____ THE
_____ MATHEMATICS ___
________ EDUCATOR _____
Volume 13 Number 2
Fall 2003
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Brian R. Lawler
Along with a team of editors, reviewers, and other colleagues that help me to think, I wish to
present to you—the ever-present reader in our work—the second and final issue to be produced
during my brief tenure as editor of Volume 13 of The Mathematics Educator. At this moment in
mathematics education, while reform remains a normalizing discourse and accountability snarls
as a threatening tyrant, the work of teachers clearly remains central in our collective efforts to
grow the field. In these welcoming words, I hope to interest you in the papers assembled here,
while provoking you to read alongside ideas and theory that may not usually be with you.
In this issue’s final essay, Adelyn Steele, a Kansas state finalist for the Presidential Award of
Excellence in Mathematics and Science Teaching, reflects on the work of the teacher. She makes
evident that this work, in which she suggests that we “just get out of the way,” is an immeasurably
artistic maneuvering amid an intent to provoke both thoughtfulness and autonomy in the learner.
Several of the papers herein review and build theory that can inform and impact our
collective efforts toward such goals in teacher education. Boris Handal, Drew Ishii, and Norene
Lowery review and build ground-level theory to help us think and act when working alongside
evolving teachers. These researchers’ work for reform in mathematics education concentrates on
pressing together the beliefs and actions of teachers, then developing reflective practitioners to
make sense of what they do and are trying to do. Melissa DeHaven and Lynda Wiest help us
consider reform by documenting effects of a particular design for a girls mathematics and
technology program.
Continuing to move in reverse order through the journal, Danny Martin opens up and
troubles calls for equity in the reform discourse. Refusing the co-optation of equity work into the
always already unjust institution of public education, he presents a view upward and into the
inequitable structures of schooling to make possible the chance to think differently about our
work for reform in mathematics education.
In considering the body of research and theory within this issue of TME, it is evident the
work of mathematics educators is intra-human relations and activity—and thus by it’s nature,
political. Invigorated to know our work is not value-free, Brian Greer and Swapna
Mukhopadhyay challenge us to think deeply about what is mathematics education for? What may
emerge in our field if we reject Chomsky’s identified goal of schooling “to keep people from
asking questions”?
I hope this issue of TME is both insightful and stimulating. I hope the research and reference
materials provide room for you, the reader, to think. I hope you are reminded to not stop
wondering along with me, “What kind of politics am I doing in my classroom?”
Associate Editors
Holly Garrett Anthony
Dennis Hembree
Zelha Tunç-Pekkan
Publication
Laurel Bleich
Advisors
Denise S. Mewborn
Nicholas Oppong
James W. Wilson
MESA Officers
2003-2004
President
Dennis Hembree
Vice-President
Erik Tillema
Secretary
R. Judith Reed
Treasurer
Angel Abney
NCTM
Representative
Holly Garrett Anthony
Undergraduate
Representative
Tiffany Goodwin
Brian R. Lawler
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@coe.uga.edu
www.ugamesa.org
About the cover
Excerpt from a speech given at Western Michigan University, December 18, 1963:
Some time ago, it was our good fortune to journey to that great country known as India. I never will forget the experience. I never will forget the marvelous
experiences that came to Mrs. King and I as we met and talked with the great leaders of India, met and talked with hundreds and thousands of people all over the
cities and villages of that vast country. These experiences will remain dear to me as long as the chords of memories shall linger. But I must also say that there
were those depressing moments, for how can one avoid being depressed when he sees with his own eyes millions of people going to bed hungry at night? How
can one avoid being depressed when he sees with his own eyes millions of people sleeping on the sidewalks at night, no beds to sleep in, no houses to go in. How
can one avoid being depressed when he discovers that out of India’s population, more than 400,000,000 people, some 380,000,000 earn less than ninety dollars a
year. Most of these people have never seen a doctor or dentist. As I notice these conditions, something within me cried out, “Can we in America stand idly by
and not be concerned?” Then an answer came, “Oh, no, because the destiny of the United States is tied up with the destiny of India and every other nation.” I
started thinking about the fact that we spend millions of dollars a day to store surplus food. I said to myself, I know where we can store that food free of charge,
the wrinkled stomachs of the millions of God’s children that go to bed hungry at night.
All I’m saying is simply this, that all life is interrelated, that somehow we’re caught in an inescapable network of mutuality tied in a single garment of destiny.
Whatever affects one directly affects all indirectly. For some strange reason, I can never be what I ought to be until you are what you ought to be. You can never
be what you ought to be until I am what I ought to be. This is the interrelated structure of reality. John Donne caught it years ago and placed it in graphic terms.
“No man is an Island, entire of itself; every man is a piece of a Continent, a part of the main.” He goes on toward the end to say “Any man’s death diminishes me
because I am involved in mankind; and therefore never send to know for whom the bell tolls; It tolls for thee.” It seems to me that this is the first challenge. This
emerging new age.
Dr. Martin Luther King, Jr.
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
Fall 2003
Volume 13 Number 2
Table of Contents
2
Guest Editorial… What is Mathematics Education For?
BRIAN GREER & SWAPNA MUKHOPADHYAY
7
Hidden Assumptions and Unaddressed Questions in Mathematics for All Rhetoric
DANNY BERNARD MARTIN
23 The Fourth “R”: Reflection
NORENE VAIL LOWERY
32 Impact of a Girls Mathematics and Technology Program
on Middle School Girls’ Attitudes Toward Mathematics
MELISSA A. DEHAVEN & LYNDA R. WIEST
38 First-Time Teacher-Researchers Use Writing in Middle School
Mathematics Instruction
DREW K. ISHII
47 Teachers’ Mathematical Beliefs: A Review
BORIS HANDAL
58 In Focus… Just Get Out of the Way
ADELYN STEELE
22 Upcoming conferences
60 Subscription form
61 Submissions information
© 2003 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2003, Vol. 13, No. 2, 2–6
Guest Editorial…
What is Mathematics Education For?
Brian Greer
Swapna Mukhopadhyay
This is a great discovery, education is politics! After that, when a teacher discovers that he or she is a
politician, too, the teacher has to ask, What kind of politics am I doing in the classroom?
-Paulo Freire
We are not experts in social and political theory,
but rather educators struggling to understand the
implications and manifestations of Paulo Freire’s
(Freire & Shor, 1987, p. 46) statement, with particular
reference to mathematics education. The views
expressed here are personal and emergent, and
intended to be provocative.
According to Apple (2000): “It is unfortunate but
true that there is not a long tradition within the
mainstream of mathematics education of both critically
and rigorously examining the connections between
mathematics as an area of study and the larger relations
of unequal economic, political, and cultural power” (p.
243). However, there are signs of change, building on a
major shift within the discipline of mathematics
education from a mainly cognitive and pedagogical
perspective towards recognition of the historical,
cultural, and social contexts of both mathematics and
mathematics education (e.g., various chapters in
Boaler, 2000). This shift is encapsulated in the phrase
“mathematics as a human activity” whence the
acknowledgment of the political situatedness of
mathematics education is a natural outgrowth
(Mukhopadhyay & Greer, 2001).
Brian Greer is in the Department of Mathematics and Statistics and
the Center for Research on Mathematics and Science Education at
San Diego State University. His PhD was in the School of
Psychology, Queen’s University, Belfast, where he worked until
2000. His interests have unfolded from a narrow focus on cognitive
aspects to recognition of the cultural embedding and political
ramifications of mathematics education.
Swapna Mukhopadhyay is in the School of Education at Portland
State University. For her PhD in Education at Syracuse University
she studied the ethnomathematics of potters and weavers in a
village in her native India. She characterizes mathematics as a
cultural construction and is committed to the view that mathematics
education should promote social justice. She serves on the
committee of the Portland branch of Rethinking Schools.
2
Mathematics, Mathematics Education, And
Mathematics In People’s Lives
The answer to the question “What is
mathematics?” is generally considered to be relatively
unproblematic, although continuing to evolve as a
result of internal developments and external factors
such as accessibility to the power of computers as
processors of symbols and images. However, having
accepted that modern mathematics is a worldwide,
unified discipline, close-knit through global
communication and networks of scholars and
institutions, there remain the questions of the
relationship between that body of knowledge and what
is taught in schools, how, and why. Given the pace at
which mathematics has been, and is being, developed,
the gap is increasing between the body of knowledge
and what can reasonably be included in school
education. At the same time, there is more and more
concern about the gap between school mathematics and
the lived experience of students and the adults that they
become.
Davis and Hersh (1981, pp. 39) composed an
imaginary dialogue between “the ideal mathematician”
and the public information officer of the University,
part of which goes like this:
P.I.O.: Do you see any way that the work in your area
could lead to anything that would be
understandable to the ordinary citizen of this
country?
I.M.: No.
P.I.O.: How about engineers or scientists?
I.M.: I doubt it very much.
P.I.O.: Among pure mathematicians, would the
majority be interested in or acquainted with
your work?
I.M.: No, it would be a small minority.
What is Mathematics Education For?
It is, then, hardly controversial to assert that
mathematics is now too big to allow school students to
be exposed to more than a fraction. So, on what basis,
and by whom, are selections made?
Both reflecting and reinforcing the highly
organized nature of mathematics as a discipline, there
is a very striking uniformity of school mathematics
curricula across the world. Usiskin (1999, p. 224)
observed students in Shanghai solving Euclidean
geometry problems exactly like those in Japanese and
American texts, even to the point of noticing the
abbreviation “SAS” (for the Side-Angle-Side
congruence condition) among the Chinese characters.
As Usiskin also pointed out, the existence of
international exercises such as the Third International
Mathematics and Science Survey (TIMSS) assumes
enough curricular commonality to make such
comparisons meaningful (e.g., p. 213).
As another example of how mathematics education
shows remarkable uniformity over time and cultures,
Fasheh (1997) describes living through four
educational systems in Palestine—British, Jordanian,
Jordanian with Israeli modification, “Palestinian”—and
then comments:
What is startling about the math curriculum
is—with the exception of some changes at the
technical level—how stubborn and unchanging it
has remained under the four completely different
realities in which I have lived, studied, and taught;
how insensitive and unresponsive it has been to the
drastic changes that were taking place in the
immediate environment! When something like this
is noticed, it is only natural to ask whether this is
due to the fact that math is neutral or that it is
actually dead! (p. 24)
However, in various parts of the world, attempts
have been made to combat global homogenization of
mathematics education combined with the predominant
mode of teaching that dissociates mathematics from
people’s lived experience. Describing “People’s
Mathematics” (Julie, 1993), Volmink (1999)
commented that it “developed independently and
indigenously rather than an attempt to embrace the
loudest fad from the West” (p. 94) and listed as
distinguishing features:
• an ability to reveal how school mathematics can be
used to reproduce social inequalities
• a rejection of absolutism in school mathematics and
its contribution towards seeing mathematics as a
human activity and therefore necessarily fallibilist
• an incorporation of the social history of mathematics
into mathematics curricula
Brian Greer & Swapna Mukhopadhyay
• a belief in the primacy of applications of
mathematics
The last point above exemplifies one trend that we
consider potentially positive. Insofar as there is greater
emphasis in curricula on applications of mathematics
and increasing incorporation of data handling into the
curriculum, these changes open up possibilities for
diversification through using mathematics to analyse
socially and culturally relevant problems. For example,
Gutstein (2003) writes about teaching in a low-income
Mexican immigrant community in Chicago:
I use ideas of social justice along with helping
students develop mathematical power (being able
to reason and communicate mathematically,
develop their own mathematical thinking, and
solve real-world problems in multiple and novel
ways)—and pass the “gatekeeping” standardized
tests. (p. 35)
Another ways of making connections between
mathematics education and the lives of students is to
break down the barriers between schools and
communities (e.g., Abreu, 2002; Civil, 2002; Moll &
Greenberg, 1992). Fasheh (2000) declared “I cannot
subscribe to a system that ignores the lives and ways of
living of the social majorities in the world; a system
that ignores their ways of living, knowing and making
sense of the world” (p. 5). By an extension of these
principles, those who research mathematics education
are separated only artificially from the social and
political realities within which they work (Vithal &
Valero, 2001).
To summarize, we are suggesting that the
relationships
between
these
three
aspects—mathematics as a discipline, mathematics as a
school subject, and mathematics as a part of people’s
lives—need serious analysis.
I see mathematics playing an important role in
achieving the high humanitarian ideals of a new
civilization with equity, justice, and dignity for the
entire human species without distinction of race,
gender, beliefs and creeds, nationalities, and
cultures, but achieving these goals depends on our
understanding of the relation between mathematics
and human behavior. Consideration of this relation
is normally untouched by mathematicians,
historians of mathematics, and mathematics
educators. (D’Ambrosio, 1999, p. 143)
What Is Mathematics Education For?
We list, and make brief comments on, a number of
answers. All have validity, so how they are evaluated is
a matter of balance and priorities, which vary with
3
experience, intellectual history, beliefs, values, and
ideologies.
1. For some mathematicians, the obvious purpose
of mathematics education is to produce more
mathematicians (and also scientists, engineers, and
others who will use substantial technical mathematics
in their work). At the extreme, this supports a
conception of mathematics education as a pyramid,
with curriculum planned primarily for the few at the
peak, and the majority left to struggle up as far as they
can manage. Some of the calls for “mathematics for
all” amount to just trying harder to push more people
further up the pyramid. There is a hint of that attitude
in the following statement by the National Council of
Teachers of Mathematics (NCTM):
NCTM challenges the assumption that mathematics
is only for the select few. On the contrary,
everyone needs to understand mathematics. All
students should have the opportunity and the
support necessary to learn significant mathematics
with depth and understanding. There is no conflict
between equity and excellence. (2000, p. 5)
2. However, recently it has become extremely
common to portray the main reason for mathematics
education as the training of a workforce able to
compete successfully in the global economy of the
information age. The following statement comes from
a spokesperson of the People’s Republic of China, but
could have come from almost anywhere in the world:
As the economy adapts to information-age needs,
workers in every sector must learn to interpret
computer-controlled processes. Most jobs now
require analytical rather than merely mechanical
skills. So most students need more mathematical
ability in school as preparation for their future jobs.
... [P]eople must deal daily with profit, stock,
market forecast, risk evaluation etc. Therefore,
mathematics relevant to these economic activities,
such as ratio and proportion, operational research
and optimization, systematic analysis and decision
theory, etc., should be a part of school mathematics
education. (Er-sheng, 1999, p. 58)
Gatto (2003) presents an argument that public
schooling in the United States was shaped by
industrialists (notably Carnegie, Morgan, Rockefeller,
and Ford) in order to produce a docile and efficient
workforce.
3. Briefly and uncontroversially, mathematics—as
much as literature or music—is part of the cultural
heritage that can make people intellectually well
rounded and creative solvers of intellectual problems.
We assert, without argument or evidence, that
4
mathematics education has mostly failed disastrously
in these respects.
4. Mathematics is also characterized as the purest
form of reasoning, embodying the highest standards of
proof; and as a training in dispassionate, objective,
rational thinking. We do not attempt here to analyse
the various critiques of this position.
5. It is often stated that mathematics is needed as
preparation for the practicalities of everyday life. Does
this statement bear scrutiny? Is it not the case that most
people handle the practicalities of daily life effectively
without benefit of school mathematics beyond simple
arithmetic and that the knowledge and skills that are
essential are acquired through learning within practices
situated outside of school? On the other hand, we argue
below that there are other aspects of people’s lives that
could and should be radically improved through access
to mathematical tools for critical analysis.
6. From the perspective of a different value system,
the most important reason for mathematics education is
to make accessible to many people powerful
mathematical ideas as conceptual and critical tools to
analyse issues relevant to their lives (e.g., Skovsmose
& Valero, 2002). For example, the application of
mathematics as a critical tool for the analysis of
American society is illustrated by an exercise
beginning with the question “If Barbie was as tall as
one of us, what would she look like?” (Mukhopadhyay,
1998).
7. According to Davis and Hersh (1986):
The social and physical worlds are being
mathematized at an increasing rate. The moral is:
We’d better watch it, because too much of it may
not be good for us. (p. xv)
Mathematics not only reflects our view of the
world, but also helps to shape it, so that “when part of
reality becomes modeled and remodeled, then this
process also influences reality itself” (Skovsmose,
2000, p. 5). What Skovsmose terms “the formatting
power of mathematics” is by no means a new
development, but it is amplified by technological
developments. It seems clear that the ratio:
accessible information
––––––––––––––––––––––––––––––––––
conceptual means for making sense of it
is accelerating, with unforeseeable consequences.
Looking Around
In sketchy and illustrative form, some prominent
features of the contemporary politico-educational scene
in the USA are the following:
What is Mathematics Education For?
Underfunded mandates: A recent entry in Harper’s
Index (Sept. 2003) reads:
Change since last year in federal spending to
implement the No Child Left Behind Act:
$1,200,000,000. (p. 13)
In case you are wondering, the change was
downwards.
A “black box” model for control of schools:
Within the black box are the teachers and the students
and the human interactions that constitute
teaching/learning. What is inside the box can be
ignored as control is exerted through the manipulation
of external levers—money, testing, and punishment
being the main ones.
Corruption: Robert Kimball, an assistant principal
in Houston, was surprised that in his high school with a
freshman class of 1,000 that was reduced to fewer than
300 by senior year, the number of dropouts reported
was zero (Winerip, 2003). When he blew the whistle,
Robert Kimball was isolated and expects to be fired in
January—you might like to track his story. Horn and
Kincheloe (2001) compiled a generally skeptical
analysis of the “Texas miracle”.
Fantasy: The first President Bush set a goal for the
USA to be number one in math and science education
by the year 2000. Now it is mandated that every child
in the USA will pass reading and math proficiency
tests by 2014. There is only one way in which this
could happen, namely by disappearing those who don’t
make it, like the dropouts of Houston.
Inequity: All of the above are contributory factors
to the failure to diminish the “performance gap”
between white students and minority groups, in
particular African-Americans and Latinos. We
attended a meeting recently where a public school
teacher spoke of a report (in English) being sent to
parents who do not speak English telling them that
their child, who also does not speak English, had
scored zero in a test written in English. There is
currently a class action suit, Williams vs. the State of
California, arguing that California provides a
fundamentally inequitable education to students based
on wealth, and based on language status. As
background for this case, Gándara, Rumberger,
Maxwell-Jolly, and Callahan (2003) have documented
seven aspects of this inequity.
Naïve expectations about the power of research:
Slavin (2002) wrote as follows:
At the dawn of the 21st century, educational
research is finally entering the 20th century. The
use of randomized experiments that transformed
medicine, agriculture, and technology in the 20th
Brian Greer & Swapna Mukhopadhyay
century is now beginning to affect educational
policy… [A] focus on rigorous experiments
evaluating replicable programs and practices is
essential to build confidence in educational
research among policymakers and educators.
However, … there is still a need for correlational,
descriptive, and other disciplined inquiry in
education. (p. 15)
The best cautionary rejoinders that we know of to
the expectations that mathematics education can be
automatically improved through evidence-based
policies generated by rigorous research (again, the
image of a black box comes to mind) are Freudenthal’s
(1978) book Weeding and sowing and Kilpatrick’s
(1981) paper The reasonable ineffectiveness of
research in mathematics education. The latter, in
particular, points out that the improvement of
mathematics education is hard because it is not an
engineering problem, but a human problem. The
endeavor rests in fundamental ways on questions that
lie beyond the powers of research to generate definitive
answers, but rather related to beliefs, values, and the
aims of education.
Intellectual child abuse: Without singling out any
example (you might like to select your own), we assert
that the most salient features of most documents that
lay out a K-12 program for mathematics education is
that they make an intellectually exciting subject boring.
Emotional child abuse: One of the really big
questions in mathematics education is: “Why do so
many people fear and dislike mathematics?” Here is
one answer, from a Bronx school (Wilgoren, 2001):
It is a morning ritual… [The teacher] stalks across
his classroom, scowls at his sixth-grade students
and barks the same simple question: “What is
this?” “This is math,” they respond. “I don’t have
to like it to pass it. I don’t have to enjoy it to learn
it. I don’t have to love it to understand it. But I
must, and I will, master it”. (p. A1)
Final Comments
Freire used the term “conscientization” to refer to a
process of critical self-consciousness. As stated in the
opening quotation, this implies reflection on the
political nature of what we are doing as teachers or
others engaged in education. During a recent meeting
with students at Portland State University, Donaldo
Macedo commented on the virtual absence from
university education courses of classes on topics such
as “ethics” or “ideology”. This comment recalls the
statement of Chomsky (2000) that “the goal [of
schools] is to keep people from asking questions that
matter about important issues that directly affect them
5
and others” (p. 24). Is it possible to turn this around, to
make schools and universities places where people do
ask such questions? How many graduate programs in
mathematics education have a class on political aspects
of mathematics education? The establishment of such
classes might be a good way to start, if change is to
occur within our field.
REFERENCES
Abreu, G. de (2002). Mathematics learning in out-of-school
contexts: A cultural psychology perspective. In L. English
(Ed.), Handbook of international research in mathematics
education: Directions for the 21st century (pp. 323–353).
Mahwah, NJ: Erlbaum.
Apple, M. W. (2000). Mathematics reform through conservative
modernization? Standards, markets, and inequality in
education. In J. Boaler (Ed.), Multiple perspectives on
mathematics teaching and learning (pp. 243–259). Westport,
CT: Ablex.
Boaler, J. (Ed.). (2000). Multiple perspectives on mathematics
teaching and learning. Westport, CT: Ablex.
Chomsky, N. (2000). Chomsky on miseducation. Lanham, MD:
Rowman and Littlefield.
Civil, M. (2002). Everyday mathematics, mathematicians’
mathematics, and school mathematics: Can we bring them
together? In M. Brenner and J. Moschkovich (Eds.), Journal
of Research in Mathematics Education Monograph #11:
Everyday and academic mathematics in the classroom (pp.
40–62). Reston, VA: National Council of Teachers of
Mathematics.
D’Ambrosio, U. (1999). Literacy, matheracy, and technoracy: A
trivium for today. Mathematical Thinking and Learning, 1,
131–154.
Davis, P. J., & Hersh, R. (1981). The mathematical experience.
Brighton, England: Harvester.
Davis, P. J., & Hersh, R. (1986). Descartes’ dream: The world
according to mathematics. Brighton, England: Harvester.
Er-Sheng, Ding (1999). Mathematics curriculum reform facing the
new century in China. In Z. Usiskin (Ed.), Developments in
mathematics education around the world, Volume 4 (pp.
58–70). Reston, VA: National Council of Teachers of
Mathematics.
Fasheh, M. (1997). Is math in the classroom neutral—or dead? For
the Learning of Mathematics, 17(2), 24–27.
Fasheh, M. (2000, September). The trouble with knowledge. Paper
presented at the meeting A global dialogue on building
learning societies—knowledge, information and human
development, Hanover, Germany.
Gutstein, E. (2003). Home buying while brown or black.
Rethinking Schools, 18(1), 35–37.
Harper’s Index (2003, September). Harper’s Magazine, p. 13.
Horn, R. A. Jr., & Kincheloe, J. L. (Eds.), (2001). American
standards: Quality education in a complex world: The Texas
case. New York: Peter Lang.
Julie, C. (1993). People’s mathematics and the applications of
mathematics. In J. de Lange, I. Huntley, C. Keitel, & M. Niss
(Eds.), Innovating in mathematics education by modeling and
applications (pp. 32–40). London: Ellis Horwood.
Kilpatrick, J. (1981). The reasonable ineffectiveness of research in
mathematics education. For the Learning of Mathematics,
2(2), 22–29.
Moll, L. C., & Greenberg, J. B. (1992). Creating zones of
possibilities: Combining social contexts for instruction. In L.
C. Moll (Ed.), Vygotsky and education: Instructional
implications of sociocultural psychology (pp. 319–348).
Cambridge: Cambridge University Press.
Mukhopadhyay, S. (1998). When Barbie goes to classrooms:
Mathematics in creating a social discourse. In C. Keitel (Ed.),
Social justice and mathematics education (pp. 150–161).
Berlin: Freie Universitat.
Mukhopadhyay, S., & Greer, B. (2001). Modeling with purpose:
Mathematics as a critical tool. In B. Atweh, H. Forgasz, & B.
Nebres (Eds.), Sociocultural Research on mathematics
education: An international perspective (pp. 295-312).
Mahwah, NJ: Erlbaum.
National Council of Teachers of Mathematics (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Skovsmose, O. (2000). Aporism and critical mathematics
education. For the Learning of Mathematics, 20(1), 2–8.
Skovsmose, O., & Valero, P. (2002). Democratic access to
powerful mathematical ideas. In L. English (Ed.), Handbook
of international research in mathematics education:
Directions for the 21st century. Mahwah, NJ: Erlbaum.
Slavin, R. E. (2002). Evidence-based education policies:
Transforming educational practice and research. Educational
Researcher, 31(7), 15–21.
Usiskin, Z. (1999). Is there a worldwide mathematics curriculum?
In Z. Usiskin (Ed.), Developments in mathematics education
around the world, Volume 4 (pp. 213–227). Reston, VA:
National Council of Teachers of Mathematics.
Vithal, R., & Valero, P. (2001). Researching mathematics
education in situations of social and political conflict.
Roskilde, Denmark: Centre for Research in Learning
Mathematics.
Freire, P., & Shor, I. (1987). A pedagogy for liberation. Westport,
CT: Bergin & Garvey.
Volmink, J. D. (1999). School mathematics and outcomes-based
education: A view from South Africa. In Z. Usiskin (Ed.),
Developments in mathematics education around the world,
Volume 4 (pp. 84–95). Reston, VA: National Council of
Teachers of Mathematics.
Freudenthal, H. (1978). Weeding and sowing. Dordrecht, The
Netherlands: Reidel.
Wilgoren, J. (2001, June 6). Repetition + Rap = Charter School
Success. New York Times, pp. A1.
Gándara, P., Rumberger, R., Maxwell-Jolly, J., & Callahan, R.
(2003). English learners in California schools: Unequal
resources, unequal outcomes. Education Policy Analysis
Archives, 11, Number 36.
Winerip, M. (2003, August 13). The ‘zero dropout’ miracle: Alas!
Alack! A Texas tall tale. New York Times, p. B7. (Reprinted in
Rethinking Schools, 18(1), 8).
Gatto, J. T. (2003). The underground history of American
education. New York: Oxford Village Press.
6
What is Mathematics Education For?
The Mathematics Educator
2003, Vol. 13, No. 2, 7–21
Hidden Assumptions and Unaddressed Questions
in Mathematics for All Rhetoric
Danny Bernard Martin
In this article, I discuss some of the hidden assumptions and unaddressed questions in the increasingly popular
Mathematics for All rhetoric by presenting an alternative, critical view of equity in mathematics education.
Conceptualizations of equity within mainstream mathematics education research and policy have, for the most
part, been top-down and school-focused in ways that marginalize equity as a topic of inquiry. Bottom-up,
community-based notions of education in mathematics education are often of a different sort and more focused
on the connections, or lack thereof, between mathematics learning and real opportunities in life. Because of
these differences, there has been a continued misalignment of the goals for equity set by mathematics educators
and policy makers in comparison to the goals of those who continue to be underserved in mathematics
education. I also argue that equity discussions and equity-related efforts in mathematics education need to be
connected to discussions of equity in the larger social and structural contexts that impact the lives of
underrepresented students. Achieving Mathematics for All in the context of limited opportunity elsewhere may
represent a Pyrrhic victory.
Portions of this paper are based on the author’s published dissertation, Martin (2000), postdoctoral work, Martin (1998), and
an earlier paper, Martin (2002a), presented at the Annual Meeting of the American Educational Research Association, New
Orleans, 2002.
In this article, I discuss some of the hidden
assumptions and unaddressed questions in the
increasingly popular Mathematics for All rhetoric by
presenting an alternative, critical view of equity
rhetoric in mathematics education. My arguments will
probably generate more questions than answers, but it
is my hope that any subsequent discussion serves as a
catalyst to move mathematics educators beyond the
rhetoric stage in this movement toward meaningful
action.
Mathematics for All is a worthy philosophical
approach to mathematics education. However,
mathematics educators should not be satisfied with
working toward equity in mathematics education
simply for the sake of equity in mathematics education
and settling for small victories like Mathematics for
All. For reasons of social justice, I also argue that
equity discussions and equity-related efforts in
mathematics education should extend beyond a myopic
Danny Bernard Martin is Professor and Chair of Mathematics at
Contra Costa College. He received his Ph.D. in Mathematics
Education from the University of California, Berkeley and was a
National Academy of Education Spencer Postdoctoral Fellow in
1998-2000. His research focuses on equity issues in mathematics
education; mathematics learning among African Americans; and
mathematics teaching and curriculum issues in middle school,
secondary, and community college contexts. He is also interested in
school-community collaborations.
Danny Bernard Martin
focus on modifying curricula, classroom environments
and school cultures absent any consideration of the
social and structural realities faced by marginalized
students outside of school and the ways that
mathematical opportunities are situated in those larger
realities (e.g., Abraham & Bibby, 1988; Anderson,
1990; Apple, 1992/1999, 1995; Campbell, 1989;
D’Ambrosio, 1990; Frankenstein, 1990, 1994;
Gutstein, 2002, 2003; Martin, 2000b; Martin, Franco,
& Mayfield-Ingram, 2003; Stanic, 1989; Tate, 1995;
Tate & Rousseau, 2002).
Empty Promises and Prior Reforms
In order to add a bit of historical context to my
critical remarks, I want to briefly revisit three
interrelated events within mathematics education, each
occurring about fifteen years ago.
An Attempt to Frame Equity and Achievement.
The first event occurred in 1988. In that year,
Reyes and Stanic (1988) published one of the most
significant pieces of literature on issues of race, sex,
socioeconomic status, achievement, and persistence to
have appeared within the field at that time. In that
article, they provided a useful, although incomplete,
theoretical framework to explain differences in
mathematics achievement. That framework served as a
foundation upon which to base future research on
7
equity issues in mathematics education. The paper
called for studies exploring relationships among the
following factors: teacher attitudes, societal influences,
school mathematics curricula, classroom processes,
student achievement, student attitudes, and student
achievement-related behaviors.
Creating Standards for Mathematics Learning.
The second event occurred in 1989. The National
Council of Teachers of Mathematics (NCTM)
published its Curriculum and Evaluation Standards for
School Mathematics (NCTM, 1989). The authors of
that document were given two charges (a) create a
coherent vision of what it means to be mathematically
literate both in a world that relies on calculators and
computers to carry out mathematical procedures and in
a world where mathematics is rapidly growing and is
extensively being applied in diverse fields and (b)
create a set of standards to guide the revision of the
school mathematics curriculum and its associated
evaluation toward this vision.
In addition to creating a vision for mathematical
literacy and setting standards for school mathematics,
the Curriculum and Evaluation Standards (NCTM,
1989) also contained very strong statements about
equity, stressing the fact that all students should learn
mathematics, not just the college-bound or (white)
males:
1.
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. Women and most
minorities study less mathematics and are seriously
underrepresented in careers using science and
technology. 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. x)
2.
Finally, in developing the standards, we considered
the content appropriate for all students…. The
mathematical content outlined in the Standards is
what we believe all students will need if they are to
be productive citizens in the twenty-first century. If
all students do not have the opportunity to learn
this mathematics, we face the danger of creating an
intellectual elite and a polarized society. The image
of a society in which a few have the mathematical
knowledge needed for the control of economic and
scientific development is not consistent either with
the values of a just democratic system or with its
8
economic needs. We believe that all students
should have an opportunity to learn the important
ideas of mathematics expressed in these standards.
(p. x)
These statements constituted the early tenets of the
Mathematics for All movement and characterized the
early discourse surrounding this movement. Along
with similar statements found in other reform-oriented
documents (e.g., National Research Council [NRC],
1989), they also alluded to the fact that African
American, Latino, Native American, female, and poor
students have traditionally trailed their White and
Asian American peers on most measures of
achievement and persistence and have lacked access to
the kind of mathematics that allows them to fully
function in school and society (Meyer, 1989; Tate,
1997; Tate & Rousseau, 2002). Further, these
statements acknowledged that both policy and
curriculum changes are needed to help reverse these
trends.
Underserving a Generation of Students.
At the time, the authors of the Curriculum and
Evaluation Standards may have believed that strong
statements about equity, in combination with the
principles outlined in that document, would lead to the
kind of reform efforts that would help alleviate
inequities in achievement and persistence for future
generations of underserved students. However,
achievement and persistence data show that African
American, Latino, Native American, and many poor
students continued to experience these inequities (e.g.,
Tate, 1997). By way of evidence for this last statement,
consider the third event, which occurred in 1990. It
was then that the students representing the Class of
2002 entered first grade. Looking back, I would argue
that many of those students were not well-served by
more than a decade of mathematics education reform
and strong statements about equity. Data from the
Third International Mathematics and Science Studies
(TIMSS) show that American students, as a group,
continued to lag behind their peers in many countries
(Schoenfeld, 2002). It can also be argued that the most
underserved students of the Class of 2002 were large
numbers of African American, Latino American,
Native American, and poor students. National
Assessment of Educational Progress (NAEP) data over
the past fifteen years reveal that although there have
been some modest gains in mathematics achievement
and persistence by these students (Schoenfeld, 2002),
disparities continue to exist and there is evidence that
Hidden Assumptions
differences in achievement may be increasing once
again (e.g., Lee, 2002).
The convergence of affairs described in the three
events above leads me to the following conclusion:
Despite strong equity-oriented discourse in the 1989
Curriculum and Evaluation Standards, the
development of equity-based frameworks such as those
outlined by Reyes and Stanic (1988) and others
(Oakes, 1990; Secada, 1989, 1992; Secada, Fennema,
& Adajian, 1995; Secada, Ogbu, Peterson, Stiff &
Tonemah, 1994) and despite increased understandings
of how students learn, how teachers teach, and
improved methods of assessing teachers and
students—math educators have yet to produce
adequate solutions to differential achievement and
persistence along ethnic lines. Equity in mathematics
education remains elusive more than a decade
following the three events described above.
Renewing the Promise
Nearly fifteen years after publication of the
Curriculum and Evaluation Standards, the architects of
mathematics education reform have produced an
updated standards document entitled Principles and
Standards for School Mathematics1 (NCTM, 2000).
According to Schoenfeld (2002), the Standards are “a
vision statement for mathematics education designed to
reflect a decade’s experience since the publication of
the [Curriculum and Evaluation Standards]” (p. 15).
They make explicit the mathematics that is valued and
describe the goals for learning this valued mathematics
(i.e., mathematics for life, mathematics as a part of
cultural heritage, mathematics for the workplace, and
mathematics for the scientific and technical
community). In the March 2002 issue of the NCTM
News Bulletin, NCTM past-President Lee Stiff
confirmed this when he stated that the 1989
Curriculum and Evaluation Standards “described the
teaching and learning that were valued. In the updated
version of this document… the teaching and learning
outcomes that we continue to value are revisited” (p.
3). Like their 1989 counterpart, the 2000 Standards
also indicate which students should learn this valued
mathematics (i.e., Mathematics for All), how they
might go about learning it, and how we should assess
both teachers and students as they attempt to teach and
learn it. Noticeably absent are references to teaching
and learning mathematics for social justice; that is,
having those who have been traditionally shut out of
the mathematics pipeline learn mathematics to help
them improve the conditions of their lives.
Danny Bernard Martin
In effect, the old and new Standards documents
describe what Apple (1992/1999, 1993) calls the
official knowledge of mathematics education. This term
is an outgrowth of Apple’s contentions that some
forms of knowledge are more valued than others and
that these preferred “forms of curricula, teaching, and
evaluation in schools are always the results of such
accords or compromises where dominant groups, in
order to maintain their dominance, must take the
concerns of the less powerful into account” (1993, p.
10). Apple indicated that these compromises “are
usually not impositions, but signify how dominant
groups try to create situations where the compromises
that are formed favor them” (p. 10). Having identified
mathematics knowledge as a form of high-status
knowledge and having invoked the questions of What
knowledge is of most worth? and Whose knowledge is
of most worth?, Apple offered a critical analysis of the
1989 Curriculum and Evaluation Standards. This
critique challenged the notions of mathematics literacy
called for in the standards. In one part of that analysis,
he stated:
The recognition that mathematical knowledge is
often produced, accumulated, and used in ways that
may not be completely democratic requires us to
think carefully about definitions of mathematical
literacy with which we now work and which are
embedded in Standards volumes…. My arguments
in this article are based on a recognition that there
is a complex relationship between what comes to
be called official knowledge in schools and the
unequal relations of power in the larger society…. I
have claimed that one of the primary reasons that
mathematics knowledge is given high status in
current reform efforts is not because of its beauty,
internal characteristics, or status as a constitutive
form of human knowing, but because of it
socioeconomic utility for those who already
possess economic capital. In order for our students
to see this and to employ mathematics for purposes
other than the ways that now largely dominate
society, a particular kind of mathematics literacy
may be required. (Apple, 1992/1999, p. 97-98)
In light of this critique, it is reasonable to ask
whether the updated Standards address the substance
of Apple’s concerns. The updated Standards are based
on six core principles: equity, curriculum, teaching,
learning, assessment, and technology. Because equity
is listed first among the core principles, there appears
to be an implied promise that Standards-based reform
will result in the kind of significant change that will be
necessary to improve achievement and persistence
among marginalized students (Martin, Franco, et al,
9
2003). Like its 1989 counterpart, the 2000 Standards
volume also indicates which students should learn
mathematics, how they might go about learning it, and
how we should assess both teachers and students as
they attempt to teach and learn mathematics.
Moreover, the language of Mathematics for All
continues to emanate from this and other recent
documents that discuss standards (e.g., RAND
Mathematics Study Panel, 2003). In writing about
standards and equity, Alan Schoenfeld, who is widely
recognized as a leader in the field of mathematics
education, recently stated “Mathematical literacy
should be a goal for all students” (2002, p. 13).
Building on the ideas of Robert Moses (Moses, 1994;
Moses & Cobb, 2001; Moses, Kamii, & Swap, 1989),
Schoenfeld also likened mathematics literacy to a new
form of civil rights, highlighting the belief that “the
ongoing struggle for citizenship and equality for
minority people is now linked to an issue of math and
science literacy” (Moses, 1994, p. 107). It is important
to accept such statements by leaders in the field as
good-faith efforts to bring attention to the inequities
faced by marginalized students in mathematics
education. However, to ensure that such statements
about equity and Mathematics for All do not amount to
another decade of empty promises and sloganizing, I
believe that continued interrogation, similar to Apple’s
(1992/1999) critique of the Curriculum and Evaluation
Standards, should be extended to the 2000 Standards
and other current reform efforts that claim to have
equity as a goal. Only through ongoing critical analysis
and reflection is it possible to ensure that attention to
the issues affecting mathematics achievement and
persistence among African American, Latino, Native
American, and poor students remain front and center
and that high quality mathematics teaching, learning,
curriculum, and life opportunities become a reality for
these students, many whom have lacked access to and
benefited very little from previous reform efforts,
despite strong pronouncements about equity (Martin,
Franco, et al, 2003).
Indeed, if one compares the discourse about equity
found in the 1989 Curriculum and Evaluation
Standards (see above) to that found in the 2000
Standards—which is limited to statements about high
expectations and strong support for all students—it is
very apparent that earlier language stressing
mathematics learning for liberatory purposes and
having marginalized students use mathematics to
critically analyze the conditions in which they live has
subsided. In fact, the Equity Principle of the Standards
contains no explicit or particular references to African
10
American, Latino, Native American, and poor students
or the conditions they face in their lives outside of
school, including the inequitable arrangements of
mathematical opportunities in these out of school
contexts. I would argue that blanket statements about
all students signals an uneasiness or unwillingness to
grapple with the complexities and particularities of
race, minority/marginalized status, differential
treatment, underachievement in deference to the
assumption that teaching, curriculum, learning, and
assessment are all that matter2 (e.g., NRC, 2002). A
recent pronouncement by the NRC (2002) involving
research on the influence of standards on mathematics
and science education held that rigorous research, by
definition, cannot be conceptualized as advocacy work.
However, the quest to make sure that equity issues are
brought to the fore, and remain there, in mathematics
education research will involve the kind of advocacy
work that some do not see as legitimate.
Mathematics for All: How Do We Get There?
In addition to the publication of the Principles and
Standards for School Mathematics, a potentially
influential paper on equity issues in mathematics
education has appeared. That paper, authored by
Allexsaht-Snider and Hart (2001)3, is entitled
“Mathematics for All”: How Do We Get There?. In it,
the authors synthesize progress on equity issues in
mathematics education over the last decade. Based on
their reviews of the research literature and of analysis
of math education reform, they suggest three areas of
focus for continued research: structural aspects of
school districts, teacher beliefs about diverse students
and the learning of mathematics, and classroom
practices.
The paper by Allexsaht-Snider and Hart is
especially timely because its appearance, against the
backdrop of persistently low achievement by minority
and poor students and critiques such as that leveled by
Apple (1992/1999), leads to questions about how far
mathematics education for under-represented students
has evolved and questions about how researchers and
policy-makers will respond to a host of complex
equity-related issues that were not of paramount
importance fifteen years ago:
• Rapidly changing demographics that will
continually challenge our definitions of equity and
diversity, both in terms of defining student
populations and determining what resources are
needed to help these students excel (e.g., Day,
1993).
Hidden Assumptions
•
•
•
Changing curriculum and course-taking policies in
many school districts that now require all students,
despite their prior preparation, to enroll in algebra
by 8th or 9th grade.
High-stakes testing in mathematics that will have a
disproportionately negative impact on underrepresented students given that many of these
students have less access to high-quality teaching
and curriculum and that accountability measures
for low test performance are often punitive in
nature (e.g., Gutstein, 2003; Tate, 1995; Tate &
Rousseau, 2002).
A changing economy that now relies on large
numbers of foreign-born workers to fill math and
science-based technical jobs and less on the large
pool of under-represented students who remain on
the periphery of mathematics and science.
Critiquing Equity and Mathematics for All Rhetoric
In my view, an analysis of equity in mathematics
education that takes into the account the issues raised
above and that contemplates the tensions that these
considerations raise for Mathematics for All will help
move mathematics educators beyond the rhetoric stage.
Below, I attempt such an analysis by focusing on four
main themes: (1) the marginalization of equity issues
within mathematics education research, (2) the
misalignment of top-down and bottom-up approaches
to equity, (3) restrictive definitions of equity, and (4)
the need to situate equity concerns within a broader
conceptual framework that extends beyond classrooms
and curricula.
Complicity and Marginalization of Equity Issues.
Echoing similar claims made by others (e.g., Cobb &
Nasir, 2002; Gutstein, 2002, 2003; Khisty, 2002;
Secada, 1989; Secada et al, 1995), I suggest that a
major reason the mathematics education community
has struggled with achievement and persistence issues
among underrepresented students, and why effective
solutions have been slow in coming, lies in how
mathematics educators have dealt with equity-related
issues, both in terms of the theoretical frameworks and
analytical methods that have been employed and the
equity-related goals that have been set.
If we examine the way that the “equity problem” in
mathematics education has been situated and defined
relative to the other research that gets done, it can be
said that, contrary to its listing at the first principle in
the Standards, equity has been a marginalized topic in
mathematics education (Meyer, 1989; Secada, 1989,
1991, 1992; Secada et al, 1995; Skovsmose & Valero,
Danny Bernard Martin
2001; Thomas, 2001). Discussions of equity within
mathematics education have typically been confined to
special sessions at conferences, special issues of
journals, or critical issues sections of books. In my
view, the status of African American, Latino, Native
American, and poor students has not been a primary
determinant driving mathematics education reform.
When discussions do focus on increasing participation
among these students, it is usually in reference to
workforce and national economic concerns. Secada
(1989) called this “enlightened self-interest.” Gutstein
(2003) stated “to discuss equity from the perspective of
U.S. economic competition is to diminish its moral
imperative and urgency” (p. 38).
Even within the context of the “math wars4,” an
intense political and philosophical debate between
those supporting traditional, skills-focused approaches
to mathematics teaching and learning and those
supporting approaches called for in the 1989
Curriculum and Evaluation Standards (i.e. a focus on
conceptual understanding, connections, mathematical
communication, multiple representations, and
analyzing data), the needs of marginalized students
have never been the center of discussion in these very
public arguments. As such arguments have raged on
among academics and politically powerful interest
groups, marginalized students have continued to suffer
low achievement and limited persistence. When a
similar debate about skills versus process approaches
to writing erupted in the field of literacy, Delpit (1995)
had the following to say:
In short, the debate is fallacious; the dichotomy is
false. The issue is really an illusion created initially
not by teachers but by academics whose worldview
demands the creation of categorical divisions—not
for the purpose of better teaching, but the for the
goal of easier analysis. As I have been reminded by
many teachers... those who are most skilled at
educating black and poor children do not allow
themselves to be placed in “skills” or “process”
boxes. They understand the need for both
approaches. (p. 46)
I would also argue that such debates are
symptomatic of a certain kind of complicity that has
been largely ignored in discussions involving equity,
accountability, and standards-setting. Despite strong
statements about equity that were included in the 1989
Curriculum and Evaluation Standards and despite the
fact that equity has been listed as the first cornerstone
principle of the 2000 NCTM Standards, one of the
great paradoxes of mathematics education reform over
the last fifteen years is that the very same community
11
that has engineered these reforms also has the dubious
distinction of overseeing the inequities in achievement
and persistence that have characterized the experiences
of many poor and minority students (Martin, Franco, et
al, 2003).
Because equity concerns have not been central to
mainstream mathematics education research, there is
also a risk that recent attention to these issues have
turned equity into the “problem of the day” in the same
way that trends in mathematics education research
have shifted from one “theory of the day” to another
whether it be cognitive analyses, constructivism, or
situated learning. The last several years have seen the
rise of cognitive and decontextualized analyses (e.g.,
Davis, 1986; Schoenfeld, 1985, 1987) followed by a
transition to situated analyses (e.g., Anderson, Reder,
& Simon, 1996; Brown, Collins, & Duguid, 1989;
Cobb, 2000; Cobb & Bowers, 1999; Cobb, Yackel, &
Wood, 1992; Lave & Wenger, 1991). The first research
approach has resulted in studies that include
marginalized students but that do not explicitly address
the social and contextual factors that contribute to their
underachievement, focusing instead on content and
problem-solving behaviors. Studies in the situated
approach have addressed issues of context but in such
limited ways that discussions of differential
socialization, stratification, opportunity structure,
ethnicity, and social class are often noticeably absent.
Misalignment of Top-Down and Bottom-Up
Approaches.
Rather than responding directly to the needs of
marginalized students and centering discussions around
what is best for these students, policy makers and
mathematics educators have decided what (valued)
mathematics should be learned, who should learn this
mathematics, and for what purposes equity in
mathematics is to be achieved. I want to suggest that
conceptualizations of equity within mathematics
education have, for the most part, been top-down and
school-focused. Very little equity research and policy
has focused on bottom-up, community-based notions
of equity (e.g., Moses, 1994; Moses & Cobb, 2001).
Class (2002), for example, has stated that such bottomup approaches are unusual among education reformers,
who typically focus on curriculum, teaching, and test
scores and who believe that equity has been achieved
“when differences among sub-groups... of students are
disappearing” (Allexsaht-Snider & Hart, 2001, p. 93)
as a result of fixing or remedying curriculum, teachers,
and funding streams.
12
On the other hand, equity in mathematics
education, as defined by marginalized students,
parents, and community members is likely to be related
as much to their day-to-day experiences in those outof-school contexts whose participation is mediated or
dictated by knowledge of mathematics as it is to their
school-based experiences (Anhalt, Allexsaht-Snider, &
Civil, 2002; Civil, Andrade, & Anhalt, 2000; Civil,
Bernier, & Quintos, 2003; Lubienski, 2003; Martin,
2000, 2003; Perissini, 1997, 1998). In my own research
with African American adults and adolescents, I have
found that a failure to benefit from mathematics
knowledge, both real and perceived, and perceptions
about limitations in the larger opportunity structure has
an impact on the desire to invest or re-invest in
mathematics learning (Martin, 2000, 2003). Because of
the differences in these top-down and bottom-up
approaches to equity, interventions formulated by
mathematics educators have remained, and are likely to
remain, out of alignment with the inequities
experienced by underrepresented students, parents, and
communities.
Defining Equity in Mathematics Education.
How has equity in mathematics education been
defined and what essential elements of these working
definitions are missing? I raise this question because if
we are to get there, it certainly helps to understand
where there is. Moreover, the definitions we use in
solving problems also serve as intellectual compasses
for the solution routes that we take. As a starting point
in their discussion, Allexsaht-Snider and Hart (2001)
define5 equity in mathematics education as follows:
Equity in mathematics education requires: (a)
equitable distribution of resources to schools,
students, and teachers, (b) equitable quality of
instruction, and (c) equitable outcomes for
students. Equity is achieved when differences
among sub-groups in these three areas are
disappearing. (p. 93)
This definition is in response to the welldocumented disparities in achievement and persistence
outcomes that have remained among between African
American, Latino, Native American, and poor students
on the one hand and many White and Asian American
students on the other. This is significant because it is
only recently that definitions of equity in mathematics
education have addressed the students to whom we
now apply them. Past concerns with educating the best
and the brightest to achieve national competitiveness
for the United States have now shifted to a concern for
mathematics for all. That is, definitions of and
Hidden Assumptions
approaches to equity in mathematics education have
ranged from being highly selective and conditional to
being as broad and non-specific as mathematics for all.
Rather than centering our discourse in mathematics
education on the relationships between mathematics
learning and the kind of mathematics that leads to real
opportunities in the lives for marginalized students,
what I call opportunity mathematics (Martin, 2003;
Martin, Franco, et al, 2003), we have continued to
norm our efforts and discussions around White,
middle-class students and the kinds of mathematics
that they have long been given access (Stanic, 1989). I
would further argue that too little of the mathematics
learned by many African American, Latino, Native
American, and poor students leads to the kinds of
opportunities that improve their conditions in life.
Enrollment patterns in high-status mathematics courses
substantiate this claim (e.g., Oakes, 1990; Tate &
Rousseau, 2002).
If we are truly interested in critically examining
issues of equity so that we can be more responsive to
the needs of students, teachers, parents, and
communities, several questions must be brought to
bear: Do our definitions of equity gloss over the deeply
embedded structures that produce inequities? Do
reform-minded equity efforts get transformed in ways
that continue to leave some groups on the outside
looking in? Do theoretical perspectives and equityoriented rhetoric take into account the collective
histories of the groups for whom equity is desired,
resisting the temptation to attribute low achievement to
race and ethnicity instead of highlighting the
devastating effects of raci s m and the way that
schooling and curriculum has contributed to
differential opportunities to learn (Apple, 1992/1999).
Most important, will we resist the temptation to accept
short-term gains (i.e. all students taking algebra) as
evidence that equity in mathematics education has been
achieved?
Rather than restricting our definitions of and goals
for equity to equal access, equal opportunity to learn,
and equal outcomes, I would like to suggest that math
educators working to eliminate inequities seek to
extend Allexsaht-Snider and Hart’s (2001) three areas
of focus. A focus on structural aspects of school
districts, teacher beliefs about diverse students, and
classroom practices is important but, in many ways,
this focus does not allow us to situate disproportionate
achievement and persistence patterns within a broader
conceptual framework of sociohistorical, structural,
community, school, and intrapersonal factors (Atweh,
Forgasz, & Nebres, 2001; Gutstein, 2003; Martin,
Danny Bernard Martin
2000; Oakes, 1990). I suggest that a fourth goal of
equity research should be to empower students and
communities with mathematics knowledge and literacy
as a powerful act of working for social justice and
addressing issues of unequal power relations among
dominant and marginalized groups (e.g., Abraham &
Bibby, 1988; Anderson, 1990; Apple, 1992/1999;
D’Ambrosio, 1990; Frankenstein, 1990, 1994;
Gutstein, 2002, 2003; Moses & Cobb, 2001).
Comments by Apple (1992/1999) are helpful in
clarifying this broader conceptual framework:
Education does not exist in isolation from the
larger society. Its means and ends and the daily
events of curriculum, teaching, and evaluation in
schools are all connected to patterns of differential
economic, political, and cultural power…. That is,
one must see both inside and outside the school at
the same time. And one must have an adequate
picture of the ways in which these patterns of
differential power operations operate. In a society
driven by social tensions and by increasingly larger
inequalities, schools will not be immune
from—and in fact may participate in
recreating—these inequalities. If this is true of
education in general, it is equally true of attempts
to reform it. Efforts to reform teaching and
curricula—especially in such areas as mathematics
that have always been sources of social
stratification, as well as possible paths of
mobility—are also situated within these larger
relations. (p. 86)
Situating Equity Within a Broader Conceptual
Framework.
Some might ask What is the marginal gain in
adding this fourth goal? I believe, as do others who
support this goal (e.g., Frankenstein, 1990, 1994;
Gutstein, 2002, 2003; Ladson-Billings, 1995; Tate,
1995), considerations of social justice force
mathematics educators to think beyond curriculum and
classrooms so as to situate mathematics learning for
marginalized students within the larger contexts that
impact their lives. Without attention to the ways in
which the arrangement of mathematical, and other,
opportunities outside of school further contributes to
the marginalization of African American, Latino,
Native American, and poor students, I believe equitybased efforts in mathematics education will continue to
fall short. Ensuring that marginalized students gain
access to quality curriculum and teaching, experience
equitable treatment, and achieve at high levels should
mark the beginning of equity efforts, not the end. If
these students are not able to use mathematics
knowledge in liberatory ways to change and improve
13
the conditions of their lives outside of school, they will
continue to be marginalized even while mathematics
educators and policy makers claim small victories like
Mathematics for All.
Recent work by Gutstein (2003) with low-income
Mexican and Mexican American students and families
is also helpful in understanding the goals of a social
justice pedagogy in mathematics education. Given the
sociopolitical context in which these students and
families lived, Gutstein stated “An important principle
of a social justice pedagogy is that students themselves
are ultimately part of the solution to injustice, both as
youth and as they grow into adulthood. To play this
role, they need to understand more deeply the
conditions of their lives and the sociopolitical
dynamics of their world” (p. 39). He set the following
goals and objectives for his teaching and his students’
learning:
Goals of Teaching for Social
Justice
Specific Mathematics-Related
Objectives
Develop Sociopolitical
Consciousness
Read the World Using
Mathematics
Develop Sense of Agency
Develop Mathematical Power
Develop Positive
Social/Cultural Identities
Change Dispositions Toward
Mathematics
Figure 1: Gutstein’s (2003) goals and objectives
for student learning
In one project, entitled Racism in Housing Data?,
Gutstein (2003) asked his students to “use mathematics
to help answer whether racism has anything to do with
the housing prices” (p. 47) in a particular county. More
specifically, he asked his students to address questions
such as the following: (a) What mathematics would
you use to answer that question?, (b) How would you
use the mathematics?, and (c) If you would collect any
data to answer the question, explain what data you
would collect and why you would collect the data.
It is clear that Gutstein is attempting to situate
mathematics teaching and learning in a context that
extends beyond curriculum and classrooms and that he
is also attempting to help his students use mathematics
to change the conditions of their lives.
I also point out that Gutstein’s work and
perspective provide evidence for another of my claims:
that there are subtle, but important differences,
between achieving equity (a goal) and eliminating
i n e q u i t y (a process) (Tate, 1995). The first
conceptualization—equity as a goal—assumes that
there is a point to be reached when all is well and the
hard work of getting there can cease. This view also
14
ignores the fact of changing demographics that will
continually challenge us to refine our definitions of
equity. Although our current conceptions of equity
often do not take into account the realities and needs of
marginalized groups, new conceptualizations of equity
concerns will have to. When those who are
marginalized in mathematics begin to exercise their
individual and collective agency and power to demand
the kind of mathematical literacy leading to real
opportunities, policy makers and mathematics
educators will have no choice but to listen to these
voices and to formulate visions of equity that move
these individuals and groups from the periphery of
mathematics to the center. The convenient
“compromises” described by Apple earlier in this paper
will no longer suffice.
However, the second conceptualization of
equity—as a process—highlights the fact that the
necessary hard work will be ongoing and even when
gains are made, a high degree of vigilance will be
necessary to ensure that needs of marginalized students
are attended to and that our definitions of equity are
responsive to who these students are, where they come
from, and where they want to go in life. In the context
of Mathematics for All: How Do We Get There?,
mathematics educators may be more focused on
achieving the goal of getting there than on the process
of how to get there. This is supported by the large
number of school districts that now require all students
to take algebra in 8th or 9th grade. In the pursuit of this
goal, the inequities faced by marginalized students are
further compounded because many of them have not
been adequately prepared in their earlier mathematical
educations due to lack of quality educational resources
(e.g., Tate & Rousseau, 2002). Because of a lack of
attention to process, the well-meaning goals of
Mathematics for All may actually contribute to the
inequities faced by underrepresented students. There is
a danger that when many of these students do not
achieve up to their potential, there will be a tendency to
either (a) locate the problem within the student (Boaler,
2002) or (b) assume that contextual forces are so
deterministic that students are incapable of invoking
agency to resist these forces. Future equity-based
research will have to more closely examine how
students and contextual forces influence each other.
In my view, working to eliminate the inequities
faced by marginalized students will require an ongoing
commitment that extends beyond simply rendering
students eligible for the opportunities that we assume
and hope will exist for them. Underrepresented
students may experience equal access to mathematics,
Hidden Assumptions
have equal learning opportunities, and quantitative data
could show equal outcomes. However, these students
may still be disempowered if they are not able to use
mathematics to alter the power relations and structural
barriers that continually work against their progress in
life.
Let us assume for the moment that the there in
Allexsaht-Snider and Hart’s question of Mathematics
for All: How Do We Get There? has been reached. The
situation is now the following: African American,
Latino American, Native American, and poor students
now complete substantially more mathematics courses
than they did before and their achievement levels have
risen to where we deem them acceptable. I pose the
same simple, but incisive, questions asked by Gloria
Ladson-Billings (2002) during a recent American
Educational Research Association symposium: “Now
what? What are we going to do for these students?”
Will more of these students be allowed to attend the
Universities of California at Berkeley and Los Angeles
or other universities that are sometimes forced to
engage in zero-sum admissions policies (Jones,
Yonezawa, Ballesteros, & Mehan, 2002), leaving many
qualified minority students on the outside looking in?
For example, a state budget crisis has recently forced
the Regents of the University of California to consider
restricting enrollment at its campuses, signaling a
reversal of the state’s commitment to guarantee
placement for the state’s top 12% of graduating
seniors. That commitment has been in place since
1960. For spring 2003, the university turned away
hundreds of mid-year applications from transfer
students and freshman. Budget reductions, fee
increases, increasing numbers of college-eligible
students, and competition for slots have, subsequently,
forced many students to the state’s community
colleges. The trickle down effect is that many students
who have traditionally attended community colleges
now find themselves in competition with top-notch
high school students. Recently, the state community
colleges eliminated 8200 classes, leading to a loss of
90,000 students (Hebel, 2003). Will students who have
long used community college as a bridge to higher
education now be squeezed out of the community
college context and back to their neighborhoods where
opportunities are often limited? Will those top high
school students now feel that the reward for all their
hard work is being taken from them when they are
directed to the community college? Where does
mathematics fit into all of this? It is well known that
mathematics serves as a gatekeeper course for high
school graduation and college admissions and many
Danny Bernard Martin
students do not gain access to the kind of mathematics
to make these graduation and admissions outcomes a
reality.
Even for those students who are successful in
navigating their way to four-year colleges and
universities and into math and science majors, it can be
asked whether hi-tech companies in Silicon Valley will
increase their efforts to recruit these students as
engineers and scientists? Will there be an increase in
the number of women and minority faculty in
mathematics and science departments at colleges and
universities?
However, such questions may be a case of putting
the cart before the horse. If we go back and start with
school-mathematics itself, we have to remember that it
does not exist in isolation of other curriculum areas.
Will marginalized students gain greater access to
quality science coursework and instruction? What
about literacy? If these students are not able to read
and write effectively, how will they be able to handle
the rigors of mathematics and science? A common
question asked by younger students about mathematics
knowledge is How am I going to use this? Convincing
students that mathematics learning is worthwhile and
can have a significant impact on their lives will be a
hard sell for many African American, Latino, Native
American, and poor students if they continue to
experience inequitable treatment and see few people in
their communities who have benefited from
mathematics learning or if they are only given access
to the kind of mathematics that limits their
opportunities in life. I reiterate my earlier point: it is
not enough for mathematics educators to work toward
equity in mathematics education simply for the sake of
equity in mathematics education. Equity discussions
and equity-related efforts in mathematics education
need to be connected to discussions of equity and in the
larger social and structural contexts that impact the
lives of underrepresented students.
The questions raised above are not intended to
throw up a white flag and accept inequities as
inevitable. Nor am I suggesting that Mathematics for
All is not a worthy goal. However, if achieving equity
as a goal in mathematics education means having all
students take algebra and, once this is done, that our
responsibilities as mathematics educators have been
fulfilled, this is, in my view, not an acceptable goal.
Mathematics Learning and Literacy in African
American Context
In advancing my overall arguments, I draw partly
from my own research with a diversity of African
15
American adolescents, community college students,
parents, and teachers of African American students in
two San Francisco Bay Area communities (Martin
2000, 2002b, 2003). For nearly ten years, my
ethnographic and participant observation research in
these communities has focused on the contextual
factors (sociohistorical, structural, community, school,
family, peer) that influence well-documented
underachievement and limited persistence issues. I
have also devoted a great deal of attention to
mathematics success and resiliency among adolescents
and adults. In particular, I have focused on issues of
mathematics socialization and mathematics identity.
Mathematics socialization refers to the experiences that
individuals and groups have within a variety of
mathematical contexts, including school and the
workplace, and that legitimize or inhibit meaningful
participation in mathematics. Mathematics identity
refers to the beliefs that individuals and groups develop
about their mathematical abilities, their perceived selfefficacy in mathematical contexts (that is, their beliefs
about their ability to perform effectively in
mathematical contexts and to use mathematics to solve
problems in the contexts that impact their lives), and
their motivation to pursue mathematics knowledge.
Mathematics socialization and the development of
a mathematics identity occur as individuals and groups
attempt to negotiate their way into contexts whose
participation is mediated or dictated by knowledge of
mathematics. Given the wide variety of mathematical
practices and contexts in which individuals participate
or are denied participation (classrooms, curriculum
units, jobs, etc.), mathematics socialization can be
conceptualized as both a mechanism for reproducing
inequities and for working toward equity in
mathematics. A focus on mathematics identity, then,
leads to a better understanding of how these
experiences operate at a psychological level and give
rise to the meanings that people develop about
mathematics. I have studied these issues within a
broader, multilevel framework that incorporates
sociohistorical, community, school, and intrapersonal
factors. For the purpose of example, the first two levels
of that framework are depicted in Figure 2.
I believe that in studying mathematics socialization
and mathematics identity issues from a multilevel point
of view, I have also gained greater insight into the
bottom-up, community-based notions of equity in
mathematics education that I mentioned earlier in this
paper.
Although studies have shown that African
American adults and adolescents hold the same folk
16
theories about mathematics as mainstream adults and
students, stressing it as an important school subject,
few studies have sought to directly examine their
beliefs about constraints and opportunities associated
with mathematics learning, both for themselves and
their children. My research has shown, for example,
that African Americans’ conceptions of equity in
mathematics education can be deeper, more
sophisticated, and even misaligned with those found in
reform documents (Martin, 2000, 2002b, 2003). For
adults, in particular, I argue that their racialized
accounts of their mathematical experiences inside and
outside of school reveal that many African American
parents situate mathematics learning and the struggle
for mathematical literacy/equity within the larger
contexts of socioeconomic, political, educational, and
African American struggle. As they attempt to become
doers of mathematics and advocates for their children’s
mathematics learning, discriminatory experiences have
continued to subjugate some of these parents while
others have resisted their continued subjugation based
on a belief that mathematics knowledge, beyond its
role in schools, can be used to penetrate the larger
opportunity structure. I often use case studies (Martin,
2000, 2002b, 2003) to exemplify these varying
trajectories of experiences and beliefs about
mathematics. Narratives embedded in these case
studies often reveal social justice concerns having to
do with mathematical opportunity.
Sociohistorical Forces
Differential treatment in mathematics-related contexts
Community Forces
Beliefs about African American status and differential treatment
in educational and socioeconomic contexts
Beliefs about mathematics abilities and motivation to learn
mathematics
Beliefs about the instrumental importance of mathematics
knowledge
Relationships with school officials and teachers
Math-dependent socioeconomic and educational goals
Expectations for children and educational strategies
Figure 2: Mathematics socialization and identity among
African Americans: Sociohistorical and community forces.6
In an excerpt from an example that I present
elsewhere (Martin, 2000), an African American father
offers an insightful opinion about the relationship
between African American students’ efforts in
Hidden Assumptions
mathematics and their perceptions of subsequent
opportunity:
DM:
Do you think [low motivation is] true for a lot
of kids now?
Father: I think that’s true for a lot of kids now, yes.
DM: It’s mainly that a lot of them don’t see the
opportunity attached to [math]?
Father: They see the opportunity…. For me, all I
wanted was an opportunity. The opportunity
wasn’t even there. So, I didn’t pursue it. But
what opportunity was there required so much
[math] and I satisfied that. Today’s kids, I
think, have the opportunity but they need more
than just the opportunity. They need the
guarantee.
DM: Can we guarantee?
Father: Yeah, we can. If we will. I mean I can
guarantee you that if you do these things, given
the way the social structure is set up, there’s a
place for you. But you’ve got to set the social
structure up first.
This view represents just one point in the
constellation of African American voices but it offers
some evidence for my claim that it will not be enough
to achieve equity in mathematics education and settle
for that as an end goal.
While student ability, teacher bias, tracking, and
inadequate curriculum are often cited as causes of low
mathematics achievement and limited persistence
among African American students (see Martin, 2000),
the comments made by this father highlight the fact
that not only do adults situate mathematics learning in
a larger socioeconomic and political context, but
marginalized students may do the same. Addressing
teacher bias, tracking, and inadequate curriculum in the
name of equity and undoing the role the of
mathematics as a gatekeeper may address school-level
issues but if students are not able to use mathematics in
the out-of-school contexts that define their lives, then
underachievement and limited persistence may be
rational responses to perceptions of the larger
opportunity structure.
In addition to my research, my fourteen years of
teaching mathematics to students who have often fallen
through the cracks and for whom mathematics
education reform has done little has convinced me that
attempts to achieve equity which focus on content and
curriculum issues, teacher beliefs, and school cultures
alone will probably have limited impact on negative
trends in achievement and persistence if, for example,
(1) community forces counteract any good that is done
Danny Bernard Martin
within schools despite the best efforts of good teachers
who use quality curriculum and exemplary (Standards
and non-Standards-based) classroom practices and (2)
no attempt is made to leverage these community forces
to support in-school efforts designed to eliminate
inequity. Eliminating inequities in access,
achievement, and persistence in mathematics is not an
issue that can be separated from the larger contexts in
which schools exist and in which students live.
Integrating Theory, Methods, and Practice
To improve the status of underrepresented students
in mathematics, mathematics educators will need to
move beyond the initial rhetoric of Mathematics for All
and any tendency to frame equity issues using only the
theory and methods of mathematics education. Clearly,
our approaches to equity need to be extended in ways
that draw on perspectives outside of mathematics
education where issues of culture, social context,
stratification, and opportunity structure receive greater
and more serious attention. Areas like critical
social/race theory (e.g., Ladson-Billings & Tate, 1995),
sociology of education, and anthropology of education
(Ogbu, 1988, 1990) come to mind. Read from one
vantage point, one could take from Allexsaht-Snider
and Hart’s (2001) definition of equity the assumption
that inequities in mathematics education are caused by
and can be remedied by fixing school-related factors.
Although Allexsaht-Snider and Hart clearly do not
assume this, some mathematics educators might. As a
result, there might be continued reluctance to analyze
the complex social issues that have an impact on
mathematics teaching, learning, and disparate
outcomes, despite the fact that these issues have been
cited in the research literature as being critically
important. To return to my preview of recent history in
our field, Reyes and Stanic (1988) stated:
In the field of mathematics education, there is little,
if any, research documentation of the effect of
societal influences on other factors in the model.
Documenting these connections is both the most
difficult and the most necessary direction for future
research on differential achievement in
mathematics education. (p. 33)
This foregrounding of the complex social issues
involved in equity are not yet taking center stage, 15
years later.
Finding a way to maintain our concern with
mathematics content, mathematics teaching, and
learning, while using powerful sociocultural analyses
to understand how the arrangement of mathematical
opportunities inside and outside of school interact and
17
further contribute to inequities continues to represent
the next difficult step in equity-focused research. A
second step involves designing meaningful
interventions, inside and outside of school, to empower
marginalized students with mathematics so that they
can change the conditions which contribute to the
inequities they face (e.g., Gutstein, 2002, 2003). If
equity research in mathematics education is to move
forward, we must recognize that inequities in
mathematics are reflections of the inequities that exist
in out-of-school contexts. Parents, teachers, and
students often recognize this parallel to the outside
world (e.g., Civil et al, 2000; Civil et al, 2003; Civil &
Quintos, 2002; Martin, 2000, 2003; Martin, Franco, et
al, 2003) as have critical and progressive mathematics
educators (e.g., Abraham & Bibby, 1988; Anderson,
1990; Atweh et al, 2001; Campbell, 1989;
D’Ambrosio, 1990; Frankenstein, 1990, 1994;
Gutstein, 2003; Hart & Allexsaht-Snider, 1996; Secada
& Meyer, 1989; Secada et al, 1995; Tate, 1995).
I would also suggest that mathematics educators be
wary of transforming equity issues into issues of
learning
mathematics
content.
Whether
underrepresented students can learn mathematics
should not be the main issue of concern. As a field, we
should be well beyond deficit-based thinking and
trying to fix students so that they conform to normative
notions of what a student should be and for what
purpose mathematics education should serve these
students.
Because so much research has been devoted to
student failure, there is also the danger that
underachievement among underrepresented students
will be accepted as the natural and normal starting
point for research involving these students. But rich
data collected across the many contexts where
underrepresented students live and learn will help us
reformulate our understanding of both failure a n d
success. As a result, we can begin to look for more
meaningful explanations and solutions to problematic
outcomes and build on what we learn about success.
By focusing on diverse contexts, we can begin to
uncover a range of solutions focused on what works,
where, when, and why, rather than trying to lump all
students together and applying one-size-fits-all
interventions. Mathematics for All will require that we
find a variety of ways to bring underrepresented
students into mathematics and a variety of
ways—working through schools and communities and
at the individual student level—to support their
continued development and empowerment.
18
Conclusion
As both a teacher and a researcher, I am a strong
advocate of ensuring that all students experience equal
access, equal treatment, achieve to their highest
potential in mathematics, and participate freely in all
forms of mathematical practices that appeal to them
inside and outside of schools. I also agree with those
who conceptualize mathematics as a gatekeeper and
filter (Sells, 1978) and who identify math literacy as a
new form of civil right (Moses, 1994; Moses & Cobb,
2001). Yet, I also advocate critical examination of
Mathematics for All rhetoric that, in my view, is
limited in its vision. By making problematic the there
in How Do We Get There?, I hope that my discussion
of the hidden assumptions and unaddressed questions
in Mathematics for All rhetoric will contribute to a
reconceptualization of our equity efforts and our
attempts to help students who are marginalized in
mathematics.
The transition from mathematics for the few to
mathematics for all will undoubtedly be an arduous
task. As the mathematics education community gives
greater attention to equity issues, we cannot assume
that Mathematics for All and Algebra for All represent
victories over the inequities that marginalized students
and their communities face inside and outside of
mathematics. Moreover, the people who comprise the
communities that we wish to help must become equal
partners in mathematics equity discussions and in
formulating solutions that address not only content and
curricular concerns but issues of social justice as well.
It is also my hope that the students who were first
graders in the year 2000, the year of the updated
Standards, will benefit from a renewed focus and a
true desire to move beyond rhetoric so that these
students fare better than the Class of 2002.
REFERENCES
Abraham, J., & Bibby, N. (1988). Mathematics and society. For the
Learning of Mathematics, 8(2), 2–11.
Allexsaht-Snider, M., & Hart, L. (2001). Mathematics for all: How
do we get there? Theory Into Practice, 40(2), 93–101.
Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated
learning and education. Educational Researcher, 25(4), 5–11.
Anderson, S. (1990). Worldmath curriculum: Fighting
Eurocentrism in mathematics. Journal of Negro Education,
59(3), 348–359.
Anhalt, C., Allexsaht-Snider, M., & Civil, M. (2002). Middle
school mathematics classrooms: A place for Latina parents’
involvement. Journal of Latinos and Education, 1(4),
255–262.
Apple, M. (1993). Official knowledge: Democratic education in a
conservative age. London: Routledge.
Hidden Assumptions
Apple, M. (1995). Taking power seriously. In Secada, W.,
Fennema, E. & Adajian, L. B. (Eds.). New directions for
equity in mathematics education (pp. 329–348). Cambridge:
Cambridge University Press.
Apple, M. (1999). Do the Standards go far enough? Power, policy,
and practice in mathematics education. Reproduced in Power,
meaning, and identity: Essays in critical educational studies.
New York: Peter Lang (Original work published 1992).
Day, J. (1993). Population projections of the United States by age,
sex, race, and Hispanic origin: 1993 to 2050. Washington,
D.C.: U.S. Department of Commerce (vii–xxiii).
Delpit, L. (1995). Other people’s children: Cultural conflict in the
classroom. New York: The New Press.
Frankenstein, M. (1990). Incorporating race, gender, and class
issues into a critical mathematical literacy curriculum. Journal
of Negro Education, 59(3), 336–347.
Atweh, B., Forgasz, H., & Nebres, B. (2001). Sociocultural
research on mathematics education: An international
perspective. Mahwah, NJ: Erlbaum.
Frankenstein, M. (1994). Understanding the politics of
mathematical knowledge as an integral part of becoming
critically numerate. Radical Statistics, 56, 22–40.
Boaler, J. (2002, April). So girls don’t really understand
mathematics?: Shifting the analytic lens in equity research.
Paper presented at the annual meeting of the American
Educational Research Association, New Orleans, LA.
Gutstein, E. (2002, April). Roads to equity in mathematics
education. Paper presented at the annual meeting of the
American Educational Research Association, New Orleans,
LA.
Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition
and the culture of learning. Educational Researcher, 18(1),
32–42.
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.
Campbell, P. (1989). So what do we do with the poor, non-white
female?: Issues of gender, race, and social class in
mathematics and equity. Peabody Journal of Education, 66(2),
96–112.
Hart, L., & Allexsaht-Snider, M. (1996). Sociocultural and
motivational contexts of mathematics learning for diverse
students. In M. Carr (Ed.), Motivation in mathematics (pp.
1–24). Creskill, NJ: Hampton Press.
Civil, M., Andrade, R., & Anhalt, C. (2000). Parents as learners of
mathematics: A different look at parental involvement. In M.
L. Fernández (Ed.), Proceedings of the Twenty-Second Annual
Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education, Vol. 2
(pp. 421–426). Columbus, OH: ERIC Clearinghouse.
Hebel, S. (2003, October 10). California’s budget woes lead
colleges to limit access. The Chronicle of Higher Education,
p. 21.
Civil, M., Bernier, E., & Quintos, B. (2003, April). Parental
involvement in mathematics: A focus on parents’ voices. Paper
presented at the Annual Meeting of the American Educational
Research Association, Chicago, IL.
Civil, M., & Quintos, B. (2002, April). Uncovering mothers’
perceptions about the teaching and learning of mathematics.
Paper presented at the annual meeting of American
Educational Research Association, New Orleans, LA.
Class, J. (2002). The Moses factor [Electronic version]. Mother
Jones, May/June. Retrieved November 14, 2003, from
http://www.motherjones.com/news/feature/2002/05/
moses.html
Cobb, P. (2000). The importance of a situated view of learning to
the design of research and instruction. In J. Boaler (Ed.),
Multiple perspectives on mathematics teaching and learning
(pp. 45–82). Westport: Ablex.
Cobb, P., & Bowers, J (1999). Cognitive and situated perspectives
in theory and practice. Educational Researcher, 28(2), 4–15.
Cobb, P., & Nasir, N. (Eds.). (2002). Diversity, equity, and
mathematics learning [Special double issue]. Mathematical
Thinking and Learning, 4(2/3).
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist
alternative to the representational view of mind in
mathematics education, Journal for Research in Mathematics
Education, 23(1), 2–33.
D’Ambrosio, U. (1990). The role of mathematics in building a
democratic and just society. For the Learning of Mathematics,
10(3), 20–23.
Jones, M., Yonezawa, S., Ballesteros, E., & Mehan, H. (2002).
Shaping pathways to higher education. Educational
Researcher, 31(2), 3–12.
Khisty, L. (2002, April). Equity in mathematics education
revisited: Issues of “getting there” for Latino second
language learners. Paper presented at the annual meeting of
the American Educational Research Association, New
Orleans, LA
Ladson-Billings, G. (1995). Making mathematics meaningful in
multicultural contexts. In W. G. Secada, E. Fennema, & L. B.
Adajian (Eds.), New Directions for Equity in Mathematics
Education (pp. 279–297). Cambridge: Cambridge University
Press.
Ladson-Billings, G. (2002, April). Urban education and
marginalized youth in an age of high-stakes testing:
Progressive responses. Symposium presented at the annual
meeting of the American Educational Research Association,
New Orleans. LA.
Ladson-Billings, G. & Tate, W.F. (1995). Toward a critical race
theory of education. Teachers College Record, 97, 47–68.
Lave, J. & Wegner, E. (1991). Situated learning: Legitimate
peripheral participation. Cambridge: Cambridge University
Press.
Lee, J. (2002). Racial and ethnic achievement gap trends:
Reversing the progress toward equity? Educational
Researcher, 31(1), 3–12.
Lubienski, S. (2003, April). Traditional or standards based
mathematics?: Parents’ and students’ choices in one district.
Paper presented at the Annual Meeting of the American
Educational Research Association, Chicago, IL.
Davis, R. (1986). The convergence of cognitive science and
mathematics education. Journal of Mathematical Behavior, 5,
321–335.
Danny Bernard Martin
19
Martin, D. B. (1998). Mathematics socialization and identity
among African Americans: A multilevel analysis of community
forces, school forces, and individual agency. Unpublished
postdoctoral project completed for National Academy of
Education/Spencer Postdoctoral Fellows program.
Ogbu, J. U. (1988). Diversity and equity in public education:
Community forces and minority school adjustment and
performance. In R. Haskins & D. McRae (Eds.), Policies for
America’s public schools: Teachers, equity, and indicators
(pp. 127–170). Norwood: Ablex.
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: Erlbaum..
Ogbu, J. U. (1990). Cultural model, identity, and literacy. In J. W.
Stigler, R. A. Shweder, & G. Herdt (Eds.), Cultural
psychology (pp. 520–541). Cambridge: Cambridge University
Press.
Martin, D. B. (2002a, April). Is there a there?: Avoiding equity
traps in mathematics education and some additional
considerations in mathematics achievement and persistence
among underrepresented students. Paper presented at the
Annual Meeting of the American Educational Research
Association, New Orleans, LA.
Perissini, D. (1997). Parental involvement in the reform of
mathematics education. The Mathematics Teacher, 90(6),
421–427.
Martin, D. (2002b, April). Situating self, situating mathematics:
Issues of identity and agency among African American adults
and adolescents. Paper presented at the annual conference of
the National Council of Teachers of Mathematics, Las Vegas,
NV.
Martin, D. (2003, April). Gatekeepers and guardians: African
American parents’ responses to mathematics and mathematics
education reform. Paper presented at the Annual Meeting of
the American Educational Research Association, Chicago, IL.
Martin, D., Franco, J., & Mayfield-Ingram, K. (2003). Mathematics
education, opportunity, and social justice: Advocating for
equity and diversity within the context of standards-based
reform. Research brief (draft) prepared for Research Advisory
Committee, National Council of Teachers of Mathematics
Catalyst conference.
Meyer, M.R. (1989). Equity: The missing element in recent
agendas for mathematics education. Peabody Journal of
Education, 66(2), 6–21.
Moses, R. P. (1994). Remarks on the struggle for citizenship and
math/science literacy. Journal of Mathematical Behavior, 13,
107–111.
Moses, R. P. & Cobb, C.E. (2001). Radical equations: Math
literacy and civil rights. Boston MA: Beacon Press.
Moses, R., Kamii, M., Swap, S., & Howard, J. (1989). The Algebra
Project: Organizing in the spirit of Ella. Harvard Educational
Review, 59(4), 423–443.
National Council of Teachers of Mathematics. (1989). Curriculum
and evaluation standards for school mathematics. Reston,
VA: Author.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Research Council. (1989). Everybody counts: A report to
the nation on the future of mathematics education.
Washington, DC: Author.
National Research Council. (2002). Investigating the influence of
standards: A framework for research in mathematics, science,
and technology education. Washington, D.C.: National
Academy Press.
Oakes, J. (1990). Opportunities, achievement and choice: Women
and minority students in science and mathematics. In C. B.
Cazden (Ed.), Review of Research in Education, Vol. 16 (pp.
153–222). Washington, DC: AERA.
20
Perissini, D. (1998). The portrayal of parents in the school
mathematics reform literature: Locating the context for
parental involvement. Journal for Research in Mathematics
Education, 29(5), 555–582.
RAND Mathematics Study Panel. (2003). Mathematical
proficiency for all students: Toward a strategic research and
development program in mathematics education. Washington,
DC: RAND.
Reyes, L. H., & Stanic, G. (1988). Race, sex, socioeconomic status,
and mathematics. Journal for Research in Mathematics
Education, 19(1), 26–43.
Schoenfeld, A. H. (1985). Mathematical problem solving. New
York, NY: Academic Press.
Schoenfeld, A. H. (Ed.). (1987). Cognitive science and
mathematics education. Hillsdale, NJ: Erlbaum.
Schoenfeld, A. (2002). Making mathematics work for all children:
Issues of standards, testing, and equity. Educational
Researcher, 31(1), 13–25.
Secada, W. (1989). Agenda setting, enlightened self-interest, and
equity in mathematics education. Peabody Journal of
Education, 66(2), 22–56.
Secada, W. (1991). Diversity, equity, and cognitivist research. In
Fennema, E., Carpenter, T. P., & Lamon, S. (Eds.),
Integrating research on teaching and learning mathematics
(pp. 17–54). SUNY: Albany.
Secada, W. (1992). Race, ethnicity, social class, language and
achievement in mathematics. In D. Grouws (Ed.), Handbook
of research on mathematics teaching and learning (pp.
623–660). New York: Macmillan.
Secada, W., Fennema, E., & Adajian, L. B. (1995). New directions
for equity in mathematics education. Cambridge: Cambridge
University Press.
Secada, W., & Meyer, M. (1989). Needed: An agenda for equity in
mathematics education. Peabody Journal of Education, 66(1),
1–5.
Secada, W., Ogbu, J. U., Peterson, P., Stiff, L. M., Tonemah, S.
(1994). At the intersection of school mathematics and student
diversity: A challenge to research and reform. Washington,
DC: Mathematical Sciences Education Board and National
Academy of Education.
Sells, L. W. (1978). Mathematics: A critical filter. Science Teacher,
45, 28–29.
Skovsmose, O., & Valero, P. (2001). The critical engagement of
mathematics education within democracy. In Atweh, B.,
Forgasz, H., & Nebres, B. (Eds.), Sociocultural research on
mathematics education (pp. 37–56). Hillsdale, NJ: Erlbaum.
Hidden Assumptions
Stanic, G. M. A. (1989). Social inequality, cultural discontinuity,
and equity in school mathematics. Peabody Journal of
Education, 66(2), 57–71.
Stiff, L. V. (2002). Reclaiming standards. NTCM News Bulletin,
38(7), 3–7.
Tate, W. F. (1995). Economics, equity, and the national
mathematics assessment: Are we creating a national toll road?
In W. Secada, E. Fennema, & L. Byrd Adajian (Eds.), New
directions for equity in mathematics in mathematics education
(pp. 191–206). Cambridge: Cambridge University Press.
Tate, W. (1997). Race, ethnicity, SES, gender, and language
proficiency trends in mathematics achievement: An update.
Journal for Research in Mathematics Education, 28(6),
652–680.
Tate, W. F., & Rousseau, C. (2002). Access and opportunity: The
political and social context of mathematics education. In L.
English (Eds.), International Handbook of Research in
Mathematics Education (pp. 271–300). Mahwah, NJ:
Erlbaum.
Thomas, J. (2001). Globalization and the politics of mathematics
education. In Atweh, B., Forgasz, H., & Nebres, B. (Eds.)
Sociocultural research on mathematics education (pp.
95–112). Hillsdale, NJ: Erlbaum.
Weinstein, R. S. (1996). High standards in a tracked system of
schooling: For which students and with what educational
supports? Educational Researcher, 25(8), 16–19.
Danny Bernard Martin
Wilson, S. (2002). California dreaming: Reforming mathematics
education. New Haven, CT: Yale University Press.
1
I will refer to this document as the Standards when a
distinction against another standards-based document is
unnecessary. Occasionally I will collectively refer to both
the 1989 and 2000 NCTM documents as the Standards. I
have taken care that in each place, the reader will know to
which document I refer.
2
These ideas are from discussions that took place in a
Working Group on the Changing Nature of Schooling and
Demographics led by William Tate and Pauline Lipman at
the National Council of Teachers of Mathematics Catalyst
Conference held in Reston, VA, September 11-13, 2003.
3
Hart (2001) and Reyes (1988) are the same person.
4
For an account tracing the history of these debates, see
Wilson (2002).
5
The prevailing notion is that equity in mathematics is
three-pronged: equal access, equal opportunity to learn, and
equal outcomes.
6
Readers are urged to see Martin (2000) for a more
detailed description of this framework.
21
CONFERENCES 2004…
22
MAA-AMS Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org/amsmtgs/2078_intro.html
Phoenix, AZ
Jan. 7-10
AMTE
Association of Mathematics Teacher Educators
http://www.amte.net
San Diego, CA
Jan. 23-24
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML
Oklahoma City, OK
Feb. 19-21
AERA
American Education Research Association
http://www.aera.net
San Diego, CA
April 12-16
Mα
The Mathematical Association
http://m-a.org.uk
York, UK
April 13-16
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
Philadelphia, PA
April 21-24
CMESG/GCEDM
Canadian Mathematics Education Study Group
http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html
Université Laval,
Québec, Canada
May 28-June1
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://igpme.org
Bergen, Norway
July 14-18
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings
Toronto, Canada
Aug. 8-12
PME-NA
North American Chapter of the International Group for the Psychology of Mathematics
Education
http://www.pmena.org
Toronto, Canada
Oct. 21-24
SSMA
School Science and Mathematics Association
http://www.ssma.org
College Park, GA
Oct. 21-23
AAMT 2005
Australian Association of Mathematics Teachers
http://www.aamt.edu.au/mmv
Sydney, Australia
Jan. 17-20
2005
The Mathematics Educator
2003, Vol. 13, No. 2, 23–31
The Fourth “R”: Reflection
Norene Vail Lowery
Research promotes reflective teaching as an important distinguishing strategy between experienced and novice
teachers and is a critical tool for developing teacher knowledge. Reflective teaching practices are supported by
national reform efforts and have the potential to affect student achievement in the mathematics classroom.
Unfortunately, reflective teaching practices are not always a component of teacher preparation and professional
experiences. This discussion highlights: (1) research concerning the importance of teacher reflection; (2) the
results of a study implementing reflective teaching practices in an elementary mathematics and science methods
course; (3) the resulting “best practices” applied to other teacher learning contexts; and, (4) the benefits of the
fourth “R”.
National standards, having emerged from
educational reform, promote learning environments
that encourage meaningful learning, rather than rote
learning, and create a different view of teaching and
learning. A component of these reform efforts is to
develop teachers who are reflective about teaching and
learning. Reflection is seen as what a teacher does
when he or she looks back at the teaching and learning
that has been experienced, and recreates the events,
emotions, and happenings of the situation (Wilson,
Shulman, & Richert, 1987). Hoberman and Mailick
(1994) believe that learning and competence are gained
by practice in performance that involves reflection
before, during, and after the action.
Research indicates that teacher reflection is a key
aspect for obtaining teacher knowledge and
pedagogical content knowledge. There exists a stage in
which teachers look back on the teaching and learning
that has occurred as a means of making sense of their
actions and learning from their experiences (Wilson,
Shulman, & Richert, 1987). Reflection is seen as a
process of reconstructing classroom enactments,
including both cognitive and affective dimensions that
involve a developmental progression through stages.
Experienced and novice teachers differ in their ability
to learn from reflection on experience. Reflective
experts are more discriminating in their perception and
more resourceful in their actions and problem solving.
Experienced teachers have a highly developed
knowledge base concerning students; notice different
Norene Vail Lowery, Ph.D., is an Assistant Professor of
Mathematics Education in the Curriculum & Instruction
Department of the College of Education at the University of
Houston, Houston, Texas. Research interests include elementary
and middle school mathematics education, preservice and inservice
teacher education, assessment, and the integration of literature and
mathematics. Her email address is nlowery@uh.edu.
Norene Vail Lowery
classroom aspects; are more selective in their use of
information during planning and teaching; and, make
greater use of instructional and management routines
(Borko & Livingston 1989; Borko & Shavelson 1990;
Carter, Cushing, Sabers, Stein, & Berliner, 1988). The
basis for instructional decisions (teacher’s practical
knowledge) is dynamic as it builds through reflective
experience (Elbaz, 1983).
The Principles and Standards for School
M a t h e m a t i c s (National Council of Teachers of
Mathematics [NCTM], 2000) advocates new ways of
teaching and learning mathematics in the classroom.
As mathematics educators begin to implement these
guidelines, it is even more crucial that teachers become
reflective in practice. However, reflection on teaching
is not a traditional component of mathematics
instruction. As it is believed that teachers tend to teach
the way they were taught, reflective teaching will only
be implemented and flourish if teachers become
knowledgeable of and are supported by “best
practices” in the classroom.
The importance of reflective teaching is a central
component for designing teaching and learning
experiences for teachers. I have implemented a variety
of strategies for encouraging reflection into all my
courses with both preservice and inservice teachers.
Effective protocol for becoming a reflective teacher
has emerged through these experiences. This
discussion describes original research findings with
preservice teachers that led to creating and
implementing reflective practices into other teacher
education courses. A collective synthesis of these
successful efforts resulting in “best practices” is
presented for creating learning experiences for teachers
that are conducive to begin and support reflective
teaching. No longer can there be traditional reliance on
just “Reading, ‘Riting, and ‘Rithmetic” as the basic
ingredients for providing a quality education. It is vital
23
for all teachers, especially teachers of mathematics, to
take a serious look at a fourth “R”: Reflection.
Reflective Research
Even though the role of reflection in teaching is
considered important, reflective action in preservice
and inservice teachers is either inhibited by isolation of
teachers or by structure of courses and schools
(Feiman-Nemser & Buchmann, 1985; Lortie, 1975;
Nisbett & Wilson, 1977). Fenstermacher (1994) has
suggested reform needed and wanted in teacher
preparation requires a tremendous effort in
understanding teaching and teacher learning. Other
advocates in the renovation of teacher preparation
programs appear to confirm this and suggest that such
new programs must understand the conditions that
promote reflection in beginning teachers. Research
findings confirm that the likelihood of long-term
success for many novice teachers is hindered by the
absence of expert guidance, support, and opportunities
to reflect (Tisher, 1978; Veenman, 1984).
Examination of the role of reflection involved in
learning how to teach can make significant
contributions in strengthening the preparation of
teachers by complementing a growing knowledge base
for teaching. The purpose of this inquiry was to
examine the construction of teacher knowledge in
learning to teach elementary mathematics and science.
Implementing teacher reflection was an integral
component of the study. A qualitative methodology
was employed and facilitated the discovery of the
importance of reflection in learning to teach by
preservice teachers in a school-based setting. The
respondents were twenty-one junior and senior level
interdisciplinary studies majors at a large university
who were enrolled in methods courses required for the
fulfillment of certification in elementary mathematics
and science instruction. The site for this contentfocused (mathematics and science only) professional
development school was located on a middle-class,
suburban, public K-5 elementary school campus in a
central Texas school district. The methods course
experience was non-traditional and innovative in: the
approach (constructivist1); the content (standardsbased2); the site (school-based with immediate access
to inservice teachers and elementary students); and the
instructional strategies (reflective practices).
Additionally, during the semester, the preservice
teachers participated in teaching and tutoring
elementary student’s mathematics and science lessons.
Groups of three or four preservice teachers were
assigned to elementary grade level teachers for creating
24
and teaching these elementary mathematics and
science lessons. They debriefed and reflected in small
groups, and then individually responded to the course
tasks of analyzing, evaluating, and synthesizing all
experiences. These activities created the opportunity
for the required reflective journals, lab entries,
classroom tasks, and summative portfolios that were
assigned course products, and consequently used as
data sources. In addition, I conducted interviews with
the students to obtain additional information. The data
were processed following the suggested steps of
synthesis of a constant comparative method adapted
from Glaser and Strauss (1967). Both evidence and
extent of the value in the use reflection were apparent.
Knowledge of self, of the learner, and of the task
(content) were prevalent themes that emerged from the
data.
Development and use of the reflective process was
an objective of the methods course. Course instructors
considered reflective thinking before, during, and after
teaching imperative for a thorough teaching
experience. The reflective process is valued in
professional growth and successful teaching (Dewey,
1933; Schön, 1987). The entire collaborative learning
environment was subjected to the reflective process.
Reflection on learning occurred in the large group
tasks and activities; while working and planning with
teachers; while teaching children mathematics and
science; while working and planning within grade level
groups; and while working and interacting with course
instructors. To aid in simplifying the communication of
data sources for the purposes of this paper, I will rely
on the codes (see Table 1) I used during analysis while
I discuss these findings below. Through reflection,
much was revealed about the learning components of
teacher knowledge and pedagogical content knowledge
in mathematics and science.
Reflection is not an easy task. Initial reflections in
notebooks and in weekly evaluations were superficial
and more descriptive in nature rather than reflective
(RJ: 12.17.96). Reflections grew in depth and quality
over the semester to culminate in the product of the
personal portfolio. The portfolio was the ultimate
expression of the acquisition of learning that the
preservice teachers experienced in this context. The
summative portfolio was a deep, elaborate reflection
that revealed construction of teacher knowledge.
Individual interviews, focus group interviews, and
other data sources confirmed and verified results
reported in portfolios. These results can largely be
Reflection
Table 1.
Codes developed to distinguish data sources.
Source Origin
Preservice Teachers (PST)
Artifacts file
Data Source
Individual interview
Overview
Focus group interview
Overview
Lab notebook
Overview
Weekly Evaluations: Question #: Date
Final Examination: Question #: Response #
Portfolio: Artifact #: Response #
Field Notes (Researcher)
Overview
Reflexive Journal (Researcher)
sorted into 4 themes for the purpose of this report: (1)
knowledge about self, autonomy, self-efficacy; (2) the
importance of confidence and competence; (3) the
importance of the value of content, coordinating lesson
planning, and questioning; and (4) expanding
knowledge about children, assessment, relevancy and
group dynamics
Knowledge about self. Personal growth was
revealed through reflective processes, such as the
portfolio. Enthusiasm and patience were among those
reported. Knowledge of self as a learner and as a
teacher was important. At the end of the semester, the
preservice teachers expressed the importance of the
reflective process in tracking their growth and
experiences.
Watching myself evolve into an educator who
desires to be a constant learner has been one of the
most important changes that have taken place
during the semester (FE: Q4: 25).
Confidence and competence. Growth in confidence
and competence were reported in the portfolios, the
final exams, in weekly evaluations, and through
interviews. The preservice teachers were continually
encouraged to develop reflective thought as a tool for
developing confidence and competence in teaching
mathematics and science.
... my attitude toward teaching math and science.
Now I know that the best way to teach these
subjects is through discovery and exploration,
connecting math and science to the real world
is vital. I feel more confident about math and
science as I enter student teaching (FE: Q4: 6).
This experience has been beneficial to my
confidence as a teacher and a person in general. ... I
have learned from my instructors that science and
math must move away from worksheets and
become hands-on, minds-on activities. The
Norene Vail Lowery
Code
PSTI
PSTIO
PSTF
PSTFO
LN
LNO
WE: Q3: 9.4.96 (sample)
FE: Q4: 6 (sample)
PF: 6: 1 (sample)
FN
FNO
RJ
children must be actively engaged in their learning.
Children grow so much deeper when they have
opportunities to discover for themselves (PF: 6: 1).
The summative evaluation task, a personal
portfolio, required the preservice teachers to confront
their own learning and the extent of that learning.
Reflecting on previously recorded reflections over the
semester was part of the process in the creation of
portfolios.
I also feel that I have gained confidence in my
knowledge of mathematics and science. I always
enjoyed these subjects, but did not feel confident
about my knowledge until now. This was such a
valuable experience because I will carry this
confidence with me during my teaching career (FE:
Q4: 19).
I have always been afraid of math and science. I
have never been good at either one because they
were boring and abstract to me. Through this
semester, I have learned ways to make math and
science relevant, fun, and interesting. I now enjoy
learning scientific things and events and look
forward to teaching them (FE: Q4: 13).
Expanding knowledge of the content. Mathematics
and science content learning was experienced as a
group and individually. Many preservice teachers had
previously had limited or no meaningful experiences
with mathematics and science content. There were
“Aha’s” while working as learners in content activities.
I really enjoyed the math we experienced today. I
love math and I am excited to hear that it is not
being taught the way I was taught (PSTWE: Q1:
4.9.1).
I learned how to explain math in a meaningful way
(PSTWE: Q3: 12.14.3).
The preservice teachers had progressed from the
lower levels of ability such as question-response
25
techniques, to a level of higher-order thinking skills.
From there, they began asking open-ended questions to
stimulate children’s thinking and responses. Through
reflection on their own deeper learning of the content,
teachers identified the value of effective questioning as
an important instructional strategy.
... go further than the teacher asking simple
questions – have children ask questions and find
their own answers (PSTWE: Q2: 3.9.2).
Today we worked with individual children who
were having trouble with borrowing and
subtraction. [The student] didn’t really understand
when to borrow and when not to borrow. I picked
up that he wasn’t understanding the concept with
the way his teacher was explaining it to him. So I
took a different approach and I could see that he
truly understood the concept after I explained it. At
this point, I could tell his confidence was boosted
and he had more enthusiasm. Seeing his excitement
made me feel great. I had actually taught him
something. When we finished with the activity, he
asked the teacher if he could work with me some
more. I felt really special when he said that (PF:
6.4)!!!
Expanding knowledge of pedagogy. During
individual interviews, preservice teachers were asked
what they saw as the most important aspects about
science/mathematics teaching that an elementary
teacher should know. Responses suggested that
teachers should work with the students using hands-on,
real-world situations; make lessons relevant and
challenging; realize that abstract concepts are hard;
have more problem solving; know the students; act as a
facilitator of learning; and integrate math and science,
as well as other areas. Through reflection preservice
teachers came to terms with their own attitudes toward
mathematics and science, as was evidenced in the data.
I also feel that I have gained confidence in my
knowledge of mathematics and science. I always
enjoyed these subjects, but did not feel confident
about my knowledge until now. This was such a
valuable experience because I will carry this
confidence with me during my teaching career.
(FE: Q4: 19).
Now I know that the best way to teach these
subjects is through discovery and exploration,
connecting math and science to the real world is
vital. I feel more confident about math and science
as I enter student teaching. (FE: Q4.6).
reflective journals, indicated that these preservice
teachers attributed the following types of learning to
interacting with mathematics and science lessons:
content knowledge; the importance of hands-on
manipulatives; relevancy; instructional strategies;
group dynamics; and student cognitive attributes,
abilities and levels. Instructional strategies were
identified and implemented, including effective
questioning, timing, lesson planning, classroom
management, preparation, and authentic assessment.
This collaborative interaction was a relevant, authentic
learning environment for preservice teachers. Through
reflection, preservice teachers were able to track,
evaluate, and project their learning.
The researcher in this study also used other modes
of reflection. A reflexive journal was maintained to
record the researcher’s learning, decision-making
processes for data collection, analysis, in report
writing, and in the embellishments of the field notes,
interviews, and others.
In this study, reflection was a binding thread for all
the experiences of the preservice teachers. Reflective
practices allowed the professors and preservice
teachers to actively assess, evaluate, and modify the
learning experiences. Data indicate that through
reflective practices, preservice teachers had a greater
sense of self, autonomy, self-efficacy, confidence and
competence in teaching mathematics and science, and
had incorporated the value of reflection into their belief
system. These research findings promoted the
implementation of reflective practices used here into
other teacher education courses. Inservice teachers
have provided similar reports in graduate classes of
valuing the use of reflective teaching. From this study
many strategies emerged that were effectively used to
promote reflection.
Best Practices
Over time working with these emerging “best
practices”, I’ve seen that a belief in the importance of
reflection and strategies for becoming a reflective
mathematics teacher are developed and progress
through three levels: understanding reflection;
implementing reflective practices; and developing a
reflective venue. In this section, I will share the
strategies used and developed in situations such as
those in the above research, to illustrate ways to utilize
this three-level plan to promote reflective teaching.
Preservice teachers confronted their own learning
and the extent of that learning through their summative
portfolio assessment. The portfolios, along with the
26
Reflection
Level One: Understanding the Importance of Reflective
Thinking
Understanding prior knowledge and beliefs
provides a foundation for becoming a reflective
teacher. Teachers are encouraged to explore the
meaning of “reflection” by actually negotiating an
operational definition individually, in small groups,
and in whole class discussion.
Prompts for Defining Reflection
•
•
•
Write your own definition of “reflection”.
What do you perceive as “reflective teaching”?
In your group, discuss your responses.
Once common ground is established teachers are
asked to put reflection into action. Reflecting on past
experiences, prior knowledge, and expectations helps
to provide more insight into a teacher’s current
perspective on teaching and learning mathematics.
Teachers are asked to write an autobiographical sketch
that visualizes their own perspectives as a learner and
as a teacher.
Reflecting on Mathematical Experiences
•
•
•
•
•
What were your experiences with mathematics at
school/home/other? What kinds of instructional
activities and practices did you experience?
Describe any influential teachers, either positive
or negative.
What are your preferred learning strategies? What
do you do to learn? What do you enjoy learning?
How did you come to be like this? What stories
reveal your roots as a learner?
What does this mean for your current instructional
strategies? What is (are) your teaching
styles/instructional strategies? How might you do
things differently?
How do you envision mathematics learning for my
students?
To validate these reflections, teachers share their
thinking in a safe, collaborative group situation only to
uncover common attitudes, apprehensions, and beliefs
about mathematics. Large group discussion offers an
even greater opportunity to connect with peers and to
deepen reflective thinking.
Many preservice and some inservice teachers are
not excited about teaching mathematics. A teacher’s
perception of the nature of mathematics greatly
influences the mathematics instruction and learning
environment in the classroom (Cooney, 1985; Hersch,
1986). The following exercise challenges teachers to
come to terms with their own perspectives of
Norene Vail Lowery
mathematics teaching and learning. Charged with the
task of developing a statement of the nature of
mathematics, teachers are encouraged with these
writing prompts.
Developing a Perspective of the Nature of Mathematics
What is mathematics? What is the nature of
mathematics?
• What are the components of mathematics?
• What is the conception of mathematics that you
believe is important for your students to know?
• What mathematics do you want your students to
learn?
•
The resulting statements are shared in small groups
and then in the large classroom group. This initial draft
is revisited periodically to modify and enhance. At the
end of the semester, each teacher reviews the first
draft, revises, re-writes, or edits as they deem
appropriate to complete a final draft to submit. Along
with this final draft, teachers submit a “reflective
rationale” for any changes that were made in the
process. Teachers explain and justify any changes. This
allows for growth and clarity of reflective thought and
teaching. Teachers need time to “reflect” on the value,
complexity, and beauty of mathematics and teaching
mathematics.
Level Two: Implementing Reflective Strategies – The
Reflective Cycle
The revelations from Level One create a
foundation for implementing the power of reflection
into practical classroom applications. Experiences in
my undergraduate and graduate classes are created for
both field-based preservice and inservice teachers to
put reflection into active classroom interaction. These
exercises, presented and elaborated below, have
successfully enabled teachers to begin reflective
thinking and teaching practices. They form a cycle of
reflective activity.
This reflective cycle includes reflective planning,
reflective teaching, and creating a guided reflection.
Teachers use prompts individually and then in small
groups for self-questioning. Teachers write responses
to establish a thinking routine. Small group discourse is
used to facilitate reflective thinking practices. As more
experience in reflective thinking and teaching
develops, many of these become automatic, and the
cycle begins to feed upon itself.
Reflective planning. A strong emphasis is placed
on the first phase of this strategy, reflective planning.
Efforts involved in this area appear to benefit all areas
of reflection. Teachers are asked to carefully and
27
thoughtfully respond in depth to three components of
planning (the lesson, the learner, and the teacher).
Reflective Planning
The Lesson
• What is the content or topic and what prior
reasoning, strategies, and thinking skills are
expected?
• What learning venues are required for learner
success? (e.g., communication and representations)
• What are expected outcomes?
The Learner
• What are the goals for the learner?
• What are some of the possible misconceptions?
• How will the learner be assessed?
The Teacher
• What must be done to plan and organize?
• What are the most effective strategies and
questions? (Write some examples.)
• What are expected outcomes? What modifications
are anticipated?
Teachers proclaim this initial step in reflective
teaching prepares them for teaching a better lesson by
creating competence and confidence.
Reflective teaching. Reactive reflection or
reflection in action is sometimes quite difficult if not
impossible to recall after the events. However,
reflection here is crucial. Experienced teachers have a
rich repertoire of exemplars from which to choose
when confronted with problematic situations in the
classroom. Experienced teachers may demonstrate an
uninterrupted flow of teaching easily adapting to the
unexpected. As more classroom experience is gained,
novice teachers develop their own similar resources.
Why was this done? What questions prompted the
course of the teaching and learning? Making mental
notes may be all that is possible to do. However,
written notes are very effective. Periodic video or/and
audio taping of teaching lessons can also be a valuable
reflective tool for all teachers and is especially
revealing for preservice teachers.
In addition to taping teaching sessions, another
best practice for reflection in action for preservice
teachers involves using small groups. Groups of three
to four preservice teachers are created as teaching
teams to experience all facets of classroom teaching.
Team members are able to collectively recall more of
the events and sequences that occurred. Some teams
even record notes or use the taping strategy for later
debriefing. As the semester progresses and a well
established reflective teaching routine is rooted, the
28
teams diminish from four members to individual
efforts as confidence and experience grows. The
following are the prompts used as the basis for
encouraging reflective teaching and frame informal
assessment of learning interactions. While instruction
is occurring, teachers try to keep these items in mind.
Reflective Teaching
Are the students on task?
Do the students appear to understand the
concept? If not, what are my alternatives/resources
for actively adapting and modifying?
•
Are my instructional strategies appropriate for
all students?
•
What are my expected outcomes?
•
What modifications are needed for reteaching?
•
•
Guided reflection. Post-teaching reflection allows
the cycle of reflection to continue and helps teachers
develop a greater repertoire of learning experiences.
Too often, teachers are not encouraged to reflect due to
time constraints. Time for reflection is imperative for
developing teacher knowledge in novice teachers and
in furthering the depth of knowledge for experienced
teachers. By looking deeper into the learning
interaction, going beyond the superficial, reflection
provides teachers with insight into their teaching
successes and failures. Reflection affords teachers the
opportunity to select best practices for specific contexts
and specific students. In methods course experiences,
debriefing after teaching with preservice teachers is an
extremely valuable strategy. Debriefing for inservice
teachers may be achieved in collaborative efforts
within graduate classes or with colleagues in schools.
Developing an individual system of responding to
the prompts discussed next is important. These
prompts are similar to the ones initially addressed in
the reflective planning section and have been used to
begin classroom debriefing experiences for teachers,
but notice the additional aspects addressed here.
Teachers are asked to use specific examples from their
teaching experiences to support, clarify, and elaborate
responses.
Guided Reflection
Write a guided reflection.
The Lesson
• What were the goals of the lesson?
• What did you do to make this lesson relevant?
• What changes did you make in the flow of the
lesson? Why and when?
Reflection
•
What strategies (if any) were used for
remediation?
The Learner
• What questions were asked, how did you respond,
and why were they asked?
• If no questions were asked, reflect and respond to
this.
• What motivated student learning the most and why
or why not?
• What did you learn about students’ prior
understanding and approaches to the content?
The Teacher
• How did you motivate and keep the learners on
task?
• What type of questioning did you use?
• How did you assess the students’ learning and the
success of the lesson?
• What would you do differently? How and why?
*Add other “teacher thinking”.
Expert teachers may regard many of these
reflective questions as normal components of
classroom instruction. It is important, however, for all
teachers to view these strategies in a new light.
Learners are better served, as reflection becomes a
habit of mind for teachers. Devoting valuable time to
writing post-teaching reflections is a positive action
towards becoming a reflective teacher. Reflective
teaching has the potential to directly impact instruction
and student achievement. This, in turn, reveals the
power of reflective thinking and teaching.
Level Three: Developing a Reflective Venue
Journal writing is a popular instructional strategy
for teachers to use in the classroom with students.
Often teachers overlook its value as a reflective tool for
teaching. To initiate and maintain a professional
reflective journal in teaching is quite rewarding. Once
preservice and inservice teachers develop through
levels one and two, level three offers teachers an
opportunity to develop a practical and individualized
approach to reflective teaching. Reflective journal
entries enable teachers to track thinking and learning,
to evaluate these processes, and to improve teaching.
Entries are more than recording and reporting events.
They are visions of teaching and learning experiences
recreated through thinking, feeling, and intuition.
Establishing a journal writing routine is revealing and
rewarding.
Effective reflective journal writing provides
evidence of and documents the details of how teachers
plan, prepare, execute, and evaluate the teaching and
Norene Vail Lowery
learning tasks. It encourages goal setting and
documents the process as well as the products of the
teaching experience. Teachers think about their own
learning and understanding of the subject matter.
Journals are the venue for implementing the reflective
teaching strategies advocated in this discussion.
Teachers are encouraged to begin journal entries at the
beginning of the semester by addressing the prompts
on reflection, on prior experiences in mathematics, and
on mathematics and teaching from the first of this
discussion.
As reflective journal writing becomes a natural
part of teaching, teachers may find a free response
format more appropriate. This format is also
encouraged as a basic venue, since journal writing can
be quite overwhelming for novice teachers.
Free Response Format
• What did the class do today?
• What did I expect students to learn?
• What did I learn?
Give some thought to what you and the students
learned from the activities. A brief description of what
was done in class that day helps to reflect on the
learning. Was it relevant? Too easy? Too hard? What
did you learn about mathematics, about teaching,
about students, and about yourself?
Levels one and two, understanding the importance
of reflective thinking and implementing reflective
strategies, are core components of both my
undergraduate and graduate courses. Reflective
practices are used with preservice teachers as they plan
mathematics lessons to be taught in an elementary,
field-based context during a semester-long,
mathematics methods course. Inservice teachers
practice these same exercises with their own
mathematics classrooms, and then report insights in the
context of their university-based graduate methods
course. As time and classroom experience progresses,
teachers find individual and unique ways to use journal
writing as a reflective tool to improve instruction and
to increase student understanding and achievement.
Reflective Summary
Although the role of reflection in teaching is
considered important, reflective action in teachers is
inhibited often by isolation of teachers or by the
structure of courses and schools. The promotion of
reflection should begin early in preservice experiences
and with novice teachers (Artzt, 1999). Expert teachers
are encouraged to share reflective practices with
colleagues. Research findings confirm that the
29
likelihood of long-term success for many novice
teachers is hindered by the absence of expert guidance,
support, and opportunities to reflect (Veenman, 1984).
Through reflective teaching, all teachers acquire
critical skills in determining the value of instructional
strategies, in assessing students’ mathematical
understanding, and in developing curricular
knowledge. School administrators, teacher educators,
and expert teachers play pivotal roles in supporting the
development of reflective teachers.
Successful classroom experiences of preservice
and inservice teachers demonstrate the value of
reflective teaching. Reflective teaching practices
promote greater student achievement and success in the
classroom. Benefits from reflective teaching include
increases in confidence, autonomy, and self-efficacy
for teachers (Lowery, 2002). More effective
questioning techniques – such as use of those that
promote higher order thinking skills and use of openended questions – are employed, and classroom
discourse is enhanced. Reflection allows teachers to
judge mathematics grade-level appropriateness, to
assess student abilities, to evaluate the use of
motivational techniques, and to design appropriate and
challenging mathematical learning activities.
Reflective teaching is an essential skill for teachers and
is a powerful component of successful teaching
(Goodell, 2000; Mewborn, 2000).
The power of reflection that blossoms from
implementing reflective strategies strengthens the
teaching and learning of mathematics. However, time
is a critical factor. Time spent on developing reflective
thinking and teaching is time well spent (Artzt &
Armour-Thomas, 1999). The depth of understanding
revealed in reflective teaching, the resulting
improvement in instruction, and the ultimate growth in
student learning far surpasses the initial sacrifice of
time. Admittedly, this discussion asks for a lot from
teachers. Yet, the best practices that have been
presented here have merit. Teachers, preservice and
inservice alike, have reported gaining much from
reflective practices (Lowery, 2002). Reflective
teaching in mathematics reaps incredible rewards.
Inservice teachers report having taken the next step in
promoting reflective practice by their mathematics
students. Having students develop reflective
perspectives on learning mathematics is promoted in
the NCTM’s (2000) principles and standards. Students
benefit by reflecting on their own learning to make
sense of mathematics. Reflection is a crucial
component that must be incorporated into every
teacher’s toolkit of instructional strategies at all levels
30
of mathematics instruction and learning. To provide
quality instruction and to increase student success and
achievement in all classrooms, the basic three “R’s”
must include a fourth—Reflection.
REFERENCES
Artzt, A. (1999). A structure to enable preservice teachers of
mathematics to reflect on their teaching. Journal of
Mathematics Teacher Education, 2(2), 143–166.
Artzt, A., & Armour-Thomas, E. (1999). A cognitive model for
examining teachers’ instructional practice in mathematics: A
guide for facilitating teacher reflection. Educational Studies in
Mathematics, 4(3), 211–235.
Borko, H., & Livingston, C. (1989). Cognition and improvisation:
Differences in mathematics instruction by expert and novice
teachers. American Educational Research Journal, 26(4),
473–498.
Borko, H., & Shavelson, R. (1990). Teachers’ decision-making. In
B. Jones & L. Idols (Eds.), Dimensions of thinking and
cognitive instruction (pp. 311–346). Hillsdale, NJ: Erlbaum.
Brooks, J. G., & Brooks, M. G. (1993). In search of understanding:
The case for constructivist classrooms. Alexandria, VA:
Association for Supervision and Curriculum Development.
[Preface, pp. vii–viii, by Catherine Twomey Fosnot].
Carter, K., Cushing, K., Sabers, D., Stein, P., & Berliner, D.
(1988). Expert-novice differences in perceiving and
processing visual classroom stimuli. Journal of Teacher
Education, 39(3), 25–31.
Cooney, T. (1985). A beginning teacher’s view of problem solving.
Journal for Research in Mathematics Education, 16, 324–336.
Dewey, J. (1933). How we think. Boston: Heath.
Elbaz, F. (1983). Teacher thinking: A study of practical knowledge.
New York: Nichols.
Feiman-Nemser, S., & Buchmann, M. (1985). Pitfalls of experience
in teacher preparation. Teachers’ College Record, 87(1),
53–65.
Fenstermacher, G. D. (1994). The knower and the known: The
nature of knowledge in research on teaching. In L. DarlingHammond (Vol. Ed.), Review of Research in Education (Vol.
20, pp. 3–56). Washington, DC: American Educational
Research Association.
Frid, S. (2000). Constructivism and reflective practice in practice:
Challenges and dilemmas of a mathematics educator.
Mathematics Teacher Education and Development, 2, 17–33.
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory.
Chicago: Aldine.
Goodell, J. (2000). Learning to teach mathematics for
understanding: The role of reflection. Mathematics Teacher
Education and Development, 2, 48–61.
Hersch, R. (1986). Some proposals for reviving the philosophy of
mathematics. In T. Tymoczko (Ed.), New directions in the
philosophy of mathematics (pp. 9–28). Boston: Birkhäuser.
Hoberman, S., & Mailick, S., (1994). (Eds.) Professional education
in the United States: Experiential learning, issues, and
prospects. Westport, CT: Praeger.
Lortie, D. (1975). Schoolteacher: A sociological study. Chicago:
University of Chicago Press.
Reflection
Lowery, N. V. (2002). Construction of teacher knowledge in
context: Preparing elementary teachers to teach mathematics
and science. Journal of School Science and Mathematics
Association (SSMA), 102(2), 16–31.
accommodating prior knowledge. The following selected
passages may help to make evident what a constructivist
learning approach means, for me as the teacher with the
intent to create such a classroom:
Mewborn, D. (2000). Learning to teach elementary mathematics:
Ecological elements of a field experience. Journal of
Mathematics Teacher Education, 3(1), 27–46.
Constructivism is not a theory about teaching. It's a
theory about knowledge and learning. Drawing on
a synthesis of current work in cognitive
psychology, philosophy, and anthropology, the
theory defines knowledge as temporary,
developmental, socially and culturally mediated,
and thus, non-objective. Learning from this
perspective is understood as a self-regulated
process of resolving inner cognitive conflicts that
often become apparent through concrete
experience, collaborative discourse and reflection
(Brooks & Brooks, 1993, p. vii).
National Council of Teachers of Mathematics (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Nisbett, R. E., & Wilson, T. D. (1977). Telling more than we can
know: Verbal reports on mental processes. Psychological
Review, 84(3), 231–259.
Schön, D. (1987). Educating the reflective practitioner: Toward a
new design for teaching and learning in the professions. San
Francisco, CA: Jossey-Bass.
Tisher, R. (Ed.). (1978). The induction of beginning teachers in
Australia. Melbourne: Monash University.
The constructivist viewpoint of learning supports
reflective thought (Frid, 2000).
Tobin, K., Tippins, D., & Gallard, A. (1994). Research on
instructional strategies for teaching science. In D. Gable (Ed.),
Handbook of research on science teaching and learning: A
project of the National Science Teachers Association (pp.
45–93). New York: Macmillan.
Prior knowledge influences knowledge
construction. As a reflective tool, constructivism
enables teachers to design appropriate learning
activities, promotes higher levels of thinking about
educational problems (reflection), and leads to
questioning, and the construction of more
knowledge (Tobin, Tippins, & Gallard, 1994).
Veenman, S. (1984). Perceived problems of beginning teachers.
Review of Educational Research, 54(2). 143–178.
Wilson, S. M., Shulman, L. S., & Richert, A. E. (1987). “150
different ways” of knowing: Representations of knowledge in
teaching. In J. Calderhead (Ed.), Exploring teachers’ thinking
(pp. 104–124). London: Cassell.
1
I choose the word constructivist here to represent
perspectives that learning is constructed by a learner who is
developing new knowledge, based upon as well as
Norene Vail Lowery
2
Here, I refer to the vision for teaching and learning set
forth in the National Council of Teachers of Mathematics
Principles and Standards for School Mathematics (2000).
31
The Mathematics Educator
2003, Vol. 13, No. 2, 32–37
Impact of a Girls Mathematics and Technology Program
on Middle School Girls’ Attitudes Toward Mathematics
Melissa A. DeHaven
Lynda R. Wiest
This research investigated the impact of an all-female, non-school-based mathematics program on middleschool-aged girls’ attitudes towards mathematics. Girls who attended a Girls Math and Technology Program for
two consecutive years completed the Modified Fennema-Sherman Mathematics Attitude Scale before and after
attending the program. Confidence scores increased significantly, whereas score increases in perceived
usefulness of mathematics and perceived teachers’ attitudes toward the girls in mathematics were not
significant. (The “mathematics as a male domain” subscale was not assessed due to a low reliability score.)
Race and community background factors did not significantly affect the girls’ scores. Implications of findings
and key program features are discussed.
During their early years, students develop the skills
and attitudes toward learning that form the basis for
future academic growth (Boland, 1995). If students
develop a negative learning pattern toward a subject, it
is extremely difficult to change.
Females’ lower mathematics achievement in
comparison with males is one area of educational
concern that appears to be attitudinally based. On the
2000 National Assessment of Educational Progress
(NAEP), males attained higher scores than females at
the three grade levels tested (fourth, eighth, and
twelfth). Males’ Scholastic Assessment
Test–Mathematics (SAT-M) scores for the 2000-2001
school year topped that of females by 35 points
(National Center for Education Statistics, 2002). Fox
and Soller (2001) point out that performance
differences on the SAT-M, which also appear on the
Graduate Record Exam (GRE), can be costly for
women in terms of college admissions and scholarship
decisions.
The research reported here investigated whether
voluntary participation in a Girls Math and Technology
Program improved middle-school-aged girls’ attitudes
toward mathematics. Results are reported and
discussed for a group of two-year program
participants’ initial and follow-up ratings of their
personal confidence in mathematics, perceived
Melissa A. DeHaven teaches third grade at Smithridge Elementary
School in the Washoe County School District in Reno, Nevada. She
recently completed her master’s degree in elementary education
with an emphasis in mathematics at the University of Nevada,
Reno.
Lynda R. Wiest is an Associate Professor of Mathematics
Education at the University of Nevada, Reno. Her professional
interests include K-8 mathematics education, educational equity,
and teacher education.
32
usefulness of mathematics, and perceptions of their
regular classroom teachers’ attitudes toward
themselves in mathematics. The data were also
examined for Whites and Non-Whites, as well as rural
and urban participants, to see if the program had a
differential impact on some participants.
Review of Related Literature
Attitudes toward mathematics, including
perceptions of how appropriate mathematics is for
females, play a prominent role in females’ lower
performance and participation in mathematics in
relation to males. Based on their analysis of NAEP
data trends, Bae, Choy, Geddes, Sable, and Snyder
(2000) contend, “Achievement gaps appear more
closely related to attitudes than to course taking” (p.
117). The data show that females are less likely than
males to like or to think they were good at
mathematics. Females also experience mathematics
anxiety to a greater degree than males (Levine, 1995).
Females’ dispositions toward—and hence
achievement and participation in—mathematics are
believed to be socialized, inculcated by a society that
tends to view mathematics as a male domain and which
perpetuates the idea that males are naturally more
mathematically inclined (Hanson, 1997). Teachers
sometimes contribute to girls’ poor self-concept in
mathematics. They may imply, for example, that girls
do not need mathematics or they may react more
negatively when girls ask questions of clarification
than when boys ask (Jackson & Leffingwell, 1999).
Jones and Smart (1995) consider lack of confidence to
be a major factor affecting girls’ low participation in
mathematics.
Much interest in single-sex educational settings has
appeared in recent years. Evidence from a variety of
Impact of a Girls Mathematics Program
researchers and educators speaks to increased
confidence, achievement, or subsequent participation
in higher-level coursework for girls in single-sex
mathematics classrooms (e.g., Streitmatter, 1997;
Wood & Brown, 1997). Participants in Streitmatter’s
(1997) two-year study of seventh- and eighth-grade
girls in all-female mathematics classes reported an
enhanced ability to learn the mathematics, an improved
view of themselves as mathematicians, and a clear
preference for this type of environment. One reason for
girls’ greater comfort level in this type of classroom
may be their expressed concerns about intimidation by
boys in mixed-gender mathematics settings, namely,
fear of being dubbed smart or fear of asking questions
that boys deem “dumb” or otherwise unacceptable
(Durost, 1996). Moreover, boys tend to dominate
classroom conversation, be called on in class, be
permitted to call out in class more often than girls, and
they receive more teacher attention, including more
useful feedback (Durost, 1996; Sadker, Sadker, Fox, &
Salata, 1993/94).
Numerous out-of-school Science, Mathematics,
Engineering, and Technology (SMET) programs for
girls, such as after-school clubs or summer programs,
have had a positive impact on their participants in
terms of knowledge acquired and—in
particular—favorable attitudes gained (e.g., Karp &
Niemi, 2000; Mawasha, Lam, Vesalo, Leitch, & Rice,
2001). Dobosenski (2001) maintains that these types of
experiences should begin in elementary or early middle
school. Common elements in successful SMET
programs include: a comfortable learning climate (e.g.,
fun, noncompetitive, open to questions); career-related
information and issues; development of SMET content
knowledge acquired experientially; academic and
social support that includes peers and adult role
models; self-concept and confidence building through
effective group work and successful performance in
SMET activities (Campbell, 1995; Mawasha et al.,
2001). Opportunities to see mathematics as femaleappropriate permeate these program features.
Girls’ interest in mathematics begins to wane at
about the middle school level, which is also the
juncture at which students make decisions about future
course enrollments and career tracks. Therefore,
middle school is a critical “make-or-break-it” point for
girls in mathematics (Campbell, Denes, & Morrison,
2000). Researchers stress the importance of offering
early intervention programs for underrepresented
groups (e.g., girls and students of color). These
programs would emphasize career preparation,
improve mathematics skills, and develop interest and
Melissa A. DeHaven & Lynda R. Wiest
positive attitudes (Trentacosta & Kenney, 1997). The
program this paper describes is one attempt to bolster
and extend middle school girls’ in-school mathematics
experiences.
The Girls Math and Technology Program
The Girls Math and Technology Program1 is
available to Northern Nevada girls who will enter
grade 7 or grade 8 the fall after they enter the program.
The main program component is a five-day, residential
summer camp held at the University of Nevada, Reno
with classes held at the College of Education. The
program includes two full-day Saturday sessions, one
held in the fall and one in the spring of the following
school year. The program began in the summer of 1998
and ran for four years by the time data analysis took
place for this research.
Currently, applications are sent to all Northern
Nevada public, private, and Native American schools.
The typical class size is 28 girls who work with others
of their own grade level. In 1998, 28 girls entering
grade 7 participated in the program, followed by 56
girls entering grades 7 or 8 in 1999, 76 girls entering
grades 7, 8, or 9 in 2000, and 57 girls entering grades
7, 8, or 9 in 2001. The girls are randomly selected to
ensure a fair selection process, so students of varied
ability, race/ethnicity, socioeconomic status, and
community background have the opportunity to
participate. Scholarships are available to participating
girls with demonstrated financial need.
The mathematics topics addressed during the Girls
Math and Technology Program include geometry,
algebra, data analysis and probability, problem solving,
and spatial skills. The girls also learn biographical
information about historical and contemporary female
mathematicians, and a guest speaker from the local
community discusses her use of mathematics on the
job.
Two to four all-female staff members are in each
classroom at all times. The staff consists of a balanced
mix of veteran teachers who are active in mathematics
education and upper-division teacher education majors
or beginning teachers. Each lesson is developed in
accordance with the Nevada Mathematics Standards
established for the grade level the girls will enter in the
fall.
Two key program components designed to impact
participants’ attitudes positively include providing
female role models and employing an instructional
approach that involves hands-on, conceptual,
collaborative learning in a non-threatening atmosphere.
Although these features can be incorporated into the
33
regular classroom, they appear to be infrequent or at
least inconsistent aspects of middle-grades
mathematics instruction, as the girls’ comments
indicated in other research on this program (Wiest,
2003). Moreover, the single-sex nature of the
program—in terms of both participants and
staff—deviates from the typical mathematics
classroom and was perceived to support the two
program components noted above. Further discussion
of critical program elements appears in the Discussion
and Summary section of this paper.
Research Purpose
The purpose of this research was to investigate the
impact of a same-sex, non-school-based mathematics
program on middle-school-aged girls’ perceptions of
their attitudes towards mathematics.
1. Did the girls’ perceptions of their personal
confidence in mathematics, the usefulness of
mathematics, mathematics as a male domain, and
their teachers’ attitudes towards themselves in
mathematics improve over time after attending the
Girls Math and Technology Program for two
consecutive years?
2. Were the girls’ attitudes influenced by their race
(White or Non-White) or community background
(urban or rural)?
Research Method
Sample
The research sample consists of 36 Northern
Nevada girls who attended the Girls Math and
Technology Program for two consecutive years. Each
girl had started the program during the summer prior to
entering grade 7 and returned a year later prior to
entering grade 8. The girls’ backgrounds are varied in
terms of mathematics ability, socioeconomic status,
and home community type. The sample includes 64%
Whites and 36% Non-Whites (5% Black, 20% Native
American, 5% Hispanic, 3% Asian, and 3% Biracial),
of which 61% come from an urban area and 39% from
a rural area.
Design and Procedures
The data-gathering instrument used in this research
was the Modified Fennema-Sherman Mathematics
Attitude Scale. This scale provides information about
girls’ attitudes toward mathematics in the following
categories: personal confidence about the subject
matter, usefulness of the subject matter, perception of
34
the subject as a male domain, and perception of
teachers’ attitudes toward the respondent in the
subject2.
The
Modified
Fennema-Sherman
Mathematics Attitude Scale contains 47 positive and
negative statements on a five-point, Likert-type scale
that ranges from “strongly agree” to “strongly
disagree.” The highest score possible is 235, with 5
points assigned to the most self- or mathematicsfavorable choice on each of the 47 items. The
confidence, usefulness, and teacher’s attitudes
subscales each contain 12 items with a highest possible
score of 60. The male domain subscale contains 11
items with a highest score of 55.
The Modified Fennema-Sherman Mathematics
Attitude Scale was given to the girls the first day (pretest) of the Girls Math and Technology Program as
well as the final day (post-test) of the week-long
summer camp they attended for the first and second
years, respectively.
Data Analysis
Pre-test and post-test scores on the FennemaSherman instrument were used for each girl in this
sample. The scores consisted of totals for each of the
four subscales.
A reliability analysis, using Cronbach’s alpha, was
conducted to test for internal consistency within each
of the four subscales for the pre- and post-tests. Means
and standard deviations were calculated for the four
subscales for the pre- and post-test. To determine if the
subscale scores improved, the means for the two tests
in each of the four subscales were compared using a
two-tailed, paired-samples t-test, with alpha set at the
.10 level.
To determine whether the girls’ attitudes were
related to their race (White or Non-White) or
community background (urban or rural), we conducted
a one-way analysis of covariance (ANCOVA). Means
and standard deviations were calculated for each level
of the variables in all subscales for both pre- and posttests.
Results
Internal consistency for each subscale was
calculated using Cronbach’s Alpha. For the pre-test,
the highest alpha (.79) was obtained for teachers’
attitudes, with .75 for confidence, .74 for usefulness,
and .37 for male domain. For the post-test, the highest
alpha (.87) was obtained for confidence, with .84 for
teachers’ attitudes, .81 for usefulness, and .61 for male
domain.
Impact of a Girls Mathematics Program
Table 1
Modified Fennema-Sherman Mathematics Attitude Scale: Pre- and Post-Test Means and Standard Deviations
Confidence
Usefulness
Teachers’ Attitudes
N
M
SD
N
M
SD
N
M
SD
Pre-Test
33
50.70
6.62
32
55.34
5.22
33
49.70
7.51
Post-Test
33
53.48
7.15
32
56.31
5.33
33
50.51
7.51
Discussion and Summary
The alphas for three of the subscales—confidence,
usefulness, and teacher’s attitudes—fell within the
acceptable range of .70 or above. However, the
reliability of the male domain subscale was below the
acceptable range, with the pre-test analysis at .37 and
the post-test at .61. Therefore, it was omitted from
further exploration.
The most influential aspect of the Girls Math and
Technology Program is the positive impact it has on
the girls’ self-confidence in mathematics (and perhaps
technology, which the instrument did not measure).
Girls’ and boys’ confidence in their mathematics
abilities do not differ in the early grades, but a lack of
confidence becomes evident for girls as they enter
middle school (Boland, 1995). This is particularly
important because mathematics becomes more
complex at this point in time (Boland, 1995).
Improvements in participants’ perceptions of the
usefulness of mathematics and their teachers’ attitudes
toward themselves were slight and were not
statistically significant. In the case of mathematics’
utilitarian value, scores were already somewhat high
and thus were less likely to show a statistically
significant increase. It may also be difficult for middle
school students in general to appreciate the usefulness
of mathematics, because they may be too old to accept
rhetoric stating that mathematics is useful and they
may be too young to associate school mathematics
with their daily lives in a meaningful manner. It is
disappointing that the girls’ greater self-confidence in
mathematics, as associated with this program, did not
translate into more positive perceptions of their
teachers’ attitudes toward themselves. This may
highlight educators’ critically important role—and
therefore the need for high-quality professional
Girls’ Attitudes Toward Mathematics
A paired-samples t-test was conducted to evaluate
if the girls’ attitudes improved over the two years they
attended the camp. Table 1 shows mean and standard
deviation scores for the two Fennema-Sherman
Mathematics Attitude Scale tests: the pre-test at the
beginning of the first year and the post-test at the end
of the second year. The results show that the increase
in the girls’ confidence level was statistically
significant (t (35)=2.65, p= .012). The girls’ scores on
the other two subscales did not increase significantly
over the two-year period.
Race and Community Background
A one-way analysis of covariance (ANCOVA) was
conducted on each subscale to evaluate if the girls’
scores were influenced by their race and by their
community background with the pre-test scores as the
covariate. Race and community background factors did
not significantly affect the girls’ scores (see Table 2).
Table 2
Modified Fennema-Sherman Mathematics Attitude Scale: Analysis of Covariance for Influence of Race and
Community Background
Confidence
Usefulness
Teachers’ Attitudes
Source
df
F
p
df
F
p
df
F
p
Race
1
0.175
0.679
1
0.184
0.725
1
0.056
0.814
Area
1
1.795
0.191
1
0.56
0.815
1
0.457
0.505
Race X Area
1
1.492
0.232
1
0.006
0.94
1
0.079
0.781
Error
28
-33.19
27
-23.01
28
-56.8
Melissa A. DeHaven & Lynda R. Wiest
35
development in gender-equitable teaching—in
mathematics instruction for middle school girls. It is
not possible to determine from these data the impact
perceptions of teachers’ attitudes had on the girls’
attitudes. Nevertheless, it is reasonable to assume that
teachers, with whom students spend a great deal of
time and who judge the value of students’ work,
influence students’ academic self-perceptions.
According to these data, differences in race or
community background did not appear to cause
differential program impact. Instead, several shared
attributes and interests predominated across the varied
individuals who participated in this program. In other
research on this program (Wiest, 2003), qualitative
data in the form of personal interviews, camp-end
surveys, and fall follow-up questionnaires showed
some of these key commonalities to be gender, interest
in mathematics, and a chance to meet new people and
experience life away from home.
Several features led to this program’s success
(Wiest, 2003). The residential nature of the summer
camp is a critically important aspect in that it allows
girls from rural towns to participate. For instance, a
group of parents drive their daughters seven hours
from a remote Native American reservation to Reno in
order to attend this program. By staying over night, the
girls bond with each other and with the staff members
over the course of the week, and they see that girls and
women who like mathematics are “normal” people
with many interests and abilities. One second-year
camper’s parent told the camp director, for example,
that one thing that surprised and impressed her
daughter the previous year was that the camp director
had played football with the girls on the campus quad
during an evening recreation time.
Another important program element, according to
Wiest’s (2003) qualitative data, is the type, amount,
and quality of the mathematics content. The topics
addressed in the program are made to be interesting
and challenging in ways many girls have never
experienced. Several girls noted that learning
mathematics without homework or the pressure of
earning a grade greatly reduced their anxiety compared
with their school experiences.
This program’s instructional approach, which
centers about group work and hands-on learning in a
supportive environment, also surfaced as a key
program element (Wiest, 2003). The program’s
methods of instruction and comfortable climate seemed
to differ from that which many of the girls encountered
in school, and they better suited the girls’ needs and
learning styles.
36
Finally, Wiest (2003) found that the all-female
staff is another strong component of the Girls Math and
Technology Program. The role-model aspect of this
program, including female instructors, a female guest
speaker, and information about accomplished female
mathematicians, helped the girls see themselves as
potentially successful mathematicians in both the
present and future. Several parents said that their
daughters began talking about the importance of
mathematics and considering mathematics-related
careers in the months after the summer camp had
ended.
This supplementary program for girls has the
luxury of several benefits that the typical school does
not. One is that instructors are chosen from among the
most highly qualified local mathematics teachers.
Another benefit not afforded to most schools—besides
the single-sex nature of the participants and
instructors—is the lower student-to-teacher ratio, with
2 to 4 instructors per 28 girls. This allows for more
individual attention than most schools are able to
provide.
Closing Comments
The National Council of Teachers of Mathematics
(1995) notes that equity is a critical factor in the
nation’s economic viability. The workplace requires
that all Americans, including minorities and women,
have the mathematics skills needed to meet the
demands of the global marketplace. Eliminating the
social injustices of past schooling practices will require
the support of policymakers, administrators, teachers,
parents, and others concerned about excellence and
equity in mathematics education. All children can learn
challenging mathematics with appropriate support and
an equitable learning environment, regardless of
ethnicity, race, gender, or social class. Until females
and other lower-achieving and underrepresented
students attain parity in mathematics, supplemental
programs such as the one discussed in this paper can
provide important support mechanisms beyond that
which schools may offer.
REFERENCES
Bae, Y., Choy, S., Geddes, C., Sable, J., & Snyder, T. (2000).
Trends in educational equity of girls and women. Education
Statistics Quarterly, 2(2), 115–120.
Boland, P. (Ed.). (1995). Gender-fair math. Newton, MA: WEEA
Publishing Center.
Campbell, G., Denes, R., & Morrison, C. (2000). Access denied:
Race, ethnicity, and the scientific enterprise. New York:
Oxford University Press.
Impact of a Girls Mathematics Program
Campbell, P. B. (1995). Redefining the “girl problem in
mathematics.” In W. G. Secada, E. Fennema, & L. B. Adajian
(Eds.), New directions for equity in mathematics education
(pp. 225–240). New York: Cambridge University Press.
Dobosenski, L. (2001). Girls and computer technology: Building
skills and improving attitudes through a girls’ computer club.
Library Talk, 14(4), 12–16.
Durost, R. A. (1996). Single sex math classes: What and for whom?
One school’s experiences. NASSP Bulletin, 80, 27–31
Fox, L. H., & Soller, J. F. (2001). Psychosocial dimensions of
gender differences in mathematics. In J. E. Jacobs, J. R.
Becker, & G. F. Gilmer (Eds.), Changing the faces of
mathematics: Perspectives on gender (pp. 9–24). Reston, VA:
National Council of Teachers of Mathematics.
Hanson, K. (1997). Gender, discourse, and technology. Newton,
MA: Education Development Center.
Jackson, C. & Leffingwell, R. (1999). The role of instructors in
creating math anxiety in students from kindergarten through
college. The Mathematics Teacher, 92(7), 583–586.
Jones, L., & Smart, T. (1995). Confidence and mathematics: A
gender issue? Gender and Education, 7, 157–166.
Karp, K. S., & Niemi, R. C. (2000). The math club for girls and
other problem solvers. Mathematics Teaching in the Middle
School, 5, 426–432.
Levine, G. (1995). Closing the gender gap: Focus on mathematics
anxiety. Contemporary Education, 67, 42–45.
Mawasha, P. R., Lam, P. C., Vesalo, J., Leitch, R., & Rice, S.
(2001). Girls entering technology, science, math and research
training (GET SMART): A model for preparing girls in
science and engineering disciplines. Journal of Women and
Minorities in Science and Engineering, 7, 49–57.
National Center for Education Statistics. (2002). Digest of
education statistics, 2001. Retrieved May 2, 2002, from
http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2002130
Melissa A. DeHaven & Lynda R. Wiest
National Council of Teachers of Mathematics. (1995). Assessment
standards for school mathematics. Reston, VA: Author.
Sadker, M., Sadker, D., Fox, L., & Salata, M. (1993/94). Gender
equity in the classroom: The unfinished agenda. The College
Board Review, 170, 14–21.
Streitmatter, J. (1997). An exploratory study of risk-taking and
attitudes in a girls-only middle school math class. The
Elementary School Journal, 98, 15– 26
Trentacosta, J., Kenney, M. (Eds.). (1997). Multicultural and
gender equity in the mathematics classroom: The gift of
diversity: 1997 yearbook. Reston, VA: National Council of
Teachers of Mathematics.
Wiest, L. R. (2003). Impact of a summer mathematics and
technology program for middle school girls. Manuscript
submitted for publication. [Contact wiest@unr.edu for the
manuscript.]
Wood, B. S., & Brown, L. A. (1997). Participation in an all-female
Algebra 1 class: Effects on high school math and science
selection. Journal of Women and Minorities in Science and
Engineering, 3, 265–277
1
Interested individuals can obtain the manuscript entitled
“Impact of a Summer Mathematics and Technology Program
for Middle School Girls” by contacting Lynda Wiest at
wiest@unr.edu.
2
Teachers’ attitudes refer to those of regular classroom
teachers rather than instructors employed in this program.
These attitudes center about teachers’ interest in, respect for,
and encouragement of the girls as present and future
mathematicians. For more information, see the FennemaSherman
Mathematics
Attitude
Scale
at
http://www.woodrow.org/teachers/math/gender/
08scale.html.
37
The Mathematics Educator
2003, Vol. 13, No. 2, 38–46
First-Time Teacher-Researchers Use Writing in
Middle School Mathematics Instruction
Drew K. Ishii
This paper is a study of 4 middle school teacher-researchers who engage in action research projects for the first
time, in which they incorporate writing activities as part of their instructional practices. Embedded in a
professional development program with an emphasis on reform mathematics efforts, the teacher-researchers
report to their research support group on their experiences with using writing. They used writing in order to
improve classroom communication and state-mandated test scores. Recordings of conversations, written
reflections, and other documents showed that they used various writing activities including journal writing,
essays, problem solving, and the writing of stories. The teacher-researchers identify the major benefits of using
writing to be the support of student thinking and the increase in student discourse. The teachers’ projects
encouraged future ideas for instructional change.
Part of this research was presented at the 2002 Annual Meeting of the Mid-Western Educational Research Association,
Columbus, OH. October 17, 2002
This qualitative study investigates the experiences
of four middle school mathematics teacher-researchers
engaged in action research as part of a professional
development program. The focus of this paper is to
examine the experiences, practices, and issues that
emerged from the teacher-researchers’ projects as they
employed non-traditional writing activities in their
mathematics classes.
Action research is a practice by which teacherresearchers have the opportunity to learn from and
about their teaching. Through this methodology
teacher-researchers can reflect, evaluate, and learn not
only about their teaching, but also from their students.
Conducting action research projects allows teacherresearchers to reflect on their teaching and to explore
issues of teaching and learning that are relevant to their
lives. Engaging in action research can benefit all those
involved in that it can bring self-renewal and increase
efficacy, morale, and student performance (Sagor,
2000). Additionally, researchers reported that action
research increases a sense of professionalism for the
teacher-researcher (Elliot, 1991; Smith, Layng &
Jones, 1996).
The teacher-researchers around whose experiences
this discussion revolves were involved in a
professional development program at a major urban
midwestern research university. This program served
as a master’s degree program for some teachers and as
Drew K. Ishii is a doctoral candidate in mathematics education
at The Ohio State University in Columbus, Ohio. His research
interests are writing in mathematics, mathematical discourse and
communication, and representations.
38
a professional development program offering graduate
credit for those either not pursuing a master’s degree,
or those who had previously obtained a master’s
degree. The premise of the program was for the
teachers to implement innovative practices in their
teaching that coincided with current educational reform
and conduct an action research project with the support
of doctoral students and faculty from the university.
This culminated in a final paper centered on their
research. Collaborative efforts between teacherresearchers and universities as well as professional
development programs such as this one serve to aid in
teachers’ pursuits of conducting research projects of
their own, and thus create a life-long process of inquiry
for the teachers (Raymond & Hamersley, 1995).
Given what research says about using writing in
mathematics, I set out to see what the use of writing
looked like in the field from these teacher-researchers’
experiences. It was important to me that I get their
perspectives on using writing in mathematics
instruction. In keeping with a grounded theory research
methodology, the data was approached without a priori
research questions that would subsequently drive the
data analysis. Instead, personal questions or inquiry
issues provided the motivation to investigate the
experiences of the teacher-researchers in this project:
1. What do the teacher-researchers hope to gain by
using writing?
2. How do the teacher-researchers use writing in their
teaching?
3. What benefits do the teacher-researchers see in
using writing?
Writing in Middle School
Why Writing?
The use of writing assignments in school
mathematics gained recognition with the “writing to
learn” movement in the ‘80s and continues today as
evidenced by the National Council of Teachers of
Mathematics’ (NCTM) standards document, Principles
and Standards for School Mathematics (2000). The call
from NCTM to make communication an important
facet in the mathematics classroom has led to an
increase in instructional activities that encourage
communication not only between teacher and student,
but also among students. The Communication Standard
(NCTM, 2000) includes being able to organize,
communicate, analyze, and evaluate thoughts using the
language of mathematics. An essential facet of
communication is writing, which is used in just about
every academic subject though rarely in mathematics
classes. When used, communication through writing in
mathematics classes generally takes two forms: (a)
journal writing, or (b) expository writing assignments
and activities. In journal writing, students reflect on
some activity or respond to a prompt given by the
teacher in order to solidify their thinking on some topic
or concept. In expository writing, students use writing
as an active part of the learning process with in-class
writing activities or prompts aimed towards
explanatory or expressive purposes. For instance, a
writing prompt may require students to solve a
mathematical problem and then explain their thinking
or problem-solving processes. Expository writing
activities allow students to use another avenue or
representation in their mathematical learning, along
with a period of reflection when deciding what to
write. Expository writing assignments can be thought
of similarly to what some (Birken, 1989; Powell &
Lopez, 1989; Rose, 1989) call transactional writing, in
which the assignment is meant to be read by someone
other than the writer, usually a teacher. It is important
to consider both uses of writing in mathematics
because each activity has its own benefits depending
on what the teacher wants to accomplish (Birken,
1989; Borasi & Rose, 1989; Cai, Jakabcsin & Lane,
1996; Drake & Amspaugh, 1994).
The writing that students do in mathematics classes
is quite different from other classes or disciplines since
mathematics is presented as a heavily symbolic
discipline. The bulk of student work in mathematics
classes consists of symbol manipulation. The symbols
are the language of mathematics and ordinary language
is used to explain the mathematics. The use of writing
to learn mathematics, however, tries to use writing in
different capacities of the learning process. Keith
Drew K. Ishii
(1989) offers several types of writing assignments such
as: assessment of material, anticipation of new
material, discussion, peer collaboration, revision, and
evaluation. Birken (1989) suggests that writing can be
used for informal in-class writing, homework problems
that interpret or analyze, essay questions, and formal
technical writing. Multiple-entry logs, another type of
writing technique, combine journal writing with
expository writing; students are asked to respond to a
prompt or problem, then revisit their writing and
thinking periodically to see how it changes over time,
if at all (Powell, 1997). In trying to articulate their
thoughts into words, students engaged in these types of
assignments reflect and internalize. This process
promotes further learning.
Two studies (Pearce & Davison, 1988; Shield &
Galbraith, 1998) approached student writing in a
discourse analytic manner where they classified
students’ writing in order to determine the elements
present in student writing. They offer a more in-depth
look into student writing in mathematics classes. In
these studies, researchers recognized that the type of
writing that occurs in mathematics classes is different
than that of other disciplines, and thus needed to be
examined further in order to assess the elements of
student mathematical writing. In effect, they examined
the writing that resulted from various types of writing
assignments and discovered how students
communicated their knowledge to the teacher. Shield
and Galbraith (1998) analyzed 8th-grade students’
writing and developed a coding scheme for content of
the writing in order to generate a model of student
writing. In addition to developing the coding scheme,
they compared the writing samples with the type of
writing that occurred in the students’ textbook. They
identified six features of the students’ writing:
exemplar, goal statement, kernel, justification, link to
prior knowledge, and practice exercises. The most
common of these was exemplar in which students gave
written descriptions of specific examples, diagrams,
conventions, and graphs (p. 39). In comparison with
the textbook, they found that studentwriting samples
heavily reflected the same type of writing style: (a) a
focus on procedures and algorithms with little
elaboration, and (b) an authoritative tone (p. 45).
Previously, in 1988 Pearce and Davison
determined the amount, kinds, and uses of writing that
teachers employ in junior high school mathematics
classes. By looking at student samples and teacher
interviews, they classified five types of writing
activities: direct use of language (copying and
transcribing information), linguistic translation
39
(translation of mathematical symbols into words),
summarizing/interpreting (summarizing, paraphrasing,
and making personal notations about material from
texts or other sources), applied use of language
(situations where a mathematical idea is applied to a
problem context), and creative use of language (using
written language to explore and convey mathematically
related language) (p. 10). They found that the direct
use of language activities were most frequently used.
Research on writing in mathematics offers not only
various methods of incorporation into instruction, but
also the benefits from using such techniques. Borasi
and Rose (1989) found that journal writing had a
therapeutic effect on students, as well as increased
learning of the material, and improved problem-solving
skills. They also found that teachers benefit from using
writing in that they are better able to provide feedback
and make better evaluations of student learning or
misconceptions. From this, there is potential for
teachers to make long-term improvements in their
instruction. Miller (1992) reported similar benefits for
teachers utilizing impromptu writing prompts. By
reacting to student writing, instructional practices were
influenced when the teachers would re-teach, delay
exams, schedule review sessions, and initiate
discussion over misconceptions.
This account of some of the research in
communication and mathematics shows how writing
can be beneficial for both the teacher and the
mathematical learning of the students. In many of these
studies, researchers partnered with schools in an effort
to study the issues concurrent with the mathematics
reform efforts. In a similar manner, based upon the
examination of the current NCTM standards, the
teacher-researchers with whom I worked in the
professional development program sought to
incorporate similar research ideas into their instruction
and thusly into their action research projects.
Methodology
The Project and the Teachers
I assisted in a professional development program1
to support 4 of the 13 mathematics teacher-researchers
enrolled. These four were in the data collection stages
of their research when I joined the effort. I provided
regular guidance in their data collection and analysis
efforts for their action research projects. I had also
served as a support person for one of the teachers
(Iris2) in the previous year. As a doctoral student in
mathematics education, I was asked to participate in
this program as part of the support team because my
research interests (communication and mathematics)
40
and experience would be useful to some of the teacherresearchers involved in the project.
As part of their participation in the program, all of
the teachers in the program were assigned a support
person, who was either a graduate student or a
university professor. The support person helped with
planning and implementing instruction and provided
support and expertise in their action research
endeavors. By the time I joined the support team, the
teacher-researchers had been in the professional
development program for just over one year with one
year left to go. Teachers joined the program in order to
learn more from and about their own teaching, and (for
some) to work towards a master’s degree in education.
Each teacher-researcher chose a topic and designed
research questions they would investigate throughout
the duration of the two-year program. The desire to
change their teaching practices drove their research
questions, which in turn provided a theme for their
instruction for the two years of the program.
For their action research projects, the four teachers
discussed in this paper chose to implement writing in
their mathematics classes using either journals or
expository writing exercises such as those mentioned
earlier. Three of the teachers taught sixth-grade
mathematics while one taught eighth-grade
mathematics. The sixth-grade teachers, Iris, Jean, and
Amber, taught in urban schools, while the eighth-grade
teacher, Joanne, taught in a suburban school. Amber
was the only teacher of the four who was working
towards her master’s degree. The other three were in
the program to obtain graduate professional
development credit. These four teachers individually
have fewer than 10 years teaching experience.
From their research proposals and from numerous
conversations, three reasons resonated between the
teacher-researchers indicating why they chose writing
in mathematics as a focus for their research. First,
writing is encouraged in mathematics education reform
efforts. To the teacher-researchers, writing in
mathematics was a practice they saw as novel and
outside the realm of the traditional mathematics
classroom. Second, writing is incorporated in the openresponse elements of the state proficiency exams. As
with most school districts across the country, student
performance on state exams is important, and these
teacher-researchers saw the utilization of writing as a
technique that would help prepare their students for the
tests. Finally, the open-response questions on state
exams were traditionally an area of the exams in which
middle school students in their districts scored very
Writing in Middle School
poorly. Thus, the teacher researchers sought to improve
students’ scores by focusing on writing tasks.
Research Design
In this study, I used qualitative methods to
examine the experiences of the four teacherresearchers. Three types of data were collected: audio
recordings of meetings and conversations, documents
collected from the teacher-researchers, and my
personal field notes. Each data analysis meeting for the
teacher-researchers’ projects was audio recorded and
field notes were taken during those meetings. Other
conversations regarding the projects were audio
recorded as well. The documents that were analyzed
included their research proposals, reflections
throughout the past year, open-ended surveys, and final
papers. I analyzed these tapes and documents using
principles of grounded theory (Charmaz, 2000). The
emergent patterns and themes in the taped
conversations were investigated further and
triangulated with the documents (Janesick, 2000). In
qualitative research methods, these types of documents
are important data sources because they catalog the
participants’ beliefs, values, and experiences (Marshall
& Rossman, 1995), as they did throughout the two
years of this program.
Although I was a support person for Iris and had
personally assisted with her action research project, for
this study I limited the scope of the audiotaped data
collected from our interactions to those that included
the other three participants in order to be fair to all of
the teacher-researchers. Since I started supporting all
of the participants approximately 10 weeks before the
end of their school year, and subsequently the end of
their data gathering and analyzing, the conversations
that were audiotaped occurred within the near-weekly
meetings of those 10 weeks. The documents, on the
other hand, were collected throughout their program by
the director and given to me once I joined the support
team.
Once the project concluded and all data for this
study was collected, I inductively analyzed all of the
documents, including my field notes, for emerging
patterns and issues. These fell within two general
categories, research issues, and issues related to the use
of writing. For the purposes of this discussion the
research issues have not been included in the findings.
I then listened to the tapes of our meetings and
conversations with the intent of finding more evidence
to support the long list of codes that were made from
the patterns and issues obtained from the documents.
After several iterations of this process, the codes that
Drew K. Ishii
boasted the most support were further examined and
developed into the theory that will be discussed. It is
important to note that in keeping consistency with the
principles of grounded theory, disconfirming data or
negative cases were sought after, but were not found.
Without discussing each teacher-researcher’s
individual project, the proceeding discussion is limited
to their experiences with implementing the writing in
their classes, including their future research directions.
Findings
What They Hoped to Gain
Each teacher-researcher began their academic year
by writing a research proposal outlining their research
plans for the year. These proposals were complete with
research questions, methods, and proposed data
collection and analysis. As mentioned before, an
important reason for the four teacher-researchers to
implement writing into their mathematics instruction
was to improve the open-ended response questions on
their students’ state proficiency tests. Joanne said, “I
hope to change the way students feel about math, help
students do better in math, and increase their
mathematical understanding.” Amber echoed this
sentiment by explaining that she wanted to supply her
students with appropriate tools for approaching the
extended response questions on the state exams. From
past teaching experiences, she noticed that her students
struggled on open response questions and sought to
improve their scores. Similarly, Jean hoped her project
would result in a change in students’ attitudes and
improve the open-ended response question scores. For
all of the teacher-researchers, seeing their students
succeed in mathematics was important. But beyond
that, seeing that problem solving is an important aspect
of daily life both inside and outside the classroom, Iris
and Joanne wanted to furnish their students with the
necessary tools to help them in the future. Iris said, “I
am looking for some way of making problem solving
less threatening in general, [and] to help increase
students problem solving capabilities. Joanne agreed
saying, “I am hoping that through writing,
communicating, students’ attitudes and conceptual
understanding will improve.”
All of the teacher-researchers not only wanted their
students to do better on their tests, but also wanted to
help their students learn the mathematics and make it
less difficult. This concern for their students provided
motivation for their projects. From their research
proposals, in addition to the current literature on
writing in mathematics (e.g., Borasi & Rose, 1989;
Johanning, 2000; Jurdak & Zein, 1998), the teacher41
researchers’ concerns and goals were both appropriate
and reasonable tasks.
Their Writing Activities
Joanne. The teacher-researchers implemented
writing in a variety of ways ranging from journal
writing to problem solving. These types of activities
were similar to those activities found in the literature.
Joanne used writing activities to start class, frequently
using them as a warm-up exercise to focus the
students’ attention on the mathematics of the day. She
used writing prompts that were problem-solving in
nature and insisted that students work individually
ensuring that everyone attempted the problem. She
often had students form small groups giving the
opportunity for sharing their strategies and solutions
with each other. This led to increased student
participation and motivation. Since students spent time
working on the problems, they were interested in
sharing their work and seeing the various ways other
students approached the problems. Even if students did
not understand how to process the problem or get the
answer, they could share how they set up their
information and attempted to solve it.
To facilitate students' writing, Joanne developed a
problem-solving format called ODEAR (see Zupancic
& Ishii, 2002), an acronym that helped the students
organize their thoughts when writing. ODEAR consists
of five elements: Organize, Define, Explore, Answer,
and Reflect. When given a problem to solve, the
students used the acronym to start and thoroughly
answer their problems.
Iris. Iris’s employment of writing in her classes
was done primarily as in-class activities. She used a
prescribed writing process similar to Joanne’s
ODEAR. Iris’s problem-solving format, called EPSE
(Explore, Plan, Solve, Evaluate), was a process
prescribed by her district’s curriculum materials. She
used a teacher’s supplement as a source for many of
the problems she assigned. Typically, Iris gave a word
problem on the board and had students solve and write
individually. Occasionally they would compare their
work with each other, but generally they worked alone.
In one of her lessons, Iris gave a problem and let
students work together in groups of four to five. The
groups then presented their work to the entire class
allowing everyone to see the different solutions. She
reported that students really liked that lesson and she
found it beneficial too because she immediately saw
what they knew about the material. Generally, Iris gave
her classes a few EPSE problems every week. The
students kept these problems along with their class
42
notes in binders that she referred to as their portfolios.
She eventually used some of the students’ portfolios as
data for her action research project.
Based upon our research meetings and from Iris’
reflective writings, she felt she had difficulties keeping
up with evaluating her students’ writing. She said, “I
wasn’t able to respond to their problems as well as I
should have. I should have given them more feedback
and let them give each other more feedback.” Time
was something with which all of the teacherresearchers struggled, but Iris felt that it was the major
struggle for her. Since Iris rarely allowed group sharing
of writing in the same way that her colleagues did, her
students received limited benefits from reflecting on
their writing after receiving feedback, whether it be
from her or from fellow students.
Amber. Amber was the only teacher-researcher
who used journal-writing activities. She used a journal
format where she asked students to write about their
feelings or attitudes, mathematical processes, and
mathematical concepts. The students kept journals or
notebooks as records of all of their writing. The
students regularly shared their writings that focused on
the mathematical procedures and content. Amber
periodically collected and provided feedback
addressing all the types of journal entries—affective,
procedural, and conceptual. She reported that students
would have benefited more from the journal-writing
assignments had she been able to collect them and
provide feedback more often. She said that keeping up
on journals was difficult especially since the process
was new to her; it was difficult to adjust to the time
constraints and reorganize time usage. Even so, Amber
did use the journal-writing assignments to have the
students share with each other and provide peer
evaluations.
Jean. Jean used a variety of writing activities,
instead of focusing on one type of activity as her
colleagues did. She used writing activities both during
class and as final thoughts or assignments that
encouraged reflection and summarization. One activity
in particular was what Jean referred to as the exit
ticket, a final activity of the class period that required
reflection or solving a problem. This activity was to be
completed either before students left the class or
moved onto science, which she also taught. She also
used prewriting assignments for expository essays to
help create assignments her students could share and
edit together. This was a way to foster thinking ahead
of time. Jean felt this along with students’ writing,
sharing, and revising, could lead to clear cohesive
pieces of expository writing. In addition to the well
Writing in Middle School
thought-out prewriting and writing assignments, Jean
used writing as a way of closing down or reflecting
upon discovery-type activities. For instance, when she
used manipulatives to model fraction arithmetic, she
included a writing activity for students to express what
they discovered. Jean also had a year-long project
where her sixth-grade students made math story books
for elementary school students from the neighboring
elementary school. At the end of the year, Jean’s
students shared their books with their partner class, and
she brought samples in for the rest of the members of
the professional development program to see.
Their Observed Benefits
In our final conversations, as well as in their
reflective writings, the teacher-researchers’ concluded
that after using the writing activities for a whole school
year, there were two aspects of the experience that
were of noteworthy benefit to the students and their
learning. The greatest benefit was that the use of
writing assignments promoted student-to-student
discourse, something that usually does not occur in the
traditional mathematics classrooms. The second benefit
the teacher-researchers identified was an observed
increase in student motivation, thinking, and
understanding from previous years of teaching. This
increase was a “perception” (sense of increase), not an
empirical increase since teachers did not perform
actual comparisons from the previous years. The
teacher-researchers acknowledged that the benefits to
students also served as benefits for themselves in that
they saw overall improvement in the very things they
sought to change.
Discussion
Improved discourse
In reform mathematics efforts (NCTM, 2000),
student discourse is an important element in the
activities of the mathematics classroom. Current
research supports the notion that social interactions
whether they be whole-class or small group discussions
benefit student learning (e.g., Cobb, Wood, & Yackel,
1993; Yackel & Cobb, 1996). Although improving
student-to-student discourse was not a specific goal for
the four teacher-researchers, they were well aware of
the importance for increasing communication in
general, and had that in mind when they chose to
implement writing. In addition, increasing classroom
communication was an overarching theme for the
entire professional development program. The
improvement in student discourse was somewhat of a
surprise to the teacher-researchers in that it was not
Drew K. Ishii
planned. For the students, however, it seemed as
though discussions naturally followed their writing.
Amber admitted that she never intended for the
writing activities to accompany discussion of it among
students. She planned to use writing as a learning tool
students could use individually, and use the journals
for personal reflection and learning. However, the
discussion of her students’ writing began by accident
when a student volunteered to read her writing aloud.
Amber indulged the student and after a couple of
instances, the student sharing of writing became a
norm and expectation of the classroom activity.
In a conversation we had about using writing and
how student-to-student discourse seemed to be a
natural consequence, Amber offered that the teacher
would have to allow it. “I don’t necessarily think that
employing a writing component in your math class is
very beneficial unless you utilize it and discuss [the
writing].” Amber also mentioned that in interviews
with her students, they indicated it was not necessarily
the writing that helped, but the sharing of the writing
and the discussions that came after. Even the students
saw the benefit of writing along with the opportunity to
discuss what they wrote with each other.
Iris commented that she agreed with what Amber
discovered about writing in her classes. Iris’ goal was
to improve students’ problem-solving skills, and she
felt that writing alone would not be sufficient, but
could when coupled with discussions of their problems
and solutions. Though Iris did not use writing activities
to promote discussions per se, she became aware that
through discussion the writing might be used as a
technique to encourage classroom discourse. Both
Joanne and Jean reported that students enjoyed
explaining their solutions to their classes and were
often eager to share their findings with others. As a
result, student participation became natural for students
instead of requiring solicitation by the teacher. Jean
responded, “The student-to-student discourse in my
class has promoted conversation and debate about
mathematical concepts.”
Having students discuss and debate mathematical
concepts is precisely the point of encouraging student
discourse. Through those discussions students are
given the opportunity to further reflect upon their own
thinking while possibly augmenting other students’
thinking to their own. With regard to writing activities,
students feel they have invested their time and effort
into something other than ordinary mathematics work,
and thus feel the natural progression to discussing their
work with each other and their teachers. These
conversations then provide the students with valuable
43
feedback about the way they are thinking about the
mathematics. The writing activities do not have to end
there however, another round of revision to the writing
students have already produced can solidify thinking
and add another layer to their understanding much the
same way multiple-entry logs enable students to revisit
their work (Powell, 1997).
Supported student thinking
Another consequence of using writing activities
along with discourse is that it supports student
thinking. Because of the reflective nature of speech and
dialogue, discussions among students can be valuable
tools for learning (Vygotsky, 1978). As mentioned
earlier, the discussions that accompany writing
activities enhance classroom communication and have
the potential to provide students with another
opportunity for reflection upon their thinking. Since
writing is a product-oriented classroom activity, the
students have a concrete record of their participation
and of their thinking, which they can refer to and revise
during discussions. The written product affords
students the opportunity for critical reflection, which
has the potential to give students control of their
learning as well as a means of monitoring progress
(Powell & Lopez, 1989). All of these steps within the
activity of writing support students’ thinking in a way
that is not usually seen in the traditional mathematics
classroom. Thus the use of writing can provide
students with extra tools for learning mathematics.
Joanne agreed with this position saying that
without discussion to “force” students to think about
their thinking, the writing activities are not meaningful.
She commented, “My students have learned many
things from each other this year, and from themselves.
Sometimes they understand better when another
student explains the mathematics.” Joanne felt that if
students really understand a concept they should be
able to teach and explain it. Iris followed with a
comment about argument and how it advances
learning; “Trying to convince someone you are right
through discourse is certainly a form of teaching and
teaching is a great way to learn.” The relationship
between learning and social interaction can be seen in
Joanne and Iris’ experiences. The cycle of doing,
thinking, and reflecting that writing promotes supports
the learning process by empowering students so that
they feel comfortable to take on peer teaching
responsibilities.
In Jean’s class, she noticed that reflection upon
mathematical material did not necessarily have to take
place in an elaborate/formal assignment, but could
44
occur as the day’s final activity. Recall that her exit
ticket activity required students to work out a problem
and/or reflect on it or that day’s lesson as a concluding
activity for the day. Jean said, “The exit ticket at the
end of the class lesson has encouraged students to think
about what has been learned in class and encouraged
discussion that sometimes does not occur in the
classroom due to time.” Jean discovered that the
students’ writing gave them a topic with which they
could think deeply. Their ideas and thinking were
pondered even after the class was over, and could
provide an opening discussion for the next time they
met.
Writing also provided support for student thinking
indirectly by supplying their teachers with feedback
they would not normally have from their students. In a
sense, student thinking was made more clear to their
teachers, which in turn allowed the teacher-researchers
to make adjustments in their teaching and acknowledge
misconceptions. To this effect, Jean explained, “I
sometimes realize that I may have not taught a concept
clearly when many of the students have come to the
same misdirected conclusion.” Amber concurred
saying that she felt that she knew her students’
mathematical ability much better than in past years. “I
know more about my kids than I ever have any other
year,” demonstrating the ability of writing activities to
transform learning experiences for students. Joanne
remarked that she was able to find out what her
students really knew, and cited an example of
discovering that a poorly achieving student - knows
more mathematics than his/her grades indicate.
Future directions
The benefits of using writing in their classes show
that the teacher-researchers learned a great deal from
their students by reading and participating in
discussion. They learned from themselves by using
different teaching techniques and deciding on better
ways to foster student learning. They also learned a
great deal from each other by participating in our
conversations and meetings about the data analysis and
debriefing of their action research projects. Another,
among the many things the teacher-researchers learned
not only about their students but also about themselves,
is what they want and/or need to do in the next school
year when they use writing. It is important to realize
that when trying out new teaching techniques,
everything might not result ideally the first time. Good
teaching techniques take years to perfect, and these
teacher-researchers have a sense of how they would
proceed in the future.
Writing in Middle School
Amber expressed that she wants to collaborate
with her language arts teacher to use writing more than
what she did this year. She also wants students to keep
a journal book in the room instead of using loose paper
as they did this year. Timely responses were a concern
for Amber and she intends to make a better effort at
responding in an appropriate amount of time. Joanne
wants to try writing activities with her learning
disability (LD) students. Seeing the benefit to her past
year’s students, using writing with her LD students
might show similar promise. She wants to have
students grade their own and each other’s writing using
the ODEAR rubric that she devised. Hearing about
improved student discourse from the other teacherresearchers, Iris plans to incorporate the use of
discourse with writing into her classroom. Next time
she wants to incorporate more discourse and re-writing
(post-writing) after they have discussed their solutions.
Jean wants to make changes to the rubric she used to
grade expository essays. This past year, she used the
district’s rubric and ended up not liking it towards the
end of the project.
Concluding Thoughts
After completing these projects with the teacherresearchers, I think they learned wonderful lessons
from their own teaching. They enjoyed the process
enough to want to continue the use of writing in their
classes, and continue to make improvements in their
teaching—one of the main goals of conducting action
research in the first place. This research surveyed the
experiences and issues that arose from first-time
teacher-researchers incorporating writing strategies
into their mathematics classrooms. Teacher-researchers
utilized several types of writing strategies including
expository writing, warm-up writing, problem solving,
journal writing, and reflective writings. They
discovered several benefits of using writing in their
practice. They found that writing was not only
advantageous to the students, but also to the teachers
themselves. These results are consistent with research
that addresses not only student benefits, but also those
for teachers (Borasi & Rose, 1989; Miller, 1992).
Students benefited from writing by increasing their
thinking and reflection, and having an opportunity to
share their writing that, in turn, led to dialogue and
discussion with each other as well as the teachers. The
teacher-researchers developed a better understanding
of their students’ knowledge and conceptions because
of the additional opportunities to discuss students’
thinking and provide feedback on their writing
samples. The ultimate benefit from writing is that it
Drew K. Ishii
enables more dialogue between all members of the
classroom, something that is often missing from the
traditional mathematics classrooms.
This project served as a great learning tool for
everyone involved. The teacher-researchers learned
about their teaching, as well as potential future
directions for their research. Action research provided
another learning arena for teachers because they
stepped back from their practice and evaluated it
systematically. Furthermore, writing can serve as a
learning tool that has the potential to be extremely
beneficial as well as enjoyable when discussions are an
integral part of the process.
My involvement in this project gave me the
opportunity to evaluate the use of writing in
mathematics in action. Engaging in this project
allowed me to see the applications of research to
classroom situations and vice versa. Working with the
four teacher-researchers highlighted the reality of
conducting action-research in a middle school setting
and all of the challenges and enjoyment that can result
from it. This experience illustrated for me, firsthand,
the issues and concerns teacher-researchers encounter
when trying new teaching and instruction techniques
for the first time.
REFERENCES
Birken, M. (1989). Using writing to assist learning in college
mathematics classes. In P. Connolly, & T. Vilardi (Eds.),
Writing to learn mathematics and science (pp. 33–47). New
York: Teachers College Press.
Borasi, R., & Rose, B. (1989). Journal writing and mathematics
instruction. Educational Studies in Mathematics, 20(4).
327–365.
Cai, J., Jakabcsin, M. S., & Lane, S. (1996). Assessing students’
mathematical communication. School Science and
Mathematics, 96(5), 238–246.
Charmaz, K. (2000). Grounded theory: Objectivist and
constructivist methods. In N. Denzin & Y. S. Lincoln (Eds.),
Handbook of qualitative research (2nd ed., pp. 509-535).
Thousand Oaks: Sage.
Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical
thinking, and classroom practice. In N. Minick, E. Forman, &
A. Stone (Eds.), Education and mind: Institutional, social, and
developmental processes (pp. 91–119). Oxford: University
Press.
Drake, B. M., & Amspaugh, L. B. (1994). What writing reveals in
mathematics. Focus on Learning Problems in Mathematics,
16(3), 43–50.
Elliot, J. (1991). Action research for educational change.
Philadelphia, PA: Open University Press.
Janesick, V. J. (2000). The choreography of qualitative research
design: Minuets, improvisations, and crystallization. In N. K.
Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative
research (2nd Ed., pp. 379–400). Thousand Oaks, CA: Sage.
45
Johanning, D. J. (2000). An analysis of writing and postwriting
group collaboration in middle school pre-algebra. School
Science and Mathematics. 100(3), 151–157.
Rose, B. (1989). Writing and mathematics: Theory and practices. In
P. Connolly, & T. Vilardi (Eds.), Writing to learn mathematics
and science (pp. 15–32). New York: Teachers College Press.
Jurdak, M., & Zein, R. A. (1998). The effect of journal writing on
achievement in and attitudes toward mathematics. School
Science and Mathematics, 98(8), 413–419.
Sagor, R. (2000). Guiding school improvement with action
research. Alexandria, VA: Association for Supervision and
Curriculum Development.
Keith, S. (1989). Exploring mathematics in writing. In P. Connolly,
& T. Vilardi (Eds.), Writing to learn mathematics and science
(pp. 134–146). New York: Teachers College Press.
Marshall, C., & Rossman, G. (1995). Designing qualitative
research. Thousand Oaks, CA : Sage.
Smith, S., Layng, J., & Jones, M. (1996). The impact of qualitative
observational methodology on the authentic assessment
process. Proceedings of Selected Research and Development
Presentations (pp. 745–842). Indianapolis, IN: Association for
Educational Communications and Technology.
Miller, L. D. (1992). Teacher benefits from using impromptu
writing prompts in algebra classes. Journal for Research in
Mathematics Education, 23(4), 329–340.
Shield, M., & Galbraith, P. (1998). The analysis of student
expository writing in mathematics. Educational Studies in
Mathematics, 36(1). 29–52.
National Council of Teachers of Mathematics (2000). Principles
and Standards for School Mathematics, Reston, VA: Author.
Vygotsky, L. (1978). Mind in society: The development of higher
psychological processes. Cambridge: Harvard University
Press.
Pearce, D. L., & Davison, D. M. (1988). Teacher use of writing in
the junior high mathematics classroom. School Science and
Mathematics, 88(1), 6–15.
Powell, A. B. (1997). Capturing, examining, and responding to
mathematical thinking through writing. The Clearing House:
A Journal of Educational Research, Controversy, and
Practices, 71(1), 21–25.
Powell, A. B., & Lopez, J. A. (1989). Writing as a vehicle to
learning mathematics: A case study. In P. Connolly, & T.
Vilardi (Eds.), Writing to learn mathematics and science (pp.
157–177). New York: Teachers College Press.
Raymond, A. M. and Hamersley, B. (1995, April). Collaborative
action research in a seventh-grade mathematics classroom.
Paper presented at the Annual Meeting of the American
Educational Research Association, San Francisco, CA.
46
Yackel, E., & Cobb, P. (1996). Sociomathematical norms,
argumentation, and autonomy in mathematics. Journal for
Research in Mathematics Education, 27(4), 458–477.
Zupancic, J. & Ishii, D. K. (2002) Writing as a tool for learning in
mathematics: A case study in eighth-grade algebra. Ohio
Journal of School Mathematics, 46(Autumn), 35–40.
1
The Teacher-Researcher Program was supported by
grants under the federally funded Dwight D. Eisenhower
Professional Development Program, administered by the
Ohio Board of Regents, and The Ohio State
University/Urban Schools Initiative funded through the
Jennings Foundation.
2
All names are pseudonyms.
Writing in Middle School
The Mathematics Educator
2003, Vol. 13, No. 2, 47–57
Teachers’ Mathematical Beliefs: A Review
Boris Handal
This paper examines the nature and role of teachers’ mathematical beliefs in instruction. It is argued that
teachers’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate
from previous traditional learning experiences mainly during schooling. Once acquired, teachers’ beliefs are
eventually reproduced in classroom instruction. It is also argued that, due to their conservative nature,
educational environments foster and reinforce the development of traditional instructional beliefs. Although
there is evidence that teachers’ beliefs influence their instructional behaviour, the nature of the relationship is
complex and mediated by external factors.
For the purpose of this paper, t e a c h e r s ’
mathematical beliefs refers to those belief systems held
by teachers on the teaching and learning of
mathematics. Educationalists have attempted to
systematize a framework for teachers’ mathematical
belief systems into smaller sub–systems. Most authors
agree with a system mainly consisting of beliefs about
(a) what mathematics is, (b) how mathematics teaching
and learning actually occurs, and (c) how mathematics
teaching and learning should occur ideally (Ernest,
1989a, 1989b; Thompson, 1991). Certainly, the range
of teachers’ mathematical beliefs is vast since such a
list would include all teachers’ thoughts on personal
efficacy, computers, calculators, assessment, group
work, perceptions of school culture, particular
instructional strategies, textbooks, students’
characteristics, and attributional theory, among others.
In this paper, the concept of progressive instruction
is associated with a socio-constructivist view of
teaching and learning mathematics. Socioconstructivism, which for the sake of brevity will be
called just constructivism, gives recognition and value
to new instructional strategies in which students are
able to learn mathematics by personally and socially
constructing mathematical knowledge. Constructivist
strategies advocate instruction that emphasises
problem-solving and generative learning, as well as
reflective processes and exploratory learning. These
strategies also recommend group learning, plenty of
discussion, informal and lateral thinking, and situated
learning (Handal, 2002; Murphy, 1997). In turn,
Boris Handal has taught and lectured in schools and universities in
Australia, Latin America and Asia. He has written extensively on
academic issues in academic journals in the United States, United
Kingdom, Australia, Latin America, and South East Asia. Boris
obtained his Bachelors of Education from the Higher Pedagogical
Institute of Peru, a Masters of Education from Edith Cowan
University and his Doctorate of Education from the University of
Sydney. In addition he has a postgraduate degree in educational
technology from Melbourne University.
Boris Handal
traditional instruction is associated with a behaviourist
perspective on education. Behaviourist practices are
said to emphasise transmission of knowledge and stress
the pedagogical value of formulas, procedures and
drill, and products rather than processes. Behaviourism
also puts great value on isolated and independent
learning, as well as conformity to established one-way
methods and a predilection for pure and abstract
mathematics (McGinnis, Shama, Graeber, &
Watanabe, 1997; Wood, Cobb, & Yackel, 1991). Leder
(1994) stated that in the behaviourist movement “the
mind was regarded as a muscle that needed to be
exercised for it to grow stronger” (p. 35).
The study of teachers’ instructional beliefs and
their influence on instructional practice gained
momentum in the last decade. Some research on
teachers’ thinking reveals that teachers hold wellarticulated educational beliefs that in turn shape
instructional practice (Buzeika, 1996; Frykholm, 1995;
McClain, 2002; Stipek, Givvin, Salmon, &
MacGyvers, 2001; Thompson, 1992). Examples of
research, as reviewed in this paper, have also shown
that each teacher holds a particular belief system
comprising a wide range of beliefs about learners,
teachers, teaching, learning, schooling, resources,
knowledge, and curriculum (Gudmundsdottir &
Shulman, 1987; Lovat & Smith, 1995). These beliefs
act as a filter through which teachers make their
decisions rather than just relying on their pedagogical
knowledge or curriculum guidelines (Clark & Peterson,
1986). In fact, these beliefs appear to be cogent enough
to either facilitate or slow down educational reform,
whichever is the case (Handal & Herrington, 1993, in
press). The literature also shows that there are internal
and external factors mediating beliefs and practice
(Pajares, 1992). This dissonance bears serious
implications for the implementation of curricular
innovations since teachers’ beliefs may not match the
belief system underpinning educational reform. Even if
47
teachers’ beliefs match curricular reform, very often
the traditional nature of educational systems make it
difficult for teachers to enact their espoused
progressive beliefs. In contrast to linear and static
approaches to curriculum implementation, modern
perspectives look at how teachers make sense of
educational innovations in order to re-appraise an
ongoing and always flexible process of implementation
(Handal & Herrington, 2003).
Theoretical Conceptualisations
Theoretical conceptualisations of teachers’
mathematical beliefs show that the range of these
beliefs can be expressed in multiple dimensions (Kuhs
& Ball, 1986; Renne, 1992; Ernest, 1991). Ernest
(1991), for example, outlined a developmental
sequence of five different mathematics-related belief
systems that are hypothesized to be found amongst
teachers: authoritarian, utilitarian, mathematics
centred, progressive, and socially aware. Ernest’s
contribution showed that it is possible to relate these
attitudinal representations to conceptions on the theory
of mathematics, learning mathematics, teaching
mathematics, and assessment in mathematics, as well
as identifying beliefs on the aims of mathematics
education. According to Ernest, the most important of
these categories is the teacher’s philosophy of
mathematics, which might vary from absolutist to
social-constructivist values. Teachers’ theories of
learning and teaching are said to relate to approaches
used in class and are fundamental because they define
the teacher’s perception of the learner’s role as active
or passive, dependent or autonomous, or as receiver or
creator of knowledge. Ernest also proposed three main
philosophical conceptions of mathematics among
teachers. In the instrumentalist view, mathematics is
seen as a collection of rules and skills that are to be
used for the attainment of a particular goal. Teachers
adhering to the Platonist view will maintain that
“mathematics is a static but unified body of certain
rules” (p. 250) that are to be discovered and are not
amenable of personal creation. The problem solving
view presents mathematics as a continuous process of
inquiry that always remains open to revision.
In turn, Kuhs and Ball (1986) characterised three
different and dominant conceptions of the ideal
teaching and learning of mathematics. The first is the
learner-focused view that stresses the learner’s
construction of mathematical knowledge through social
interaction. The second is the content-focused view
with an emphasis on conceptual understanding. The
third is the content-focused approach with an emphasis
48
on performance which values performance as the key
goal whose attainment depends on the mastery of rules
and procedures.
Furthermore, Renne (1992) proposed a Purpose of
Schooling/Knowledge matrix to conceptualise four
different teachers’ conceptions of teaching and
learning mathematics. Two groups of teachers are
identified in the purpose of schooling category,
namely, school-knowledge oriented and childdevelopment oriented. Teachers within the schoolknowledge group believe that teaching is an act of
passing information on to others while learning
involves the process of reproducing that information.
At the same time, school-knowledge oriented teachers
place great emphasis on the syllabus and curricular
guidelines to guide their instruction. In turn, childdevelopment oriented teachers are more likely to
consider children’s needs and characteristics as the
primary factors in instructional decision making. The
second category in the matrix relates teachers’ beliefs
to the way teachers perceive knowledge itself. Schoolknowledge oriented teachers design activities that
emphasise acquisition of knowledge in terms of “what”
is going to be learned. As such, this type of knowledge
is concerned more with rules, procedures, and drill.
This type of knowledge is very fragmentary because it
does not help the learner relate isolated pieces of
knowledge to the whole framework. In contrast, childdevelopment oriented teachers are more concerned
with learning of mathematical concepts within an
interrelated knowledge structure that is holistic and
meaningful.
These three different conceptualisations of
teachers’ beliefs about the nature and pedagogy of
mathematics (Ernest, 1991; Kuhs & Ball, 1986; Renne,
1992) constitute an analytical framework to discuss
teachers’ mathematical belief systems. In general, it
can be argued that teachers’ belief systems are
complex networks of smaller sub-systems operating
contextually. The following section attempts to explain
the origin of these belief systems within the context of
present and past educational environments that appear
very traditional and resistant to change.
The Cycle of Teachers’ Mathematical Beliefs
How do teachers’ mathematical beliefs originate?
In part, teachers acquire these beliefs symbiotically
from their former mathematics school teachers after
sitting and observing classroom lessons for literally
thousands of hours throughout their past schooling
(Carroll, 1995; Thompson, 1984). This process
parallels in many respects the apprenticeship style of
Teachers’ Mathematical Beliefs
learning that takes place while learning a trade.
Traditionally, tradesmen learn by observing a master
doing a particular job (Buchmann, 1987; Lortie, 1975).
In the schooling process, students learn not only
content-based knowledge but also instructional
strategies as well as other dispositions. By the time the
aspirant is admitted to a teacher education program,
these beliefs about how to teach and learn are deeply
embedded in the individual, and very often are
reinforced by the traditional nature of some teacher
education institutions which may not have positive
effects on preservice teachers’ mathematical beliefs
(Brown & Rose, 1995; Day, 1996; Foss & Kleinsasser,
1996; Kagan, 1992; McGinnis & Parker, 2001).
There is evidence that, in some cases, teacher
education programs are so busy concentrating on
imparting pedagogical knowledge that little
consideration is given to modifying these beliefs
(Tillema, 1995). Consequently, teacher education
programs might have little effect in producing teachers
with beliefs consistent with curriculum innovation and
research (Kennedy, 1991). For example, Marland
(1994) found that reasons given by inservice teachers
regarding their classroom strategies were not related to
what was actually taught in their college training.
There is also some evidence confirming that teachers’
decision making does not rely solely on their
pedagogical knowledge but also on what they believe
the subject-matter is and how it should be taught
(Brown & Baird, 1993; Laurenson, 1995; Prawat,
1990). These beliefs are also difficult to change
(Borko, Flory, & Cumbo, 1993) and very often conflict
with educational innovations, threatening educational
change (Brown & Rose, 1995; Fullan, 1993). As
discussed in the next sections, there are also a number
of external factors influencing teachers’ beliefs.
The Constraining Nature of Educational
Environments
The context of school instruction obliges practising
elementary and secondary teachers to teach traditional
mathematics even when they may hold alternative
views about mathematics and about mathematics
teaching and learning. Parents and professional
colleagues, for example, expect teachers to teach in a
traditional way. Teachers are also expected to focus on
external examinations, to adhere to a textbook, and to
keep a low level of noise and movement in their
classrooms. In such environments, even teachers with
progressive educational beliefs are forced to
compromise and conform to traditional instructional
styles (Handal, 2002; Perry, Howard, & Tracey, 1999;
Boris Handal
Sosniak, Ethington, & Varelas, 1991). Other
accountable factors are ethnic background, social class
origins, experience living in other cultures, gender
issues, and prior styles of teaching experience (Butt &
Raymond, 1989; Raymond, Butt & Towsend, 1991).
Thompson (1984) argued that teachers, in the exercise
of their practice, and because of the large number and
diversity of interactions, tend to develop quick
responses to types of episodes, which in time become
patterns in their instructional repertoire.
McAninch (1993) reviewed a body of literature
showing that teachers are very practical in their
approach to pedagogical tasks. Jackson’s (1968)
interviews revealed that teachers tend to be “confident,
subjective, and individualistic in their professional
views” (cited by McAninch, 1993, p. 7). In addition,
Doyle and Ponder (1977) and Lortie (1975), both cited
by McAninch (1993), described “teachers as pragmatic
in their decision making…and intuitive in their
approach to problem solving” (p. 7). Moreover,
teaching is seen as a highly practical and utilitarian
profession where teachers quickly label innovations as
practical or impractical, depending on whether the
teacher considers that the proposal will work for him or
her. Success of innovations was also found to be
related to a teacher’s personality and teachers were
found to emphasise the peculiarities of their classroom
over the generalizations of innovations.
Nespor (1987) adds that, given the unpredictability
and uniqueness of classroom events, teachers have to
resort to their own beliefs, particularly in pedagogical
situations when formal knowledge is not available, is
disconnected, or cannot be retrieved. In Nespor’s
words, “When people encounter entangled domains or
ill-structured problems, many standard cognitive
processing strategies such as schema-abstraction or
analytical reduction are no longer viable” (p. 325).
This type of situation is characteristic of classroom
teaching. In general, teaching is a decision-making
based activity in which teachers have to make an
interactive decision every two minutes (Brown and
Rose, 1995; Clark & Peterson, 1986; Lovat and Smith,
1995).
In brief, the teaching job places great external
demands on decisions that teachers have to make
rapidly, in isolation, and in widely varied
circumstances. These demands put teachers in the
position of resorting to practicability and intuition as
indispensable resources for survival in the profession.
These demands in turn favour the development of
beliefs about what works and what does not in a
classroom. At the same time, it seems that teachers
49
generate their own beliefs about how to teach in their
school years and these beliefs are perpetuated in their
teaching practice. Thus, educational beliefs are passed
on to the students.
Teachers’ Instructional Practice
If, as the adage says, “teachers teach the way they
have been taught” (Frank, 1990, p.12), we need to ask
ourselves: what type of mathematics teaching have our
and past generations been exposed to? Studies
conducted in American mathematics classrooms by
Cuban (1984), Mewborn (2001), Sirotnik (1983), and
Romberg and Carpenter (1986), Gregg (1995) indicate
that most mathematics lessons follow a pattern of
whole-class lecturing and “show and tell” style of
teaching. Work in small groups is not common and
students do not participate actively. Teacher
questioning emphasizes right or wrong answers and
students are often allocated to passive seatwork. Too
much emphasis is given to rote learning, procedures,
and facts. It was also found that excess teacher talk
dominates in classroom communication and desks
usually are arranged to face the teacher’s desk. In sum,
this pattern of lessons in American classrooms can be
characterised as traditional oriented. Furthermore, the
Third International Mathematics and Science Study
(TIMSS) identified a similar pattern in Australian
classrooms, “one of what might be called ‘traditional
approaches’
dominating
classroom
instruction…particularly in relation to lesson
sequencing and types of activities undertaken” (Lokan,
Ford, & Greenwood, 1997, p. 231).
Based on the above arguments it is possible to
suggest that the educational system may act as a
vehicle to reproduce traditional mathematical beliefs.
Teachers seem to pass on these beliefs in subtle ways
in school classrooms. By the time candidates enroll in
a teacher education program, these ideas are so
solidified and entrenched in their personal philosophy
that they will be passed on to their students once the
candidates commence their teaching careers, thus
carrying on a cycle. The following section attempts to
explore the character, intensity, and diversity of these
mathematical beliefs as conveyed by schoolteachers.
Teachers’ Beliefs about Mathematics
and the Learning and Teaching of Mathematics
Teachers’ mathematical beliefs are personal and
are therefore mental constructs peculiar to each
individual (Brown & Rose, 1995). A number of studies
have been conducted to obtain “typical” teachers’
mathematical beliefs. Teachers’ mathematical beliefs
50
have been analysed statistically and in many instances
judgements were passed on a right-and-wrong criteria
by researchers. Although patterns are identifiable
within representative samples, these studies have at the
same time revealed a broad diversity in the direction
and intensity of these beliefs (Carpenter, Fennema,
Loef, & Peterson, 1989; Moreira, 1991; Schmidt &
Kennedy, 1990). This fact led some researchers to
think that these differences could be alternatively
interpreted either as stages of a developmental process,
individual cognitive differences, or simply due to
differences in socio-economic status, educational
systems, or cultural environments (Moreira, 1991;
Stonewater & Oprea, 1988; Thompson, 1991;
Whitman & Morris, 1990).
The studies described below show that a large
population of teachers still believe that teaching and
learning mathematics is more effective in the
traditional model, thus suggesting a historical
correspondence between teachers’ mathematical
beliefs and the teaching practices described in the
previous section. What follows is a summary of the
main studies conducted to explore mathematical beliefs
in preservice and inservice teachers.
Mathematical Beliefs of Preservice Teachers
A growing body of literature suggests that
preservice teachers, that is, student teachers attending
teacher education institutions, hold sets of beliefs more
traditional than progressive with respect to the teaching
of mathematics. Research findings reveal that
preservice teachers bring into their education program
mental structures overvaluing the role of memorization
of rules and procedures in the learning and teaching of
school mathematics. For example, Benbow (1993)
found that preservice elementary teachers thought of
mathematics as a discipline based on rules and
procedures to be memorized, and that there is usually
one best way to arrive at an answer. Most of the
teachers also saw mathematics as dichotomized into
“completely right or completely wrong” (p. 10). A
similar conservative trend in teachers’ beliefs was
reported by Nisbert and Warren (2000), who surveyed
398 primary school teachers with regard to their views
on mathematics as a subject, and on teaching and
assessing mathematics. Civil (1990) interviewed four
prospective elementary teachers and found that they
believed that mathematics required neatness and speed,
and that there is usually a best way to solve a problem.
Frank (1990) surveyed the mathematical beliefs of
preservice teachers and found a high level agreement
in items such as: (a) “Some people have a
Teachers’ Mathematical Beliefs
mathematical mind and some don’t”, (b) “Mathematics
requires logic not intuition”, and (c) “You must always
know how you got the answer” (p. 11). Moreover, Foss
and Kleinsasser (1996, p. 438) surveyed, observed, and
interviewed preservice elementary teachers and found
that the participants placed great emphasis on practice
and memorization. Teachers also were of the opinion
that ability in mathematics was innate. Southwell and
Khamis (1992) surveyed 71 preservice teachers and
found that most participants perceive that mathematics
learned in school should be based on memorization of
facts and rules. Lappan and Even (1989) and Wood
and Floden (1990) report similar findings.
Mathematical Beliefs of Inservice Teachers
Results from research on inservice teachers show a
broader spectrum of responses than with preservice
teachers. This is partially the result of more flexible
research designs allowing the collection of a broader
set of responses in the samples. A number of these
studies also show a more varied scope of research
questions rather than just simply characterizing
teachers’ mathematical beliefs in a dichotomy.
The Third International Mathematics and Science
Study (TIMSS) (Beaton, et al., 1996), conducted in
selected countries around the world, revealed that most
teachers believe mathematics is essentially a vehicle to
model the real world, that ability in mathematics is
innate, and that more than one representation should be
used in explaining a mathematical concept. With
respect to the emphasis on drill and repetitive practice,
teachers around the world did not show a consistent
response. Anderson (1997) surveyed and interviewed
25 primary teachers and found that the majority of the
participants believe in the value of whole-class
discussion, teacher’s modelling, and the use of
manipulatives in the classroom. However, it was found
that teachers were of the opinion that calculators
should not be an important component in teaching
mathematics in the primary school. Grossman and
Stodolsky (1995) surveyed and interviewed 399
teachers of mathematics, sciences, social studies, and
foreign languages. The authors found that mathematics
teachers, compared with those of the other subjects,
consider their subject highly sequential, static, and
have stronger consultation within their faculty for
coordinating course content and common exams. The
findings also showed that mathematics teachers prefer
students to be grouped by prior academic achievement
in order to get better benefits from instruction.
Schubert (1981), quoted by Brown and Rose (1995), in
questioning 123 educators, found that most teachers
Boris Handal
believe that pupils learn “in a passive manner by
reacting to forces external to them, rather than in an
active manner as producer of their own knowledge” (p.
21), a conclusion also supported by Desforges and
Cockburn (1987).
Finally, Howard, Perry and Lindsay (1997)
surveyed 249 secondary mathematics teachers in
Sydney, Australia, and found two different patterns of
beliefs. The first is identified with the “transmission”
profile, that is, a traditional categorization of teaching
and learning as the transmission and verification of
information in which memorization of rules and
procedures is fundamental. This group was larger in
number than the constructivist profile, where teachers
believe that students are capable enough of
constructing their own mathematical knowledge in an
atmosphere of negotiation and relevance. The evidence
that a large number of inservice teachers hold a diverse
collection of mathematical beliefs associated with
traditional instruction is also documented in studies
conducted by Handal, Bobis, and Grimison (2001),
Kifer and Robitaille (1992), Middleton (1992), Perry,
Howard, and Conroy (1996), and Perry et al. (1999).
Teachers’ Mathematical Beliefs And Instructional
Practice
Studies on the relationship between pedagogical
beliefs and instructional behaviour have reported
different degrees of consistency (Frykholm, 1995;
Thompson, 1992). While the nature of this relationship
seems to be dialectical in nature (Wood et al., 1991) it
is not clear whether beliefs influence practice or
practice influences beliefs (McGalliard, 1983). It is in
fact a complex relationship (Thompson, 1992) where
many mediating factors determine the direction and
magnitude of the relationship. This section reports a
number of studies that have explored the relationship
between teachers’ mathematical beliefs and
instructional practice.
Benbow (1995) conducted an intervention program
to deliberately modify the beliefs and instructional
practices of 25 preservice mathematics elementary
teachers. Findings showed that there was no change in
teachers’ mathematical beliefs at the end of the
program. However, the researcher stated that
instructional behaviour in terms of selection of
curriculum content and learning activities, teacher’s
role, and teachers’ beliefs on self-efficacy were
modified as a result of the program. Lack of
pedagogical knowledge and subject-based content were
found in some cases to be an obstacle to transfer
progressive oriented beliefs into practice.
51
Brown and Rose (1995) conducted an interview
study with 10 elementary mathematics teachers in
order to determine their theoretical orientations.
Teachers’ responses showed a varied range of theories
of teaching and learning mathematics. Teachers also
said that these orientations influenced their
instructional behaviour. The analysis of data revealed
that teachers do not implement fully their ideal
conceptions of mathematics education because of
perceived pressure from parents and school
administrators to implement traditional teaching. Other
identified mediating factors were the need for more
preparation time to satisfy instructional and curricular
demands, and the challenges of mixed ability classes.
Erickson (1993), in a study with two experienced
middle school mathematics teachers, concluded that
teachers’ ideal beliefs have a strong influence on their
instructional practice. However, obstacles to fully
implement their ideals included lack of preparation
time and lack of collaboration among peers; size of
room; availability of technology, materials, and
money; non-supportive administration and parents;
need for lengthened class periods; and personal
opportunity for growth.
Foss and Kleinsasser (1996) studied the behaviour
and instructional practice of 20 elementary
mathematics preservice teachers. At the end of a onesemester methods course participants had not changed
their beliefs about teaching and learning mathematics,
which were found to be traditional-oriented and
heavily influenced by previous traditional learning
experiences in diverse educational settings.
Participants’ instructional behaviour replicated or
modelled activities learned in the methods course, but
not to the extent that reflected an adoption of
innovative approaches to teaching and learning
mathematics in an articulated and consistent way. In
addition, Cooney (1985) studied a beginning
mathematics teacher who was committed in belief and
in practice to problem solving instruction. The author
described the conflict between the teacher’s struggle to
teach problem solving and students who preferred a
more content-based instruction, a friction that
sometimes led to classroom management problems.
Perry et al. (1999) studied the beliefs of Australian
head secondary mathematics teachers and classroom
secondary mathematics teachers as independent
samples. Head teachers said that curriculum demands
were an obstacle to implementing innovative teaching.
In the respondents’ words:
We try to make the work relevant but we are
constrained by the syllabus. Sometimes, I feel,
52
pressure of the syllabus tends to force us to cut
corners with the kids…If I sound cheesed off, it’s
just that I may be a disillusioned mathematics
teacher. (p. 14)
Raymond (1993) investigated beliefs and practices
of six beginning elementary mathematics teachers and
found diverse degrees of consistency. Two teachers
displayed a high degree of correspondence between
belief and practice, two teachers showed a moderate
level, while the other two showed a low level. Reasons
for the inconsistencies were found to be lack of
resources, time limitations, discipline, and pressure to
conform to standardized testing. The author concluded
that there is a dialectical relationship between beliefs
and practice. According to the researcher, teachers’
mathematical beliefs influenced their practice more
than their instructional practices influence their
mathematical beliefs. The researcher also found that
previous school experiences, teachers’ current practice,
and, importantly, teacher education courses also
influence teachers’ mathematical beliefs. Teachers also
identified their own mathematical beliefs, students’
abilities, the particular topic to be taught, the school
culture, as well as the mathematics curriculum as
factors that influenced their instructional practice.
Taylor (1990) attempted to assist a high school
teacher to modify his beliefs through a process of
conceptual change. However, there were conflicting
beliefs, such as the teacher’s belief that he had to teach
for constant assessment and for covering the syllabus
given that he did not want to jeopardize students’
learning with alternative strategies. Consequently,
change in instructional behaviour was restricted.
Van Zoest, Jones, and Thornton (1994)
interviewed and observed six elementary preservice
mathematics teachers participating as students in an
intervention program to enhance their teachers’
mathematical beliefs. The authors found that
participants acquired beliefs consistent with socioconstructivist views of learning and teaching
mathematics, although they were not able to translate
these views into practice in the early stages of
instructional episodes. The reason for this
inconsistency was found in teachers’ lack of
pedagogical skill to guide students through the whole
problem solving process, time needed to go through a
task, teachers’ and students’ tension on how to go
about a problem solving situation, and teachers’
concerns about students’ ability to solve the problem.
Other studies not showing consistency include Grant
(1984) studying secondary mathematics teachers,
Kessler (1985) investigating four senior high school
Teachers’ Mathematical Beliefs
mathematics teachers, Brosnan, Edwards, and Erickson
(1996) researching four middle school mathematics
preservice teachers, and Desforges and Cockburn
(1987) studying seven experienced mathematics
primary school teachers.
Thompson (1985) studied two relatively
experienced mathematics teachers in their teaching of
problem solving and found a high level of consistency
between their beliefs and instructional practice. Phillip,
Flores, Sowder, and Schapelle (1994) reached the same
conclusion while studying four “extraordinary”
mathematics teachers. Other studies reporting a strong
relationship between teachers’ beliefs and practices
have been conducted by McGalliard (1983)
investigating senior high school mathematics teachers,
and Steinberg, Haymore, and Marks (1985) studying
novice teachers. Shirk (1973) working with preservice
elementary teachers and Stonewater and Oprea (1988)
working with inservice teachers also reported similar
consistencies.
In general, inconsistencies between teachers’
beliefs and practices are due to constraining forces out
of a teachers’ control, such as parental and
administrative pressure to follow traditional oriented
methods of instruction. Other factors include the
traditional oriented mathematical learning style of the
students as well as a lack of time and materials. These
factors seem to act as major barriers for some teachers
in implementing their progressive beliefs, constraints
that current approaches in mathematics education do
not take into account (Nolder, 1990).
Incongruities Between Teachers’ Beliefs And
Practice
The incongruity between beliefs and practice can
also be explained through the agitation and
unpredictability of classroom life and the external
pressures put on teachers. Thompson (1985) affirmed
that these incongruities might be due to the frequency
of unexpected occurrences which teachers face in the
classroom. The high frequency of these incidents does
not permit the teacher to reflect on alternative
responses; rather, teachers have time only to react.
Jackson (1968) suggested that elementary teachers
engage in more than one thousand interactions with
students in a single day.
Another source of incongruity lies in the personal
resolution of conflicting beliefs. Orton (1991)
suggested that teachers’ commitment to progressive
beliefs is not always a guarantee that these beliefs are
going to be translated into practice because sometimes
teachers have to compromise their progressive beliefs
Boris Handal
for the crude reality of traditional oriented educational
environments. For example, a teacher might be
motivated to provide rote-learning activities in class
when that teacher knows that his or her students will be
tested on basic skills in a district proficiency exam. In
this case, the teacher might perceive that drill and
repetitive practice is the best strategy to attain a
temporary goal. Consequent to this strategy, the
teacher suspends his or her own progressive beliefs for
others that are more central at that particular time.
Teacher’s resistance to adopting new approaches in
the teaching of mathematics may be part of a defense
mechanism that teachers adopt to avoid changes in
their own mental structures (Clarke, 1997) because
“changing beliefs causes feelings of discomfort,
disbelief, distrust, and frustration” (Anderson &
Piazza, 1996, p. 53). Orton (1991) stated that it is not
easy to change a long-cherished mathematical belief
since this belief proved before to be rewarding and
useful to the teacher in the performance of his or her
professional duties. Furthermore, changing a particular
belief implies a re-structuring of the whole network of
one’s belief system, a feeling that might cause anxiety
and emotional pain (Rokeach, 1968). Concerning
teachers’ resistance to change, it has been observed
that teachers holding more relativistic orientations to
teaching mathematics are more likely to consider and
adopt new ideas (Arvold & Albright, 1995).
School cultures also influence teachers’
mathematical beliefs (Anderson, 1997). This is
particularly true when teachers are found holding
beliefs different from the school culture in which they
work. For example, a certain school environment might
effectively foster values associated with progressive
practices and this influence might be stronger than in
other schools. In many instances, teachers are caught in
a conflict of interest between their “technicalpositivist” and their “constructivist” beliefs and
therefore they compromise (Taylor, 1990). Moreover,
teachers know that although administrators and
supervisors promote reform efforts, professional
assessment is in terms of the traditional paradigm and
therefore they tend to conform to the status quo to
minimize disturbance and professional risk in an
ethical-practical way (Anderson & Piazza, 1996; Doyle
& Ponder, 1977).
Research also shows that teachers may not hold
consistent belief systems. Sosniak et al. (1991)
analysed mathematical beliefs and self-perceptions of
practice of US teachers representing 178 typical eighth
grade classes. Based on those responses, the
researchers attempted to profile teachers in either a
53
traditional or progressive orientation to the curriculum.
However, it was found by statistical analysis that
teachers lack a consistent theoretical orientation
towards the curriculum. According to the authors,
within each teacher’s belief system there are beliefs
that appear to be ideologically incompatible with the
others. Andrews and Hatch (1999), working mainly
with secondary mathematics teachers in the United
Kingdom, and Howard et al. (1997) in Australia,
reached similar conclusions.
Finally, Richardson (1996) adds that in some cases
teachers cannot articulate a particular belief because
they are unfamiliar with a specific educational
innovation. According to Richardson (1996):
… it cannot be assumed that all changes in beliefs
translate into changes in practices, certainly not
practices that may be considered worthwhile. In
fact, a given teacher’s belief or conception could
support many different practices or no practices at
all if the teacher does not know how to develop or
enact a practice that meshes with a new belief. (p.
114)
Summary
This paper argued that despite many educational
reforms, a large number of teachers still perceive
mathematics in traditional rather than in progressive
terms; that is, as a discipline with a priori rules and
procedures, “out-there,” that has to be mechanically
discovered rather than constructed. As such, students
have to learn mathematics by rote and removed from
human experience. The discussion also shows that the
relationship between teachers’ mathematical beliefs
and their instructional practice is dialectical in nature
and is mediated by many conflicting factors. Teachers’
beliefs do influence their instructional practice;
however, a precise one-to-one causal relationship
cannot be asserted because of the interference of
contingencies that are embedded in the school and
classroom culture. Even teachers holding progressive
beliefs find it difficult to render their ideas into practice
due to mediating factors such as the pressure of
examinations, administrative demands or policies,
students’ and parents’ traditional expectations, as well
as the lack of resources, the nature of textbooks,
students’ behaviour, demands for covering the
syllabus, and supervisory style, among many others. In
addition, the teaching profession appears to mould the
nature of beliefs because teachers have to make
decisions and make meaning of situations quickly, in
solitude, with a diversity of subjects, based on
empirical knowledge, and under the pressure of
external factors. Pedagogical knowledge therefore is
54
not a total predictor of instructional behaviour because
beliefs appear to mediate between theory and practice
as a powerful interface. Teachers’ mathematical beliefs
are seen as self-perpetuating within the atmosphere of
a system that promotes progressive teaching but in fact
helps in maintaining traditional beliefs and practices. It
was also argued that by the time an individual enters a
teacher education program, these traditional
conceptions are so solidified and entrenched in their
personal philosophy that change to alternative beliefs is
difficult although not impossible.
REFERENCES
Anderson, J. (1997). Teachers’ reported use of problem solving
teaching strategies in primary mathematics classrooms. In F.
Biddulph & K. Carr (Eds.), People in mathematics education.
Proceedings of the 20th Annual Conference of the
Mathematics Education Research Group of Australasia (pp.
50–57). Rotorua, NZ: MERGA.
Anderson, D. S., & Piazza, J. A. (1996). Teaching and learning
mathematics in constructivist preservice classrooms. Action in
Teacher Education, 18(2), 51–62.
Andrews, P., & Hatch, G. (1999). A new look at secondary
teachers’ conceptions of mathematics and its teaching. British
Educational Research Journal, 25(2), 203–223.
Arvold, B., & Albright, M. (1995). Tensions and struggles:
Prospective secondary mathematics teachers confronting the
unfamiliar. Proceedings of the Annual Conference of the
North American Chapter of the International Group for the
Psychology of Mathematics Education. (ERIC Document
Reproduction Service No. ED 389608.)
Beaton, A. E., Mullis, I. V. S., Martin, M. O., Gonzalez, E. J.,
Kelly, D. L., Smith, T. A. (1996). Mathematics achievement in
the middle school years. Boston: Center for the Study of
Testing, Evaluation, and Educational Policy, Boston College.
Benbow, R. M. (1993). Tracing mathematical beliefs of preservice
teachers through integrated content-methods courses.
Proceedings of the Annual Conference of the American
Educational Research Association. (ERIC Document
Reproduction Service No. ED 388638.)
Benbow, R. M. (1995). Mathematics beliefs in an “early teaching
experience”. Proceedings of the Annual Conference of the
North American Chapter of the International Group for the
Psychology of Mathematics Education. (ERIC Document
Reproduction Service No. ED 391662.)
Borko, H., Flory, M., & Cumbo, K. (1993). Teachers' ideas and
practices about assessment and instruction: A case study of the
effects of alternative assessment in instruction, student
learning and accountability practices. Proceedings of the
Annual Conference of the American Educational Research
Association, Atlanta, Ga. (ERIC Document Reproduction
Service No. ED 378226.)
Brosnan, P. A., Edwards, T., & Erickson, D. (1996). An
exploration of change in teachers’ beliefs and practices during
implementation of mathematics standards. Focus on Learning
Problems in Mathematics, 18(4), 35–53.
Teachers’ Mathematical Beliefs
Brown, A. B. & Baird, J. (1993). Inside the teacher: Knowledge,
beliefs and attitudes. In P. Wilson (Ed.), Research ideas for
the classroom: High school mathematics (pp. 245–259). New
York: Macmillian.
Brown, D. F., & Rose, T. D. (1995). Self-reported classroom
impact of teachers’ theories about learning and obstacles to
implementation. Action in Teacher Education, 17(1), 20–29.
Buchmann, M. (1987). Teacher knowledge: The lights that teachers
live by. Oxford Review of Education, 13, 151–164.
Butt, R. L., & Raymond, D. (1989). Studying the nature and
development of teachers’ knowledge using collaborative
autobiography. International Journal of Educational
Research, 13, 403–419.
Buzeika, A. (1996). Teachers’ beliefs and practice: The chicken or
the egg? In P.C. Clarkson (Ed.), Technology in mathematics
education. Proceedings of the 19th Annual Conference of the
Mathematics Education Research Group of Australasia (pp.
93–100). Melbourne: MERGA.
Carpenter, T. P., Fennema, E., Loef, M., & Peterson, P. L. (1989).
Teachers’ pedagogical content beliefs in mathematics.
Cognition and Instruction, 6(1), 1–40.
Carroll, J. (1995). Primary teachers’ conceptions of mathematics.
In B. Atweh & S. Flavel (Eds), Galtha. Proceedings of the 18th
Annual Conference of the Mathematics Education Research
Group of Australasia (pp. 151–155). Darwin: MERGA.
Civil, M. (1990). A look at four prospective teachers’ views about
mathematics. For the Learning of Mathematics, 10(1), 7–9.
Clark, C. M., & Peterson, P. L. (1986). Teachers’ thought
processes. In M. C. Wittrock (Ed.), Handbook of research on
teaching (pp. 255–296). New York: Macmillan.
Clarke, D. M. (1997). The changing role of the mathematics
teacher. Journal for Research in Mathematics Education,
28(3), 278–308.
Cooney, T. J. (1985). A beginning teachers’ view of problem
solving. Journal for Research in Mathematics Education,
16(5), 324–336.
Cuban, L. (1984). Constancy and change in American classrooms,
1890-1980. New York: Longman.
Day, R. (1996). Case studies of preservice secondary mathematics
teachers’ beliefs: Emerging and evolving themes. Mathematics
Education Research Journal, 8(1), 5–22.
Desforges, C., & Cockburn, A. (1987). Understanding the
mathematics teacher: A study of practice in first schools.
London: Falmer.
Doyle, W., & Ponder, G. (1977). The practicality ethic in teacher
decision-making. Interchange, 8(3), 1–12.
Erickson, D. K. (1993). Middle school mathematics teachers’ view
of mathematics and mathematics education, their planning and
classroom instruction, and student beliefs and achievement.
Proceedings of the Annual Conference of the American
Educational Research Association. Atlanta, GA. (ERIC
Document Reproduction Service No. ED 364412.)
Ernest, P. (1989a). The impact of beliefs on the teaching of
mathematics. In P. Ernest (Ed.), Mathematics teaching: The
state of art (pp. 249–254). New York: Falmer.
Ernest, P. (1989b). The knowledge, beliefs and attitudes of the
mathematics teacher: A model. Journal of Education for
Teaching, 15, 13–34.
Boris Handal
Ernest. P. (1991). Mathematics teacher education and quality.
Assessment and Evaluation in Higher Education, 16(1),
56–65.
Foss, D. H., & Kleinsasser, R. C. (1996) Preservice elementary
teachers’ views of pedagogical and mathematical content
knowledge. Teaching and Teacher Education, 12(4), 429–442.
Frank, M. L. (1990). What myths about mathematics are held and
conveyed by teachers? The Arithmetic Teacher, 37(5), 10–12.
Frykholm, J. A. (1995). The impact of the NCTM Standards on
preservice teachers’ beliefs and practices. (ERIC Document
Reproduction Service No. ED 383669.)
Fullan, M. (1993). Changing forces: Probing the depths of
educational reform. London: Falmer.
Grant, C. E. (1984). A study of the relationship between secondary
mathematics teachers beliefs about the teaching-learning
process and their observed classroom behavior. Dissertation
Abstracts International, 46, 919A.
Gregg, J. (1995). The tensions and contradictions of the school
mathematics tradition. Journal for Research in Mathematics
Education, 26(5), 442–66.
Grossman, P., & Stodolsky, S. S. (1995). Content as context: The
role of school subjects in secondary school teaching.
Educational Researcher, 24(2), 5–11.
Gudmundsdottir, S., & Shulman, L. S. (1987). Pedagogical content
knowledge in social studies. Scandinavian Journal of
Educational Research, 31(2), 59–70.
Handal, B. (2002, September). Teachers’ beliefs and gender,
faculty position, teaching socio-Economic area, teaching
experience and academic qualifications. Proceedings of the
2002 International Biannual Conference of the UWS Self
Research Centre, Sydney.
Handal, B., Bobis, J., & Grimison, L. (2001). Teachers'
Mathematical beliefs and practices in teaching and learning
thematically. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.),
Numeracy and Beyond. Proceedings of the Twenty-Fourth
Annual Conference of the Mathematics Education Research
Group of Australasia (pp. 265–272), Sydney: MERGA.
Handal, B., & Herrington, T. (2003, in press). Mathematics
teachers’ beliefs and curriculum reform. Mathematics
Education Research Journal.
Howard, P., Perry, B., & Lindsay, M. (1997). Secondary
mathematics teachers’ beliefs about the learning and teaching
of mathematics. In F. Biddulph & K. Carr (Eds.), People in
mathematics education. Proceedings of the 20th Annual
Conference of the Mathematics Education Research Group of
Australasia (pp. 231–238). Rotorua, NZ: MERGA.
Jackson, P. W. (1968). Life in classrooms. New York: Holt,
Rinehart & Winston.
Kagan, D. M. (1992). Professional growth among preservice and
beginning teachers. Review of Educational Research, 62(2),
129–169.
Kennedy, M. (1991). An agenda for research on teacher learning.
East Lansing: National Center for Research on Teacher
Learning. Michigan State University.
Kessler, R. J. (1985). Teachers’ instructional behavior related to
their conceptions of teaching and mathematics and their level
of dogmatism: Four case studies. Unpublished doctoral
dissertation. University of Georgia, Athens.
55
Kifer, E. & Robitaille D. F. (1992). Attitudes, preferences and
opinions. In D.F. Robitaille & R. A. Garden (Eds.), The IEA
study of mathematics II: Contexts and outcomes of school
mathematics (pp. 178–208). New York: Pergamon.
Mewborn, D. S. (2001). The role of mathematics content
knowledge in the preparation of elementary teachers in the
United States. Journal of Mathematical Teacher Development
3, 28-36.
Kuhs, T. M. & Ball, D. L. (1986). Approaches to teaching
mathematics: Mapping the domains of knowledge, skills, and
dispositions (Research Memo). East Lansing, MI: Michigan
State University, Center on Teacher Education.
Middleton, J. A. (1992). Teachers’ vs. students’ beliefs regarding
intrinsic motivation in the mathematics classroom: A personal
construct approach. Proceedings of the Annual Conference of
the American Educational Research Association, San
Francisco, CA. (ERIC Document Reproduction Service No.
ED 353154.)
Lappan, G., & Even, R. (1989). Learning to teach: Constructing
meaningful understanding of mathematical content. East
Lansing, MI: National Center for Research on Teacher
Education.
Laurenson, D .J. (1995). Mathematics and the drift towards
constructivism: Are teacher beliefs and teaching practice
following the beat of the same drummer? National
Consortium for Specialized Secondary Schools of
Mathematics, Science, and Technology Journal, 1(2), 3–7.
Leder, G. C. (1994). Research in mathematics education –
constraints on construction? In G. Bell (Ed.), Challenges in
mathematics education: Constraints on construction.
Proceedings of the 17th Annual Conference of the Mathematics
Education Research Group of Australasia (pp. 31–48).
Lismore: MERGA.
Lokan, J., Ford, P., & Greenwood, L. (1997). Maths & science on
the line: Australian middle primary students’ performance in
the Third International Mathematics and Science Study.
Melbourne: ACER.
Lortie, D. C. (1975). Schoolteacher: A sociological study. Chicago:
University of Chicago.
Lovat, T. J., & Smith, D. (1995). Curriculum: Action on reflection
revisited. Australia: Social Science Press.
Marland, P. W. (1994). Teaching: Implicit theories. In T. Husen, &
T. N. Postlewaite (Editors-in-chief), The international
encyclopaedia of education (pp. 6178–6183). New York:
Pergamon.
McAninch, A. M. (1993). Teacher thinking and the case method.
New York: Teachers College Press.
McClain, K. (2002). Teacher’s and students’ understanding: The
role of tools and inscriptions in supporting effective
communication. The Journal of the Learning Sciences,
11(2&3), 217–249.
McGalliard, W. A. Jr. (1983). Selected factors in the conceptual
systems of geometry teachers: Four case studies. (Doctoral
dissertation, University of Georgia). Dissertation Abstracts
International, 44, 1364A.
McGinnis, J. R., Shama, G., Graeber, A., & Watanabe, T. (1997).
Development of an instrument to measure teacher candidates’
attitudes and beliefs about the nature of and the teaching of
mathematics and sciences. Proceedings of the Annual
Conference of the National Association for Research in
Science Teaching, Oak Brook, Illinois. (ERIC Document
Reproduction Service No. ED 406201.)
McGinnis, J. R., & Parker, C, (2001, March). What Beliefs and
Intentions Concerning Science and Mathematics and the
Teaching of Those Subjects Do Reform-Prepared Specialist
Elementary/Middle Level Teachers Bring to the Workplace?
Paper presented at the annual meeting of the National
Association for Research in Science Teaching, St. Louis, MO.
56
Moreira, C. (1991). Teachers’ attitudes towards mathematics and
mathematics teaching: perspectives across two countries.
Proceedings of the Annual Conference of the International
Group for the Psychology of Mathematics Education. Assissi,
Italy. (ERIC Document Reproduction Service No. ED
413164.)
Murphy, E. (1997). Characteristics of constructivist learning and
teaching. Universite Laval, Quebec. Retrieved November 19,
2003, from http://www.stemnet.nf.ca/~elmurphy/
emurphy/cle3.html.
Nespor, J. (1987). The role of beliefs in the practice of teaching.
Journal of Curriculum Studies, 19(4), 317–328.
Nisbert, S., & Warren, E. (2000). Primary school teachers’ beliefs
relating to mathematics teaching and assessing mathematics
and factors that influence these beliefs. Mathematics
Education Research Journal, 13(2), 34–47.
Nolder, R. (1990). Accommodating curriculum change in
mathematics teachers’ dilemmas. Proceedings of the Annual
Conference of the Annual Meeting of the Psychology of
Mathematics Education Group, Mexico. (ERIC Document
Reproduction Service No. ED 411137.)
Orton, R. K. (1991). Summary. In P. L., Peterson & E. Fennema
(Eds.) (1989). Mathematics teaching and learning:
Researching in well-defined mathematical domains.
Proceedings of the Michigan State University Conference.
East Lansing, Michigan Elementary Subjects Center Series,
No. 40. (ERIC Document Reproduction Service No. ED
341545.)
Pajares, M. F. (1992). Teachers’ beliefs and educational research:
Cleaning up a messy construct. Review of Educational
Research, 62(3), 307–332.
Perry, B., Howard, P., & Conroy, J. (1996). K-6 teacher beliefs
about the learning and teaching of mathematics. In P. C.
Clarkson (Ed.), Technology in mathematics education.
Proceedings of the 19th Annual Conference of the Mathematics
Education Research Group of Australasia (pp. 453–460).
Melbourne: MERGA.
Perry, B., Howard, P., & Tracey, D. (1999). Head mathematics
teachers’ beliefs about the learning and teaching of
Mathematics. Mathematics Education Research. Journal, 11,
39–57.
Phillip, R. A., Flores, A., Sowder, J. T., & Schappelle, B. P. (1994).
Conceptions of extraordinary mathematics teachers. Journal of
Mathematical Behavior, 13(2), 155–180.
Prawat, R. (1990). Changing schools by changing teachers’ beliefs
about teaching and learning (Elementary Subjects Center
Series, No. 19). East Lansing: Michigan State University,
Institute for Research on Teaching.
Teachers’ Mathematical Beliefs
Raymond, A. M. (1993). Unraveling the relationships between
beginning elementary teachers’ mathematics beliefs and
teaching practices. Proceedings of the 15th Annual Conference
of the International Group for the Psychology of Mathematics
Education, Monterey, CA. (ERIC Document Reproduction
Service No. ED 390694.)
Raymond, D., Butt, R., & Towsend, D. (1991). Contexts for teacher
development: Insights from teachers’ stories. In A. Hargreaves
& M. Fullan (Eds.), Understanding teacher development (pp.
196–221). London: Cassells.
Renne, C. G. (1992). Elementary school teachers’ view of
knowledge pertaining to mathematics. Proceedings of the
Annual Conference of the American Research Association,
San Francisco, CA. (ERIC Document Reproduction Service
No. ED 353143.)
Richardson, V. (1996). The role of attitudes and beliefs in learning
to teach. In J. Sikula (Ed.), The handbook of research in
teacher education (pp. 102–119). New York: Macmillan.
Rokeach, M. (1968). Beliefs, attitudes and values. San Francisco:
Jossey-Bass.
Romberg, T. A., & Carpenter, T. P. (1986). Research on teaching
and learning mathematics: Two disciplines of scientific
inquiry. In M. C. Wittrock (Ed.), The handbook of research on
teaching (3rd ed., pp. 850–873). New York: Macmillan.
Steinberg, R., Haymore, J., & Marks, R. (1985, April). Teachers’
knowledge and structuring content in mathematics.
Proceedings of the Annual meeting of the American
Educational Research Association, San Francisco.
Stonewater, J. K., & Oprea, J. M. (1988). An analysis of in-service
teachers’ mathematical beliefs: A cognitive development
perspective. In M. J. Behr, C. B. Lacampagne, & M. M.
Wheeler (Eds.), Proceedings of the 10th Annual Conference of
the North American Chapter of the International Group for
the Psychology of Mathematics Education (pp. 356–363).
DeKalb, Il: Northern University.
Taylor, T. (1990). Mathematical attitude development from a
Vygotskian perspective. Mathematical Education Research
Journal, 4(3), 8–23.
Thompson, A. G. (1984). The relationship of teachers’ conceptions
of mathematics and mathematics teaching to instructional
practice. Educational Studies in Mathematics, 15, 105–127.
Thompson, A. G. (1985). Teachers' conceptions of mathematics
and the teaching of problem solving. In E. A. Silver (Ed.),
Teaching and learning mathematical problem solving:
Multiple research perspectives (pp. 281–294). Hillsdale, NJ:
Erlbaum.
Schmidt, W. H., & Kennedy, M. M. (1990). Teachers’ and teacher
candidates’ beliefs about subject matter and about teaching
responsibilities. (ERIC Document Reproduction Service No.
ED 320902.)
Thompson, A. G. (1991). The development of teachers conceptions
of mathematics teaching. Proceedings of the 13th Annual
Conference of the North American Chapter of the
International Group for the Psychology of Mathematics
Education. (ERIC Document Reproduction Service No. ED
352274.)
Schubert, N.A. (1981). Educators' perceptions of the degree that
their students learn according to selected principles of
learning. University of Southern Mississippi. (ERIC
Document Reproduction Service No. ED 222491).
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A
synthesis of research. In D. A. Grouws (Ed.). Handbook of
research on mathematics teaching and learning (pp.
127–146). New York: Macmillan.
Shirk, G. B. (1973). An examination of conceptual frameworks of
beginning mathematics teachers. Unpublished doctoral
dissertation, University of Illinois at Urbana-Champaign.
Tillema, H. H. (1995). Changing the professional knowledge and
beliefs of teachers: A training study. Learning and Instruction,
5(4), 291–318.
Sirotnik, K. A. (1983). What you see is what you get—consistency,
persistency, and mediocrity in classrooms. Harvard
Educational Review, 53, 16–31.
Van Zoest, L., Jones, G. A., & Thornton, C. A. (1994). Beliefs
about mathematics teaching held by pre-service teachers
involved in a first grade mentorship program. Mathematics
Education Research Journal, 6(1), 37–55.
Sosniak, L. A., & Ethington, C. A., & Varelas, M. (1991).
Teaching mathematics without a coherent point of view:
Findings from the IEA Second International Mathematics
Study. Journal of Curriculum Studies, 23, 119–131.
Southwell, B. & Khamis, M. (1992). Beliefs about mathematics
and mathematics education. In K. Owens, B. Perry, & B.
Southwell (Eds.) Space, the first and final frontier.
Proceedings of the 15th Annual Conference of the Mathematics
Research Group of Australasia (pp. 497–509). Sydney:
MERGA.
Stipek, D., Givvin, K., Salmon, J. & MacGyvers, V. (2001).
Teachers’ beliefs and practices related to mathematics
instruction. Teaching and Teacher Education. 17 (2), 213 –
226.
Boris Handal
Whitman, N. C., & Morris, K. L. (1990). Similarities and
differences in teachers’ beliefs about effective teaching of
mathematics: Japan and Hawaii. Educational Studies in
Mathematics, 21, 71–81.
Wood, E. F., & Floden, R. E. (1990). Where teacher education
students agree: Beliefs widely shared before teacher
education. (ERIC Document Reproduction Service No. ED
331781.)
Wood, T., Cobb, P., & Yackel, E. (1991). Change in teaching
mathematics. American Educational Research Journal, 28(3),
587–616.
57
The Mathematics Educator
2003, Vol. 13, No. 2, 58–59
In Focus…
Just Get Out of the Way
Adelyn Steele
Not too long ago during a typical planning period,
I was in my classroom working on lesson plans when
in walked my colleague in the math department. As a
first year teacher, he is full of questions and ideas and
has the ability to see with fresh eyes those situations
that some of us just take for granted by now.
His question this time had to do with filling out
papers for the special education department. After we
talked about what he should do and why, he remarked,
“You know, at some point I hope to get back to doing
what I am supposed to do.” I smiled the sympathetic
smile of a paperwork weary comrade and watched him
walk out the door.
But his statement haunts me and leaves me to
wonder: What is it that a teacher is supposed to do?
I have asked the question to many people: students,
colleagues, principals, parents, and friends; and their
answers are startling. Everything from inspire, guide,
and rescue; to show up, present facts, and record scores
are mentioned. All I know is at the end of every one of
these conversations; I feel frustrated, confused, and
down right exhausted.
So I really don’t know what a teacher is supposed
to do (which alarms me slightly as I show up everyday
to do it), but maybe the trouble is that I am trying to
find a single phrase or idea. A slogan of sorts that
could keep me focused and put everything into
perspective. Teaching is much too complex for that.
There is one idea that keeps coming back to me,
however, and that is that the job of a teacher is to set
up the task and then get out of the way. Sounds simple,
but I assure you it is not.
A teacher must either design or find tasks that will
allow students to engage in the mathematics. The task
must be rich enough to give students something to talk
about and wrestle with. It ought to have some
significance and build toward an understanding of
Adelyn Steele is a mathematics instructor and K-12 mathematics
chairperson for Cheney High School in Cheney, KS. She received a
BSE in Mathematics from Emporia State University and a Masters
in Teaching from Friends University. Adelyn has been named a
state finalist in Kansas for the Presidential Award of Excellence in
Mathematics and Science Teaching.
58
mathematics in a way that will be powerful and lasting.
I use to think that this was the hard part, but I have
since come to understand that selecting the tasks is
much easier than deciding what to do with them.
Earlier I said that a teacher should get out of the
way, and I mean that, but not in the sense of heading to
the teacher’s lounge or reading the newspaper in the
back of the room. I mean it in the sense of letting the
students do as much of it on their own as possible. A
teacher should watch the interaction between students
and guide what is happening, be on the lookout for
evidence of both correct and incorrect observations and
understanding, and absorb what is happening in the
classroom.
Now, most people who have a stake in education
seem to say things very similar to what I am saying;
yet we disagree at every turn. That is, we disagree with
how this is done. The reason, I think, is that we start
with different beliefs. See, I believe that understanding
and making sense is internal. So if I want that to
happen in my students, I need to allow them the time to
see that through. If they don’t “get it” in a specific
amount of time and I rush in and tell them, then I have
robbed them of the opportunity to construct an
understanding for themselves. I may have left them
with the ability to say back what someone might expect
to hear, but they do not own the idea. They are
repeating mine.
To that end, I need to make the observation that
thinking is hard. It can require concentration and
awareness to be able to synthesize content. Then a
person has to find the words to express what they are
thinking. I watch and listen to students, and find that
throughout much of their lives people are speaking at
them rather than to or with them. If they do not have
the opportunity to think and discuss what they are
thinking, I believe that those skills are stunted. If,
however, students are encouraged to try to find words
and are given time to synthesize and articulate then
they create within themselves the confidence to wrestle
with increasingly complex ideas.
I also believe that learning is social. Therefore, my
classroom needs to be a social environment where
students both talk and listen. They need to respect
Just Get Out of the Way
ideas and people and learn to manage themselves in a
conversation. (What is this business about having to
raise your hand anyway? Where, outside of a
classroom, do you see people raising their hands for
the opportunity to speak? We owe kids better.) This
takes practice and can be messy at the beginning, but
as a vehicle to true discourse it is imperative.
Core to my beliefs is also the learners’ ability to
ask and answer their own questions if given the
chance. So often teachers, upon observing and noticing
a misconception, jump in to “fix it,” often before a
child has the opportunity to really understand that they
have that question. I had an algebra teacher who would
say to us, “By now you all are thinking….” It was so
strange because sometimes we were not thinking
whatever he said we were. Once, when this was
pointed out to him, he said, “Well, if you are not
thinking it now, you should be.” Which brings me to
my next belief.
People ought to have the right to believe what they
want to. This one, ironically, causes all sorts of trouble
for me from my colleagues. “So if I want to believe
that the sum of 5 and 3 is 7 then that is ok with you?”
they have asked. But my point is a real one. So often
adults tell students not only what, but also how to
think. This creates a dependence that is problematic for
years. It is much more powerful to have students
defend their own beliefs with data and/or proof. As
students do this they develop skills as autonomous
learners and profit much more than might be imagined.
So what does this have to do with staying out of
the way? Well first, if I believe what I say that I do,
then my actions will reflect that. So by staying out of
the way, I again mean that I let students do as much on
their own as possible. I wait. I listen. I probe. I ask
Adelyn Steele
questions. I wait. I listen. I give them an opportunity to
think. I am silent. I wait. I listen. I encourage them to
test their ideas. I encourage them to talk to each other. I
wait. I listen. My intent is to have their thinking on the
table for examination by themselves and others, not to
dictate what I think. As I listen, I gather information
about student understanding (and lack thereof) and
look for opportunities for students to build
understanding and make sense of the mathematics for
themselves. I ask questions much more often than I
make statements.
I wish to make it clear that I am not passive in the
classroom. I am silent much of the time, but that does
not equate to passivity. I am actively participating in
what students are doing and saying by listening. I will
certainly make observations, give relevant information
and guide conversations if needed. I just strongly
believe that students need very little intervention in
their thinking and certainly much less than what they
usually encounter. I do not just leave students to falter
and stumble, but I only help them up (so to speak) if
they demonstrate an inability to do it themselves.
The result is not chaos as many might predict. Nor
does it take too much time. Certainly there is a trade
off in the beginning, but the reward (that of students
who think for themselves and articulate and defend
their knowledge) far surpasses it in just a few weeks.
The beauty of mathematics is that it does make sense
and can be explained. Students are just as capable of
making sense and building an understanding as anyone
else is. The bonus is that if they become confident
learners who value their ideas and the ideas of others in
addition to the discipline of mathematics at the level
they are in, they will have some interest in a future
study of mathematics and/or in learning in general.
59
Mathematics Education
Student Association
MESA
The Mathematics Education Student Association is an official
affiliate of the National Council of Teachers of Mathematics.
MESA is an integral part of The University of Georgia’s
mathematics education community and is dedicated to serving
all students. Membership is open to all UGA students, as well
as other members of the mathematics education community.
Visit MESA online at http://www.ugamesa.org
TME Subscriptions
TME is published both online and in print format. The current issue as well as back issues are available online at
http://www.ugamesa.org, then click TME. A paid subscription is required to receive the printed version of The Mathematics
Educator. Subscribe now for Volume 14, issues 1 & 2, to be published in the spring and fall of 2004.
If you would like to be notified by email when a new issue is available online, please send a request to
tme@coe.uga.edu
To subscribe, send a copy of this form, along with the requested information and the subscription fee to
The Mathematics Educator
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
___ I wish to subscribe to The Mathematics Educator for Volume 14 (Numbers 1 & 2).
___ I would like a previous issue of TME sent. Please indicate Volume and issue number(s):
Name
___________________
Amount Enclosed ________________
subscription: $6/individual; $10/institutional
each back issue: $3/individual; $5/institutional
Address
Email _________________________
60
The Mathematics Educator (ISSN 1062-9017) is a biannual publication of the Mathematics Education Student Association
(MESA) at The University of Georgia. The purpose of the journal is to promote the interchange of ideas among students, faculty,
and alumni of The University of Georgia, as well as the broader mathematics education community. The Mathematics Educator
presents a variety of viewpoints within a broad spectrum of issues related to mathematics education. The Mathematics Educator
is catalogued in ERIC and abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical
Education).
The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in
mathematics education including:
•
•
•
•
•
•
•
•
reports of research (including experiments, case studies, surveys, philosophical studies, and historical studies), curriculum
projects, or classroom experiences;
commentaries on issues pertaining to research, classroom experiences, or public policies in mathematics education;
literature reviews;
theoretical analyses;
critiques of general articles, research reports, books, or software;
mathematical problems;
translations of articles previously published in other languages;
abstracts of or entire articles that have been published in journals or proceedings that may not be easily available.
The Mathematics Educator strives to provide a forum for a developing collaboration of mathematics educators at varying levels of
professional experience throughout the field. The work presented should be well conceptualized; should be theoretically grounded;
and should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and researchers.
Electronic Submission (preferred):
Submit an attachment of your manuscript saved as a Microsoft Word or Rich Text Format document. The manuscript should be
double-spaced and written in a 12 point font. It must conform to the style specified in the Publication Manual of the American
Psychological Association, 5th Edition, and not exceed 25 pages, including references and footnotes. Pictures, tables, and figures
should be in a format compatible with Word 95 or later. The author’s name and affiliation should appear only on the e-mail
message used to send the file to ensure anonymity during the reviewing process. If the manuscript is based on dissertation
research, a funded project, or a paper presented at a professional meeting, the e-mail message should provide the relevant facts.
Send manuscripts to the electronic address given below.
Submission by Mail:
Submit five copies of each manuscript. Manuscripts should be typed and double-spaced, conform to the style specified in the
Publication Manual of the American Psychological Association, 5th Edition, and not exceed 25 pages, including references and
endnotes. Pictures, tables, and figures should be camera ready. The author’s name and affiliation should appear only on a separate
title page to ensure anonymity during the reviewing process. If the manuscript is based on dissertation research, a funded project,
or a paper presented at a professional meeting, a note on the title page should provide the relevant facts. Send manuscripts to the
postal address below.
To Become a Reviewer:
Contact the Editor at the postal or email address below. Please indicate if you have special interests in reviewing articles that
address certain topics such as curriculum change, student learning, teacher education, or technology.
Postal Address:
The Mathematics Educator
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
Electronic address:
tme@coe.uga.edu
In this Issue,
Guest Editorial… What is Mathematics Education For?
BRIAN GREER & SWAPNA MUKHOPADHYAY
Hidden Assumptions and Unaddressed Questions in Mathematics for All Rhetoric
DANNY BERNARD MARTIN
The Fourth “R”: Reflection
NORENE VAIL LOWERY
Impact of a Girls Mathematics and Technology Program
on Middle School Girls’ Attitudes Toward Mathematics
MELISSA A. DEHAVEN & LYNDA R. WIEST
First-Time Teacher-Researchers Use Writing in Middle School
Mathematics Instruction
DREW K. ISHII
Teachers’ Mathematical Beliefs: A Review
BORIS HANDAL
In Focus… Just Get Out of the Way
ADELYN STEELE