the mathematics educator - university of...
TRANSCRIPT
This publication is supported by the College of Education at The University of Georgia.
Editorial Staff
Editor
Catherine Ulrich Kevin LaForest
Associate Editors
Amber G. Candela
Tonya DeGeorge Erik D. Jacobson
David R. Liss, II Allyson Thrasher
Advisor Dorothy Y. White
MESA Officers
2011-2012
President
Tonya DeGeorge
Vice-President
Shawn Broderick
Secretary
Jenny Johnson
Treasurer
Patty Anne Wagner
NCTM
Representative
Clayton N. Kitchings
Colloquium Chair Ronnachai
Panapoi
A Note From the Editor
Dear TME readers,
On behalf of the editorial staff and the Mathematics
Education Student Association of The University of
Georgia, I am pleased to present the first issue of Volume
22 of The Mathematics Educator. This will be my final
issue as Co-Editor. I have learned so much about
communication and writing from my interactions with the
TME authors, editors, and reviewers I have worked with over the years. Thank you to everyone who has
volunteered their time and talents to helping me publish
this unique journal. I am sad to leave my post as Co-Editor
after three incredible years, but I am glad to hand over the
reigns to Kevin LaForest, my Co-Editor for this issue, and
Amber Candela, a long-time Associate Editors.
Anna Sfard graciously agreed to write an editorial in
which she urges mathematics educators to critically
reexamine the privileged place of mathematics in the
school curriculum. She also discusses the merits of
thinking of mathematics as a discourse. Dovetailing with
this theme, Valerie Sharon applies discourse analysis methods to study prospective elementary teachers’ varying
roles during mathematical learning, and Aria Rafzar gives
a wonderful example of using discourse analysis methods
to help pre- and in-service teacher examine their practices.
In contrast to qualitative methods of these two articles,
Lemire, Melby, Haskins, and Williams use a quantitative
approach to examine how student perception of the
appropriateness of the difficulty level and teacher support
in mathematics classes correlates with mathematical
performance. Amanda Ross and Anthony J. Onweugbuzie
tie these articles together in their piece exploring prevalence rates of mixed methods research and describing
how using qualitative and quantitative approaches in
concert can enhance mathematics education research.
I hope you enjoy this issue as much as we have
enjoyed putting it together. Thank you especially to our
authors for their contributions to the field.
All the best,
Catherine Ulrich
Cover by Kylie Wagner.
© 2012 Mathematics Education Student Association All Rights Reserved
THE MATHEMATICS EDUCATOR
An Official Publication of The Mathematics Education Student Association
The University of Georgia
Summer 2012 Volume 22 Number 1
Table of Contents
3 Guest Editorial… Why Mathematics? What
Mathematics? ANNA SFARD
17 The Roles They Play: Prospective Elementary
Teachers and a Problem -Solving Task
VALERIE SHARON
39 Dicoursing Mathematically: Using Discourse
Analysis to Develop a Sociocritical
Perspective of Mathematics Education
ARIA RAZFAR
63 The Devalued Student: Misalignment of
Current Mathematics Knowledge and Level
of Instruction
STEVEN D. LEMIRE, MARCELLA L. MELBY,
ANNE M. HASKINS, & TONY WILLIAMS
84 Prevalence of Mixed Methods Research in
Mathematics Education
AMANDA ROSS & ANTHONY J.
ONWUEGBUZIE
114 A Note to Reviewers
The Mathematics Educator
2012 Vol. 22, No. 1, 3–16
Anna Sfard is a Professor of Education and head of the Mathematics Education department at the University of Haifa, Israel. She served as Lappan-Philips-Fitzgerald Professor at Michigan State University and has been affiliated with the University of London. For her research work, devoted to the study of human thinking as a form of communication and of mathematics as a kind of discourse,
she has been granted the 2007 Freudenthal Award.
Guest Editorial…
Why Mathematics? What Mathematics? Anna Sfard
“Why do I have to learn mathematics? What do I need it for?”
When I was a school student, it never occurred to me to ask these
questions, nor do I remember hearing it from any of my
classmates. “Why do I need history?”—yes. “Why Latin?” (yes, as a high school student I was supposed to study this ancient
language)—certainly. But not, “Why mathematics?” The need to
deal with numbers, geometric figures, and functions was beyond doubt, and mathematics was unassailable.
Things changed. Today, every other student seems to ask why
we need mathematics. Over the years, the quiet certainty of the
mathematics learner has disappeared: No longer do young people take it for granted that everybody has to learn math, or at least the
particular mathematics curriculum that is practiced with only
marginal variations all over the world. The questions, “Why mathematics? Why so much of it? Why ‘for all’?” are now being
asked by almost anybody invested, or just interested, in the
business of education. Almost, but not all. Whereas the question seems to be bothering students, parents and, more generally, all
the “ordinary people” concerned about the current standards of
good education, the doubt does not seem to cross the minds of
those who should probably be the first to wonder: mathematics educators, policy makers, and researchers. Not only are
mathematics educators and researchers convinced about the
importance of school mathematics, they also know how to make the case for it. If asked, they will all come up with a
Anna Sfard
4
number of reasons, and their arguments will look more or less the
same, whatever the cultural background of its presenter. Yet these common arguments are almost as old as school mathematics itself,
and those who use them do not seem to have considered the
possibility that, as times change, these arguments might have
become unconvincing. Psychologically, this attitude is fully understandable. After all,
at stake is the twig on which mathematics education community
has weaved its nest. And yet, as the wonderings about the status of school mathematics are becoming louder and louder, the need for
a revision of our reasons can no longer be ignored. In what
follows, I respond to this need by taking a critical look at some of the most popular arguments for the currently popular slogan,
“Mathematics for all.” This analysis is preceded by a proposal of
how to think about mathematics so as to loosen the grip of clichés
and to shed off hidden prejudice. It is followed by my own take on the question of what mathematics to teach, to whom, and how.
What Is Mathematics?
To justify the conviction that competence in mathematics is a condition for good citizenship, one must first address the question
of what mathematics is and what role it has been playing in the life
of the Western society.1 Here is a proposal: I believe that it might
be useful to think about any type of human knowing, mathematics
included, as an activity of, or a potential for, telling certain kinds
of stories about the world. These stories may sometimes appear far removed from anything we can see or touch, but they nevertheless
are believed to remain in close relationship to the tangible reality
and, in the final account, are expected to mediate all our actions
and improve the ways in which we are going about our human affairs. Since mathematical stories are about objects that cannot be
seen, smelled, or touched, it may be a bit difficult to see that the
claim of practical usefulness applies to mathematics as much as to physics or biology. But then it suffices to recall the role of, say,
measurements and calculations in almost any task a person or a
society may wish to undertake to realize that mathematical stories are, indeed, a centerpiece of our universal world-managing toolkit.
And I have used just the simplest, most obvious example.
So, as the activity of storytelling, mathematics is not much
different from any other subject taught in school. Still, its
Why Mathematics?
5
narratives are quite unlike those told in history, physics or
geography. The nature of the objects these stories are about is but one aspect of the apparent dissimilarity. The way the narratives are
constructed and deemed as endorsable (“valid” or “true”) makes a
less obvious, but certainly not any less important, difference. It is
thus justified to say that mathematics is a discourse – a special way of communicating, made unique by its vocabulary, visual
means, routine ways of doing things and the resulting set of
endorsed narratives – of stories believed to faithfully reflect the real state of affairs. By presenting mathematics in this way (see
also Sfard, 2008), I am moving away from the traditional vision of
mathematics as given to people by the world itself. Although definitely constrained by external reality, mathematics is to a great
extent a matter of human decisions and choices, and of
contingency rather than of necessity. This means that
mathematical communication can and should be constantly monitored for its effects. In particular, nothing that regards the
uses of mathematics is written in stone, and there is no other
authority than ourselves to say what needs to be preserved and what must be changed. This conceptualization, therefore, asks for
a critical analysis of our common mathematics-related educational
practices.
Why Mathematics? Deconstructing Some Common Answers
Three arguments for the status of mathematics as a sine qua
non of school curricula can usually be heard these days in response to the question of why mathematics: the utilitarian, the
political, and the cultural. I will call these three motives “official,”
so as to distinguish them from yet another one, which, although
not any less powerful than the rest, is never explicitly stated by the proponents of the slogan “mathematics for all.”
The Utilitarian Argument: Mathematics Helps in Dealing
With Real-Life Problems
Let me say it again: Mathematics, just as any other domain of
human knowledge, is the activity of describing—thus
understanding—the world in ways that can mediate and improve our actions. It is often useful to tell ourselves some mathematical
stories before we act, and to repeat them as we act, while also
forging some new ones. With their exceptionally high level of
Anna Sfard
6
abstraction and the unparalleled capacity for generalization,
mathematical narratives are believed to be a universal tool, applicable in all domains of our lives. And indeed, mathematics
has a long and glorious history of contributions to the well-being
of humankind. Ever since its inception, it has been providing us
with stories that, in spite of their being concerned with the universe of intangible objects, make us able to deal with the reality
around us in particularly effective and useful ways. No wonder,
then, that mathematics is considered indispensable for our existence. And yet, whereas this utilitarian argument holds when
the term “our existence” is understood as referring to the life of the
human society as a whole, it falls apart when it comes to individual lives.
I can point to at least two reasons because of which the utility
claim does not work at the individual level. First, it is enough to
take a critical look at our own lives to realize that we do not, in fact, need much mathematics in our everyday lives. A university
professor recently said in a TV interview that in spite of his sound
scientific-mathematical background he could not remember the last time he had used trigonometry, derivatives, or mathematical
induction for any purpose. His need for mathematical techniques
never goes beyond simple calculation, he said. As it turns out, even those whose profession requires more advanced
mathematical competency are likely to say that whatever
mathematical tools they are using, the tools have been learned at
the job rather than in school. The second issue I want to point to may be at least a partial
explanation for the first: People do not necessarily recognize the
applicability of even those mathematical concepts and techniques with which they are fairly familiar. Indeed, research of the last few
decades (Brown et al., 1989; Lave, 1988; Lave & Wenger, 1991)
brought ample evidence that having mathematical tools does not
mean knowing when and how to use them. If we ever have recourse to mathematical discourse, it is usually in contexts that
closely resemble those in which we encountered this discourse for
the first time. The majority of the school-learned mathematics remains in school for the rest of our lives. These days, all this is
known as a manifestation of the phenomenon called situatedness
of learning, the dependence of the things we know on the context in which they have been learned. To sum up, not only is our
Why Mathematics?
7
everyday need for school mathematics rather limited, the
mathematics we could use does not make it easily it into our lives. All this pulls the rug from under the feet of those who defend the
idea of teaching mathematics to all because of its utility.
The Political Argument: Mathematics Empowers
Because of the universality of mathematics and its special
usefulness,2 the slogan “knowledge is power,” which can now be
translated into “discourses are power,” applies to this special form of talk with a particular force. Ever since the advent of modernity,
with its high respect for, and utmost confidence in, human reason,
mathematics has been one of the hegemonic discourses of Western
society. In this positivistically-minded world, whatever is stated in mathematical terms tends to override any other type of argument
(just recall, for instance, what counts as decisive “scientific
evidence” in the eyes of the politician), and the ability to talk mathematics is thus considered as an important social asset,
indeed, a key to success. But the effectiveness of mathematics as a
problem-solving tool is only a partial answer to the question of
where this omnipotence of mathematical talk comes from. Another relevant feature of mathematics is its ability to impose linear order
on anything quantifiable. Number-imbued discourses are perfect
settings for decision-making and, as such, they are favored by many, and especially by politicians (and it really does not matter
that all too often, politicians can only speak pidgin mathematics;
the lack of competency is not an obstacle for those who know their audience and are well aware of the fact that numbers do not have
to be used correctly to impress).
The second pro-math argument, one that I called political, can
now be stated in just two words: Mathematics empowers. Indeed, if mathematics is the discourse of power, mathematical
competency is our armor and mathematical techniques are our
social survival skills. When we wonder whether mathematics is worth our effort, at stake is our agency as individuals and our
independence as members of society: If we do not want to be
pushed around by professional number-jugglers, we must be able to juggle numbers with them and do it equally well, if not better.
Add to this the fact that in our society mathematics is a gatekeeper
to many coveted jobs and is thus a key to social mobility, and you
cannot doubt the universal need for mathematics any longer.
Anna Sfard
8
Now it is time for my counter-arguments. The claim that
“mathematics empowers” is grounded in the assumption that mathematics is a privileged discourse, a discourse likely to
supersede any other. But should the hegemony of mathematics go
unquestioned? On a closer look, not each of its uses may be for the
good of those whose well-being and empowerment we have in mind when we require “mathematics for all.” For example, when
mathematics, so effective in creating useful stories about the
physical reality around us, is also applied in crafting stories about children (as in “This is a below average student”) and plays a
decisive role in determining the paths their lives are going to take,
the results may be less than helpful. More often than not, the numerical tags with which these stories label their young
protagonists, rather than empowering the student, may be raising
barriers that some of the children will never be able to cross. The
same happens when the ability to participate in mathematical discourse is seen as a norm and the lack thereof as pathology and a
symptom of a general insufficiency of the child’s “potential.” I
will return to all this when presenting the “unofficial” argument for the obligatory school mathematics. For now, the bottom line of
what was written so far is simple: we need to remember that by
embracing the slogan “mathematics empowers” as is, without any amendments, we may be unwittingly reinforcing social orders we
wish to change. As I will be arguing in the concluding part of this
editorial, trying to change the game may be much more
“empowering” than trying to make everybody join in and play it well.
The Cultural Argument: Mathematics Is a Necessary
Ingredient of Your Cultural Makeup
In the last paragraph, I touched upon the issue of the place of
mathematics in our culture and in an individual person’s identities.
I will now elaborate on this topic while presenting the cultural argument for teaching mathematics to all.
Considering the fact that to think means to participate in some
kind of discourse, it is fair to say that our discourses, those discourses in which each of us is able to participate, constitute
who we are as social beings. In the society that appreciates
intellectual skills and communication, the greater and more
diverse our discursive repertoire, the richer, more valued, and
Why Mathematics?
9
more attractive our identities. However, not all discourses are
made equal, so the adjective “valued” must be qualified. Some forms of communicating are considered to be good for our
identities and some others much less so. As to mathematics, many
would say that it belongs to the former category. Considered as a
pinnacle of human intellectual achievement and thus as one of the most precious cultural assets, it bestows some of its glory even on
peripheral members of the mathematical community. Those who
share this view believe that mathematical competency makes you a better person, if only because of the prestigious membership that
it affords. A good illustration of this claim comes from an Israeli
study (Sfard & Prusak, 2005) in which 16-year-old immigrant students, originally from the former Soviet Union, unanimously
justified their choice of the advanced mathematics program with
claims that mathematics is an indispensable ingredient of one’s
identity, saying, for example, “Without mathematics, one is not a complete human being.”
But the truth is that the attitude demonstrated by those
immigrant students stands today as an exception rather than a rule. In the eyes of today’s young people, at least those who come from
cultural backgrounds I am well acquainted with, mathematics does
not seem to have the allure it had for my generation. Whereas this statement can be supported with numbers that show a continuous
decline in percentages of graduates who choose to study
mathematics (or science)—and currently, this seems to be a
general trend in the Western world3—I can also present some first-
hand evidence. In the same research in which the immigrant
students declared their need for mathematical competency as a
necessary ingredient of their identities, the Israeli-born participants spoke about mathematics solely as a stepping stone for whatever
else they would like to do in the future. Such an approach means
that one can dispose with mathematics once it has fulfilled its role
as an entrance ticket to preferable places. For the Israeli-born participants, as for many other young people these days,
mathematical competency is no longer a highly desired ingredient
of one’s identity. Considering the way the world has been changing in the last
few decades it may not be too difficult to account for this drop in
the popularity of mathematics. One of the reasons may be the fact that mathematical activity does not match the life experiences
Anna Sfard
10
typical of our postmodern communication-driven world. As aptly
observed in a recent book by Susan Cain (2012), the hero of our times is a vocal, assertive extrovert with well-developed
communicational skills and insatiable appetite for interpersonal
contact. Although there is a clear tendency, these days, to teach
mathematics in collaborative groups—the type of learning that is very much in tune with this general trend toward the collective and
the interpersonal—we need to remember that one cannot turn
mathematics into a discourse-for-oneself unless one also practices talking mathematics to oneself. And yet, as long as interpersonal
communication is the name of the game and a person with a
preference for the intra-personal dialogue risks marginalization, few students may be ready to suspend their intense exchanges with
others for the sake of well-focused conversation with themselves.
In spite of all that has been said above, I must confess that the
cultural argument is particularly difficult for me to renounce. I have been brought up to love mathematics for what it is. Born into
the modernist world ruled by logical positivism, I believed that
mathematics must be treated as a queen even when it acts as a servant. Like the immigrant participants of Anna Prusak’s study, I
have always felt that mathematics is a valuable, indeed
indispensable, ingredient of my identity—an element to cherish and be proud of. But this is just a matter of emotions. Rationally,
there is little I can say in defense of this stance. I am acutely aware
of the fact that times change and that, these days, modernist
romanticism is at odds with postmodernist pragmatism. In the end, I must concede that the designation of mathematics as a cultural
asset is not any different than that of poetry or art. Thus, however
we look at it, the cultural argument alone does not justify the prominent presence of mathematics in school curricula.
The Unofficial Argument: Mathematics Is a Perfect Selection
Tool
The last argument harks back to the abuses of mathematics to
which I hinted while reflecting on the statement “mathematics
empowers.” I call it “unofficial,” because no educational policy maker would admit to its being the principal, if not the only,
motive for his or her decisions. I am talking here about the use of
school mathematics as a basis for the measuring-and-labeling
practices mentioned above. In our society, grades in mathematics
Why Mathematics?
11
serve as one of the main criteria for selecting school graduates for
their future careers. Justifiably or not, mathematics is considered to be the lingua franca of our times, the universal language, less
sensitive to culture than any other well-defined discourse. No
intellectual competency, therefore, seems as well suited as
mathematics for the role of a universal yardstick for evaluating and comparing people. Add to this the common conviction that
“Good in math = generally brilliant” (with the complement being,
illogically, “not good in math = generally suspect”), and you begin realizing that teaching mathematics and then assessing the results
may be, above all, an activity of classifying people with “price
tags” that, once attached, will have to be displayed whenever a person is trying to get access to one career or another. I do not
think that an elaborate argument is needed to deconstruct this kind
of motive. The very assertion that this harmful practice is perhaps
the only reason for requiring mathematics for all should be enough to make us rethink our policies.
What Mathematics and Why? A Personal View
It is time for me to make a personal statement. Just in case I have been misunderstood, let me make it clear: I do care for
mathematics and I am as concerned as anybody about its future
and the future of those who are going to need it. All that I said above grew from this very genuine concern. By no means do I
advocate discontinuing the practice of teaching mathematics in
school. All I am trying to say is that we should approach the task in a more flexible, less authoritarian way, while giving more
thought to the question of how much should be required from all
and how much choice should be left to the learner. In other words,
I propose that we rethink school mathematics and revise it quite radically. As I said before, if there is a doubt about the game being
played, let us change this game rather than trying to play it well.
These days, deep, far-reaching change is needed in what we teach, to whom, and how.
I do have a concrete proposal with regard to what we can do.
But let me precede this discussion with two basic “don’t”s. First, let us not use mathematics as a universal instrument for selection.
This practice hurts the student and it spoils the mathematics that is
being learned. Second, let us not force the traditional school
Anna Sfard
12
curriculum on everybody, and, whatever mathematics we do
decide to teach, let us teach it in a different way. In the rest of this editorial, let me elaborate on this latter issue,
which, in more constructive terms, can be stated as follows: Yes,
let us teach everybody some mathematics, the mathematics whose
everyday usefulness is beyond question. Arithmetic? Yes. Some geometry? Definitely. Basic algebra? No doubt. Add to this some
rudimentary statistics, the extremely useful topic that is still only
rarely taught in schools, and the list of what I consider as “mathematics for all” is complete. And what about trigonometry,
calculus, liner algebra? Let us leave these more advanced topic as
electives, to be chosen by those who want to study them. But the proposed syllabus does not, per se, convey the idea of
the change I had in mind when claiming the need to rethink school
mathematics. The question is not just of what to teach or to whom,
but also of how to conceptualize what is being taught so as to make it more convincing and more learnable. There are two tightly
interrelated ways in which mathematics could be framed in school
as an object of learning: we can think about mathematics as the art of communicating or as one of the basic forms of literacy. Clearly,
both these framings are predicated on the vision of mathematics as
a discourse. Moreover, a combination of the two approaches could be found so that the student can benefit from both. Let me briefly
elaborate on each one of the two framings.
Mathematics as the Art of Communicating
As a discourse, mathematics offers special ways of
communicating with others and with oneself. When it comes to the
effectiveness of communication, mathematics is unrivaled: When
at its best, it is ambiguity-proof and has an unparalleled capacity for generalization. To put it differently, mathematical discourse
appears to be infallible—any two people who follow its rules must
eventually agree, that is, endorse the same narratives; in addition, this discourse has an exceptional power of expression, allowing us
to say more with less.
I can see a number of reasons why teaching mathematics as the art of communicating may be a good thing to do. First, it will
bring to the fore the interpersonal dimension of mathematics: the
word communication reminds us that mathematics originates in a
conversation between mathematically-minded thinkers, concerned
Why Mathematics?
13
about the quality of their exchange at least as much as about what
this exchange is all about. Second, the importance of the communicational habits one develops when motivated by the wish
to prevent ambiguity and ensure consensus exceeds the boundaries
of mathematics. I am prepared to go so far as to claim that if some
of the habits of mathematical communication were regulating all human conversations, from those that take place between married
couples to those between politicians, our world would be a happier
place to live. Third, presenting mathematics as the art of interpersonal communication is, potentially, a more effective
educational strategy than focusing exclusively on intra-personal
communication. The interpersonal approach fits with today’s young people’s preferences. It is also easier to implement. After
all, shaping the ways students talk to each other is, for obvious
reasons, a more straightforward job that trying to mould their
thinking directly. Fourth, framing the task of learning mathematics as perfecting one’s ability to communicate with others may be
helpful, even if not sufficient, in overcoming the situatedness of
mathematical learning. Challenging students to find solutions that would convince the worst skeptic will likely help them develop the
life-long habit of paying attention to the way they talk (and thus
think!). This kind of attention, being focused on one’s own actions, may bring about discursive habits that are less context-
dependent and more universal than those that develop when the
learner is almost exclusively preoccupied with mathematical
objects. There may be more, but I think these four reasons should suffice to explain why teaching mathematics as an art of
communication appears to be a worthy endeavor.
Mathematics as a Basic Literacy
While teaching mathematics as an art of communicating, we
stress the question of how to talk. Fostering mathematical literacy
completes the picture by emphasizing the issues of when to talk mathematically and what about.
Although, nowadays, mathematical literacy is a buzz phrase, a
cursory review of literature suffices to show that there is not much agreement on how it should be used. For the sake of the present
conversation, I will define mathematical literacy as the ability to
decide not just about how to participate in mathematical discourse
but also about when to do so. It is the emphasis on the word when
Anna Sfard
14
that signals that mathematical literacy is different from the type of
formal mathematical knowledge that is being developed, in practice if not in principle, through the majority of present-day
curricula. These curricula offer mathematics as, first and foremost,
a self-sustained discourse that speaks about its own unique objects
and has little ties to anything external. Thus, they stress the how of mathematics to the neglect of the when. Mathematical literacy, in
contrast, means the ability to engage in mathematical
communication whenever this may help in understanding and manipulating the world around us. It thus requires fostering the
how and the when of the mathematical routines at the same time.
To put it in discursive terms, along with developing students' participation in mathematical discourse, we need to teach them
how to combine this discourse with other ones. Literacy
instruction must stress students’ ability to switch to the
mathematical discourse from any other discourse whenever appropriate and useful, and it has to foster the capacity for
incorporating some of the meta-mathematical rules of
communication into other discourses. My proposal, therefore, is to replace the slogan “mathematics
for all” with the call for “mathematical literacy for all.”
Arithmetic, geometry, elementary algebra, the basics of statistics—these are mathematical discourses that, I believe,
should become a part and parcel of every child’s literacy kit. This
is easier said than done, of course. Because of the inherent
situatedness of learning, the call for mathematical literacy presents educators with a major challenge. The question of how to teach for
mathematical literacy must be theoretically and empirically
studied. Considering the urgency of the issue, such research should be given high priority.
***
In this editorial, I tried to make the case for a change in the
way we think about school mathematics. In spite of the constant talk about reform, the current mathematical curricula are almost
the same in their content (as opposed to pedagogy) as they were
decades, if not centuries, ago. Times change, but our general conception of school mathematics remains invariant. As
mathematics educators, we have a strong urge to preserve the kind
of mathematics that has been at the center of our lives ever since
Why Mathematics?
15
our own days as school students. We want to make sure that the
new generation can have and enjoy all those things that our own generation has seen as precious and enjoyable. But times do
change, and students’ needs and preferences change with them.
With the advent of knowledge technologies that allow an
individual to be an agent of her own learning, our ability to tell the learner what to study changes as well. In this editorial, I proposed
that we take a good look at our reasons and then, rather than
imposing one rigid model on all, restrict our requirements to a basis from which many valuable variants of mathematical
competency may spring in the future.
References
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42.
Cain, S. (2012). Quiet—The power of introverts in a world that can't stop talking. New York, NY: Crown Publishers.
Economic and Social Research Council, Teaching and Learning Research
Programme (2006) Science education in schools: Issues, evidence and proposals. Retrieved from http://www.tlrp.org/pub/documents/TLRP_Science_Commentary_FINAL.pdf
Garfunkel, S. A., & Young, G. S. (1998). The Sky Is Falling. Notices of the AMS, 45, 256–257.
Lave, J. (1988). Cognition in practice. New York, NY: Cambridge University Press.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York, NY: Cambridge University Press.
Organization for Economic Co-operation and Development, Global Sciences Forum (2006). Evolution of student interest in science and technology studies. Retrieved from http://www.oecd.org/dataoecd/16/30/36645825.pdf
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.
Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22.
Anna Sfard
16
1 I am talking about the Western society because due to my personal background, this is the only one I feel competent to talk about. The odds
are, however, that in our globalized world there is not much difference, in this respect, between the Western society and all the others.
2 Just to make it clear: the former argument that mathematics is not necessarily useful in every person’s life does not contradict the claim about its general usefulness!
3 As evidenced by numerous publications on the drop in enrollment to mathematics-related university subjects (e.g. Garfunkel & Young, 1998; Gilbert, 2006; OECD, 2006) and by the frequent calls for research projects that examine ways to reverse this trend (see e.g. TISME initiative in UK, http://tisme-scienceandmaths.org/), the decline in young
people's interest in mathematics and science is generally considered these days as one of the most serious educational problems, to be studied by educational researchers and dealt with by educators and policy makers.
The Mathematics Educator
2012 Vol. 22, No. 1, 17–38
Valerie Sharon is an Assistant Professor in the Department of Mathematics and Statistics at Sam Houston State University. Her research interests include
classroom discourse and mathematics teacher preparation.
The Roles They Play: Prospective Elementary
Teachers and a Problem-Solving Task
Valerie Sharon
The transition from learner to teacher of mathematics is often a difficult
one for prospective elementary teachers to negotiate. Learning to teach
necessitates the opportunity to practice the discourse of teacher of
mathematics. The undergraduate mathematics content classroom
provides a setting for prospective teachers to practice the discourse of
teacher through their interactions with each other while also learning the
mathematical concepts presented in class. This qualitative study sought
to examine what roles prospective teachers adopt while engaged in a cooperative problem-solving task. Discourse analysis was applied to
analyze the verbal interactions between three participants in a
mathematics content course. Key disruptions in the conversation
revealed instances of the fluid relationship between learner and teacher
of mathematics in the roles they adopted while solving an application
problem: self as learner-in-teacher, collaborator as learner-in-teacher,
and unlikely learner-in-teacher. The presence of this fluid relationship
led to the proposal of a model of learner-in-teacher-in-learner of
mathematics. This proposed model suggests that prospective teachers
have the opportunity to learn how to teach in and through each other
when given the opportunity to engage in dialogue with one another.
The shift from learner of mathematics to teacher of mathematics usually begins in the prospective elementary
teacher’s mathematics content classroom. Up to this point, the
prospective elementary teacher has taken part in the mathematics community as a learner of mathematics and now hopes to take on
the role as teacher of mathematics. In the mathematics content
classroom, the prospective teacher is expecting to learn both
mathematical concepts and how to teach them effectively. The individual in this transitory space is “learning about
becoming…by participation in practices” (Lerman, 2001, p. 88).
Valerie Sharon
18
The process of acquiring a new identity may be complicated by
past experiences with mathematics, especially if these experiences were not positive (Jones, Brown, Hanley, & McNamara, 2000).
Jones, et al (2000) described the transitions between the identities
of learner and teacher as a “means of reconciling the past with the
present and the future” (p. 2). It is important that mathematics teacher educators understand how prospective teachers form their
own identities as teachers of mathematics to develop an
efficacious curriculum that supports this reconciliation. Sfard (2003) viewed identity as a process of becoming part of
a community of discourse. This is in agreement with Gee’s notion
of discourse as an established set of social practices, including language, gestures, beliefs and ways of acting within the society
(Gee, 1989). This set of norms make up what he intentionally
referred to as Discourse, with a capital D. Our ways of being are
mirrored in our Discourse, which Gee referred to as our “identity kit”. This identity kit comes “complete with the appropriate
costume and instructions on how to act, talk, and often write, so as
to take on a particular role that others will recognize” (p. 7). The roles of teacher and learner in a mathematics classroom would
each have their own Discourse, with overlapping language and
ways of being, but distinct in ways that others recognize which role is being played. For example, the Discourse of Teacher often
differs from the Discourse of Student in regards to the intent of an
inquiry. Teachers tend to pose questions that they already know
the answer to, whereas students’ questions usually arise from a lack of knowledge. Both teacher and student may respond to each
others’ questions with explanations but the reasons for asking the
questions are unique to the role being played. Gee asserted that Discourse cannot be explicitly taught to the players, but must be
acquired “by enculturation (“apprenticeship”) into social practices
through scaffolded and supported interaction with people who
have already mastered the Discourse” (p. 7). The mathematics content classroom provides a setting for
prospective teachers to practice the Discourse of Teacher through
their interactions with each other. However, within this setting, the prospective teacher is also using the Discourse of Student to learn
the mathematical concepts presented in class. These two processes
of learning often result in conflicting identities as the individual pushes to become a teacher (Gee, 1989). The ongoing process of
The Roles They Play
19
becoming a teacher of mathematics is imbedded in the process of
learning mathematics, both of which take place within the individual. The process moves back and forth within the
individual, manifesting these two identities in the discourses of the
individual. The constant flux of these two identities leaves us
unable to extricate one from the other (Wang, 2004). Therefore, I propose we examine this transition in movement using the learner-
in-teacher-in-learner as our unit of analysis. In this manner,
perhaps we can catch a glimpse of the ongoing process of becoming a teacher while preservice teachers are learning and
participating in the mathematics community. The purpose of this
paper is to present a glimpse into how this transition might begin in the prospective elementary teachers’ mathematics classroom by
listening to the voices of prospective teachers engaged in a peer
problem-solving task. Analysis of the conversations will be used
to answer the research question: What roles do prospective teachers assume while involved in cooperative problem-solving?
Background Information
Sociocultural Theory
The foundation of this research study is entrenched in the
sociocultural theories of Vygotsky, who asserted that the process
of meaning making is mediated through the use of the symbolic tools of language and other cultural artifacts (Vygotsky,
1934/1986). According to Bruner (1997), this meaning making is
situated within the cultural context we find ourselves in and is facilitated by our social interactions with one another. The
transferability of cultural ways of knowing takes place in the
semiotic space between teacher and learner. Vygotsky described
this space as the zone of proximal development in which the discourse of a more knowledgeable person supports the learner’s
growth in knowledge (Lerman, 2001). This zone of proximal
development may emerge through the interactions between the teacher and the learner, but it may also arise through the
interactions between members of collaborative learning groups
(Goos, Galbraith, & Renshaw, 2002). Goos et al. analyzed transcripts of the conversations between secondary students
assigned to a group problem-solving task. They noted the
availability of a collaborative zone of proximal development when
students with complementary abilities monitored each other’s
Valerie Sharon
20
thinking. In other words, learning can take place whenever the
learner and a knower of a concept have the opportunity to interact together.
Opportunities for interactions between apprentices and ones
who have mastered the skills of a craft are situated within social
settings referred to as communities of practice (Greeno, 2003; Lave, 1991; Lave & Wenger, 1991; Wenger, 1999, 2000). Within
these communities, the learner is able to practice the skills of the
knower and gradually acquire the competencies that define the members of the community. The process of gaining these skills is
enveloped in the process of becoming a member of the community
(Lave, 1991; Lave & Wenger, 1991; Wenger, 1999, 2000). Lave (1991) asserted, “…without participation with others, there may be
no basis for lived identity” (p. 74).
The Discourse of Mathematics
The social semiotic perspective taken by Morgan (2006) used
a critical lens to describe the relationship between the learner, his
or her culture, and the discourses the learner participates in.
Morgan asserted that context consists of both the immediate realm of interaction and the broader culture in which the learner
participates. Careful consideration of the influences of the multiple
discourses a learner participates in may open “a crucial window for researchers on to the processes of teaching, learning, and doing
mathematics” (Morgan, 2006, p. 219). Morgan illustrated this
approach with examples of how the critical lens of social semiotics could be applied to student writing, especially in open-ended
questions on high-stakes tests. However, written text, Morgan
warned, provides only a partial image of the identity of the author,
leaving it up to the reader to create the rest. Morgan stated that the discursive interactions between two or more people are a richer
source of information concerning how individual identities are
formed. Through the process of collaborating and/or jockeying for positions, participants manage to negotiate their own identities in
relation to each other.
Kieran (2001) examined the discourse between pairs of adolescents assigned to work together to solve a series of
problems. Drawing from the field of applied linguistics, Kieran
created an interactivity flow chart to indicate the direction and the
presumed intent of the utterances spoken during the event (2001,
The Roles They Play
21
p. 202). The flow chart then was analyzed under the umbrella of
Vygotskian theory on the relationship between language and thought. This combined approach “makes explicit the integration
of the two in that both talking and thinking are considered
examples of communication – communication with others and
communication with self” (Kieran, 2001, p. 190). Sfard (2001) used the metaphor of learning-as-participation to
describe a pedagogical model that focuses specifically on the
interactions between individuals within a community of practice. The researcher working within this framework is concerned with
analyzing how the artifacts of individual learning are manifested
in the communications between members of a group. For instance, the use of a newly introduced mathematical term or procedure is
an indication that the student is learning how to use the tool
(Lerman, 2001). Sfard illustrated the application of a discursive
approach to analysis through an investigation into the benefits of collaborative efforts in learning mathematics. Utilizing the same
type of interactivity chart as Kieran (2001), Sfard exemplified this
illustration with two contrasting examples of non-productive discourse. Her analytical approach considered the focus, or
intended focus, of the discourse and the position of each
participant in response to that utterance. For example, seeking to learn mathematics by questioning or challenging the thinking of
others signals one’s intent to become part of the mathematics
community. By considering this interplay between the what, why,
and for whom features of an utterance, Sfard was able to explain why the tools that people use to communicate and the meta-rules
of discourse shape how we listen and learn in the classroom. Sfard
claimed that “careful analyses of diverse classroom episodes can be trusted to provide a good idea of what could be done in order to
make mathematical communication, and thus mathematical
learning, more effective” (p. 44). Discourse analysis can also be
used to explore how participants in the mathematics community co-create the identities of teacher and learner as they interact in the
classroom (Sfard, 2003).
Greeno (2003) recommended that researchers study how small group conversations contribute to the formation of identities in the
mathematics classroom. He detailed examples of how situated
research such as focusing on the conversations of cooperative problem-solving groups may reveal how students develop their
Valerie Sharon
22
identities as learners and knowers of mathematics. In the
mathematics content classroom, prospective teachers have opportunities to engage in problem-solving experiences while
working in cooperative groups. These group experiences create a
space for prospective teachers to practice communicating their
mathematical thinking and develop an understanding of how others learn mathematics. Within this space, there is a potential
curriculum for the mathematics teacher educator to immerse with
in an attempt to understand the formation of the Discourse of Teacher.
Learning to Teach
Nicol and Crespo (2003) explored how teacher educators can enable prospective teachers to learn how to teach through the
critical self-examination of initial field experiences. Nicol and
Crespo based their qualitative study on Wenger’s theory of learning and his ideas on identity formation, stating Wenger
maintained “…that learning involves the development of identity,
the changing of who we are, in the context of the communities of
practice that we participate in” (p. 374). Participants in the study conducted by Nicol and Crespo shared their positive and negative
experiences in the classroom, discussing their personal struggles
with mathematics and what they learned about how their students learn mathematics. For these prospective teachers, their identities
as learners of mathematics were deeply connected to their image
of themselves as teachers of mathematics by the desire to deepen their own understanding of the subject (Nicol & Crespo, 2003).
Jones, Brown, Hanley, and McNamara (2000) interviewed a
group of prospective elementary teachers in order to examine their
experiences as they were learning how to teach mathematics. The researchers’ analysis of interview data keyed in on how these
prospective teachers assimilated past and present encounters with
mathematics in order to describe themselves as future teachers. For example, teachers with negative experiences with mathematics
were able to reconcile the past with the future by using these
experiences as models for how not to teach. Jones, et al. stated that the interactions between past, present, and future perceptions of
mathematics in relation to the self play a major role in the
development of identity as teacher of mathematics. Amato’s
(2004) work on developing a liberating mathematics curriculum
The Roles They Play
23
for prospective elementary teachers was based on this same
interplay between past, present, and future. Amato used activities designed to build conceptual understanding of elementary school
mathematics as a way to change pre-service teachers’ beliefs and
attitudes toward mathematics. He asserted that prospective
elementary teachers needed to have meaningful experiences in mathematics to become effective teachers.
Perhaps the most compelling explanation of how individuals
learn how to teach was proffered by Freire (1970/2007) in his seminal piece, Pedagogy of the Oppressed. Freire described how
teachers who engaged in open dialogue, or praxis, with their
students escape the idea that teaching is merely the unidirectional transmission of knowledge. Instead, the teacher who engages in
praxis “…is no longer merely the-one-who-teaches, but one who is
himself taught in dialogue with the students, who in turn while
being taught also teach” (1970/2007, p. 80). Borrowing from Vygotsky’s model of mind-in-society-in-mind, the idea of learning
how to teach through the act of teaching can be described
metaphorically as learner-in-teacher-in learner. The question arises then, how can mathematics teacher educators facilitate the
transition from learner to teacher of mathematics before the
prospective teacher enters the elementary classroom? What lessons can prospective teachers learn about teaching mathematics
while they are learning mathematics content?
Theories of how the discourse of mathematics is learned
within the classroom culture dominated the literature on the teaching and learning of mathematics examined for this study
(e.g., Kieran, 2001; Morgan, 2006). Analyses of classroom
discourse focused primarily on pedagogical concerns dealing with the teaching and learning of mathematics in the primary and
secondary classroom. Literature on prospective teachers’
experiences in the undergraduate mathematics classroom
concentrated on how positive experiences with mathematics can change beliefs and attitudes toward mathematics (e.g., Amato,
2004). However, little research has been done on the
apprenticeship of prospective teachers into the Discourse of teacher of mathematics. As a mathematics teacher educator, I
recognize the need for communication in my classroom by
inviting my students to participate in the discourse of mathematics. Besides attempting to model productive discourse during whole
Valerie Sharon
24
class discussions, I also provide multiple opportunities for my
students to engage in peer problem-solving activities. I believe that these experiences fulfill the dual purpose of learning how to teach
mathematics while also learning mathematical concepts. However,
as a researcher in mathematics education, I wonder if my attempts
to promote communication in my classroom are sufficient to enable my students to develop their self-identity as a teacher of
mathematics. The purpose of this study was to analyze the roles
prospective teachers assume while engaged in problem-solving tasks and, in turn, shed light on how prospective teachers negotiate
the transition from learner to teacher of mathematics within the
culture of the mathematics content classroom.
Methodology
Framework
Ethnomethodology provided the framework for studying the interactions of the prospective teachers participating in this study.
According to Roulston (2001, 2004), the focus of
ethnomethodology has historically been on the analysis of the
ordinary discourse that takes place between individuals in everyday situations. This is in contrast to the usual ethnographic
approach of interviewing participants to ascertain what has taken
place in the past. Ethnomethodological approaches allow the researcher to witness the interactions between group members in
real-time versus relying on the memory and interpretation of
participants after the fact. Roulston (2004) explained that “researchers using ethnomethodological approaches to research
are keenly interested in how members’ knowledge is constructed
in and through talk and text” (p.140). Traditionally, researchers
adhering to this methodology have focused on interactions which take place in natural setting such as the work place or the
classroom. For this research inquiry, the conversations of one of
the cooperative learning groups in my mathematics classroom were recorded and analyzed to investigate the roles prospective
elementary teachers assume while engaged in a problem-solving
task.
The Setting and the Participants
This study took place on a satellite campus of a regional
university in the Midwest near the end of the spring semester of
The Roles They Play
25
2008. The participants in the study were all enrolled in a
mathematics content course for prospective elementary teachers taught by the researcher. This three-hour credit course dealt
primarily with rational number concepts. Two groups of students
volunteered to participate in this study by recording the audio of
the conversation shared while working collaboratively on a mathematics activity. One group, consisting of two female
students in the class, tended to request help from the
teacher/researcher whenever they struggled to answer a question. The second group, a triad of females, sought help from each other
when they could not solve a problem. For this paper, I have chosen
to discuss my analysis of the significant moments embedded in the conversations of the triad.
The group consisted of three female students who had worked
together on problem-solving tasks in the past. Cindy and Brooke
were both nontraditional students in their early thirties. The third student, Jenny, was in her mid-twenties at the time of this
investigation.. Brooke appeared to be the least confident in the
group of the three students and often voiced her frustration with mathematics during our whole-class discussions. The other two
had comparable abilities in mathematics which would seemingly
open a space for the emergence of a collaborative zone of proximal development as described by Goos, Gailbraith, and
Renshaw (2002). The presence of this zone of proximal
development may make it possible these two to support each
other’s thinking and learn from each other, much like the scaffolding a teacher provides for their students.
The activity involved counting and sorting M & M® candies
to examine the connections between ratios, decimals, and percents. The light-hearted nature of the activity hopefully eased the tension
students might have experienced about being recorded. However,
the triad encountered difficulties with the contextual problems
they were required to complete after sorting the candies. These disruptions in the flow of talk and how the speakers resolved
misunderstandings provided pieces to the puzzle of how
participants (re)negotiate self-identities and roles during the course of a conversation (Ten Have, 1999).
Valerie Sharon
26
Methods of Data Analysis
I used a modified version of an interactivity flowchart created (See Table 1) by Kieran (2001) to create a visual representation of
the mechanics of the conversation. The flow chart consisted of
arrows pointing up or down, depending on the intent of the speaker. If an utterance appeared to be in response to a prior
statement, then an upward pointing arrow was used to represent
the utterance. If the intent appeared to be soliciting a response, then a downward pointing arrow was used. These arrows could
point to self (personal channel) or to other (interpersonal channel).
According to Kieran the researcher bases these classifications on
the apparent intent of the speaker. I modified Kieran’s flowchart so that I could apply it to triadic conversations and omitted
additional classifications she had used.
In the excerpt displayed in Table 1, the participants were responding to a question in which they needed to find 30% of 86.
Jenny had decided to solve the problem by multiplying 86 by 0.3
instead of using a proportion. Although Cindy recognized that
Jenny’s procedure would yield the same answer, she suggested to Brooke (line 187), “let’s do it this way.” In line 189, Jenny is
speaking softly to herself as she works the problem her way;
therefore the upward pointing arrow is located in her personal channel, labeled J. Brooke and Cindy are working together to
solve the problem using proportions when Brooke stops Cindy on
line 190 to ask her “…how did you get that?” Since this statement was directed at Cindy, the arrow appears in the column labeled
BC with the arrow’s beginning located on the right to
symbolize the statement was made by Brooke. The statement is
labeled proactive, as indicated by the downward pointing arrow,
since Brooke is soliciting a response from Cindy. On line 191,
Jenny offers her answer up for approval. The statement is directed at both Brooke and Cindy, therefore downward pointing arrows
are placed in both interpersonal channels, JB and CJ .
Jenny redirects the question to Cindy (line 193) and the two
engage in an exchange that excludes Brooke until line 199. Their
responses (lines 194 and 195) to each other are labeled reactive as indicated by the upward point arrows in the far right column.
The Roles They Play
27
Table 1
Example of Flow Chart
Statement C BC B JB J CJ
187
C: Let’s do it this
way.
188 B: Yeah.
189
J: …times 86 (softly)
190
B: Wait..how did you get that?
191 J: 25.8?
192
B: Part? What’s the part?
193
J: Did you get 25.8 Cindy?
194
C: Hold on. I’m not there yet.
195 J: Okay. Sorry…
Using Kieran’s (2001) recommendation I then began to focus
on the action implied in the words of the utterance. Proactive
statements generally fell under the categories of seeking information in the form of help, verification, or justification.
Reactive statements were categorized as helping, justifying, or
simply responding with information. After characterizing the
actions of each utterance, I examined both the flow of the conversation and the inferred actions of each utterance in order to
focus on the nature of talk in terms of turn taking, corresponding
threads, topic management, disagreements, and repair as Zhou (2006) recommended. For example, there were times when a
reactive statement was made that also solicited a response, such as
Valerie Sharon
28
when a participant responded to an unexpected solution with a
request for an explanation of how the answer was obtained. These dual-coded statements usually resulted in a disruption in the
progress on the task while members of the group worked to
resolve the issue. Prior to the disruption, the conversation focused
on verifying solutions to the problems they were working on. The participants moved to the next problem at hand as long as their
solution pathways and/or solutions were the same. However, when
differences in their pathways or solutions became apparent, the conversation focused on resolving those differences. Examining
how the participants resolved these differences brought insight
into how this group of prospective teachers negotiated the roles of learner and teacher while engaged in problem solving.
Discussion
The triad spent approximately 27 minutes on the problem-solving task. Cindy initiated the activity by asking the other
students how many of each color candy they had in their
individual samples and determining the total counts for each color.
Throughout the conversation, Cindy played this role of leader by directing attention to the next problem on the page once issues
with the previous one were resolved. The mathematics was
relatively simple at first; converting ratios to decimals and percents. All three worked independently as they verified answers
and questioned the reducibility of a fraction.
Self as Learner-in-Teacher-in-Learner
The first conversational disruption occurred when Jenny
supplied an unexpected answer while the students were
simplifying the fractions they wrote for each color of candy as part of the total and converting each fraction to decimal and percent
form. The interactivity flow chart of the utterances prior to this
sequence showed arrows pointing up and down in all three interpersonal channels (see Appendix). All three students were
involved in the conversation as they worked in tandem, blurting
out answers to one another for verification.
88 J: I got like… 15 out of a hundred
89 C: Huh?
90 J: I got like point one five which is like fifteen percent.
The Roles They Play
29
91 B: (Oh yeah?)
92 C: Oh…for the next column?
93 J: Well…
94 B: Where are we at?
95 J: No…if…I took 13 …divided by 86. And I got
point one five one or something like that
96 C: Yess…for the decimal
97 J: Yes…
98 J: So …yeah…if you…
99 J: Oh it’s just ratio as a fraction…
100 J: No that’s right!
101 J: It would be 13 over 86.
102 J: I see what you’re saying …
103 J: I see!
104 J: Yes, as a decimal.
105 C: Okay!
106 J: Sorry.
107 C: That’s okay.
108 B: So what’s the decimal?
109 J: Point one five.
Jenny’s request for verification (line 88) resulted in a reactive
statement from Cindy that served the dual purpose of soliciting a response (line 89). Cindy’s statement was labeled with both up
and down arrows on the interactivity chart (see Appendix) and
signified a disruption in the flow of talk. Note that immediately
following the unexpected answer given by Jenny (line 88), Brooke is excluded from the repair of the disruption. She tries to break in
(lines 91 and 94), but neither Cindy nor Jenny respond to her
queries. Once the issue is repaired, Jenny responds to Brooke’s request by simply supplying the answer without explanation (line
109). This scenario repeated itself whenever Cindy and Jenny
came up with conflicting answers. Cindy and Jenny tended to rely
on each other for verification of their solutions, indicating that the two were confident in each other’s ability to solve these types of
problems. On the other hand, their apparent exclusion of Brooke
Valerie Sharon
30
from the verification process seems to indicate a lack of
confidence in Brooke’s ability. The other interesting story in this particular sequence is one of
metacognition. Notice the string of utterances Jenny makes after
Cindy made the comment “Yesss…for the decimal” (line 96).
Jenny was supposed to simplify the fraction first and then record the decimal form of the quantity in the next column. She goes
back and forth between the right and wrong answers, reacting to
her own statements, until she finally convinces herself that she was mistakenly finding the decimal instead of the fraction form of
the quantity. Jenny seeks help from self as learner and replies back
to self as teacher. Through this series of utterances we see a story of self as learner-in-teacher-in-learner.
Collaborating as Learner-in-Teacher-in-Learner
According to NCTM (2000), an effective teacher of mathematics is able to “analyze what they and their students are
doing and consider how those actions are affecting students'
learning” (p. 18). Both Cindy and Jenny took on the identity of
teacher by monitoring each other’s work, as well as Brooke’s. However, there were also instances in which the roles of teacher
and learner merged as Cindy and Jenny supported each other’s
thinking. One such instance began when the triad encountered a rather long disruption. The students were attempting to solve a
problem in which they had to deduct ten percent from the total
number of candies (86) and then take another thirty percent off of the remaining amount. Jenny explained to the others they could
eliminate an extra step by calculating 90 percent of the total
instead. Brooke seemed confused by the plan, stating that she had
“…no idea obviously what she does.” Although Cindy initially suggested that Jenny “…do it that way and then we’ll see if we
come up with the same answers…,” she decided to follow suit and
proceeded to calculate ninety percent of 86. However, Cindy did not quite understand how to complete the problem once this issue
was resolved.
274 C: Minus 86…right?
275 J: No…I didn’t do it that way.
276 B: So…now you take 86 minus 77.4…So is that what
you’re saying?
The Roles They Play
31
277 J: I got 77.4…okay? That’s what you have left…and
you saved thirty percent of that to take home.
278 C: So we ate 8.6. Is that what you got?
279 J: No…this is what we have left (77.4)…and we’re
taking part of it home.
280 C: So we are taking thirty percent of the 77.4?
281 J: So what we could have eaten was 54.8. I’ll show
you what I did.
282 C: So we have to figure out what 30 percent of 77.4 is?
283 J: Yes…which is 23.22. So that’s what you’re taking
home to your husband…or to your kids…or to a
friend.
284 C: Thirty percent of …got it…..23.22?
285 J: That’s right.
286 C: Okie dokie.
287 J: So then…so then…then it wants to know how much you could have eaten. Okay. You had 77.4 and you
take 23.22 home…so what would…what could
you…?
288 C: So you have to subtract it.
289 J: We subtract it.
Brooke appeared to be an outsider during most of this first sequence, but she did manage to interrupt the discussion with a
question concerning the final answer. Jenny seemed about to
respond to Brooke when Cindy asked for verification of the next
step (line 274). Cindy wanted to subtract their previous answer from the total, which would have negated the advantage of taking
ninety percent of the total instead of ten percent. During this
sequence, Jenny explained the rationale behind each step in her procedure. Her approach seemed to support Cindy’s thinking,
enabling her to understand how to solve the problem before Jenny
had finished explaining the procedures. In fact, near the end of the
sequence, Cindy was explaining the steps and Jenny was confirming them (lines 284 – 289). This series of back and forth
responses illustrates what Goos, Galbraith, and Renshaw (2002)
referred to as mediated thinking. Within this collaborative zone of proximal development Cindy and Jenny shared, Cindy was able to
Valerie Sharon
32
correct her own error. The scaffolding approach taken by Jenny is
part of the repertoire of an effective teacher (NCTM, 2000). Brooke re-entered the conversation soon after by repeating the
same error as Cindy had on line 274. Jenny offered a quick
explanation, but apparently noticed that Brooke was even more
confused than before (line 307). Instead of simply supplying the answer and moving on to the next problem on the sheet, Jenny
tried a more dialogical approach by asking supportive questions.
307 J: That really seems to confuse you even more.
308 B: Well, umm…
309 J: This is what you have…you’re taking that
home…so how much did you eat in class?
310 J: If this is your total and you took that part
home…how much is left for you?
311 J: (long pause) …you know how you got there..
312 B: But if you add those…if you add all those up
together it doesn’t equal
313 C: (it adds up to 86)
314 J: Yes…
315 B: It doesn’t add up to 86.
319 B Yeah, but 54.2 plus 77.4 doesn’t add up to 86…
320 C: Because….we didn’t do the ten percent. Right? We didn’t do the ten percent. Right?
Cindy briefly re-entered the conversation (line 320) by
offering a possible explanation for why the quantities (54.2 +
23.22 + 77.4) did not add to 86. However, Jenny began to doubt her answer (line 328). Neither she nor Brooke seemed able to
explain why it might be incorrect.
328 J: Do you guys think I did it wrong?
329 B: Well…I just…no….I don’t…understand
330 J: Well…if you do…just tell me what I…I may
have…I may have done it wrong.
331 B: I don’t know…I don’t know…Well…I just don’t
understand.
332 J: That’s the way I understood it.
The Roles They Play
33
333 C: If we gave away 8.6 of them…cause that would be
ten percent. So we’re going to save thirty percent of…77.4…Which is…twenty three point two two
334 J: MmmHmm
335 C: How many could we eat in class today? So…23.22
plus…
336 J: I see what you’re saying
337 J: I’m not sure why it doesn’t add up…8.6 and 77.4
should add up to 86
338 C: (8.6)
339 C: Right…so
340 J: Not the 23
341 C: Right
342 B: B…but if you had…
343 J: Cause the 23.22 is already included in your 77.4
344 B: Oh…okay…hold on
345 C: So this is what we took home…No what we gave
her
346 J: Because you took thirty percent of a different total
347 C: Yeah…
348 J: That’s why it’ not adding up…
This time Cindy supplied the scaffolding to support Jenny’s
thinking. Through the scaffolding provided by their collaborative
zone of proximal development, Cindy and Jenny were learning how to teach and learning how to communicate their mathematical
thinking. Together they were learning the discourse of
mathematics.
The Unlikely Learner-in-Teacher-Learner
Throughout this experience, Brooke seemed to remain frozen
within the position of learner of mathematics. At times she was an outsider to the conversations around her despite attempts to join in
the conversation. For example, while examining the interactivity
chart I noted five instances where her attempts to seek help were
ignored by the other two members of the group. Several other utterances she made were incomplete, cut off by one of the other
Valerie Sharon
34
two speakers. The fact that she seemed to struggle more with the
mathematics than the other two may explain why she was responsible for less than one- fourth of the total statements made
during the conversation. One possible explanation for her lack of
engagement in the conversation could be due to the silencing
effect mathematics may have over those who do not understand its discourse (Walkerdine, 1988, 1997; as cited by Forman, 2003).
Walkerdine (1985) described the effects anxiety imposes on many
women in academic settings, stating that the person may come to believe “…that if they open their mouth, they will ‘say the wrong
thing’…” (p. 226). However, as I listened to the conversations of
these three students, I began to explore the possibility that the subject labeled as ‘learner’ was teaching the other how to teach.
What lessons was Brooke teaching to Jenny as she pushed for an
understanding of why the quantities on hand did not add up as
expected? Her inability to understand forced Jenny to think of another way to explain the mathematics and it also forced her to
think about her mathematical thinking. As Wang (2004) stated, the
“subject-in-process is intricately related to subject-in-relation because the fluidity of self is enabled by responding to the other”
(p. 120). Within the culture of the mathematics classroom, the
prospective teacher has the opportunity to learn how to teach through her interactions with others.
Conclusion
This study is limited by both the duration and the number of participants. Although the analysis of their conversation supports
the view that prospective teachers are able to practice the
Discourse of Teacher while learning mathematics content, this
desired outcome may not always come to fruition. Other groups of prospective teachers may only engage in the Discourse of Student,
depending on the official teacher in the classroom for explanations
instead of asking/supplying explanations to each other. Mathematics educators need to encourage cooperative learning
and provide opportunities for the prospective teacher to practice
communicating his or her own mathematical thinking in order for the mathematics content classroom to serve as an apprenticeship
into the mathematics community. The discourse of a college
mathematics classroom is a place where prospective teachers can
learn to talk the talk of teacher of mathematics, thus assembling
The Roles They Play
35
the identify kit Gee (1989) refers to in his definition of Discourse.
However, further research needs to be done on ways to initiate the transition from learner to teacher of mathematics during the time
prospective teachers are participating in the mathematics content
classroom. For example, what types of tasks will encourage
prospective teachers to share their mathematical thinking with each other? In what ways can mathematics educators foster
collaboration and create an environment where participants feel
safe to justify their answers to mathematical problems? This research study was an attempt to understand how
prospective teachers negotiate the transition from teacher to
learner of mathematics. The socio-cultural theories of Vygotsky (1934/1986) assert that all learning takes place through the use of
language within a cultural setting. Lerman (2001) suggested
applying Vygotsky’s mind-in-society-in-mind unit of analysis to
the learning that takes place in the mathematics classroom. He proposed we view this learning within the framework of learner-
in-mathematics-in-classroom-in-learner. The ongoing process of
becoming a teacher of mathematics is imbedded in the process of learning mathematics, both of which take place within the
individual engaging in the discourse of mathematics. The process
moves back and forth within the individual, manifesting these two identities in the discourses of the subject, as illustrated in the
conversation of these three pre-service teachers. The overlapping
movement of these identities leaves us unable to extricate one
from the other. Therefore, I propose we examine this transitory formation of identity using the learner-in-teacher-in-learner as our
unit of analysis. In this manner, perhaps we can catch a glimpse of
the ongoing process of becoming a teacher while prospective teachers are learning and participating in the mathematics
community.
References
Amato, S. A. (2004). Improving student teachers’ attitudes to mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics, 2, 25–32.
Bruner, J. (1997). Celebrating divergence: Piaget and Vygotsky. Human Development, 40(2), 63-73.
Valerie Sharon
36
Forman, E. (2003). A sociocultural approach to mathematics reform: Speaking, inscribing and doing mathematics within communities of practice. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 333–352). Reston, VA: National Council of Teachers of Mathematics.
Freire, P. (1970/2007). Pedagogy of the oppressed. New York, NY: Continuum.
Gee, J. P. (1989). Literacy, discourse, and linguistics: Introduction. Journal of Education, 171(1), 5–17.
Goos, M., Galbraith, P., & Renshaw, R. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49, 193–223.
Greeno, J. G. (2003). Situative research relevant to standards for school
mathematics. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 304–332). Reston, VA: National Council of Teachers of Mathematics.
Jones, L., Brown, T., Hanley, U., & McNamara, O. (2000). An enquiry into transitions: from being a ‘learner of mathematics’ to becoming a ‘teacher of mathematics’. Research in Education, 63, 1–10.
Kieran, C. (2001). The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges. Educational
Studies in Mathematics, 46, 187–228.
Lave, J. (1991). Situated learning in communities of practice. In L. B. Resnick, J. M. Levine, & S. D. Teasley (Eds.). Perspectives on socially shared cognition (pp. 63–82). Washington, DC: American Psychological Association.
Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University.
Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to
studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46(1/3), 87–113.
Morgan, C. (2006). What does social semiotics have to offer mathematics education research? Educational Studies in Mathematics, 61, 219–245.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.
Nicol, C. & Crespo, S. (2003). Learning in and from practice: Pre-service teachers investigate their mathematics teaching. International Group for the Psychology of Mathematics Education. (ERIC Document Reproduction No.
501041)
Roulston, K. (2001). Data analysis and ‘theorizing as ideology’. Qualitative Research, 1(3), 279–302.
The Roles They Play
37
Roulston, K. (2004). Ethnomethodological and conversation analytic studies. In K. deMarris, & S. D. Lapan (Eds.) Foundations for research: Methods of inquiry in education and the social sciences (pp. 139–160). Mahwah, NJ: Erlbaum.
Simon, M. (1994). Learning mathematics and learning to teach: Learning cycles
in mathematics teacher education. Educational Studies in Mathematics, 26(1), 71–94.
Sfard, A. (2001). There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical thinking. Educational Studies in Mathematics, 46, 13–57.
Sfard, A. (2003). Balancing the unbalanceable: The NCTM standards in light of theories of learning mathematics. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for
school mathematics (pp. 332–392). Reston, VA: National Council of Teachers of Mathematics.
Ten Have, P. (1999). Doing conversation analysis. London, England: Sage.
Vygotsky, L. (1986). Thought and language (A. Kozulin, Ed). Cambridge, MA: MIT Press. (Original work published 1934).
Walkerdine, V. (1985). On the regulation of speaking and silence: Subjectivity, class and gender in contemporary schooling. In C. Steedman, C. Urwin, & V. Walkerdine (Eds.), Language, gender and childhood (pp. 203–241).
London, England: Routledge & Kegan Paul.
Walkerdine, V. (1988). The mastery of reason: Cognitive development and the production of rationality. New York, NY: Routledge.
Walkerdine, V. (1997). Redefining the subject in situation cognition theory. In D. Kirschner, & J. A. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives (pp. 57–70). Mahweh, NJ: Erlbaum.
Wang, H. (2004). The call from the stranger on a journey home: Curriculum in a third space. New York, NY: Peter Lang.
Wenger, E. (1999). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University.
Wenger, E. (2000). Communities of practice and social learning systems. Organization, 7(2), 225–246.
Zhou, Q. (2006). Application of a discourse approach to speaking in teaching of conversation. US-China Education Review, 3(3), 57–63.
Valerie Sharon
38
Appendix
Interactivity Flow Chart for Triadic Communication
Line # C BC B JB J CJ
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
The Mathematics Educator
2012 Vol. 22, No. 1, 39–62
Discoursing Mathematically: Using Discourse
Analysis to Develop a Sociocritical
Perspective of Mathematics Education
Aria Razfar
This article explores how the concepts of discourse and its
methodological extension discourse analysis can help mathematics
educators re-conceptualize their practices using a sociocultural view of
learning. It provides conceptual and methodological tools as well as
activities that can be helpful in mathematics methods courses and professional development sessions aimed at developing a more situated
and social view of mathematical discourse and its relationship to student
learning, particularly how mathematical discourse relates to Language
Minority Students (LMS). In this article, I discuss the main features of
discourse as a framework for mathematics educators and how
participants in a cross-site research center collectively engaged and
developed a more robust understanding of the significance of discourse
and discourse analysis for understanding mathematics as a sociocultural
practice. This article describes learning activities whose instructional
goal is to develop a sociocritical understanding of language and
mathematics. The activities presented here can be adopted as a model for engaging mathematics teacher educators and mathematics teachers to
deepen their understanding of the inextricable link between language
and mathematics, and of mathematics as a cultural and political activity.
The preparation of teachers for linguistically and culturally
diverse populations has been the subject of a growing body of
research and discussion over the last two decades (Brisk, 2008; Cochran-Smith, Fieman-Nemser, McIntyre, & Demers, 2008). The
relatively recent emphasis on this issue in the research community
has taken place primarily because of the rapidly changing
Aria Razfar is an Associate Professor of Literacy, Language and Culture at the University of Illinois at Chicago. His research is grounded in sociocultural and critical theories of language, learning, and development. He teaches courses on linguistics for teachers and currently directs several nationally funded projects aimed at training teachers of English learners to develop academic literacy
practices through mathematics, science, and action research.
Aria Razfar
40
demographics in the nation’s student population accompanied by
the persistent disparities that exist in educational achievement, resources, and life opportunities between Language Minority
Students (LMS) and their majority counterparts. According to a
report from the National Center of Educational Statistics (NCES,
2010), in 2008, 21% of all children aged 5 to 17 spoke a language other than English at home. As it stands now, there are an
insufficient number of teachers who are adequately and
appropriately prepared with the skills and knowledge to teach LMS (Lucas & Grinberg, 2008). Since achievement in
mathematics is highly dependent on teachers’ capabilities, the
under-preparedness of teachers does not bode well for LMS who are not receiving the support they need to perform well in
mathematics (Gutiérrez, 2002).
Although research has pointed to the importance of
linguistically responsive learning environments for LMS in mathematics (e.g., Khisty, 2002; Moschovich, 1999a) and to
practices teachers can use to facilitate LMS learning of
mathematics (e.g., Moschovich, 1999b), there still remains a question of how to prepare and support teachers in creating such
learning environments. In fact, almost no research has been
conducted on the preparation of teachers to teach LMS (Lucas & Grinberg, 2008; Zeichner, 2005). This is particularly true in the
domain of mathematics, as most research on mathematics teacher
preparation has focused on preservice teachers’ knowledge and
beliefs about mathematics, their applications of constructivist principles, and understanding of problem-solving processes and
skills (Lester, 2007; Llinares & Krainer, 2006). Discussions in
mathematics education have not given sufficient attention to developing teacher knowledge related to teaching LMS, and most
mathematics teacher educators do not have the background
knowledge necessary to prepare teachers to teach mathematics to
LMS. As a result, preservice teachers enter the profession having little knowledge about the needs, resources, and supports required
to effectively teach mathematics to LMS (Chval & Pinnow, 2010).
Teachers must have a deep knowledge of the linguistic and cultural demands that are unique to the teaching and learning of
mathematics. This becomes more important when students speak
(or are learning) more than one language (Valdés, Bunch, Snow, Lee, & Matos, 2005). Although the importance of language and
Discoursing Mathematically
41
mathematical discourse1 in the process of teaching and learning
mathematics has gained considerable attention in recent years both in monolingual (e.g., Cobb, Yackel, & McClain, 2000) and
bi/multilingual (e.g., Moschovich, 2007; Setati, 2005) contexts, it
has not been given sufficient attention in teacher preparation
programs. Substantial language and discourse content is absent in most mathematics teaching courses for preservice teachers
because language is typically treated as a subject in teacher
education and is separated from the content subjects. In addition, mathematics teacher educators need professional development in
order to include language and discourse issues in their teacher
preparation courses. In this article, I provide conceptual and methodological tools
as well as activities that can be helpful in mathematics methods
courses and professional development sessions aimed at
developing a more situated and social view of mathematical discourse and its relationship to student learning, particularly how
mathematical discourse relates to LMS. I explore how the
concepts of discourse and its methodological extension discourse analysis can help mathematics educators re-conceptualize their
disciplinary field and student learning.
First, I provide the context in which these methodological and conceptual tools were developed. Next, I outline some of the main
features of discourse as a framework for mathematics educators
and teachers. Drawing on Gee’s definitions of primary and
secondary discourses (Gee, 1996) as well as the material, activity, semiotic, and sociocultural (MASS) dimensions of discourse
analysis and learning (Gee & Green, 1998), I show how the
concepts of discourse and discourse analysis are particularly relevant in mathematics education and, more specifically,
mathematics teacher preparation. I conclude with implications for
mathematics teacher preparation and directions for future research.
Context
In 2004, as part of NSF’s Centers for Learning and Teaching
(CLT) initiative, the Center for Mathematics Education of Latinas/os (CEMELA) received a five-year grant to train doctoral
students across four campuses who would focus on the
intersections of mathematics, language, and culture especially in
the context of bilingual, Latina/o children in urban settings. At two
Aria Razfar
42
of the sites, CEMELA conducted after-school mathematics clubs
to develop mathematics literacy based on the learning principles described later in this article and community expertise. As part of
their training/socialization, doctoral students, faculty across
disciplines (mathematics, mathematics education, literacy), and
practitioners participated in summer intensives dedicated to the topics of mathematics and discourse (total of 65 participants). It
was the central subject of a six day intensive “school” held at the
University of Illinois at Chicago in the summer of 2007. Over 97% of the participants reported that the school “helped develop skills
to analyze discourse processes” (LeCroy & Milligan, 2007, p. 28)
especially as they relate to the mathematics education of bilingual students. One participant commented, “I learned about discourse
analysis and aspects of bilingualism that apply in the classroom”
(p. 29) According to other students, the activities were “well-
developed to learn difficult concepts such as Gee’s discourse” (p. 32), they were “useful” (p. 33), and “more” (p. 33) activities like
this should be done. More specifically, the Baseball Language
Learner (BLL) activity, which I will discuss in more detail later, was discussed as the most effective for making the distinction
between language and discourse clear, “most helpful were the
[Baseball Language Learner] activity combined with the ideas of diverse communities that consider language and cultural context.”
In this article, I will provide a detailed account of how the
participants and I engaged in discussions of discourse, discourse
analysis, and mathematics through the BLL activity. I and other doctoral fellows, who are now in faculty positions,
have continued to use these learning activities in a variety of
teacher education and bilingual/ESL courses for the purposes of developing teacher awareness about the relationship of language
and mathematics. In the following sections, I discuss the four
fundamental tenets of discourse and discourse analysis that drive
this professional development and illustrate how the issues were discussed at the summer intensive.
From Language to Discourse: Four Fundamental Questions
Many teachers, including doctoral students, came to the
discussion of language and mathematics with “folk theories” of
what counts as language. When asked to define “language” there
was unanimous agreement in that language is either the spoken or
Discoursing Mathematically
43
written word for the purpose of communication. In the following
section, I show the activities and process that the participants undertook in order to reframe this intuitively yet deceptively
“correct” view of language and how they progressively moved
towards less intuitive yet more profound and critical notions of
language as “discourse.” I show how the participants and I moved from “what people say” to critical issues of “values, beliefs, and
power relations.” Given that my research questions and projects
are situated in Latina/o urban settings with large populations of LMS, the importance of teacher beliefs about the nature and
function of language in relation to mathematics has significant
implications for student learning, instruction, and ultimately outcomes (Razfar, 2003).
1) What do people say?
In examining the salience of discourse and discourse analysis for the mathematics education communities, it is important to
consider some of the fundamental principles and questions that
guide discourse analysts as they look at transcripts of talk
irrespective of their field or discipline. Of central concern to practitioners and researchers is that discourse analysis is one of the
most important tools for organizing and assessing learning and
development especially from a cultural historical perspective. The first and perhaps most obvious question is, what do people say?
Linguists have traditionally referred to this as the code or the more
formal and explicit features of language, namely the structure. While for linguists these utterances do not typically take place in
naturalistic situations, the idea that this is the most descriptive
aspect of language form applies, and all discourse analysis
necessarily accounts for this dimension. More specifically, this refers to the most apparent features of language such as sounds,
pronunciation (phonetic and phonological aspects), words (lexical
choice), morphology, and grammar (syntax). If this dimension were extended to typical interactions, this would include the
spoken utterances attributed to each speaker and the obvious turns
that speakers take within an episode of talk. In order to make this point I provided a transcript of talk to all participants. The first
snippet of discourse that is presented is strictly transcribed based
on spoken words (code) and all performative aspects are missing.
It is an interaction between Juan (denoted J in the transcript), one
Aria Razfar
44
of the kids in one of the after-school clubs, a graduate assistant
(denoted G in the transcript), and a mechanic (denoted M in the transcript) about the hydraulics of a car:
1
2
3
G: So if you wanted to make a car a low-rider?
[0.5 second pause] Like make it so that it is
lower.
4
5
6 7
8
9 10
11
12
13 14
15
16 17
18
19 20
M: On a regular car you would actually have to
do a lot of suspension work. One of the first
things that you want to do- there are different things that you want to do. You can start with
airbags where you compress the air. You
know, and then they’re actually bags itself where you just compress the air, it deflates
‘em and increases the air and that’ll make the
car go up and down. The other one hydraulics
and that’s actually based on fluid. Fluid is actually what’s going to go through there. It’s
going to actually put pressure on the cylinder.
Once the fluid puts pressure on the cylinder, the cylinder will go up. [inaudible] makes the
cylinder go down. So basically you have
those two. Do you want to go with airbags or do you want to go with hydraulics?
While Juan is present in the interaction, he is not visible in
the transcript. After reviewing this clip, and discussing it in
small groups, participants drew conclusions based on the code available in the transcript. When this brief exchange was
analyzed, participants concluded that there were only two
speakers: One was asking a question, and the other was responding. One speaker is or appears to be clarifying the
initial question (line 1) where the concept of “low rider” is
extended, “like make it so that it is lower” (lines 2–3).
Structurally, everybody agreed that the words being used were English and followed normative rules of English morphology
and syntax. Some even used the transcript to identify various
parts of speech (nouns, verbs, prepositions, etc.), word order, subject/object functions, modals, and even the logical
connectors. Participants arguably used the more common/folk
approach to what counts as language and drew typical and
Discoursing Mathematically
45
uncontroversial conclusions from the code. The following
sections illustrate why this approach is not sufficient and how the transcription exercise made this visible to participants.
How do people say what they say?
If the analysis were to stop here, it would clearly be insufficient in terms of the second and third questions that are
central to discourse analysis which are: How do people say what
they say? And what do they mean? The second question has historically been the domain of applied linguists and sociolinguists
and is traditionally referred to as performance. In general, this is
actual language use in real communicative situations and is
concerned with how speakers draw on contextual cues to communicate. In addition, performance also consists of prosodic
dimensions of language use like tone, intonation, loudness, pitch,
and rhythm. This can also include gestures, facial expressions, and other non-verbal acts which make transcription quite challenging
and impossible without video. Prosody offers an initial glimpse
into the affective stances speakers assume within discourse
frames. Participants were then asked to reflect on a different transcription of the same speech event that took into account the
performative qualities. Lines (1-3) from the previous transcript are
“re-presented” below (G=Graduate Assistant; J=Juan):2
G: So if you wanted to make a ca:::r (.5 sec) a (.5 sec) a
low rider (rapid voice, falling intonation), li:ke (.5 sec)
ma:ke it so that it is lower.
J: [Juan nodding] [yeah]
After reflection and discussion, several issues became clear. First, what initially looked like a question followed by a
clarification for the mechanic appears to be some type of
scaffolding directed at Juan, a student in the after-school club. In comparing the first transcript with the second, everybody noticed
the invisibility of Juan in the first transcript, which was strictly
code. As the discussion moved from an analysis of code to an
analysis of performance, Juan’s role in the interaction became more apparent. One participant made the following observation,
“in the first transcript there were only two speakers, but in the
second there are three…we couldn’t see the non-verbal.” Several talked about the importance of video, but even video can be
Aria Razfar
46
limited as I discuss in the next section on meaning. The
overlapping talk whereby the graduate student assumes the floor interspersed with non-verbal acknowledgements from Juan is
critical to the analysis. Furthermore, there is clear hedging
(deliberate pause followed by a rapid voice and falling intonation)
surrounding the word “low rider.” As the participants moved in this direction, there were more questions about the meaning and
functions of the words described in the initial phase of the
analysis. The main question that was raised was, “If Juan had already acknowledged the use of the term low-rider and from
previous turns and interactions all participants use the term freely,
what is the purpose of the ‘clarification’?”
3) What do people mean?
This question led us to the central and arguably most contested
interpretive question for discourse analysts and that is, what do people mean? If one assumes that meaning is fixed, absolute, and
independent from the situation in which it occurs, then there is
little argument; however, meaning is situated and necessarily
dependent on the footing of the participants within a particular frame (Goffman, 1981).
3 The question that arises: Does the
graduate assistant in the interaction, using the term “low rider,”
share the same footing with the other participants? In addition, participants invoke intentions and purposes that are often hidden
from the immediate and apparent discourse. It is essential for us to
historically locate the term “low rider” as used by the immediate participants and well beyond, in order to grapple with issues of
purpose and intention. Speakers often draw on multiple signs and
symbols in multiple modalities available to them in order to
achieve higher degrees of shared meaning or what Bakhtin called intersubjectivity (Holquist, 1990).
4
From the above example, one might argue that the hesitation
surrounding the word “low rider” is not about referential meaning or shared understanding, but more about speech rights and
identities indexed by the use of the term. Does the speaker feel a
right to freely use the term “low rider”? Does the speaker have a discourse affinity with the term? One participant noted, “I don’t
think she is comfortable using the term [low rider]…maybe she is
nervous.” The issue of speech rights has serious implications for
discourse and identity. It impacts the what, who, and how of
Discoursing Mathematically
47
allowable discourse. In this case, the graduate student is a White
female, who although fluent in Spanish and having lived in a Latin American country for a long period of time, appeared to be
hesitant and aware that she could be encroaching upon implicit
cultural boundaries. This conversation proved to be the most
unsettling in terms of participants’ assumptions about language, discourse, and identity; nevertheless, it made issues of meaning,
intention, and identity more visible. One participant commented,
“Discourse is more than just words, it is who we are and who we get to be.” Thus, meaning-making is necessarily embedded within
the values, beliefs, and historical relations of power; an aspect of
discourse that Gee has often referred to as Discourse (Gee, 1996). This dimension is often beyond the apparent text and requires
deeper ethnographic relations between the researcher and
participating community members in order to conduct more
authentic analysis of meaning-making. This leads to the final premise of what constitutes discourse.
4) How do values, beliefs, social, institutional relations of
power mediate meaning?
This question constitutes the critical dimension, and its
importance with respect to discourse analysis cannot be
underscored enough especially vis a vis mathematical discourses. It is the central question when it comes to understanding how
some practices are more valued, privileged, and attributed greater
legitimacy than others. This is particularly salient when dealing with non-dominant dialects, languages, and cultures that are
prevalent in urban settings. Issues of racial, economic, and gender
inequity and access are no longer variables that can be placed on
the periphery of analysis, but rather take on a central role. Identities and ideologies become fore-grounded in the analysis of
talk and text. Street and Baker (2005) call this the ideological
model of numeracy which is an extension of Street’s ideological approach to literacy. In the context of the questions posed by
researchers and others looking at mathematical and scientific
practices in non-classroom settings, it is particularly salient when one considers what gets counted as legitimate mathematics.
The process of interpreting the meaning-making of people is
continuous, subject to constant revision, and dependent on how
much of an ethnographic perspective the analysis presumes. A
Aria Razfar
48
teacher as an ethnographer (Gonzalez, Moll, & Amanti, 2005) is a
powerful metaphor that brings together the aims of discourse analysis and the practitioner in the classroom. Given the emphasis
on meaning-making, mathematical practices are also viewed in
this light. In the remainder of this article, I will explore how
discourse analysis can be a valuable tool for understanding mathematical practices as situated problem solving that largely
depend on local cultural contexts and symbol systems.
Learning as Shifts in Discursive Identities: Primary versus
Secondary Discourses
At this point in the discussion within the professional
development, an argument in favor of “discourse” versus narrow conceptions of “language” had emerged. In external evaluations
conducted after the session, nearly all participants “strongly
agreed” that the transcript exercise was an effective tool for this purpose. When participants considered the four
dimensions/questions of discourse analysis raised above, it became
evident that the notion of discourse (as opposed to “language”)
afforded a more holistic view of human meaning-making. Yet, the connection to learning, teaching, and instruction is not self-
evident. One participant commented, “So we analyze all of this
discourse, but how does it help a teacher in the classroom…and where’s the math?” Discourse analysts have long argued that
learning itself is best understood as shifts in discourse over time,
especially the appropriation of discursive identities (Brown, 2004; Rogoff, 2003; Wortham, 2003). The critical point here is “over
time” and according to Brown, Reveles, and Kelly (2005),
“research in education needs to examine identity development,
learning, and affiliation across multiple timescales.” (p. 783). Understanding how discursive identities change over time is
difficult for participants to appreciate in a short course or
professional development session (however intensive). Doctoral fellows and practitioners, however, were able to develop such a
perspective over the course of four years of ethnographic work in
the after-school clubs. As practitioners and researchers embrace the notion of
learning as shifts in discursive identities, a couple of questions
remain: What kinds of discourse constitute mathematics? More
generally, where do formalized discourses (i.e., those that are
Discoursing Mathematically
49
learned in schools) fit in relation to everyday discourses?
Although human beings undergo a life-long process of language socialization, not all discourses are equivalent both in terms of the
process and purpose of appropriation. Discourses that seem more
natural or are appropriated as a result of spontaneous interaction
are distinct from those that are appropriated through participation in formalized institutional settings. For example, the learning of
one’s native, national language (e.g., Spanish, English, etc.) is
different from learning biological nomenclatures or geometric theorems.
With regards to this distinction there is a clear delineation
between primary discourses and secondary discourses (Gee, 1996). In the fields of cognition and second language acquisition
(SLA), one of the most contentious arguments has been the
distinction between learning and acquisition (Krashen, 2003;
White, 1987). Learning is generally conscious, formal, and explicit, while acquisition is subconscious, informal, and implicit.
In contrast to most cognitivists and SLA perspectives who locate
both processes within the individual, Gee takes a more situated and sociocultural view on the issue; he argues that acquisition, or
primary discourse, is good for performance, and learning is good
for meta-level knowledge (secondary discourse). This distinction is important as one considers the features of what constitutes
mathematical discourse in relation to learning in informal and
formal settings. According to Gee (1996), primary discourses “are
those to which people are apprenticed early in life during their primary socialization as members of particular families within
their socio-cultural setting” (p. 137); and secondary discourses are
“those to which people are apprenticed as part of their socialization within various local, state and national groups and
institutions outside early and peer group socialisation, for
example, churches, schools, etc.” (p. 133). Secondary discourses
have the properties of a more generalizable cultural model, are more explicitly taught, and are less dependent on the immediate
situation for access by a larger audience.
If algebraic discourse is considered as an example of discourse appropriated through school, then the symbol x in x+2=7 is
understood by algebraic discourse community members as
representing the unknown within an equation as opposed to an arbitrary letter. Members of this community may also assume that
Aria Razfar
50
in this case x has a single value and they must follow certain rules
to find the answer (all school like practices). Furthermore, for those who have appropriated geometric discourses such as the
Pythagorean Theorem ( 222 zyx ), the x and the y represent the
two adjacent sides that form the right angle (or legs) and the z represents the hypotenuse. Thus, mathematical symbols gain
specialized meanings within multiple domains of mathematics.
These literacies serve as mediational tools in novel problem-
solving situations, and literate discourses tend to be more generalizable problem-solving tools (Sfard, 2002).
These types of “formal” mathematics discourses would qualify
as secondary discourses. This does not, however, mean that primary discourses (especially informal numeracy and
mathematical practices) are separate and unrelated to the
development of secondary discourses (formal and specialized
mathematical practices). Given that learning from a sociocultural point of view is historically continuous, all secondary discourses
are either formally or informally connected to the learner’s
primary discourses. However, this does not mean that primary discourses are always optimally leveraged to develop secondary
discourses, especially in formal, “school-like,” instructional
settings. Ideally, secondary discourses would be explicitly developed through primary discourses, which require a greater
understanding of learners’ primary discursive identities.
Mathematics could be considered a specialized secondary
discourse developed by people for specific purposes. It is important to explicitly define the discursive markers of each in
order to have such a phenomenon as mathematics or to have a
conversation about what counts as mathematics. For example, one possible definition is that mathematics is a special type of
discourse that deals with quantities and shapes (i.e., a secondary
discourse); however, there are many ways in which this can be done depending on the context as many studies have shown (e.g.,
Cole, 1996; Lave, 1988; Scribner & Cole, 1981). Although this
definition (or any definition) of a domain of knowledge is not
without contestation and would undoubtedly be considered a narrow view of what counts as mathematics, it is an example of
one way that mathematics discourse distinguishes itself from other
forms of talk. I now turn to how the connection between discourse and learning is made more explicit in the context of professional
Discoursing Mathematically
51
development.
Connecting Discourse to Learning and Development
In connecting sociocultural views of learning and development
(especially CHAT5) with the discourse analysis issues discussed
above, there are five issues to consider: (a) activity goals, (b) mediational tools (symbolic/visual), (c) the action/object to
meaning ratio, (d) situated versus literate discourses, and (e)
“transfer” or cross-situational discourses. As far as mediation is concerned, it is well established within Vygotskian and neo-
Vygotskian traditions that learning proceeds from the
interpersonal plane toward the intrapersonal plane through the
active use of symbolic and visual artifacts. The material and ideational tools that human beings draw on are historically and
socially constituted and become organized as Discourses across
generations of actors. According to Wertsch (1998), all human meaning-making is
purposeful, goal driven, and rule governed. These factors are
assumed features of discursive practices regardless of the setting.
In his work on children in play situations, Vygotsky (1978; 1987) argued that one of the primary measures of development are the
shifts in the action to meaning ratio. In the early stages of learning,
the object(s)/action(s) dominate the child’s ability to make meaning. For example, the presence of a cup filled with some type
of liquid would prompt a child to say “water” because the set of
object(s)/action(s) dominate the use of signs and symbols which are highly context dependent in the early stages of development.
However, over time the meaning of the phonetic sounds for the
word “water” (/wɔtər/) become less dependent on the presence of
object(s)/action(s). Through the mediation of more expert others and the use of symbolic tools, learners develop the ability to
regulate meaning without relying on context (see Figure 1):
MEANING
ACTION
ACTION
MEANING
Figure 1. The shift in Action/Meaning Ratio.
The appropriation of primary and secondary discourses happen in much the same way with one difference: secondary
MEDIATION
Aria Razfar
52
discourses represent a greater level of abstraction which means the
ratio of action to meaning is slanted toward meaning. This gives secondary discourses the added utility of having cross-situational
applicability. However, when mathematical and scientific
practices (i.e., the disciplinary activities of a community of
scholars) are conceptualized as “discourse” or more precisely a secondary discourse, then it follows that one cannot reach more
abstract levels without the mediation of objects and actions. A
clear implication of this point is how sometimes mathematics learning in formal instructional settings is organized as discrete
activities in the form of text-based lessons or reductive
worksheets. These types of activities serve to present mathematics practices as a set of isolated skills devoid of culturally situated
purposes. The following table illustrates how primary and
secondary discourses compare with respect to development, the
types of mediation, durability, and ranges of applicability (Table 1).
Table 1
Comparison of Primary and Secondary Discourses
Characteristics Primary Discourse Secondary Discourse
Development Spontaneous Through reflection, that
is, at meta-level with
respect to the primary
Mediation Predominantly Physical
Predominantly symbolic
Durability Transient Lasting
Applicability Highly Restricted Universal
(Sfard, 2002)
Discourse and Learning: The MASS System
Gee and Green (1998) offer a framework for discourse
analysis for educators in any setting that effectively integrates the key elements of discourse analysis and sociocultural theories of
learning and development. The MASS system has four
components: material, activity, semiotic, and sociocultural. Each of these dimensions of meaning-making can occur in one of two
scenarios: (a) situated types of meaning and (b) more abstracted
Discoursing Mathematically
53
cultural models. Social languages are distinct from other types of
language (i.e., national languages) in that they immediately draw attention to the context and purpose of language use. Gee (1999)
compares two language samples that basically convey the same
information; yet, have very distinct purposes and thus count as two
social languages (p. 27):
1. Experiments show that Heliconius butterflies are less
likely to ovipost on host plants that possess eggs or egg-
like structures. These egg mimics are an unambiguous example of a plant trait evolved in response to a host-
restricted group of insect herbivores. (professional
journal)
2. Heliconius butterflies lay their eggs on Passiflora vines.
In defense the vines seem to have evolved fake eggs that make it look to the butterflies as if eggs have already been
laid on them. (popular science)
Participants were asked to describe the difference between the two social languages. Many would describe sample 1 as being
more “academic” or more “scientific.” When pushed a little
further to identify the discourse markers that index academic or
scientific values, some pointed to extra-textual issues such as the genre of the publications (popular science vs. professional
journal), thus, the differing discourse communities. Others noted
that the language used in sample 1 requires a greater degree of abstraction from the situation. For example, the choice of subject
“experiments show” versus “butterflies lay” transforms a single
observation into a more generalizable proposition. The lexical
choice in sample 1 refers to classes of plants and insects. It is no longer about what a single instance of Heliconius butterflies do,
but what can be concluded about all Heliconius butterflies. Some
pointed out that there is an unnecessary formality to sample 1 especially when you compare “egg mimics” to “fake eggs.” One
of the participants compared this example with children’s
tendency to use informal units of measurement as opposed to formal units of measurement. For example, a child might describe
the length of the floor in terms of his or her “red shoes” rather than
using more generalizable conventions such as meters, feet, or
inches. This might be indicative of the nominalization tendency of mathematics discourse to use nouns rather than adjectives and
Aria Razfar
54
nouns (Pimm, 1987; 1995; Morgan, 1998). Sample 1 is also better
suited for predicting future behavior which is a value of scientific discourse. Sample 2 is more descriptive and observable and does
not require additional inductive reasoning beyond the situation.
Examining the two samples showed not only the linguistic
difference between them but also that they represent differentiated learning and thinking (i.e., higher order cognition). Both samples
can be considered part of the scientific process with the discursive
form of sample 1 representing a more durable and universal type of discourse (secondary discourses). If the importance of
discursive identities is considered in learning, the empirical
question one might ask is, Which form would a child have more affinity with? This is a critical question for discourse researchers
and practitioners because discursive identity, who a person
projects themselves to be socially through discourse, is a powerful
purveyor of learning and development.
From Language to Discourse Proficiency: The Baseball
Language Learners
Using the MASS system as the central unit of analysis for understanding learning and development has four parts:
1. Material: The who and what in an interactional frame
(the actors, place, social space, time, and objects present (or referred to) during an interaction.
2. Activity: What’s happening and how is it organized?
3. Semiotic: What are they using to make sense and communicate? (This includes gestures, images, or
other symbolic systems)
4. Sociocultural: What are participants thinking, feeling,
and being?
In order to make these ideas more concrete, participants were asked to answer the following questions:
1. What discourses have you partially or fully mastered?
2. Describe features of the discourse that marked
membership.
3. Which discourses do you consider “primary” and which ones do you consider “secondary”?
Discoursing Mathematically
55
After discussing various discourses and features that marked membership within those communities, I decided to focus the discussion on a typical scenario that is grounded in the baseball
discourse community. I divided the participants into three
homogenous (self-selected) groups with respect to expertise in that
community: the experts, the casual fans, and the “BLLs” (Baseball Language Learners). A list of discrete words and phrases were
placed on the board that each group had the task of defining: bat,
ball, strike, diamond, base, steal, hit and run, stealing home, batting three hundred, triple crown, run, out, balk, save, and bean
ball.
As expected, the expert group and those who consider baseball to be a primary discourse were easily able to define these terms.
However, the novice group (our affectionate term “BLLs”)
struggled to accurately make sense of the terms within a baseball
context. The point of the activity was clear as many of the members of this group expressed how for the first time they
experienced what it was like to be an English Language Learner
(ELL).6 Of course, they all spoke English, but they didn’t speak
baseball. As a result, “bat” was more like a bird than a stick, and
“ball” was a spherical object instead of a pitch that isn’t good to
hit, etc. Levinson (1983) argued that it doesn’t make sense to talk about any kind of meaning without an activity system that frames
meaning. Even apparently discrete meaning-making is predicated
on situated and action based participation. The activity system, in
this case baseball, is governed by explicit and implicit rules that discourse members know in order to successfully make sense.
(This does not necessarily mean they play or are good players, but
rather that they are good sense makers within the activity). The activity system mediates meaning with respect to the
other three dimensions of Gee and Green’s (1998) framework.
There are implications for mathematical problem solving. I gave
the following simple arithmetic problem to the participants:
Barry Bonds, one of the most prolific home run hitters of the
modern era, slugged over eight-hundred in one season. If he had
six hundred at bats, how many total bases did he get?
This problem is not complicated for someone who is a
baseball discourse community member; however, it illustrates how mathematical meaning-making can be situated. All of the
Aria Razfar
56
“baseball novices” were stumped by this problem; of course, the
experts were able to solve it right away and the homogenous grouping was intended to make this point visible to all the
participants rather than a model of “best practice” (although it
made the point in favor of heterogeneous grouping of language
learners). In fact, simple, straightforward and seemingly universal numerical representations like “hundred” have two different
meanings within the same question stem. The first instance “eight-
hundred” represents a percentage where the whole is not referred to as 100% but rather 1000%. The second instance of “hundred” is
the more accustomed usage (the value 100). As shown below, the
language load of the math problem can be virtually eliminated by providing the formula for slugging percentage, and hence anyone
with the knowledge of how to employ formulas could derive the
answers (although “eight hundred” might still be a stumbling
block).
Barry Bonds, one of the most prolific home run hitters of the
modern era, slugged over eight-hundred in one season. If he
had six hundred at bats, how many total bases did he get?
Slugging Percentage=Total Bases/At Bats
1. Total Bases/At Bats=.800
2. Total Bases/600=.800
3. Total Bases=600*.800
4. =480
However, this type of modification presumes math to be free
from linguistic and discursive issues and does not always work, especially in high-stakes mathematical assessments.
One of the school participants, who was a doctoral fellow at
the time and is now a mathematics teacher educator, thought that
this type of activity would be ideal to use in a mathematics methods course. After the conclusion of the session, she reflected
upon the BLL activity,
I think this would be a great example to use with the preservice
teachers to have them get in the shoes of those ELLs who have
acquired conversational fluency in English but not academic—mathematical—fluency in English. Most people, including
teachers, tend to think of ELLs as those who have difficulty
speaking in English or have a heavy foreign accent. If a child
Discoursing Mathematically
57
speaks English fluently or has a native-like American-English
accent then, in their minds, that child is not an ELL.
The activities that are typically used with (monolingual)
preservice teachers to have them experience what ELLs experience in the classroom, and to perhaps model strategies that
can be used to accommodate ELLs are often in a language that
none of the preservice teachers speak. Such activities, for example, include a mathematics problem written in a language the
preservice teachers are not familiar with, or a health video giving
instructions in Farsi (Harding-DeKam, 2007). While these activities can be useful to have preservice teachers experience
what it feels like to be an ELL who has recently moved to the U.S.
and speaks no English, the majority of the ELLs that preservice
teachers will be teaching will not fall into that category. In fact, most ELLs have some level of conversational fluency in English,
and many of them might not have an easily detectable foreign
accent, making it difficult for teachers to classify them appropriately as ELLs. According to Cummins (1981)
conversational fluency in English is acquired within 2 years, while
it takes 5 to 7 years to acquire academic (including mathematical) fluency in English. Teachers need to be aware of this important
distinction, and they need to understand its implications for
teaching mathematics to ELLs. Preservice teachers are often
taught this distinction in their coursework but do not necessarily make connections with what this means for teaching mathematics
to ELLs (Vomvoridi-Ivanovic & Khisty, 2007).
Conclusion
In this article, I provided conceptual and methodological tools
as well as activities that can be used for the preparation and
professional development of both mathematics teacher educators and mathematics teachers to aid their development of a more
situated and social view of mathematical discourse and its
relationship to student learning, particularly how mathematical discourse relates to LMS. The concrete examples discussed in this
article help make the discursive nature of mathematics more overt
for those who believe that mathematics is a universal language. As
the field considers the mathematics education of LMS, mathematics teacher educators as well as mathematics teachers
can draw on the notions of primary and secondary discourses to
Aria Razfar
58
move beyond static views of development, especially vis a vis
mathematics learning. To improve the mathematics education of LMS, mathematics
teacher educators should receive professional development that
supports them in including issues of language and discourse in
their mathematics teacher preparation courses and in professional development settings with in-service mathematics teachers. This,
in turn, will help mathematics teachers begin to develop
knowledge that is required to support the mathematics learning of LMS. Teacher educators need more research that examines what
preservice teachers learn when they participate in activities
designed to build critical awareness about issues in language learning and develop an emic perspective of the challenges
encountered by ELLs and other members of non-dominant
populations who engage in non-orthodox forms of mathematical
meaning-making (e.g., Saxe, 1988). Although the activities presented in this article have great potential to move preservice
teachers towards these critical understandings of discourse,
language, and learning, it is important for teacher educators to develop new activities that are suited to the needs of their
preservice teachers.
Acknowledgements
This work is based on work conducted with the Center for Mathematics
Education of Latinas/os (CEMELA). CEMELA is a Center for Learning
and Teaching supported by the National Science Foundation, grant number ESI-0424983. Any opinions, findings, and conclusions or
recommendations expressed in this article are those of the author and do
not necessarily reflect the views of the National Science Foundation.
References
Brisk, M. E. (Ed.). (2008). Language, culture, and community in teacher education. New York, NY: Erlbaum.
Brown, B.A. (2004). Discursive identity: Assimilation into the culture of science
and its implications for minority students. Journal of Research in Science Teaching, 41, 810–834.
Discoursing Mathematically
59
Brown, B. A., Reveles, J. M., & Kelly, G. J. (2005). Scientific literacy and discursive identity: A theoretical framework for understanding science learning. Science Education, 89, 779-802.
Chval, K. B., & Pinnow, R. (2010). Preservice teachers’ assumptions about Latino/a English language learners. Journal of Teaching for Excellence and
Equity in Mathematics, 2(1), 6–12.
Cobb, P., Yackel, E., & McClain, K. (Eds.) (2000). Communicating and symbolizing in mathematics: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Erlbaum.
Cochran-Smith, M., Fieman-Nemser, S., McIntyre, D. J., & Demers, K. E. (Eds.). (2008). Handbook of research on teacher education: Enduring questions in changing contexts (3rd Ed.). New York, NY: Routledge.
Cole, M. (1996). Cultural psychology. Cambridge, MA: Belknap.
Cummins, J. (1981). The role of primary language development in promoting educational success for language minority students. In Schooling and language minority students: A theoretical framework. Los Angeles, CA: Evaluation, Dissemination, and Assessment Center, California State University.
Gee, J. P. (1996). Social linguistics and literacies: Ideology in discourses (2nd Ed.). London, UK: Taylor & Francis.
Gee, J. P. (1999). An introduction to discourse analysis: Theory and method.
New York, NY: Routledge.
Gee, J. P., & Green, J. (1998). Discourse analysis, learning, and social practice: A methodological study. Review of Research in Education, 23, 119–169.
Goffman, E. (1981). Forms of talk. Philadelphia, PA: University of Pennsylvania.
Gonzalez, N., Moll, L. C., & Amanti, C. (2005). Funds of knowledge: Theorizing practices in households and classrooms. Erlbaum.
Gutiérrez, R. (2002). Beyond essentialism: The complexity of language in teaching mathematics to Latino students. American Educational Research
Journal, 39, 1047–1088.
Harding-DeKam, J. L. (2007). Foundations in ethnomathematics for prospective elementary teachers. Journal of Mathematics and Culture, 2(1), 1–19.
Holquist, M. (1990). Dialogism: Bakhtin and his world. New York, NY: Routledge.
Khisty, L. L. (2002). Mathematics learning and the Latino student: Suggestions from research for classroom practice. Teaching Children Mathematics, 9, 32–35.
Krashen, S. D. (2003). Explorations in language acquisition and use.
Portsmouth, NH: Heinemann.
Lave, J. (1988). Cognition in practice. Cambridge, MA: Cambridge University.
Aria Razfar
60
LeCroy & Milligan Associates, Inc. and Mekinak Consulting. (2007). Center for the Mathematics Education of Latinos/as (CEMELA) Evaluation Summary Year 3, July 2007. Tucson, AZ.
Lester, F. K. (Ed.). (2007) Second handbook of research on mathematics teaching and learning (pp. 39–68). Charlotte, NC: Information Age.
Levinson, S. (1983). Pragmatics. Cambridge, MA: Cambridge University.
Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future. Rotterdam, NE: Sense.
Lucas, T., & Grinberg, J. (2008). Responding to the linguistic reality of mainstream classrooms: Preparing all teachers to teach English language learners. In M. Cochran-Smith, S. Feiman-Nemser, J. D. McIntyre & K. E.
Demers (Eds.), Handbook of research on teacher education: Enduring questions in changing contexts (pp. 606–636). New York, NY: Routledge.
Morgan, C. (1998). Writing mathematically: The discourse of investigation. London, UK: Falmer.
Moschovich, J. (1999a). Understanding the needs of Latino students in reform-oriented mathematics classrooms. In L. Ortiz-Franco, N. G. Hernandez & Y. De La Cruz (Eds.), Changing the faces of mathematics: Perspectives on Latinos. Reston, VA: National Council of Teachers of Mathematics.
Moschovich, J. (1999b). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19.
Moschovich, J. (2007). Examining mathematical discourse practices. For The Learning of Mathematics, 27(1), 24–30.
National Center for Educational Statistics (NCES) (2010). The condition of education 2010. Retrieved from: http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2010028
Pimm, D. (1987). Speaking mathematically. New York, NY: Routledge.
Pimm, D. (1995). Symbols and meanings in school mathematics. New York, NY: Routledge.
Razfar, A. (2003). Language ideologies in ELL contexts: Implications for Latinos and higher education. Journal of Hispanic Higher Education, 2, 241–268.
Rogoff, B. (2003). The cultural nature of human development. New York, NY: Oxford University.
Saxe, G. (1988). Candy selling and math learning. Educational Researcher, 17, 14–21.
Scribner, S., & Cole, M. (1981). The psychology of literacy. Cambridge, MA: Cambridge University.
Discoursing Mathematically
61
Setati, M. (2005). Teaching mathematics in a primary multilingual classroom. Journal for Research in Mathematics Education, 36, 447–466.
Sfard, A. (2002). Learning mathematics as developing a discourse. In R. Speiser, C. Maher, C. Walter (Eds.), Proceedings of 21st Conference of PME-NA (pp. 23–44). Columbus, OH: Clearing House for Science, Mathematics, and
Environmental Education.
Street, B., Baker, D., & Tomlin, A. (2005). Navigating numeracies: Home/School numeracy practices. London, UK: Springer.
Valdés, G., Bunch, G., Snow, C., Lee, C., & Matos, L. (2005). Enhancing the development of students’ language(s). In L. Darling-Hammond, & J. Bradsford (Ed.), Preparing teachers for a changing world: What teachers should learn and be able to do (pp. 126–168). San Francisco, CA: Jossey-Bass.
Vomvoridi-Ivanovic, E., & Khisty, L. (2007). The multidimentionality of language in mathematics: The case of five prospective Latino/a teachers. In Lamberg, T., & Wiest, L. R. (Eds.), Proceedings of the Twenty Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 991–998). Stateline (Lake Tahoe), NV: University of Nevada, Reno.
Vygotsky, L.S. (1978). Mind in society: The development of higher psychological processes. Cambridge: MA: Harvard University.
Vygotsky, L. S. (1987). Thought and language (A. Kozulin, Trans.). Cambridge, MA: Cambridge University.
Wertsch, J. V. (1998). Mind as action. New York, NY: Oxford University.
White, L. (1987). Against comprehensible input: The input hypothesis and the development of second language competence. Applied Linguistics, 8, 95–110.
Wortham, S. (2003). Accomplishing identity in participant-denoting discourse. Journal of Linguistic Anthropology, 13, 1–22.
Zeichner, K. M. (2005). A research agenda for teacher education. In M. Cochran-Smith & K. M. Zeichner (Eds.), Studying teacher education: The report of the AERA panel on research and teacher education (pp. 737–759). Mahwah, NJ: Erlbaum.
Aria Razfar
62
1 “Language” refers to the structural aspects of language (i.e., code) and/or the use of national languages (e.g., Spanish, English). “Discourse” refers to the specialized and situated language of mathematics (e.g., quantitative and symbolic language). The distinction between “language” and “discourse” will be elaborated later in the
paper.
2 The [ ] are a transcription convention used to indicate overlapping talk; colons (:::) indicate prolongation of sound. All names are pseudonyms.
3 Footing refers to how the mode and frame of a conversation is determined by participants in an interaction, and how speakers empower and/or disempower each other through various linguistic practices that invoke power relations, social status, and legitimacy.
4 Intersubjectivity is an interdisciplinary term used to describe the agreement between speakers on a given set of meanings, definitions, ideas, feelings, and social relations. The degree of agreement could be
partial or sometimes divergent as in the case of deception, sarcasm, irony, or lying.
5 Cultural Historical Activity Theory (CHAT) is a more recent term used by neo-Vygotskians to emphasize the historical dimensions of learning (e.g., Rogoff, 1995; Sfard, 2002).
6 English Language Learner (ELL) is a subgroup of Language Minority Students (LMS). It is the common term used in U.S. public schools to classify students for whom English is either their second language or come from bilingual homes.
The Mathematics Educator
2012 Vol. 22, No. 1, 63–83
Steven LeMire teaches statistics and educational research at the University of North Dakota, Grand Forks.
Marcella Melby teaches mathematics and mathematics education courses at the University of Minnesota, Crookston.
Anne Haskins teaches occupational therapy at the University of North Dakota,
Grand Forks.
Tony Williams teaches management at Auburn University Montgomery.
The Devalued Student: Misalignment of
Current Mathematics Knowledge and Level
of Instruction
Steven D. LeMire, Marcella L. Melby , Anne M.
Haskins , and Tony Williams
Within this study, we investigated the association between 10th-grade
students’ mathematics performance and their feelings of instructional
misalignment between their current mathematics knowledge and
educator support. Data from the 2002 Education Longitudinal Study,
which included a national sample of 750 public and private high schools
in the United States, was used for the investigation. Our findings indicate
that student perceptions of both instructional alignment and educator
support are associated with mathematics performance. Students who
reported receiving misaligned instruction in mathematics and felt
devalued by educators had lower mathematics performance than students who reported aligned mathematics instruction and who felt valued by
teachers. A key implication for practitioners of this work is that
mathematics educators should consider cognitive and affective elements
of student development. Specifically in addition to cognitive factors, the
affective elements of student capacity to receive, respond to, and value
whole-group mathematics instruction in academically diverse classrooms
should be considered in curriculum planning.
Learning is not just the acquisition and manipulation of content;
how and how well we learn is influenced by the affective realm –
our emotions and feelings – as well as by the cognitive domain.
(Ferro, 1993, p. 25)
It is well known that not all students reach their full mathematics potential in high school. According to Tomlinson et
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
64
al. (2003), one potentially important reason for this is a lack of
instructional level alignment. In such cases, teachers fail to adjust their instruction effectively to accommodate academically diverse
student abilities. If instruction does not accommodate students’
varied readiness levels, students will have inequitable learning
opportunities (Tomlinson et al., 2003). Instructional level alignment, in which instruction is given at a level that is beneficial
to the student, depends upon aspects of the cognitive domain.
Effective instruction that is aligned with a student’s ability level in mathematics could lead to cognitive growth in the student’s
knowledge, comprehension, and critical thinking. Failure to align
instruction in a way that may be beneficial to a given student could lead to a sense that the educational process does not value him or
her. Feeling valued in an educational process is another important
factor in students reaching their full potential and can be viewed as
an affective domain. A key affective element would be a student’s inability to respond to the misaligned instruction (Bloom,
Englehart, Furst, Hill, & Krathwohl, 1956). For example, if a
student is unable to understand a difficult mathematics class because it is at a level above their ability to respond to the
instruction, the student may not progress to the affective level of
valuing the instruction. The inability of the student to reach a valuing state could have substantial negative consequences and
may cause the student to affectively shut down (Hackenberg,
2010). What is understood to a lesser degree is the impact that
instructional level misalignment and not feeling valued in the educational process can have on high school students’
mathematics success.
Further investigation of the potential impact of these two issues is needed to better understand instructional level alignment
as it relates to school policy issues such as instructional level
grouping (Paul, 2005) and whole-group or differentiated
classroom delivery of instructional content (Lawrence-Brown, 2004). Instructional grouping is in part motivated to reduce student
ability level diversity so more students will be aligned with the
delivery of whole-group instruction. Differentiated instruction attempts to create different levels of instruction alignment for
students’ diverse ability levels within a group of learners
(Lawrence-Brown, 2004).
The Devalued Student
65
To address this need to understand more about the role of the
affective domain in mathematics education, we investigated the educational performance of 10th-grade mathematics students
coupled with their perceived experience of instructional level
alignment—based on their perceived ability to understand a
difficult mathematics class—and their impression of not feeling valued by teachers. While a multitude of variables (including
student and educational factors) may influence student success and
engagement in academic settings, we focused on the direct interactivity between the students’ mathematics performance and
both their sense of being valued and their perception of
understanding a difficult mathematics class.
Literature Review
Student Diversity
Although the diversity of students’ current subject knowledge can be a challenge for teachers of mathematics, it is often a desired
classroom characteristic (Kennedy, Fisher, Fontaine, & Martin-
Holland, 2008). Diversity may be characterized by factors that
include students’ learning styles, gender, age (Bell, 2003), racial or ethnic backgrounds (Kennedy et al., 2008), life experience,
personality, educational background (Freeman, Collier, Staniforth,
& Smith, 2008), or current subject knowledge. For the purposes of the study, we were most concerned with students’ reported
perception of their ability to understand a difficult mathematics
class. Furthermore, we feel that this factor is closely related to the other aspects of diversity mentioned above.
The Cognitive Domain
How students learn mathematics. Mathematics is an interconnected discipline comprised of different topical strands:
number sense and operation, algebra, geometry, measurement, and
data analysis and probability (National Council of Teachers of Mathematics [NCTM], 2000). According to the NCTM, a school
mathematics curriculum should be coherent and organized in such
a way that the important fundamental ideas form an integrated
whole. Students need to be able to comprehend how ideas build upon and connect with other ideas. In mathematics, a student may
understand new material when he or she can make connections
with his or her existing mathematical knowledge. Those students
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
66
with sufficient prerequisite mathematical knowledge are more
likely to be able to build upon that knowledge and progress to a deeper understanding.
Research in cognitive learning theory, pioneered by such
researchers as Piaget and Vygotsky, has provided valuable insights
for mathematics educators concerning the ways in which children learn and understand mathematics (Fuson, 2009; Kilpatrick, 1992;
Ojose, 2008). The work of Ojose (2008) is particularly important
because he applied Piaget’s four stages of cognitive development (sensorimotor, preoperational, concrete operational, and formal
operational) directly to the mathematical development of children.
He concluded that when students are grouped solely by chronological age, their developmental levels can vary drastically.
Ojose emphasized the need for teachers to discover their students’
current cognitive levels and adjust their mathematics teaching
accordingly. Vygotsky also provided insight into the development of
cognitive learning theory and the understanding of how children
learn mathematics. According to Vygotsky (as cited in Carter, 2005), learning happens when an individual is working within his
or her zone of proximal development (ZPD). The ZPD is at a level
above independence. Independence is defined as the stage where a student already knows the material and could perform that task
without assistance. On the other hand, when material is in a
student’s ZPD, he or she is capable of performing tasks with help
from a teacher or more able peer (Carter, 2005; Smith, 2009; Van de Walle & Lovin, 2006).
Whole-group instruction contributes to misalignment. In
the dominant model of whole-group instruction, in which one teacher provides instruction to a group of students, educators often
attempt to target a central prior knowledge level of the group.
Furthermore, as stated by Tomlinson et al. (2003), organizational
restraints restrict teachers from meeting the needs of students who “diverge markedly from the norm” (p. 120). This approach may be
utilized for a variety of reasons and has been linked to the
availability of faculty as well as increased class sizes (Ochsendorf, Boehncke, Sommerlad, & Kaufmann, 2006). Targeting a central
ability level of a large group of students allows the instruction to
be presented at a level that would facilitate effective learning for a majority of students in the group. For these students, the
The Devalued Student
67
instruction is expected to be beneficial because it is at a level that
their current knowledge can support. However, students near the ends of the spectrum of background knowledge may not benefit
from instruction if it is above or below their ZPD, possibly causing
them to disconnect from the learning process. The NCTM’s (2000)
Equity Principle maintains that all students should have the opportunity and support needed to learn mathematics with
understanding. The principle states, “equity does not mean that
every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be
made as needed to promote access and attainment for all students”
(p. 12). When whole-group instruction is used, the unit of instruction
is the group. The unit of instructional interest, however, is the
student. This represents a mismatch of instructional unit versus
learner unit. When this mismatch occurs, important elements of the instructional environment to consider are the variability of
between-student current knowledge levels and the hierarchical and
cumulative nature of the content. Variability of between-student current knowledge. The
goal of a successful educational experience is to form an
alignment between instruction and the current knowledge of individual students. Atkinson, Churchill, Nishino, and Okada
(2007) described alignment as a coordinated interaction. They
asserted that learning should be aligned with the socio-cognitive
environment. Using Atkinson’s et al. (2007) proposition, one could then view alignment in the context of this work as
coordinated interaction between the student and the instructor.
This would imply coordination, which results in successful alignment, and has been described as “the novice and the expert
functioning as a cross-cognitive organism—rather than as
cognitive nomads involved in the same activity” (p. 177).
When an instructor is presenting content that is not aligned with the student’s current knowledge level, the instructor and the
student can be in different and unconnected cognitive locations. If
instruction is beyond a student’s ZPD, the student might perceive that he or she is unable to understand material or that the
information is too difficult to comprehend. Conversely, when
instruction is given below a student’s current knowledge level, the curriculum does not challenge him or her, possibly leading to
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
68
boredom and the risk of slipping into underachievement status.
The hierarchical and cumulative nature of mathematics. Instructional level misalignment is more likely when the nature of
the content is hierarchical. Nonhierarchical subject content is
where instructional learning units are based on the knowledge
level associated with Bloom’s Taxonomy of the Cognitive Domain. The goal of this type of instruction would be for the
student to remember specific declarative or procedural facts
(Bloom, et al., 1956; Hopkins, 1998). This recall requirement is the first stage in Bloom’s Taxonomy of the Cognitive Domain and
therefore the learner requires few knowledge prerequisites. In this
type of learning, a student whose knowledge is less than that required by the current instruction level may be able to make
substantial gains from the instruction. In contrast, learning
requiring higher order abilities such as comprehension and
analysis rests on the foundation of lower order knowledge and hence is more hierarchal (Booker, 2007). The hierarchical nature
of mathematics learning, for example, may require mastery of
basic skills to facilitate the attainment of higher order conceptual understanding (Siadat, Musial, & Sagher, 2008; Wu, 1999). In this
case, successful learning of the current unit of instruction may
require translation, interpretation, and extrapolation of previous learning units’ material. In the absence of prerequisite knowledge,
it is assumed that students will have difficulty transitioning to
higher levels of learning and understanding within the subject.
The Affective Domain
The application of the levels of Bloom’s cognitive domain of
the educational taxonomy can be seen readily throughout
education in the United States (Booker, 2007). The affective domain, however, has received less attention, and there is limited
research on the affective learning of the student (Porter & Schick,
2003). Despite its lack of prevalence, a student’s affective response to instruction might play a significant role in a student’s
interest in a given course. This is supported by Subban (2006),
who found that students who enjoyed a task at an early age continued to seek the cognitive stimulation related to the task
which helps even marginalized students in the classroom.
Categories of the affective domain. The affective domain of
the Taxonomy of Educational Objectives includes the emotional
The Devalued Student
69
engagement of the student with the topic and is linked inextricably
to the cognitive domain (Krathwohl, Bloom, & Masia, 1964). The major categories of this domain are hierarchically organized from
lowest to highest behavior processes. The first is receiving
phenomena, which requires a learner’s awareness of an idea and
his or her willingness to acknowledge that idea (Maier-Lorentz, 1999). For example, a student busy texting during a mathematics
class is unlikely to receive the teacher’s definition of a
mathematical idea. The next level is responding, which refers to the learner’s ability to act on or respond to the idea they are
receiving (Maier-Lorentz, 1999). A student that is in the
responding state may be receiving and understanding the topic enough to be able to participate in a discussion or answer a
teacher’s question about the topic. It is here that we assert that
students who are being exposed to instruction that is not aligned
with their own current knowledge level can affectively disconnect from the learning process. This prevents them from reaching
valuing, which is the next level in the affective domain. In the
valuing state, a student may see worth in the learning even if the topic does not interest them (Deci, Vallerand, Pelletier, & Ryan,
1991). For example, they may be able to see where they can use
their learning in their daily life or to get a better grade on an exam. On the other hand, those students who are unable to respond to a
learning task due to a lack of alignment between the instructional
level and current knowledge may start to become unwilling to
consider new information. Thus, those who do not reach the valuing level in the affective domain because they were in a
cognitively misaligned instructional experience may then feel that
the teacher does not value them or that they are being put down (Krathwohl et al., 1964).
The transition from not being able to reach the valuing stage
(Krathwohl et al., 1964) because of misaligned instruction to not
feeling valued by a teacher can be viewed through the lens of self-determination theory (Deci & Ryan, 2000). A component of this
theory is motivation, which can be related to valuing, competence,
autonomy, and relatedness (Deci et al., 1991). In order for students to be motivated to see themselves as valued in the educational
effort, they need to have some level of competency and autonomy
of control of an outcome through some strategy for success (Deci et al., 1991). Competency and autonomy pertain to the student’s
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
70
ability to have some independent success at a task. An example of
this might be that a student could self-initiate and self-regulate the undertaking and completion of a set of homework problems which
were based on a teacher’s effective instruction in that day’s
mathematics class. This is possible when a student has a sense of
relatedness that pertains to a developed, secure, and satisfying connection to significant adults (Deci et al., 1991), such as
mathematics teachers. We contend that if a student lacks
relatedness to a teacher because of a misaligned instructional level educational interaction, which leaves the student without a feeling
of competency or autonomy, the student may not be motivated to
feel valued by the teacher. This connection between learning engagement and a sense of feeling valued by the teacher is also
supported by Wentzel (1997).
A causal framework for the affective consequences of
inaccessible misaligned instruction was presented by Boshier (1973), who described congruence and incongruence. He proposed
that “when an individual is not threatened, and manifests intra-self
and self/other congruence he is open to experience” (p. 260). The idea of intra-self and self/other congruence is related to the
condition of harmony with self or with others. However, when an
individual feels devalued or threatened, a condition of incongruence may occur. Incongruence of intra-self or self/other
“leads to anxiety, which is a subjective state of uneasiness,
discomfort, or unrest. Anxiety causes the individual to adopt
defensive strategies which induce a closing of cognitive functioning to elements of experience” (p. 260). Receiving
instruction above the level of a student’s current knowledge can be
viewed as a form of incongruence caused by instructional misalignment.
Engaging the affective domain in the learning of
mathematics. Mathematics is a unique subject in the school
curriculum because typically there is only one answer accepted to be correct (Chinn, 2009). Coupled with the cultural view that
mathematics should be completed quickly, it could be argued that
a student’s willingness to learn mathematics involves taking a risk (Chinn, 2009). The fear of failure induced by risk taking is an
affective dynamic that can cause anxiety, which may lead to low
mathematics achievement (Chinn, 2009). Hackenberg’s (2010) work on mathematical caring relations
The Devalued Student
71
(MCRs) addresses the importance of involving the affective
domain in the teaching and learning of mathematics. Hackenberg defines an MCR as “a quality interaction between a student and a
teacher that conjoins affective and cognitive realms in the process
of aiming for mathematical learning” (p. 237). In her study on
MCRs, Hackenberg took on the dual role of teacher and researcher for four 6th-grade students. When Hackenberg posed problems
that one of her students could not solve, she witnessed the
emotional shutdown of the student. The interactions that took place to bring her student back to a state of operating put a heavy
burden on not only the student but Hackenberg as well,
demonstrating that MCRs include the needs of both teachers and students. When a student perceives his or her teacher as someone
who understands, values, and challenges them with mathematical
tasks within their ZPD, trust builds and he or she is more likely to
take the risks that are involved in learning mathematics.
Purpose of Study
The purpose of the study was to assess the association of
mathematics performance with students’ feelings of being valued and their sense of instructional alignment. Specifically, we sought
to answer whether there was an association between students’
general feelings that teachers valued them and their standardized mathematics performance. We hypothesized that students who felt
that teachers valued them would have higher scores in
mathematics than students who felt that teachers did not value them. Secondly, we asked if there was an association between
understanding a difficult mathematics class and students’ feelings
of being “put down” by teachers (devalued) in relation to
standardized mathematics scores. We hypothesized that students who felt valued through instructor interest and perceived that the
instruction was aligned with their knowledge (i.e., they were able
to understand it) would demonstrate significantly higher performance in mathematics than students who did not.
Methods
Participants
The data for this study came from the National Center for
Education Statistics (Bozick & Ingels, 2008; NCES, 2006) and
resulted from the Education Longitudinal Study of 2002 (ELS:
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
72
2002/04). This study included a national sample of 750 public and
private high schools and 17,590 10th-grade students and obtained 15,360 returned surveys, for a response rate of 87%. Of these
15,360 students, 14,540 had completed cognitive assessments in
mathematics.
Instrument
Four variables were used from the ELS: 2002 base year
instrument (three independent variables and one dependent variable). The dependent variable for both of the research
questions was the standardized mathematics achievement score
(Bozick & Ingels, 2008). The mathematics test standardized score
was a T-score created by a transformation of the IRT (Item Response Theory) theta (ability) estimate from the cognitive
assessments in ELS: 2002. The first research question’s
independent variable was: “teachers are interested in students.” The independent variables for the second research questions were:
“in class often feels put down by teachers” and “can understand
difficult math class” (Bozick & Ingels, 2008).
Analysis
A one-way and a two-way analysis of variance (ANOVA)
were used for the analysis. For the one-way ANOVA, the independent variable was derived from the statement, “Teachers
are interested in students.” Students choose from the following
responses: strongly agree, agree, disagree, and strongly disagree.
For the purpose of analysis, the options were collapsed into some form of agreement (strongly agree, agree) and some form of
disagreement (disagree, strongly disagree). These options were
then compared with the students’ standardized mathematics score as the dependent variable.
For the two-way ANOVA, the dependent variable was the
students’ standardized mathematics scores. The independent variables were derived from the following two ELS: 2002 survey
items: “In class often feels put down by teachers” and “can
understand difficult math class.” The options for the students in
answering the item “in class often feels put down by teachers” were strongly agree, agree, disagree, and strongly disagree. For
analysis, the options were collapsed into some form of agreement
(strongly agree, agree) and some form of disagreement (disagree,
The Devalued Student
73
strongly disagree). The options for the students in answering the
question, “can understand difficult math class” were almost never, sometimes, often, and almost always. For the purpose of analysis,
the options were collapsed into two groups. The first group of
students responded with almost never or sometimes, and the
second group responded with often or almost always. These two groups represented students who were likely to struggle or were
not likely to struggle with mathematics instruction based on their
current knowledge levels. This collapsing of groups was informed by the ZPD as
discussed by Tomlinson et al. (2003). We contend that a student
that can often or almost always understand the instruction is effectively operating in the ZPD or at independence. A student
that never or even sometimes understands the instruction is not
operating in their ZPD and is not receiving effective instruction.
Although we are not aware of any mathematics education research that attempts to quantify these categories, there is an example in
the writing literature that does. Parker, McMaster, and Burns
(2011) discuss operational levels for reading which were developed by Gickling and Armstrong (1978). If a student can
read 97% or more of the words in a passage, they would be
considered to be operating at independence. A student reading 93% to 97% is at a level at which reading instruction should take
place, which represents the ZPD. A student reading below 93% of
the words would be operating at a frustration level (Parker et al.,
2011). We assert that a student that never or only sometimes understands difficult mathematics classes is operating at the
frustration level, which is categorically different than operating in
their ZPD or at independence.
Results
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
74
For our first research question we explored the association
between students’ general feelings that teachers were interested in them and standardized mathematics performance. The mean
standardized mathematics score for students who had some form
of agreement that teachers are interested in students was M = 51.5
(n = 10,948) and for students who indicated some form of disagreement was M = 48.6 (n = 3,423). This was found to be
statistically significant, F(1, 14,369) = 222.44, p < .05, with a
standardized effect size of d = 0.29. For our second research question we explored the association
between understanding a difficult mathematics class and students’
feelings of being put down by teachers (devalued) in relation to standardized mathematics scores. The results of the second
analysis indicated that both main effect factors of students feeling
put down by teachers (devalued) and students feeling that they
could understand difficult mathematics classes were associated with standardized mathematics scores. The means for these four
conditions are shown in Table 1.
Table 1
Means for Two-way ANOVA for Feels Put Down by Teachers and
Can Understand Difficult Math Class (MSE = 89.5)
In class often feels
put down by
teachers
Can understand
difficult math class N M
Some form of agreement
Never, Sometimes 935 46.9
Some form of
agreement Often, Always 542 50.9
Some form of
disagreement Never, Sometimes 5,140 49.8
Some form of disagreement
Often, Always 4,399 55.1
The main effect for “in class often feels put down by teachers”
was M = 3.5 with a standardized effect size of d = 0.37, F(1, 11,012) = 165.2, p < .05. The main effect for “can understand
difficult math class” was M = 4.6 with a standardized effect size of
The Devalued Student
75
d = 0.49, F(1, 11,012) = 286.2, p < .05. The interaction effect for
“feels put down by teachers” and “can understand difficult math class” was M = 1.5 with a standardized effect size of d = 0.15, F(1,
11,012) = 6.32, p < .05. A plot of the means is shown in Figure 1.
Strikingly, mathematics scores for those students who often “feel
put down by teachers” were lower even if they often or always understood a difficult mathematics class.
Figure 1. Interaction plot for the factors of “Can understand difficult math class” (never, sometimes or often, always) and “In
class often feels put down by teachers” (some form of agreement or some form of disagreement).
As shown in Figure 1, students who performed the best
(average math score of M = 55.1) indicated that they could often or always understand a difficult mathematics class and disagreed
that they often feel “put down” by teachers. Students who
performed the worst (average mathematics score of M = 46.9), indicated that they never or sometimes understand a difficult
mathematics class and agreed that they often felt “put down” by
teachers. The standardized effect size for this simple effect
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
76
difference is d = 0.87. This difference represents a large effect
(Cohen, 1988).
Discussion
Our findings revealed associations of students’ ability to
understand difficult mathematics classes and feeling devalued by teachers with standardized mathematics scores. Students who felt
they were “often or always” put down (devalued) by teachers in
class and “never or sometimes” could understand a difficult mathematics class had the overall lowest success in the
standardized tests. This could be explained partially by Boshier’s
(1973) definition of congruence as an event in which students
demonstrated greater likelihood of being open and accepting to new experiences in learning. A student who could not understand
a mathematics class and felt put down by the teacher could
experience a state of incongruence. Boshier’s stance was similar to that of Krathwohl et al.’s (1964) affective category of receiving in
which the student, through a sense of being devalued through not
understanding a difficult mathematics class, does not accept the
new learning content. Once a student drops out of the learning process, it can be difficult to bring him or her back, as Hackenberg
(2010) experienced when the inability of her student to solve a
variety of problems led to emotional shutdown. To avoid students’ perceptions of not understanding a difficult
mathematics class and a sense of being put down, high quality
instruction is necessary. Gamoran and Weinstein (1998) wrote, “conditions that support high-quality instruction in a
heterogeneous context include small class sizes and extra
resources that permit a highly individualized approach to
instruction” (p. 385). According to Gamoran and Weinstein, resources that support individualized attention can lead to high-
quality instruction. This is also a goal of reform-oriented
mathematics teaching which embraces creating instruction aligned with current knowledge and abilities of students (Superfine, 2008).
While our findings indicate that only 11.8% (935/11,016) of
the students from the analysis shown in Table 1 fell into the group that had the overall lowest success in mathematics (in class often
feels put down by teachers and cannot understand difficult math
class), we contend, with support from the NCTM’s Equity
Principle (2000), that is 11.8% too many. As stated by Chamberlin
The Devalued Student
77
and Powers (2010), all students should participate in respectful
work, and teachers should challenge students at a level attainable for them, which promotes individual growth. Whatever factors are
associated with inhibiting a student’s opportunity to meet the
expectations set forth by the NCTM must be addressed in
mathematics education literature and practice. The decisions made concerning mathematics curriculum and instruction in each
educational system have important consequences for not only
students but society as well. Furthermore, these decisions should not only deal with the cognitive aspects of the curriculum but the
affective as well.
Lack of instructional level alignment and students’ consequential feelings of being devalued by the educational
process could also be an influential factor in achievement gaps. In
a 2009 study, House found a correlation between Native American
students’ beliefs and attitudes towards learning mathematics and their score on the eighth-grade Trends in International
Mathematics and Science Study (TIMSS) conducted in 2003. As
one might suspect, those with positive self-beliefs about mathematics tended to score higher, whereas those with more
negative self-beliefs scored lower. This illustrates how both the
cognitive and affective state of the student could matter in mathematics education. A goal of the NCTM’s (2000) Equity
Principle is to increase students’ beliefs about their ability to do
mathematics. Clearly, this is a significant challenge, but essential
to attaining equity in mathematics education. Gregory, Skiba, and Noguera (2010) argued that
disproportionate rates of disciplinary sanctions on minority
children, which include exclusion from the classroom, could have a negative impact on student success. We contend that any
substantial exclusion from a whole-group instruction mathematics
classroom could be detrimental to the student’s success. This is
because the instruction would continue to progress without the student. Upon returning to the classroom, the student could face an
even more misaligned instruction level; an occurrence that may
enhance the likelihood of further disengagement and related consequences (Ireson & Hallam, 2001). The returning student’s
exposure to a misaligned level of instruction could lead to poor
performance in the class and lower academic achievement. Choi (2007) found that academic performance was a significant
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
78
predictor of delinquent behavior. This connection between poor
achievement and behavior was supported by Miles and Stipek (2006) who found that poor literacy achievement in the early
grades predicted high aggressive behavior in the later grades. A
label of lower achievement implies that a student’s knowledge is
below the current unit of instruction, which is appropriate for the comparison instructional group. In essence, instructional level
misalignment could potentially induce poor behavior that induces
exclusion and produces even larger misalignment. Consequently, this may lead to an affective sense of devaluation by the student in
the educational process.
It is imperative that educators continue to explore the influence of instructional level alignment on students’
comprehension, emotional and cognitive well-being, and
identification of being valued by the educational system. This
proposition is congruent with Hallinan’s (1994) assertion that there is a growing need for “rigorous empirical research on the
effects of homogeneous and heterogeneous grouping in schools
that vary in the several dimensions of school context to determine the impact of the organization of students on learning” (p. 91).
Testerman (1996) emphasized the need to consider the
affective domain when working with high school students. We agree that the affective domain can no longer be ignored, and
“schools must deal with the head and the heart” (para. 1). Our
results lend support to Testerman’s claims. Although more than 16
years have passed since Testerman’s proposition, few studies have examined the connection between student achievement within the
cognitive learning domain and the affective achievement domain.
Petrilli (2011) argues that the greatest current challenge to U.S. schools is the enormous variation in academic ability level of
students in any given classroom. He states that some variation is
good, but it is not uncommon to have variation in ability levels as
high as six grade levels in one classroom. Whole-group instruction with this much ability level variability is likely to result in a
sizable percentage of students who do not understand a difficult
mathematics class and who possibly do not feel valued. Overall, our findings support the integrative influence of
cognition and affective processes in relation to 10th-grade
mathematics performance. Results of this work support a need for educators to further examine instructional planning and the
The Devalued Student
79
delivery of content to heterogeneous prior-knowledge-level groups
of students. Minimally, we hope these findings will stimulate further conversation regarding student grouping policies,
instructional practices (such as whole-group and differentiated
instruction), and repercussions of those items in relation to a
students’ sense of understanding a difficult mathematics class and feeling valued by teachers within both the cognitive and affective
domain.
Implication for Practice and Future Research
The best performance of this national sample of 10th-grade
mathematics students was associated with students who could understand a difficult mathematics class and did not feel “put
down” by their teachers. While student understanding and
instructional alignment has long been considered a cognitive issue, this work demonstrates a possible link with mathematics
performance and the affective domain. Further research involving
qualitative methods may help make this link more clear and
provide insight for practitioners as to what can be done differently in the classroom. Specifically, student interviews or focus groups
could provide valuable insight about what leads students to feel
“put down” by teachers and what contributes to feeling valued in the classroom. Within the structure of planning and implementing
mathematics instruction, plans for improvement of both cognitive
and affective domains should be considered by practitioners. We feel that at least the first three student affective components of
receiving phenomena, responding to phenomena, and valuing
(Krathwohl et al., 1964) should inform the design of mathematics
instruction for all students.
Acknowledgement
We would like to thank Angela Holkesvig for her assistance with the preparation of this work.
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
80
References
Atkinson, D., Churchill, E., Nishino, T., & Okada, H. (2007). Alignment and interaction in a sociocognitive approach to second language acquisition. The Modern Language Journal, 91, 169–188.
Bell, J. A. (2003). Statistics anxiety: The non-traditional student. Education, 124(1), 157–162.
Bloom, B. S., Englehart, M. D., Furst, E. J., Hill, W. H., & Krathwohl, D. R. (1956). Taxonomy of educational objectives: The classification of educational goals. Handbook 1: Cognitive domain. New York, NY: McKay.
Booker, M. J. (2007). A roof without walls: Benjamin Bloom’s taxonomy and the misdirection of American education. Academic Questions, 20, 347–355.
Boshier, R. (1973). Educational participation and dropout: A theoretical model. Adult Education Quarterly, 23, 255–282.
Bozick, R., & Ingels, S. J. (2008). Mathematics course taking and achievement at the end of high school: Evidence from the Education Longitudinal Study of 2002 (ELS: 2002) (NCES 2008-319). U.S. Department of Education, Institute of Education Sciences. Washington, DC: National Center for Education Statistics.
Carter, C. (2005). Vygotsky & assessment for learning (AfL). Mathematics Teaching, 192, 9–11.
Chamberlin, M., & Powers, R. (2010). The promise of differentiated instruction for enhancing the mathematical understandings of college students. Teaching Mathematics and Its Applications, 29, 113–139.
Chinn, S. (2009). Mathematics anxiety in secondary students in England. Dyslexia, 15, 61–68.
Choi, Y. (2007). Academic achievement and problem behaviors among Asian
Pacific Islander American adolescents. Journal of Youth and Adolescence, 36, 403–415. doi:10.1007/s10964-006-9152-4
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (3rd ed.). Hillsdale, NJ: Erlbaum.
Deci, E. L., & Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. Psychological Inquiry, 11, 227–268.
Deci, E. L., Vallerand, R. J., Pelletier, L. G., & Ryan, R. M. (1991). Motivation
and education: The self-determination perspective. Educational Psychologist, 26, 325–346.
Ferro, T. R. (1993). The influence of affective processing in education and training. New Directions for Adult and Continuing Education, 59, 25–33.
Freeman, J. V., Collier, S., Staniforth, D., & Smith, K. J. (2008). Innovations in curriculum design: A multi-disciplinary approach to teaching statistics to
The Devalued Student
81
undergraduate medical students. BMC Medical Education, 8(28), doi:10.1186/1472-6920-8-28
Fuson, K. C. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching? Cognitive Development, 24, 343–361.
Gamoran, A., & Weinstein, M. (1998). Differentiation and opportunity in restructured schools. American Journal of Education, 106, 385–415.
Gickling, E. E., & Armstrong, D. L. (1978). Levels of instructional difficulty as related to on-task behavior, task completion, and comprehension. Journal of Learning Disabilities, 11, 559–566.
Gregory, A., Skiba, R. J., & Noguera, P. A. (2010). The achievement gap and the discipline gap: Two sides of the same coin? Educational Researcher, 39(1), 59–68. doi:10.3102/0013189X09357621
Hackenberg, A. J. (2010). Mathematical caring relations in action. Journal for Research in Mathematics Education, 41, 236–273.
Hallinan, M. T. (1994). Tracking: From theory to practice. Sociology of Education, 67, 79–84.
Hopkins, K. D. (1998). Educational and psychological measurement and evaluation (8th ed.). Boston, MA: Allyn and Bacon.
House, J. D. (2009). Mathematics beliefs and achievement of a national sample of Native American students: Results from the Trends in International
Mathematics and Science Study (TIMSS) 2003 United States assessment. Psychological Reports, 104, 439–446.
Ireson, J., & Hallam, S. (2001). Ability grouping in education. London, England: Sage.
Kennedy, H. P., Fisher L., Fontaine D., & Martin-Holland, J. (2008). Evaluating diversity in nursing education. The Journal of Transcultural Nursing, 19, 363–370. doi:10.1177/1043659608322500
Kilpatrick, J. (1992). A history of research in mathematics education. In D. A.
Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 3–38). New York, NY: Macmillan.
Krathwohl, D. R., Bloom, B. S., & Masia, B. B. (1964). Taxonomy of educational objectives. The classification of educational goals, Handbook II: Affective domain. New York, NY: Longman.
Lawrence-Brown, D. (2004). Differentiated instruction: Inclusive strategies for standards-based learning that benefit the whole class. American Secondary Education, 32(3), 34–62.
Maier-Lorentz, M. M. (1999). Writing objectives and evaluating learning in the
affective domain. Journal for Nurses in Staff Development, 15, 167–171.
Miles, S. B., & Stipek, D. (2006). Contemporaneous and longitudinal associations between social behavior and literacy achievement in a sample
Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
82
of low-income elementary school children. Child Development, 77, 103–117.
National Center for Education Statistics. (2006). Educational Longitudinal Study: 2002/04 Data Files and Electronic Codebook System. ECB/CD-ROM obtained from the U.S. Department of Education Institute of Educational
Sciences NCES 2006-346.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Ochsendorf, F. R., Boehncke, W. H., Sommerlad, M., & Kaufmann, R. (2006). Interactive large-group teaching in a dermatology course. Medical Teacher, 28, 697–701.
Ojose, B. (2008). Applying Piaget’s theory of cognitive development to mathematics instruction. The Mathematics Educator, 18(1), 26–30.
Paul, F. G. (2005). Grouping within Algebra I: A structural sieve with powerful effects for low-income, minority, and immigrant students. Educational Policy, 19, 262–282.
Parker, D. C., McMaster, K. L., & Burns, M. K. (2011). Determining an instructional level for early writing skills. School Psychology Review, 40, 158–167.
Petrilli, M. (2011). All together now? Educating high and low achievers in the same classroom. Education Next, 11(1), 49–55.
Porter, R. D., & Schick, I. C. (2003). Revisiting Bloom’s taxonomy for ethics and other educational domains. The Journal of Health Administration Education, 20, 167–188.
Siadat, M. V., Musial, P. M., & Sagher, Y. (2008). Keystone method: A learning paradigm in mathematics. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 18, 337–348.
Smith, S. S. (2009). Early childhood mathematics (4th ed.). Boston, MA: Pearson.
Subban, P. (2006). Differentiated instruction: A research basis. International Education Journal, 7, 935–947.
Superfine, A. C. (2008). Planning for mathematics instruction: A model of experienced teachers’ planning processes in the context of a reform mathematics curriculum. The Mathematics Educator, 18(2), 11–22.
Testerman, J. (1996). Holding at-risk students: The secret is one to one. Phi Delta Kappan, 77, 364–365.
Tomlinson, C. A., Brighton, C., Hertberg, H., Callahan, C. M., Moon, T. R., Brimijoin, K., Conover, L. A., & Reynolds, T. (2003). Differentiating
instruction in response to student readiness, interest, and learning profile in academically diverse classrooms: A review of literature. Journal for the Education of the Gifted, 27, 119–145.
The Devalued Student
83
Wentzel, K. R. (1997). Student motivation in middle school: The role of perceived pedagogical caring. Journal of Educational Psychology, 89, 411–419.
Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3), 14–52.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3-5. Boston, MA: Pearson.
The Mathematics Educator
2012 Vol. 22, No. 1, 84–113
Prevalence of Mixed Methods Research in
Mathematics Education
Amanda Ross and Anthony J. Onwuegbuzie
In wake of federal legislation such as the No Child Left Behind Act of
2001 that have called for “scientifically based research in
education,” this study examined the possible trends in mixed methods research articles published in 2 peer-reviewed mathematics
education journals (n = 87) from 2002 to 2006. The study also
illustrates how the integration of quantitative and qualitative
research enhances the findings in mathematics education research.
Mixed methods research accounted for 31% of empirical articles
published in the 2 journals, with a 10% decrease over the 5-year
span. Mixed methods research articles were slightly more
qualitatively oriented, with 59% constituting such a design. Topics
involving mathematical thought processes, problem solving, mental
actions, behaviors, and other occurrences related to mathematical
understanding were examined in these studies. Qualitative and quantitative data were used to complement one another and reveal
relationships between observations and mathematical achievement.
In recent years there have been renewed calls in the United
States for reform in mathematics education research as a result
of federal legislation such as the No Child Left Behind (NCLB)
Act of 2001 (NCLB, 2001) and the Education Sciences Reform
Act (ESRA) of
2002 (ESRA, 2002) that have called for
Amanda A. Ross is an educational consultant and president of A. A. Ross Consulting and Research, LLC. She currently writes and reviews mathematics curriculum and assessment items, creates instructional design components, performs standards alignments, writes preparatory standardized test materials, writes grant proposals, and serves as external
evaluator.
Anthony Onwuegbuzie is a professor in the Department of Educational Leadership and Counseling at Sam Houston State University, where he teaches doctoral-level courses in qualitative research, quantitative research, and mixed research. With a h-index of 47, and writing extensively on qualitative, quantitative, and mixed methodological topics, he has had published more than 340 works, including more than 270 journal articles, 50 book chapters, and 2 books.
Prevalence of Mixed Methods
85
“scientifically based research in education.” In particular, much
of the ensuing debate has revolved around whether or not the purpose of research should be to determine what works.
Moreover, guidelines and review procedures of the Institute of
Education Sciences (U.S. Department of Education) and its
influential What Works Clearinghouse (see www.whatworks.ed.gov) have led some researchers and
policymakers to imply that randomized controlled trials
represent the gold standard for research and that designs associated with qualitative research and mixed methods
research are inferior to quantitative research designs in general
and experimental research designs in particular (Patton, 2006). The current debate in the United States regarding the gold
standard is in stark contrast to the controversy that prevailed
40 years ago when calls abounded to make mathematics
education research more scientific (Lester, 2005). Lester and Lambdin (2003) noted that the use of experimental and quasi-
experimental techniques in mathematics education research
during that time was criticized as being inappropriate for addressing questions of what works. Advocating the need for a
journal devoted solely to mathematics education research, Joe
Scandura (1967), a prominent researcher in the United States during the 1960s and 1970s, concluded:
[M]any thoughtful people are critical of the quality of research
in mathematics education. They look at tables of statistical
data and they say “So what!” They feel that vital questions go
unanswered while means, standard deviations, and t-tests pile
up. (p. iii)
Over the last several decades, mathematics education researchers and policy makers have struggled to agree upon
what represents the most appropriate research approach to use
for research in mathematics education, leading to a form of
research identity crisis. This struggle has been complicated further by federal legislation such as NCLB and ESRA wherein
“scientifically based research in education” has been a
contested phrase in many education fields (cf. McLafferty, Slate, & Onwuegbuzie, 2010). Indeed, little is known about the
effect of this federal legislation on articles published in
mathematics education research journals. In particular, little
Amanda Ross & Anthony J. Onwuegbuzie
86
information appears to exist regarding the extent to which the
published research in mathematics education journals includes what is commonly known as mixed methods (or mixed)
research (Johnson & Onwuegbuzie, 2004).
For the purposes of this paper, we view qualitative
research, quantitative research, and mixed methods research as representing the three major research or methodological
paradigms. We define qualitative research as relying on the
collection, analysis, and interpretation of non-numeric data that naturally occur (Lincoln & Guba, 1985) from one or more of
the sources identified by Leech and Onwuegbuzie (2008): talk,
observations, drawing/photographs/videos, and documents. We define quantitative research as involving the collection,
analysis, and interpretation of numeric data, with the goals of
describing, explaining, and predicting phenomena. We follow
Johnson, Onwuegbuzie, and Turner (2007) in their definition of mixed methods research:
Mixed methods research is an intellectual and practical
synthesis based on qualitative and quantitative research…. It
recognizes the importance of traditional quantitative and
qualitative research but also offers a powerful third paradigm
choice that often will provide the most informative, complete,
balanced, and useful research results. Mixed methods research is the research paradigm that (a) partners with the philosophy
of pragmatism in one of its forms (left, right, middle); (b)
follows the logic of mixed methods research (including the
logic of the fundamental principle and any other useful logics
imported from qualitative or quantitative research that are
helpful for producing defensible and usable research findings);
(c) relies on qualitative and quantitative viewpoints, data
collection, analysis, and inference techniques combined
according to the logic of mixed methods research to address
one’s research question(s); and (d) is cognizant, appreciative,
and inclusive of local and broader sociopolitical realities, resources, and needs. (p. 129)
In addition, mixed methods research can be further classified as
quantitative-dominant, qualitative-dominant (Johnson et al.,
2007), or equal-status mixed methods (where the emphasis
between quantitative and qualitative approaches is evenly split), termed by Morse (1991, 2003) as QUAN-Qual, QUAL-
Quan, and QUAN-QUAL respectively.
Prevalence of Mixed Methods
87
Although both quantitative and qualitative research
methods have many strengths and, if conducted with rigor, can inform mathematics education policy, they each contain unique
weaknesses. Quantitative research is well suited to “answering
questions of who, where, how many, how much, and what is
the relationship between specific variables” (Adler, 1996, p. 5). However, quantitative research studies typically yield data that
do not explain the reasons underlying prevalence rates,
relationships, or differences that have been identified by the researcher. That is, quantitative research is not apt for
answering questions of why and how. In contrast, the strength
of qualitative research lies in its ability to capture the lived experiences of individuals; to understand the meaning of
phenomena and relationships among variables as they occur
naturally; to understand the role that culture plays in the
context of phenomena; and to understand processes that are reflected in language, thoughts, and behaviors from the
perspective of the participants. However, as noted by
Onwuegbuzie and Johnson (2004), “Qualitative research is typically based on small, nonrandom samples…which means
that qualitative research findings are often not very
generalizable beyond the local research participants” (p. 410). Thus, because of the strengths and weaknesses inherent in
mono-method research, in recent years, an increasing number
of researchers from numerous fields have advocated for
conducting studies that utilize both quantitative and qualitative research within the same inquiry—namely, mixed methods
research.
Collins, Onwuegbuzie, and Sutton (2006) have identified four common rationales for mixing quantitative and qualitative
research approaches: participant enrichment, instrument
fidelity, treatment integrity, and significance enhancement.
According to these methodologists, participant enrichment refers to the combining of quantitative and qualitative
approaches for the rationale of optimizing the sample (e.g.,
increasing the number of participants, improving the suitability of the participants for the research study). Instrument fidelity
refers to a combination of quantitative and qualitative
procedures used by researchers to maximize the appropriateness and/or utility of the quantitative and/or
Amanda Ross & Anthony J. Onwuegbuzie
88
qualitative instruments used in the study. Treatment integrity
pertains to the combining of quantitative and qualitative techniques for the rationale of assessing the fidelity of
treatments, programs, or interventions. And, finally,
significance enhancement involves the use of qualitative and
quantitative approaches to maximize the interpretation of the results.
Each of these four rationales can come before, during,
and/or after the study. With respect to participant enrichment, for example, mathematics education researchers could increase
both the quantity and quality of their pool of participants of
either a quantitative or qualitative study by interviewing participants who already have been selected for the study
before the actual investigation begins (i.e., pre-study phase) to
ask them to identify potential additional participants and to
collect (additional) qualitative and quantitative information that establishes their suitability and willingness to participate in the
study (Collins et al., 2006). Alternatively, interviews or other
data collection tools (e.g., survey, rating scale, Likert-format scale) could be used during the study (i.e., study phase), for
instance, to determine each participant’s suitability to continue
in the study or to determine whether any modifications to the design protocol are needed. Further, these tools could be used
after the study ends (i.e., post-study phase) as a means of
debriefing the participants or to identify any outlying, deviant,
or negative cases (Collins et al., 2006). With regard to instrument fidelity, mathematics education researchers might
conduct a pilot study either to assess the appropriateness (e.g.,
score reliability, score validity, clarity, potential to yield rich data) and/or utility (e.g., cost, accessibility) of existing
qualitative and/or quantitative instruments with the goal of
making modifications, where needed, or developing a new
instrument. Alternatively, in studies that involve multiple phases, mathematics education researchers could assess
instrument fidelity on an ongoing basis and make
modifications, where needed, at one or more phases of the study. In addition, mathematics education researchers could
assess the validity/legitimation of the qualitative and/or
quantitative information yielded by the instrument(s) in order to place the findings in a more appropriate context.
Prevalence of Mixed Methods
89
With respect to treatment integrity, mathematics education
researchers could assess the intervention used in a study either quantitatively (e.g., obtaining a fidelity score that indicates the
percentage of the intervention component that was
implemented fully or the degree to which the treatment or
program was implemented) or qualitatively (e.g., via interviews, focus groups, and/or observations). The use of both
quantitative and qualitative techniques for assessing treatment
integrity yields “the greatest insights into treatment integrity” (Collins et al., 2006, p. 82). Finally, with regard to significance
enhancement, mathematics education researchers could use
qualitative data to complement statistical analyses, quantitative data to complement qualitative analyses, or both. Moreover,
using both quantitative and qualitative data analysis techniques
either concurrently or sequentially within the same study can
fulfill one or more of Greene, Caracelli, and Graham’s (1989) five purposes for integrating quantitative and qualitative
approaches: triangulation (i.e., comparing results from
quantitative data with qualitative findings to assess levels of convergence), complementarity (i.e., seeking elaboration,
illustration, enhancement, and clarification of the findings from
one method with results from the other method), initiation (i.e., identifying paradox and contradiction stemming from the
quantitative and qualitative findings), development (i.e., using
the findings from one method to help inform the other method),
or expansion (i.e., expanding the breadth and range of a study by using multiple methods for different study phases). Thus,
using mixed methods research approaches to fulfill one or more
of these four rationales strengthens the design of some research studies.
Although there is a lack of knowledge about the prevalence
of mixed methods research in mathematics education, an
increasing number of researchers regard mixed methods research as representing scientifically based research. For
example, in response to the narrow guidelines and review
procedures of the Institute of Education Sciences, the American Evaluation Association (2003) adopted an official
organizational policy response that included the statement,
“Actual practice and many published examples demonstrate that alternative and mixed methods are rigorous and scientific.
Amanda Ross & Anthony J. Onwuegbuzie
90
To discourage a repertoire of methods would force evaluators
backward” (§ 6). Even members of the National Research Council (NRC), who entered the dispute with a published
consensus statement, Scientific Research in Education (NRC,
2002), supported the utilization of mixed methods research. For
instance, Eisenhart and Towne (2003) noted that the NRC report supports the inclusion of “a range of research designs
(experimental, case study, ethnographic, survey) and mixed
methods (qualitative and quantitative) depending on the research questions under investigation” (p. 31).
Mixed methods research, the integration of qualitative and
quantitative approaches in research studies, began in the 1960s. Campbell and Fiske (1959) are credited with providing the
impetus for mixed methods research by introducing the idea of
triangulation, which was extended further by Webb, Campbell,
Schwartz, and Sechrest (1966). This research approach quickly is becoming prominent in the field of educational research
(e.g., Bazeley, 2009; Denscombe, 2008; Greene, 2007; Happ,
DeVito Dabbs, Tate, Hricik, & Erlen, 2006; Jang, McDougall, Pollon, & Russell, 2008; Johnson & Gray, 2010; Johnson &
Onwuegbuzie, 2004; Johnson et al., 2007; Leech, Dellinger,
Brannagan, & Tanaka, 2010; Molina-Azorín, 2010; O'Cathain, 2010; O'Cathain, Murphy, & Nicholl, 2008; Pluye, Gagnon,
Griffiths, & Johnson-Lafleur, 2009; Teddlie & Tashakkori,
2009, 2010).
The prevalence of mixed methods research in other academic fields and disciplines (e.g., school psychology,
counseling, special education, stress and coping research) has
been investigated (e.g., Alise & Teddlie, 2010; Collins, Onwuegbuzie, & Jiao, 2006, 2007; Collins, Onwuegbuzie, &
Sutton, 2007; Fidel, 2008; Hanson, Creswell, Plano Clark,
Petska, & Creswell, 2005; Hurmerinta-Peltomaki & Nummela,
2006; Hutchinson & Lovell, 2004; Ivankova & Kawamura, 2010; Niglas, 2004; Onwuegbuzie, Jiao, & Collins, 2007;
Powell, Mihalas, Onwuegbuzie, Suldo, & Daley, 2008;
Truscott et al., 2010). In particular, with respect to the field of school psychology, Powell et al. (2008) examined empirical
studies (n = 438) published in the four leading school
psychology journals (i.e., Journal of School Psychology, Psychology in the Schools, School Psychology Quarterly, and
Prevalence of Mixed Methods
91
School Psychology Review) between 2001 and 2005. These
researchers found that 13.7% of these studies were classified as representing mixed methods research. Of these mixed methods
studies, 95.65% placed emphasis on the quantitative
component (i.e., quantitative-dominant mixed methods
research; Johnson et al., 2007), whereas only 4.35% were primarily qualitative in nature (i.e., qualitative-dominant mixed
methods research; Johnson et al., 2007). Similarly, with regard
to the field of special education, Collins, Onwuegbuzie, and Sutton (2007) undertook a content analysis of empirical studies
(n = 131) published in the Journal of Special Education
between 2000 and 2005. These researchers reported that 11.5% of these studies were classified as representing mixed methods
research. Of these mixed methods investigations, 55.6%
represented quantitative-dominant mixed methods research,
22.2% represented qualitative-dominant mixed methods research, and 22.2% represented equal-status mixed methods
research (i.e., the emphasis between quantitative and
qualitative approaches was approximately evenly split). With respect to the field of counseling, Hanson et al. (2005) searched
for mixed methods research studies that had been published in
counseling journals prior to May 2002. These researchers identified only 22 such studies that were published in
counseling journals, with the majority of these articles (40.9%)
being published in the Journal of Counseling Psychology.
Building on the work of these researchers, Leech and Onwuegbuzie (2006) investigated the prevalence of mixed
methods research published in the Journal of Counseling and
Development (JCD) from late 2002 (Volume 80, Issue 3) through 2006 (Volume 84, Issue 4). Of the 99 empirical articles
published in JCD during this period, only 2% represented
mixed methods research. Finally, Onwuegbuzie et al. (2007)
examined the prevalence of mixed methods research related to the area of stress and coping by selecting five major electronic
bibliographic databases (i.e., PsycARTICLES[(EbscoHost];
PsycINFO[(EbscoHost]; Wilson Education Full-Text; CSA Illumina-Psychology; Business Source Premier [EbscoHost])
that represented the fields of psychology, education, and
business. Using the keywords “stress and coping,” these
Amanda Ross & Anthony J. Onwuegbuzie
92
researchers noted that, of the 288 empirical articles that were
identified, 5% represented mixed methods research.
Purpose of this Study
Although researchers have documented the prevalence rate
of mixed methods research in other fields, few articles have been published examining the prevalence of mixed methods
mathematics education research. Recently, Hart, Smith, Swars,
and Smith (2009) examined the prevalence of mixed methods research in mathematics education articles published in six
journals from 1995 to 2005. These researchers documented that
29% of the articles used both approaches in some way. Ross
and Onwuegbuzie (2010) compared the prevalence of mixed methods in a flagship mathematics education journal, Journal
for Research in Mathematics Education (JRME), to the
prevalence in an all-discipline flagship education journal, American Educational Research Journal (AERJ), from 1999 to
2008. Mixed methods research accounted for 33% of all
articles published in these two journals, whereas mixed
methods was found to be more prevalent in JRME. With so few studies of the prevalence of mixed methods research in
mathematics education, this study is important because it
provides additional information regarding the extent to which mathematics education is keeping abreast of the latest
methodological advances in incorporating mixed methods
approaches. We focused on mathematics education articles published in JRME and The Mathematics Educator (TME) to
(a) determine the prevalence of mixed methods research in
mathematics education from 2002-2006, (b) to investigate the
context associated with the use of mixed methods in mathematics education, and (c) to document possible reasons
for using mixed methods in mathematics education research.
This time period was chosen for investigation because it includes articles published after the passage of NCLB and the
publication of the classic mixed methods textbooks (i.e.,
Bryman, 1988; Creswell, 1995; Greene & Caracelli, 1997; Newman & Benz, 1998; Reichardt & Rallis, 1994; Tashakkori
& Teddlie, 1998). Additionally, we compared the prevalence of
mixed methods articles in mathematics education journals to
those in other disciplines. Because previous studies examining
Prevalence of Mixed Methods
93
the prevalence rates of mixed methods research articles in
different disciplines and fields have revealed different distributions according to which component—qualitative or
quantitative—was more dominant (e.g., Alise & Teddlie, 2010;
Powell et al., 2008), this study also examined whether the
articles were QUAN-Qual, QUAL-Quan, or QUAN-QUAL. Finally, to reveal a more complete picture of the research
findings, we analyzed an exemplar mixed methods
mathematics education article to demonstrate how qualitative and quantitative research approaches complement one another.
In particular, we sought to answer the following research
questions:
(i) How has the use of mixed methods research in two
peer-reviewed mathematics education journals,
JRME and TME, changed from 2002 to 2006 and
how does the prevalence of mixed methods research in mathematics education compare to the prevalence
in other academic disciplines?
(ii) Of the articles that utilize mixed methods research in
JRME and TME:
(a) What is the context of the research?
(b) What are the reasons cited in the articles for
the utilization of mixed methods?
(c) What reasons are cited in the articles for their
particular composition of methods (QUAN-
Qual, QUAL-Quan, or Quan-Qual)?
(iii) How can qualitative and quantitative methods
complement one another in providing good educational research in mathematics education?
Method
Sample
This study examined 87 journal articles published in JRME
(n = 60) or TME (n = 27), two peer-reviewed mathematics
education journals. We chose the two journals because of their relatively low acceptance rates (11-20% for JRME and 10-25%
for TME). JRME is widely regarded as the premier
mathematics education journal in the United States and TME
Amanda Ross & Anthony J. Onwuegbuzie
94
provides a publication venue for research conducted by those
new to the field, graduate students and recently minted PhDs. This sample represented all empirical articles published in
these two journals between 2002 and 2006. Non-empirical
articles (n = 75), such as editorials, reviews of the related
literature, and commentaries, were not included in the study. It should be noted that neither journal encouraged the use of
mixed methods research in their mission statements.
Additionally, a mathematics education article (i.e., Wood, Williams, & McNeal, 2006) that exemplified a mixed methods
research design was selected, not only because of the quality of
the study but because it has been highly cited (i.e., 55 citations at the time this article took place; cf. Hirsch, 2005).
Data Collection
We determined if each of the 87 articles in our sample included mixed methods research. Articles were identified as
using a mixed methods design if both qualitative and
quantitative methods were utilized to any meaningful extent.
For example, studies had to include one or more quantitative and qualitative data (such as frequency count and quotations) to
be considered mixed methods. Attempts to classify actual
published studies into distinct categories necessitated the addition of seven categorization rules (see Appendix). For each
article, the particular emphasis used (QUAN-Qual, QUAL-
Quan, QUAN-QUAL), the reasons for using more than one approach, and the context of the study were recorded. The
example mixed methods research article was read closely to
determine the qualitative and quantitative approaches utilized
and the way that each approach provided a more comprehensive understanding of the results.
Data Analysis
After determining the number of articles utilizing mixed
methods research for each journal over the 5-year span, we
calculated the annual and total percentages of mixed methods
usage for each journal, as well as both journals combined, for the years 2002 to 2006. We used these values to describe how
the prevalence of mixed methods research in mathematics
education research has changed over time and to compare these
Prevalence of Mixed Methods
95
rates with those in other academic disciplines. A series of chi-
square tests of homogeneity (cf. Leech & Onwuegbuzie, 2002) was used to compare the prevalence rates (i.e., percentages)
between the number of mixed methods research articles
published in the two mathematics education journals and the
number published in other disciplines for which the sample size and group sizes were reported clearly. A 5% level of statistical
significance was used. Also, effect sizes, as measured by
Cramer’s V, were reported for all statistically significant findings. Also, we computed odds ratios as a second index of
effect size.
After each mixed methods research article was coded according to the emphasized research orientation (QUAN-
Qual, QUAL-Quan, or QUAN-QUAL), the annual total and
percentage for each journal were calculated. In most cases, it
was easy to determine which approach was dominant. However, in some cases, we had to re-examine the purpose of
the article and research questions to determine the emphasis.
Constant comparison analysis (Glaser & Strauss, 1967) was used to determine the reasons for using mixed methods.
Specifically, each identified reason was given a code. Also,
each reason was compared with previous reasons to ensure that similar reasons were labeled with the same thematic code. Each
emergent theme contained one or more reasons that were each
linked to a formulated meaning of significant statements. Thus,
the themes emerged a posteriori, and, in contrast, classification of the utilization of designs occurred a priori using the
predetermined codes, QUAN-Qual, QUAL-Quan, or QUAN-
QUAL. Additionally, we determined the contextual frame of each mixed methods article by identifying the topic.
To demonstrate how combining both quantitative and
qualitative approaches within one research study can provide
more rigorous educational research we chose one mixed methods journal article as an exemplar. We described the
results and inferences stemming from the use of each approach
and then compared these to the overall results and inferences from combining both approaches.
Amanda Ross & Anthony J. Onwuegbuzie
96
Results and Discussion
Mixed methods research constituted approximately one third (31%) of all empirical articles accepted for publication in
JRME and TME from 2002 to 2006; yet the rate of mixed
methods research decreased from 2002 to 2006 from 40% to 30% (Table 1). From 2002 to 2006 the percentage of mixed
methods research articles published in JRME went from 55% to
23%, with 2006 having the lowest percentage. On the other hand, the percentage of mixed methods articles published in
TME increased from 0% in 2002 to 43% by 2006. Over the 5-
year period, JRME actually published more than twice the
percentage of mixed methods research articles than did TME, with 38% and 15%, respectively. Interestingly, no articles
specifically contained the phrase “mixed methods” but two
articles did specify the use of both quantitative and qualitative approaches.
Table 1
Percentages of Mixed Methods Research Studies in JRME and
TME
Year JRME TME Annual Total
2002 6/11 = 55% 0/4 = 0% 6/15 = 40%
2003 6/13 = 46% 1/7 = 14% 7/20 = 35%
2004 3/11 = 27% 0/4 = 0% 3/15 = 20%
2005 5/12 = 42% 0/5 = 0% 5/17 = 29%
2006 3/13 = 23% 3/7 = 43% 6/20 = 30%
Total 23/60 = 38% 4/27 = 15% 27/87 = 31%
The combined 31% prevalence rate found in the current
study for the two selected mathematics education research
journals over a 5-year span is similar to the 29% prevalence of mixed methods in mathematics education journal articles
documented by Hart et al. (2009) from 1995 to 2005. However,
the 31% prevalence rate was much higher than those reported
in other academic disciplines (e.g., Collins, Onwuegbuzie, & Sutton, 2007; Hanson et al., 2005; Leech & Onwuegbuzie,
Prevalence of Mixed Methods
97
2006; Onwuegbuzie et al., 2007; Powell et al., 2008). Lower
prevalence rates for other disciplines have been reported for a similar time span, the two highest rates at 13.7% and 11.5% in
school psychology journals (Powell et al., 2008), and special
education journals, respectively (Collins, Onwuegbuzie, &
Sutton, 2007). Both of these rates are less than one half of the rate of mixed methods research identified in JRME and TME.
For other disciplines, mixed methods research studies are
published with even less frequency, with such studies accounting for only 2% of the published empirical studies in
counseling journals (Leech & Onwuegbuzie, 2006) and only
5% in various research journals that publish stress and coping research (Onwuegbuzie et al., 2007).
More specifically, the 31% prevalence rate is statistically
significantly higher than the prevalence rate observed by
Powell et al. (2008) for leading school psychology journals (Χ
2[1] = 10.30, p < .0013, Cramer’s V = .13), the prevalence
rate observed by Leech and Onwuegbuzie (2006) for a leading
counseling journal (Χ2[1] = 21.62, p < .0001, Cramer’s V =
.32), the prevalence rate observed by Collins, Onwuegbuzie,
and Sutton (2007) for a leading special education journal
(Χ2[1] = 5.97, p < .01, Cramer’s V = .17), and the prevalence
rate observed by Onwuegbuzie et al. (2007) for a the field of
stress and coping (Χ2[1] = 35.60, p < .0001, Cramer’s V = .29).
Moreover, mixed methods research articles were more than
twice as likely to be published in the selected mathematics education journals than in the leading school psychology
journals (Odds ratio = 2.27, 95% Confidence Interval [CI] =
1.36, 3.77) and a leading special education journal (Odds ratio = 2.69, 95% CI = 1.19, 6.07), more than 6 times as likely to be
published in the mathematics education journals than in the
field of stress and coping (Odds ratio = 6.88, 95% CI = 3.40,
13.90), and more than 15 times as likely to be published in the mathematics education journals than in a leading counseling
journal (Odds ratio = 15.36, 95% CI = 3.55, 66.47).
Of the mixed methods articles in both journals over the 5-year period, 59% were qualitative-dominant, whereas
quantitative-dominant articles constituted 33% and equal-status
mixed research articles constituted only 7% (Table 2). Given the increase in qualitative approaches used in mathematics
Amanda Ross & Anthony J. Onwuegbuzie
98
education articles over the past 20 years, it is not surprising that
a qualitative-dominant approach constituted the highest percentage of articles overall, as well as in each of JRME and
TME individually, (cf. Table 3a and 3b). It is also
understandable that fewer articles would constitute a balanced
quantitative-qualitative design.
Table 2
Percentages of Mixed Methods Research Study Emphasis in
JRME and TME Combined
Year n QUAN-Qual QUAL-Quan QUAN-QUAL
2002 6 50% 33% 17%
2003 7 0% 86% 14%
2004 3 67% 33% 0%
2005 5 40% 60% 0%
2006 6 33% 67% 0%
Total 27 33% 59% 7%
Table 3a
Percentages of Mixed Methods Research Study Emphasis in
TME
Year n QUAN-Qual QUAL-Quan QUAN-QUAL
2002 0
2003 1 0% 100% 0%
2004 0
2005 0
2006 3 33% 67% 0%
Total 4 25% 75% 0%
Prevalence of Mixed Methods
99
Table 3b
Percentages of Mixed Methods Research Study Emphasis in JRME
Year n QUAN-Qual QUAL-Quan QUAN-QUAL
2002 6 50% 33% 17%
2003 6 0% 83% 17%
2004 3 67% 33% 0%
2005 5 40% 60% 0%
2006 3 33% 67% 0%
Total 23 35% 57% 9%
Constant comparison analysis provided interesting
information regarding reasons behind researchers’ use of mixed
methods research in mathematics education journals, as well as the emphasis of the mixed methods research designs. Specific
reasons documented throughout the mixed methods research
articles included examination of relationships, ideas, beliefs,
strategies, mental actions, abilities, conceptions, reflections, reasoning development, experiences, self-reports,
understanding, behaviors, determination of differences, effects
of pictorial representations on success, practices, descriptions of courses, performance as ascertained via a variety of
outcomes, and problem solving. All articles involving both
qualitative and quantitative research approaches examined
actions, behaviors, relationships, ideas, and/or understanding. In other words, ideals and outcomes involving more than mere
achievement scores and closed-ended effects required evidence
ascertained from both approaches to support one another. The researchers of these mixed methods articles did not
simply examine outcomes of various independent factors on
student success measured solely quantitatively. Researchers in these studies also did not simply rely solely on analysis of
transcribed or summarized interview data or observations to
determine student knowledge and understanding. Examination
of these two mathematics education journals revealed that their use of mixed methods research was needed to delve deeper into
Amanda Ross & Anthony J. Onwuegbuzie
100
teachers’ and students’ behaviors, actions, and understandings.
Articles utilizing both methods required data that supported ideas that could be understood via description and statistical
techniques—whether categorical data, or achievement scores.
The high percentage of mixed methods research published
in these journals indicates a growing desire of mathematics education researchers to include thought processes,
occurrences, actions, and behaviors as related to student
achievement outcomes and successful instruction. No longer are mathematics education researchers only collecting and
analyzing either quantitative or qualitative data, they are
realizing the value in combining description, narration, summaries, comparisons, patterns, and so on, as they impact
mathematical understanding. Noteworthy is the fact that most
mixed methods research studies involve mathematical
understanding, not simply knowledge or skills. The National Council of Teachers of Mathematics (2000) advocates the
combined attainment of conceptual and procedural
understanding in mathematics. The findings revealed the importance of mixed methods, qualitative, and quantitative
approaches in these two mathematics education journals.
Mixed methods research again constituted 31% of all empirical articles, whereas qualitative and quantitative research
accounted for 39% and 21%, respectively. It should be noted
that qualitative studies accounted for the highest proportion of
empirical articles and that qualitative-dominant mixed methods research designs accounted for the highest percentage of mixed
methods research. With the movement towards overall
mathematical literacy (Hiebert & Carpenter, 1992; Van de Walle, 2001), constructivist approaches (von Glasersfeld,
1997), and standards-based curriculum (National Council of
Teachers of Mathematics [NCTM], 2000), the findings might
suggest that mathematics education researchers are interested in revealing a big picture associated with mathematics teaching
and learning, with high emphasis on thinking patterns,
behaviors, understanding, and the relations thereof, providing justification for a higher proportion of qualitative-dominant
mixed methods research articles.
The constant comparison analysis revealed reasons behind orientation of mixed methods research articles. Articles labeled
Prevalence of Mixed Methods
101
as QUAN-Qual were designed to investigate levels of
understanding, levels of correctness, classification, correlations, categorization, significance, and accuracy—to
name a few research objectives. Articles labeled as QUAL-
Quan were designed to depict actions and behaviors via
detailed descriptions and pictorial representations of thinking patterns, problem solving, and social discourse. Researchers
who used qualitative-dominant studies also sought to examine
processes underlying understanding, instead of merely identifying relationships between a priori variables and levels
of understanding. Specifically, researchers of qualitative-
dominant mixed methods research studies examined mental actions, discourse, verbal justifications, beliefs, correlations
between observations and scores, conceptions, social
interactions, task descriptions, observed qualities, and problem
solving processes. These researchers reported richer data than would have been the case if data from only one strand (e.g.,
quantitative findings) had been reported. Thus, findings from
both the quantitative and qualitative components of quantitative-dominant mixed methods research studies and
qualitative-dominant mixed methods research studies provided
justification for the use of each approach. Analysis of the 27 mixed methods research studies
revealed the following five major contextual themes in
mathematics education research: relationships, thought
processes, pedagogy, representations, and understanding (Table 4). Exemplars whose topics of study were these themes
included levels of abstraction, levels of representation,
understanding of fractions, teachers’ ideas, arithmetic and algebraic problem-solving skills, thinking patterns, and beliefs
about fairness—to name a few research areas.
Amanda Ross & Anthony J. Onwuegbuzie
102
Table 4
Contexts Associated with Mathematics Education Mixed Methods Articles
Contextual
Themes
Exemplars
Relationships Japanese students’ level of abstraction and level of
representation
Ethnicity, out-of-school activities, and arithmetical
achievements of Latin American and Korean
American students
Normative patterns of social interaction and children’s
mathematical thinking
Third graders’ use of reference point and guess-and-
check strategies and accuracy
Thought Processes
Math majors’ reflections on proofs
Inservice teachers’ conceptions of proof
Students’ conception of mathematical definition
Preservice teachers’ conceptions of how materials
should be used
Preservice teachers’ arithmetic and algebraic problem-
solving strategies
Mental actions involved in covariational reasoning
Reasoning development and thinking patterns of
middle school students
Beliefs about fairness of dice
Sixth and seventh graders’ problem- solving strategies
Seventh and eighth graders’ problem- solving
strategies, specifically in algebra
Pedagogy Third-grade teacher’s efforts to support the
development of students’ algebraic skills
Formal evaluative events across courses in a range of
institutions in South Africa
Japanese and U.S. teachers’ ideas on teaching
strategies
Extent to which teachers implement mathematics
education reform
Compatibility of teaching practices of fourth-grade
teachers with NCTM Standards
Prevalence of Mixed Methods
103
Representations High school students’ calculus diagrams
Use of representations in write-ups
Understanding Fourth and fifth graders’ understanding of
fractions
Two low-performing first-grade students’
understanding
Preservice teachers’ understanding of prime
numbers
Above-average high school students’ calculus
and algebra skills and understanding
Middle school students’ understanding of the
equal sign and performance in solving algebraic
equations
Performance of NCTM-oriented students on standardized tests
Sample Mixed Methods Research Article
Qualitative and quantitative research approaches can be
used to complement one another in mathematics education
research articles. We used a sample mixed methods research article to illustrate how mixed methods techniques can be used
in mathematics education, as well as to illustrate the factors
influencing such a complementary design. The sample article, entitled, “Children’s Mathematical Thinking in Different
Classroom Cultures,” published in 2006, was taken from
JRME. This article (Wood et al., 2006) focused on
investigating effects of social interaction on children’s mathematical thinking. Using what they referred to as a
“quantitative-qualitative research paradigm” (p. 229), they
observed 42 classroom lessons, in order to investigate children's mathematical thinking as articulated in class
discussions and their interaction patterns. The analysis used
two coding schemes—one for interaction patterns and one for mathematical thinking. Classroom observation transcription
notes were used to reveal qualitative and quantitative findings,
related to both coding schemes. Qualitative research
approaches included transcription of classroom dialogue, identification of classroom cultures, identification of consistent
Amanda Ross & Anthony J. Onwuegbuzie
104
patterns of interaction within segments and across lessons, and
the provision of examples of interaction patterns described per classroom culture. Quantitative research approaches included
calculation of percentages of interaction patterns and
mathematical thinking by class culture (conventional textbook,
conventional problem solving, strategy reporting, and inquiry/argument) and calculation of percentages of children’s
levels of spoken mathematical thinking (via coding of
dialogue). Transcripts of dialogue for each mathematical problem were coded as a particular interaction pattern (e.g., a
hinted solution, inquiry, exploration of methods). The
percentages of occurrences for 17 interaction patterns were calculated for each of the class cultures. Types of mathematical
thinking (recognizing, building-with, and constructing) were
also examined quantitatively, via calculation of percentages of
each type that occurred at the following levels: comprehension, application, analyzing, synthetic-analyzing, evaluative-
analyzing, synthesizing, and evaluating.
Many articles necessitate both types of data collection approaches in order to answer the underlying research
questions. In this study, simply providing the transcribed
dialogue and/or segmenting the dialogue into pieces would not have illustrated the frequency of types of interactions or the
level of student understanding. Such data collection called for
quantitative coding of the data. In fact, with the ability to
segment the classroom cultures, interaction patterns, and mathematical thinking, the study required numerical data to
support the qualitative-dominant study. Frequency scores
allowed the researchers to explore the relationship among interaction types, expressed mathematical thinking, and
classroom culture. Reform-oriented class cultures revealed
more student-dominated participation, as well as higher
percentages of higher level thinking. The transcribed student solutions showed the exact facets of such higher level thinking
and discourse. In summary, both approaches used together
revealed that social interaction does, in fact, increase children’s thinking.
Prevalence of Mixed Methods
105
Conclusion
The results of this study have revealed the increasing role of mixed methods designs in mathematics education research
studies. Despite federal legislation such as NCLB and ESRA
that have called for scientifically based research in education—wherein randomized controlled trials were deemed
to represent the gold standard for research—approximately one
third of all empirical articles published in these two mathematics education research journals over a 5-year span
represented mixed methods research studies. Bearing in mind
the utility of mixed methods research (Collins et al., 2006;
Greene, 2007; Johnson & Onwuegbuzie, 2004; Johnson et al., 2007), this finding is very encouraging because the present
study provided evidence that mixed methods research is being
used by a significant proportion of mathematics education researchers whose articles are published in these journals. Such
articles provide in-depth descriptions of tangible and intangible
variables, as related to improvement of students’ mathematical
understanding. However, it remains for mathematics education researchers to optimize their mixed methods research designs;
they will decide how to design and to modify such studies to
best meet their needs. This can be accomplished by utilizing frameworks for conducting mixed methods research that have
been developed for many disciplines belonging to the health or
social and behavioral science fields. For instance, Collins, Onwuegbuzie, and Sutton’s (2006) framework could be used to
help researchers determine their rationale for mixing
quantitative and qualitative research approaches. These
researchers conceptualized that four rationales (participant enrichment, instrument fidelity, treatment integrity, and/or
significance enhancement) can be addressed before, during,
and/or after the study. For example, in mathematics education a researcher might administer a quantitative measure and
conduct interviews or observations to assess the fidelity of an
instructional treatment, program, or intervention, thereby using mixed methods to establish treatment fidelity. Using these
methods at different points in the research process can support
the researchers’ goals in different ways. Assessing treatment
fidelity at the outset of the study can help assess the feasibility of the treatment protocol being implemented in a rigorous and
Amanda Ross & Anthony J. Onwuegbuzie
106
comprehensive manner; during the study it can provide
formative evaluation of the fidelity of the treatment to determine whether mid-course adjustments are needed; and
after the study it can provide summative evaluation of the
treatment to determine the extent to which fidelity occurred.
This example highlights that using such a framework could help mathematics education researchers view the combining of
quantitative and qualitative approaches as a fluid process that
can occur at any stage of the research. Perhaps because of the uniqueness of mathematics education, a framework needs to be
developed for utilizing mixed methods techniques in
mathematics education research. In any case, determining appropriate frameworks for mathematics education researchers
should be the subject of future research.
.
References
Adler, L. (1996). Qualitative research of legal issues. In D. Schimmel (Ed.), Research that makes a difference: Complementary methods for examining legal issues in education, NOLPE Monograph Series (No. 56) (pp. 3-31). Topeka, KS: NOLPE.
Alise, M. A., & Teddlie, C. (2010). A continuation of the paradigm wars? Prevalence rates of methodological approaches across the social/behavioral sciences. Journal of Mixed Methods Research, 4, 103–126. doi:10.1177/1558689809360805
American Evaluation Association. (2003). American Evaluation Association
Response to U. S. Department of Education Notice of proposed priority, Federal Register RIN 1890-ZA00, November 4, 2003: Scientifically based evaluation methods. Retrieved February 19, 2008, from http://www.eval.org/doestatement.htm
Bazeley, P. (2009). Mixed methods data analysis. In S. Andrew & E. J. Halcomb (Eds.), Mixed methods research for nursing and the health sciences (pp. 84–118). Chichester, UK: Wiley-Blackwell.
Bryman, A. (1988). Quantity and quality in social research. London,
England: Unwin Hyman.
Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56, 81–105. doi:10.1037/h0046016
Prevalence of Mixed Methods
107
Collins, K. M. T., Onwuegbuzie, A. J., & Jiao, Q. G. (2006). Prevalence of mixed methods sampling designs in social science research. Evaluation and Research in Education, 19, 83–101. doi:10.2167/eri421.0
Collins, K. M. T., Onwuegbuzie, A. J., & Jiao, Q. G. (2007). A mixed methods investigation of mixed methods sampling designs in social and
health science research. Journal of Mixed Methods Research, 1, 267–294. doi:10.1177/1558689807299526
Collins, K. M. T., Onwuegbuzie, A. J., & Sutton, I. L. (2006). A model incorporating the rationale and purpose for conducting mixed methods research in special education and beyond. Learning Disabilities: A Contemporary Journal, 4, 67–100.
Collins, K. M. T., Onwuegbuzie, A. J., & Sutton, I. L. (2007, February). The role of mixed methods in special education. Paper presented at the
annual meeting of the Southwest Educational Research Association, San Antonio, TX.
Creswell, J. W. (1995). Research design: Qualitative and quantitative approaches. Thousand Oaks, CA: Sage.
Denscombe, M. (2008). Communities of practice: A research paradigm for the mixed methods approach. Journal of Mixed Methods Research, 2, 270–283. doi:10.1177/1558689808316807
Education Sciences Reform Act. (Pub. L. No. 107-279). (2002). Retrieved
February 18, 2008, from http://www.ed.gov/policy/rschstat/leg/PL107-279.pdf
Eisenhart, M., & Towne, L. (2003). Contestation and change in national policy on “scientifically based” education research. Educational Researcher, 32(7), 31–38. doi:10.3102/0013189X032007031
Fidel, R. (2008). Are we there yet? Mixed methods research in library and information science. Library &Information Science Research, 30, 265–272.
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago, IL: Aldine.
Greene, J. C. (2007). Mixed methods in social inquiry. San Francisco, CA: Jossey-Bass.
Greene, J. C., & Caracelli, V. W. (1997). Advances in mixed-method evaluation: The challenges and benefits of integrating diverse paradigms. New Directions for Evaluations, 74. San Francisco, CA: Jossey-Bass.
Greene, J. C., Caracelli, V. J., & Graham, W. F. (1989). Toward a conceptual
framework for mixed-method evaluation designs. Educational Evaluation and Policy Analysis, 11, 255–274.
Amanda Ross & Anthony J. Onwuegbuzie
108
Hanson, W. E., Creswell, J. W., Plano Clark, V. L., Petska, K. S., & Creswell, J. D. (2005). Mixed methods research designs in counseling psychology. Journal of Counseling Psychology, 52, 224–235. doi:10.1037/0022-0167.52.2.224
Happ, M. B., DeVito Dabbs, D. A., Tate, J., Hricik, A., & Erlen, J. (2006).
Exemplars of mixed methods data combination and analysis. Nursing Research, 55(2, Supplement 1), S43–S49. doi:10.1097/00006199-200603001-00008
Hart, L. C., Smith, S. Z., Swars, S. L., & Smith, M. E. (2009). An examination of research methods in mathematics education. Journal of Mixed Methods Research, 30, 26–41. doi:10.1177/1558689808325771
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A Grouws (Ed.), Handbook of research on
mathematics teaching and learning (pp. 65–97). New York, NY: Macmillan.
Hirsch, J. E. (2005). An index to quantify an individual’s scientific research output. Proceedings of the National Academy of Sciences, 102, 16569–16572.
Hurmerinta-Peltomaki, L., & Nummela, N. (2006). Mixed methods in international business research: A value-added perspective. Management International Review, 46, 439–459. doi:10.1007/s11575-006-0100-z
Hutchinson, S. R., & Lovell, C. D. (2004). A review of methodological characteristics of research published in key journals in higher education: Implications for graduate research teaching. Research in Higher Education, 45, 383–403.
Ivankova, N. V., & Kawamura, Y. (2010). Emerging trends in the utilization of integrated designs in the social, behavioral, and health sciences. In A. Tashakkori & C. Teddlie (Eds.), Sage handbook of mixed methods in social and behavioral research (2nd ed., pp. 581–611). Thousand Oaks,
CA: Sage.
Jang, E. E., McDougall, D. E., Pollon, D., & Russell, M. (2008). Integrative mixed methods data analytic strategies in research on school success in challenging environments. Journal of Mixed Methods Research, 2, 221–247. doi:10.1177/1558689808315323
Johnson, R. B., & Gray, R. (2010). A history of philosophical and theoretical issues for mixed methods research. In A. Tashakkori & C. Teddlie (Eds.), Sage handbook of mixed methods in social and behavioral research (2nd ed., pp. 69–94). Thousand Oaks, CA: Sage.
Johnson, R. B., & Onwuegbuzie, A. J. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14–26. doi:10.3102/0013189X033007014
Prevalence of Mixed Methods
109
Johnson, R. B., Onwuegbuzie, A. J., & Turner, L. A. (2007). Toward a definition of mixed methods research. Journal of Mixed Methods Research, 1, 112–133. doi:10.1525/sp.1960.8.2.03a00030
Leech, N. L., Dellinger, A., Brannagan, K. B., & Tanaka, H. (2010). Evaluating mixed research studies: A mixed methods approach. Journal
of Mixed Methods Research, 4, 17–31.
Leech, N. L., & Onwuegbuzie, A. J. (2002, November). A call for greater use of nonparametric statistics. Paper presented at the annual meeting of the Mid-South Educational Research Association, Chattanooga, TN.
Leech, N. L., & Onwuegbuzie, A. J. (2006, November). Mixed methods research in counseling research: The past, present, and future. Paper presented at the annual meeting of the Mid-South Educational Research Association, Birmingham, AL.
Leech, N. L., & Onwuegbuzie, A. J. (2008). Qualitative data analysis: A compendium of techniques and a framework for selection for school psychology research and beyond. School Psychology Quarterly, 23, 587–604. doi:10.1037/1045-3830.23.4.587
Lester, F. K. (2005). On the theoretical, conceptual, and philosophical foundations for research in mathematics education. ZDM, 37, 457–467. doi:10.1007/BF02655854
Lester, F. K., & Lambdin, D. V. (2003). From amateur to professional: The
emergence and maturation of the U. S. mathematics education research community. In G. M. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (pp. 1629–1700). Reston, VA: National Council of Teachers of Mathematics.
Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage.
McLafferty, C. L., Slate. J. R., & Onwuegbuzie, A. J. (2010). Transcending the quantitative–qualitative divide with mixed methods: A
multidimensional framework for understanding congruence, coherence, and completeness in the study of values. Counseling and Values, 55, 46–62.
Molina-Azorín, J. F. (2010). The use and added value of mixed methods in management research. Journal of Mixed Methods Research, 5, 7–24. doi:10.1177/1558689810384490
Morse, J. (1991). Approaches to qualitative-quantitative methodological triangulation. Nursing Research, 40, 120–123. doi:10.1097/00006199-199103000-00014
Morse, J. (2003). Principles of mixed methods and multimethod research design. In A. Tashakkori & C. Teddlie (Eds.) Handbook of mixed methods in social and behavioral Research (pp. 189–208). Thousand Oaks, CA: Sage.
Amanda Ross & Anthony J. Onwuegbuzie
110
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Research Council. (2002). Scientific research in education. Washington, DC: National Academy Press.
Newman, I., & Benz, C. R. (1998). Qualitative-quantitative research
methodology: Exploring the interactive continuum. Carbondale, IL: Southern Illinois University Press.
Niglas, K. (2004). The combined use of qualitative and quantitative methods in educational research. Unpublished doctoral dissertation, Tallinn Pedagogical University, Tallinn, Estonia.
No Child Left Behind Act of 2001. (2001). Pub. L. No. 107-110.
O’Cathain, A. (2010). Assessing the quality of mixed methods research: Towards a comprehensive framework. In A. Tashakkori & C. Teddlie
(Eds.), Handbook of mixed methods in social and behavioral research (2nd ed., pp. 531–555).Thousand Oaks, CA: Sage.
O’Cathain, A., Murphy, E., & Nicholl, J. (2008). The quality of mixed methods studies in health services research. Journal of Health ServicesResearch Policy, 13, 92–98. doi:10.1258/jhsrp.2007.007074
Onwuegbuzie, A. J., Jiao, Q. G., & Collins, K. M. T. (2007). Mixed methods research: A new direction for the study of stress and coping. In. G. Gates (Ed.), Emerging thought and research on students, teacher, and
administrator stress and coping (Research on Stress and Coping in Education, Vol. 4) (pp. 215-243). Greenway, CT: Information Age.
Onwuegbuzie, A. J., & Johnson, R. B. (2004). Mixed method and mixed model research. In R.B. Johnson & L. B. Christensen (Eds.), Educational research: Quantitative, qualitative, and mixed approaches (pp. 408–431). Needham Heights, MA: Allyn & Bacon.
Patton, M. Q. (2006). Foreword: Trends and issues as context. Research in the Schools, 13(1), i–ii.
Pluye, P., Gagnon, M., Griffiths, F., & Johnson-Lafleur, J. (2009). A scoring system for appraising mixed methods research, and concomitantly appraising qualitative, quantitative, and mixed methods primary studies in mixed studies reviews. International Journal of Nursing Studies, 46, 529–546.
Powell, H., Mihalas, S., Onwuegbuzie, A. J., Suldo, S., & Daley, C. E., (2008). Mixed methods research in school psychology: A mixed methods investigation of trends in the literature. Psychology in the Schools, 45, 291–309. doi:10.1002/pits.20296
Reichardt, C. S., & Rallis, S. F. (1994). (Eds.). The qualitative-quantitative debate: New Perspectives. New Directions for Evaluation, 61. San Francisco, CA: Jossey-Bass.
Prevalence of Mixed Methods
111
Ross, A., & Onwuegbuzie, A. J. (2010). Mixed methods research design: A comparison of prevalence in JRME and AERJ. The International Journal of Multiple Research Approaches, 4, 233–245.
Scandura, J. M. (Ed.). (1967). Research in mathematics education. Washington, DC: National Council of Teachers of Mathematics.
Tashakkori, A., & Teddlie, C. (1998). Mixed methodology: Combining qualitative and quantitative approaches. Applied Social Research Methods Series (Vol. 46). Thousand Oaks, CA: Sage.
Teddlie, C., & Tashakkori, A. (2009). Foundations of mixed methods research: Integrating quantitative and qualitative techniques in the social and behavioral sciences. Thousand Oaks, CA: Sage.
Teddlie, C., & Tashakkori, A. (2010). Overview of contemporary issues in mixed methods research. In A. Tashakkori & C. Teddlie (Eds.),
Handbook of mixed methods in social and behavioral research (2nd ed., pp. 1–41). Thousand Oaks, CA: Sage.
Truscott, D., Swars, S., Smith, S., Thornton-Reid, F., Zhao, Y., & Dooley, C. (2010). A cross-disciplinary examination of the prevalence of mixed methods in educational research: 1995-2005. International Journal of Social Research Methodology, 13, 317–328. doi:10.1080/13645570903097950
Van de Walle, J. A. (2001). Elementary and middle school mathematics:
Teaching developmentally (4th ed.). New York, NY: Longman.
von Glasersfeld, E. (1997). Amplification of a constructivist perspective. Issues in Education, 3, 203–211.
Webb, E. J., Campbell, D. T., Schwartz, R. D., & Sechrest, L. (1966). Unobtrusive measures. Chicago, IL: Rand McNally.
Wood, T., Williams, G., & McNeal, B. (2006). Children’s mathematical thinking in different classroom cultures. Journal for Research in Mathematics Education, 37, 222–255.
Amanda Ross & Anthony J. Onwuegbuzie
112
Appendix
Decision Rules for Classifying Articles Published in
Selected Mathematics Education Journals
Rule 1. Studies were not coded as representing mixed methods
research if the addition of the qualitative information was not
systematic and/or planned. For example, reporting spontaneous, anecdotal comments from study participants in
the discussion section of a quantitative study did not result in a
mixed methods research designation.
Rule 2. Mere use of interview methods during data collection
did not automatically result in a mixed methods research
designation. Furthermore, structured or semistructured interviews that generated solely or predominantly quantitative
data, such as frequency counts or a list of target behaviors,
were not considered as being representative of qualitative
research.
Rule 3. In studies that used small sample sizes to evaluate
quantitatively intervention effectiveness, detailed background
information about participant(s) was not coded as representing a qualitative component.
Rule 4. Reporting planned collection of qualitative data for the
purpose of assessing or verifying the appropriateness of an intervention resulted in a mixed methods research designation
(assuming that quantitative data were collected solely for the
purpose of evaluating treatment outcomes). Even intervention
studies that reported only quantitative analyses in the results section were still coded as mixed methods research if the brief
discussion of treatment integrity included qualitative data.
Rule 5. Mixed methods research studies in which the qualitative component was essential in order for the remainder
of the study to be conducted, and those studies that reported
and analyzed both qualitative and quantitative data were coded
as mixed methods research. For example, studies employing qualitative methods (e.g., focus groups, open-ended
questionnaires) in order to develop the measurement tool that
was used in the remainder of the study were designated as
Prevalence of Mixed Methods
113
mixed methods research because the completion of the study
was contingent on the creation of the instrument.
Rule 6. Content analyses were coded as quantitative if the
results of the content analysis were reported numerically (e.g.,
this study). If the content analysis yielded themes that were not
quantified in any way, the study was coded as representing qualitative research.
Rule 7. Highlighting case examples from a larger quantitative
study did not result in a mixed methods research designation unless the case example section was augmented by new
qualitative data (as opposed to simply an in-depth examination
of the quantitative data yielded from the case examples who were participants in the larger quantitative study).
114
REVIEWERS FOR
THE MATHEMATICS EDUCATOR, VOLUME 22, ISSUE 1
The editorial board of The Mathematics Educator would like to take
this opportunity to recognize the time and expertise our many
volunteer reviewers contribute. We have listed below the reviewers
who have helped make the current issue possible through their invaluable advice for both the editorial board and the contributing
authors. Our work would not be possible without them.
Kimberly Bennekin
Behnaz Rouhani
Georgia Perimeter College
Stephen Bismarck
Keene State College
Laurel Bleich
The Westminster Schools
Margaret Breed
RMIT University
Rachael Brown Knowles Science. Teaching
Foundation.
Günhan Çağlayan Columbus State University
Samuel Cartwright
Fort Valley State University
Alison Castro-Superfine Danny Martin
Mara Martinez
University of Illinois at Chicago
Lu Pien Cheng
National Institute. of Singapore
Nell Cobb
DePaul University
Jill Cochran
Texas State University
Shawn Broderick
Tonya Brooks
Victor Brunaud-Vega Amber Candela
Nicholas Cluster
Anna Marie Conner
Zandra DeAraujo Tonya DeGeorge
Richard Francisco
Christine Franklin Jackie Gammaro
Erik Jacobson
Jeremy Kilpatrick Brenda King
David Liss, II
Anne Marie Marshall
Kevin Moore John Olive
Ronnachai Panapoi
Laura Singletary Ryan Smith
Denise A. Spangler
Leslie P. Steffe
Dana TeCroney Kate Thompson
Patty Wagner
James Wilson The University of Georgia
Michael McCallum
Georgia Gwinnett College
115
Mark Ellis
California. State University,
Fullerton
Kelly Edenfield Filyet Asli Ersoz
Kennesaw State University
Ryan Fox Pennsylvania. State University,
Abington
Brian Gleason
University of New Hampshire
Eric Gold
University of Massachusetts,
Dartmouth
Sibel Kazak
Pamukkale Üniversitesi
Hulya Kilic Yeditepe University
Hee Jung Kim
Louisiana State University
Yusuf Koc Indiana University, Northwest
Terri Kurz
ASU, Polytechnic
Ana Kuzle
Universität Paderborn
Carmen Latterell University of Minnesota, Duluth
Brian Lawler
California State University, San
Marcos
Soo Jin Lee
Montclair State University
Norene Lowery
Houston Baptist University
Anderson Hassell Norton, III
Virginia Tech
Molade Osibodu African Leadership Academy
Drew Polly
University of North Carolina
Charlotte
Ginger Rhodes
University of North Carolina
Wilmington
Kyle Schultz
James Madison University
Jaehong Shin Korea National University of
Education
Ann Sitomer
Portland Community College
Susan Sexton Stanton
East Carolina University
Erik Tillema Indiana University-Perdue
University Indianapolis
Andrew Tyminski Clemson University
Catherine Vistro-Yu
Ateneo de Manila University
If you are interested in becoming a reviewer for The Mathematics
Educator, contact the Editors at [email protected].
In This Issue,
Guest Editorial… Why Mathematics? What
Mathematics? ANNA SFARD
The Roles They Play: Prospective Elementary Teachers
and a Problem-Solving Task
VALERIE SHARON
Discoursing Mathematically: Using Discourse Analysis
to Develop a Sociocritical Perspective of Mathematics
Education
ARIA RAZFAR
The Devalued Student: Misalignment of Current
Mathematics Knowledge and Level of Instruction
STEVEN D. LEMIRE, MARCELLA L. MELBY, ANNE
M. HASKINS, & TONY WILLIAMS
The Prevalence of Mixed Methods Research in
Mathematics Education
AMANDA ROSS & ANTHONY J. ONWUEGBUZIE
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 ht tp://w w w .ugamesa.org