ࡱ> 5@ Ebjbj22 1XX)|>>>8$#)"222222""""8<"4p%4($)R+,(22(22( d22" " !!2 @ > P!!$(0#)!,ZH,!,!2<n tl222(($  X MATHEMATICS AND CURRICULUM INTEGRATION SEQ CHAPTER \h \r 1: CHALLENGING THE HIERARCHY OF SCHOOL KNOWLEDGE Elizabeth de Freitas Adelphi University, New York < HYPERLINK "mailto:edefreitas@upei.ca" edefreitas(at)upei.ca> Introduction This paper explores the complex ways that mathematics teachers develop their identity within schools. I discuss a curriculum integration project in which untenured mathematics teachers attempted to cross discipline borders and create an authentic learning experience in a traditional middle school in Canada. I first discuss the procedure for curriculum integration, and explain how such integration is often problematic if too quickly implemented in traditional school contexts. I also discuss the theoretical literature about curriculum integration and explore the role of school mathematics in knowledge integration. I focus my analysis on the conflicts between untenured and veteran teachers in the mathematics department as the project was implemented, pointing out how tensions between these teachers can be interpreted through the lens of critical theory. Integrated and trans-discipline curriculum James Beane (1997) argues that integrated curriculum is politically controversial for three important reasons: firstly, integrated curriculum is designed from the bottom up, because content is meant to be responsive to student lives and contexts; secondly, integrated curriculum is collaboratively designed by school teachers and not by curriculum consultants or textbook writers, thus thwarting interest groups who seek to centralize authority over teaching; thirdly, integrated curriculum honours alternative ways of knowing that trouble the borders between traditional disciplines. Since discipline knowledge is a reflection of cultural interests, the dominant culture has a vested interest in maintaining these traditional discipline borders. Disrupting the borders and hierarchies that structure school subjects makes integrated curriculum difficult to implement in traditional schools. The first stage of integration usually involves finding affinity between disciplines and then collapsing specific course outcomes into more essential cross-curricular outcomes. This stage can be characterized as interdisciplinary, a preliminary stage in the attempt to fully integrate curriculum (Drake, 1999). Interdisciplinary stages tend to sustain discipline borders while searching for common ground, and are usually developed as thematic explorations calling on skills from various subjects. Trans-discipline curriculum, in contrast, disrupts previous knowledge categories and opens up the field of inquiry to a more radical form of integration. At the trans-discipline stage, students play a far more active role in the generating of content outcomes, tapping any and all resources and strategies, without concern for their traditional discipline source. This stage is very difficult to implement within a standards based national curricula, and requires a great deal of institutional independence. Teachers with a certain "liberal" philosophy can often make immediate sense of how to integrate subjects, as they often see themselves as generalists first, and specialists second (Lear, 1992). Despite strong commitment to holistic instruction, many curriculum integration projects have come to naught when teachers return to the classroom, close the door, and teach to the familiar text. Perhaps the most daunting obstacle to curriculum integration (at various stages) is the current emphasis on standardized assessment, which always functions to entrench the traditional borders between disciplines. The strategic and sustained implementation of integration projects often falters, even amongst the most committed (Beane, 1997). Mathematics as a school discipline is perceived as a highly sequential and linearly structured field of expertise (Stevens, 2000). The linear cumulative acquisition of mathematics knowledge is a well entrenched aspect of school mathematics. But it is precisely this model of linear and sequential knowledge acquisition that is under attack by the advocates of integrated curriculum (Drake, 1999). Integrated curriculum modifies the linear model of knowledge acquisition by suggesting that particular content be introduced according to students needs in solving authentic problems, instead of introducing content according to a set agenda. Such a differentiated approach to instruction is more democratic in the face-to-face encounters between students and teachers, and seems more organic and authentic in the way it responds to student needs. In practice, integrated curriculum might mean introducing more advanced mathematical concepts earlier to students, depending on the context, if complex real world problems demand it. It may also mean not covering basic skills at a particular grade, and letting students proceed to the next year without mastering particular algorithms. In many ways, the mathematics teacher has the most to risk in embracing an integrated curriculum. No other subject seems to invest so much in a curriculum arranged in terms of increasing logical complexity. The interruption of sequential linear development threatens the very infrastructure of school mathematics. Indeed, such a radical breaching of protocol around procedural and conceptual development might endanger the discipline by naming the ways in which it reproduces itself as a dominant cultural discourse (Skovsmose, 2005). In The Sociology of Mathematics Education: Mathematical Myths/Pedagogic Texts Paul Dowling (1998) describes a number of myths associated with school mathematics, arguing that each myth functions as a way of centering mathematics within our culture. These are not myths in the sense of being untrue or religious, but rather cultural myths that people often take for granted. School mathematics plays a hugely significant role in structuring social life because of the way it functions as a critical filter. It is a high status discipline, impacting the lives of most youth, both negatively and positively (Ahlquist, 2000). Dowling is interested in the socio-cultural framing of school mathematics, offering a way of critically interpreting the role of mathematics within an integrated curriculum. In this case, myths refer to beliefs about mathematics that frequently circulate inside and outside of schools. In this paper, I use two of Dowlings myths, the myth of reference and the myth of utility. The myth of reference refers to mathematics as a system of exchange-values, a currency(Dowling, 1998 p. 6) that underwrites concrete reality. According to this myth, the mathematician casts a penetrative gaze upon the non-mathematical world and is able to decode or translate the surface appearances into mathematical terms. The myth of utility, in contrast, is more utilitarian, constructing mathematics as a reservoir of use-values or a tool box (ibid). In this case, mathematics serves other domains and other inquiries, its meaning and value implicated in the ways in which it works for that which is outside its borders. Neither of these myths are politically neutral. They participate in the material structuring of society and are themselves consequences of power relations. Both myths, claims Dowling, read culture, community and context as incomplete until inscribed by/within mathematics. In the following case study, the two myths are evidenced in the actions and stated beliefs of teachers involved in a school wide curriculum integration project. The case study Our need to revise our middle school curriculum sprang from two parallel problems. The first was the shrinking of a five-year high school program into four, due to a government decision to eliminate grade 13 in Ontario, with the consequent rush to cover content at earlier grades. The second was a recognition that our courses were already overcrowded with too many performance outcomes, and that students were unable to draw connections between subjects. In an attempt to solve the first problem and ease student course loads at the senior school level, we decided to integrate four half credits into the already crowded grade seven and eight curriculum, thereby giving students a two-credit head start on their four-year high school diploma. In order to accommodate these new credits, which focussed on learning strategies, teachers were asked to integrate the new outcomes into their existent units. In an attempt to address the second problem, teachers were asked to create interdisciplinary units that addressed shared expectations. As one of two mathematics teachers working in both the middle and senior schools, I was asked to re-evaluate the arrangement of our mathematics curriculum. We began with a five day planning session attended by all participating teachers. Teachers were initially concerned that there would now be less time to cover more material. Although duplication of content was occurring across disciplines, teachers were initially unaware of this redundancy. Curriculum had developed over the years to meet changing ministry guidelines, but no dialogue had occurred across discipline borders. Like so many students in all schools, ours were feeling the strain of a fragmented learning experience. Students devised coping mechanisms that involved slotting knowledge into pre-ordained compartments without any feel for the connections between subjects. For instance, they learned bar and line graphing techniques in at least three different courses, but each course framed the knowledge differently, and students failed to comprehend that they were performing the same task in different courses. The first two planning days focussed on the four half-credit courses, for which government guidelines dictated, in no uncertain terms, the specific skills to be demonstrated. Administrators responsible for collecting the evidence of our having designed the appropriate performance indicators played the role of auditor, through no fault of their own, demanding lesson plans and rubrics as documentation, which were a necessary part of the legal process for obtaining a credit rating. The usual lists of outcomes, all carefully composed using the mandatory active verb tenses, circulated amongst the departments as we tried to accommodate the new content to be embedded in our courses. Major chunks of English and Art courses were then appropriated in the name of the new integrated curriculum. Representatives from these departments felt as though they were sacrificing the most, and indeed they were. At the end of day-two, relations between disciplines were already souring, as the humanities began to realize that mathematics and science were not being asked to sacrifice their instructional time. As a mathematics teacher, I altered a few units so as to address the new course expectations, but it was the other courses to which the administration repeatedly turned when asking for additional teaching time. The insidious hegemony that ranked certain kinds of knowledge over other kinds was felt immediately upon launching the project. Tacit assumptions that ranked mathematics and science above the arts were operating at every level. Once the rather clerical task of documenting how we would address the new material was complete, and the necessary templates were composed, and our administration stamped it with approval, we then moved on to the issue of cross-curricular integration. We began by presenting detailed course outlines to each other which allowed us to identify the areas of overlap and the areas of connection. Many teachers were surprised to learn about course content in other fields and found the exercise exhilarating. We truly began to grasp the connectedness of what we were doing. A more holistic understanding of the student emerged. We were able to get outside of our own subject expertise, and began to comprehend the richness and diversity of student learning. The shear volume of learning outcomes was overwhelming. There were fourteen teachers working in the middle grades, spanning nine subjects, many with their Masters degrees. Most taught higher grades as well as seven and eight, which meant we invariably imposed our higher-level standards of behaviour and learning on the middle school students. As we went around the room, and the course descriptions were shared, the diverse personalities and teaching styles became evident, and I imagined a thirteen year old student trying to juggle all our different expectations, both explicit and implicit - our pedagogic assumptions, our class-management strategies and classroom dynamic, our personal histories, senses of humour, and our own learning styles. This was a potentially powerful affiliation of distinct identities brought together precisely because of our different disciplines. But our collective purpose was to transcend those discipline borders and create a new space for learning. It seemed suddenly very urgent that we recognize the importance of the personal, that we dwell on our diverse teacher identities, and realize that we teach who we are. As the week progressed, and the traditional barriers were undone, teachers shared more of their personal styles. Casting aside some of the conventions of their discipline, teachers began to discuss their own idiosyncratic styles. The luxury of a full week of collaboration allowed us to delve deeply into our assumptions about learning. The other mathematics teacher and I discussed beliefs about education that we felt had been buried beneath the implicitly imposed discipline paradigm, to which we felt we had capitulated. Government guidelines, school expectations, and parental pressure had figured prominently in the shaping of our middle school mathematics curriculum. In discussing our curriculum within this new community, and away from the rest of the mathematics department, we were able to name the ways in which we felt coerced into complying with the dominant departmental practices. Department meetings often focussed on the careful alignment of sequentially organized learning outcomes, ensuring that specific skills and concepts could be traced from the earlier grades to the later grades. In disrupting that affiliation and demanding that we invest in this laterally associated community, where the focus moved from linear development to the co-temporal lateral associations that comprise a saturated learning experience, the integration project seemed to create a space where both students and teachers diverse ways of knowing might be heard. What emerged during the week were our marginalized beliefs about mathematics and education which had been silenced by the entrenched practices of the status-quo. In this new community we were given a distinct voice and able to question the common sense and naturalized practices that were taken for granted by the mathematics department. The expert knowledge of veteran mathematics teachers had imposed strict role-playing rules on both of us as beginning mathematics teachers, to such an extent that any possibility of actively and critically interrogating the dominant paradigm was contravened. But as we troubled our discipline subject positions, if only slightly, we remembered these once silenced idiosyncratic beliefs about learners and how they experience mathematics. In breaking open the massive structure of mathematics curriculum, I was able to put the student back at the centre of my planning. After the initial sharing of outlines, teachers from each discipline met with each other, discussed points where their two curricula might meld into one, and developed lesson plans and a concrete schedule of collaboration. We were ensconced in a large meeting room where lunch and coffee were delivered each day. We circulated about the room, arranging for one-hour brainstorm sessions involving pairs of different discipline teachers. Every discipline teacher sat down with every other discipline teacher, found some affinity between courses, and then collapsed their specific goals into more essential cross-curricular expectations. As a mathematics teacher, I found that my conversations were primarily driven by the others course content. I chose not to open with my own curriculum objectives. Instead, I asked that they explain in more detail some of their major topics, and I then suggested mathematical tools that might be useful in exploring their topics. Whether it was understanding human anatomy, designing Egyptian art, learning musical forms, or analysing global demographics, there was always a way to insert mathematics. Without intending to, we moved from subject to subject and proffered mathematics as an explanatory foundation of all knowledge. The other discipline teachers accepted the premise that mathematics was a tool for exploring other forms of knowledge. In each subject, we suggested that mathematics might be integrated by seeing it as an abstraction within the other content, a hidden language of pattern and rule, a means of verifying the truth of the phenomena under study. Data management and pattern recognition were the two most often used topics in this regard. Although one might have imagined a scenario where the mathematics was framed by the other disciplines, it often seemed as though teachers began to see the mathematical nature of their own subjects. When we returned to our classrooms, we implemented some of the integrated lesson plans during the first two weeks of the year, but it quickly became evident that teachers were being drawn back into their discipline borders by their departments. Communication between disciplines broke down, and no new integration emerged. There was no time for planning, and department meetings took priority over cross-discipline meetings. As many of the teachers in the middle grades were untenured, issues of power inhibited the capacity of these teachers to disrupt the traditional practices in the discipline departments. As the year progressed, I found myself more and more concerned with covering the textbook material so as to ensure that my students were prepared for mathematics courses offered the following year by other more veteran teachers. As the days passed, and I discussed issues with other department members, I found it difficult to resist the monumental coercive forces that define legitimate mathematics. In the math department meetings, the veteran teachers who taught only senior students expressed disdain and disapproval for an integrated curriculum that would result in eliminating mathematics content. One teacher suggested that the students future was at risk because of the integrated curriculum, and that disrupting the sequential acquisition model would ruin their chances of obtaining the high grades they needed to enter university. Although such an argument seems initially motivated by concern for student success, it also functions as an argument for maintaining the current discipline borders for the sake of the status quo. Department meetings became sites where normalization of practice occurred. The veteran teachers were very resistant to change. The integrated model was perceived as threatening the unitary and cohesive community of the mathematics department. It was as though an integrated curriculum betrayed the discipline by dispersing the authority of mathematics across other disciplines. My attempt to be partially affiliated with the mathematics department and partially affiliated with the integration project created a conflict of loyalties. Teachers were unable to sustain the multiple affiliations, as departments demanded complete identification with the discipline, and made integration awkward and eventually impossible. In the mathematics department, I often felt coerced into abandoning the integration project, as though I had to choose between mathematics or that other project. The implicit refrain might have been youre either with us or against us. Senior mathematics teachers were particularly negative about the prospect of an open-ended activity as a form of summative assessment, in lieu of an exam. They were concerned that it would consume precious time without meeting many concrete expectations. There was a fear that the culminating activity would be an organizational nightmare, and that take-home work would inevitably be completed by parents instead of students. Parents investment in their childs success, within this extremely traditional school, was far-reaching and had huge impact on school culture and teacher identity. Exams had come to function as a way of assessing mathematics teacher accountability. The mathematics department was wedded to exams as a principal form of assessment. Beyond the concern that abandoning the logical sequence of mathematical development in favour of a more organic approach would cause a definite loss of content, there was another concern about how to assess simple knowledge and algorithmic skills within an integrated curriculum. Veteran mathematics teachers worried that the students would not have proven their understanding if they hadnt done so in isolation from other disciplines. Because this concern was further reinforced by parents concerns over the prospect of no objective assessment, the middle-school students were given in-class exams, and their performance in mathematics was documented and recorded separately, as though nothing had been integrated. Concluding Remarks Exams were crucial in repeatedly re-establishing the legitimacy of mathematics as a discipline, validating both the content and the teachers credibility. The exam, argues Carrie Paechter, controls the work of the teacher, who is constrained by the pressure for examination success to remain within the boundaries of the examination-defined subject.(Paechter, 2000: 136). She draws from the work of Foucault, suggesting that disciplinary power works through the teachers, The examiner both carries out the scrutiny and is disciplined by its demands. (151) In our case, assessment and accountability interrupted all attempts at integration because school disciplines were wedded to their specific forms of assessment, and because teachers were ultimately bound by those same forms of assessment. My attempts at generating cross-discipline connections contributed to what Dowling describes as the myth of utility which functions to legitimate mathematics in schools (Dowling, 1998). Again, Dowling does not dispute the utility of mathematics, nor suggest that utility is in itself problematic, but advocates for an increased awareness of how these habits of legitimation are enacted in schools, so as to raise teachers critical consciousness around the structuring of the institution. By repeatedly declining to define my own needs during the collaborative stage, and inviting the other to specify her own context, I set in motion a power dynamic that inscribed my knowledge into that of all the other disciplines, whether it be geography, art or gym. Although one might see mathematics as a hand maiden to the other disciplines in this instance, Dowling argues that the myth of service further centers mathematics within education, establishing itself as foundational to all knowledge. This notion of foundational knowledge presumes the existence of an essential explanatory system and contravenes the more hermeneutic and dynamic epistemology at the heart of integrated curriculum. The disapproval expressed by the veteran mathematics teachers represents a re-inscribing of the disciplinary power of school mathematics. Their strategic use of students at risk in sanctioning sequential curriculum, and their disdain for alternative assessments must be read as acts that re-produce and regulate the status quo. Although their disapproval was prohibitive, it was also a structuring of what was possible, and thereby enacted the disciplinary power that produces and governs subjectivities in schools. Disciplinary power circulates through forms of instruction and assessment, sometimes so tacitly that it is difficult to resist, and always in ways that validate some voices and practices while dismissing others. Speakers often unknowingly participate in the further enculturation of dominant instructional practices, although there is always space for altering and troubling these same practices if we are willing to risk our own comfort. As a member of the mathematics department, I felt an obligation to share in the emergent collective identity; my resistance to well-entrenched master narratives about curriculum engendered a feeling of alienation which, in turn, made me more vulnerable. It was this vulnerability which became the site for the co-construction of my teacher identity in conformance with the expectations of the department. These moments of identity construction point to the regimes of experience and accountability (Wenger, 1998) that regulate membership in communities of practice. Similarly, the government curriculum documents functioned as technologies of surveillance that were all too easily internalized. Despite my original commitment to cross discipline borders, I felt the deep structural impact of an external regulating body. The government curriculum guidelines addressed each teacher or rather each discipline separately, and in doing so, we were hailed as subject teachers first, and only obliquely recognized as potentially cross-discipline collaborative educators through the documents reference to connections. The consequent sense of belonging which emerged was grounded once more within the discipline, where the documents officially (mis)recognized me as a member. The case study highlights the complex ways in which teachers develop their identities in affiliation with others. We can see that mathematics teacher identity is structured as a singular or unitary social entity with little allowance for diverse or multiple affiliations. This notion of the loyal mathematics teacher whose interest is to serve the discipline by continuing to defend its border and status needs further examination. It may be that such habits of enculturation impact all aspects of instruction and impose a sameness on mathematics departments that might need some disrupting. The case study suggests that there is little tolerance for a multiply affiliated teacher identity that isnt exhaustively defined within the discipline. Further research might examine ways of recognizing multiple affiliations within mathematics teacher identity, and explore the school and department practices that neutralize difference and demand identification. Mathematics departments may rely on a notion of teacher identity that idealizes sameness and consensus while demoting difference and dissent. Homi Bhabha urges us to re-think identity as a process of affiliation rather than autonomy, a site of partial solidarity rather than sovereign mastery, where the complementarity and reciprocity of multiply performed identities might be more ethically recognized (Bhabha, 2003: 6). Re-thinking identity in terms of contingency and provisional commitment suggests that we move away from a paradigm of consensus where contradictions are resolved and difference is sutured over with self-censorship, and towards, possibly, the prospect of a coalition that allows for divergent positions. Amongst mathematics teachers, then, there is a need to confront the notion of identity wholeness or cohesion, to reconfigure identity as an ongoing process that resists closure, so that collaborative or collective enterprises can become sites where mutual alterity is recognized. REFERENCES Appelbaum, P.M. (1995). Popular culture, educational discourse, and mathematics. Albany, NY: State University Press. Beane, J. (1997). Curriculum integration: Designing the core of democratic education. New York: Teachers College press. Bernstein, B. (2000). Pedagogy, Symbolic Control and Identity: Theory, Research, Critique, Revised edition. Oxford, UK: Rowman & Littlefield Publishers. Bhabha, H. 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