled in
the elementary mathematics methods course. Since
students take the Math for Elementary Teachers
courses in their freshman or sophomore year, FMFN is
a good way to get preservice teachers thinking about
the content they are going to teach. Also, the
experience supports the developmental view of
mathematics learning presented in the content course
and provides an experiential background for students in
the methods courses. The preservice teachers use
activities from the Family Math books we keep on
reserve, and we encourage students to devise or find
activities from other sources. The preservice teachers
in the methods courses are especially encouraged to
examine professional journals and databases in
preparation for their projects.
Parent-teacher groups at schools provide a small
amount of funding (usually about $25) to purchase
stickers, pencils, erasers, etc. for prizes; though some
pre-service teachers buy their own, and many preservice teacher groups do not give out prizes at all. The
lack of prizes does not seem to affect the popularity of
the activities for most children. For past FMFNs, we
have received small grants from NASA to devise
activities that have a space theme. We have not
designated a theme for the event since, but have found
that a theme often emerges. For example, we have had
FMFNs whose activities revolve around sports and
FMFNs whose activities relate to voting.
Reflecting on our version of FMFN raises points
of interest that are worth sharing: (1) the types of
35
activities that are presented at the events and what
determines their quality, (2) what considerations are
necessary for coordinating a FMFN, and (3) what can
be learned as a result of the experiences. In the
following sections, I will attend to each of these
categories.
Types of Activities
For each school site, the university organizers
provide two activities in addition to those the preservice teachers present. The first activity uses jars
containing snacks that are taken to the school a week
prior to the FMFN. Jars of varying shape are used for
different age levels. Each class within an age level
estimates the number of snacks in the jar and records
its estimate. During the FMFN, individual students or
parents can make estimates and enter them for a
particular class. The class with the closest estimate
receives the snacks. This activity serves two purposes.
The first purpose is to generate interest in and
awareness of the event and encourage participation.
The second purpose is to support NCTM’s efforts
(NCTM, 2000) by emphasizing estimation skills. The
second activity provided by the university requires a
school representative to greet children and parents at
the door and ask them to add a sticker to his or her
birth month on a pre-designed bar graph. This helps
take attendance for the evening and also helps children
see the process of data collection and how a graph
evolves from the process.
Preservice teachers design all other activities, and
their activities must involve mathematics concepts
covered in their math classes (Math For the
Elementary Teachers I—numeration, whole number
and fraction operations, problem solving; Math For the
Elementary Teacher II—geometry, measurement,
probability, and statistics). The types of projects the
preservice teachers choose to present usually fall into
the categories of drill and practice, problem solving, or
estimation. I will discuss types of activities that fall
into each category and then discuss two exceptional
activities that do not fall into any of the three
categories.
Drill and Practice
Although students are charged (and monitored) to
do more than BINGO or flash cards as the essence of
the activity, drill and practice may be part of the
activity. Pre-service teachers’ initial attempts at
creating these activities are generally weak but with
coaching or feedback, they develop more thoughtprovoking activities. Rich activities designed to
36
incorporate drill and practice are usually presented in
the context of a game. For example, one student group
used a plastic bowling set to practice: addition and
subtraction facts with younger children, how to keep a
running total with slightly older children, and how to
identify fractions and percents for upper elementary
children. Other examples of drill and practice activities
are educational video games in which correct responses
help students reach a goal (fuel for the spacecraft,
money to buy souvenirs, moving closer to a target,
etc.). These activities allow children to pick the
difficulty of the task and move through different levels
of calculation, building their self-confidence and
knowledge. We encourage preservice teachers to
broaden their activities to include topics such as
geometry, estimation, logic, patterns, graph
interpretation, and computation since all are important
to review. Board games are yet another way to support
drill and practice activity. The board is laid out on the
floor so that students walk around it landing on spaces.
When a student is on a space, he or she is asked a
mathematics question that varies depending on the age
of the student.
Problem Solving
Examples of problem solving activities are games
from which preservice teachers create adaptations.
Preservice teachers like to challenge themselves with
games that incorporate mathematical ideas and skills
and then adapt them to the skill level of the children.
Adaptations of games such as Yahtzee® Equations® or
24® help children plan and carry out different
strategies. Memory games, similar to Concentration®
are used to match fractions to decimals, operations to
results, or various representations of numbers. These
games1 are inexpensive to produce, easy to explain,
and easily adaptable for different ages and grade levels.
A second example of a problem solving activity is
asking children to identify or copy patterns in beads,
pictures, tessellations, or shapes. Bead stringing is
commonly used to demonstrate patterns. The youngest
children describe and extend simple patterns while
somewhat older children choose a preset pattern and
string beads to illustrate the pattern. The oldest group
of students designs bead strings that contain multiple
patterns such as combining patterns of color with
patterns of shape or size. This activity is more
expensive because children keep the materials they use
to make the bracelets or necklaces.
Family Math
Estimation
In addition to the introductory snack estimation
activity, almost every FMFN has at least one
preservice teacher designed activity that asks children
to estimate capacity, weight, area, and/or quantity. One
popular activity requires children to estimate through
the use of indirect measurement. In this activity, there
are approximately 15 objects to measure and the
characteristics of objects vary in difficulty according to
children’s differing abilities.
In recent years, we have seen a growing number of
activities that use estimation to help students develop
probability concepts. These activities illustrate our
preservice teachers’ increased awareness of the
importance of estimation and probability as well as
their increased confidence in students’ abilities to do
such activities. In these activities, children are asked
how frequently an event happens or how close an
estimated answer is to the correct solution.
Exceptional Activities
Two exceptional activities from the past do not fall
into any of the above categories. They are exceptional
because they are unique and demonstrate the creativity
of the preservice teachers who made them. The first
was presented in one of the first FMFNs we ran.
Preservice teachers, with the help of the students, used
math symbols to represent letters of each child's name
on a nametag. Children were then told to see if they
could figure out other people’s names by equating the
letter of a name with a math symbol. For example,
Anne's name might be + φ ≠ = (add, null set, not equal,
equal) and she would then know the math symbol that
corresponded to the letter “a”, “n”, and “e” and could
use this to deduce the names of other people.
The second exceptional activity had three pictures
made up of geometric shapes. Children were given a
paper shape and asked to match their paper shape with
the shape in one of the pictures. The youngest children
had shapes that were congruent to shapes in the
picture, while older students were asked to find shapes
similar to their shape but that differed in size, color, or
orientation. The preservice teachers prompted children
to name the shape and describe its attributes. This
activity proved quite challenging for children but was
extremely popular.
Coordinating an FMFN
In organizing FMFNs, we have discovered that
communication among all the parties involved is
essential. We have developed guidelines and a timeline
to facilitate communication, to give schools and
Melissa R. Frieberg
preservice teachers a clear understanding of
expectations, and to detail past problems we have
faced. Since incorporating FMFN into our curriculum,
we have identified objectives and assessments assuring
that FMFN activities are mathematically sound (see
Appendix A and Form A). The most frequent problems
we encounter revolve around logistics such as
coordinating transportation to schools, advertising the
event in the community, and setting up the school
space. The following steps are used to conduct our
FMFN events and might be helpful for those who want
to organize similar work:
1. Contact schools that might be interested in hosting
the event. We contact school districts through direct
mailings or use various connections our department has
to area schools. After several years of conducting three
FMFNs each semester, most schools contact us to
schedule the event.
2. Information about FMFN is given to our preservice
teachers with their class syllabus. The preservice
teachers are allowed to choose the topic around which
they will make their activity (within guidelines
mentioned earlier). Groups may be made up of students
from different classes requiring FMFN or from classes
that offer it as an optional activity.
3. The preservice teachers turn in a description of their
activities (see Form A and Evaluation Form) indicating
how it will be adjusted for various ages/grades, how
parents will be involved, and how they will assess the
success of their activities. This allows the faculty to
assess the activities for mathematical integrity and
avoid redundancies in activities. It also gives students a
foundation for writing their reflections on the event
(See number 8).
4. We assign our preservice teachers to specific dates
and schools based on preferences and class schedules.
Groups are usually made up of three to four people and
about twenty groups are assigned to each school.
5. We confirm who is assigned to each school and
allow groups to indicate special set-up needs (see Form
B). The preservice teachers indicate if they are able to
provide transportation to the schools so car pools can
be established.
6. A faculty member or preservice teacher visits each
school to determine space and resource availability, to
discuss the role of the school staff, and to give
suggestions for advertising the event. We suggest the
school connect FMFN with a regularly scheduled PTA
meeting. Sending reminders home with school
37
children, having the event on the school calendar, and
writing an article in the school or local newsletter
explaining the event are ways that have been effective
in bringing FMFN to the attention of parents.
7. The night of the event, university and school
personnel monitor the preservice teacher groups and
the families attending. At the close of the event, we
announce the winning class for the estimation exercise
and leave activity kits at the school for classroom use.
8. Each university preservice teacher group turns in a
written reflection of impressions of the event. This
report is not only helpful in assessing the university
students' learning, but also helps us identify problems
that might need to be addressed in the future. This
report focuses on the content and success of the
activity, how students handled problems and questions
that arose, how students interacted with parents and
teachers, and how they collaborated with their groups.
9. An individual report is also required from each
preservice teacher. This report is focused on how the
student felt the group process worked, what was
learned about mathematics, and a self-reflection about
one's ability as a teacher.
Conclusions
In the introduction, we stated that we found this
activity to be beneficial to university preservice
teachers, university faculty and staff, school staff,
parents, and especially children. Although this paper is
not intended to present a research study on FMFN, we
believe that we have seen beneficial results for those
involved.
The university students have consistently, and
almost unanimously, responded positively to their
participation in FMFN both in their reports and in class
discussions. Even students who described themselves
as poor math students found the experience to be
enjoyable and uplifting. They appreciated the chance to
work with a small group of elementary students. As
one student said, "I found that helping them [the
children] out with solving a problem was an interesting
and rewarding experience...this is what teaching is all
about." Many university students were especially
surprised and buoyed by the fact that they were able to
adjust questions, offer hints and assistance, or explain
mathematical ideas more easily than they anticipated.
They also learned how to share responsibilities, ask for
help, and make changes to their activities as needed.
Too often preservice teachers believe these things are a
sign of weakness rather than a sign of collaboration.
FMFN helps change that perception.
38
One of the most rewarding results of this
experience for the university faculty and staff is the
opportunity to work collaboratively across departments
and colleges. College of Education faculty/staff who
teach the elementary mathematics methods courses
assist faculty and staff from the Mathematics
Department in planning, implementing, assessing, and
revising the program. Additionally, the experience
provides an opportunity for the Mathematics
Department members to visit local elementary schools
with teachers and children. Education faculty and staff
who do regular supervision of student teachers in
schools get to see students' abilities to teach to a
variety of ages and abilities, which requires flexibility
and instant adaptations that might be missed in single
grade level settings.
Teachers, administrators, and parents are effusive
in their praise for the event. The university students
mention that they often have classroom teachers
waiting “like vultures” to pick up the activity at the end
of the night. Alternately, classroom teachers give
university students ideas for improving or adapting the
projects for different children’s needs or abilities.
Administrators find that the turnout for this event is
higher than for other school sponsored programs and,
interestingly, draws more fathers. We average about
200 participants at each event, even in schools where
there are fewer than 300 students.
Parents have a varied level of involvement in
activities from merely standing and waiting to sitting
down and participating with their children in the
activity. Many times parents mention that they are
surprised at how well their children performed on a
given task or how well they thought through a
problem. In rare instances parents appear to be
impatient or negative with respect to their children’s
efforts, and the university students get their first chance
to try out their mediating skills. Although not a benefit
to children, parent outbursts do give university students
an opportunity to see how parents influence children’s
learning.
Most importantly, it appears that the elementary
children who attend FMFN come away satisfied.
University and school faculty have observed that
students almost universally leave the event feeling
successful and empowered in math. Certainly children
fearful in math are less likely to attend, but we have
watched children start out very tentatively and soon
find themselves immersed in an activity. Virtually all
the children at each event try every activity, but they
return to certain activities—and these are rarely the
easiest activities. This behavior indicates that students
Family Math
are motivated by activities that challenge them and
make them think rather than simple mastery.
In conclusion, we have found that all of us have
gained from the experiences. As university instructors
we continually need to listen to our students in order to
adapt and refine the expectations and requirements for
FMFN. As prospective mathematics teachers, our
students have the chance to devise and carry out
activities in a low-stress, supportive atmosphere.
Schools and teachers are provided with examples
of activities that complement classroom instruction.
Parents see how their children’s active involvement in
activities enhances their learning, and parents may
come away with a better understanding of the
mathematics curriculum. Finally, children always seem
to walk away feeling successful and eager to move on
to the next level in mathematics.
REFERENCES
Carlson, C. G. (1991). Getting parents involved in their children’s
education. Education Digest, 57(10), 10–12.
Kyle, D. W., McIntyre, E., & Moore, G. H. (2001). Connecting
mathematics instruction with the families of young children.
Teaching Children Mathematics, 8, 80–86.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Pagni, D. (2002). Mathematics outside of schools. Teaching
Children Mathematics, 9, 75–78.
Stenmark, J. K., Thompson, V., & Cossey, R. (1986). Family math.
Berkeley: University of California-Berkeley,
Wood, J. (Ed.). (1992). Variations on a theme: Family math night.
Curriculum review, 32(2), 10. Retrieved May 17, 2004, from
Galileo database (ISSN 0147-2453; No. 9705276559).
Wood, J. (Ed.). (1991). “Family math” teaches English as well as
math. Curriculum review, 31(1), 21. Retrieved May 17, 2004,
from Galileo database (ISSN 0147-2453; No. 9705223330).
Internet site for student information on FMFN:
http://facstaff.uww.edu/whitmorr/whitmore/FMFN.html
1
Equations® is a game in which a specific number of cards are
drawn. The cards have whole numbers on them, and students are to
arrange the cards and determine operations that will create an
equation. 24® is a similar game in which each card has four whole
numbers that, when using different operations on the numbers, will
equal 24. Concentration® is a game in which a set of cards is placed
face down in an array, and players take turns turning up two cards
at a time looking for pairs. In commercially made games these are
usually identical pictures; however, in educational games these may
be two equivalent numbers using different symbols or
representations.
Melissa R. Frieberg
Family Math Fun Night Project Requirements
Your grade for this project is based on 80 points. The numbers
following the due dates below indicate the points that can be earned
on each portion of the project.
Jan 31 (5 points) Form A – Group Membership and Activity Idea
Hand in one copy of Form A to each instructor of members of your
group. Your Group ID Code will be assigned when returned.
February 17 for District #1 (School A), February 19 for all others
(20 points) Activity Description
Typed descriptions of your FMFN project should include:
Names of group members with leader indicated, Group ID Code,
name of activity, date and location of presentation.
Procedures and/or instructions you will be giving for the activity.
What the child is to do and learn from your activity? Include
sample problems and activities for each level.
If adults accompany children at the event, how will the adult
participate in your activity?
If prizes are used, how will they be awarded? Who will supply the
prizes?
How will you evaluate different aspects of your activity? Refer to
the attached evaluation sheet used by faculty and questions listed
below.
Feb 24 (School A), Feb 26 (School B), March 5 (School C), March
12 (School D) Form B – Needs List
Hand in one copy of Form B to each instructor of members of your
group.
Mar 7 (School A), March 17 (School B), March 31 (School C),
April 7 (School D) (5 pts) FMFN Evaluation Form
Hand in two copies of FMFN Evaluation Form to your group
leader’s instructor with answers completed for the questions on the
right side of the form.
Mar 11 (School A), March 20 (School B), April 3 (School C), April
10 (School D) (30 points) FMFN Event
Run your activity (6:30-8:00 p.m. in School A & B, 6:00 to 7:30 in
School D) and have fun.
Arrive at school 30 to 60 minutes prior to start.
Set up your activity.
Try to find time to visit and play the activities of other groups
during the evening.
Mar 19 (School A), Apr. 4 (School B), April 11 (School C), April
16 (School D) (20 points) Individual Evaluation
Sorry, no evaluations will be returned until all evaluations have
been graded.
Your individual evaluation of the group learning activity (2 to 3
pages) should include:
Your name, Group ID Code, your activity name, school attended,
group members’ names and their instructor, if other than your
instructor, and method(s) used to evaluate your activity.
Did your group work well together? Why or why not? How well
did you work within your group? What part of the project did you
do?
Briefly state what the math concepts were that you were integrating
into your activity. Was this activity an effective means to convey
these concepts to the student? How could your activity be adapted
for use in a classroom?
What strategies did you see students use? What strategies did you
use to help them succeed?
39
Did things go as planned during FMFN? What did you not
anticipate?
How did you modify/adjust your activity during the evening to
meet the needs of the students/parents? Include specific examples
of difficulties and adaptations.
What would you do differently if you did a similar activity again?
What did you learn about yourself and the grade(s) you are
planning to teach? Is teaching at this level still your goal? Why or
why not?
Grammar and other English mechanics will count.
Form A
Group Membership and Activity Idea (Spring 2003)
Due: Friday, January 31, 2003
If the table has attached benches, our group will need only __
additional chairs.
Our group would also like the following to be supplied by the host
school:
Our group would prefer to be located (please check one and give
your reasoning in the space to the right)
___so we can hang things on a wall behind us
___in a corner of the room
___in the center of the room
___it doesn't matter
___near a power source
Our group (please check one)
___doesn't plan to use prizes
___will supply its own prizes
___is counting on having the school supply prizes
Value: 5 point
Appendix A
Assigned Group ID Code:
Please turn in one copy of this form to each teacher of a member of
your group. (Group ID Code will be assigned after you submit
Form A. Use it on all subsequent submissions.)
Materials will be returned the group via the leader.
Group Leader’s Name, Phone, Email Address, Course/Section,
Teacher's Name, Other Members’ Names:
Brief Activity Description:
Indicate your choice for FMFN presentation. Consider evening
classes, sports schedules, previous commitments, and work
schedules of all members of the group in making your selections. If
your group requires a particular time, please explain the
circumstances. You will not be allowed to switch assignments after
they have been made unless you can find a group able to exchange
with you.
___Our group has no preference of night presentation; any night
will work for us.
___Our group would prefer the following nights: (Please circle first
and second choices, and give reasons in space to the right.)
Does your group have transportation for FMFN? yes
no
Could your group provide transportation for others the
evening of FMFN?
yes
no
Form B
Family Math Fun Night: Needs List
Instructor(s)
Group ID Code
Due:
Please turn in one copy of this form to each teacher of a member of
your group.
Name of Activity:
Brief description of activity:
Group Leader:
Other group members:
Things you may need for your activity:
Tables - Limit your project to one table. These may be lunch tables
with attached benches
Chairs - remember most elementary teachers do not sit down
Tape, scissors, pencils, paper, scrap paper, markers, etc - please
bring your own !!
Our group will need to have (please indicate how many)
Table (zero or one):
Chairs:
40
This semester you will work with elementary students and their
parents/guardians in a project called Family Math Fun Night
(FMFN). This project is designed to show children and parents that
mathematics is an essential part of their everyday life and can be
FUN!! Most importantly, it provides the opportunity for you to be
involved with elementary children as they do mathematics in
enjoyable problem solving activities.
As a member of a group, you will be presenting an activity for
Family Math Fun Night (FMFN) at one of four elementary schools:
School A (PK-5, 300 students); School B (K-5, 280 students; and
School C (K-3, 400 students). All children from these schools and
their families will be invited to attend from 6:30 - 8:00 p.m. (6:00 to
7:30 in one school). The fourth elementary presentation is from 1:30
to 3:00 at School D. We will run all events like a carnival having
booths (tables) set up with various activities. There will not be a
whole group presentation. You should plan to be at your school at
least one half hour early. This will allow you time to set up your
activity, and to visit and enjoy the activities of other groups before
the children and parents arrive. You should be cleaned-up and out of
the school 30 minutes after the closing time.
FORMING GROUPS. Who will design the activities for this
carnival? Your group will select, make, and present your activity at
FMFN. Form a group of four; a group with 3 or 5 students must be
approved by your instructor(s). Group members may be from any
section of the course you are taking. As you are selecting groups,
think about class and work schedules for all of the members of your
group: work on this project will be done outside of class. Also, be
sure that each member of your group can be at the school to present
the activity. You may indicate your group's preference for evening
of presentation. VERY IMPORTANT: After groups have been
assigned an evening, you will not be able to change assignments
unless you can find a group willing to switch with you.
SELECTING AN ACTIVITY. Your group should select an
activity that is accessible and meaningful to the full range of
students in attendance. If you are unsure what is taught at various
grade levels, do some research in the LMC on the lower level of the
library. Your activity should be fun and challenging for students
and parents and need not be competitive. It should involve problem
solving, not merely mechanics or facts. Flash card type drill is not
usually fun, and is not appropriate for a FMFN activity. Be sure to
involve parents in your activity; parents should be doing not just
watching. “Helping by giving hints and encouragement” is not
sufficient adult involvement. Your activity will need to be planned
with space limitations in mind. Plan on setting up on one six to
eight-foot table. Please also realize there will be about twenty
activities in a gym-sized room; consider how your activity and its
Family Math
sounds and lights will affect others. You are not to present an
activity with music, popping balloons or other distractions for
neighboring groups. Be aware of copyright laws! For example, the
latest cartoon characters may attract elementary students, but may
be an infringement of copyright. Invent a clone! Be creative! Don't
just take an activity from a book or off a shelf; put something of
yourself into it. Don’t just use the activity you, or a friend, used last
semester. Math 148 and 149 students should develop an activity
that involves math topics they will be covering in class. Realizing
that there are many connections between the mathematics in the
two courses, this does not exclude presenting a topic from your
course in an activity that also uses a topic from the other course.
Take this opportunity to develop an activity you could use in your
future classroom. Please do not use TWISTER activities.
The book Family Math has been placed on reserve (2 hour, no
overnight) in the library. You will need to ask for it by name at the
main circulation desk. This book has over 100 Family Math
activities. You may wish to use one of these, combine a couple,
modify one, or come up with an idea on your own. You could also
check Teaching Children Mathematics, other periodicals, and the
Internet for ideas. Make this a fun learning experience for you!
WHAT YOU WILL NEED. Your group must have a sign with
the name of your activity. You may need to make some equipment
to be used at your booth such as markers, counters, game board,
etc. Other things such as pencils, scissors, ruler, scrap paper, and
manipulatives are also useful. The LMC has some equipment that
can be checked out. If they cannot meet your needs, your instructor
may have some ideas. You may also want to have copies of
handouts, problems, or puzzles available for parents/teachers to
take home. Remember these are activities for the children and
parents, so make sure they have plenty to DO.
If you feel that prizes would be appropriate for your activity,
please indicate this on Form B that is due 2 weeks before your
FMFN. The PTO's of the various schools have given us some
money with which to purchase small prizes - pencils, erasers,
stickers, etc. These will be divided among the groups requesting
them. There will not be a large number of prizes per group. Please
limit the candy your group plans to use; not all children are allowed
candy, especially after supper. Many groups in the past have
presented very successful activities without prizes. Do not spend a
lot of money purchasing prizes. The students should be having fun
doing math -- NOT seeing who can accumulate the most/best
prizes!
EVALUATION. Three-quarters of your grade will be assigned
through group work. If your group contains members from more
than one class, some written work must be submitted to each
instructor involved. Your group will supply two copies of the
FMFN Evaluation Form a few days prior to your activity night. A
copy of this form is attached. On the night of your presentation,
faculty attending FMFN will evaluate your project. A week after
your FMFN, a typed individual reflective evaluation is to be
submitted. Select a method to help you evaluate your activity. You
may get written evaluations from students and parents; keep a
journal of student/parent reactions during the evening, etc.
March 11, the first FMFN, is only SEVEN weeks away! It is
time to get started selecting a group and an activity NOW. The
deadline for forming groups and selecting an activity is January
31st.
Melissa R. Frieberg
Evaluation Form
Submit two copies to your group leader’s instructor
Activity Date:
Instructor(s):
Activity Name:
Group Members:
Faculty evaluators will use the following portion (and rate between
1 and 5).
Math content: (Problem solving, concept development, more than
mechanics)
Adaptability of project: (Grade level, special needs, mental, written
and manipulative capabilities)
Materials: (Quality, durability and economy of materials)
Appeal and Creativity: (Attract and retain participants)
Interaction: (With students and adults, where possible)
Professionalism: (Dress, group demeanor, setup on time,
enthusiasm)
Total Points (out of 30):
Average number of points:
(Based on
evaluations)
Groups are to provide the following information in the space
provided:
Describe your activity’s math content and how you emphasized it.
How did you adapt your activity to meet all students’ capabilities?
Describe the quality, durability and economy of your
materials.
41
The Mathematics Educator
2004, Vol. 14, No. 1, 42–46
Solving Algebra and Other Story Problems with Simple
Diagrams: a Method Demonstrated in
Grade 4–6 Texts Used in Singapore
Sybilla Beckmann
Out of the 38 nations studied in the 1999 Trends in International Mathematics and Science Study (TIMSS),
children in Singapore scored highest in mathematics (National Center for Education Statistics, NCES, 2003).
Why do Singapore’s children do so well in mathematics? The reasons are undoubtedly complex and involve
social aspects. However, the mathematics texts used in Singapore present some interesting, accessible problemsolving methods, which help children solve problems in ways that are sensible and intuitive. Could the texts
used in Singapore be a significant factor in children’s mathematics achievement? There are some reasons to
believe so. In this article, I give reasons for studying the way mathematics is presented in the elementary
mathematics texts used in Singapore; show some of the mathematics problems presented in these texts and the
simple diagrams that accompany these problems as sense-making aids; and present data from TIMSS indicating
that children in Singapore are proficient problem solvers who far outperform U.S. children in problem-solving.
Why Study the Methods of Singapore’s
Mathematics Texts?
What is special about the elementary mathematics
texts used in Singapore? These texts look very
different from major elementary school mathematics
texts used in the U.S. The presentation of mathematics
in Singapore’s elementary texts is direct and brief.
Words are used sparingly, but even so, problems
sometimes have complex sentence structures. The page
layout is clean and uncluttered. Perhaps the most
striking feature is the heavy use of pictures and
diagrams to present material succinctly—although
pictures are never used for embellishment. Simple
pictures and diagrams accompany many problems, and
the same types of pictures and diagrams are used
repeatedly, as supports for different types of problems,
and across grade levels. These simple pictures and
diagrams are not mere procedural aids designed to help
children produce speedy solutions without
understanding. Rather, the pictures and diagrams
appear to be designed to help children make sense of
problems and to use solution strategies that can be
justified on solid conceptual grounds. Because of this
pictorial, sense-making approach, the elementary texts
Sybilla Beckmann is a mathematician at the University of
Georgia who has a strong interest in education. She has
developed three mathematics content courses for prospective
elementary teachers and has written a textbook, Mathematics for
Elementary Teachers, published by Addison-Wesley, for use in
such courses. In the 2004/2005 academic year, she will teach a
class of 6th grade mathematics daily at a local public middle
school.
Sybilla Beckman
used in Singapore can include problems that are quite
complex and advanced. Children can reasonably be
expected to solve these problems given the problemsolving and sense-making tools they have been
exposed to.
Thus the strong performance of Singapore’s
children in mathematics may be due in part to the way
mathematics is presented in their textbooks, including
the way simple pictures and diagrams are used to
communicate mathematical ideas and to provide sensemaking aids for solving problems. If so, then teachers,
mathematics educators, and instructional designers in
the U.S. will benefit from studying the presentation of
mathematics in Singapore’s textbooks, so that they can
help children in the U.S. improve their understanding
of mathematics and their ability to solve problems.
Using Strip Diagrams to Solve Story
Problems
One of the most interesting aspects of the
elementary school mathematics texts and workbooks
used in Singapore (Curriculum Planning and
Development Division, Ministry of Education,
Singapore, 1999, hereafter referred to as Primary
Mathematics and Primary Mathematics Workbook) is
the repeated use of a few simple types of diagrams to
aid in solving problems. Starting in volume 3A, which
is used in the first half of 3rd grade, simple “strip
diagrams” accompany a variety of story problems.
Consider the following 3rd grade subtraction story
problem:
42
Mary made 686 biscuits. She sold some of them. If
298 were left over, how many biscuits did she sell?
(Primary Mathematics volume 3A, page 20,
problem 4)
A farmer has 7 ducks. He has 5 times as many
chickens as ducks….How many more chickens
than ducks does he have? (Primary Mathematics
volume 3A, page 46, problem 4)
The problem is accompanied by a strip diagram like
the one shown in Figure 1.
(Note: The first part of the problem asks how many
chickens there are in all, hence the question mark about
all the chickens in Figure 3 below.)
Figure 1: How Many Biscuits Were Sold?
Figure 3: How Many More Chickens Than Ducks?
On the next page in volume 3A is the following
problem:
Meilin saved $184. She saved $63 more than Betty.
How much did Betty save? (Primary Mathematics
volume 3A, page 21, problem 7)
This problem is accompanied by a strip diagram like
the one in Figure 2.
Figure 2: How Much Did Betty Save?
These two problems are examples of some of the
more difficult types of subtraction story problems for
children. The first problem is difficult because we must
take an unknown number of biscuits away from the
initial number of biscuits. This problem is of the type
change-take-from, unknown change (see Fuson, 2003,
for a discussion of the classification of addition and
subtraction story problems). The second problem is
difficult because it includes the phrase “$63 more
than,” which may prompt children to add $63 rather
than subtract it. This problem is of type compare,
inconsistent (see Fuson, 2003). The term inconsistent is
used because the phrase “more than” is inconsistent
with the required subtraction. Other linguistically
difficult problems, including those that involve a
multiplicative comparison with a phrase such as “N
times as many as”, are common in P r i m a r y
Mathematics and are often supported with a strip
diagram. Consider the following 3rd grade problem,
which is supported with a diagram like the one in
Figure 3:
Sybilla Beckman
Although the strip diagrams will not always help
children carry out the required calculations (for
example, we don’t see how to carry out the subtraction
$184 – $63 from Figure 2), they are clearly designed to
help children decide which operations to use. Instead
of relying on superficial and unreliable clues like key
words, the simple visual diagram can help children
understand why the appropriate operations make sense.
The diagram prompts children to choose the
appropriate operations on solid conceptual grounds.
From volume 3A onward, strip diagrams regularly
accompany some of the addition, subtraction,
multiplication, division, fraction, and decimal story
problems. Other problems that could be solved with the
aid of a strip diagram do not have an accompanying
diagram and do not mention drawing a diagram.
Fraction problems, such as the following 4th grade
problem, are naturally modeled with strip diagrams
such as the accompanying diagram in Figure 4:
David spent 2/5 of his money on a storybook. The
storybook cost $20. How much money did he have
at first? (Primary Mathematics volume 4A, page
62, problem 11)
Without a diagram, the problem becomes much
more difficult to solve. We could formulate it with the
equation (2/5)x = 20 where x stands for David’s
original amount of money, which we can solve by
dividing 20 by 2/5. Notice that the diagram can help us
see why we should divide fractions by multiplying by
the reciprocal of the divisor. When we solve the
problem with the aid of the diagram, we first divide
$20 by 2, and then we multiply the result by 5. In other
words, we multiply $20 by 5/2, the reciprocal of 2/5.
43
Raju and Samy shared $410 between them. Raju
received $100 more than Samy. How much money
did Samy receive? (Primary Mathematics volume
5A, page 23, problem 1)
Figure 4: How Much Money Did David Have?
The problems presented previously are arithmetic
problems, even though we could also formulate and
solve these problems algebraically with equations. But
starting with volume 4A, which is used in the first half
of 4th grade, algebra story problems begin to appear.
Consider the following problems:
1. 300 children are divided into two groups. There
are 50 more children in the first group than in the
second group. How many children are there in the
second group? (Primary Mathematics volume 4A,
page 40, problem 8)
2. The difference between two numbers is 2184. If
the bigger number is 3 times the smaller number,
find the sum of the two numbers. (P r i m a r y
Mathematics volume 4A, page 40, problem 9)
3. 3000 exercise books are arranged into 3 piles.
The fist pile has 10 more books than the second
pile. The number of books in the second pile is
twice the number of books in the third pile. How
many books are there in the third pile? (Primary
Mathematics volume 4A, page 41, problem 10)
These problems are readily formulated and solved
algebraically with equations, but since the text has not
introduced equations with variables, the children are
presumably expected to draw diagrams to help them
solve these problems. Notice that from an algebraic
point of view, the second problem is most naturally
formulated with two linear equations in two unknowns,
and yet 4th graders can solve this problem.
The 5th grade Primary Mathematics texts and
workbooks include many algebra story problems which
are to be solved with the aid of strip diagrams. Some
do not have accompanying diagrams, but others do,
and some include a number of prompts, such as a
diagram like the one in Figure 5 which accompanies
the following problem:
44
Figure 5: Raju and Samy Split Some Money
Notice that the manipulations we perform with
strip diagrams usually correspond to the algebraic
manipulations we perform in solving the problem
algebraically. For example, to solve the previous Raju
and Samy problem, we could let S be Samy’s initial
amount of money. Then,
2S + 100 = 410
as we also see in Figure 5. When we solve the problem
algebraically, we subtract 100 from 410 and then
divide the resulting 310 by 2, just as we do when we
solve the problem with the aid of the strip diagram.
Strip diagrams make it possible for children who
have not studied algebra to attempt remarkably
complex problems, such as the following two, which
are accompanied by diagrams like the ones in Figure 6
and Figure 7 respectively:
Encik Hassan gave 2/5 of his money to his wife
and spent 1/2 of the remainder. If he had $300 left,
how much money did he have at first? (Primary
Mathematics volume 5A, page 59, problem 6)
Raju had 3 times as much money as Gopal. After
Raju spent $60 and Gopal spent $10, they each had
an equal amount of money left. How much money
did Raju have at first? (Primary Mathematics
volume 6B, page 67, problem 1)
Solving Problems with Simple Diagrams
Penny had a bag of marbles. She gave one-third of
them to Rebecca, and then one-fourth of the
remaining marbles to John. Penny then had 24
marbles left in the bag. How many marbles were in
the bag to start with?
A. 36
B. 48
C. 60
D. 96
(Problem N16, page 19. Overall percent correct,
Singapore: 81%, United States: 41%)
Figure 6: How Much Money Did Encik Hassan Have at
First?
Figure 7: How Much Did Raju Have at First?
Performance of 8th Graders on TIMSS
In light of the complex problems that children in
Singapore are taught how to solve in elementary
school, the strong performance of Singapore’s 8th
graders on the TIMSS assessment is not surprising.
Among the released TIMSS 8th grade assessment
items in the content domain “Fractions and Number
Sense” classified as “Investigating and Solving
Problems,” Singapore 8th graders scored higher than
U.S. 8th graders on all items. These released items
included the following problems (see NCES, 2003):
Laura had $240. She spent 5/8 of it. How much
money did she have left? (Problem R14, page 29.
Overall percent correct, Singapore: 78%, United
States: 25%).
Sybilla Beckman
These problems are similar to problems in Primary
Mathematics. The strong performance of Singapore 8th
graders on these problems indicates that the instruction
children receive in solving these kinds of problems is
effective. Similarly, among the released TIMSS 8th
grade assessment items in the content domain
“Algebra” classified as “Investigating and Solving
Problems,” Singapore 8th graders scored higher than
U.S. 8th graders on all items.
But the strong problem-solving abilities of
Singapore’s 8th graders in fractions and number sense
and in algebra does not necessarily result in factual
knowledge in other mathematical domains in which the
children have not had instruction. For example, U.S.
8th graders scored higher than Singapore 8th graders
on the following item in the content domain “Data
Representation, Analysis and Probability” classified as
“Knowing”:
If a fair coin is tossed, the probability that it will
land heads up is 1/2. In four successive tosses, a
fair coin lands heads up each time. What is likely
to happen when the coin is tossed a fifth time?
A. It is more likely to land tails up than heads up.
B. It is more likely to land heads up than tails up.
C. It is equally likely to land heads up or tails up.
D. More information is needed to answer the
question.
(Problem F08, page 74. Overall percent correct,
United States: 62%, Singapore: 48%)
The mathematics texts used in Singapore through
8th grade do not address probability. Thus the
difference in performance in fraction, number sense,
and algebra problem-solving versus knowledge about
probability can reasonably be attributed to effective
instruction.
45
Conclusion
The mathematics textbooks used in elementary
schools in Singapore show how to represent quantities
with drawings of strips. With the aid of these simple
strip diagrams, children can use straightforward
reasoning to solve many challenging story problems
conceptually. The TIMSS 8th grade assessment shows
that 8th graders in Singapore are effective problem
solvers and are much better problem solvers than U.S.
8th graders. Although cultural factors probably also
affect the strong mathematics performance of children
in Singapore, children in the U.S. could probably
strengthen their problem-solving abilities by learning
Singapore’s methods and by being exposed to more
challenging and linguistically complex story problems
early in their mathematics education.
46
REFERENCES
Curriculum Planning and Development Division, Ministry of
Education, Singapore (1999, 2000). Primary Mathematics (3rd
ed.) volumes 1A–6B. Singapore: Times Media Private
Limited. Note: additional copyright dates listed on books in
this series are 1981, 1982, 1983, 1984, 1985, 1992, 1993,
1994, 1995, 1996, 1997, thus 8th graders who took the 1999
TIMSS assessment used an edition of these books.
Curriculum Planning and Development Division, Ministry of
Education, Singapore (1999, 2000). Primary Mathematics
Workbook (3rd ed.) volumes 1A–6B. Singapore: Times Media
Private Limited.
Fuson, K. C. (2003). Developing Mathematical Power in Whole
Number Operations. In J. Kilpatrick, W. G. Martin, and D.
Schifter, (Eds.), A Research companion to principles and
standards for school mathematics (pp. 68–94). Reston,VA:
National Council of Teachers of Mathematics.
National Center for Education Statistics (2003). Trends in
international mathematics and science study. Retrieved May
3, 2004, from http://nces.ed.gov/timss/results.asp and from
http://nces.ed.gov/timss/educators.asp
Solving Problems with Simple Diagrams
The Mathematics Educator
2004, Vol. 14, No. 1, 47–51
Book Review…
Diverse Voices Call for Rethinking and Refining Notions of
Equity
Amy J. Hackenberg
Burton, L. (Ed.). (2003). Which way social justice in mathematics education? Westport, CT:
Praeger. 344 pp. ISBN 1-56750-680-1 (hb). $69.95.
Editor Leone Burton remarks that the title of this
book reflects a “shift in focus from equity to a more
inclusive perspective that embraces social justice as a
contested area of investigation within mathematics
education” (p. xv). What’s interesting is that the
question in the title lacks a verb—is the question
“which are ways to social justice in mathematics
education?” Or more tentatively, “which ways might
bring about social justice in mathematics education?”
Or perhaps the focus is more on research, either up to
now or in the future: “which ways have research on
social justice in mathematics education taken? Or
“which ways could (should?) research on social justice
in mathematics education take?” Each of the thirteen
chapters in the volume addresses at least one of those
four questions. Overall, this book responds to its title
question through diverse voices that call for expanding
work on gender issues into broader sociocultural,
political, and technological contexts; rethinking and
refining key notions such as equity, citizenship, and
difference; and considering how to conduct studies that
reach beyond school and university boundaries toward
families, communities, and policy-makers.
The collection is the third volume in the
International Perspectives on Mathematics Education
series for which Burton has served as series editor.1 In
her introduction she describes the origin of the book in
the activities of the International Organization of
Women in Mathematics Education (IOWME) at the
Ninth International Congress of Mathematics
Amy Hackenberg is at work on her doctoral dissertation on the
emergence of sixth graders’ algebraic reasoning from their
quantitative reasoning in the context of mathematically caring
teacher-student relations. In addition to her fascination with
mathematical learning and the orchestration of it, she is
compelled by issues of social justice, the nature and
consequences of social interaction, and the relationship between
the “social” and the “psychological” in mathematics education.
47
Education (ICME9) in Tokyo, Japan, in 2000. Perhaps
this context explains why approximately half of the
chapters focus primarily on gender, while other
chapters include issues related to differences in race,
class, language, and thinking styles. Burton notes that
this book, as the fourth publication of IOWME,
“reflects the development of the group’s interests that
have evolved over 16 years from a sharp focus on
gender issues to its present wider interest in social
justice” (p. xiii).
In the introduction Burton also outlines the process
by which the book developed. After a general call for
papers, an international review panel of mathematics
educators reviewed submissions. Chapter authors were
then paired to give feedback to each other on their
work in order to promote dialogue as well as “crossreferencing possibilities” (p. xv). As perhaps is always
the case in an edited book without summary pieces to
highlight connections between chapters, the crossreferencing of concepts in this volume could be
expanded. Burton does a nice job of drawing some
connections in her introduction, but otherwise such
resonance is largely left to the reader. Fortunately, as I
hope to demonstrate in this review, there is ample
opportunity to draw connections between chapters (and
also occasionally to wish that an author had heeded
another author’s points or ideas!)
Organization of the Book
The thirteen chapters in the book are organized
into three sections. The four chapters in the first
section focus on definitional work, conceptual
frameworks, and reviews of and recommendations for
research, thereby “setting the scene” (p. 1). The authors
of this section are from Australia (Brew), Germany
(Jungwirth), the United Kingdom (Povey), and the
United States (Hart). The second section consists of
seven chapters primarily about studies that take place
in classrooms and address the question “what does
Book Review
social justice mean in classrooms?” (p. 101). The
authors of this section are from Australia (Forgasz,
Leder, and Thomas; Zevenbergen), Germany (Ferri
and Kaiser), Malawi (Chamdimba), the United States
and Peru (Secada, Cueto, and Andrade), and the United
Kingdom (Mendick; Wiliam). The last section includes
two chapters focused specifically on “computers and
mathematics learning” (p. 261) with regard to social
justice. The authors (Wood, Viskic, and Petocz; Vale)
come from Australia and Eastern Europe, but all now
practice mathematics education in Australia.
The placement of chapters within this organization
is a little puzzling. Wiliam’s illuminating chapter on
the construction of statistical differences and its
implications is included in the second section on
classroom studies, but since it grapples with definitions
and conceptual ideas (and is not a classroom study), it
might have been better placed in the first more
theoretically-oriented section. Brew’s chapter, a study
about reasons that mothers return to study
mathematics, is included in the first section but seems
to fit better in the second, despite the fact that the study
does not take place in mathematics classrooms.
Support for changing the placement of Brew’s chapter
is provided by the position of Mendick’s: Her report of
young British men’s choices to study mathematics
beyond compulsory schooling is only peripherally
located in classrooms and was still placed in the second
section.
The other weak organizational aspect of the book
is the inclusion of only two chapters in the third section
on computers and mathematics learning. One wonders
if there were intentions for a more substantial section
but some papers did not make the publication deadline.
In any case, because both chapters in this section report
on studies set in classrooms, it seems that they could
have been included in the second section—or that
perhaps two sections about studies might have been
warranted, one that focused directly on studies in
mathematics classrooms and one that included research
on mathematics education outside of immediate
classroom contexts.
Conceptually-Oriented Chapters: What Is Equity?
What Is Social Justice?
Organizational difficulties aside, I focus first on
the more conceptually-oriented chapters, which are
contained in the first three chapters of the first section
of the book as well as in Wiliam’s chapter from the
second section. These authors engage in definitional
and conceptual work that forms a foundation for
research on social justice. All four authors ponder the
48
nature of equity and justice within different contexts: a
typology of gender-sensitive teaching, previous and
current research on equity and justice in mathematics
education, citizenship education in the United
Kingdom, and statistical analyses of gender differences
in mathematics education.
Jungwirth describes a typology of gender-sensitive
teaching that consists of three types distinguished by
modifications made according to gender, the degree to
which gender groups are identified and treated as
monolithic, and corresponding conceptions of equity.
In Type I teaching, teachers are “gender-blind” and
make no modifications according to gender since they
believe that boys and girls can do math equally well. In
Type II teaching, teachers adjust practices based on
gender but tend to treat students of a single gender as
monolithic (i.e., tend to essentialize.) Jungwirth
believes that in the third (and implicitly most
advanced) type, the concept of equity “no longer
applies…Equity here refers to the individual, with
respect to learning arrangements and, somewhat
qualified, to outcomes” (p. 16). Teachers engaging in
Type III teaching attend to individual differences
within gender groups and tailor teaching to individuals.
Although Jungwirth’s typology offers a conceptual
framework for examining the equitable implications of
teachers’ orientations toward mathematics teaching and
mathematics classrooms, her dichotomizing of groups
and individuals is problematic. For example, in their
attention to individuals, might not Type III teachers
create classrooms in which mathematics could be
devoid of women, which Jungwirth sees as
considerably less evolved than even Type I teaching?
The problem seems to be in characterizing equity
based on group-individual dichotomies—to adhere too
strongly to group identities can result in essentializing,
while to focus primarily on the individual can leave out
trends and broad characteristics of groups that are
important considerations in work toward equity and
social justice (cf. Lubienski, 2003).
These issues are reflected in Hart’s review of
scholarship on equity and justice in mathematics
education over the last 25 years. Her chapter is notable
for explicit discussion about different ways researchers
have used equity and justice (and equality); for her
clearly stated choice to use equity to mean justice; and
for her formulation of calls for future research. In
particular, she calls for research on pedagogies that
contribute to justice; self-study of educators’ own
practices; and more research that explores student
motivation, socialization, identity, and agency with
respect to mathematics. Hart highlights Martin’s
Book Review
(2000) study on factors contributing to failure and
success of African American students in mathematics
as an exemplar for future research because of its
multilevel framework for analyzing mathematics
socialization and identity. Although her points about
his work are well taken, the considerable space she
gives to this relatively recent study seems odd given
her aims to review 25 years of research.
Povey continues Jungwirth’s and Hart’s
definitional work by considering the complex and
contested notion of citizenship in relation to social
justice and mathematics education. She describes how
recent mandates for citizenship education in England
reinforce a conservative perspective by focusing on
political and legal citizenship (the right to vote, for
example), without questioning the nature and character
of social citizenship, let alone its connections to “the
(mathematics) education of future citizens” (p. 52).
Povey believes that for citizenship to be a useful
concept in democratizing mathematics classrooms the
concept “will have to be more plural, more active, and
more concerned with participation in the here and
now” (p. 56).
Perhaps the strongest chapter of these four (and
one of the strongest in the collection) is Wiliam’s on
the construction of statistical differences in
mathematical assessments. He demonstrates that in
gender research in mathematics education, effect sizes
of standardized differences between male and female
test scores are relatively small, and the variability
within a gender is greater than between genders. Based
on this analysis, Wiliam concludes that differences
between genders depend on what counts as
mathematics on assessments. In particular, what counts
as mathematics may be maintained because it supports
patriarchal hegemony.
As an implication of his argument, Wiliam
proposes “random justice” (p. 202) to produce equity
in selection based on test scores. Wiliam calls the
percentage of the population that reaches a certain
standard (for, say, entrance to medical school) a
recruitment population. Usually, selecting from a
recruitment population (i.e., creating a selection
population) involves choosing a small top percentage
of it. This mode of selection perpetuates selecting more
males than females, largely because males show
greater variability in their test scores compared to
females (males produce more highs and lows.) Wiliam
proposes that a random sample of the recruitment
population that sustains the gender (or racial, class,
etc.) make-up of it is “the only fair way” (p. 204) of
creating a selection population. Although this proposal
Amy J. Hackenberg
may seem counterintuitive (and certainly differs from
typical U.S. selection processes!), Wiliam makes a
compelling argument that is worth reading.
Chapters on Studies in or Surrounding
Mathematics Classrooms
In these chapters—Brew’s chapter from the first
section as well as the other 8 chapters in the book—the
diverse voices in the volume become quite apparent,
not only because of the different geographical locations
or ethnic heritages of the authors but because of the
diverse ways in which the authors focus on issues of
social justice in relation to mathematics classrooms and
mathematical study. These nine chapters can also be
loosely grouped as exemplifying, supporting,
informing, or aligning with the more conceptuallyoriented chapters.
In particular, two chapters that focus specifically
on teaching practices in relation to social justice may
exemplify and inform Jungwith’s typology. The
authors of these chapters attend to how teachers
approach students who belong to disadvantaged
groups. Chamdimba, whose research took place in the
southern African country of Malawi, studied the year
11 students of a Malawian teacher who agreed to use
cooperative learning to potentially promote a “learnerfriendly classroom climate” (p. 156) for girls. As a
researcher, Chamdimba might exemplify a Type II
orientation out of her concerns over Malawian girls’
lack of representation and achievement in mathematics
and subsequent Malawian women’s lack of bargaining
power as a group for social and economic resources in
the country. Chamdimba’s conclusion that female
students experienced largely positive effects might
help Jungwirth refine her typology so that recognizing
students as part of disenfranchised groups and acting
on that recognition to address the group is seen as
legitimate and useful (i.e., not necessarily less evolved
than Type III teaching.) However, Chamdimba’s study
is also subject to scrutiny over whether a particular
classroom structure can bring about improvements in
all Malawian females’ educational, social, and
economic status.
Perhaps a better example of the subtlety involved
in the group-individual distinctions with regard to
social justice is found in Zevenbergen’s study.
Zevenbergen used Bourdieu’s tools as a frame for
understanding teachers’ beliefs about students from
socially disadvantaged backgrounds in the South-East
Queensland region of Australia. Eight of the 9 teachers
interviewed expressed views of students as deficient
due to poverty and cultural practices. Stretching
49
Jungwirth’s typology beyond gender-sensitivity, the
ninth teacher had more of a Type III orientation in her
respect for these students as individuals. However, by
expressing an understanding of how parents’ lack of
cultural capital prevented them from challenging the
ways in which schools (under)served their children,
this teacher did not ignore these students as belonging
to a disadvantaged group. This teacher’s ability to
understand and value students as both individuals and
part of a group might allow Jungwirth to amplify and
further articulate her typology.
These two chapters and three others exhibit work
that aligns with Hart’s call for research on pedagogies
that contribute to social justice and on one’s own
teaching in relation to social justice. Vale’s two case
studies of computer-intensive mathematics learning in
two junior secondary mathematics classrooms focus on
how teachers’ practices with technology impede (but
might facilitate) more just classroom environments.
Vale’s work is complemented by the three university
classroom studies presented by Wood, Viskic, and
Petocz. In studying their own computer-intensive
teaching of differential equations, statistics, and
preparatory mathematics classes, these three
researchers found positive attitudes toward the use of
technology across gender. Finally, Ferri and Kaiser’s
comparative case study on the styles of mathematical
thinking of year 9 and 10 students (ages 15-16) has
implications for developing pedagogies that recognize
differences other than due to gender, race, or class, and
that thereby contribute to justice and diversity in
classrooms.
However, Secada, Cueto, and Andrade’s largescale, comprehensive study of the conditions of
schooling for fourth and fifth-grade children who speak
Aymara, Quechua, and Spanish in Peru may be the
strongest example of work toward Hart’s
recommendation of multilevel frameworks in research
on social justice. These researchers intended to create a
“policy-relevant study” (p. 106). To do so they
articulated their conceptions of equity as distributive
social justice (opportunity to learn mathematics is a
social good and should not be related to accidents of
birth) and socially enlightened self-interest (it is in
everyone’s interest for everyone to do well so as not to
cause great cost to society). In addition, the researchers
took as a premise that equity must come with both high
quality and equality (i.e., lowering the bar does not
foster equity). Thus they contribute to definitional
work while formulating “practical” conclusions and
recommendations for Peruvian governmental policy.
50
Finally, the remaining three chapters in the book
connect with Povey’s chapter in exploring a particular
contested and complex concept or relate to Wiliam’s
work on considering the construction of difference.
Brew’s study entails rethinking aspects of the complex
concept of mothering in the context of mathematical
learning of both mothers and their children. By
including voices of the children in the study, Brew is
able to show the fluid roles of care-taking between
studying mothers and their children (e.g., children
sometimes acted as carers for their mothers) and “the
pivotal role that children can play…in providing not
only a consistent motivating factor but also enhancing
their mother’s intellectual development” (p. 94).
What Povey does for citizenship and Brew does for
mothering, Mendick does for masculinity in the
context of doing mathematics. In a very strong and
thoughtful chapter, she describes stories of three young
British men who have opted to study mathematics in
their A-levels even though they do not enjoy it.
Mendick’s smart use of a poststructuralist perspective
that deconstructs the classic opposition between
structure and agency allows her to argue that taking up
mathematics is a way for the men to “do masculinity”
in a variety of ways: to prove their intelligence to
employers and others as well as to secure a future in
labor market. The stories of the three males prompt the
question: “why is maths a more powerful proof of
ability than other subjects?” (p. 182). To respond,
Mendick contrasts the men’s stories with young
women’s stories (part of her larger research project.)
This artful move is not intended to draw
dichotomies between how men and women “do maths”
differently—Mendick cautions against such simplistic
conclusions and notes that some females use
mathematics the way these three males do. Instead the
contrast allows her to demonstrate and deepen her
theorizing of masculinity as a relational configuration
of a practice, as well as to argue for more complexity
in gender reform work. Thus for her, “maths and
gender are mutually constitutive; maths reform work is
gender reform work” (p. 184). By examining gender in
this way, like Wiliam, she calls into question
differences between males and females in relation to
mathematics and supports his contention that what
counts as mathematics (and, Mendick would add, as
masculine and feminine) is the basis for these
differences.
Differences between males and females are also
the subject of the chapter by Fogasz, Leder, and
Thomas. They used a new survey instrument to capture
the beliefs of over 800 grade 7–10 Australian students
Book Review
regarding gender stereotyping of mathematics. Their
findings revealed interesting reversals of expected
(stereotyped) beliefs. For example, their participants
believed that boys are more likely than girls to give up
when they find a problem too difficult, and that girls
are more likely than boys to like math and find it
interesting. However, through an examination of
participation rates and achievement levels of male and
female grade 12 mathematics students from 1994 to
1999 in Victoria, Australia, the researchers refute
recent, media-hyped contentions (see, e.g., Conlin,
2003; Weaver-Hightower, 2003) that males are now
disadvantaged in mathematics. Frankly, Fogasz and
colleagues might have benefited from Wiliam’s advice
on examining effect size—it is hard to know how much
significance to give to the differences they found.
Nevertheless, their work supports the notion that
mathematics may be maintained as a male domain
despite certain advances of females.
Overall, I agree with Burton that the chapters in
this volume achieve the goal of providing “an
introduction for new researchers as well as stimulation
for those seeking to develop their thinking in new or
unfamiliar directions” (p. xiii). Although the
organization is a bit puzzling and some chapters are
clearly stronger than others, the book is a useful read
for researchers in mathematics education. More
important, the diversity of voices—and the connections
that readers can draw among this diversity—gives a
complex and layered picture of how resources,
sociocultural contexts, governmental policy, teacher
and student practices, human preferences and
expectations, and researchers’ theorizing and
interpretations, all contribute to “…who does, and who
does not, become a learner of mathematics” (p. xviii).
Amy J. Hackenberg
REFERENCES
Conlin, M. (2003, May 26). The new gender gap. Business Week
online. Retrieved September 1, 2003, from
http://www.businessweek.com
Lubienski, S. T. (2003). Celebrating diversity and denying
disparities: A critical assessment. Educational Researcher,
32(8), 30–38.
Martin, D. B. (2000). Mathematics success and failure among
African-American youth: The roles of sociohistorical context,
community forces, school influence, and individual agency.
Mahwah, NJ: Lawrence Erlbaum.
Weaver-Hightower, M. (2003). The “boy turn” in research on
gender and education. Review of Educational Research, 73(4).
471–498.
1
The first volume was Multiple Perspectives on Mathematics
Teaching and Learning (2000) edited by Jo Boaler; the second
volume was Researching Mathematics Classrooms: A Critical
Examination of Methodology (2002) edited by Simon Goodchild
and Lyn English.
51
CONFERENCES 2004…
CMESG/GCEDM
Canadian Mathematics Education Study Group
http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html
Universite Laval
Quebec, Canada
May 28–June 1
HIC
The 3rd Annual Hawaii International Conference on
Statistics, Mathematics and Related Fields
http://www.hicstatistics.org/index.htm
Honolulu, Hawaii
June 9–12
EDGE Symposium
Graduate School Experience for Women in Mathematics:
From Assessment to Action
http://www.edgeforwomen.org/symposium.html
Atlanta, Georgia
June 25–26
AMESA
Tenth Annual National Congress
http://www.sun.ac.za/MATHED/AMESA/AMESA2004/Index.htm
Potchefstroom,
South Africa
July 1–4
ICOTS7
International Conference on Teaching Statistics
http://www.maths.otago.ac.nz/icots7/layout.php
Salvador, Brazil
July 2–7
ICME – 10
The 10th International Congress on Mathematics Education
http://www.icme-10.dk
Copenhagen, Denmark
July 4–11
HPM
History & Pedagogy of Mathematics Conference
http://www-conference.slu.se/hpm/about/
Uppsala, Sweden
July 12–17
PME-28
International Group for the Psychology of Mathematics Education
http://home.hia.no/~annebf/pme28/
Bergen, Norway
July 14–18
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings
Toronto, Canada
August 8–12
CABRI 2004
Third CabriGeometry International Conference
http://italia2004.cabriworld.com/redazione/cabrieng2004
Rome, Italy
September 9–12
GCTM
GCTM Annual Conference
http://www.gctm.org/georgia_mathematics_conference.htm
Rock Eagle, Georgia
PME-NA
North American Chapter
International Group for the Psychology of Mathematics Education
http://www.pmena.org
Toronto, Canada
October 21–24
SSMA
School Science and Mathematics Association
http://www.ssma.org
College Park, Georgia
October 21–23
AAMT 2005
Australian Association of Mathematics Teachers
http://www.aamt.edu.au/mmv
Sydney, Australia
January 17–20
2005
52
October 14–16
Book Review
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In this Issue,
Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways Respected by
the Mathematics Education Community?
CHANDRA HAWLEY ORRILL
Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim Toward
Empowering All Children With a Key to the Gate
DAVID W. STINSON
The Characteristics of Mathematical Creativity
BHARATH SRIRAMAN
Getting Everyone Involved in Family Math
MELISSA R. FREIBERG
In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a Method
Demonstrated in Grade 4–6 Texts Used in Singapore
SYBILLA BECKMANN
Book Review… Diverse Voices Call for Rethinking and Refining Notions of Equity
AMY J. HACKENBERG