[Paper presented at the = 22nd Conference of the International Group for the Psychology = of Mathematics Education, Stellenbosch, South Africa, July 1998]
LINKING HOME AND SCHOOL:
IN PURSUIT OF A TWO-WAY MATHEMATICAL DIALOGUE
University of Arizona
This paper is based on a research project that = has as one of its main goals the development of mathematics = instructional innovations in classrooms composed of predominantly = minority working-class students. These innovations aim to engage = students and teachers in mathematically rich situations through the = creation of learning modules that capitalize on students' (and their = families') knowledge and experiences in their everyday life. Our = research project relies on a model that has four key inter-related = components: 1) Household Visits; 2) Teacher-Researcher Study Groups; 3) = Classroom Implementation; and 4) Parents as Learning Resources. = A key research question is to which extent we can combine school = mathematics and everyday mathematics to create powerful learning = environments.
This paper is based on a research project that has = as one of its main goals the development of mathematics instructional = innovations in classrooms composed of predominantly minority = working-class students. In our local context, minority students = (mostly Mexican American) from poor backgrounds tend to fall further = behind in their mathematical learning as they move up through the = grades. Quite often minority students from working-class backgrounds = receive a mathematical education that stresses basic skills and rote = learning (Porter, 1990). Unfortunately, these students are often = exposed to an education approach based on a deficit model in school = teaching. Such a model presupposes that the households of minority, = working-class children are at the root of "the problem." That is, this = model assumes that students lack adequate experiences and background for = formal schooling. In this model, students are often cast in a passive = role and perceived as "lacking something." They are given remedial or = watered down work and labeled as slow, learning disabled, or any other = such label. Because a certain aura of helplessness and hopelessness is = created regarding the academic development of these children, they are = in turn treated with pity, as if the "problem" were an inherent one, not = a socially imposed one.
However, findings from a prior project--the Funds = of Knowledge for Teaching Project (Moll, 1992.)-- show a wealth of = resources and information in these childrens households that often = is untapped in school (examples of funds of knowledge include ranching = and farming; budgets; construction; folk medicine). Furthermore, that = project also gathered evidence that at home and in their community, = these children are often active participants in the functioning of the = household (e.g., language interpreters for parents and other relatives; = assist in the child care of younger siblings; help out in the economical = development of the household (e.g., helping in the repair of appliances, = cars); play an active role in traditional ceremonies (e.g., Yoeme = Easter)).
What are the implications for the mathematical = education of these children, if we take their experiences and = backgrounds as resources for learning in the classroom? What may the = learning environment look like if we try to develop a more participatory = approach towards their learning of mathematics (similar to what these = children experience in their out-of-school lives)? A key research = question underlying our work is whether we can develop a teaching = innovation which successfully combines in-school and out-of-school = mathematics. What do we mean by a successful combination? And = furthermore, what do we mean by these two kinds of mathematics? We take = the position that there are many different kinds of mathematics, each of = which is characterized by its set of values, its associated discourse, = its motivation. By in-school mathematics we refer here to the = mathematics needed for the students to successfully advance through the = school system. It may be "reform" oriented, "traditional," or yet = something else. The point is that in our work we are constantly = confronted with these differing interpretations about school mathematics = (and thus differing expectations for student achievement).
Two large bodies of literature were at the = basis of our research: studies on the social construction of mathematics = in the classroom (Cobb, 1991; Ernest, 1998; Lampert, 1990) and studies = on everyday mathematics and on the gap between in-school and = out-of-school learning (Abreu, 1995; Bishop, 1994; Lave, 1988; Nunes, = Schliemann, & Carraher, 1993). Furthermore, our work is heavily = influenced by a prior project which was based on a sociocultural = approach to instruction (Moll, 1992). Rogoff's (1994) description of an = instructional approach that characterizes learning as a process of = transformation of participation guides our current thinking for the = classroom implementation. Finally, Secada's (1989) question "how should = we educate individuals from groups that have suffered discrimination to = live in a world in which they are likely to be subjected to similar = treatment?" (p. 46) is particularly relevant to us since our work is = primarily with these individuals. Thus, we have turned to research in = critical mathematics education to further inform our work (Apple, 1995; = Frankenstein & Powell, 1994; Mellin-Olsen, 1987; Skovsmose, 1994; = Vithal & Skovsmose, 1997). In particular, since our work relies so = much on the idea of the background of the students, I find Vithal and = Skovsmose's (1997) discussion of students' foreground very compelling as = it adds another twist to our work.
The Research Project
Our research project relies on a model that has = four key inter-related components: 1) Household Visits; 2) = Teacher-Researcher Study Groups; 3) Classroom Implementation; and 4) = Parents as Learning Resources. These components will be described next = including a brief sampling of some of the activities in each of them. = This will be followed by a reflection in an attempt to engage in a = self-critical appraisal of our work (Apple, 1995). The first three = components emerge from the prior project but have been refined to take = into account the mathematical focus of our current work. The last = component is new and particularly dwells on our concept of a two-way = dialogue. Due to the complexity of the research endeavor, we rely on a = variety of sources of data including: field notes (household visits; = study group meetings; classroom observations; parents' mathematical = workshops); interviews (project teachers, students, parents); portfolios = of some students; audio and video tapes of selected activities. Data = analysis takes place at different levels with individual researchers = tackling some aspects of the project and producing working papers that = are then discussed by several of us and further elaborated. Our = disciplines are quite varied including anthropology, bilingual = education, elementary education, and mathematics education, and thus we = each bring a different perspective to the data analysis. However, we = all share a basic orientation towards an ethnographic research = methodology. Maybe due to this diversity in our backgrounds, an issue = that keeps coming up is that of finding a language that describes our = research to everybody's satisfaction. We realize that there are certain = expressions that evoke very different images, sometimes even = contradictory ones, and this presents a constant challenge as we write = about our work.
Household visits. The project teachers = receive training in ethnographic research methods. They then visit the = homes of some of their students to learn about the funds of = knowledge in these households. Questionnaires on the family structure, = parental attitudes towards child-rearing, labor history, and household = activities are used to provide some structure to these home visits. = These questionnaires include several questions aimed at uncovering the = mathematical potential in the households. A key effect of these = household visits is on the teachers' perceptions of their students. By = seeing and learning about their students' experiences, they develop a = firsthand understanding of such experiences, as opposed to being told = generalities about the "minority culture." They also learn about = networking practices in the community and about the children's = participation in household and community activities. All the teachers = have commented very positively on the impact that these household visits = have had on them, as well as often on the students whom they have = visited.
Based on the one home visit, I know what the = student does in her family. And what the family does. It makes me more = sensitive to asking questions that I know she knows the answers to. It = is great. She is now participating more in class. (Teacher's interview = #1)
Teachers visit the same family at least three times = and through these in-depth visits, they learn about the wealth of = knowledge that exits in every household, but that may go unnoticed = otherwise. This opens up the door for a different look at parental = involvement: parents are invited to come to the classroom to share their = knowledge and expertise. This is what another teacher says in response = to the effect of the household visits on her teaching:
It has allowed me to think about ways to involve = parents in a different way. Literally involve the parents. [She then = describes how one parent drew the plans for how to build a burner for a = hot air balloon, a theme she was exploring with her students.]
As far as [the household visits] impacting my = teaching (...) first is has to impact my thinking and then at some point = it will impact my teaching. (Teacher's interview #1)
Teacher-researcher study groups. This is = the key component in our research approach. In these 2 to 3 hours = after-school sessions (every 3 weeks), we all get together to discuss = the different aspects of the project. Here is where the pedagogical = transformation of the findings from the household visits takes place. = What are the curricular implications of these household findings? Since = our focus is on mathematics, these sessions have led us to a constant = examination of what we mean when we say that a certain activity or = practice has mathematics in it. There seems to be a tendency in the = current rhetoric of reform to make statements such as "mathematics is = everywhere." Some of this happened in our initial discussions of the = household visits: some saw mathematics everywhere, some did not. For = example, in analyzing the daily routine of one of the families, the = teacher observed a high level of organization in terms of schedules = (particularly in the morning) and also in their approach to grocery = shopping. This led some of the members of the study group to describe = these routines as very mathematical. I challenged that notion by asking = "When you say, 'I use math to get to work,' what do we mean by this? = What kind of math?" As one of the project participants said, "many = activities have math concepts in them. But we need the analysis and = reflection to bring out the mathematics" (Study Group Fieldnotes = 2/12/97).
For a more focused pedagogical transformation of = the findings from the household, we hold "curriculum retreats" in which = we spend one whole day looking at possible curriculum themes that emerge = from these visits. Besides the possible classroom implications that = such retreats have, an important aspect is that it engages all of us = into looking at what mathematics is. It forces us to examine our values = and beliefs about what we count as mathematics. At our last retreat, = three topics (cooking; home business; gardening) that emerged from the = household visits were analyzed with an eye on possible curricular = implications. Household activities related to each of these topics were = listed. These activities were then analyzed in terms of their = mathematical content and in terms of the required curriculum in the = schools. Something that became clear in the debriefing of household = visits and in the curriculum retreat is the difficulty in characterizing = mathematical practices. For example, one of the mothers makes crafts = that she then sells for additional income. A brief analysis of the = craft samples that the teacher brought highlighted a series of = geometrical ideas in connection to the making of these objects, but we = also realized that we needed more focused interviews in which the = individuals describe their approach to the task. Thus, we are now = developing protocols for what we call "occupational interviews," in = which we will look in detail at the mathematics in specific practices = (at home or in the work-place). In one pilot experience, one of the = researchers in our team interviewed a seamstress (González, = Civil, Andrade, & Fonseca, 1997). We then proceeded to analyze her = practice in terms of mathematical content (e.g., angles; from 2D to 3D = and viceversa; area; estimation) and dispositions (e.g., persistence; = enjoyment of challenge). During this process, and all throughout this = project, we are faced with Millroy's (1992) paradox of whether we can = see mathematics that may look very different from the kind of = mathematics that we learned in classrooms. Through these "occupational = interviews" and their subsequent analysis, we want to open up a dialogue = in our study group meetings centered around what we view as "valid" = mathematics and how we can use the results from these interviews in = curriculum development. Also, by exploring the mathematics embedded in = practice (carpentry, sewing, construction, design, etc.), we are = addressing another component of these study groups. In our prior = project, as we started talking about mathematics, a teacher remarked = that she knew how to let students play with language but that she did = not know how to let them play with mathematics. Thus, in our current = work, we include different kinds of explorations of mathematics. So far = most of these have leaned towards "academic" mathematics but grounded on = everyday experiences. For example, an exploration of what area is and = how to compute it, was grounded on a discussion of the different methods = to compute area used by the Landless People's Movement in Brazil = (Knijnik, 1996). More recently, the work on the garden theme led us to = an exploration of area and volume and how the scale factor affects them. = Through these explorations we are not only furthering our learning of = mathematics, but also experiencing a collaborative sense-making in = mathematics and working on mathematical discourse--areas that the = teachers are particularly interested in as they reflect on their = classroom practice.
Classroom implementation. The different = teachers in the project have expressed a variety of reasons for their = wanting to take part in this research. Some of these are certainly = related to "the teacher's understanding of the concept of culture and = the reasons for focusing on learners' cultural backgrounds" (Vithal = & Skovsmose, 1997, p. 145). Besides their different reasons, they = also have different school constraints. The project tries to adapt to = their different interests and needs, as well as to these differing = constraints. For example, one of the teachers wants to explore whether = "rigorous" mathematics can be developed from everyday mathematics. She = is currently developing her curriculum around the theme of planting a = garden. She strongly believes that her students have to co-construct = the curriculum with her, and this shows in her approach to teaching. = She enjoys having one of the mathematics educators in her classroom as a = support to do "that collaborative kind of teaching that honors what I'm = learning and doing, and what she [the mathematics educator] already = knows" (teacher's int. #2). To be better able to push her students' = mathematical inquiry, she herself is taking more mathematics courses. = The garden theme has led her class to an exploration of geometry, an = area that otherwise may not have received much attention (the focus = often being on arithmetic topics). This is a very confident teacher who = is comfortable sharing her thinking as she ventures in uncharted = territory:
I feel like sometimes I'm limited in my own = knowledge as far as what I want to do mathematically. And so, I have to = go to books and say, "now, is that really were I want to go with my 4th = and 5th graders? Or do I want to go in that direction? And would this = be considered rigorous math? And will it work when my kids get tested on = [a district standardized test]? Will they have learned something that = will transfer over?" And that's threatening, really threatening. = (teacher's interview #2)
Another teacher is particularly interested in = understanding the current reform movement in terms of implications for = the mathematics discourse in her classroom. Hence, she asked to be = videotaped to be better able to work on her discourse. This teacher = became particularly interested in Lampert's (1990) article, which we = read in one of our study group meetings. She is using the approach to = discourse presented there as a general framework to guide her = reflection.
Two teachers are using a reform oriented curriculum = that is quite demanding on their time. Thus, for them, the idea of = developing themes based on the children's experiences and knowledge does = not seam feasible at this point. However, this adopted curriculum seems = to embrace some of the same pedagogical principles that we are promoting = in our work. Hence, one challenge for us is to explore how the project = plays up in this situation.
In the first interview, one of these teachers = expressed her concern about how to reach out to all her students if she = was only visiting one or two families (since the household visits are = very time intense, it is not realistic to do many more). The children's = interviews that we implemented have addressed her concern by actually = helping her conceptualize the project in a way that seems to go in the = opposite direction from when it was conceived, but that actually is = reinforcing this image of a "two-way dialogue":
How they use math, more, instead of home to school, = school to home. At home, from our little survey you have so many = diverse backgrounds, that I would rather take what they need to know for = school, to progress in school. There are certain things that they need = to have no matter what their background is. I would rather take those = things and use them at home, learn how to apply them at home. So, I am = not sure how that relates to Bridge's [the project name] philosophy, it = is kind of backwards. (...) It seems like this would mean getting the = kids to apply at home what they have learned in school, to generalize it = for home. (...) My philosophy is that kids should be generalizing what = they learn in math class and taking it home and applying it to their = everyday life. (Teacher's interview #2)
Parents as learning resources. The prior = project already made very clear the wealth of knowledge that parents = have. We now want to look at the mathematical aspect of this knowledge, = but we want to do this through a discussion with the parents themselves. = For example, as we piloted some of this work, we had the opportunity to = listen to a woman describe a step by step procedure of how to make a = dress. She used the teacher as her model and walked us through the = whole process. Throughout her presentation, the other women in the = group shared their knowledge of sewing, exchanging "tricks" and asking = each other how to do certain things. In a very collegial atmosphere, = comments about the mathematics they saw in sewing, as well as comments = about their own experiences as learners of mathematics were naturally = shared. From my point of view, there was a richness in mathematical = content in the whole process of making the dress (e.g., her technique to = draw a quarter of a circle; her use of symmetry all throughout; a = possible arena for exploration of theorem of Pythagoras). At this = point, this was mostly an exercise for me as I attempted to uncover the = mathematics in the making of the pattern for a dress. I did not really = discuss the embedded mathematics (from my point of view) with the = seamstress (see Vithal & Skovsmose, 1997 on this issue). However, = because we do think that it is important that such discussions take = place, we felt that we needed to engage in a two-way dialogue about = mathematics with a core group of parents (at this point, it is mostly = mothers). On one hand we want to learn about their everyday uses of = mathematics as well as their beliefs about mathematics in particular as = they relate to their children's education. On the other hand, we want = to engage them in mathematical investigations similar to those that = their children are likely to do at school. Through regular workshops we = are developing a "two-way" dialogue to learn from each other. A key = motivation for many of these parents for their wanting to participate in = these workshops is to enhance their own understanding of mathematics to = then be better able to help their children. We believe that by = exploring connections between their experiences and topics of school = mathematics, we can help reach this goal.
Our recent workshops have been on exploring = fractions through activities typical from reform oriented curricula = (building the fraction concept through hands-on activities). We decided = to do this because fractions appear to be one of the main stumbling = blocks in school mathematics in the grades we are focusing on. One of = these workshops in which we had asked the parents to bring examples of = their everyday uses of fractions, brought up the issue of everyday = language and mathematical language. The situation had to do with the = terms "fraction" and "portion" (the exchange took place in Spanish, = where "porción" is a term commonly used to refer to part-- of a = cake, for example). But, are portions really fractions? That is, what = mathematical concepts do we associate to each of these terms? Can we = say, "oh, OK, let's use 'portion' instead of 'fraction'" and still = proceed with school mathematics fractions? In fact, should we be looking = at fractions from the school mathematics point of view in these = workshops with the parents? How do these parents (most of whom come from = a "traditional" approach to the in-school learning of mathematics in = which they were told how to do mathematics) appreciate an open-ended = approach in which closure is not necessarily reached at the end of each = individual workshop? These questions lead to the last part in this = paper--a brief personal reflection on our work, mostly in the form of = questions or dilemmas that I have.
Key questions that apply to our work with the = students in the classroom as well as with the parents are: what kinds of = mathematics can we extract from their experiences and practices? and how = can we relate them in a truthful manner to the content and the ways of = school mathematics? How can we develop learning situations that while = capitalizing on their experiences outside school allow them to "advance" = in their academic mathematics? Extracting the mathematics from the = everyday practices can be particularly difficult -- in part due to our = views about mathematics but also to our knowledge of the subject. In = our project, one of the most successful curriculum innovations was = developed by a middle-school mathematics teacher who, in addition to an = extensive teaching experience at most age levels, has a very solid = background in mathematics (graduate level) and in civil engineering. = (Through an architectural theme, he embedded most of the required = curriculum in mathematics.)
Besides the mathematical content per se, our desire = is to develop learning environments in which students engage in = mathematical inquiry. How can we ensure that as the discourse moves = towards "academic" mathematics, we do not fall back into the old = patterns of participation that seem to bring into the conversation only = a few students (usually always the same)? We are becoming quite good at = giving tasks that allow for multiple points of entry, but are we = allowing multiple points of exit?
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