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American Educational Research Journal

Different Goals, Similar Practices: Making Sense of the Mathematics and Literacy Instruction in a Standards -Based Mathematics Classroom Roni Jo Draper and Daniel Siebert Am Educ Res J 2004; 41; 927 DOI: 10.3102/00028312041004927 The online version of this article can be found at:

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American Educational Research Journal Winter 2004, Vol. 41, No. 4, pp. 927–962

Different Goals, Similar Practices: Making Sense of the Mathematics and Literacy Instruction in a Standards-Based Mathematics Classroom Roni Jo Draper and Daniel Siebert Brigham Young University This article describes both the process and products of a cooperative inquiry project between two educational researchers—one from literacy education and one from mathematics education. The collaboration took place in an undergraduate, inquiry-based mathematics classroom in which the researchers sought to develop a shared vision of learning and literacy. The researchers discovered that they each used a different learning model to make sense of mathematics instruction, and that both of these models obscured important aspects of learning in a Standards-based mathematics classroom. An alternative model of learning and literacy in mathematics that takes into consideration both models is presented, as well as the process through which the researchers negotiated this shared perspective. KEYWORDS: content-area literacy, cooperative inquiry, literacy, mathematics education, text.


iteracy educators have repeatedly issued calls for the teaching of reading and writing across the curriculum. Underlying these calls for literacy is the belief that the goal of all content-area teachers should be to help their students learn to read and write in their fields. From the perspective of literacy educators, these calls have gone largely unheeded. Literacy researchers have consistently reported that content-area teachers, including mathematics teachers, believe that (a) it is someone else’s responsibility to teach reading and writing, (b) they lack the ability or training to teach reading and writing, and/or (c) they do not have the time to provide literacy instruction along with their full content curriculum (Vacca & Vacca, 2002). To literacy educators, it appears that content-area teachers seldom explicitly address literacy in their classes. Researchers and teachers in the content areas may in fact be much more sympathetic to literacy educators’ concerns for reading and writing across Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert the curriculum than what is reflected in literacy education research. For example, in mathematics education, teachers and researchers have long recognized the importance of reading, writing, symbolizing, and communicating. The Standards documents (National Council of Teachers of Mathematics [NCTM], 1989, 2000) in mathematics education tout the importance of reading and writing, and in fact encourage teachers to engage students in reading and writing mathematics. Nevertheless, these calls in the Standards for reading and writing take place within the larger context of encouraging students to communicate their thinking with others so that students can develop a deep understanding of important mathematical concepts and ideas. In these documents, reading and writing appear to be merely tools for learning and understanding mathematics, with little information about how to develop or hone these tools. This orientation toward reading and writing as tools for learning and understanding, as well as a lack of explicit detail concerning how to help students to read and write better, is not unique to the Standards, but in fact permeates much of the literature in mathematics education. The depiction in mathematics education literature of reading and writing as unproblematic tools for learning of mathematics suggests that mathematics and literacy educators have different beliefs about literacy in the mathematics classroom. These differing views of literacy reflect a larger problem of communication between the two fields of mathematics and literacy education. Literature in mathematics education seldom draws upon the rich research literature from the field of literacy education. Even mathematics education research that studies how students represent their ideas by writing and symbolizing fails to draw upon literacy research (Cobb, Yackel, & McClain, 2000). When literacy research and practices are acknowledged by mathematics educators, they are often connected to peripheral activities in mathematics, such as studying historical, philosophical, or popular writings about mathematics (e.g., Borasi & Siegel, 2000; Siegel, Borasi, & Fonzi, 1998), not to the everyday activity of doing mathematics. Conversely, literacy researchers seldom specifically address mathematics education or provide examples of what good literacy instruction looks like in the mathematics classroom. When literacy educators do attempt to address mathematics teaching, they often cite dated studies or practices that do not fit with mathematics educators’ changing notions of what constitutes mathematical activity and understanding RONI JO DRAPER is an Assistant Professor in the David O. McKay School of Education at Brigham Young University, 206-Q MCKB, Provo, UT 84602; e-mail Her research interests include content-area literacy, specifically in the areas of mathematics and science, and teacher education. DANIEL SIEBERT is an Assistant Professor in the Department of Mathematics Education at Brigham Young University, 257 TMCB, Provo, UT 84602; e-mail His research interests include literacy and discourse in mathematics teaching and learning.

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Standards-Based Mathematics and Literacy Instruction (e.g., Manzo, Manzo, & Estes, 2001; Rudnitsky, Etheredge, Freeman, & Gilbert, 1995). Proponents of mathematics reform value reading and writing and thus would benefit from what literacy educators know about effective literacy instruction. Likewise, literacy educators care about reading and writing in mathematics classrooms, but typically do not have a solid background in mathematics, and thus would benefit from consulting with mathematics educators. However, before literacy and mathematics educators can work together effectively, they must establish a bridge of communication between the two fields that recognizes and respects the discipline-specific goals of each field and that attempts to build a common ground from which meaningful literacy instruction in a mathematics classroom can emerge. In this article, we examine the problem of how to establish communication between the fields of mathematics and literacy education. By examining a collaboration between a mathematics education researcher and a literacy researcher, we provide insight into how mathematics and literacy educators can begin to communicate with each other and establish a common ground from which a form of literacy instruction can emerge that meets the goals of both fields of inquiry. Viewed more broadly, the description of our collaboration suggests promising methods for building lines of communication and a shared vision between collaborators from a wide variety of educational specializations.

Views From the Field In this section, we offer a brief discussion of the views from each field so as to further highlight and explain the chasm that exists between mathematics and literacy educators. As we present these descriptions, we acknowledge that neither field is homogeneous in ideas and theories. However, rather than try to capture the complexity of each field, we merely attempt to present the main issues in each discipline so as to further establish that the two fields attend to different phenomena. Also, our intent is not to establish where we fall within these ongoing debates, because our perspectives toward the main issues in these fields represent a part of the process by which we came to establish a common vision, and thus will be explained in detail in the results section. Mathematics Education The field of mathematics education is concerned with helping students develop mathematical understanding and the ability to engage in authentic mathematical activity. Given the seemingly objective nature of mathematics, one might assume that there is consensus about what constitutes mathematical knowledge, understanding, and activity. However, there are in fact at least two different commonly held views concerning what constitutes mathematics and mathematical activity (Ernest, 1989; Thompson, 1984). The view underlying the current reform movement in mathematics education is often referred to in the literature as the problem-solving view of mathematics (Ernest, 1989). According to this perspective, mathematics is created by 929 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert humans to organize and interpret their experience and to solve problems. It is a collection of powerful ideas that is continually expanding and open to revision. People are doing mathematics when they engage in reasoning about situations involving quantities. Thus, mathematical activity includes bounding and solving problems involving quantities and relationships between quantities; forming, testing, and proving conjectures; and communicating mathematical ideas and solutions. Although these activities naturally lead to the development of well accepted procedures for solving problems, as well as widely accepted definitions and “facts,� it is the powerful ideas and concepts that are most crucial in mathematics (Hiebert et al., 1997). Consequently, proponents of the problem-solving view of mathematics advocate the teaching of mathematical concepts and ideas, and believe that students can best learn these by reasoning about and making sense of meaningful contexts that involve quantities and quantitative relationships. The problem-solving view of mathematics differs radically from the instrumentalist view of mathematics, the perspective that currently underpins much of what is traditional mathematics instruction in U.S. public schools. According to the instrumentalist view, mathematics is a set of facts, algorithms, and skills used to complete mathematical exercises (usually those found at the end of each section in the textbook) (Ernest, 1989). The act of doing mathematics consists of selecting the appropriate skill from one’s mathematical toolbox and performing a series of computations or symbol manipulations to produce the same answer as the teacher or in the answer section in the back of the textbook. Because doing mathematics consists of quoting facts and correctly performing computations, memorization and skill practice are of utmost importance in learning mathematics. Proponents of the instrumentalist view of mathematics learning give little attention to understanding the important mathematical concepts that underlie facts and procedures, because they assume that the procedures are the concepts, or that the concepts will become selfapparent once enough facts and procedures have been mastered. The instrumentalist view of mathematics has long been in disfavor with mathematics education researchers and professional mathematics teacher organizations. Starting at least as far back as Skemp (1978), mathematics education researchers have acknowledged the difference between these two views of mathematics, and have advocated for the problem-solving view because it champions understanding and sense making. In fact, understanding and sense making have become the battle cry of mathematics education researchers as more and more studies have uncovered just how little understanding children gain from traditional mathematics instruction that focuses primarily on procedures and facts (Boaler, 1998; Erlwanger, 1973; Sowder, 1988). The Standards documents prepared by the National Council of Teachers of Mathematics (NCTM, 1989, 1991, 1995, 2000) have also expressed concern over the lack of understanding that results from instrumentalist approaches to instruction. Current reform in mathematics education fits well with the problem-solving view of mathematics and its emphasis on understanding, meaning, sense making, and reasoning in authentic mathematical activity. 930 Downloaded from by Armando Loera on October 31, 2009

Literacy Instruction The primary aim of literacy instruction is to support students’ literacy development. Literacy scholars, however, often disagree about the nature of literacy and the reading and writing processes (see Harris & Hodges, 1995). From one viewpoint, reading and writing can be conceptualized narrowly as decoding and encoding text, where text is simply understood as print material. Within this perspective, texts are viewed as vessels of meaning and the act of decoding the symbols into utterances provides the reader with access to meaning. Narrow definitions of literacy, which give preference to reading and writing traditional print material, have been rejected by many literacy researchers as not honoring the role of listening, speaking, and experiencing in the comprehension and understanding of texts. A philosophically different concept of literacy expands it from the ability of an individual to read and/or write to include multiple activities (reading, writing, listening, speaking, viewing, symbolizing, etc.) with multiple associated texts (print, digital, video, symbolic, images, diagrams, graphs, conversations, etc.) (Moje, Dillon, & O’Brien, 2000; Neilson, 1998; Wade & Moje, 2000). Broad definitions of literacy have consequences for what counts as text and what counts as literacy instruction. Conceptions of text that fit with broad definitions of literacy arise from a variety of theoretical viewpoints. From a social constructivist view of learning and knowing, people do not have the ability to convey meaning directly to other people—mind to mind. Instead, human beings endow objects with meaning, and negotiate meanings as they try to make sense of their world and to communicate with one another (Cole, 1990; Moll, 1990). The meanings that individuals create and attach to objects are socially mediated. From a semiotic perspective, the objects that people use to represent, convey, and negotiate meaning are texts (Sebeok & Danesi, 2000). This definition of text expands it to include visual images, symbols, film, and conversations— virtually any object that can be endowed with meaning or that can serve as a tool to mediate meaning making (Vygotsky, 1986). Recently, some literacy educators have considered broader meanings for text, which fit with social constructivist and social semiotic views of communication and consider print as well as nonprint objects as text (Neilson, 1998; Wade & Moje, 2000). Social constructivist and semiotic views of text are useful for literacy educators wishing to honor the variety of texts and literacy in content-area classrooms. Broadened definitions of literacy and text have consequences for literacy instruction, especially in relation to content-area classrooms. There has been a dichotomy within content-area literacy between two views of literacy instruction for content-area classrooms—reading and writing to learn versus learning to read and write. The general notion has been that the elementary grades provide children with the instruction and skills they need to learn to read and write (especially when reading and writing are activities performed with traditional print material). Beyond the elementary grades, then, literacy instruction should focus on using reading and writing as tools 931 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert for learning (Chall, Jacombs, & Baldwin, 1990; Vacca, 2002). In fact, Vacca and Vacca state that, “to help students become literate in a content area does not mean to teach them how to read or write. Instead, reading and writing are tools that they use to think and learn with text in a given subject area” (2002, p. 17, italics in original). Contrast this view with that of Hynd and Stahl (1998), who call for content-area teachers to assist students in making sense of diverse texts—or helping students learn to read and write. They state that, “[teachers] must focus a significant portion of education on accessing, evaluating, and using information if [they] are to develop citizens who truly contribute to the work and worth of society” (Hynd & Stahl, 1998, p. 37). Given the variety of texts within and across disciplines, helping students gain facility with the texts particular to a given discipline becomes the responsibility of a content-area teacher. This implies that content-area teachers should support students’ facility with diverse texts (which may not be limited to print material). Indeed, the motto every teacher a literacy teacher requires that all teachers, regardless of content area, support students’ literacy development rather than simply use literacy as a tool for learning.

Research Questions Our brief description of the theories and perspectives of the fields of mathematics and literacy education suggests that the fields are indeed different in their focus of study and in the theoretical and practical issues they address. At its core, mathematics education is about mathematical learning and thinking, while literacy education is about supporting students’ facility with text. These differences help explain the difficulty that literacy and mathematics educators have had in communicating. However, it is unclear how insurmountable these differences are and what common ground might exist on which literacy and mathematics educators can begin a conversation. In this study, we address the issues of irreconcilable differences and common ground by examining our own research collaboration as we attempted to establish a conversation across our two disciplines. We sought to answer the following two research questions: (a) What are the similarities and differences in instructional goals and practices as they are represented by mathematics and literacy educators? and (b) What is the common ground where literacy and mathematics educators can simultaneously consider literacy and mathematics issues that occur in mathematics classrooms? The first question specifically addresses the degree to which the differences between mathematics and literacy educators are surmountable. The second question aims at locating a common ground from which constructive dialogue between mathematics and literacy educators can emerge.

Methodology The methodology for this study can best be described as a form of cooperative inquiry (Reason, 1994). In cooperative inquiry, coresearchers cycle through phases of (a) immersion in a shared experience and (b) hypotheses 932 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction generation and testing to negotiate a shared perspective. In order to move through these phases, those involved in the activity must act as both corespondents and coresearchers, and they must locate their activity in a shared experience. The shared experience serves as a location to engage in joint action as well as a point for discussion that involves both a description and a critique of the experience. The activity and the dialogue act in discursive ways that allow those involved to bring to the shared experience their perspectives and to cocreate theories that take into account those perspectives. Cooperative inquiry was an ideal methodology for addressing our two research questions, because our first question required a description from our various perspectives and the second question required a critique of our perspectives. Furthermore, this methodology allowed us to take advantage of the multidisciplinary perspectives we each brought to the study by requiring each of us to function both as coresearchers and corespondents (HajdukowskiAhmed, 1998). As we moved back and forth between these two roles, we were not only compelled to listen to one another, but also to broaden our perspectives to accommodate the views and beliefs of our coresearcher. This type of discursive activity led to the establishment of a common ground from which literacy and mathematics educators can think about literacy and learning in a mathematics classroom, which was the purpose of our study. Our use of cooperative inquiry deviated somewhat from the literature in that our focus was not social injustice. Traditionally, cooperative inquiry as a research method has been used in clinical medical research, social work research, and in locations where practitioners and researchers work together to seek understanding and critique of problems related to practice. Cooperative inquiry, along with other forms of participatory or collaborative research, has typically been described as research done in conjunction with individuals who have traditionally not been part of the theory generation done by researchers (see Kremmis & McTaggert, 2000; Reason, 1994; Schensul & Schensul, 1992). In the examples mentioned earlier, people like nurses, tenants in low-income housing projects, and other clinical practitioners typically not members of research teams would be invited to participate in theory and solution generation. Because cooperative inquiry often involves the consideration and legitimatization of marginalized voices, it has become associated with research for social change. However, cooperative methodology was also a legitimate methodology for a study such as ours, where we acted as coresearchers and corespondents to coconstruct theories that honored both our perspectives. Cooperative inquiry is a methodology that allows researchers to conduct research with people, rather than on people (Reason, 1999). Often content-area literacy research is conducted by educational researchers with literacy or even language arts backgrounds who seek the assistance of a content specialist (usually a practicing teacher) as an informant or participant rather than a research collaborator. In our case, we chose cooperative inquiry because of the opportunity it provided us to collaborate with each other rather than simply take advantage of the assistance of the other. This sort of collaboration requires that researchers move beyond describing the 933 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert objective world, which seeks to describe the world from an objective observer’s point of view. Furthermore, the cooperative researcher moves beyond the subjective, which attempts to describe the world from the participant’s point of view (Blumer, 1969). Rather, cooperative research attempts to locate the intersubjective, which requires participants and observers to come to know together (Habermas, 1976). Coresearchers/Corespondents Like other forms of qualitative research, cooperative inquiry relies on the experiences, knowledge, and epistemologies of the researchers to make sense of the data collected from the joint activity. Because researchers serve as a lens through which data are viewed, it is necessary to share our backgrounds and perspectives. In this section, we provide a brief summary of our professional backgrounds and perspectives. The details of our knowledge, beliefs, and working epistemologies will emerge in the results section as they became issues in our struggle to cocreate a shared perspective. Roni Jo Draper came to this study as a literacy researcher and educator with a background in mathematics. Prior to completing her graduate studies, she worked as a high school mathematics teacher. Roni Jo pursued graduate degrees to find ways to help her students think about and learn mathematics. She discovered that many of the ideas she found in her literacy classes resonated with her ideas about cognition and the necessary conditions for supporting students’ thinking and learning. Although she was not completely satisfied with the explanations provided in the literature about how teachers might support literacy in mathematics classrooms, she believed that literacy held some potential to address some of the problems she had faced as a mathematics teacher. However, beyond using literacy to support mathematics instruction, she did not have a clear sense of what kind of mathematics instruction would support literacy. Dan Siebert came to this study as a mathematics education researcher and educator. He briefly taught mathematics in public schools prior to completing his advanced degree in mathematics education. While Dan valued literacy and literacy instruction in general, he did not see the value of literacy instruction in mathematics classrooms. Like other mathematics teachers who espouse a problem-solving view of mathematics, he sought to create a mathematics classroom steeped in discourse that arose naturally as students engaged in solving interesting, complex, and nonroutine problems. Dan believed that the discourse surrounding such activities facilitated students’ construction of mathematical understanding. He saw little connection between discourse and literacy, and thus did not feel that literacy was important in mathematics teaching. Immersion in a Shared Experience Our exchange of ideas would not have been possible without a mathematics class to serve as a context in which we could immerse ourselves in a shared experience. The shared experience we chose for our study was Dan’s 934 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction mathematics classroom, which served as a foundation for discussion. Without the shared experience, our discussions would simply have been another opportunity for two researchers, from two different disciplines, to talk past one another. Instead, the shared experience grounded our conversations and provided us with specific objects around which we could share our related meanings. We both immersed ourselves in Dan’s classroom—Dan as the instructor and Roni Jo as a participant/observer. Dan’s mathematics class was the first in a series of three mathematics and mathematics methods courses for preservice elementary teachers. The major goals of the class were to (a) help students develop rich conceptions of and images for the mathematical topics of fractions, probability, and geometry; (b) challenge students’ traditional beliefs that mathematics is about memorizing facts and procedures and performing tedious calculations; and (c) model Standards-based mathematics instruction so that students could experience the type of instruction that they should engage in when they teach mathematics. Twenty-five students—most of them female (23), most of them juniors, and all of them majoring in elementary or early childhood education—were enrolled in Dan’s class. The three-credit class met twice a week, for a total of 3 hours, for the 15-week semester. We collected various forms of data from Dan’s classroom that would allow us to consider his mathematics instruction from our individual perspectives. First, both of us were present for all sessions. Each session, except the first, was videotaped to provide us with a means for returning to Dan’s instruction to test our interpretations and conjectures. At the same time, Roni Jo created field notes during each session wherein she described Dan’s instruction and the nature of his attention to literacy. The field notes also served as a catalog of the videotaped class sessions, allowing us to locate easily specific episodes in the instruction. Finally, we collected student artifacts (homework, quizzes, and tests) along with other teaching materials (syllabus, overhead transparencies, and handouts) as they would inform our understanding of Dan’s mathematics teaching. Dan’s classroom served as an ideal location because, as a mathematics teacher, he had both (a) an awareness and appreciation of the issues and goals related to reform mathematics (i.e., Standards-based mathematics instruction) and (b) a knowledge of how to scaffold experiences to help students construct images and deep understanding of the mathematical concepts under study. He closely aligned his instruction with the problem-solving view of mathematics teaching as advocated by recent Standards documents (NCTM, 1989, 2000). Students were organized into groups of four to six, and spent the majority of class time working in their groups on inquiry-based tasks designed to help them elicit new images and key concepts for the topics they were studying. Dan carefully scaffolded tasks so students themselves could develop many of the mathematical ideas and concepts. They worked with manipulatives, drew pictures, created and reasoned about appropriate realworld situations, wrote explanations, and read and discussed each others’ solutions. Dan devoted the majority of class time to small-group and whole935 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert class discussion so students could share ideas and negotiate the correctness of their reasoning and solutions. Negotiation of a Shared Perspective Our work to come to know together or to negotiate a shared perspective required us to engage in a process that consisted of four activities: (a) building trust, (b) sharing our personal perspectives, (c) identifying dissatisfaction, and (d) negotiating a shared perspective. The four activities of our collaborative process served both as locations for data production and data analysis. The data sources for these activities included transcriptions of research meetings, notes created during research meetings, and drafts of manuscripts. These activities occurred in a simultaneous rather than in a chronological fashion. Indeed, our shared perspective, what we consider the product of our collaboration, emerged in response to the specific challenges we encountered as we progressively built trust between each other, gained insight into the other’s perspective, and worked to resolve our growing dissatisfaction with our personal perspectives. Thus, it is difficult to describe meaningfully the process of negotiation without also presenting the results that dictated our choice of what to do. In this section, we briefly outline the four activities of the process that defined our interaction, and leave many of the details of these interactions to the results section. Building trust was fundamental to our work together as our growing trust with each other allowed us to stay in the conversation and supported our honest sharing. Trust building took place throughout our collaboration, but began during the planning of the study and the first of four meetings that took place after the class began. During the initial conversations (which were audio taped and the tapes transcribed), we were intentionally vague about the purpose of the study and engaged in careful conversations that revealed that we shared fundamental beliefs about learning, which helped establish a basic level of trust and understanding. Sharing our perspectives served to help us build trust with one another, while it enabled us to consider our partner’s perspective of Dan’s mathematics instruction. The sharing began during the remaining three discussions that we had during the semester. During these meetings, we shared our sense of what was happening in Dan’s classroom. In addition to sharing, we questioned each other in order to clarify and understand the other’s perspective and how it differed from our own. We discussed specific instructional episodes and returned to field notes and the videotapes in order to keep the discussion focused on Dan’s instruction. These discussions (which usually lasted about an hour) were audiotaped and the tapes were transcribed. Through these discussions, we were able to create descriptions of the instruction that reflected both our perspectives, thus answering our first research question. Furthermore, the transcriptions of these meetings provided data to which we could return to understand how we were making sense of the literacy and mathematics instruction that we observed in Dan’s classroom. 936 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction Sharing our perspectives also allowed us to identify our growing dissatisfaction with our personal perspectives, thus leading to an answer for our second research question. Like building trust, this activity was present throughout our collaboration; however, it intensified after the semester course had ended. At this point in the data analysis, we each reviewed the field notes and videotapes and identified three episodes that we believed exemplified what we valued in Dan’s mathematics instruction. Episodes varied in length, with one representing one day of instruction and others representing a series of related lessons over several days. As we discussed these three episodes and shared our perspectives concerning the underlying goals of instruction, the discussions moved from sharing our personal perspectives to viewing our personal perspectives from the point of view of our partner. This forced us to reconsider our own perspectives critically, and ultimately led to our growing dissatisfaction with the perspective we originally espoused. The fourth activity of our collaborative process consisted of negotiating a shared perspective. Again, this activity occurred throughout our collaboration as we sought to build trust and resolve our growing dissatisfaction; however, the negotiation of a shared perspective was most intense during the coauthoring of our article. To accomplish this purpose, we returned to the theoretical frameworks underlying each of our perspectives of learning and literacy found in the transcripts of the research meetings and in notes we created during our meetings together. Other forms of qualitative research rely on the use of member checks as a way of ensuring that emerging theories represent the essence of observed experiences. An advantage of the cooperative inquiry and our serving as correspondents was our frequent access to each other during theory generation and the opportunities we had to question each other about his or her theories of learning, literacy, and instruction throughout this process. Our burgeoning theories were recorded in meeting notes and drafts of our manuscript. Through intensive questioning and negotiation of meaning and ideas, we were able to arrive at a theoretical model that acknowledged and honored both of our perspectives. As the model began to emerge in subsequent drafts, we shared it with colleagues from our two fields. We received helpful feedback, which led to further revisions of the model. We present this model, as well as additional details of the dialogic process that resulted in the creation of this model, at the end of the results section. Protecting Against Threats to Validity Engaging in the kind of critical dialog that is required by cooperative inquiry can be an anxiety-provoking experience (Reason, 1994). Rather than seek new understanding, coresearchers may respond to growing anxiety by choosing to entrench themselves in their own views and retreat from opportunities to consider alternative views, values, and goals. Reason (1994) suggested that coresearchers can entrench themselves in their own views in two ways. First, coresearchers may deceive themselves and project what they want to see onto the world they are studying. Second, coresearchers may unconsciously 937 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert choose to protect their own perspectives by unknowingly uniting in a way that does not allow them to critique and question those perspectives. This situation is exacerbated if there is an unequal distribution of power between researchers. Reason referred to these two responses as unaware projection and consensus collusion, respectively. Both of these responses pose serious threats to the validity of the cocreated theory generated. We protected our data analysis against threats to validity in two ways. First, we began our inquiry on relatively equal standing as first-year, assistant professors, from two different colleges on campus. Our relative equal position as researchers and educators created a balanced distribution of power between us, which allowed us to challenge each other and push the conversation beyond surface agreements. Second, we checked our analysis with colleagues from our departments who served as critical friends who had no vested interest in the outcome of the data analysis, except that it remain consistent to their respective fields of expertise. Ultimately, we sought to cocreate a perspective of mathematics and literacy learning that was compatible with each of our disciplinary perspectives.

Results We set out to answer two questions. The first question—what are the similarities and differences in instructional goals and practices as they are represented by mathematics and literacy educators?—was answered through forming individual descriptions of Dan’s mathematics instruction. The second question—what is the common ground where literacy and mathematics educators can simultaneously consider literacy and mathematics issues that occur in mathematics classrooms?—was answered through negotiating a shared perspective that resolved our growing dissatisfaction with our personal models. Because we sought answers to our research questions through collaboration, in this section we describe our results as well as provide details of the process through which we arrived at those results. These details help render the conclusions comprehensible. Forming Descriptions In order to form a description of the similarities and differences in instructional goals and practices valued by mathematics and literacy educators, we had to build trust and share our perspectives of Dan’s mathematics instruction. In this section, we describe our process for building trust, which was a prerequisite for honestly sharing our personal perspectives of Dan’s mathematics classroom. Then we compare our two perspectives in order to highlight the similarities and differences in the perspectives we each brought to our collaboration. Building Trust We met for the first time at a university function for new faculty members from all across campus. Both of us were first-year assistant professors and 938 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction knew no one else at the workshop series. We naturally sought each other out because we both had a common background in education. During our initial conversation, we talked about our research interests. Roni Jo inquired about the possibility of visiting Dan’s class to see how literacy was addressed in a mathematics classroom. Dan politely explained that literacy was not an issue he addressed in his teaching of mathematics and suggested that his classroom would offer little of interest to a literacy researcher. However, Roni Jo persisted, and out of professional courtesy, and a hunch that the research might lead to tenure-securing publications, Dan agreed to participate in a study. Initially, we both had serious misgivings about collaborating. Despite Roni Jo’s reassurances, Dan continued to suspect that Roni Jo would be disappointed by his inquiry-based mathematics course, where no textbook was being used. Dan could not see how literacy was an issue worth studying in his class. Also, Dan was unsure whether Roni Jo would value his conceptually oriented mathematics instruction. He was particularly concerned that she would wonder why he did not provide more direct instruction on computation and algorithms. He was worried that she would judge his teaching to be poor mathematics instruction. For her part, Roni Jo was also pessimistic about the collaborative effort, suspecting that Dan taught a traditional mathematics class and would be unreceptive to the need to help students learn to read and write mathematics texts. In fact, she initially thought the study would consist of documenting how traditional mathematics instruction neglects literacy. Her biggest concern was how she was going to report the findings of the study without offending Dan. Despite our misgivings, we went ahead and planned the study, meeting a few times before data collection to design it. During these meetings, we tentatively shared only a little about what we valued in instruction and what we hoped to achieve through our collaboration. At that time, both of us were wary that the other would misunderstand and devalue the perspectives of our respective disciplines. Each of us was aware of how the perspectives of our fields had been misinterpreted and devalued by others outside of our fields, and thus we were cautious about sharing ideas that we felt were hard to understand and easily misinterpreted by outsiders. Also, we were concerned about offending or troubling the other by expressing beliefs or ideas that we assumed might be in direct conflict with the other’s beliefs and understandings. Therefore, we were cautious when sharing our perspectives during these initial meetings. It became clear to us after the first few class meetings and our informal discussions following these classes that our misgivings about the collaboration were unfounded. In fact during our first research meeting, Dan expressed pleasure and relief to see that Roni Jo enjoyed attending his class and was excited about the instructional activities he planned. Roni Jo explained that she appreciated his reform-oriented instructional approach, and she valued Dan’s focus on mathematical concepts (research meeting transcript). This set Dan at ease. For her part, Roni Jo recognized that Dan’s mathematics class was a text-rich environment. Furthermore, Roni Jo was 939 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert able to identify several instructional activities in each class where Dan purposefully helped his students negotiate and create texts. Roni Jo went so far as to inform Dan on several occasions that he was actually a literacy teacher, which both intrigued and annoyed him (research meeting transcript). During the course of the semester, we met four times to discuss Dan’s instruction. We discovered early in the collaboration that we shared common beliefs about teaching and learning based on social constructivism. However, we did not spend a great deal of time exploring the differences in our conceptualization of social constructivism. It was sufficient for us to know that we both embraced the following two basic tenets of social constructivism. First, people do not have direct access to reality; rather people build their knowledge of reality or construct it (von Glasersfeld, 1991). Second, people’s experiences that lead to knowledge creation are mediated through social interaction (Fosnot, 1996). As a consequence of this perspective toward knowledge, we both viewed learning as a constructive, meaningmaking process that occurs as an interaction between the learner and the cultural, societal, and political context in which the learner is situated (Cole, 1990; Luke, 2000; Moje et al., 2000; O’Brien, Stewart, & Moje, 1995; Vygotsky, 1978). This common view of learning was not unique to us, but was a reflection of the prominence of social constructivism in both of our disciplines (for example, see Cobb and Bauersfeld [1995] in mathematics education and Moje et al. [2000] in literacy education). Our common beliefs about learning were crucial to the success of our research. First, because of this common ground, we had a shared foundation of beliefs and some shared language that enabled us to describe, discuss, and evaluate observations and ideas. Second, as we came to see the other as a trustworthy researcher of learning and teaching, we became more respectful of each other and of the value of one another’s disciplinary knowledge. Because we esteemed the other as an intelligent and thoughtful person, we were willing to temporarily suspend judgment in order to understand better what the other valued from his or her disciplinary perspective. Third, because we ascribed to constructivism, we recognized that our understanding of teaching and learning in the mathematics classroom was inevitably imperfect and incomplete. Because we were acutely aware of the possibility that the other could provide valuable insights about teaching and learning in the mathematics classroom, we actively shared our perspectives and attempted to understand the perspective of the other. Sharing Perspectives Once we had established some common ground, we were prepared to discuss what we saw as important in Dan’s instruction. We discovered that we valued many of the same instructional practices, although usually for different reasons as demonstrated by our selecting the same instructional episodes as examples of exemplary mathematics and literacy instruction. To illustrate these findings, we share our perspectives of an episode from Dan’s class940 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction room that took place during the first two class periods of a unit on geometry, approximately three-fourths of the way through the semester. We each created our personal perspective from the video transcripts of Dan’s instruction, which we initially identified from the field notes. Roni Jo’s perspective. Dan regularly incorporated literacy into his teaching of mathematics, using literacy in both invisible and visible ways (Vacca, 2002). Invisibly, he used literacy as a tool to support his students’ learning of the content—or what has been termed by some literacy educators as reading and writing to learn. Using literacy as a tool to support student learning includes engaging students in discussions (Alvermann, Dillon, & O’Brien, 1987), allowing them to write to clarify and support their thinking (McIntosh & Draper, 2001), and supporting their reading and study skills so they can learn and retain information (Vacca & Vacca, 2002). In Dan’s classroom, students used discussions to facilitate their thinking and learning of mathematical concepts daily. In fact, the majority of each class period was devoted to small-group discussions about mathematical problems that he posed. Additionally, Dan required the students to write extensive explanations to accompany solutions to the problems from class, homework assignments, quizzes, and tests. He also encouraged his students to use nontraditional texts, such as manipulatives, drawings, and diagrams, to develop, represent, and communicate mathematical ideas and justifications. However, as the following episode suggests, Dan also engaged in visible literacy instruction—instruction that explicitly addressed his students’ facility with the texts in his classroom. During the first two class sessions of the geometry unit, I observed what I regarded as exemplary vocabulary instruction. By vocabulary instruction I mean instruction that focuses on students’ acquisition of words and the meanings for those words (Nagy & Scott, 2000). Vocabulary instruction represents a visible form of literacy instruction in that the instructor focuses instruction on symbolic representations (e.g., words) in conjunction with their associated meaning (e.g., the mathematics). Literacy educators have advocated vocabulary instruction that allows students to (a) participate in multiple and rich experiences with the vocabulary terms and their corresponding concepts, (b) engage in conversations and activities that allow students to experience the words in context, and (c) build their own definitions of the words based on these experiences and conversations (Blachowicz & Fisher, 2000; Carr & Wixson, 1986; Nagy, 1988). Dan’s teaching of the terms for various polygons was consistent with the type of vocabulary instruction recommended by literacy educators. Dan introduced his students to various polygons by supplying his students with a set of drawings representing assorted polygons, having the students sort the drawings into groups based on criteria established by the students, and having students play several games with the polygon drawings. In order for students to play the games, they needed to have a shared vocabulary (both words and meanings) for the various polygons. Consequently, Dan led the students in a whole-class discussion in which students volunteered terms for the different shapes and held up examples from their polygon sets to illustrate each term. 941 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert Dan continued to engage his students in vocabulary instruction even after they had generated common terms for the geometric shapes. He asked his students to work in groups to make lists of all the properties they could think of for each of the six special quadrilaterals (kite, rectangle, parallelogram, square, rhombus, and trapezoid). When they finished, students shared their lists in a whole-class discussion. Dan then asked the students to work with their groups to compile minimal lists of properties for each of the six special quadrilaterals by deleting as many “extra” properties as possible from each list. As students shared their minimal lists with the class, they negotiated certain properties of the shapes (e.g., whether a trapezoid could have more than one set of parallel sides). Dan took advantage of these disputes to point out that some aspects of mathematics are conventions that have been adopted by the larger mathematical community. As homework, students were asked to create a set of examples and nonexamples for each of the six special quadrilaterals. Finally, in the following class period, students worked in groups to create hierarchy charts that showed the relationship between different shapes (e.g., that all squares are rectangles, and all rectangles are parallelograms). After sharing their charts with the class, students were allowed to refine their charts by adding, deleting, or reorganizing to clarify the relationships among the shapes. Dan provided multiple and rich experiences for the students to engage with the words and the concepts represented by the words through games and discussions. Furthermore, Dan’s insistence that students create hierarchies and review hierarchies made by other groups helped them not only to learn the terms associated with each shape, but to understand the relationship between and among the various concepts associated with each term. This is important as an individual’s understanding of words is mediated by the relationship between the various concepts that different words represent (Carr & Wixson, 1986; Nagy, 1988; Nagy & Scott, 2000). Dan’s perspective. When Roni Jo mentioned to me that she was impressed with the way that I had taught vocabulary in this episode, I was somewhat taken aback. I had never thought of these instructional activities in terms of teaching vocabulary. In fact, when Roni Jo suggested this episode was an excellent example of how vocabulary should be taught in a mathematics class, I felt that she had missed the whole point of the 2 days of instruction. Vocabulary instruction seemed to me but a trivial part of what I had been trying to accomplish. I had three major goals for this sequence of instruction, none of which was to teach vocabulary. The first goal was to have students develop rich, flexible, dynamic conceptions of the properties associated with geometric shapes. I wanted students to see that there were a variety of properties associated with each shape that we were studying, and that a shape could be determined by its properties. Students needed to understand the effect of adding or removing properties associated with a shape, and the influence these additions or deletions had on the possible forms that a shape could take. 942 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction The second goal of instruction was to allow students to have the experience of engaging in doing mathematics the way in which a mathematician would. The sequence of activities that the students engaged in—namely, discovering and listing properties, forming minimal lists of properties, and identifying relationships between different shapes—is similar to the processes that mathematicians engage in when studying new mathematical structures or phenomena. Rather than having the mathematics handed to them, students were allowed to organize the shapes in a way meaningful to them, and to justify their organization based on the properties of shapes. I thought it was critical that my students have this opportunity of creating mathematics, instead of always being consumers of mathematics. My third goal for the students was to have them think about geometric shapes from the perspective that a child might take when learning geometry. In particular, I wanted them to experience a progression of activities that would move them through the first few van Hiele levels of reasoning about geometric shapes (Burger & Culpepper, 1993; Fuys, Geddes, & Tischler, 1988). According to the van Hiele model, children progress through a series of stages when reasoning about shapes. Children first begin to recognize shapes based on their appearance as a whole (e.g., a shape is a triangle if it looks like a mountain). With more experience, children can recognize the properties of shapes. They can eventually reason about these properties, deducing and justifying relationships between properties and relationships between shapes. In class instruction, I allowed the students to experience the types of activities I would use to move children through the first few stages of the van Hiele model—namely, sorting shapes, noticing and listing properties, and creating minimal lists and shape hierarchies. Note that the first two goals of this episode of instruction—mathematical understanding and insight into the nature of mathematical activity—were goals I typically have for most of my mathematics lessons, regardless of who my students are. Furthermore, these goals are common in Standards-based mathematics instruction (see NCTM, 1989, 2000), However, the third goal— to help my students gain insight into children’s thinking about geometric shapes—was specific to the student population I was teaching. Because my students were preservice elementary teachers and needed to know mathematics for the purpose of teaching children, the first two instructional goals were insufficient. I also needed to provide them with a framework for thinking about how they might engage children in learning mathematics with understanding, which is why I later engaged them in a discussion of the van Hiele model and related those levels to the mathematical activities they had done with geometric shapes. Comparing perspectives. As the individual descriptions of our shared experience above suggest, our initial success in building common ground by uncovering a shared belief in social constructivism did not avert subsequent difficulties in continuing a dialogue about instruction in Dan’s classroom. Throughout the semester-long course, we frequently differed drastically as to how we perceived, described, and explained classroom events. We were 943 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert initially shocked at how differently we each saw the events, and often wondered how the other could so consistently miss the entire point of an instructional episode. However, we continued our conversations for two reasons. First, by this time, we had invested in the personal relationship and had come to trust that the other’s perspective would provide us with new, valuable insights. Second, despite our use of different lenses to view instructional episodes, we typically agreed on which episodes represented particularly good instruction. For example, both of us identified the previous episode as exemplary instruction, despite our different interpretations of what was important in this instructional segment. Both of these reasons motivated us to attempt to uncover and understand what lens the other was using to view classroom events. What we discovered was that our disagreements reflected real differences in our beliefs about learning and teaching in mathematics classrooms and were not merely due to differences in the terminology used in our respective fields. We began the process of understanding the lenses we were using to view Dan’s mathematics instruction by exploring and sharing our underlying assumptions of the important components and interactions that comprise mathematics learning. This helped both of us clarify what we attended to during instruction, and helped the other to begin to view classroom events from an alternative perspective. These descriptions eventually led us to distill our personal models for learning in a mathematics classroom—models we had brought to the collaboration—which we recorded in the notes we made during our research meetings. We refer to these as personal models because they are the result of our individual efforts to make sense of the literature from our fields, personal interactions with other scholars, and our experiences in learning and teaching mathematics. While these models were oversimplified and certainly did not account for everything that we thought was important to mathematics learning, they were robust in two senses. First, they accounted for and addressed what we considered was important from our own experience with mathematics teaching and learning, and they were compatible with our understanding of our respective fields. Second, these models could be successfully used by the other as a lens to view instruction so that the other came to feel that he or she was able to see the same things we were. Because of the personal nature of these models, we present them here in the results section as data rather than at the beginning of this article as summaries of the literature review. Indeed, these models surfaced as texts we created during our research meetings and in initial drafts of our manuscript to explain how we each made sense of the activities we observed in Dan’s mathematics classroom. Roni Jo’s literacy model of learning focused on the meaningful use of text, because she believed that texts and literacy events play a crucial role in teaching and learning. Roni Jo viewed literacy as an interaction between the reader, the text, and the context (Moje et al., 2000; Ruddell & Unrau, 1994), as illustrated in the right-hand side of Figure 1. Under this model of literacy, the reader makes meaning through an interaction between what the reader 944 Downloaded from by Armando Loera on October 31, 2009

Mathematics Education

Literacy Education Learner




Context (includes text)

Context (includes mathematics)

Figure 1. Separate personal models of mathematics learning and literacy.

takes to the literacy event, the text, and the context in which the event takes place. Furthermore, by defining texts broadly as any object used to convey meaning (Wade & Moje, 2000), Roni Jo perceived all communicative actions in the classroom as literacy events involving spoken, written, or enacted texts. Thus, for Roni Jo, the activities of learning and teaching in a mathematics classroom are permeated with texts and literacy events. In order for students to acquire mathematical knowledge and participate meaningfully in mathematical activity, students must become adept at creating, negotiating, and consuming texts. Furthermore, she believed that it is the responsibility of mathematics teachers to explicitly teach students how to create and negotiate texts, because mathematics teachers are the literacy experts in their classrooms. Thus, Roni Jo’s model of learning naturally led her to view the above teaching episode largely in terms of Dan fulfilling his responsibility to help his students gain access to texts and the nature of literacy events in his mathematics classroom. Dan did not initially understand texts and literacy events in the same way that Roni Jo did. He equated text with traditional print materials or lists of vocabulary words. Consequently, the only segment in the entire instructional episode above that he consciously equated with helping students develop facility with text was when he led students through a discussion of the different terms for geometric shapes. Even then, Dan’s sole purpose for engaging in this literacy instruction was so that his students would be able to participate in the subsequent activities of creating and negotiating mathematics (research meeting transcript). He did not see the rest of the instructional episode as having a strong connection to texts or vocabulary development. Because the discussion of terms for geometric shapes only comprised a few minutes of two class periods of instruction, he resented Roni Jo’s implication that this episode was a lesson on vocabulary. His mathematical focus obfuscated the role of text or even the need for students to become facile in their 945 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert ability to use texts as part of their mathematical activity. Thus, he had difficulty perceiving the literacy instruction in his teaching. Dan’s own model for learning centered on mathematical meaning and understanding, not texts and literacy. He viewed mathematics learning as the interaction between the learner, context, and the mathematical ideas and practices of the discipline (see the left-hand side of Figure 1). Dan used his own knowledge of mathematics and his knowledge of students’ current mathematical understandings and misconceptions to create inquiry-based instructional activities that would guide his students toward powerful conceptions of mathematics (Simon, 1995). He then orchestrated the classroom setting and social interactions so as to provide students’ with the necessary support to develop the mathematical ideas through participation in the inquiry-based activities. Dan believed that a reflexive relationship existed between individuals’ mathematical understandings and the class-sanctioned mathematical ideas and practices (Yackel & Cobb, 1996). Indeed, he thought that as mathematical ideas and practices were developed in the class, students would develop mathematical understanding, which in turn would influence subsequent classroom discourse and practices. This can be seen in the above episode, where activities that helped students develop an understanding of the properties of different shapes enabled a subsequent discussion of minimal lists of properties and shape hierarchies. In general, Dan focused his attention on selecting appropriate inquiry-based tasks and then guiding classroom social interactions so that authentic mathematical discourse and practices emerged, thus providing students with opportunities to develop rich understandings of mathematics. Roni Jo was often unaware of how the nature and structure of mathematics as a discipline influenced Dan’s selection of the understandings, discourse, and practices he attempted to foster in his class. Although she perceived that Dan’s students were developing mathematical understanding, she often did not know why those particular understandings, and not others, were important. In addition, she was often unaware of how Dan purposely guided classroom interactions so that discipline-authentic discourse and practices emerged. For example, although Roni Jo believed that the instructional activities in the above episode were sufficient to help the students develop rich meanings for shape vocabulary, she did not appreciate the discipline-specific goals that guided Dan in the selection of these particular activities and meanings. She did not notice that Dan was teaching his students about how meaning and ideas are created, negotiated, and established in the discipline of mathematics. She was also unaware of the explanatory and organizational power inherent in the particular properties Dan sanctioned (i.e., rewarded, emphasized, and encouraged) during whole-class discussions. For Roni Jo, the properties associated with a shape were meanings that could be associated with a particular text—in this case, a vocabulary word. She did not perceive properties as socially negotiated tools upon which classification and argumentation in the discipline are based. Indeed, Roni Jo’s literacy framework did not acknowledge the crucial role that the 946 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction discipline of mathematics itself plays in determining the meanings, texts, and literacy acts appropriate for reasoning and communicating. Although we used different personal models to explain learning in a mathematics classroom, and despite often disagreeing about the purposes we saw in instructional activities, we were nonetheless surprised to find that we typically agreed about what constituted good instruction. For example, both of us singled out the above episode as a particularly good example of mathematics teaching, albeit for very different reasons. Despite the different goals we perceived in the episode, we both felt that Dan’s selection of instructional activities were more than adequate to address both of our perceived instructional goals. Furthermore, students’ responses to Dan’s informal assessments and our observation of their responses in group and whole-class discussions confirmed to us that students were meeting the goals that we perceived for this instructional episode. Thus, we both valued the instruction in this episode, and we both felt that it represented good teaching from the perspective of our respective disciplines. Our consistent agreement about what constituted good instruction led us to investigate the compatibility of the instructional goals that we perceived in Dan’s teaching. We discovered that our perceived goals were often complementary and mutually supporting (manuscript draft). In the above episode, for example, to achieve Dan’s goals of having his students participate in creating and negotiating mathematics, the students needed to have rich meanings for geometric vocabulary. Without knowledge of and facility with relevant vocabulary, Dan’s goals would have been impossible to achieve. At the same time, the many activities in which Dan’s students participated to create and negotiate mathematical meaning supported Roni Jo’s perceived goal of learning vocabulary, because the activities increased the richness of the meanings that students associated with the geometric terms. The findings presented from this particular episode—that we valued the same kinds of instructional activities, and that we saw different but complementary goals in Dan’s instruction—permeated our discussions of Dan’s teaching (research meeting transcripts, meeting notes, and manuscript drafts). At first these findings surprised us. We were somewhat shocked to find that someone else could consistently be enthusiastic about the same episode of instruction as ourselves, but for very different reasons. Initially, we felt that the other had misinterpreted the situation and was missing the main point of what was happening in the classroom. However, when these results repeated themselves over and over again, we began to acknowledge that perhaps neither one of us had a complete view of what was happening. Each of us became quicker to acknowledge the value of the other’s perspective. By the end of Dan’s course, we both had come to value the perspective of the other, and in fact could often see things from the other’s point of view. However, at this stage in our collaboration, neither of us were able to combine both views into one integrated whole. While we were able to switch back and forth between perspectives, we had yet to develop a conceptual framework that allowed us to perceive and value both perspectives simultaneously. 947 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert Thus, we had yet to establish a shared perspective from which to view literacy as an essential part of mathematics instruction. Negotiating a Shared Perspective Had we been satisfied with simply describing Dan’s mathematics instruction from our individual perspectives, our collaboration would have ended once we had shared and compared our perspectives. Indeed, we could have simply disregarded the other’s view or agreed to disagree. However, cooperative inquiry also required that we take advantage of the shared experience and the surrounding discussion to critique our perspectives. In fact, the sharing and comparing of our perspectives was necessary because they enabled us to view our personal perspectives and models from the point of view of our partner, thereby enabling us to critique our perspectives. The process of critique and resolution of our disparate perspectives necessarily resulted in a shared perspective from which we could consider mathematics instruction. We would not characterize what resulted as compromise, because compromise implies that each of us gave up something we valued in order to meet the other halfway. Instead, we truly cocreated a shared view of literacy and mathematics learning, which answered our second research question. In this section, we share the process in which we developed our growing dissatisfaction with our personal perspectives. We also describe the shared perspective we negotiated in order to resolve our dissatisfaction. Roni Jo’s Dissatisfaction My dissatisfaction with my model was gradual and caused by the model’s inability to anticipate and address the discipline-specific goals of instruction in a mathematics classroom. Prior to our collaboration, I was certain that I could rely on literacy instructional methodologies to solve instructional problems in mathematics classrooms. For instance, if Dan had asked my advice for teaching vocabulary, and particularly the terms associated with various polygons, I could have provided him with some wonderful activities that would have helped his students develop powerful meanings for these terms. In particular, I would have suggested that he focus his instruction directly on Greek and Latin word roots to help his students make connections between the meanings of the roots and the qualities of the polygons. My rationale for this advice would be that a focus on word roots would help students make a connection between the root meanings and the attributes of the various polygons. For instance, students would consider the roots poly and gonas (the principal roots of polygon) and consider how knowing that poly means “many” and gonas means “angle” helps them see that a polygon consists of many angles. Furthermore, my recommended instructional activities could also have been structured to fit the inquiry-based nature of Dan’s class by allowing students to discover the meanings of the roots and their relationship to terms used to represent the various polygons (see McIntosh, 1994). 948 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction Students could begin by working in groups with real objects to discover the connection between roots and their meanings, and then conclude by reflecting on their experiences through writing what they had learned. As I watched Dan teach the vocabulary lesson and later listened to him describe his instructional goals, I became aware that the instructional activities I would have recommended failed to address certain discipline-specific goals that Dan thought were important. Certainly the word-root approach to learning vocabulary terms would have helped Dan’s students develop meaning for shape vocabulary. However, this approach would not have provided students with opportunities to create definitions that consist only of the minimal lists of the properties necessary to define the object, nor allowed students to negotiate definitions and come to agreement as a group. Both of these activities—creating minimal lists and negotiating shared definitions— do not represent merely differences in methodological approaches to teaching vocabulary or mathematics. Rather, they represent an attention to how the mathematics as a discipline, with particular ways of knowing, must be created in classrooms in order to help students participate in authentic mathematics activity. The major shortcoming in my learning model was the unanticipated effect that the discipline had on what counted as texts and literacy practices in a mathematics classroom. Before I began my observations in Dan’s class, I anticipated that many of the literacy instructional practices that I had learned could be used in a mathematics classroom with only moderate modifications. Furthermore, I assumed that I would recognize what modifications needed to be made, and what the goal of literacy instructional activities should be. However, after several weeks of watching Dan engage in literacy instructional practices that were aimed at addressing discipline-specific issues that I had not previously considered, I realized that my model accounted for only part of what was important in a mathematics classroom. In order to understand how literacy should be addressed in mathematics, I would have to attend more carefully to the influence of the discipline on what counts as text and literacy acts. Dan’s Dissatisfaction Approximately 1 month into the semester, I began to suspect that I needed to reconsider my model of learning and address literacy more explicitly in my mathematics instruction. My students had just completed the first exam, which was comprised entirely of items that required them to explain and justify their thinking. Many of the students had not done as well as they had anticipated. Approximately half of my students claimed that they understood the concepts much better than their test scores reflected. They complained that their scores were artificially low because they had not been taught what they needed to include in the explanations to receive full credit. In past courses, I had always discredited such claims by assuming that if students had really understood the mathematical ideas and the connections between those ideas, then they 949 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert would have no difficulty explaining and justifying their reasoning. However, because I was engaged in conversations about literacy in my classroom, and because Roni Jo was constantly talking about literacy events in my instruction, I began to wonder for the first time if it was possible for my students to have a solid understanding of the mathematics and still be unable to write good explanations. Perhaps writing conceptually oriented explanations and justifications was not a straightforward process that could be equated solely with understanding the mathematics, as I had previously assumed. In actuality, I had not treated my students’ writing of conceptually oriented explanations as something that just naturally occurred as they came to understand mathematics. In fact, I had done several things to help my students write better explanations. First, I provided my students with many opportunities to practice writing conceptually oriented explanations. Indeed, nearly every problem they worked on in class and on homework assignments involved writing conceptually oriented explanations and justifications. Second, students had frequent opportunities to observe the creation and negotiation of conceptually oriented explanations in their small-group and whole-class discussions. Conversations about mathematics frequently included discussions about what needed to be included in a particular explanation and why. Third, I regularly graded students’ written solutions to homework assignments, and provided feedback directly on their written work in the form of detailed comments about the strengths and weaknesses of their explanations. Lastly, when there were widespread difficulties with the writing of certain homework problems, I would compile several disparate solutions from the students’ homework and place them on an overhead for the entire class to read. I would then ask the students to discuss each solution in their groups and decide whether or not it was a good explanation. Then we would discuss the solutions as a whole class, and I would give them feedback about how I viewed each explanation. Despite the many things I did to help my students learn how to write conceptually oriented explanations, my main instructional focus during these activities was on helping my students develop an understanding of the mathematical ideas. In fact, I actually perceived all of the above literacy activities as instruction on mathematical understanding and activity, and not as literacy instruction. My purpose in requiring my students to write explanations was to force them to think deeply about mathematics and make their thoughts explicit. I anticipated that explicit discussions about conceptually oriented explanations and justifications would address students’ beliefs about the nature of mathematics by encouraging them to see mathematics in terms of concepts, images, and reasoning. I also believed these discussions would improve their knowledge of mathematics, because it would highlight the important mathematical ideas and the connections between them. Thus, I viewed helping my students write good, conceptually oriented explanations merely as a means to an end, and not a goal in and of itself. The problem with this perspective, as I came to realize through my discussions with Roni Jo, was that it equated mathematical understanding with 950 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction communicating that understanding. Because my focus was explicitly on my students’ mathematical conceptions, I did not recognize the need to help them become fluent in the genre of conceptually oriented explanations. I never engaged them in a general discussion about the nature and content of conceptually oriented explanations. All of the discussions we had about explanations revolved around specific examples in which I was mostly concerned about the mathematical ideas. Thus, I failed to discuss the following issues with them: intended audience, purpose, general format, acceptable methods of argumentation, acceptable evidence, common misconceptions or mistakes in writing conceptually oriented explanations, the role of diagrams and pictures, and the ways in which this type of explanation differed from other forms of mathematical argumentation. The recognition that these were important issues came only after I began to focus my attention on my students’ ability to read and write texts. This new lens provided me with an alternate way of viewing my students’ learning of mathematics. Negotiating a Shared Perspective As we grew in our dissatisfaction with our own models and as our trust and interest in the other’s perspective increased, we became motivated to develop a shared perspective of learning and literacy in the mathematics classroom that incorporated both our models (meeting notes). This shared perspective had to meet three criteria. First, it had to be credible to both of us and members of our respective fields. Consequently, the shared perspective had to respect, incorporate, and address the key issues and ideas in both our fields. Second, the shared perspective had to address and compensate for the perceived weaknesses we each saw in our own models; otherwise, neither one of us would be motivated to adopt the new perspective. Third, the model had to truly be a shared vision. It could not be merely the juxtaposition of the two perspectives, placing the two lenses side by side so that one could look through one lens and see the mathematics, then quickly switch to the other to see literacy. Instead, we sought a shared perspective that allowed us to perceive simultaneously both the literacy and the mathematics learning and teaching. We felt that if we were able to develop a shared perspective that met these three criteria, we would have developed common ground on which we and others from our discipline could collaborate. We began the negotiation of a shared perspective by comparing and discussing our individual models. We started by laying our two models side-byside and noting that we seemed to agree on two of the three components in each model, namely the learner/reader and the context (see Figure 1). Clearly we did not agree on the third components, text for Roni Jo and mathematics for Dan. After a brief discussion, we realized that both of us perceived the other’s third component as part of what we thought of as context. This immediately led to a long, heated debate about whether the other’s third component really deserved the privileged status of being separated from the context. Both of us were surprised that the other did not immediately see the 951 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert relevance of privileging the particular third component that we espoused. Equally surprising was the realization that another aspect of the context might be worth attending to as much as our own privileged third component. As our discussion continued, we realized that the privileged status that we granted to our third component was what made us the particular educational specialists we were. Consequently, neither of us was willing to give up our privileged third component, nor did we feel it was fair to force the other to give up his or her third component. Instead, we chose to end the argument by agreeing on a live-and-let-live stance that consisted of the four-component model shown in Figure 2 (manuscript draft). This new model included the two components we had initially shared, namely the learner/reader and the context, as well as our individual third components, text and mathematics. Initially the four-component model represented a truce, not a shared perspective. However, as we continued our collaboration, it served as a text around which we were able to negotiate a shared perspective. The development of this shared perspective was a result of several different activities. First, we began to read the literature from each other’s fields so that we could better understand how members of the other’s discipline might view the four components of the model. This activity not only enriched our own views of the model, but also helped us identify shortcomings in how our disciplines viewed learning and literacy. Second, we began to share our story and model with people from our fields, both in professional settings and personal conversations. This allowed us to test our model for validity in both fields. The

Learner/ Reader




Figure 2. Four-component model of mathematics learning and literacy from a shared perspective.

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Standards-Based Mathematics and Literacy Instruction reactions of members of our discipline also helped point us to parts of the model that needed further development and explanation. Finally, we began to coauthor papers together. The permanency of written texts required an even more careful coordination of our perspectives and the perspectives of our respective fields, because these texts could be more carefully scrutinized by others than spoken texts. In fact, it was not until we began producing manuscripts together that we became convinced that we had constructed a shared perspective that met our three criteria above. Through our collaboration, we have been able to construct the following shared interpretation of the four-component model (see Figure 2) concerning how learners come to understand in a mathematics classroom. As with our initial models of literacy and mathematics learning, we continue to acknowledge the interaction between the individual learner and the context in which the learning takes place. However, our new model also acknowledges the role of both texts and mathematics for learning and participating in a mathematics classroom. Texts play a crucial role in the learning of mathematics because it is through written, spoken, and enacted texts that mathematics is created, communicated, and negotiated. Even individuals acting alone cannot engage in mathematical activity without the use of some form of text. Thus, texts must be given a prominent position in the model, because mathematics learning is not possible without texts. However, texts cannot be elevated above the mathematical content, because the mathematics influences what texts are appropriate, how those texts should be used, and what those texts mean. Furthermore, for an individual to make sense of the texts in a mathematics classroom, the individual must be acquainted with the mathematics that the texts represent. Thus, we perceive text and mathematics as having equal status in our model. For us, an inescapable implication of the four-component model is that mathematics learning and literacy are invariably intertwined in a mathematics classroom. As students learn new mathematical ideas and practices, they must engage with the texts used to create, convey, and negotiate those mathematical ideas and practices. To facilitate the students’ learning, the teacher must engage in literacy instruction that helps students create and consume those texts; otherwise, students will not have access to the mathematical ideas and practices. Consequently, a mathematics lesson that helps students develop new understandings and practices also involves some form of literacy instruction to help students develop facility with the texts associated with the mathematical ideas and practices. Conversely, literacy instruction in a mathematics classroom will inevitably involve mathematics learning, because literacy instruction on the creation and consumption of mathematics texts cannot be content free. As students engage in the creation and consumption of texts, they inevitably deepen their understanding of the ideas being communicated. Because mathematics learning and literacy are inseparably intertwined, we have come to see that every mathematics learning event is also a literacy event, and every literacy event in a mathematics classroom is a mathematics learning event. 953 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert We feel that our negotiated perspective of the four-component model adequately addresses the three criteria for a shared perspective. First, we believe the model is credible to both us and members of our respective fields. Not only does the model capture the components and interactions that we feel are important, but it also has earned positive feedback from colleagues in our respective fields. Second, the four-component model addresses the perceived weaknesses in our original models. The presence of text in the model addresses Dan’s concern that he was neglecting literacy in his instruction; the presence of mathematics in the model addresses Roni Jo concerns about the way the discipline influenced what counts as text and literacy events. Lastly, the four-component model not only allows one to perceive mathematics learning and literacy events simultaneously, but actually requires that one do so. As stated above, the mathematics cannot be separated from the texts in which meanings are created, conveyed, and negotiated, just as literacy cannot be separated from the content that defines how the texts are to be read and written.

Discussion and Implications Taking our four-component model of literacy and mathematics learning seriously has implications for how one views the activities in mathematics classrooms. First, the model implies broad definitions of text and literacy in order to incorporate the objects used in mathematics classrooms to reason and communicate mathematics. Second, the model implies that students in mathematics classrooms must learn both the mathematics and how to participate with the texts used to reason and communicate mathematics. And finally, the model implies that literacy instruction for mathematics classrooms must honor mathematics, both in terms of mathematics as a body of knowledge represented by particular texts and mathematics as a community of practice (Wenger, 1998) in which individuals use those texts in particular ways. We will now examine how these three implications sharpen and clarify issues within the current scholarly discussion as found in the research literature. Broadening Definitions of Text and Literacy As stated before, our model reflects a broad definition of text that includes any object used to represent, convey, or negotiate meaning (Sebeok & Danesi, 2000; Wade & Moje, 2000). We recognize that not all literacy educators or linguists agree on such broad definitions of text (e.g., Halliday, 1996). However, the notion of text limited to traditional print material is not a particularly useful construct for thinking about the communicative and meaning-making processes that occur in the mathematics classroom, and thus is of little use to either mathematics or literacy educators for thinking about and addressing learning and literacy in mathematics instruction. On the other hand, when the notion of text is broadened to include all of the objects that are naturally used by participants in mathematical learning and 954 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction activity, it becomes a powerful construct for thinking about cognition and practice in the mathematics classroom. A broadened definition of texts for mathematics classrooms naturally requires a broadened definition of literacy. Our model implies that literacy in a mathematics classroom is the disciplineappropriate creation and use of texts to engage in mathematical practices and to understand, represent, and communicate mathematical ideas. Note that a broadened definition of text and literacy does not mean that any use of any text counts as literacy in a mathematics classroom. Although our model allows for children’s idiosyncratic forms of representation, these texts should nonetheless be created and used in ways that are mathematically appropriate. What counts as mathematically appropriate is determined by the conventions associated with the discipline, both as a body of knowledge and as a community of practice (Wenger, 1998). Thus, our model implies that the discipline not only dictates what the content of texts should be, but also what counts as texts, reading, and writing. The notion of a content-specific definition of texts and literacy has antecedents in the research literature. As noted earlier, some literacy educators have defined texts and literacy in a broad sense to include all objects that people use to create, convey, and negotiate meaning (Sebeok & Danesi, 2000; Wade & Moje, 2000). We modify their definition by emphasizing the constraint that text creation and use be done in discipline-appropriate ways. Others have noted the importance of discipline constraints on text use and creation. For example, diSessa’s (2000) notion of material intelligence, or the appropriate use of “materially based signs, symbols, depictions, or representations” (p. 6), implies a need for the individual to understand the disciplinary constraints imposed on the appropriate use of materials for communication in a discipline. Likewise, Roth and McGinn’s (1998) analysis of students’ mistakes with inscriptions acknowledges the important role that disciplinespecific social practices play in determining whether or not students’ text use is correct. Our definition of literacy also acknowledges the importance of the discipline, but widens the focus of attention to include all objects that people use to create and convey meaning, not just objects that are represented physically. Value of Considering Literacy Instruction in Mathematics Classrooms Once text is defined broadly, then literacy must be considered an integral part of mathematics teaching and learning. As mentioned earlier, the current reform movement in mathematics education is focused on helping learners develop mathematical understanding (NCTM, 2000). However, understanding cannot be achieved without fluency with texts. Because of the key role that communication, discourse, and representation play in the learning of mathematics, students who do not develop fluency with texts will not have access to the meanings that are being developed and negotiated in the mathematics classroom. Thus, literacy is essential to the process of developing understanding. At the same time, literacy is also at least partially what is 955 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert entailed in understanding mathematics. First, it is impossible to conceive of someone understanding mathematics who is not at the same time able to create, consume, and critique certain types of texts. Second, assessment of understanding is based largely on observing how well students create and use texts. Therefore, text creation and use is a crucial issue for mathematics educators. At first glance, it may appear that literacy using a broad definition of text has already been addressed in the field of mathematics, particularly by mathematics education researchers. For example, the extensive work done on communication and discourse has brought to the fore the importance of spoken texts and other forms of representation in the learning of mathematics (Lampert & Cobb, 2003). Other researchers have adopted a semiotic perspective to examine the interrelationship between symbolizing, meaning making, and mathematical activity (Cobb et al., 2000; Rotman, 1988). Research has been conducted on the use of many types of texts, including inscriptions (Lehrer, Schauble, Carpenter, & Penner, 2000; Roth & McGinn, 1998), computer environments (Clements & Battista, 2000; Shaffer & Kaput, 1999), graphing calculators (Doerr & Zangor, 2000), and graphs (Leinhardt, Zaslovsky, & Stein, 1990; Monk, 2003). Thus, one might claim that mathematics educators have already addressed the literacy of mathematics-specific texts in their research and practice. Nevertheless, a closer look at the research that has been done on different types of texts reveals that the focus has been on studying the interrelationship between meaning creation, learning, and texts, rather than an explicit focus on how to help students learn to create, consume, and critique texts. For example, Cobb, Wood, and Yackel (1993; Yackel & Cobb, 1996) have developed a theoretical framework that coordinates the social creation of knowledge, norms, and practices with the individual construction of knowledge and beliefs. As part of this framework, they discuss “talking about talking about mathematics” (Cobb et al., 1993, p. 99), or the negotiation of social and sociomathematical norms and practices in the classroom. Part of this meta-discourse focuses on literacy, because it addresses the creation and consumption of mathematics-specific texts, such as what counts as a mathematical explanation or justification. However, the goal of this meta-discourse is to support the development of students’ mathematical dispositions—their mathematical knowledge, beliefs, and values; improved discourse or literacy is not an end goal in and of itself. This view of discourse, communication, and literacy as merely a means to an end is even more evident in the following quote by Lampert and Cobb (2003): Even though current reform efforts focus on increasing the amount of mathematical communication that occurs among students, discourse is not a goal in and of itself. . . . Specific discussions must be justified in terms of what students might be learning as they participate in them, whether it be learning to communicate mathematically or communicating to learn mathematics. (p. 244)

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Standards-Based Mathematics and Literacy Instruction The problem with this emphasis on reading and writing to learn over learning to read and write is that it privileges cognition over literacy. When literacy is perceived as merely a means—perhaps one of many means—to develop understanding, then it is easy to lose sight of the inseparable connection between understanding and text use and creation. This separation can result in learners’ difficulties being attributed solely to the lack of content knowledge and understanding, rather than also considering whether or not the learners know how to create and consume the texts that are being used to develop and assess the understanding. This oversight is exactly the initial response given by Dan when faced with his students’ difficulties in writing conceptually oriented explanations. Roth and McGinn (1998) claim that the same error has been made by mathematics and science educators when considering their students’ mistakes with graphs. Their work suggests that certain difficulties that students have with graphs are due to their lack of experience with how graphs are used in social practice, rather than their lack of understanding of science or mathematics concepts. Unless equal weight is granted to reading and writing to learn and learning to read and write, mathematics educators are not likely to consider problems with text use and creation when faced with students’ errors. A second problem with granting lesser status to learning to read and write mathematics texts is that there is no perceived need for students to create and consume texts as an end goal in and of itself. Instead, text creation and use are thought of as part of the larger process of learning mathematics. However, texts that are created and used for the purpose of learning are often different than texts one would create as an end product. In particular, texts that are created for learning can be idiosyncratic to the user or local classroom environment. Furthermore, once the desired understanding is achieved, the learner may not necessarily feel the need to improve the texts associated with the learning event. In contrast, when the end goal is creating a suitable text, then the focus changes to modifying the text so that it conveys the desired meaning to the desired audience. This type of text refinement is common in other subjects, where students are often expected to create a text, polish, and eventually “publish” it. The purpose of this activity is to prepare students to participate in the larger disciplinary discourse that transcends the immediate classroom. Mathematics students are seldom required to produce texts that would grant them the status of a contributor to the mathematical discourse outside of their classrooms. Furthermore, unlike other subjects, texts from the larger discourse community outside of the classroom are seldom ever introduced to be read critically by students. Without such experiences, students are forever students of mathematics, rather than members of the mathematical discourse community. Literacy Instruction That Honors Mathematics At first glance, our model appears to only confirm the position of literacy educators’ stance on content-area literacy instruction. Literacy educators have 957 Downloaded from by Armando Loera on October 31, 2009

Draper & Siebert long acknowledged that literacy is a crucial part of the teaching of any content area. Unfortunately, literacy educators may be ill-prepared to provide mathematics educators with guidance in how to address literacy in a mathematics classroom. This is because literacy educators typically are unfamiliar with the texts that naturally arise as students engage in creating and communicating mathematics, as well as how these texts should be read and written. Indeed, literacy educators have not always considered what texts are discipline-appropriate when attempting to incorporate and study literacy in the mathematics classroom. In their zeal to support literacy across the curriculum, some literacy educators have introduced new types of texts rather than help students use the texts that are already present in the mathematics classroom. For example, some literacy educators have suggested engaging students in writing stories, poems, songs, and plays about mathematics (e.g., Martinez, 2001). This does little to help them learn to read and write mathematics, because mathematics is seldom communicated through these particular genres. Such activities can actually undermine students’ understanding of what constitutes mathematical knowledge, practice, and communication, and thus hinder students from future participation in authentic mathematics learning and practice. Even when literacy educators have addressed the reading and writing of texts that are common in mathematics classrooms, they often select texts that are being given less emphasis and importance in reform-oriented classrooms, and then suggest literacy activities that are not consistent with the norms and practices of mathematics as a discipline. For example, literacy educators often focus literacy instruction on story problems, and sometimes recommend the use of key words to identify appropriate operations and set up equations (see Dahmus, 1970; Manzo et al., 2001). This approach to reading story problems is problematic because it not only leads to incorrect solutions on some problems, but it also fails to match the way that mathematically literate people read and make sense of story problems by reasoning about the quantities and the relationships between the quantities in the story problem. Furthermore, mathematics educators are moving away from story problems that can be solved by using particular formulas or procedures to rich problem contexts which are open-ended and require students to create their own solution methods, not merely replace English words with mathematical symbols (Hiebert et al., 1997). This shows that unless literacy educators understand the nature of mathematics as a discipline, their recommendations for literacy instruction may actually do more harm than good.

Conclusions While we have gained much through our collaboration, this study is not without limitations. Indeed, our research collaboration allowed us to describe Standards-based mathematics instruction from our disparate perspectives and to understand the perspective of our research partner. Furthermore, through collaboration we cocreated a shared perspective that enabled us to critique the 958 Downloaded from by Armando Loera on October 31, 2009

Standards-Based Mathematics and Literacy Instruction fields of literacy and mathematics education. We expect that this shared model will prove helpful to other literacy and mathematics educators as they reconsider their perspectives of literacy and learning in mathematics classrooms. Our collaboration, in fact, is only a first step at understanding the complexity of teaching in mathematics classrooms that honors both literacy and mathematics learning. More research is needed to test the validity of this model and to locate the specific texts and literacies required to use those texts in mathematics classrooms. Furthermore, the shared model does not identify all the important aspects of mathematics and literacy learning. For instance, we anticipate that additional cooperative inquiry could reveal how factors like motivation, culture, and human development influence the context, texts, mathematics, and the learner as teachers work to help their students develop rich mathematical understanding and the ability to participate in mathematical activities. Finally, we have described a method of collaboration that can be extended to other fields. The promise of cooperative inquiry for multidisciplinary research is that it allows researchers a critical perspective as they view their field from the perspective of another. This critique may be threatening to a researcher whose entire work has been guided by a particular perspective. Indeed, cooperative inquiry requires patience, trust, and commitment on the part of researchers; it requires that researchers “bracket all motives save that of a cooperative search for truth” (Habermas, 2001, p. 100). However, as cooperative researchers move toward high levels of mutual trust and commitment, communication can take place that allows them to locate new perspectives and to find solutions to the most complex problems they face. Note The authors contributed equally to the research and authorship of this manuscript. The authors wish to express gratitude to Steven R. Williams and Robert V. Bullough, Jr. for extensive, helpful comments on previous drafts of this article.

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Stadrads based math instruction