T
M
E
Volume 22 Number 1
Summer 2012
MATHEMATICS EDUCATION STUDENT ASSOCIATION

Editorial Staff A Note From the Editor
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
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.
This publication is supported by the
College of Education at The University of Georgia.

T
M
E
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Summer 2012 Volume 22 N umber 1
3
17
39
63
Guest Editorial … Why Mathem atics? What
Mathem atics?
A N N A SFA RD
The Roles They Play: Prosp ective Elem entary
Teachers and a Problem -Solving Task
V A LERIE SHA RON
Dicou rsing Mathem atically: Using Discou rse
Analysis to Develop a Sociocritical
Persp ective of Mathem atics Ed u cation
A RIA RA Z FA R
The Devalu ed Stu d ent: Misalignm ent of
Cu rrent Mathem atics Know led ge and Level of Instru ction
STEV EN D. LEM IRE, M A RCELLA L. M ELBY ,
A N N E M . HASKIN S, & TON Y W ILLIA M S
84 Prevalence of Mixed Methods Research in
Mathem atics Edu cation
A M A N DA ROSS & A N THON Y J.
ON W UEGBUZ IE
114 A N ote to Review ers
© 2012 Mathematics Education Student Association
All Rights Reserved

The Mathematics Educator
2012 Vol. 22, No. 1, 3–16
“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 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.

Anna Sfard 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
4

Why Mathematics? 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
5

Anna Sfard 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
6

Why Mathematics? 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.
7

Anna Sfard
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
8

Why Mathematics? 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 world
3 —I can also present some firsthand 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
9

Anna Sfard 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
10

Why Mathematics? 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
11

Anna Sfard 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
12

Why Mathematics? 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 contextdependent 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
13

Anna Sfard 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
14

Why Mathematics? 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.p
df
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.
15

Anna Sfard
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.
16

The Mathematics Educator
2012 Vol. 22, No. 1, 17–38
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 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.

Valerie Sharon
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
18

The Roles They Play 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 learnerin-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
19

Valerie Sharon 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,
20

The Roles They Play 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
21

Valerie Sharon 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
22

The Roles They Play 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
23

Valerie Sharon 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
24

The Roles They Play
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).
25

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
C

B 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, B

J and J

C .
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
Table 1
Example of Flow Chart
Statement
187 C: Let’s do it this
way.
C C

B B B

J J J

C
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
27

Valerie Sharon 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 problemsolving 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.
28

The Roles They Play
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
29

Valerie Sharon 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?
30

The Roles They Play
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
31

Valerie Sharon 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.
32

The Roles They Play
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
33

Valerie Sharon 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
34

The Roles They Play 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 learnerin-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 28 th
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.
35

Valerie Sharon
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.
36

The Roles They Play
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.
37

Valerie Sharon
Appendix
Interactivity Flow Chart for Triadic Communication
Line #
88
89
C C

B B B

J J J

C
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
38

The Mathematics Educator
2012 Vol. 22, No. 1, 39–62
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 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
40

Discoursing Mathematically mathematical discourse
1
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
41

Aria Razfar 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 “welldeveloped 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
42

Discoursing Mathematically 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
43

Aria Razfar 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.
12
13
14
15
16
17
18
8
9
10
11
4
5
6
7
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
19
20 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
44

Discoursing Mathematically 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
45

Aria Razfar 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
4 intersubjectivity (Holquist, 1990).
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
46

Discoursing Mathematically 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
47

Aria Razfar 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 selfevident. 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
48

Discoursing Mathematically 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
49

Aria Razfar 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 ( x 2
 y 2
 z 2 ), 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 problemsolving 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
50

Discoursing Mathematically development.
Connecting Discourse to Learning and Development
In connecting sociocultural views of learning and development
(especially CHAT
5
) 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 inter personal plane toward the intra personal 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):
ACTION
MEANING
MEDIATION
MEANING
ACTION
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
51

Aria Razfar 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
Mediation
Spontaneous Through reflection, that is, at meta-level with respect to the primary
Predominantly symbolic
Durability
Applicability
Predominantly
Physical
Transient
Highly Restricted
Lasting
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
52

Discoursing Mathematically 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.
2.
Experiments show that Heliconius butterflies are less likely to ovipost on host plants that possess eggs or egglike structures. These egg mimics are an unambiguous example of a plant trait evolved in response to a hostrestricted group of insect herbivores. (professional journal)
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
53

Aria Razfar 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.
2.
Material: The who and what in an interactional frame
(the actors, place, social space, time, and objects present (or referred to) during an interaction.
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”?
54

Discoursing Mathematically
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
55

Aria Razfar
“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 “eighthundred” 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
56

Discoursing Mathematically 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
57

Aria Razfar 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.
58

Discoursing Mathematically
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.
59

Aria Razfar
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 reformoriented 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.
60

Discoursing Mathematically
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.
61

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
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 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.

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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).
64

The Devalued Student
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
65

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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
66

The Devalued Student 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
67

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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
68

The Devalued Student 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 selfdetermination 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
69

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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
70

The Devalued Student
(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:
71

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
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,
72

The Devalued Student 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
73

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
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
Some form of agreement
Some form of agreement
Some form of disagreement
Some form of disagreement
Can understand difficult math class
Never, Sometimes
Often, Always
Never, Sometimes
Often, Always
N
935
542
5,140
4,399
M
46.9
50.9
49.8
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
74

The Devalued Student 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
75

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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 highquality 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
76

The Devalued Student 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
77

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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
78

The Devalued Student 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.
79

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams
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
80

The Devalued Student 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
81

Steven D. LeMire, Marcella L. Melby, Anne M. Haskins, & Tony Williams 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.
82

The Devalued Student
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.
83

The Mathematics Educator
2012 Vol. 22, No. 1, 84–113
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
“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 quasiexperimental 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
85

Amanda Ross & Anthony J. Onwuegbuzie 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.
86

Prevalence of Mixed Methods
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
87

Amanda Ross & Anthony J. Onwuegbuzie 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.
88

Prevalence of Mixed Methods
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.
89

Amanda Ross & Anthony J. Onwuegbuzie
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
90

Prevalence of Mixed Methods
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
91

Amanda Ross & Anthony J. Onwuegbuzie 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
92

Prevalence of Mixed Methods 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
93

Amanda Ross & Anthony J. Onwuegbuzie 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
94

Prevalence of Mixed Methods rates with those in other academic disciplines. A series of chisquare 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.
95

Amanda Ross & Anthony J. Onwuegbuzie
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 5year 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
2002
2003
2004
2005
2006
JRME
6/11 = 55%
6/13 = 46%
3/11 = 27%
5/12 = 42%
3/13 = 23%
TME
0/4 = 0%
1/7 = 14%
0/4 = 0%
0/5 = 0%
3/7 = 43%
Annual Total
6/15 = 40%
7/20 = 35%
3/15 = 20%
5/17 = 29%
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,
96

Prevalence of Mixed Methods
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 (Χ 2
[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
(Χ 2
[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 (Χ 2
[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 5year 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
97

Amanda Ross & Anthony J. Onwuegbuzie 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
2003 7
2004 3
2005 5
2006 6
Total 27
50%
0%
67%
40%
33%
33%
33%
86%
33%
60%
67%
59%
17%
14%
0%
0%
0%
7%
Table 3a
Percentages of Mixed Methods Research Study Emphasis in
TME
98
Year n QUAN-Qual QUAL-Quan QUAN-QUAL
2002 0
2003 1
2004 0
2005 0
2006 3
0%
33%
25%
100%
67%
75%
0%
0%
0% Total 4

Prevalence of Mixed Methods
Table 3b
Percentages of Mixed Methods Research Study Emphasis in
JRME
Year n QUAN-Qual QUAL-Quan QUAN-QUAL
2002 6
2003 6
50%
0%
33%
83%
17%
17%
2004 3
2005 5
2006 3
67%
40%
33%
33%
60%
67%
0%
0%
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
99

Amanda Ross & Anthony J. Onwuegbuzie 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
100

Prevalence of Mixed Methods 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 qualitativedominant 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.
101

Amanda Ross & Anthony J. Onwuegbuzie
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-andcheck 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 problemsolving 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
102

Prevalence of Mixed Methods
Representations
Understanding
High school students’ calculus diagrams
Use of representations in write-ups
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
103

Amanda Ross & Anthony J. Onwuegbuzie 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, evaluativeanalyzing, 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.
104

Prevalence of Mixed Methods
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
105

Amanda Ross & Anthony J. Onwuegbuzie 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
106

Prevalence of Mixed Methods
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.
107

Amanda Ross & Anthony J. Onwuegbuzie
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
108

Prevalence of Mixed Methods
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.
109

Amanda Ross & Anthony J. Onwuegbuzie
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.
110

Prevalence of Mixed Methods
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.
111

Amanda Ross & Anthony J. Onwuegbuzie
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
112

Prevalence of Mixed Methods 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).
113

R EVIEWERS FOR
T HE M ATHEMATICS E DUCATOR , V OLUME 22 , I SSUE 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
Shawn Broderick
Tonya Brooks
Victor Brunaud-Vega
Amber Candela
Keene State College
Nicholas Cluster
Anna Marie Conner
Laurel Bleich
The Westminster Schools
Zandra DeAraujo
Tonya DeGeorge
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
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
Jill Cochran
Texas State University
Michael McCallum
Georgia Gwinnett College
114

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 tme@uga.edu.
115

,
Guest Edit orial … Why Mathematics? What
Mathematics?
A N N A SFA RD
The Roles They Play: Prospective Elementary Teachers and a Problem-Solving Task
V A LERIE SHA RON
D iscoursing Mathematically: Using D iscourse Analysis to D evelop a Sociocritical Perspective of Mathematics
Education
A RIA RA Z FA R
The D evalued Student: Misalignment of Current
Mathematics Know ledge and Level of Instruction
STEV EN D. LEM IRE, M A RCELLA L. M ELBY , A N N E
M . HASKIN S, & TON Y W ILLIA M S
The Prevalence of Mixed Methods Research in
Mathematics Education
A M A N DA ROSS & A N THON Y J. ON W UEGBUZ IE
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.