Linking Home and School: A Bridge to the Many Faces of Mathematics (BRIDGE)

Our work takes place in a number of elementary and middle schools, within major school districts in Tucson, Arizona. The general population of students that the project works with are 70% Latino, 10% Native American, 10% African American, and 10% European American. Teacher participation in the project has fluctuated year-to-year, in great part due to lifecourse events (birth of a child, change of school, other professional commitments and more pressing professional needs and interests). Impact of the project is not to be measured by teachers’ participation alone, for example, of the nine teachers on the project thus far this year, four are in middle school and potentially impact 150 students each, the other five are in elementary school and together impact up to 150 students through curriculum modules and other project related activities and support.

Further, the scope of the work is significant in that it is dealing with the inherent nature of schools and the contradictions of and limitations placed on curriculum and pedagogy. As such, this work is a model that attempts to remedy the current state of teaching and learning given those and other constraints, and speaks a great deal more about what needs changing in the infrastructure of higher education (preservice and inservice teacher training) and schools. These points will be discussed later in the conclusions section to this summary. To facilitate a reading of this summary, we recall those early major research questions that guided us. They were:

These questions have been quite important in initiating the project and guiding our work to the present. In the process, however, and as anticipated, we began to find that while project participants (teacher-researchers, students, mothers) were asking similar questions, they were also raising other more challenging questions which begged to be addressed. This integrative summary will, therefore, explore the nature of the developing project from inception to date by addressing the above questions, while including the contributions of participants; then conclude, as we have previously noted, with where we are now, at this midpoint; and what the anticipated direction of the project will look like. These questions will each be addressed within the framework of one of the three project activities, either the teacher-researcher study groups, household visits or parent mathematics workshops.

Teacher-Researcher Study Groups

Our collaboration as a project begins where the joint activity of the project takes place: the study group. The study group is both the structure and the reason for our meetings. We meet during the school week (every two to three weeks) in the late afternoon. It is one of the few opportunities that teachers have to squeeze in extracurricular activities of their own choosing. Though we are all generally tired by that time, the topic of each meeting and the discussion of the group prove intellectually energizing, and thus a very important reason for continuing our meetings. The site where each meeting takes place rotates from classroom to classroom, from school to school, and from district to district. Though Tucson, Arizona may not be considered a large metropolitan city, the southwest is renown for its geographical vastness, and so this is no exception in our local travel, where this opportunity allows us to travel to schools and communities we may not otherwise visit or be familiar with.

What is fundamental to the study group is the establishment of relationships amongst one another. These relationships are fostered as we share interests and experiences. It is in this way that we begin to routinely bring forth everything else that informs our practice as teachers and researchers. It is in the study group that we are able to retell our respective experiences in the classroom or on a household visit and where we are able to engage one another with vivid retellings of our experiences in making a household visit (e.g., what we have seen, smelled, heard, but most importantly learned, especially through the benefit of reflection). In sharing, these retellings and subsequent discussions become a part of our collective knowledge and experiences. Each consecutive study group meeting leaves us with a new found awareness of how others experience and interpret day-to-day events and in turn, challenges how we each think and feel about those same experiences. We deliberately speak of the broader experience of the study group, because there is no one event that captures it, it is a process: a holistic sense of being involved in the activity of making sense of what we see, come to know and what we practice within the contexts of the classroom or the household visit.

The impact of teacher-researcher study groups is made in a number of ways, many of which challenge the current structure and focus of teacher-researcher professional development as well as question teachers’ pedagogical practices. It is important to note, that these challenges are being made by teachers themselves as they engage in readings in ethnography and research on home-school collaborations and mathematics provided by the project (see Reader in Portfolio) and discussions of those and other readings (shared during the course of the work) in light of classroom practices and experiences.

Unlike other content-oriented teacher study groups where, for example, participants sense and demonstrate a greater familiarity and comfort with creating and practicing innovative curriculum changes, a mathematical study group, because of its focus on mathematics, is a great deal more onerous. The mathematical study group is a place where we come to listen to others and share our own thinking as we generate questions that prod previously held conceptions further. Mathematics is a subject that befuddles us all as we begin to question and ask:

These are some of the ongoing questions posed within the group and which we have been struggling with. When we speak of "we" within the context of the teacher-researcher study group, we speak of the collective: teacher-researchers, graduate research assistants, visiting principals and school colleagues, and ourselves (Marta Civil, Norma González, Rosi Andrade).

Though we have responded and postulated on these and other matters in the study group, those have been but preliminary attempts at finding answers. Our respective responses reflect a variety of backgrounds and experiences that grant some of us certain understandings and not others. The study group has been instrumental because it has placed us all on a much different plane as we have had to reflect on our individual thinking and understandings.

Additionally, these are difficult questions to respond to, especially when our interest is to find meaning and purpose in the response with respect to teaching and learning mathematics in context. Context is elusive we are finding, and must be mutually constructed. Defining context, as it has been within our own project, has been another area for continued and passionate debate in search of a response. The exercise of challenging one’s and others thinking has afforded us to move far beyond a simplistic understanding of teaching and learning mathematics as adults, much less children.

The Ethnographic Study of Households

The foundational work of this project is based on the idea that household and community knowledge can provide significant resources for classroom practice. Central to that work has been the ethnographic study of households (Moll, 1992; González, 1995) -- an approach that has been used to investigate and analyze the family history of the households of language-minority students, in particular their labor history. The ethnographic study of households has repeatedly revealed the accumulated bodies of knowledge and the array of skills, information and strategies found within households, what we have referred to as "Funds of Knowledge." The relevance of such findings to classroom practice is that classrooms are often isolated from the social world of the community, as well as its resources. Interestingly, within that world, when pertinent knowledge or resources are not readily available within the household, relationships with outside resources are activated. This contrasts sharply with what often happens within the classroom where, for example, teachers rarely have the opportunity to access networks and draw upon resources found within the community. Additionally, while in classrooms, children may be made passive bystanders, yet within their households they are expected to participate in a broader range of activities in the context of these social relationships. In some cases, their participation is central to the household’s functioning, as when children contribute to the home’s economic production or use their knowledge of English to facilitate the communications with outside institutions (Andrade, 1998; Vásquez, Pease-Alvarez, & Shannon, 1994).

In what follows we will share active examples of new ways in which teacher-researchers are working to bridge these and other disparities in our present work. Unlike the earlier focus on literacy, the BRIDGE project brings the added twist of focusing specifically on mathematical knowledge in the home and community and linking it to pedagogical innovations in the classroom, in an effort to constantly bridge the gap between students’ home and school lived realities (see Civil, 1995). In what follows we outline key components of our work, they include a discussion on funds of knowledge and the nature of our ethnographic experiences as well as the practical applications of the same, before sharing three examples of the application of ethnographic experiences in the curriculum and pedagogy.

How do we find Funds of Knowledge?

Building on students' strengths and on local knowledge is a commonsensical way to approach pedagogy. At first thought, this seems like a remarkably simple task. Yet, in practice we soon encounter more complex questions: How do we know what our students' strengths and funds of knowledge are? How do we approach the dynamic processes of the lived experiences of students without falling into assumptions about what we suppose their out-of-school experiences to be? As our entire project is geared toward process rather than product, our approach to exploring funds of knowledge has involved an ethnographic process. Rather than rely on specific techniques, we have attempted to see the familiar through an anthropological lens. Our answer to these questions focuses on the talk born of ethnography: respectful talk between people who are mutually engaged in a constructive conversation. Throughout this process, ethnography has been both the focus, as in, for example, the household visit stage, and backdrop, as the foundation for building mathematical discourses in classrooms.

What about "culture?"

Because the term "culture" is loaded with expectations of group norms and often static ideas of how people view the world and behave in it, we have purposely avoided reference to ideas of "culture." The term "culture" presumes a coherence within groups which may not exist. Instead, we have focused on PRACTICE, that is what households actually DO. In this way, we open up a panorama of the interculturality of households, that is, how households draw from multiple cultural systems, and use these systems as strategic resources.

How do we make sense of the ethnographic experience?

One key element in jointly coming to understand the ethnographic process has been the format of the study groups. Recall that these meetings take place after school and rotate at one of the several participating school sites. The study groups provide a safe space for BRIDGE teacher-researchers to come together and share experiences in visiting households, classroom mathematical connections, and the constraints of teaching contextualized mathematics within a prescribed curriculum. The study groups also provide the glue which holds together the different facets of mathematical discourses. These discourses center on constructing knowledge around how children learn best, and how mathematical contexts are created. In this safe space, we can also explore our own ideas about what we have found in the households, how it may connect to mathematical practices, and further our own mathematical understanding. As we become both learners and teachers of mathematics, our insights and connection to mathematical pedagogies expand.

Students’ households and activities outside of the school are little understood, but frequently discussed by teachers and school administrators. Yet, when school faculty and staff have had the opportunity to visit homes for reasons other than to report the shortcomings of the student, for example, they will find that the experience is not only enlightening, but mostly positive. The ethnographic component of the project continues to be a vital yet missing link in the teaching profession. Teacher-researchers who have undertaken initial and follow-up household visits never cease to point to the number of revelations made to them by meeting with parents in a non-threatening way (for both sets of parties); in listening to parents share their experiences, interests and aspirations for the children (which families are eagerly prone to do); and of rethinking previous conceptions of students and their families, and in turn begin to wonder what else lies behind the doors of other homes in the community. The experience is equally positive for parents as it brings a human face to "teacher" and the school, and a sense of greater accessibility to the institution.

The mathematical effects of the teacher study group are, therefore, best discussed in the following question.

In response to this question, we are brought to the experiences of teacher-researchers in the project. We direct the reader to the Portfolio, specifically to "Teacher-Researchers Reflections on Funds of Knowledge Research Activity" where we highlight the experiences of three teacher-researchers (Caroline Carson, Leslie Kahn and José David Fonseca) and their respective responses to the task. Each is an example of the process, and at that a focus on varying aspects of the work of taking local knowledge and everyday experiences to an abstract level for its potential in pedagogy and curriculum.

These teacher-researchers share an interest which they personalize based on their students and their families and their own pedagogical orientations. Each is valuable and unique in its approach. Thus, taking local knowledge to an abstract level is an exercise in questioning one’s practice coupled with the ingenuity to break with rigid notions of pedagogy and curriculum. Additionally, the added burden of creating curriculum in mathematics brings with it another challenge, returning to the earlier subject of what is mathematics and is it unidirectional in its discovery? That is, is it sufficient to transform mathematical-looking practices to an abstract level for use in mathematics or should it also include a return to its original source -- in sharing that transformative experience? This question becomes another reason for propelling the parent workshops as a vehicle to explore mathematical knowledge and experiences, as will be discussed in the next section.

Parent Mathematics Workshops

The role of parents in influencing teaching practice and education is for the most part unchartered territory. However, before parents can influence teacher practice, they must change their own roles from solely that of parent as caregiver to parent as intellectual. In this respect the use of the term parents requires clarification. Mothers are generally the ones entrusted or burdened with the day-to-day work of caring and following-up on children’s educational needs. Therefore, the use of parents in this section refers collectively to mothers as parents. This reconceptualization of parents carries with it several premises, which are the subject of the following discussion.

The nature of home-school relationships for language minority and working class communities remains largely unchallenged, in that those relationships have not been examined sufficiently to preclude a discussion on the problematic and conflicting history between them. Consequently, reform initiatives at the level of implementation often tend to focus on reshaping the outward appearance of home-school collaborations without redressing underlying beliefs and tensions. In practice, there are, however varying degrees of success and lessons to be learned, as found in a number of research reports and articles (e.g., California Tomorrow, 1997; Carlson, 1991; Clemons-Brower, 1997; Pryor, 1995; Rutherford & Billig, 1995; Smrekar, 1996).

Another related phenomenon has to do with those deeply ingrained perceptions of parental roles in relation to the question of partnerships. The breadth of language minority and working class parents’ participation is well defined, especially for mothers. While earlier generations were raised with the notion that it is customary practice for mothers to come to school to support classroom activities (e.g., cutting and pasting, and cafeteria, cross-street, field trip and playground monitoring, etc.), this tradition is both hierarchical and patriarchal, and therefore problematic at various levels. This form of collaboration does not allow for participation in children’s education, it is servile in its dimensions.

At present, there is a greater urgency for the intellectual engagement of parents, especially mothers in language minority and working class communities. For them, there is much more at stake and they cannot afford to buy their way into fulfilling prescribed parental roles such as those discussed above, while advocating for children’s best interests (see Lareau, 1989; Henry, 1996 for discussions on the influences of culture and socioeconomic factors on the nature of home-school relationships, where families from dominant culture, sharing a higher economic status, extol greater reign and control of the school in advocating for their own children’s needs). Asking or requiring mothers to fulfill those prescribed roles is also not a solution, nor partnership to the current call for collaborations.

However, these discrepancies do not rest on the shoulders of schools, researchers and policy makers alone, hegemonic practices have lulled these parents into a particularly precarious state. The current shape of collaborations does not easily lead into the more crucial role for parents in the intellectual response to educating and rearing children. Parents must sense an authentic need for their intellectual engagement, not as auxiliaries, but as teacher and family liaison, H. González Le Denmat suggests, with "confianza, trust, and necessity," vital to the home-school collaborations we are all in search of (personal communication, December 17, 1997).

Acknowledging the problematic in the nature of these collaborations, there are other instrumental points to be made. First, we (those of us in the educational theater) must personally engage ourselves in the process of change, reshaping our understandings through the diligent work set by others, where possible. Secondly, we must move to facilitate the process of change in parents as well as teachers, as they move to meet the role of intellectual partners in collaboration. How then do we begin to change the pattern of interactions so steeped in tradition and that we have been socialized into accepting without question? Pedersen (1988) confides that in her own work, the role of researcher becomes one of facilitator, where

Similarly, Sarason (1994) has called for engaging the minds, hearts and voices of parents, students, and teachers alike, if collaboration is to take place. This work of facilitating partnerships is complex, requiring a great many resources, but most of all reflection on the process. Further, it is not mechanistic, for it requires giving of oneself and taking each of the interactants into account. As Sarason has suggested, it is about engaging minds, hearts and voices, but above all, it is about the unspoken eloquence of humility, where everyone comes to learn, no one better than the other, a dictum presently practiced in another local collaboration with community women pressing leadership through literacy related activities of which they are co-authors and collaborators (Leadership for Literacy project). McTaggart (1997) warns that among other things, it is also about,

Further, the significance of this approach is that it "is not just learning," but more importantly, "it has knowledge production and action aspects as well as constituting new ways of relating to each other to make the work of reform possible" (p. 36).

The Process of Fomenting Collaborations.

In developing the parental involvement component of BRIDGE we began with establishing rapport with four schools, a middle school and three feeder pattern elementary schools), from principals and teacher and family liaisons, to community representatives, while at the same time engaging a core group of parents. Also, it has been important to invite the participation of parents at the school, to look at the school as a place for parents as well as students. In earnest, however, the work began long before that as we established rapport and repeatedly met with a group of mothers to discuss issues of knowledge, learning and teaching, much of it within the realm of mathematics. This collective, "we," has thus grown to include all participants in this endeavor. Each ensuing relationship has developed at its own pace, not unlike those in the teacher-researcher mathematics study group. Our understanding is that we must continuously work toward establishing and maintaining a rapport of respect and mutual understanding.

We have presented this aspect of our work with mothers and the school, not as research, but as a collaboration wherein we are not only involved in a mutual endeavor and goal, what McTaggart (1997) has referred to as a "thematic concern," to improve the education and success of students in mathematics, to support schools and parents with the resources at our collective disposal. As participants, we all bring something to the table (be it knowledge, experience or resources) and participate with the intent of learning from one another in moving closer to finding solutions. This type of action research reaps the benefits of authentic interaction and engagement. Additionally, this component of the project is a model which we intend to posit as a plausible alternative consideration to home-school collaborations.

Engaging Parents as Intellectuals: A Model

Parental involvement has taken the form of monthly mathematics workshops. By asking parents what concerns they have with respect to their children’s education, particularly in the area of mathematics, we have generated topics for the workshops, while at the same time directly eliciting parental involvement in discussing their own household practices in which one of the topics, fractions, for example, has been found to be deeply embedded in the many activities of the household (e.g., cooking, shopping, knowledge of lunar phases, etc.). The two-hour workshops also incorporate hands-on activities for parents as learners as well as for their later application at home, with their own children.

Earlier workshops have been a general invitation for parents to explore topological puzzles and dedicated to discussions on logic and its uses in mathematics. A later workshop was the culmination of the investment in facilitating parent ownership, as it were, of the workshops. Parents, from one school took the lead in teaching another group of women the activity on the multiplication of fractions and whole numbers. Allowing these women to take leadership, as they did, in the workshop, created a sharing and intellectual intimacy that struck all those participating. Further, the participants each took home a copy of the activity designed as a board game, to utilize at home with children, spouses and others. This take-home activity and others are shared in the project’s portfolio.

Another BRIDGE sponsored event took place on a Saturday morning, unlike the other workshops which take place in the mornings during the school-week, a time that participants find more suitable to their schedules. The event was comprised of various activities that students may be familiar with in the math reform-oriented classroom (e.g., hands-on activities including problem solving, logic, etc.), but which their parents may not be familiar with, unless they are able to observe classroom activities. It included other valuable information for parents and students regarding statistics on the significance of mathematics to higher education and the preparation for futures in mathematics-related careers. Those in attendance also received a SOMA cube puzzle (see Portfolio). Though the event was not as highly attended as hoped, due in great part to preparations for year-end activities outside of the school (vacations, graduation), the 15 who did attend, including the assistant principal at the school, found the event not only intellectually stimulating, but also highly informative given that it was an opportunity to explore the mathematics of the classroom in a non-threatening atmosphere, while dispelling some of the myths surrounding mathematics and who it is for. To the evaluation question, "What would you like to see in future workshops? -- one participant responded: "Más gente, sobre todo para que se dieran cuenta de todo lo que se estan perdiendo y les sirva para ayudar a nuestros hijos" [More people, above all so that they could realize what they are missing and that it serves them to help our children].

As this group has gained momentum, there have been teachers interested in inviting these parents into their own classrooms, to teach students what they are learning in the workshops. We anticipate that we will soon have parents sharing their expertise in more formal ways, because they have already begun to do so informally, with others (parents, teachers, students, and other family members).

A. Relationship of Project Activities to Original Plan of Work

The relationship of project activities to the original plan of work remain quite unchanged with respect to the three major activities outlined (i.e., teacher-researcher study groups, ethnographic study of households, parent mathematics workshops). However, within each of these activities we have had to make adjustments and expand some of our initial plans. Within the teacher-researcher study groups we have had to be flexible with respect to the shape and direction that each group of teachers has given the activity. For example, we have found that the dynamics between teachers’ knowledge, experiences and interests greatly influences the direction of the work. With respect to the ethnographic study of households, we have found that while the initial ethnographic experience remains vital to participating teachers, the focus has moved from abstracting from households exclusively to bringing the household, in the form of parents, into the school. And also, finding that teachers themselves need to explore their own funds of knowledge in relation to that of their students and their families. This, of course has refocused our vision of parent participation and the need to include parents in the creation and dissemination of knowledge from home to school and school to home, for the academic benefit of students. By keeping a pulse, as we have, on project activities, we are able to adjust and improve the direction and ultimate outcomes of our work.

B. Relationship of the Project to Program Unifying Themes

The relationship of our project to Program 4: Instruction in Context is linked to its three central unifying themes: 1) bridging home, school, and community, 2) creating learning communities, and 3) integrating teacher learning, teaching practices and student learning. The essence of our project is to bridge between home and school (and the communities of both), by focusing on teachers and parents we directly and indirectly envelop the lives of students in dynamic collaborations, be they in the classroom with teachers or outside the classroom with parents. Each of the project activities has in mind to respect participants as intellectual partners, be it in the teacher-researcher study groups, parent mathematics workshops, or classrooms. By treating participants as intellectuals we necessarily encourage the development of mathematical learning communities, where the pursuit of mathematical knowledge and understanding are key. This is what teacher-researchers, students and parents are pursuing through their involvement in BRIDGE. In validating household knowledge and experiences, and teacher knowledge and experiences, we also validate children’s understandings and growing knowledge by promoting experiences that integrate pedagogical practice with less formal ways of knowing and doing mathematics.

C. Relationship of the Project to CREDE Unifying Themes

The relationship of our project to the program’s unifying themes is salient throughout project activities. The teacher-researcher study groups and the ethnographic study of households together reinforce our abilities to learn through the joint productive activities of teachers and researchers, teachers and students, and teachers and families, and eventually through the parent mathematics workshops, parents and students. All three project activities have the underlying intent of developing the language of mathematics and furthering understandings of mathematical concepts. As is evidenced in the ethnographic study of households, through teacher-researcher household visits and reflections coupled with opportunities to discuss them and build upon them for purposes of pedagogical implications (classroom innovations as found in the development of curriculum), we are contextualizing teaching and curriculum within the experiences and skills of home and community. The purpose of our activities in developing and implementing curriculum units in exploring teacher-researchers’ professional growth, as found in Caroline Carson’s reflections on the discourse of mathematics, Leslie Kahn’s "Weaving and Gardening" integrated unit, and José David Fonseca’s "Build Your Dream House" (found in the Portfolio), moves to challenge students’ and teachers’ own cognitive complexities. Finally, in the classroom, where our activities come to rest, the test of our successes is made in how we cognitively engage students in dialogue about mathematical discourse and mathematical concepts in innovative ways that build on their own home and community and experiences.

V. Reflections

A. Project contributions to increase knowledge of practice and theory.

Our belief is that as a project we are contributing to practice and theory of not only mathematics education, but also contributing what we believe is a unique perspective of what parent participation can mean. The various activities of our project pursue a common goal - contributing to the academic success of students in mathematics and their engagement to the life world of the school through meaningful and connected activities that respect home knowledge and experiences. Our contributions to practice and theory are evidenced in the numerous papers developed and the presentations made in conference and meetings (see Portfolio).

B. The use of effective methods.

As the Funds of Knowledge concept has evolved in our work, the approach to ethnographic training has shifted as we have learned more about what works and what does not. Not surprisingly, what works is exactly what our basic assumption is predicated on: the more that participants can engage and identify with the topic matter, the more interest and motivation are generated. What does not work is a top/down classroom style approach in which teacher-researcher participants can learn methodological technique, but which strips away the multidimensionality of a personal ethnographic encounter. In other words, we learn ethnography by doing ethnography.

It is difficult to reduce a complex process to formulaic terms, because anything called ethnography is always in jeopardy of reductionistic misuse. However, there are certain points that are key in adopting an anthropological lens. First, it is important to read ethnographic literature. Teachers in our project have always been provided with a reader (see Portfolio) which contains numerous examples of ethnographic work relating to educational settings as well as other readings we consider of importance to our project goals. Second, it is important to role-play or discuss a non-evaluative, non-judgmental stance to the fieldwork they will be conducting. We may not always agree with what we hear, but our role is to understand how others make sense of their lives. Sense-making processes may be contradictory or ambiguous, but in one way or another, understanding what "makes sense" to others is what we are about. Third, it helps to be a good observer and even more, to pay attention to detail.

The household visit begins long before the actual entrance into the home. Driving down the street, we can observe the neighborhood, the surrounding area, the external markers of what identifies this as a neighborhood. We can look for material clues to possible funds of knowledge in gardens (botanical knowledge?), patio walls (perhaps someone is a mason?), restored automobiles (mechanical knowledge?) or the nature of ornaments displayed (made by whom?). During our initial training session, (although we recoil from the word "training," since ethnography is not something one can be trained in, but we have failed to find an adequate substitute...exposure to ethnographic methods?) we show a video, from an earlier project, which contains two short segments of everyday community scenes and ask participants to talk about what they notice. The first video segment is that of a family yard sale and shows a great deal of activity going on at once. We stress that this is usually what happens on a household visit: life does not stand still so that we can otherwise document it in all its detail. The vignette usually elicits comments on what is being sold (wooden doll furniture might indicate carpentry skills), the interactions involved (the older siblings are caring for toddlers, indicating cross-age caretaking), and language use (code-switching between Spanish and English is evident throughout). It is fascinating to notice how our own interests and our own funds of knowledge often color and filter what we observe. One teacher commented that he noticed a fountain in the backyard because he was installing one himself. The second video segment is particularly rich for our BRIDGE project since it shows a nine-year old boy in a backyard workshop, working with his father to build a barbecue grill. The scene is replete with measurement, estimation, geometry, and a range of other household mathematical practices. Because we do not often think of routine household activities as containing mathematics, this "slice of life" helps to "mathematize" the household visits.

Fourth, and perhaps most important, we ask respectful questions and learn to listen to answers. As a part of the process of entering the households, we have a series of questionnaires (see Portfolio for Interview Protocols) which explore household networks, labor histories and daily activities. These topics have proved to be fertile areas for tapping into funds of knowledge. The questionnaires are not prescriptive and are meant to be used only as a guide to indicate the breadth and depth of possible questions. Most teachers have found that they can ask only one or two questions about, for instance, family histories, and find that long, evocative narratives about how families came to be where they are now, are forthcoming. In our context, in many cases, these networks reveal strong cross-border ties with families in the northern Mexican state of Sonora, thus linking children with activities and funds of knowledge on both sides of the border. Questions on labor history have also served as one of the main sources of information on possible funds of knowledge, as skills are often embedded in work contexts. However, it is important to remember that funds of knowledge are not limited to the formal market sector. Informal market strategies, such as yard sales, selling tortillas, sewing, car repairs, etc., can yield rich areas which can be tapped for pedagogical wealth. One outcome of respectful ethnographic talk is an increased sense of "confianza" or mutual trust, as parents and teachers come to view each other in multidimensional terms.

C. Contributions of current research in the field.

Our activities in this area have maintained an active profile in sharing our work in appropriate venues and learning from others in the field of mathematics education as well as from practitioners, as we highlight the three central activities of the work. Participating teachers are actively involved in this process as they present to their own peers and other professionals. Parents are also active in sharing at a local level as they engage other parents in a mutual pursuit of mathematical knowledge. In the dissemination section that follows later in this text, we outline many of our contributions.

D. Benefits of the work in relation to cost.

This question is best addressed by others outside the project. However, in all modesty, we must say that as Principal Investigators we are utilizing many of our other university-related activities to complement the BRIDGE project, as well as tapping into many resources that we collectively have access to (i.e., speakers, materials, and other insights). In a sense, we are utilizing our own funds of knowledge to further the activities of BRIDGE. One other item of significance, our work on the project is labor intensive with respect to building relationships of mutual respect and rapport, as such while these relationships go beyond monetary compensation, they are also long-lasting, going well beyond the life of the project. The benefits of participation in BRIDGE are not superficial, they go to enhance professional development of teachers and parents and creating learning opportunities for students.

E. Connections of the project to other programs.

Along with sister projects we have contributed a book chapter to a practitioner-oriented book on contextualizing instruction (Ellen McIntyre, Ed.). We also have ties with sister projects in other programs. For example, we will be showcasing our collective work with parents in a symposium at the 1999 Annual Meetings of the American Educational Research Association. The rationale for the symposium is that together these sister projects, across programs, present a holistic portrait of ways in which CREDE projects are grappling with new ways of promoting the participation of parents, treating parents as intellectuals (see Portfolio).

VI. Implications

The implications of our work are necessarily discussed in three sections, with respect to: 1) Teacher-Researcher Study Groups, 2) Ethnographic Study of Households, and 3) Parent Mathematics Workshops.

A. Implications of Teacher-Researcher Study Groups

It would be all too easy to answer some of the questions raised within the Study Group sessions by quickly dismissing what some do as mathematics by validating formal application over informal practices. But, we have the burden to understand the social and cultural practices in which mathematics are embedded. We do, however, agree that mathematics is a gatekeeper. In doing so, we have begun to realize the ways in which we often measure others’ knowledge of mathematics to the yardstick of testing, which values certain knowledge and experiences as superior, while qualifying as inferior (via remuneration and prestige, for example) other less palpable expressions of mathematical knowledge. Interestingly, but not surprisingly, we are also finding that some of these perceptions are closely related to gender and class specific knowledge and experiences of individuals -- this is an area that we are just now beginning to explore. That mathematics is not gender or culturally or socially neutral as a subject should be of no surprise, but we are usually taught that it is not and presented with it as a neutral subject (see Walkerdine, 1988). This is again an area that we will continue exploring and delving into as we pursue the activities of the study group. The significance of the study-group notwithstanding, is that it is first a forum for the process of consciousness about pedagogy, curriculum and knowledge in the teaching and learning of mathematics, then a vehicle for forging the links between home and school knowledge and experience in the creation of innovative curriculum and pedagogy in mathematics.

B. Implications of the Ethnographic Study of Households

Despite our better sense to do otherwise, somewhere, sometime during preservice teacher training and teacher professional development, myths and stereotypes about parents and especially minority, language minority and working class families are woven and reinforced time and time again (Flores, Tefft Cousin & Díaz, 1991). This is not an exception even within our study group. However, the very nature of the ethnographic experience and later analysis of the data collected, coupled with the ongoing teacher-researcher study groups challenges and transforms those counterproductive and misguided perceptions which have served to limit the ebb and flow of knowledge and experiences between home and classroom.

One particular series of household visits by Teacher-Researcher Caroline Carson makes the case for the importance of the ethnographic experiences of the project. During the second year of the project, Caroline chose to switch the focus of her household visits to single parents given their growing numbers within her classroom and school. She visited one particular family headed by a single-mother with four children. Initially, the visit was typical in that is was informative and enjoyable, the student was proud to have his teacher visit his home. During the course of the visit Caroline learned much about the routine that the family had, duties and responsibilities of the children, among other things. She also began to learn about the family’s labor and educational history. However, this and subsequent visits eventually revealed that the family had been homeless, living in a car and moving from shelter to shelter for a period following the father’s profound substance abuse and later during his subsequent separation from the family. Eventually the parents divorced. However, from the early history of the family, Caroline culled out important details, for example, the father and mother, children of Mexican immigrants, had been raised in the Midwest and had come from homes where the education and work ethic were important, as it has been in all homes we have visited. Both parents graduated high school and had some post secondary education. The father had a number of experiences (e.g., construction) which reinforced his own mathematical knowledge and abilities as was evidenced in how he taught the older children mathematics, while the mother taught other subjects during an earlier period in which they were home-schooling the children.

While there are many more details shared by Caroline about her visits, the point of this brief vignette is that despite the numerous factors for at-riskness (poverty, homelessness, divorce, minority status, etc.) this was a family that challenged any labeling. That this family history was not known to the school, may have been significant in averting any labeling. At present the children are doing well academically, with the exception of the youngest one, a 2nd grader who among other things did not have the benefit of both parents home-schooling but was caught in that awkward period of transition, he had not been enrolled in school prior to this but was making quick progress. The older daughters were enrolled in alternative academies and also doing well, and helping at home.

During a study group meeting Caroline expressed the impression this family had left on her, she shared that though they were materially speaking not wealthy, there was a wealth of love, respect and discipline. This, suggested Caroline, was a family for us all to learn from -- they had also taught her a great deal about respecting families and about a family’s own resilience during difficult periods. This is the nature of experiences that are shared by teacher-researchers following household visits, other household knowledge and experiences begin to build on this foundation.

We have learned that before we can begin discussing the academic potential for household

knowledge, it is necessary to cull out, like Caroline, the social, cultural and historical context of each family in order to appreciate the struggles they face. At the same time, in sharing the stories of a family, as in Caroline’s case, it is inevitable to bring one’s own experiences and knowledge to the table, in this way, teachers themselves are exploring their own funds of knowledge.

C. Implications of the Parent Mathematics Workshops

While there are many strides made in our work with parents, there have been numerous drawbacks to our work, some unanticipated. For example, there have been school personnel that have found that other scheduled volunteer parent activities (e.g., monitoring the cafeteria, crossing guard duty and serving as teacher aides) take greater precedence in terms of home-school collaboration, than do the workshops, which we find are similar to professional development workshops for teachers. This is one of the more salient findings we have made: if we wish to have parents involved in different and meaningful ways in the education of their children, then should we not also facilitate that transition as we do for teachers, through workshops and other opportunities for professional development?

Additionally, at different levels and junctures there have been misunderstandings as to the roles to be played and our goals, for example, an insistence on playing a researcher/observer role when such issues as establishing rapport, respecting the participants as individuals in search of intellectual stimulation, not as subjects for observation, remain in the balance. This has been an especially difficult lesson to learn. While we are keenly aware of respecting and promoting the teacher-research role (i.e., teachers as intellectuals), we do not readily apply the same lessons learned to our interactions with parents. This can be taken as another example of how hegemonic practices have become part and parcel of our professional lives as researchers, where we must, as Pedersen (1988) suggests, continually seek out more knowledgeable others while pursuing self-education. This disjuncture has, however, been necessary, for as McTaggart (1997) has suggested, it is "contingent upon authentic participation: it is research through which people work toward the improvement of their own practices (and only secondarily the improvement of other people’s practices)" (p. 34).

Evaluations of the workshops have become an integral aspect of our workshops. They serve to guide the form and direction of our work. The following evaluation by an Adult Educator reinforces the significance of this work, additionally it makes the case of current reforms in mathematics with respect to problem solving: there is more than one "correct" way of going about finding a solution.

Early on, the workshop was baptized with the name Rompecabezas, "because," as one mother suggests, "of the way they [mathematics] make you think. You split your head [te rompes la cabeza] trying to figure it out." Which is, not surprisingly, the same dilemma we are faced with in facilitating collaborations and finding our way in this mutual endeavor of challenging home-school collaborations as they define parent participation.

VII. Impact

We have found many implications in the activities of our work. Many of those implications have direct application to how schools are structured and the roles of teachers and parents within the current framework of home-school collaborations. We believe in the importance of this work, however, there are limitations to how we can extend our findings if schools as institutions and preservice teacher training and teacher professional development continue to look to the individual as the problem without reforming their own ranks and institutions. When we choose to engage teachers in meaningful reflective practice of their authorship and parents as intellectuals interested in the pursuit of knowledge and activities that extend to home activities, then we can begin the work of integrating these pursuits in authentic collaboration.

On another level, the impact on those participating in BRIDGE is made on a professional level through experiences that are propelling participants to challenge their pedagogical practices and the ways in which they take notice of the knowledge and experiences of students, their families and the local community, as well as looking to their own funds of knowledge. For students and their families, the experience is one that is "humanizing" the school and the curriculum of mathematics, while at the same time including them as critical partners in creating discourse and knowledge of mathematics in ways that bridge home and school practices.

VIII. Deliverables

Deliverables include the annual reports with summaries of activities and findings. These reports are envisioned as a handbook in progress, to be completed in Year 5 of the project. The final report, to be completed at the completion of the project, will be a manuscript suitable for publication in a professional journal. A video documentary of the project will also be completed at the end of Year 5. For now, the nature of other deliverables consists of articles, and chapters, and presentations of our ongoing work and findings.

IX. Next Steps

While we are satisfied with the significance of our present work these past two and one-half years, we have also reflected on the findings and directions for purposes of fine-tuning the impact of the project. In light of this, we propose to adjust the present direction of the project in order to maximize our collaborations. There are two significant changes that we anticipate making, as follows: 1) the teacher-researcher study groups will take on a more structured format, with readings to guide the discussion on mathematics culture, teacher practice, and student learning, and 2) the Parent Math Workshops which have proven successful in the past, will develop further to encompass a more frequent (weekly or twice a month) Parent Study Group in which parents, much like teacher-researchers in the project, will explore issues in mathematics, classroom practice, and the learning of students. Eventually, we anticipate collaboration between teacher-researcher and parent study groups. These changes are the result of our efforts to maintain a pulse on the varied activities of the project and thus respond in a manner that is informed by our experiences while at the same time attempting to link home and school practices in the teaching and learning of mathematics. We anticipate this new direction with great excitement.

Interestingly, Gil Garcia’s recent article in Talking Leaves (1998) seems to have predicted a similar direction for the funds of knowledge work by CREDE projects. Garcia’s article suggests the need for funds of knowledge to cross over into teachers’ knowledge and experience as well as that of family and community. Additionally, he emphasizes the importance of a more active role for parents as investigators (e.g., Parents as Researchers), something that we have been exploring these past two years.

X. Collaboration

Our project has a long history of collaboration with sister project 4.1 Teaching Science to At-Risk Students: Teacher Research Communities as a Context for Professional Development and School Reform dating to 1988. We share insights as well as seek out support from one another. We have also collaborated with sister project 5.5 Study of Children’s Academic and Social Development in Nongraded Primary Schools: Model Programs for Children of Poverty in sharing our own protocols for interviewing households, and more recently in two endeavors: contributing a chapter to a book being edited by Ellen McIntyre to be published by Heinemann (see Portfolio), and the BRIDGE organized symposium titled "Parents as Intellectuals, Parents as Experts" for the 1999 Annual Program Meetings of the American Educational Research Association. Additionally, this AERA symposium is a collaboration between ten researchers in several CREDE sister projects, across programs -- each contributing to the unifying theme of parents as intellectuals.

XI. Products

Awards

Outstanding Educator Award, to Leslie Kahn

Teacher of the Year, José David Fonseca

Best of Show, Student Showcase, Melanie Ayers

Chapters

"Creating Links Between Home and School Mathematics Practices," ed. Ellen McIntyre, Creating Diversity: Connecting Students’ Cultures to Instruction, Heinemann.

In the News...

"Building a better math class", Tucson Citizen, Spring 1997.

"Reports on Ethnomathematics Research, International Study Group on Ethnomathematics Newsletter, November 1997.

"Working Group on Cultural Aspects in the Learning of Mathematics, Psychology of Mathematics Education Newsletter, 1997

"Paths to Faculty Success," University of Arizona Report on Research, Spring 1998.

"Linking Home and School: A Bridge to the Many Faces of Mathematics," Talking Leaves, Summer 1998.

Papers and Presentations

A Bridge to the Many Faces of Mathematics: Exploring the Household Mathematical Experiences of Bilingual Students. March 1997, AERA Annual Meetings, Chicago, IL

Build your Dream Home: An Ethno-Mathematical Approach to Motivate the Learning of Mathematics. October 1997, Psychology of Mathematics Education, Bloomington, IL

Bridging in-School Mathematics and Out-of-School Mathematics. April 1998, AERA Annual Meetings, San Diego, CA

Linking Students’ Own Cultural Mathematics and Academic Mathematics.

Linking Home and School: In Pursuit of a Two-Way Mathematical Dialogue.

July 1998, Psychology of Mathematics Education-22, Stellenbosch, South Africa

Parents as Resources for Mathematical Instruction. July 1998, Adults Learning of Mathematics-5, Ultrecht, Netherlands.

Connecting Students’ Everyday Mathematics and School Mathematics. October 1998, Psychology of Mathematics Education, North Carolina

Exploring Children’s Understanding of Measurement through a Garden Project. April 1999, National Council of Teachers of Mathematics Annual Meetings, San Francisco, CA.

Changing the Classroom Discourse in an Elementary School Mathematics Classroom for Minority Students. April 1999, National Council of Teachers of Mathematics Annual Meetings, San Francisco, CA.

Connecting Home and School Mathematics for Minority Students. April 1999, National Council of Teachers of Mathematics Annual Meetings, San Francisco, CA.

Parents as Intellectuals, Parents as Experts (Symposium). April 1999, AERA Annual Meetings, Montreal, Canada.

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