“Quaderni di Ricerca in Didattica (Mathematics)”, n. 22 Supplemento n.1, 2012
G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy)
Learning Mathematics As A “Practically Able” Learner: An Instance Of Institutional Denial Of Access Hauke Straehler-Pohl and Uwe Gellert Freie Universität Berlin, Germany E-mail:
[email protected],
[email protected] Abstract: The article focuses on the kind of mathematical knowledge provided to underprivileged learners of mathematics in lower streams within the German school system and, respectively, on the enacted pedagogy. We argue that the students ‘mathematical experience’ in these contexts results in both a mathematical and a social displacement. Résumé : C’est article se concentre sur le genre de connaissances mathématiques qui est fourni aux apprenants des mathématiques défavorisés dans le type d’enseignement inférieur du système scolaire allemand et, respectivement, sur la pédagogie édictée. Nous argumentons que des « expériences mathématiques » des élèves dans ce contexte mènent au déplacement mathématique et social.
Aim of the study The theme of CIEAEM 63 acknowledges the growing interest in the issue of equity in the teaching of mathematics and equity through the teaching of mathematics. This interest has a long tradition. For instance, in the introductory part to the report on the Theme Group I “Mathematics for All” at ICME 5 (Damerow et al., 1984), at-that-time “new and urgent questions have been raised” (p. 3) about what kind of mathematics curriculum is adequate to the needs of the majority and what modifications are needed for special groups of learners. In that volume it has been argued that the influence of social and cultural conditions, the influence of the organisational structure of the school system, and the influence of classroom practice and classroom interaction constitute three very different dimensions that need to be taken into account when studying the issue of unequal distribution of mathematical education. These dimensions often tend to translate into cultural selectivity, selectivity of the school system and selectivity in classroom interaction (Damerow et al., 1984, pp. 8-10). Our knowledge about the multiple ways in which learners of mathematics might face inequity and selectivity has been substantially developed since the ICME 5 report and continues growing. Particularly, there is a still small but growing body of research investigating how the selectivity of the school system translates into selectivity in classroom interaction or how selectivity in classroom interaction reinforces the selectivity of the school system (e.g., Atweh, Bleicher & Cooper, 1998; Hoadley, 2007; Boaler & Wiliam, 2001). In this paper, we want to add to this growing body of knowledge. As the research quoted above shows, any discussion of issues of educational inequality, access to valued forms of knowledge and social justice needs to make its particular social and political context explicit. Our context is the strictly streamed German school systems and our focus is on those children and youth that are underprivileged in multiple ways. We investigate: •
What kind of mathematical knowledge is provided to underprivileged learners of mathematics in the lower streams of the German school system?
•
What kind of pedagogy is enacted in the classroom for underprivileged learners of mathematics?
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G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy) The social and political context In Germany, a differential distribution of access to different forms of knowledge is deeply rooted in the state’s official educational agenda. In general, the German school system separates students into different forms of schools at the age of 10 (12 in the special case of Berlin and surrounding area), according to their ‘ability’ and performance, in order to create more homogeneous learning groups. While this mechanism was established in the 19th century to explicitly separate social classes (Rösner, 2008), the system has survived until today, with just a change in its justification: “Each child shall be provided access – without taking account for the social class or income of its parents – to the type of school, which corresponds to its educational ability“. (Standing Conference of the Ministers of Education and Cultural Affairs of the Länder in the Federal Republic, 2003) “Educational ability” is a concept that has never been defined officially. In the political and public discourse it is mostly connected to the concept of intelligence and to the associated belief in practical and theoretical forms of innate ability (Rösner, 2008). Those who seem to be open to theoretical knowledge are directed to the Gymnasium (and, after that, to university studies). Those who seem to be more practically able are directed to schools which combine a provision of general education with an introduction to the values and particularities of the blue- or white-collar job market. As a matter of fact, professions based on university studies and white- and blue-collar jobs are differentially valued in society. As a consequence, learners from privileged milieus tend to gather at the Gymnasium; and the more underprivileged the learners, the more is their “educational ability” considered practical. Methodology As we are particularly interested in the mathematical education of underprivileged learners, our field of study are those schools that are located in particular inner-city districts. The social structure of these districts is characterised by several social indicators: low incomes, high percentages of social welfare receivers, no academic backgrounds, many children with German as their second language, high immigration rate (and immigrants mostly without higher formal education). Schools in these areas have been labelled schools with a “difficult milieu” (Baumert, Stanat & Watermann, 2006). In this article we focus on one group of learners and their two mathematics teachers (teacher1 and teacher2) in one of these ‘schools with a difficult milieu’. We videotaped completely the first three weeks of mathematical lessons in a secondary ‘school with difficult milieu’ (grade 7, students are 12 to 14 years old). For the learners, it is the first year in this school. Our assumption is that these first three weeks are particularly important as the teacher and the students negotiate the way classroom life is to be organised. We interviewed the responsible teacher extensively after the videotaping. Data analysis follows ‘key incident analysis’ and ‘international triangulation’ (Kroon & Sturm, 2000). This methodology is used for comparative case study research in an empirical-interpretative perspective. A key incident is key in that it represents “concrete instances of the working of abstract principles of social organization” (Wilcox, 1980, p. 9). When key incidents are presented in papers, they often are displayed in form of vignettes: “The vignette is a[n] (…) elaborated, literarily polished version of the account found in the [data]. (…) [It] does not represent the original event itself; (…) [it] is an abstraction; an analytical caricature (of a friendly sort)” (Erickson, 1986, p. 150). International triangulation might aim at, first, accumulating perspectives which make the familiar strange and, second, looking for structure in diversity. In our case, we triangulate classroom data from Canada, Germany, Spain and Sweden. A vignette The observed mathematics teacher describes the school’s environment as “ghetto-ised” with a mainly Turkish and Arabic infrastructure. Her students are mostly of Sinti and Roma origin and according to her “absolutely licentious … absolutely. … They do whatever they want, they come whenever they 196
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G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy) want to and they leave whenever they want to.” The first three weeks of the school-year have been spent explicitly and solely on classroom management. In a test on basic arithmetic facts, written during the three weeks of classroom management only, none of the students exceeded a “Fail”. Hence the two mathematics teachers decided to spend the first four weeks of mathematics class on a revision of basic operations. This decision seems to be a sensible and justified response; we are going to discuss the ways in which the revision of basic operations is didactically and methodically organised. We show how the students’ mathematical activities are constantly framed in a way that is not appropriate in terms of their age and the “practical” orientation of this school form. Vignette: “Reference to practical activities: Whose activities?” “So, uhm, subtraction, addition, multiplication, devision, that are the four basic operations, that you all have learnt … for years … in primary school.” In this way, teacher1 introduces the next mathematical activity. Although having been learnt “for years”, the teachers doubt that all students have mastered these basic operations: “So, who of you has still really, really big problems with subtraction?” No student reacts, nonetheless teacher1 hands out a worksheet “for those who still have problems” (Fig. 1) to all students. While Miranda passes out the sheet, a girl shouts out “that’s primary”. After handing out a copy to the researcher, Miranda reaches teacher2 and wants to give her a sheet as well, but the teacher refuses: “No, thank you, I already can do this.” After working a little while, Mirhiban asks “can we paint it in one colour only?” and teacher1 negates “uhm, no, look, it says what you are supposed to do”, and then takes a look at the sheet. Though it is impossible for her eyes Figure 1: worksheet to find this detail in the assignment, it still remains a demand for the activity. Mirhiban quotes the instruction “colour out” and Jolanda now joins resistance “I am not colouring this, I am not an Opfer40.” Jolanda is a student who is often in conflict with the social norms of the classroom. However, she seems to meet the mathematical requirements quite well in this scene. After working on the problems for six minutes Jolanda borrows a crayon of one of her peers and addresses publicly: “but I paint everything in just one colour.” Teacher1 responds immediately: “No, you are not doing that and if you do so, you should not wonder about getting a bad mark in the end. I just want that you – for only one time - do it properly and without painting over the lines.” Discussing the vignette This example shows one of a few activities found in that classroom that were of a “practical” nature. Right after the vignette stops, the teacher approached the researcher and commented on her “practical” aims of that activity. She and her colleague decided to start an “interdisciplary approach” including language and fine arts to meet the rather “practical” requirements of their students’ everyday life. They considered one of these requirements to precisely read and follow instructions. As the dispute about the number of colours required to complete the task illustrates, this ends in an almost absurd situation where the requirement is to follow spontaneous accounts that are not derived from the activity itself, but from the struggle to maintain hierarchical positions. Hence the “practical” competence to be
40
Opfer is the German word for “victim”. In areas of “difficult milieu” it is very commonly used amongst teenagers and children. “Du Opfer” is expressing a despiteous subordination of the other. It is assumed that this swearword came up in muslim dominated peer-groups and then got established in youth talk in general.
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G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy) acquired here is to take and follow demands without questioning. The students are positioned as objects and prepared for the carrying out of non-reflected and even pointless routine activities. The kind of “practicality” of that activity is highly problematic. “Colouring out” is an activity that is borrowed from early learners pedagogy. In the first years of primary school, teachers use an everyday and supposedly joyful activity of younger children in order to motivate a self-checking of mathematical solutions. Whether younger learners really get motivated by colouring out or not: for seventh-graders the effect can be described as infantilisation. Not only the mathematical requirements are constructed as those of primary students, but also the student’s “practical” activity. Taking into account that school is meant to reflect not only its students’ present status, but shall be oriented towards their future life this communicates a rather sad and hopeless message of what society expects of them. The students are not only – even explicitly – constructed as on the level of primary students by the teacher, but some seem to recognize and try to resist this positioning, as Jolanda expresses, when she denies to participate in that activity. For her, being a primary school child is equivalent to being an Opfer, which is not surprising taking into account the “practical” everyday life in the borough. Discussion The classroom observed is not representative. It is a special case where both institutional segregation by a streaming system and spatial segregation of social disadvantaged boroughs meet and accumulate. However, as Baumert et al. (2006) state, 60 % of all low-streaming schools in Berlin can be considered as ‘schools with a difficult milieu’. The pedagogy enacted is not just a case of unfortunate teaching but a result of almost 30 years of work in an institution that fails to meet students’ realistic needs. In the teacher interview, teacher1 showed awareness of the inadequateness of classroom practice for both mathematical and “practical” requirements. She formulated that over time she has found her role rather in being some kind of substitutive parent - an adult who listens and cares but also calls for rules. In this combination of pre-assumptions and apparent confirmation (e.g. the test results, disrespectful behaviour) and often contradictory stances of curricular “practicality” and realistic “practicality”, a pedagogy emerges that – in order not to overcharge – infantilizes students and – in order to enable classroom management – objectifies students. The mathematics involved is subordinated and restricted by this pedagogy. Here, mathematics has neither the potential to elaborate students’ thinking nor to empower their “practical” life. For the students, the mathematics class leads to an experience of both mathematical and social displacement. Learning in such mathematics classrooms adds to the underprivileged conditions that these learners face. Acknowledgement This study has been facilitated by a grant from Alexander von Humboldt Foundation. References Atweh, B., Bleicher, R.E., Cooper, T.J. (1998). The construction of the social context of mathematics classrooms: a sociolinguistic analysis. Journal of Research in Mathematics Education, 29(1), 63-82. Baumert, J., Stanat, P., Watermann, R. (Eds.) (2006). Herkunftsbedingte Disparitäten im Deutschen Schulwesen. Vertiefende Analysen im Rahmen von PISA 2000. Wiesbaden: Verlag für Sozialwissenschaften. Boaler, J., & Wiliam, D. (2001).’We’ve still got to learn!’ Students’perspectives on ability grouping and mathematics achievement. In P. Gates (Ed.), Issues in mathematics teaching (pp. 77-92). London: RoutledgeFalmer. 198
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G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy) Damerow, P., Dunkley, M.E., Nebres, B.F., Werry, B. (Eds.) (1984). Mathematics for All. Paris: Unesco. Erickson, F. (1986). Qualitative methods in research on teaching. In M.C. Wittrock (Ed.), Handbook of Research on Teaching (3rd ed.) (pp. 119-161). New York: Macmillan. Hoadley, U. (2007). The reproduction of social class inequalities through mathematics pedagogies in South African primary schools. Journal of Curriculum Studies, 39(6), 679-706. Kroon, S., & Sturm, J. (2000). Comparative case study research in education: methodological issues in an empirical-interpretative perspective. Zeitschrift für Erziehungswissenschaft, 3(4), 559-576. Rösner, E. (2008). Hauptschule am Ende. Ein Nachruf. Münster: Waxmann. Standing Conference of the Ministers of Education and Cultural Affairs of the Länder in the Federal Republic (2003). Übergang von der Grundschule in die Schulen des Sekundarbereichs I. Informationsunterlage (II A 1 / Fu – 2411). Bonn: KMK. Wilcox, K. (1980). The Ethnography of Schooling: Implications for Educational Policy-Making. Stanford (mimeo ED 199 809).
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