ࡱ> 5@ Ubjbj22 (PXX708'8'8'8p'(0ND)L)")))***MMMMMMM$MPRR\N;8**;8;8N))#NP===;8))M=;8M= ==I|J)8) `E )8'8(IMsNTNI~R;lR$J00JRJ*<.=1l;4***NN00dW=X00MATH EDUCATION AND SOCIAL JUSTICE: GATEKEEPERS, POLITICS AND TEACHER AGENCY Peter Appelbaum and Erica Davila Arcadia University, USA Appelbaum(at)arcadia.edu Davilae(at)arcadia.edu Introduction This article addresses several urgent mathematics education conversations through the lens of critical educators. As teacher educators, we see and hear the experiences pre-service and in-service teachers undertake while teaching math that is grounded in social justice, or when thinking through teaching math using pedagogies rooted in emancipatory frameworks. We deconstruct the resistance to teaching math that is grounded in social justice education. The two overarching conversations we address are: (1) The gatekeepers that surface for teachers who teach mathematics with an emphasis on social inequity. Gatekeepers are those individuals or groups of people who are perceived as potentially expressing dissatisfaction with the teachers choice of social justice pedagogies, e.g., a principal or parents. For many pre-service and practicing teachers, the conversation regarding gatekeepers is more about perceptions they have regarding the struggles they will face when faced with gatekeepers within the structure of the school system than with actual realities. (2) The curriculum politics that determine who decides what is taught in K-12 mathematics, and how these political forces connect to the implementation of socially just curricula and pedagogy, specifically for mathematics. We highlight the experiences of early childhood educators because of the recurring question posed to us as teacher educators: How can I teach about social justice to the younger children? The deconstruction of conversations that pre-service and practicing teachers pursue can empower the participants to be intellectuals and social agents in their classrooms and schools, and to make critical decisions around curriculum and pedagogy in mathematics. Keywords: mathematics education, social justice, critical race theory, resistance, curriculum design, teacher collaboration Theoretical framework Critical race theory offers a way to understand how ostensibly race-neutral structures in education--knowledge, merit, objectivity, and "good education"--in fact help form and police the boundaries of white supremacy and racism (Parker, Deyhle & Villenas 1999; Ladson-Billings & Tate 1995). Specifically for this research project, the perception of math as a neutral subject is analyzed through the experiences of K-12 educators. In particular, critical race theory is used to deconstruct the meaning of "educational achievement," to recognize that the classroom is a central site for the construction of social and racial power. There are reoccurring conversations unfolding in our university classrooms about social justice math versus real math education; this juxtaposition speaks to the meaning our students, future and current teachers, are making through the use of the phrase, educational achievement. As we analyze this dichotomy between social justice math and real math, the lens of critical race theory helps us to interpret this concern. When our students begin to understand the integral role teachers must take to empower their students, they shift to discourses in which teaching math for social justice is teaching real math. We also reference the Standards of the National Council of Teachers of Mathematics (NCTM 2000), local State Standards (PDE undated), and the Manifesto of the International Commission for the and Improvement of Mathematics Education (CIEAEM 2000), in order to place our analysis in the context of current curriculum reform efforts. We theorize teachers participation as change agents in the reform process as central to the success of social justice curriculum development. Leu (2005), Remillard (1999), and Drake and Sherin (2006) indicate that teacher participation in the theorizing of mathematics education reform in is critical to our understanding of the reform process. Teacher interpretations of curriculum materials, the philosophy of education supported by specific curriculum materials, and the meanings and purposes of classroom activities, are the nodes of multiple networks of social and cultural discourses in educational reform. Modes of inquiry Analyses of university classroom discussions, online discussions, class assignments and presentations serve as our modes of inquiry. We analyze the processes students encounter while exploring two assignments: Each of us shares an example of an assignment we currently implement in our teacher education course work at the graduate level. Both of these projects help authorize teachers to become empowered teachers who can teach social justice and real math at the same time. One project is part of a course on mathematics and the curriculum; this project requires students to develop a piece of curriculum centered in social justice themes, and to demonstrate that their lesson/unit idea can cover enough of their current curriculum to justify using it as a replacement unit. The second project is part of a foundations course on culture and education and requires students to review and critique a lesson they have taught through a sociological perspective; students must subsequently design a modified lesson. Both projects also include a reflection paper in which students identify two potential gatekeepers; they are required to explain how they would convince each gatekeeper to use this new curriculum. The ideas students develop through these two learning activities highlight the experiences teachers are sharing in terms of teaching math for social justice, or thinking math through social justice. The analysis of these experiences deconstructs the struggle to transform curriculum and construct hope. Point of view As teacher educators who are experiencing the intersection of teaching and researching, we have gathered data from our classroom dialogues for this article. The conversations unfolding in our classrooms are the catalyst for this research project. The perceptions and practices that pre-service and in service K-12 teachers share in class need to be deconstructed if we are to carve new spaces of inquiry in their teaching. The students in our courses teach one another. Their lived experiences in their classrooms and field work sites provide many opportunities for them to reflect and analyze. More specifically, in the area of math education, many of these teachers are struggling with the same dissonances between learning how to implement best practices and actually implementing them. The rationale for the dissonances varies from student to student. Many fear that teaching math for social justice does not fit with the mandated scope and sequence of their school/districts curriculum; they perceive various gatekeepers that serve as barriers. Others fear that they will bring issues to a group of children who are innocent and nave about social inequity. This latter fear has been shared by many teachers and future teachers who teach in the early grades. Another common reaction among our students is the perspective that teaching math for social justice is too political. We have also experienced students who dont share the dissonance because they hold the perception that they are implementing best practices and do not see the urgency of integrating social justice in their curriculum. Students who do not see the urgency often offer explanations about their curriculum being multicultural (and therefore they are teaching for social justice). However, most of the teachers who practice teaching math for social justice still feel the urgency and are usually eager for resources and support. A Peak at the Implications of Our The notions of urgency, dissonance, and best practices can be interpreted as the ongoing social construction of ethics. In this sense, teaching is an ethical stance one takes with the world (Block 2003). Through ones social actions, a person continually invents, or reinvents, an ethical self, in each and every moment. According to Jim Neyland (2004), the ethical self is prior to all codification, and instead founded on the direct, face-to-face ethical encounter of responsibility between persons. In contrast to currently dominant approaches, mathematics education should ensure that legislative protocols do not override the ethical primacy of the direct encounter. (p.55) This would imply that teachers must understand their practices in ways that take heed of Standards and other protocols, but only insofar as these protocols do not displace direct encounters between people. However, we see our work as contributing to a postmodern sensibility that avoids a reproduction of reform versus authentic dualities: in our work, teachers create reform projects and critique existing curriculum in ways that do not set up Standards and Innovation in opposition, and instead see both as mutually supporting the other. Standards are used to justify the social justice mathematics activities; and the social justice mathematics activities are used to accomplish the standards as well as other goals. The various perceptions shared in our teacher education courses are critical to explore in order to make effective changes in curriculum reform. We need to understand the resistance to transformative math education. In order for social justice math to be valued, teachers must first become aware of the social injustices around them. Teacher educators should encourage K-12 teachers to infuse socially just curriculum and resist the perspective that schools and teachers are apolitical. Teachers should be empowered and be conscious of their beliefs and values and reflect on who they are and what they believe in and most importantly how this impacts their teaching. During this reflection process many teachers can begin to see the moral responsibility of teaching and thinking math through social justice. Altogether this research project has served to highlight the experiences teacher educators and teachers are having around the need for infusing our classroom with socially just curricula. The two specific learning activities that empower teachers to rethink their practice in teaching mathematics in K-12 classrooms are discussed to bridge our theory and practice. Creating a Replacement Unit The Mathematics and the Curriculum course included eleven students: one first-grade teacher, one fifth-grade teacher, four middle school mathematics teachers, three secondary school mathematics teachers, one alternative high school special education teacher, and one pre-service secondary mathematics teacher. We began the course with an overview of contemporary State, National, and International Standards documents, and compared the content with the curriculum innovations in Gutstein and Petersons (2005) Rethinking Mathematics. Student reactions were varied, but each spoke in one way or another of perceived barriers to implementing the ideas presented in Rethinking Mathematics. Examples of activities appeared initially to focus on content that is not central to the given mathematics curriculum found in Standards documents and textbooks that are provided by the schools in which they teach. Organization of classroom learning experiences also appeared to require very different forms of teacher and student behaviors from what these teachers usually encountered themselves in their own work. It became clear to us that this perception of social justice mathematics as alien to common practice needed to be confronted head-on. What was originally required in the course as a curriculum design project became a mutually negotiated trajectory paper, in which students outlined the trajectory of their mathematics curriculum for an extended period of time, incorporating social justice ideas from the Rethinking Mathematics book in ways that supported the achievement of expected outcomes and Standards. One particular feature of the assignment that presented a unique challenge was to transform the teachers fears and anxieties into a productive task. We discussed the underlying reasons why we were not able to directly implement the social justice mathematics ideas in our current work, and determined that we unconsciously constructed gatekeepers whom we imagined as standing in the way of innovative curriculum development and mathematics education practices. These gatekeepers seemed very real as the teachers attempted to describe concrete reasons for why they would not be able to implement social justice mathematics. However, the gatekeepers were in some sense imaginary symbolic tools of resistance to change. It may very well be the case that other teachers and/or the school principal would worry about what is going on in a teachers classroom if the activity is very different from the usual practice in the school. But the fear that such a reaction would occur is no more real than the possibility that an innovative teacher would receive accolades or simply be ignored by others. It may very well be that parents of students might worry that their children are not learning what should be learned, or that the Standards are not being met; yet this too is no more than a fiction created to concretize fears of the unknown. One such interesting fiction that arose in our collaborative work was the teacher herself as gatekeeper: a teacher may worry herself that trying something new without previous practice, or approaching mathematics from alternative perspectives, could be unsuccessful when judged from the position of whether or not students have mastered expected skills and developed particular conceptual understandings. One clear requirement for introducing social justice mathematics into the school curriculum was thus made evident: we needed to confront potential gatekeepers directly and proactively. Whether fictional tools of resistance to change or actual constituents in the school community, gatekeepers should be recruited as supporters of innovation before they can begin to formulate concerns. With the notion of gatekeepers in mind, the group decided that the best proactive stance would be to demonstrate convincingly that the planned social justice activities not only contributed to successful mastery of Standards-based curriculum, but that the social justice curriculum would seem to promise better achievement of these objectives. In an ancillary examination of particular kinds of experiences within a mathematics curriculum, we also identified the location of a classroom activity at the beginning, middle, or end of a unit of study as an important planning decision for a teacher and/or curriculum developer. Placement at the beginning of a unit works well for initial pre-assessment of skills and concepts that students already know and would therefore be available for elaboration or as the basis for the construction of new knowledge. Placement in the middle of a unit is useful for a teachers ongoing assessment of what has been learned so far, and for making decisions about whether any materials needs to be re-taught or supported with supplementary topics. An activity in the middle can also introduce new perspectives on content, or redirect students attention to critical ideas. Sometimes such an activity can allow for reflection on new concepts or skill practice. Placement at the end of a unit can provide a culminating experience, can serve as a more definitive assessment tool for the teacher, and can serve as the context for concept integration; such an activity might also enable students to use skills in new contexts, or to provide a transition to the following unit. Appelbaum (2008) describes this curriculum planning analysis as the placement of the activity within a trajectory. Through such considerations, we created the trajectory assignment, which involved the following criteria for materials and networking tools: Incorporate one or several ideas from the Rethinking Mathematics book in order to plan an extended unit or multi-day lesson. Demonstrate that this lesson/unit idea can cover enough of my current curriculum to justify using it as a replacement unit. Describe how the larger curricular context would change if this lesson/unit were at the beginning, middle, or end of the design. Demonstrate working knowledge of NCTM, CIEAEM, and PA Standards/goals. Identify 2 gatekeepers who will need to be convinced that it would be OK for you to use this and describe how you will convince them. Create any materials that are needed in order for this lesson/unit to be immediately implementable (for example, create any student handouts that would be needed, schedule appointments with colleagues and community members, compose and deliver letters and memos that must be written, and so on). The teacher-designed units and lessons that were developed in this project were highly successful in the sense that each design incorporated social justice mathematics ideas from Gutstein and Peterson while also establishing explicit ways that these units and lessons were carefully constructed to meet more traditional content standards. A common form of resistance is to label the social justice content as other than mathematics, so that one might distance oneself from social justice mathematics by declaring it more relevant to social studies or other curriculum areas. By carefully outlining the traditional content that is addressed by the social justice curriculum, these teachers were not only able to make a reasonable argument for the social justice curriculum as a replacement unit, but were moreover better prepared to make the replacement unit more successful than the commonplace curriculum at achieving the expected content objectives. More powerfully, we note how the teachers relationships with social justice mathematics was transformed through the design experience. Each teacher was convinced that their unit or extended lesson would in fact easily meet the Standards and curriculum objectives that they believed were assigned to them by higher authorities. They were comfortable explaining how and why their unit would do a better job than the already existing set of daily lessons based on their textbooks. Furthermore, because the group supported each other throughout the design process through in-class collaborative workshops, they were convinced of the efficacy of each of the other designs as well as their own. Because each participant had given feedback and offered ideas to the others in small working groups over the course of four class meetings, they were each personally committed to each others designs, seeing something of their own response to a concern in the others final materials. The gatekeeper materials served a unique psychoanalytic function. When a teacher imagines a potential gatekeeper e.g., principal, parent, colleague, community member, student, themselves they are projecting their own anxieties fears onto an imaginary construct. As noted above, it is possible that any one of these real people may in fact display certain anticipated reactions. However, in the proactive process of working with gatekeepers, the teacher has an opportunity to interact with their own projections and concerns. Whether a teacher presents herself or her principal as worried about the lack of attention to the assigned content for the next month of the school year, the worry about not meeting obligations is the most important concern that the teacher must address. Whether explicitly addressing their own concern or not, the proactive composition of a letter to parents, or the preparation of a proposal for a meeting with an administrator, allows the teacher to work through any possible fears and worries that they may even be having trouble consciously articulating. For example, a letter that lists twenty standards that are being addressed by the forthcoming unit, along with specific suggestions for ways that skills can be practiced at home simultaneously helps a teacher recognize that the unit indeed addresses the twenty standards and indeed provides adequate coverage of particular skills. Furthermore, the composition of such a letter prepares a teacher to better accomplish these goals now that they are explicit, while also making clear to the teacher what must be assessed on an ongoing basis in order to meet the goals. The trajectory notion serves to highlight both the importance of the social justice aspects of the unit/lesson and the ways that the social justice mathematics and the traditional Standards-based mathematics mutually support each other. Our trajectory designs were really three different unit/lesson designs, and the juxtaposition of each with the others helped to clarify our goals for the designs themselves. The main social justice activities from the Rethinking Mathematics book were placed at the beginning, middle, or end of the plans. The consideration of how this placement would change the skills and concepts that would be developed through the designed curriculum helped the participants to determine what skills and concepts were possible in the first place, and in the process, helped the teachers to better understand the mathematics of the social justice experiences more deeply. Teachers rarely have the opportunity to reflect on the kind of mathematical thinking that they can spark for their students through the introduction of a specific experience. In this project, each teacher pursued the subtle differences that would be enacted by the choice that they needed to make about when and where to place a particular experience within the trajectory of the unit or lesson. They also were participants in the discussion of nine other unit/lesson designs that were also carefully analyzing these issues. A maxim of psychoanalytic thought is that resistance is essential for critical learning to occur (Appelbaum 2008). In this assignment, the teachers resistance to new curriculum ideas became the explicit focus of the task. By labeling the fears and anxieties as external gatekeepers, we were able to distance ourselves from our own resistance, objectify it, analyze it, and move through it to a new understanding of a social justice mathematics curriculum. Social justice mathematics was no longer set up as an alternative to real mathematics, and no longer positioned as oppositional to accepted practices. What was initially othered became an ally, a tool for accomplishing our own and others goals for school mathematics. Cultural Foundations and Curriculum The Cultural Foundations of Education course is also a graduate level foundations course. In this course students are empowered to be introspective, reflective and action oriented teachers. Many students are practicing teachers working towards their masters degree or teacher certification. Several are pre-service teachers, some straight out of undergraduate teacher preparation programs; others are career changers who spent significant parts of their lives working in various career tracks. The course is taught from a theoretical stance that is rooted in an emancipatory framework that empowers teachers and future teachers to be activists and advocates for their students. The students work towards developing conscientization as defined by Freire (1972), which is a level of consciousness that evokes the power to transform reality. The course begins with introspection and reflection, and then moves to the lived experience of privilege and oppression; the final portion of the course is designed to explore the implications of the previous conversations within the context of K-12 schools. The students read and explore ideas on the interconnectedness of multicultural education and social justice. The experience the students have while developing their culminating project in this course is what has peaked this inquiry in mathematics education and social justice. Review & Critique of Curriculum or Lesson Plan Individually or in pairs, students will find either a textbook chapter/unit or a lesson plan to: Evaluate and critique from a multicultural lens (e.g., What is missing from the material that would make it more multicultural? What is in the material that is stereotyped, Eurocentric and/or biased in specific ways?); and Create an alternative text that articulates your ideal multicultural lesson and reflects issues raised in the course (e.g., What could be changed or added to enhance the material? To make it more inclusive and reflective of multicultural ideals?). Presentations can be structured in a variety of ways. You are encouraged to use this as an opportunity to conceptualize and practice critiquing educational materials from a multicultural perspective as well as developing creative and critical material to use in your classroom. You will be evaluated based on the thoroughness of your critique and your ability to develop practical materials/methods that are inclusive and culturally responsive. Curricula and lesson plans as well as any additional material for presentations should be copied so that all class members have a copy on the day of your presentations. For this article, we focus on several students who are teaching or hope to teach mathematics. One of the students is a kindergarten teacher in the Philadelphia public schools who struggled to see the bridge from theory to practice when reading Skilton-Sylvesters (1994) Elementary School Curricula and Urban Transformation and Gutstein and Petersons (2005) Rethinking Mathematics. She kept saying, but theyre only in kindergarten. It was not until she chose a specific lesson to review that she had taught for several years on currency in the United States (US), specifically pennies and dimes, that she not only saw the bridge from theory to practice, but built her own. The Pennsylvania state standard she was aiming to meet was: PA- Pennsylvania Standards for Kindergarten Key Learning Area:Mathematics Standard:2.1 Numbers, Number Systems and Number Relationships Content:K. Count pennies and dimes Her initial idea was to include international coins in her lesson, and to have the kids compare and contrast US coins to other coins. However, after one of her peers asked a simple question - Will you tell them the US is the wealthiest country? - she was empowered to teach these very young children about the power of US currency, and to begin a critical conversation on global inequities. As she presented this lesson to her peers she shared the experience she had with her students, during which she came to a realization: she held the power to subsequently empower her students. She knew that presenting them with a chance to think of global inequity in kindergarten was a critical component in developing their sense of agency. Furthermore, she described this new lesson on coins and power to be the catalyst in her transformation. She described social justice as a moral issue; she shared her sense of urgency for teaching and thinking about math and social justice. This experience was rooted in a shared dialogue between a teacher and her peers; together they wrestled with heavy questions regarding teaching kindergarteners about social inequity. During the same semester (spring 2007), another student preparing to become an elementary school teacher decided to create an interdisciplinary lesson for 4-5th grade that combined teaching perimeter and geography. The state math standards she was aiming to meet were: PA- Pennsylvania Academic Standards Subject:Mathematics Area 2.3: Measurement and Estimation Grade 2.3.5: Grade 5 Standard A.: Select and use appropriate instruments and units for measuring quantities (e.g., perimeter, volume, area, weight, time, temperature). Standard E.: Add and subtract measurements This future math teacher decided to review a lesson on perimeter she found in a widely-used American textbook series, Everyday Mathematics (UCSMP 2004). She expanded the concept of perimeter to include man-made, natural and political perimeters in addition to the mathematical concept of perimeter offered in the text. She critiqued the real world application provided in the text, which discussed kitchen layouts. As her own replacement context of study, she explored the designated borders within the United States Native American Reservations, focusing on the 21 federally recognized tribes in Arizona. This future teacher was helping her students with making meaning of political borders while teaching them math skills in measuring perimeters. Similar to the kindergarten teacher, this students project was rooted in class dialogue. Although she came to the class knowing that she wanted to teach math for social justice, and knew she must given her teaching philosophy, she struggled with the how as well. Her peers supported her; it was clear through discussions in class and on an online discussion board that the project grew and evolved as the teachers and future teachers worked together to rethink teaching perimeter. The final example of a curriculum project in mathematics was developed by a secondary math teacher who teaches algebra in the suburbs of Philadelphia. In rethinking a lesson on linear programming, she had her students think about price discrimination. The lesson facilitated the application of linear programming within this context. The state standards she was aiming to meet are: PA- Pennsylvania Academic Standards Subject:Mathematics Area 2.8: Algebra and Functions Grade 2.8.11:Grade 11 Standard D.: Formulate expressions, equations, inequalities, systems of equations, systems of inequalities and matrices to model routine and non-routine problem situations. Standard S.: Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic). She shared with the other teachers and future teachers her experience re-teaching this lesson, and how she used the airline industry as an example of price discrimination. First, she grabbed the attention of her students with an opening activity. She posed some critical questions, such as. Have you ever been the victim of discrimination? After a discussion on discrimination, they moved to connecting those lived experiences to an application of linear programming to think about profit and cost. When she shared this lesson with her peers, an empowering conversation developed regarding the digital divide as well as the difference in peoples experiences around having the flexibility to make travel plans in advance. The three examples of curriculum critique projects exemplify the process teachers and pre-service teachers may undergo when thinking about transforming mathematics curriculum. Most of the students in this course were struggling with rethinking their own pedagogy. The curriculum critique assignment helped them to reach a level of conscientization (Freire 1988) in two ways: through dialoging with their peers, and through rethinking a piece of curriculum they have themselves already taught. Interpretation The experiences we discuss above highlight the critical component of dialogic teaching while offering a glimpse of the current condition of mathematics education within the framework of struggle and hope in teacher preparation. As we prepare teachers to be the most effective educators for their students, we work to model teaching practices that are rooted in emancipatory frameworks. This in turn provides the real application which the teachers are craving. They can see what these practices look like and they can build bridges between their theories and practices regarding the process of teaching and learning. The power of classroom dialogue must be instrumental within the teaching for social justice. The sharing of lived experiences both inside and outside classrooms is an authentic way to think about ones own values, beliefs and perceptions of teaching, and more importantly, of their students. Nieto (2004) offers an insightful thought about the role of teachers, asserting that teachers are students of their students. Without dialogic teaching and the perception of students being integral to the design and delivery of the course, we cannot fully learn about our students, and they cannot learn about one another. This is relevant for the university course as well as the K-12 teaching contexts that our students are grappling with. Through the sharing and subsequent analysis of personal experience and observation, the teachers and future teachers in our classrooms are constructing meaning. This is not a smooth or linear process. Our students struggle through this. For many, they are questioning their own reality, and that takes work. As teacher educators we see the role of teachers as transformative intellectuals. On the role of teaching, Giroux (1988) states If we believe that the role of teaching cannot be reduced to merely training in the practical skills, but involves, instead, the education of a class of intellectuals vital to the development of a free society, then the category of intellectual becomes a way of linking the purpose of teacher education, public schooling and in-service training to the very principles necessary for developing a democratic order and society (1988, p. 126). Giroux (2002) pushes for student empowerment, and for human agency rooted in the belief of teachers as transformative intellectuals. He offers insight on educators contemplating the role that public schools might play in facilitating an alternative discourse grounded in a critique of militarism, consumerism, and racism. Our experiences with teachers and pre-service teachers highlight the importance of working with and through sites of resistance rather than attempting to weave an alternative discourse or struggling to circumvent anxieties. Most professional development resources offer alternative pictures of what is possible in mathematics education. They are often accompanied by philosophical arguments that claim ethical or moral grounding. While these pictures and ethical arguments are important, they do not do the work of empowerment for teachers. The work of empowerment is a form of learning that is first indicated as potential by the statements and actions of resistance that teachers exhibit in their own learning contexts. Directly working to understand the meaning and sources of these resistances instigates dialogue with peers and forms of self-reflection that can lead to new forms of understanding. Implications for further research and for further innovations in teacher education Schools function as sites of struggle where teachers can critically explore their emancipatory potential as educators and public intellectuals. The political nature of classrooms entrenched in traditional notions of power, knowledge and truth that reproduce social categories of inequality, evokes the need for transformative intellectuals in schools at every level, from early childhood through higher education. Thus, critical educators at all levels of schooling must create/re-create and question the discourse around public intellectuals by reflecting on ideas that can rupture the anti-intellectualism that is woven in the current structure of public schools. When a teacher hears her- or himself questioning the appropriateness of social justice mathematics for his or her students, or when he or she must collaborate with colleagues who cannot yet see the mathematics in the social justice curriculum, these are moments of opportunity for teacher leadership. Teachers can provide gatekeepers with professional articulations of how and why the standards and tested objectives can be placed in symbiotic relationships with curricula that exceed the minimum goals that such bureaucratic expectations hold for our youth. Furthermore, Freires thorough discussion of the political nature of classrooms entrenched in traditional notions of power, knowledge and truth that reproduce social categories of inequality actualizes the context for the call for transformative intellectuals in schools. As teachers participate in collaborative curriculum design, they are no longer subjects of power but are active constructors of knowledge, working within and across regimes of truth and power. Such active design may be taken as an articulation of the ethical stance that teaching embodies. In our work with teachers, we find a focus on assessment helps to raise issues of critical race theory. Given the ways that power is implicated in the determination of the good student, notions of whether or not a student is displaying evidence of conceptual growth or understanding, for example, are grounded in cultural and historical legacies of discourse and power. Since a learners trajectory through the differentiated tracks that constitute the school is determined partially by his or her assessed performance at key branching points, any sociology of learning, broadly conceived, will need to address assessment (Cooper 2007). When the teachers and pre-service teachers created dialogues around the efficacy of an activitys placement at the beginning, middle or end of a lesson/unit trajectory in our classes and on-line, they were able to confront the complexities of this seemingly commonplace and undertheorized teacher role. Because assessment is so central to decision-making and to the judgment of success both of the teacher and the students, assessment is a critical site of power/knowledge and thus of the important work of learning through resistance. Much of the gatekeeper work is focused on assuring the gatekeepers (whether this be the teacher him- or herself and/or supervisors of their teaching) that students will perform well on standardized paper and pencil achievement tests. As teachers broaden their comprehension of assessment within the critique and design of curriculum, they are better able to place such assessment within a broader field of possibilities, each of which defines mathematical understanding and ability in different ways. Because learning is currently defined in this pragmatic way, as based on the performance of the learning by the student, the ability to imagine possible forms of performance and their potential interpretation enables teachers to explore the ways that varying assessments can mutually support as well as potentially conflict with one another. Conclusion The call for transformative intellectuals speaks to the struggles that have been manifested in a society that leaves families living in poverty, people of color, linguistic minorities, women and children on the margins of full citizenship and denies us equal educational opportunities. However, the rays of hope and victory pierce the struggle from multiple angles and burn through it. Unfortunately, there are many people who are still left to struggle in the shadows missed by the rays of hope. Schools continue to be organized and structured to perpetuate inequality. More importantly, our children continue to be the products of this inequality. It is time for schools to change these inequalities through providing spaces of resistance, coupling the discourse of critique with that of possibility and helping teachers play their role as transformative intellectuals who witness the urgency in teaching for social justice. The call for transformative intellectuals is not a well packaged recommendation that can be achieved with a workshop or even a course; instead, it is an ideological shift that is occurring within the way we think about learning and teaching within the institution of schools. Thus, if whole communities tap into their standpoint, thoughts, perspectives and ideas of teaching and learning, we can begin conversations with more than just scholars at the table. We can build collective conversations with the people that are in the lives of children every day: family members, child care providers, neighbors, teachers, counselors, secretaries, cafeteria workers, coaches, tutors, and bus drivers. Critical Mathematics Education demands a critical perspective on both mathematics and the teaching/learning of mathematics. In doing so, it takes one step further in questioning our assumptions about what critical thinking could mean and what democratic participation should mean. As Ole Skovsmose (1994) describes a critical mathematics classroom, the students (and teachers) are attributed a critical competence. A century ago (see, e.g., Fawcett 1938), we moved from teaching critical thinking skills to using the skills that students bring with them. We accepted that students, as human beings, are critical thinkers, and would display these skills if the classroom allowed such behavior. It seemed that we were not seeing critical thinking simply because we were preventing it from happening; through years of school, students were unwittingly trained not to think critically in order to succeed in school mathematics. So we found ways to lessen this dumbing down of thinking through school experiences. Now we understand human beings more richly as exhibiting a critical competence, and because of this realization, we recognize that decisive and prescribing roles must be abandoned in favor of all participants having control of the educational process. In this process, instead of merely forming a classroom community for discussion, Skovsmose suggests that the students and teachers together must establish a critical distance. What he means with this term is that seemingly objective and value-free principles for the structure of the curriculum are put into a new perspective, in which such principles are revealed as value-loaded, necessitating critical consideration of contents and other subject-matter aspects as part of the educational process itself. (See also Skovsmose 2005) Keitel, Klotzman and Skovsmose (1993) together offer a new way for teachers to think about the mathematics that is being taught. New ideas for lessons and units emerge when teachers describe mathematics as a technology with the potential to work for democratic goals, and when they make a distinction between different types of knowledge based on the object of the knowledge. The first level of mathematical work, they write, presumes a true-false ideology and corresponds to much of what we witness in current school curricula. The second level directs students and teachers to ask about right method: are there other algorithms? Which are valued for our need? The third level emphasizes the appropriateness and reliability of the mathematics for its context. This level raises the particularly technological aspect of mathematics by investigating specifically the relationship between means and ends. The fourth level requires participants to interrogate the appropriateness of formalizing the problem for solution; a mathematical/technological approach is not always wise and participants would consider this issue as a form of reflective mathematics. On the fifth level, a critical mathematics education studies the implications of pursuing special formal means; it asks how particular algorithms affect our perceptions of (a part of) reality, and how we conceive mathematical tools when we use them universally. Thus the role of mathematics in society becomes a component of reflective mathematical knowledge. Finally, the sixth level examines reflective thinking itself as an evaluative process, comparing levels 1 and 2 as essential mathematical tools, levels 3 and 4 as the relationship between means and ends, and level 5 as the global impact of using formal techniques. On this final level, reflective evaluation as a process is noted as a tool itself and as such becomes an object of reflection. When teachers and students plan their classroom experiences by making sure that all of these levels are represented in the groups activities, it is more likely that students, and teachers, can be attributed the critical competence that we envision as a more general goal of mathematics education. In formulating a democratic, critical mathematics education, it is also essential that teachers grapple with the serious multicultural indictments of mathematics as a tool of post-colonial and imperial authority. What we once accepted as pure, wholesome truth is now understood as culturally specific and tied to particular interests. Philip Davis and Reuben Hersh (1987) and David Berliner (2000), for instance, have described some aspects of mathematics as a tool in accomplishing a fantasy of control over human experience. They use the examples of math-military connections, math-business connections, and others. Critical mathematics educators ask why students, in general, do not see mathematics as helping them to interpret events in their lives, or gain control over human experience. They search for ways to help students appreciate the marvelous qualities of mathematics without adopting its historic roots in militarism and other fantasies of control over human experience. Arthur Powell and Marilyn Frankenstein (1997) have collected valuable essays in ethnomathematics and the ethnomathematical responses that educators can make to contemporary mathematics curricula. Ethnomathematics makes it clear that mathematics and mathematical reasoning are cultural constructions. This raises the challenge to embrace the global variety of cultures of mathematical activity and to confront the politics that would be unleashed by such attention in a typical North American school. That is, ethnomathematics demands most clearly that critical thinking in a mathematics classroom is a seriously political act. One important direction for critical mathematics education is in the examination of the authority to phrase the questions for discussion. Who sets the agenda in a critical thinking classroom? Stephen Brown and Marion Walter (1999) lay out a variety of powerful ways to rethink mathematics investigations through The Art of Problem Posing, and in doing so they give us a number of ideas for enabling students both to talk back to mathematics and to use their problem solving and problem posing experiences to learn about themselves as problem solvers and posers. In the process, they help us to frame yet another dilemma for future research in mathematics education: Is it always more democratic if students pose the problem? The kinds of questions that are possible, and the ways that we expect to phrase them, are to be examined by a critical thinker. Susan Gerofsky (2001) has recently noted that the questions themselves reveal more about our fantasies and desires than about the mathematics involved. Critical mathematics education has much to gain from her analysis of mathematics problems as examples of literary genre. And Mark Boylan (2007) has offered insights on the micro-politics of teacher questioning as constitutive of the larger political context for mathematics teaching and learning. His work, too, must be integrated into discussions of innovative social justice mathematics curricula. And finally, it becomes crucial to examine the discourses of mathematics and mathematics education in and out of school and popular culture (Appelbaum 1995). Critical thinking in mathematics education asks how and why the split between popular culture and school mathematics is evident in mathematical discourse, and why such a strange dichotomy must be resolved between mathematics as a commodity and as a cultural resource. Mathematics is a commodity in our consumer culture because it has been turned into stuff that people collect (knowledge) in order to spend later (on the job market, to get into college, etc.). But it is also a cultural resource in that it is a world of metaphors and ways of making meaning through which people can interpret their world and describe it in new ways. Critical mathematics educators recognize the role of mathematics as a commodity in our society; but they search for ways to effectively emphasize the meaning-making aspects of mathematics as part of the variety of cultures. In doing so, they make it possible for mathematics to be a resource for political action References (CIEAEM) International Commission for the and Improvement of Mathematics Education (2000). Manifesto 2000 for the Year of Mathematics.  HYPERLINK "http://www.cieaem.net/50_years_of_c_i_e_a_e_m.htm" http://www.cieaem.net/50_years_of_c_i_e_a_e_m.htm. (NCTM) National Council of Teachers of Mathematics (2000). Principles and Standards.  HYPERLINK "http://standards.nctm.org/" http://standards.nctm.org/. (PDE) Pennsylvania Department of Education. Undated. Academic Standards for Mathematics. http://www.pafamilyliteracy.org/k12/lib/k12/MathStan.doc (UCSMP) University of Chicago School Mathematics Project (2004; 2007). Everyday Mathematics. Second edition 2004; third edition 2007. DeSoto, TX: McGraw-Hill/Wright. Appelbaum, Peter (1995). Popular culture, educational discourse, and mathematics. Albany, NY: State University of New York Press. Appelbaum, Peter. (2008, in press). Embracing Mathematics: On Becoming a teacher and Changing with Mathematics. NY: Routledge. Berlin, David (2000) The advent of the algorithm: The idea that rules the world. NY: Harcourt Brace. Block, Alan. (2003). They Sound the Alarm Immediately: Anti-intellectualism in Teacher Education. Journal of Curriculum Theorizing, 19 (1): 33-46. Boylan, Mark (2007). Teacher Questioning in Communities of Political Practice. Philosophy of Mathematics Education Journal 20.  HYPERLINK "http://www.people.ex.ac.uk/PErnest/pome20/index.htm" http://www.people.ex.ac.uk/PErnest/pome20/index.htm. Brown, Stephen and Walter, Marion (1999). The art of problem posing. Mahwah, NJ: Erlbaum. Clandinin, D. J. & Connelly, F. M. (2002). Personal Experience Methods. In Denzin, N. K & Lincoln, Y. S. (Eds.). Collecting and Interpreting Qualitative Materials: 150-178. London: Sage Publications. Cooper, Barry (2007). Dilemmas in Designing Problems in Realistic School Mathematics: A Sociological Overview and some Research Findings. Philosophy of Mathematics Education Journal 20.  HYPERLINK "http://www.people.ex.ac.uk/PErnest/pome20/index.htm" http://www.people.ex.ac.uk/PErnest/pome20/index.htm. Davis, Philip, and Hersh, Reuben (1986). Descartes' dream: The world according to mathematics. San Diego: Harcourt, Brace, Jovanovich. Denzin, N. K& Lincoln, Y.S. (Eds.) (1998).The Landscape of Qualitative Research: Theories and Issues. California: Sage Publications. Drake, Coreyand Miriam Gamoran Sherin. 2006. Practicing Change: Curriculum Adaptation and Teacher Narrative in the Context of Mathematics Education Reform. Curriculum Inquiry 36, no. 2: 153-18. Fawcett, Harold (1938/1995). The nature of proof (NCTM 1938 Yearbook). Reston, VA: National Council of Teachers of Mathematics. Freire, P. (1988). Introduction: Teachers as Intellectuals: Critical Educational theory and the Language of Critique. In Giroux Teachers as Intellectuals: Toward a Critical Pedagogy of Learning. Massachusetts: Bergin & Garvey Publishers, Inc. Gerofsky, Susan (2001). Genre analysis as a way of understanding pedagogy n mathematics education. In John Weaver, Peter Appelbaum, and Marla Morris (eds.) (Post) modern science (education): propositions and alternative paths: 147-176. NY: Peter Lang. Giroux, Henry (1988). Teachers as Intellectuals: Toward a Critical Pedagogy for the Opposition South Hadley, MA: Bergin & Garvey. Gudmundsdottir, S. (2001) Narrative Research on School Practices. In Richardson, V. (Ed.) Handbook of Research on Teaching (4th Ed). Washington D.C.: American Educational Research Association. Gutstein, Eric, and Bob Peterson. 2005. Rethinking Mathematics: Teaching Social Justice by the Numbers. Milwaukee, WI: Rethinking Schools. Gutstein, Eric. 2007. Reading and Writing the World with Mathematics: Toward Pedagogy for Social Justice. NY: Routledge. Hones, D. F. (1998). Known in part: the transformational power of narrative inquiry. Qualitative Inquiry 4(2), pp.225-249. Keitel, Christine, Klotzmann, Ernst, and Skovsmose, Ole (1993). Beyond the tunnel vision: Analyzing the relationship between mathematics, society and technology. In Christine Keitel and Kenneth Ruthven (eds.), Learning from computers: mathematics education and technology, 243-279. NY: Springer-Verlag. Ladson-Billings, G. (1995). Just what is critical race theory and what's it doing in a nice field like education? International Journal of Qualitative Studies in Education, 11(1), 7-24. Ladson-Billings, Gloria and William F. Tate. 1995. Toward a Critical Race Theory of Education. Teachers College Record 97:1 (Fall 1995): 47-68. Leu, Yuh-Chyn. 2005. The Enactment and Perception of Mathematics Pedagogical Values in an Elementary Classroom: Buddhism, Confucianism, and Curriculum Reform. International Journal of Science and Mathematics Education 3, no. 2 (2005): 175-212. Neyland, Jim. 2004. Toward a Postmodern Ethics of Mathematics Education. In Mathematics Education within the Postmodern, edited by Margaret Walshaw: 55-73. Greenwich, CT: Information Age Publishing. ODaffer, Phares G. and Bruce Thomquist (1993). Critical thinking, mathematical reasoning, and proof. In Patricia S. Wilson (ed.), Research Ideas for the Classroom: High School Mathematics, NY: Macmillan/NCTM. Parker, Laurence, Donna Deyhle, and Sophia Villenas (eds). 1999. Race Is...Race Isn't: Critical Race Theory and Qualitative Studies in Education. Boulder, CO: Perseus Books. Powell, Arthur, and Frankenstein, Marilyn (1997) Ethnomathematics: Challenging eurocentrism in mathematics education. Albany, NY: State University of New York Press. Remillard, Janine T.1999. Curriculum materials in mathematics education reform: a framework for examining teachers' curriculum development. Curriculum Inquiry 29(3):315-342. Skilton-Sylvester, Paul (1994). Elementary School Curricula and Urban Transformation. Harvard Educational Review, 64, 309 - 329. Skovsmose, Ole (1994). Toward a philosophy of critical mathematics education. Dordrecht, Netherlands: D. Reidel. Skovsmose, Ole (2005). Traveling through education: uncertainty, mathematics, responsibility. Rotterdam: Sense Publishers.     PAGE   PAGE \* MERGEFORMAT 12 KLMnoͼzpcpP@hHl>*B*OJQJ^Jph$hhM>*B*OJQJ^JphhYhZ'OJQJ^Jh2 rOJQJ^JhYh2 rOJQJ^JhAOJQJ^JhYhYOJQJ^JhYhAOJQJ^J h2 rh2 rCJOJQJ^JaJ h2 rhACJOJQJ^JaJhYhY5OJQJ^J#h2 rhr5CJOJQJ^JaJ#h2 rh2 r5CJOJQJ^JaJLMo 0$@&gdjGC$Eƀgd,Ngdjdhgd.}@&gd< $@&a$gdYT & 7 : N O  1 2 ʹuguYYYH!hh*B*OJQJ^Jphhr>*B*OJQJ^Jph$hhr>*B*OJQJ^Jph2 4 D { +    O P [ h (RS ,KLQT̪ݙݙ݈zhC<B*OJQJ^Jph!hhC<B*OJQJ^Jph!hhAB*OJQJ^Jph!hhMTB*OJQJ^Jph!hhO_B*OJQJ^Jph!hhr]B*OJQJ^Jph!hhvB*OJQJ^Jph!hho(B*OJQJ^Jph-/01T+ͼsfYLY?Yf?fY?YhhUOJQJ^JhhOJQJ^Jhh\9OJQJ^JhhWOJQJ^J$hhh>*B*OJQJ^Jphhr>*B*OJQJ^Jph$hhr>*B*OJQJ^Jph$hh,N>*B*OJQJ^Jph!hhrB*OJQJ^Jph!hhjB*OJQJ^Jph$hjhj>*B*OJQJ^JphhjB*OJQJ^Jph01 " "u&v&w&&&,,00gdC@ dh@&gd.}@&gd<^gdOdhgd.}$@&gdj+-@b67;HT[r{|7CWk̤|nahhYOJQJ^JhhD6OJQJ^Jhhv6OJQJ^JhhC'OJQJ^JhhDOJQJ^JhhvOJQJ^Jhh9~OJQJ^Jhh`6OJQJ^Jhh\9OJQJ^JhhUOJQJ^JhhMTOJQJ^Jhh`OJQJ^J"nztuny̻taTTGhhxOJQJ^JhhWiWOJQJ^J$hhh>*B*OJQJ^Jphhr>*B*OJQJ^Jph$hhr>*B*OJQJ^Jph$hh,N>*B*OJQJ^Jph!hhOB*OJQJ^Jph!hh,NB*OJQJ^JphhhvOJQJ^Jhh7{GOJQJ^JhhYOJQJ^Jhh4#OJQJ^J$:YbcvHJO\̼󡓡ylylllly[!hh5B*OJQJ^Jphhh{OJQJ^JhhQQOJQJ^Jhh5OJQJ^JhhQOJQJ\^JhhQOJQJ^Jhhx6OJQJ^Jhh4#h4#OJQJ^Jhh4#OJQJ^Jhh^MOJQJ^Jhh7{GOJQJ^JhhxOJQJ^J!(q̻̻̪̪̝yfVC2!hh{B*OJQJ^Jph$hhh>*B*OJQJ^Jphhr>*B*OJQJ^Jph$hhr>*B*OJQJ^Jph$hh{>*B*OJQJ^Jph!hhB*OJQJ^JphhhWiWOJQJ^J!hh5B*OJQJ^Jph!hh4#B*OJQJ^Jph!hhxB*OJQJ^Jph!hhQQB*OJQJ^Jph!hhWiWB*OJQJ^JphY c t!u!!!!+","""""##O$P$^$_$%%%%%%%%w&&&&'̻ݪݪݙݙݙݙݙݙ݆vcVhhZOJQJ^J$hhh>*B*OJQJ^Jphhr>*B*OJQJ^Jph$hhC_>*B*OJQJ^Jph!hhngB*OJQJ^Jph!hhl5B*OJQJ^Jph!hhlB*OJQJ^Jph!hhZB*OJQJ^Jph!hh{B*OJQJ^Jph!hh&IB*OJQJ^Jph!'8'@'_'g'm'',%-&-B-C-0000 0 000000000ȺȺȩȘzpzpzOAhhE>*OJQJ^J@hhC@h-B*OJQJ^JcHdhdhdhphhQOJQJ^Jhh-OJQJ^J!hh-B*OJQJ^Jph!hh-B*OJQJ^Jph!hhngB*OJQJ^JphhQB*OJQJ^Jph!hhC@B*OJQJ^JphhhC@OJQJ^JhhZOJQJ^Jhh{OJQJ^J000(7)7==@@8F9FFpppppppphdhgd.}IdhEƀgd.}FEƀgdC@ 000022r33666'7(7)799d<IAoApAE7F9FFFFDGI俲̿~q_M#hh>6CJOJQJ^JaJ#hhd[6CJOJQJ^JaJhhd[OJQJ^JhhY OJQJ^Jhh-OJQJ^Jhh%OJQJ^Jhh>ϴϴ'ϴCϴ6Oϴϴ:>*ϴFٳGHX]]`pppppppIdhEƀgd.}FEƀgd_```````accRdWddde;f]fjfȸsesesP9,hh^t56>*CJOJQJ]^JaJ)hh^t56CJOJQJ]^JaJhh15OJQJ^Jhh^t5OJQJ^J*hh^t5B*OJQJ\]^Jphhh15OJQJ]^J!hh^t56OJQJ]^Jhh^t5OJQJ]^J!hh^t5>*OJQJ]^Jhh9NOJQJ^Jhh7#QOJQJ^Jhhhz|OJQJ^J````dfJdhC$Eƀgd.}gd^tIdEƀgdzdd;fjffgh kg^MM^ & F h88^8gd1^gd1N$dhEƀ^gd.}IdhEƀgd.}jf kk klkmkkkkkllPlsllbmcmmn(n-n.n9n*CJOJQJ\^JaJhh*5OJQJ^Jhh^t56OJQJ^Jhh^t5OJQJ^Jhh*5OJQJ\^Jhh^t5OJQJ\^J)hh^t56CJOJQJ\^JaJ k kmn:n|nnn0s1s>tIdhEƀgdg^gd*dhgdg nnnnnFogooooooopppprrrr0s1s}ssss>tbtctltztttttt'uBuOu~~~jj&hhse56CJOJQJ^JaJ)hh^t56CJOJQJ\^JaJ&hh^t56CJOJQJ^JaJ,hh^t56>*CJOJQJ\^JaJhh^tH*OJQJ^Jhhx-OJQJ^JhhseOJQJ^Jhh*OJQJ^Jhh^tOJQJ^J'>tbtxtttPugN LEƀ^ `LgdseJEƀ^gdseOuPu]u{u|u}uuuuuuuvv vvvvv#vvvvv%w5wwwMxlxxx,y}odTh.hse56OJQJ^Jh]5OJQJ^Jh]h]5OJQJ^Jh]56OJQJ^Jhh^t56OJQJ^Jh(Q5OJQJ^Jhhse5OJQJ^Jhh^t5OJQJ^J&hh^t56CJOJQJ^JaJ)hh^t56CJOJQJ\^JaJ)hhse56CJOJQJ\^JaJ Pu|u}uRzSzo&&IdhEƀgdgFEƀgd^tJ@ Eƀ^@ gdse,y6ygyhykylyyyyyyyyyzzszzz4{z{{{{||||2|D|L|M|ֲucucQ#hhse6CJOJQJ^JaJ#hh^t6CJOJQJ^JaJ&hh^t6CJOJQJ\^JaJ)hh^t56CJOJQJ\^JaJ&hh^t56CJOJQJ^JaJ,hh^t56>*CJOJQJ\^JaJhhseOJQJ^Jhh^tOJQJ^Jhh^t5OJQJ^Jhhse5OJQJ^JSzz{{m#JEƀ^gdseIdhEƀgdgIdhEƀgdg{|4|M|kkJEƀ^gdseJEƀ^gdseM|Z|{||||||| }/}K}}}}}}}}'JKNMcv 8<Kڻu'hh^tB*OJQJ\]^Jphhh^tOJQJ]^JhhQVCOJQJ^Jh2OJQJ^JhhseOJQJ^Jhh^tOJQJ^J#hhse6CJOJQJ^JaJ#hh^t6CJOJQJ^JaJ&hh^t6CJOJQJ\^JaJ'M||}}bcThhhhIdhEƀgdgN Eƀ^ `gdseЁ 8KSTUcdeuʂɼɼɼɼzm`mSF9Fhh}OJQJ^JhhN-OJQJ^Jhhx-OJQJ^Jhhk OJQJ^JhhHOJQJ^Jhhd>*OJQJ^Jh_>*OJQJ^JhhE>*OJQJ^JhhEOJQJ^Jhh^t>*OJQJ^Jhh^tOJQJ^JhhQVCOJQJ^J'h2h2B*OJQJ\]^Jph*hh^t5B*OJQJ\]^JphTUdeeqq&J$dhEƀgdgG$Eƀgd% FEƀgdC@Ƀ4τ ,pޅ߅78Rq 7LJЇ0OXx͈defششششششششششΧx hhCJOJQJ^JaJ hh_CJOJQJ^JaJhhHOJQJ^Jhh_OJQJ^JhhOJQJ^JhhDYOJQJ^JhuhOJQJ^Jhhh]OJQJ^Jhh}6OJQJ^Jhh}OJQJ^J/Oo$%'STY]jklA(.JKLpɒѺĭĭĖ}oaah [CJOJQJ^JaJhhd>*OJQJ^JhE>*OJQJ^JhhE>*OJQJ^Jhhk OJQJ^JhuhOJQJ^Jhh9OJQJ^JhgOJQJ^Jhhh]OJQJ^JhhOJQJ^J hh_CJOJQJ^JaJ hhCJOJQJ^JaJ#e%'jlKLIJ23>? ,h8 xHX (#dhgdg $d1$^gdk  $dh1$^gdg$gdngdgdh] dhgdgdhgdgdhgdggd_#C&HIJЙԙݙ12䴦vevTFh[YCJOJQJ^JaJ h&h,XCJOJQJ^JaJ h&h&CJOJQJ^JaJh&CJOJQJ^JaJ h&hdCJOJQJ^JaJ h [hdCJOJQJ^JaJhdCJOJQJ^JaJhECJOJQJ^JaJ hhk CJOJQJ^JaJ hh_CJOJQJ^JaJh [CJOJQJ^JaJhZ'CJOJQJ^JaJ23=>?B]/0\zˡ֣de xF`ʩݩ˾{ggg'hh6B*OJQJ]^Jph333hB*OJQJ^Jph333!hhB*OJQJ^Jph333hhZ'OJQJ^JhgOJQJ^JhhOJQJ^Jhh^OJQJ^Jhhd>*OJQJ^JhE>*OJQJ^Jhg>*OJQJ^J h&h[YCJOJQJ^JaJ&? depqtuMNhdd[$\$^h`gdnG h^h`gdnG$gdd h^h`gd2 rdhdd[$\$gdgdhgdg>??@iyż޼ cdeopqӿӿӿӿӿӱ䖉yl\hAhd5>*OJQJ^Jhyp5>*OJQJ^JhAhT]5>*OJQJ^Jhh2 rOJQJ^Jhgh1_uOJQJ^Jh$e}B*OJQJ^Jph333hgB*OJQJ^Jph333'hh6B*OJQJ]^Jph333!hhB*OJQJ^Jph333hhOJQJ^JhDoB*OJQJ^Jph333q>?pqstuHjLʻʱ۱}rh]6OJQJ^Jh0whd0JOJQJ^JjhdOJQJU^JhhdOJQJ^JhypOJQJ^JhdOJQJ^Jhhyp0JOJQJ^J!jhhypOJQJU^Jhhyp6OJQJ^Jh]OJQJ^JhhypOJQJ^J)LMf2Nc5F zm_mUGUmUGjh$e}OJQJU^Jh$e}OJQJ^Jh$e}h$e}6OJQJ^Jh$e}h$e}OJQJ^JhAhyp6OJQJ^JhAhyp6OJQJ]^JhAhypOJQJ^JhhZ'6OJQJ^Jhhyp6OJQJ^JhhypOJQJ^J'hhyp6B*OJQJ]^Jph333!hhypB*OJQJ^Jph333h]h]OJQJ^JFG@bchy$h^h`a$gdnGOhC$Eƀ^h`gdnGhdd[$\$^h`gdnGgdnG h^h`gdnG <=?@j <=>̻ՍS;0J jHhh)>;0JUhFjhFUhMhMOJQJ^JhMOJQJ^JhMhM6OJQJ^J!hMhMB*OJQJ^Jph333hMB*OJQJ^Jph333h}B*OJQJ^Jph333'hhyp6B*OJQJ]^Jph333!hhypB*OJQJ^Jph333hhypOJQJ^Jh2 rOJQJ^J"#%&()234QRSTh]hgdHp$a$gdHpod&h]hUC$Eƀgd;od&&`#$ 245LMOPSTUhMOJQJ^JhFhmHnHujh)>;Uh)>; TUNhEƀ^h`gdnG&1h:p,N/ =!"#$%@@@ rNormalCJ_HaJmH sH tH R@R $e} Heading 1dd@&[$\$5CJ0KH$\aJ0DA@D Default Paragraph FontViV  Table Normal :V 44 la (k(No List H @H r0Footer !1$7$8$H$CJaJVOV r paragraph1)56CJOJQJ\]^JaJo(phH@H r] Balloon TextCJOJQJ^JaJ4U@!4 W0 Hyperlink >*<O1< C' matchterm01 q f.)@A. Hp Page NumberZYRZ < Document Map-D M CJOJQJ^JaJB^@bB 50 Normal (Web)dd[$\$4@r4 EEHeader  H$*O* EE CharCJaJZC@Z -Body Text Indent1$7$8$G$5OJQJ\^Je@ HTML Preformatted7 2( Px 4 #\'*.25@9B*CJOJQJ^JaJphO _ Body Text InF ) p@ P !1$7$8$H$^B*CJaJph4O4 $e} Char5CJ0KH$\aJ0*W@* $e}`Strong5\"O" Y0 CharUPLMo01    uvw$$(((((/)/55888>9>>??@PUUXXXX\\;^j^^_` c cef:f|fff0k1k>lblxlllPm|m}mRrSrrsst4tMttuubxcxTzUzdzeze%'jlKLIJ23>?? depqtuMNFG@bchZ[$%Y1 "#&)4QRSV000000000000000000000000000000000000000s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$ 0s$ 0s$ 0s$ 0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$0s$ 0s$ 0s$00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000Oy00O9000Oy00O900O900O900Oy00dd,@0@0Oy00dd,O900hLMo01    uvwubxcxTzUzdzez'jlKLIJ23>? depq1"%(VM90M90M90M90M90M90M90M90M90M90M90 xM90 M90 M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90M90",ZM90"M90"M90M90M90M90M90)pM90)M90)M90M90M90M90M900M900M90M90M90M90M900M90M90\aWM90M90M90M90M900 My000@0 0@0 0XJ  66692 +'0I_jfnOu,yM|2qL B2Uorstvwxyz{}00Fʳ{ѴՂ">ʾ<aUXXXX /29!!8@0(  B S  ?Mc$u cu cdu c$u cu cu c$u cdu cu cu c$u cdu cu cu cdu c$u cu cu cu cu cdu d$u d$u du ddu d$u du ddu du du  du  du  du  du  ddu du d$u ddu ddu du du du d$u du du du d$u dgw d$hw ddhw dhw dhw d$iw  ddiw !diw "diw #d$jw $ddjw %djw &djw 'd$kw (ddkw )dkw *dkw +d$lw ,ddlw -dlw .dlw /d$mw 0ddmw 1dmw 2dmw 3d$nw 4ddnw 5dnw 6dnw 7d$ow ooee}g}gggoorrſ--5NN[[GG77E==FV      !"#$%&')(*+,.-/0132465789;:<>=?A@BDCEFGIHJKLeeggggoorrĿ˿˿377TTaaMMCGGDHH V   !"#$%&')(*+,.-/0132465789;:<>=?A@BDCEFGIHJKL9I*urn:schemas-microsoft-com:office:smarttagsStatehL*urn:schemas-microsoft-com:office:smarttagsCity0http://www.5iamas-microsoft-com:office:smarttagsVM*urn:schemas-microsoft-com:office:smarttagsplacehttp://www.5iantlavalamp.com/Z?*urn:schemas-microsoft-com:office:smarttags PlaceNamehttp://www.5iantlavalamp.com/Z>*urn:schemas-microsoft-com:office:smarttags PlaceTypehttp://www.5iantlavalamp.com/_K*urn:schemas-microsoft-com:office:smarttagscountry-regionhttp://www.5iantlavalamp.com/ MLKMIMKMKMIMLM?>MLIMLIM>>MIMIMLMLIMLMLMIMLIMIMLIMLMLIMLIMLMLIMLIMLIM>>MIMLKMLrx "#%&()SVK!O!)*6|7:: FQF__ ``dd]eheffggrryz{}zѽҽ|ɾ>?GHk,fcf\@ij >@%9:yz)%+Z >H~ "#%&()SV3333333333333333333333333333333333333333333333333333333333=eoMh?xA:*%H "#%&()SV "#%&()SV Paul Ernestt |.v.xrn2/"haGN<|.'S ݎ.U0|^`o() ^`hH. pL^p`LhH. @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PL^P`LhH.^`.^`.pp^p`.@ @ ^@ `.^`.^`.^`.^`.PP^P`.^`.^`.pp^p`.@ @ ^@ `.^`.^`.^`.^`.PP^P`.^`o() ^`hH. pL^p`LhH. @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PL^P`LhH.hh^h`o(()h ^`hH.h ^`hH.h pL^p`LhH.h @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PL^P`LhH.2/t aGN<3v..U'S                            SK_06-I!OjzR$ ??J.tM|;\Ojz=bR$OjzyQru!'W;C@6@?@7vAQVCDEEhFnG7{GH$I&IjJA$L9NQ7#Q(QFSMTUWiW,Xd[r]N^O_Ya{[c?evf}'i[nDo0p:pip2 rr?s1_um0vx=y[y{]|hz|}$e} ~~_~*OWF=Zgj){]2nk*MA1IM(Q.C`N-zMg]EDY[W5T]9\9gC_JfQQ6 M21DMd_-Eh Oypv<h]YlJ\^9~% ;ng4#A^MM,NAY}>;iHp5 [.}&seXHlduh{;Z|WlV9@Up@UnknownappelbapDavilaE Paul ErnestGz Times New Roman5Symbol3& z Arial5& zaTahomaY CG TimesTimes New Roman?5 z Courier New"1h۹݊rOlOl!4d 2qHP ?( rErica RDavilaE Paul Ernest$      Oh+'0p  , 8 DPX`hErica RricDavilaEaviavi Normal.dot Paul Ernest5ulMicrosoft Word 10.0@,;3@@fo )O՜.+,D՜.+,D hp  Arcadia UniversitylA Erica R Title 8@ _PID_HLINKSA@"h 4http://www.people.ex.ac.uk/PErnest/pome20/index.htm"h4http://www.people.ex.ac.uk/PErnest/pome20/index.htm>5http://standards.nctm.org/B2http://www.cieaem.net/50_years_of_c_i_e_a_e_m.htm  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry F0AS )Data 1TableSWordDocument(PSummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q