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] 8 ! L | F e! L $ $ $ $ & & & r| t| t| t| t| t| t| $ 9 ہ . | & & & & & | i i $ $ A | C C C & i 8 $ $ Fy , C & r| C C n M N , $ œ˹ 9 n N 2y | 0 | 0N h z> @ N N & N H* & & C & & & & & | | FC & & & | & & & & & & & & & & & & & : STRUCTURE AND CLOSURE OF SCHOOL MATHEMATICAL PRACTICE - THE EXPERIENCES OF KRISTINA
Ann-Sofi Rj-Lindberg
Abo Akademi University
Aroj(at)abo.fi
In the paper I present some preliminary results from a story-telling case study, the main focus of which is to give an account of the school mathematical practice at one particular Finland-Swedish lower secondary school. The school mathematical practice is looked upon through the voice of the student Kristina. Kristina told me about her experiences of participation in school mathematics in several qualitative interviews during the three school years 7, 8 and 9. The analysis of the interviews has so far resulted in the creation of several preliminary themes indicating her experiences. In this paper I limit the presentation to one theme, Structure and Closure. The paper meets the requests for more studies where researches and teachers listen to the voices of students in order to understand and learn from their school mathematical experiences and expectations (see e.g. Burton, 1994; Corbett & Wilson, 1995).
School mathematics as an individually experienced socio-cultural practice
Many researchers within the field of mathematics education nowadays accept a relational view of learning within which the knowing and identity of persons can be seen as social constructions and as results of the persons experiences in and contributions to social and cultural contexts. The processes of creating, teaching and learning mathematics and the outcomes of the processes can from this viewpoint be considered as social, cultural and interdependent phenomena ADDIN EN.CITE Nunes1995417Nunes, T1995Cultural practices and the conception of individual differences: Theoretical and empirical considerationsGoodnow, J.JMiller, P. JKessel, FCultural practices as contexts for developmentSan FransiscoJossey-Bass91-103Lerman2000487Lerman, S2000The social turn in mathematics education researchBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing19-44Atweh20014see e.g. 9Bill AtwehHelen ForgaszBen Nebres2001Sociocultural research on mathematics educationLondonLawrence Erlbaum(see e.g. Atweh, Forgasz, & Nebres, 2001; Lerman, 2000; Nunes, 1995). Hence, school mathematical practice as well as mathematics as a scientific subject are seen as existing within social systems of thought and culture ADDIN EN.CITE Ernest199851Paul Ernest1998Social constructivism as a philosophy of mathematicsAlbanyState University of New York Press(Ernest, 1998).
In this paper the level of analysis is the school mathematical practice in a particular school where some mathematics teachers intended to change their teaching practices with the support of colleagues. Elsewhere I have pointed at constraints the teachers experienced in trying to free themselves from the safety of habitual social and cognitive patterns (Rj-Lindberg, 2003, 2006). Practice is conceptualized in different ways by different researchers within social constructivist and socio-cultural perspectives. Paul Cobb defines practice as an emergent aspect of the communication in the classroom. Within a lesson design that focuses negotiation in communicative interactions Cobb is researching how students mathematical conceptions emerge and become established as a taken-as-shared mathematical practice in the classroom. ADDIN EN.CITE Cobb2000517Cobb, P2000Constructivism in social contextL. P. SteffeP. W. ThompsonRadical constructivism in action. Building on the pioneering work of Ernst von GlasersfeldLondonRoutledgeFalmer(Cobb, 2000). Jeppe Skott describes classrooms as communities of mathematical practice and discusses practice as an emergent phenomena influenced by the teachers school mathematical images, which are unique for each teacher. School mathematical images are in Skotts study defined as teachers personal interpretations of and priorities in relation to mathematics, mathematics as a school subject and the teaching and learning of mathematics in schools. Skott proposes that the school mathematical images contributes to the development of students learning opportunities ADDIN EN.CITE Skott20001342Skott, J2000The images and practice of mathematics teachersDepartment of MathematicsThe Royal Danish School of Educational Studies213Ph. D. Diss.(Skott, 2000). Marilyn Goos and colleagues also discuss the mathematics classroom as a community of practice, but with a focus on interactions of teachers and students to describe the social conduct in a collaborative classroom. Taken together these interactions reveal the emerging classroom culture and can be taken as indications of whether the mathematical practice in the classroom makes sense to students. Goos and her colleagues further argue that both the teachers and the students experiences of schooling can act as potential barriers to reform of school mathematical practice ADDIN EN.CITE Goos1999457Goos, MGalbraith, PRenshaw, P1999Establishing a community of practice in a secondary mathematics classroomBurton, LLearning mathematics. From hierarchies to networksLondonFalmer Press36-61(Goos, Galbraith, & Renshaw, 1999). In researching students goals Simon Goodchild approaches practice as the context of routine tasks that students are engaged in a mathematics classroom ADDIN EN.CITE Goodchild2001321Goodchild, S2001Students' goals. A case study of activity in a mathematics classroomOlsoCaspar Forlag(Goodchild, 2001). From a philosophical viewpoint Anna Sfard discusses meta-rules as her lens into mathematical practice and uses Wittgenstein to illuminate her points of view. According to Wittgenstein the person who follows a rule has been trained to react in a given way. Through this training the person learns to respond in conventional ways and thus enters into practice ADDIN EN.CITE Sfard2000170Sfard, A2000On reform movement and the limits of mathematical discourseMathematical thinking and learning23157-189http://web.ebscohost.com/ehost/detail?vid=5&hid=103&sid=ebc15d72-b7a6-432d-8f71-cda19ad6de53%40sessionmgr103(Sfard, 2000). Miller and Goodnow propose an epistemological definition of practice as actions that are repeated, shared with others in a social group and invested with normative expectations and with meanings or significance that go beyond the immediate goals of the action ADDIN EN.CITE Miller199539, 77Miller, P. JGoodnow, J. J1995Cultural practices: Toward an integration of culture and developmentGoodnow, J.JMiller, P. JKessel, FCultural practices as contexts for developmentSan FransiscoJossey-Bass5-16(Miller & Goodnow, 1995, 7). I find their definition of practice as satisfactory if action is considered to comprise both cognitive and socio-cultural activities and meaning as referring to individually experienced as well as socio-culturally constructed and valued meanings.
The word practice is used with different connotation in common language and it is difficult to avoid a certain ambiguity in its use also in this paper. I sometimes refer to practice as the actions of teacher and students in the classroom. Admittedly, most of the individually experienced and socially valued meanings of school mathematics are related to actions in the classroom. However, when I use school mathematical practice my intention is to cast the conceptual net more widely and include as well the images and pedagogical intentions of the mathematics teachers as a social group within the collaborative reform work.
My main focus in the paper is on the student Kristina and her experienced meanings of school mathematical practice during lower secondary school. A basic assumption is that the story she tells in interviews is the meaning she imposes on her world. Her story is the truths of her experiences, not an objective reflection of an ontologically real reality ADDIN EN.CITE von Glasersfeld1991729von Glasersfeld, E1991Radical constructivism in mathematics educationLondonKluwer(von Glasersfeld, 1991). Thus, it is important to note that her experienced meanings might differ from meanings constructed by other students and by the teachers who were participants in the same mathematics classroom. For example, Ben-Chaim, Fresko and Carmeli ADDIN EN.CITE Ben-Chaim1990310Ben-Chaim, DFresko, BCarmeli, M1990Comparison of teacher and pupil perceptions of the learning environment in mathematics classesEducationals Studies n Mathematics21415-429(1990) write as one conclusion from a survey to teachers and students that the teachers saw the classroom environment as more diverse than the students. Another remark of importance is the use of the concepts knowing and knowledge. Knowing refers to personal meanings of socially and culturally constructed knowledge, like for instance of mathematics in textbooks, of activities in school or of what it is to do mathematics and be a learner in a classroom.
Theoretical considerations
In the theoretical frame I am inspired by socio-cultural perspectives on human learning and development and a cultural-psychological view on the human mind ADDIN EN.CITE Rogoff2003201Rogoff, B2003The cultural nature of human developmentNew YorkOxford University PressRogoff199327Rogoff, B1993Children's guided participation and participatory appropriation in sociocultural activityWozniak, R HFisher, K WDevelopment in context: Acting and thinking in specific environmentsHillsdaleErlbaum121-153Rogoff1994220Rogoff, B1994Developing understandig of the idea of community of learnersMind, Culture, and Activity14209-229Bruner1996/2002101J Bruner1996/2002Kulturens väv. Utbildning i kulturpsykologisk belysningGöteborgDaidalosBruner1990291Bruner, J1990Acts of meaningLondonHarvard University Press(Bruner, 1990, 1996/2002; Rogoff, 2003).
Within the socio-cultural perspective that I adapt, learning is participation and development is a function of ongoing transformation of roles and understandings in the socio-cultural activities of the communities in which the indivi dual par ticipates. I argue that the students and teachers within the school mathe matical practice that is the focus of my study can be interpreted as forming developing com munities of learners in this sense. An essential theoretical argument in this perspective is that culture, including mathematics and school mathematical practice, is seen neither as something static that adds on or is gradually internalized by the indivi dual mind nor as a surrounding to the individual which is gradually changed to fit the developing mind. Culture is the common ways that participants in a community share even if they may contest them ADDIN EN.CITE Rogoff1994220Rogoff, B1994Developing understandig of the idea of community of learnersMind, Culture, and Activity14209-229Rogoff2003201Rogoff, B2003The cultural nature of human developmentNew YorkOxford University Press(Rogoff, 2003). From a cultural-psychological viewpoint Bruner argues that we should think of culture as something that is in the mind of persons ADDIN EN.CITE Bruner1996/200210, 2001J Bruner1996/2002Kulturens väv. Utbildning i kulturpsykologisk belysningGöteborgDaidalos(Bruner, 1996/2002, 200).
This perspective on learning and development is radically different from one that describes learning as transmission of knowledge from authorities outside the indivi dual or as acquisition or discovery of knowledge by the individual. More over, it challenges the idea of a boundary between internal and external pheno mena, as for example between a students knowing of school mathematical prac tice and the cultural tools used in this practice. Among cultural tools Roger Slj in clu des intellectual tools like systems of ideas and discourses and physical tools like textbooks, mathematical symbols and diagrams ADDIN EN.CITE Säljö2005211Säljö, R2005Lärande & kulturella redskap. Om lärprocesser och det kollektiva minnetFalunNorstedts Akademiska Förlag(Slj, 2005).
If this perspective is accepted, a researcher may place an individual students knowing or the school mathematical practice in the foreground without assuming that they are actually separate elements. It also makes sense to discuss an individuals school mathematical experiences in terms of situated knowing or situated understanding and to acknowledge the importance of context for the development of these experiences. The research studies by for instance Boaler ADDIN EN.CITE Boaler1997251Boaler, J1997Experiencing school mathematics. Teaching styles, sex and settingBuckinghamOpen University Press(1997), Stigler and Hiebert ADDIN EN.CITE Stigler1999141Stigler, J WHiebert, J1999The teaching gapNew YorkThe Free Press(1999), Boaler and Greeno ADDIN EN.CITE Boaler2000227Boaler, JGreeno, J G2000Identity, agency and knowing in mathematics worldsBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing171-200(2000) and Nardi and Steward ADDIN EN.CITE Nardi200380Nardi, ESteward, S2003Is mathematics T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroomBritish Educational Research Journal293(2003) are well grounded examples of the situated nature of school ma the matical experiences: it is impossible to separate the knowing of indivi duals from the social and cultural practices in which the individual participates.
Barbara Rogoff ADDIN EN.CITE Rogoff199327Rogoff, B1993Children's guided participation and participatory appropriation in sociocultural activityWozniak, R HFisher, K WDevelopment in context: Acting and thinking in specific environmentsHillsdaleErlbaum121-153(2003) uses the concept participatory appropriation instead of interna lization or appropriation to refer to the change and development in knowing resulting from a persons guided participation in an activity. As people participate they are making a process their own, they are not taking something from other persons like you take a thing and add it on to something you already possess. But they are creatively taking for their own use and changing how they may treat future situations that they see as related. Processes of guided participation are defined to incorporate guidance in the sense of direction of a shared endeavour. It is participation in meaning that is the key issue, not necessarily in shared actions of the moment. A person who is observing and following without direct contribution to the decisions made by other people is also participating in the activity. Moreover, a person who acts alone is also participating in a shared endeavour as he or she follows and builds on community traditions for the activity. For instance, a student doing homework is participating with guidance provided as well by the teacher and textbook writers, who may have set the tasks and approaches to be used, as by classmates, and family members, who may support some approaches and suppress others. Also, a mathematician like Andrew Wiles, who grappled eight years with proving the Last Theorem of Fermat and without direct involvement from peers, was building on socially validated traditions for mathematical activities.
Within a theory of development as transformation of participation people are seen as contributing to the creation of socio-cultural processes and socio-cultural processes as contributing to the creation of people ADDIN EN.CITE Rogoff200320, 511Rogoff, B2003The cultural nature of human developmentNew YorkOxford University Press(Rogoff, 2003, 51). An example of how this kind of mutuality and interdependence might operate within school mathematical practices can for example be seen in what homework means to a student. The students might perhaps imagine it as a questioning activity in the classroom where teacher wants me to show that I know the correct answer. Then by displaying the correct answer, she is participating in the creation of this activity as one where the student is expected to give the one answer that is correct. When the teacher evaluates the answer as a good answer the teacher fulfils her expectations and they both contribute to the emerging implicit agreements about homework actions, about responsibilities for actions, about when to do what and how to do it in this particular classroom. In this process the students and the teacher together create normative expectations for the kind of mathematical competence valued, what it means to be a good student and an effective teacher in relation to homework actions. Or the student might imagine homework as questioning activity where teacher wants me to argue mathematically for my answers. If this is the case a homework review in the classroom is an activity constituted by a very different participation structure with different expectations and implicit agreements. At the level of formal engagement the student may be doing the same thing in both examples, i.e. answering questions set by a teacher about homework. These examples indicate that the socially valued meaning of homework review might be constructed very differently in different school mathematical practices. They also indicate that a participation perspective on learning and development requires considerations of not only what it is that the individual is participating in but also the experienced meanings of these activities.
Jerome Bruner states that a culturally sensitive psychology should be based not only upon what people say caused them to do what they did. It is also concerned with what people say others did and why. And above all, it is concerned with what people say their worlds are like ADDIN EN.CITE Bruner199029, 161Bruner, J1990Acts of meaningLondonHarvard University Press(Bruner, 1990, 16). Bruner further states that cultural psychology seeks out the rules that human beings bring to bear in creating meanings in cultural contexts of practice (p. 118). Thus, I argue, it becomes important to foreground school mathematical practices as seen through the eyes of the students. Moreover, this research perspective is needed if students are considered as legitimate participants in, not only as beneficiaries of school mathematical practice ADDIN EN.CITE Corbett1995330Corbett, DWilson, B1995Make a difference with, not for, students: A plea to researchers and reformersEducational ResearcherJune/July12-17(Corbett & Wilson, 1995).
Methodological considerations
Methodology is interpreted by Wellington as the activity when the researcher chooses, reflects upon, evaluates and justifies the research methods she uses to answer the research questions ADDIN EN.CITE Wellington200026, 221Wellington, J2000Educational researchLondonContinuum(Wellington, 2000, 22). As already indicated the topic of this paper has emerged from a case study which involved a group of mathematics teachers who participated in reform work to develop their mathematics teaching. I took part in the process as research-assistant within a commissioned research work including a complex set of tasks, among them to bring the voices of students into the reform work. I approached the research work from the viewpoint of being a mathematics teacher myself and I used a mixture of research methods to build up a case record. I got to know the teaching traditions and the teachers pedagogical intentions from interviews with teachers and students, from surveys and from being a participant as well as a non-participant observer and note-taker at action research meetings and observer of lessons. It is thus obvious that I cannot act as an outsider detached from the story of school mathematical experiences that is the focus of this paper. As the research process started with a very open bottom-up approach it is natural that a retrospective format is used for reporting the research. Bruner ADDIN EN.CITE Bruner199029, 1191Bruner, J1990Acts of meaningLondonHarvard University Press(1990, 119) considers retrospective reporting as viable when personal meanings of activities is the focus for research inquiry.
The interest in gaining insights to the views of students was developed into more specific research questions of which I in this paper will discuss the following:
How does Kristina experience her participation in school mathematical practice during school years 7, 8 and 9?
To answer the research question I will rely on interpretations of interviews as my main method to provide evidence. The interview format offered possibilities to gather students accounts of their experiences of school mathematical practice, and make inquiries into their values, feelings, expectations and views, issues that are difficult to reach with other methods, like classroom observations or surveys. The research interview gave the students a voice and provided them with a platform and a chance to make their viewpoints heard and read ADDIN EN.CITE Wellington2000261Wellington, J2000Educational researchLondonContinuum(Wellington, 2000). With a psychological constructivist view of knowing in mind I conjectured that every student would conceive the school mathematical practice differently and thus each one of them would have a unique story to tell. The interview format seemed as a reasonable way for me to monitor the relationships between the students experiences of school mathematical practice and their emerging meanings of these experiences. As interviewing all students was impossible, and to be assured of variation in perspectives, I chose four to five students as key informants from each reform class. As instruments for selection of students I used a test for mathematical achievement and a self-concept questionnaire (see Linnanmki, 2002 for a discussion of these instruments). The key informants represent as big variation in level of mathematical achievement and self-concept as possible. Kristina is one of these key informants.
Semi structured interviews with Kristina were conducted five times: in September and December school year 7, in December school year 8 and in January and May school year 9. The interviews with her comprise a total of 2 hours and 26 minutes transcribed into appro ximately 45 pages of text (12pt, sp 1). With the double aim of both vali dat ing findings from the lower secondary interviews and shedding more light upon Kristinas relation to school mathematical practice, I had a conversation with Kristina where she as a grown up looked back on her school mathematical expe riences. Implications from this interview are included in the final coda of the paper
The theme and aspects discussed in the next section have emerged out of an open coding approach with attention both to theories regarding school mathe ma tical practice in literature and to new aspects emerging out of repeated listening to the interviews and reading of the transcripts ADDIN EN.CITE Bryman2002111Bryman, A2002Samhällsvetenskapliga metoderStockholmLiber(Bryman, 2002). The transcripts were imported into a computer program for qualitative research (NVivo) which I used as a help when I wanted to focus particular aspects of the emer ging themes. Reading research literature in parallel with coding made me more theoretically sensitive, at the same time I tried to stay as close to the voice of the students as possible by reading and listening to the interviews over and over again. The ana lytic approach I take falls within the interpretivist tradition. The interpretivist researchers aim can be to explore peoples perspectives and to develop insight in situations, e.g. schools and classrooms ADDIN EN.CITE Wellington200026, 161Wellington, J2000Educational researchLondonContinuum(Wellington, 2000, 16). Next I describe the contextual frame which is essential for the readers emerging understanding of Kristinas school mathematical experiences ADDIN EN.CITE Lincoln198527, 3601Lincoln, Y SGuba, E G1985The naturalistic inquiryLondonSage(Lincoln & Guba, 1985, 360).
The context for Kristinas story
The very same autumn as Kristina came to her first day at lower secondary school five mathematics teachers at the school committed themselves to reform work and to a redesign of their pedagogy. The format for the reform work was inspired by collaborative action research ADDIN EN.CITE Elliott19915see e.g. 1Elliott, J1991Action research for educational changeMilton Keynes: Open University PressRöj-Lindberg200629 see also 7Röj-Lindberg, A-S2006”Jag satt fast i mönster” – Metaforer i lärares berättelser om matematikundervisning i förändringHäggblom, LBurman, LRöj-Lindberg, A-SPerspektiv på kunskapens och lärandets villkorVasaPedagogiska fakulteten vid Åbo Akademi113-124(see e.g. Elliott, 1991; see also Rj-Lindberg, 2003, 2006). The teachers came together regularly to discuss and evaluate aspects of their pedagogical approaches. The driving force for the reform was of both a theoretical and a pragmatic nature.
Firstly, constructivist theories of learning had during early 1990ies gained a strong foothold in pedagogy and in the theoretical rhetoric of reform in mathematics education ADDIN EN.CITE Björkqvist1993470Ole Björkqvist1993Social konstruktivism som grund för matematikundervisningNordisk Matematikkdidaktikk118-17Kupari19931289Kupari, PHaapasalo, L1993Constructivist and curriculum issues in school mathematics education. Yearbook 1992-1993.Publication Series B 82JyväskyläUniversity of Jyväskylä. Institute for Educational ResearchHansén199467Hansén, S-EMyrskog, G1994Läroplansreform och inlärningssyn.Wickman-Skult, A.Grundskolan förändras. Läroplansarbetet i blickpunktenHelsingforsUtbildningsstyrelsen(Bjrkqvist, 1993; Hansn & Myrskog, 1994; Kupari & Haapasalo, 1993). In their pedagogical intentions the teachers were strongly inspired by constructivism, which they also saw as a new theoretical norm for their teaching of mathematics. As persons representing the culture of mathematical knowledge in the classroom they felt troubled by the difficulties of communicating this knowledge to their students. Thus the rhetoric of each student constructing his or her own mathe matics felt exactly like the situation they met in their classrooms each day. The epistemological dilemma of talking about constructivism as if it was a teaching practice was of no concern at the time.
Secondly, they involved themselves in the reform work because they wanted to leave unproductive teaching patterns and create new traditions ADDIN EN.CITE Röj-Lindberg2006297Röj-Lindberg, A-S2006”Jag satt fast i mönster” – Metaforer i lärares berättelser om matematikundervisning i förändringHäggblom, LBurman, LRöj-Lindberg, A-SPerspektiv på kunskapens och lärandets villkorVasaPedagogiska fakulteten vid Åbo Akademi113-124(Rj-Lindberg, 2003, 2006). The statements of one of the teachers can serve as an example. In an interview half ways into the first year of reform this teacher describes how he had become more and more unsatisfied with the formalistic nature of school mathematics and with his own pedagogy. In his opinion school mathematical practice was too theoretical and too much of formulas bandied about. In many ways the teachers expressed a conviction of the superiority of more process-based and student-centred environments for the learning of mathematics. Statements similar to the pedagogical intentions articulated by the teachers were expressed in the national mathematics curriculum which stated that in comprehensive school such learning situations should be organized where you discuss, experiment and as often as possible solve problems that originates from the students own everyday experiences and that problem solving and the internal logic of mathematics are the most important principles for mathematics teaching. ADDIN EN.CITE 19943, my translation7Alexandersson, M1994Den fenomenografiske forskningsansatsens fokusStarrin, B Svensson, PGKvalitativ metod och vetenskapsteoriLundStudentlitteratur111-136(1994, my translation)
Kristina was all three years of lower secondary schooling in a group of 15 students, 8 girls and 7 boys. The school mathematical practice was tuition in this mixed-ability group up to the second half of school year 9. At the action-research meetings the group was referred to as a group of high-achievers. Out of the 15 students 11 were positioned as high-achievers in the achievement test used for selecting key informants. During the three school years the average test results of the group was usually above the average results in the other two reform groups. At the beginning of school year 7 the teacher describes how the students in the group willingly explain to each other and help each other. Especially boys classified by the teacher as smart are described as very active while the girls are described as silent and cautious There are students in the group that the teacher classifies as weak. The teacher complains about lack of time for individual students and that he doesnt manage to keep up with all students. As a solution some students are scaffolded with remedial teaching out of the classroom. The teacher describes pedagogy where he monitors the social interaction in plenary session by setting simple questions first and by trying to find positive aspects also in answers classified as wrong answers.
The main tools the teachers used for pedagogical planning was mainly the textbook but also the national curriculum and the schools own mathematics curriculum. The schools curriculum stated discovery of mathematical structures as the most important aim in relation to students with qualifications and personal interest to study mathematics. This striving for structural clarity is indicated in the curriculum as a norm for the mathematics teaching at all stages in school. Among the new learning activities the teachers introduced was project work as homework and different types of mathematical problems in combination with and a stronger focus on written explanations by the students mathematical work.
Textbook tasks and teacher made tasks were used in classroom activities on a regular basis. One type of teacher made task was the so-called mini-problems. These problems the students solved individually during lessons within 10-15 minutes and handed in to the teacher for grading. Mini-problems were in most frequent use in Kristinas class during school year 8.
Once or twice each term the students worked with project work as homework, mostly on an individual basis. During school year 9 the students responsibility for developing the mathematical content of the projects increased and the students were organized into smaller groups to cooperate in the planning and execution of the projects.
The written tests used for assessment of mathematical achievement in the reform groups consisted of both traditional tasks and less traditional tasks of multi-choice type. Special types of task introduced only in tests used in the reform groups were called explanation tasks. The process of doing explanation tasks demanded the students to explain in detail each step in the written solution. During year 9 the use of explanation tasks diminished.
During the autumn term in year 9 the students in the reform groups were regrouped and new groups were formed according to the students plans for his or her academic future after lower secondary school. Three groups of students were formed: students planning for vocational school, students planning for upper secondary school and a long course in mathematics, students planning for upper secondary school and a short course in mathematics.
Kristina
In all interviews Kristina showed a narrative zest and expressed her views easily. In comparison to interviews with the other key informants from the same year group the five interviews with Kristina were the longest. A comparison of the average amount of words spoken in the five interviews show that the interviews with her comprise 60% more words than an average of the other interviews. At the first interview she played a little bit of a waiting game, a usual reaction among the key informants. But already then her answers were more narrative and reflective than the answers of most of the other key informants. A general feature in all interviews is that her answers to my questions include statements and argumentative as well as emotional and spontaneous utterances.
Kristina began lower secondary school as a high-achieving student in mathematics but with a low self-concept in relation to mathematics. In the achievement test at beginning of school year 7 she belonged to the highest achieving third of the group. In the midst of high school she was evaluated by her teacher with the grade 9 in mathematics when 10 is the best possible. Her self-concept was also generally classified as good, disregarding the aspects in the survey that related specifically to mathematics and mathematics teaching. All students in the reform classes were included in a self-concept survey in the beginning of school year 7. Kristina participated in the same questionnaire in May school year 9. According to Bruner (1996/2002, 54) a persons self-concept is a question of empowerment and sense of agency: it originates in a sense of being able to initiate and carry out actions on ones own or by ones own effort. At the end of school year 9 Kristina is still distinguished by the same strong general self-concept. In fact she seems to identify even more positively with school practice than as a newcomer. But the renewed survey of her self-concept also indicated that her self-concept in relation to mathematics and mathematics teaching did not develop positively during lower secondary school. At the end of school year 9 it was at the same low level as at the beginning of school year 7.
In the first interview Kristina describes mathematics as a useful but dull school subject. After one and a half year in lower secondary school she again indicates a negative identification with school mathematics and states that mathematics has never been a favourite subject of hers. She occasionally compares her negative identification with school mathematics to that of her mother, but more often she relates her negative identification to sources within herself, to her insufficient personal competencies, to her lack of energy to listen actively enough to the teacher and to her deficient ability to concentrate. She even says that she most of all would like to avoid mathematics in school all together because it often is such a boring activity. But on the other hand she is focused on learning and tells about how important it is to know mathematics. In December school year 9 she declares: You cannot just think of those things that are amusing, mathematics is important you know. Kristinas negative identification with school mathematics has not emerged from any general resentment towards engagement in school mathematical practice. She does like to engage herself mathematically and she has, at least in the beginning of lower secondary school, confidence in the explanation that school mathematical knowledge is valuable in the long run. At the first interview she expresses a hope that engagement in school mathematical practice will provide her with a knowledge base that can support her during all her life. However, talking with her later on I am left with an expression that she has started to doubt the argument of lifelong usefulness as a sufficient motivational basis for school mathematics. At the third interview following she is more doubtful: I dont really know, everybody you ask, if you ask someone they say that it is important to know (mathematics) in professions, you have never got any real explanation, I dont know myself what I think, everybody just tells me it is for your profession you need it, and then I say so myself too. She seems to have developed awareness that the argumentation about mathematics as a useful subject, the learning of which gives her rewards in the future, may have the form of a persuasive ritual with an unclear connection to the school mathematical practice she has experienced so far. You could thus ask whether the argument of usefulness the teachers use over and over again is seen by Kristina as part of the learning process as a whole and which social function the argument is serving.
Kristina achieved good results in mathematics during all the three years in lower secondary school. Nevertheless, she occasionally figured that a good grade in a test situation was a surprise to her and expected it would fall as a result of next graded evaluation. Interestingly she showed more confidence in being able to get good grades from project work. The dilemma was that she considered project work as outside school mathematical practice. Moreover, she did not experience herself as ever becoming a member of a community where school mathematical knowing is playing any significant role. This is not unusual for successful students who relates negatively to mathematics in school ADDIN EN.CITE Boaler200022, 1847Boaler, JGreeno, J G2000Identity, agency and knowing in mathematics worldsBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing171-200(Boaler & Greeno, 2000, 184). During school year 9 she is unsure about her choice of school after lower secondary school and goes into the group for students planning for a short course in mathematics. She has applied for studies in upper secondary school but refers to insufficient competence and energy for any deeper studies in mathematics. She argues that there is a significant difference between those that understand mathematics and those that dont understand. She labels those students that plan for a long course in mathematics as fairly smart students and declares that she needs more time to think of what she wants to do as a grown-up. The change in grouping resulted in change in teacher twice during lower secondary school. Kristina experienced extended teaching periods with two teachers: slightly more than two school years with the first, some weeks with the second and about three-fourth of school year 9 with the third teacher. The changes in teacher obviously implied some changes in the classroom discourse, but behind the changes Kristina sees a structure, a pedagogical system, which from her viewpoint lies outside the boundaries of socially valued school mathematics. The design of mathematics teaching at school simply is something the students have to cope with, not something they can transform. If all students would say how they like their lessons, it would not serve any good end, I do think it is best that the teachers have like some own system they follow. In the last interview Kristina indicates clearly how she conceptualizes such a system.
Well, surely from the beginning it is rather like, if we start with something new, it shall be something rather simple. Or that you start from something simple, that he looks to that we all from the beginning, and we say how we think ourselves, then, while it is rather simple. Then it gets more and more difficult so that something small is added all the time, so that we then, when we have got that most simple, not that anything is really simple, but, in the simpler, that we can, that we have understood everything, so that from there, some little more can be added all the time, so then it is easier to understand those more difficult tasks. (Kristina, Int5)
Structure and closure one theme created from Kristinas story
The preliminary themes created so far out of Kristinas story can be organized under four headlines. The first, Structure and Closure, together with the second, Tempo and Linearity, capture Kristinas reports of aspects of the classroom as a teaching and learning environment. The third, Alienation and Marginality, focus on her emotional reactions and identifications toward participation and engagement in school mathematics practice. Under the fourth headline, Tensions and Dilemmas I note issues that penetrate the other three themes. Due to the paper format I limit myself to the five aspects of the classroom as a teaching and learning environment which comprise the theme Structure and Closure. All themes will be presented in my forthcoming doctoral thesis. The included extracts and utterances are chosen either because they typify Kristinas descriptions or views on different occasions within or across interviews, or because they stand out as significant, critical or contradictory in some sense.
Stability of the lesson design
At each interview Kristina described the pedagogical practice of recent lessons or of lessons she conceived as ordinary mathematics lessons. Mostly her description focused on what the teacher did and what the students did or were expected to do and was a response to a direct question like in the following extract from the first interview. Sometimes it was spontaneously included in a conversation about some related issue.
I: Can you tell me something about the last lesson?
Kristina: Do you mean the last one we had?
I: Yes. Was it an ordinary lesson?
Kristina: Yes, it was ordinary. What did we do? We learnt about powers. Isnt that the name of it? I think so. An ordinary lesson.
There are activities like project work or problem-solving tasks that occasionally break the routine but through all interviews she describes the ordinary lessons as following roughly the same pattern of activities. The activities Kristina describes are very similar to the traditional design for mathematics teaching at the lower secondary level ADDIN EN.CITE Bodin199619see 7Bodin, ACapponi, B1996Junior secondary school practicesBishop, A JClements, KKeitel, CKilpatrick, JLaborde, CInternational Handbook of Mathematics EducationDordrechtKluwerRöj-Lindberg1999301Röj-Lindberg, A-S1999Läromedel och undervisning i matematik på högstadiet. En kartläggning av läget i SvenskfinlandVasaSvenskfinlands läromedelscenterHiebert1999111Hiebert, JStigler, J. W.1999The teaching gapNew YorkThe Free Press(see Bodin & Capponi, 1996; Hiebert & Stigler, 1999; Rj-Lindberg, 1999) including review of homework or mathematical contents from earlier lessons; answering questions set by the teacher at plenary sessions; listening to teacher explaining mathematical methods and solution procedures at plenary sessions; writing down methods and solution procedures; working with exercises or problems set by teacher on an individual basis and teacher assigning homework for next lesson.
In the third interview she described an ordinary lesson in the following utterance:
We go through the homework, or first we come into class and then if we have had homework, we do it, or he just asks if there is something someone has not understood, then he can take it on the board, if someone has not understood you can copy it. Then, if we dont continue training the old stuff and calculate from the book, we usually take a new thing and write some points in the theory booklet about how it is all right to calculate it and then we can try ourselves or he says, he asks, if we are going to calculate, you can raise your hand if you have some good idea about how to calculate the exercise, and then if it is correct he writes it down. There can be several alternatives then and we write down all the alternatives in the theory booklet. Then we are supposed to solve some tasks. We usually get homework, if we dont have a project work going, but anyhow, next lesson we go through that homework. Or we have a written test on the homework. (Int3)
After a change of teacher in year 9 Kristina describes a move to a more accentuated discovery approach to teaching with more independent student work. It is less telling of answers to the whole class, more finding out yourself what is important, she says. But nevertheless Kristina doesnt hesitate when asserting that as she has got used to the new teacher she has noticed that it is pretty much the same. From her descriptions of classroom practice it is obvious the lesson design includes similar features also after the change of teacher in grade 9.
Teacher does a task on the board, it is of course like to continue what we had before, and then he asks us to find a solution by ourselves or asks how we could do to calculate it. Like think forward from what we have had before and then we can go through some examples on the board and write something in the theory booklet or exercise booklet and then we will do a review the next day and look up if we have understood (Int4)
The textbook and especially the theory booklet stand out as very important tools within the ordinary lessons. Kristina describes the theory booklet as a smaller book. This booklet parallels the textbook as both contain rules and examples, the difference is in authoring. Kristina herself is the author of the booklet even though she doesnt choose herself what to write in it and is left unsure why certain things are more important to write down than others. Kristina experiences a need to legitimize the content of the booklet by the authority of the teacher as can be seen in her utterance on p. 12. The theory booklet seems to act like a security system where she catches the mathematics she is supposed to learn and which she can return to between lessons, as she does homework or prepares herself to an assessment activity. She likes lessons where she is explicitly told which rules and examples that have to go into the booklet, when the teacher not only tells things, but you write it up in the theory booklet (Int3). The telling of important mathematics has, she thinks, to be accompanied by writing it up on the board in order to secure the chance of copying into the theory booklet. As the use of the theory booklet diminished during school year nine it meant a lack of sense of order to Kristina. You always had order in your theory booklet and you new exactly, mathematics was not easier, but you new where you were and that was a good thing (Int5).
Purpose of and division of responsibility within school mathematical practice
To Kristina learning mathematics in school is about learning rules and systems, what we are busy with in the theory booklet (Int3), things to remember, different formulas and ways of calculating (Int5).
In the theory booklet Kristina writes down all those things that the teacher during plenary session indicates to her as important. There can be different alternatives and then we write all the alternatives in the theory booklet (Int3). The indication of importance is mainly of three types: (1) the teacher is explicitly telling what is important write down this, this belongs to the theory; (2) the indication emerges in discovery plenary sessions where the teacher guides a dialogue with the students. Kristina describes the dialogue as a guessing game where the students can win by discovering the mathematical content of the teachers thoughts: everybody can guess and the teacher tells you if you are closer to or further away from the correct answer and then you write a rule (Int2). If the students can win this game without understanding the content and conclusions of the dialogue, it shows a remarkable resemblance with dialogues characterized by the so called Jourdain-effect. The Jourdain-effect is described by Brousseau as a situation where the teacher acts as if he recognized evidence of knowledge in an answer despite that the response may actually be motivated by very trivial causes and meanings ADDIN EN.CITE Brousseau198487Brousseau, G1984The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematicsH SteinerTheory of Mathematics Education, Occasional Paper 54BielefeldIDM110-119(Brousseau, 1984); (3) the indication of importance is established from patterns in particular examples and solution models that are discussed at plenary sessions and written on the board.
We write something on the board and then you have to yourself, yes, this is maybe important and the you have to write it ... not as much rules and systems, it is only examples, examples, examples, and then when you go through the theory booklet you have to figure out yourself how you have done it. And if we ask him he says: Yes, those belong to the theory. (Int4)
The division of responsibility between the teacher and Kristina within the ordinary lesson design is such that the teacher presents and explains procedures; she memorizes the procedures, retrieves the right procedure(s) for solving a task and shows that she knows how to perform the procedure to find the answer to the task. She relates learning school mathematics to a persons ability to memorize rules and to discover procedural similarities in tasks. She thus accepts it as natural that she shows the teacher how she solves tasks because the teacher has to know how you do it to be able to teach you in a manner that you understand. If Kristina knows how to do it it means that she is able to perform the correct steps in a solution process. But Kristina finds it difficult to grasp the mathematical logic behind these steps unless the teacher has given a clear explanation. On the one hand she emphasizes her need for careful and non-ambiguous explanations by the teacher to maintain some order in the growing amount of things to remember like rules, formulas, ways of calculating. On the other hand, she finds the acts of active listening, memorizing and rule following as demanding an ever-increasing level of concentration, self-discipline and acceptance of dullness and boredom. Kristinas earlier positive experiences of participating in school mathematical practice, because she then felt that she knew everything, develops during the school years into a negative experience where she, because of perceived personal shortcomings have a sense of not gaining access to the understanding she strives for. To begin with she secures her mathematical understanding relying both on her thinking and intuition and on the teachers explanations. Later she withdraws more and more from her own cognitive capabilities and shows a stronger dependency on outside resources; teacher, textbook, theory booklet, classmates. To learn the rules and systems of school mathematics she has to very be good at following and catching up, however, her learning goal is impeded by the rules and systems of the classroom interaction of which her own actions are a constitutive part.
/.../ I do also remember a time in primary school when I found mathematics to be super easy and then it was fun to be at lessons, because then you new everything and you was able to raise your hand when they asked something and there was not the same need to be alert all the time, to be, like, you could take it a little bit easy. Not that you should do it then, but it just gets like So then it is fun. But, now it is like, you have to listen all the time to what he is saying and you get so tired. Like, you cannot always go on any longer. Directly as you talk just a little bit with your seatmate then it goes, oh, now I dont understand anything again. It is rather strenuous. You should have a better ability to concentrate. (Int5)
Nature of classroom interaction
Social constructivist studies have shown the significance of social interaction for the meanings individuals develop in school mathematical practice ADDIN EN.CITE Cobb199235see e.g. 0Cobb, PWood, TYackel, EMcNeal, B1992Characteristics of classroom mathematics traditions: An interactional analysisAmerican Educational Research Journal293573-604Yackel20011077Yackel, E2001Explanation, justification and argumentationvan den Heuvel-PanhuizenProceedings of the 25th conference of the international group for the PMEUtrechtFreudenthal Institute19-24(see e.g. Cobb, Wood, Yackel, & McNeal, 1992; Yackel, 2001). On the other hand, student-teacher or student-student interaction in the classroom is no guarantee for interactions to be mathematical or for significant mathematical knowing to emerge as the student might appropriate other aspects of the mathematical practice than the teacher intended ADDIN EN.CITE Wood199877Wood, T1998Alternative patterns of communication in mathematics classes: Funneling or focusing?Steinbring, HBartolini Bussi, M GSierpinska, ALanguage and communication in the mathematics classroomRestonNCTM167-178Moschkovich200435see e.g. 0Moschkovich, J N2004Appropriating mathematical practices: A case study of learning to use and explore functions through interaction with a tutorEducational Studies in Mathematics5549-80Boaler1997251Boaler, J1997Experiencing school mathematics. Teaching styles, sex and settingBuckinghamOpen University Press(Boaler, 1997; see e.g. Moschkovich, 2004; Wood, 1998). Thus, one way to understand Kristinas experiences of participating in school mathematical practice is through her utterances about the social conduct and interaction in the classroom.
Kristina states that collaborative discussions in groups are rare within the ordinary lesson design. She further declares that the school mathematical practice is different from the practice in other school subjects in this regard.
// we never are in groups as in other school subjects or, like, whole-class discussions about what we shall calculate. You just ask those you sit next to, you dont do any greater collaboration. (Int3)
When prompted about the effects of more student collaboration she visualizes problem solving activities as a possible setting for explaining and comparing solutions in groups. This is a feature of the interaction commonly found in inquiry classrooms ADDIN EN.CITE Cobb1992350Cobb, PWood, TYackel, EMcNeal, B1992Characteristics of classroom mathematics traditions: An interactional analysisAmerican Educational Research Journal293573-604(Cobb et al., 1992). But, referring to constraints of the usual task setting, Kristina asserts that working on an individual basis has to be the normal feature of social conduct. In another interview she mentions the shame she feels when she gives a wrong answer in public. A more collaborative involvement in whole-class interaction besides public exchanges between individual students and the teacher might thus perhaps prevent this to happen. However, utterances in all interviews show that the individualist feature of school mathematical practice prevails during lower secondary school.
According to Kristina cooperation with other students is generally a matter of informally checking procedures and answers, a usual mode of working in mathematics classrooms ADDIN EN.CITE Goos199945see e.g. 7Goos, MGalbraith, PRenshaw, P1999Establishing a community of practice in a secondary mathematics classroomBurton, LLearning mathematics. From hierarchies to networksLondonFalmer Press36-61(see e.g. Goos et al., 1999). The students are not responsible for explaining solutions to each other but Kristina finds it easier to follow a class-mate who explains why her answers are wrong, than to follow the teacher explaining the same thing. In every interview she mentions one classmate who is especially important to her. Also, to resolve issues by themselves seem perfectly natural to Kristina as they otherwise, at worst, wouldnt be doing anything as the teachers time to intervene with each student is limited. But the final helper and dispenser of knowledge is anyhow the teacher, to whom Kristina turns as he walks around in the classroom, and when classmates have not been able to give her the support she needs to get the answer she is reaching for.
We do work alone all of us. But it automatically gets like if you dont understand something you ask the one who sits besides you. So it usually becomes pair-work or group-work anyway, like that you ask for some help from everybody and then it is like that if you dont understand, havent yet got an answer, then it is (teacher) you ask. It is anyhow like that, that he has to come to everybody, so it is the fastest way to ask the one you sit next to, so you talk to everybody. (Int5)
Hand raising is a silent and visible sign dynamically used as a tool by both students and teachers in school practice ADDIN EN.CITE Sahlström200160Sahlström, F2001The interactional organization of hand raising in classroom interactionJournal of Classroom Interaction372(Sahlstrm, 2001). The meaning of to discuss in the mathematics classroom is for Kristina first of all to raise her hand and deliver an answer to a question set by the teacher. In year 8 Kristina explains when he (the teacher) asks then you can raise your hand and tell how you think you could calculate it. And she continues with a remark that the possibilities to participate in this type of discussion are higher if the tasks allow quick calculating in the head:
// but we havent had as much as in 7th grade those tasks where you like only tell the answer, calculate in your head, but in 7th grade we had more such that we kind of circled, everybody in class got a task from the textbook and was supposed to calculate in the head and tell, but we havent had so much of that now, it is because we havent had so much that we can calculate in the head, so thats why. (Int3)
From Kristinas utterances I can conclude that hand raising serves different functions depending on how Kristina experiences the situation. Usually hand raising is a sign of willingness to contribute in public with a suggestion for a solu tion or a valuable idea. Sometimes it is a signal from Kristina to the teacher that help is needed. But it is also an activity marker and a way to please the teacher. Being an active hand-raiser is the same as being a good student which may pay off immediately in public praise from the teacher or later on in better grades. Hand raising is definitely a visible indication of appropriated under standing from Kristinas point of view.
I just think that before I understood much more and then I raised my hand much more. But now I dont understand really well and I have not raised my hand that much and then maybe I havent done homework as much so my grade will drop I am sure (Int3)
When prompted about reasons for a drop in grades and how she thinks she can prevent this to happen, she refers to the complex relationship between hand raising, understanding, being able to do homework and getting better test results.
Well, I have thought that I will sharpen up and do my homework real properly and in that way you are of course more active during lessons and then you can raise your hand and say and try and understand. Then you dont need to read and swot up so much to the test, because you can manage better then. (Int4)
In addition to the discovery type of activity already discussed there is very little indication in Kristinas utterances of communicative activities where students are responsible for developing the mathematical content of the lessons until year 9 when there seem to be a shift. An indication of how Kristina experiences this shift was focused above (p. 14-15). Participation in public interaction in the class room is monitored by the teacher while the students are given the role of answering the teachers questions. The goal of the interaction is as a rule set by the teacher and with reference to delivery of answers or solutions to set tasks or disco vering mathematical structures from examples. The dilemma from Kristinas point of view is that she often seems to find herself in a position where she doesnt have the ability to participate fully, and remains in the peri phery of the communicative process where the mathematical content of the inter action emerges. Thus she experiences the school mathematical practice as pro ble matic because it pushes her into a position where her strategy of partici pation becomes cognitive disengagement.
Sometimes you can hear from the teacher that you should know this from primary school and then you sit there and you should figure out the answer or the solution. But as you cant find it you arent able to concentrate and then you sit and wait until somebody says something and when they start to talk you dont understand what they are talking about (Int4).
Another aspect of the classroom interaction that creates problems for Kristina is the taskrelated meta-rule that the teacher starts a plenary session with explaining tasks at the board when sufficiently many students have individually finished a cer tain amount of tasks. This creates a dilemma for Kristina as she experiences a lack of possibility to understanding and possibly a feeling of being ignored by the teacher.
Dominance of tasks
Students working with set tasks, mostly with textbook tasks, is a widely spread and contested approach in school mathematical practice ADDIN EN.CITE Röj-Lindberg1999301Röj-Lindberg, A-S1999Läromedel och undervisning i matematik på högstadiet. En kartläggning av läget i SvenskfinlandVasaSvenskfinlands läromedelscenterBoaler1998170Boaler, J1998Open and closed mathematics: Students experiences and understandingsJournal for Research in Mathematics Education29141-62Törnroos2001133Törnroos, J2001Finnish mathematics textbooks in grades 5-7.Second Scandinavian Symposium on Research Methods in Science and Mathematics Education, June 18-19HelsinkiJohansson2006120Johansson, M2006Textbooks as instruments. Three teachers' way to organize their mathematics lessonsNordisk Matematikk Didaktikk1135-30(Boaler, 1998; Johansson, 2006; Rj-Lindberg, 1999; Trnroos, 2001). Students usually work with set tasks in order to display knowledge, not because they have a personal interest in getting the answers or making inquiries into mathematical ideas. Traditionally each task forms a restricted whole and the student does one task after the other. The choices of methods to do the tasks are usually limited and set by the subject area presented in the book. A solved task, finding an answer, leads the student to the next task or to the next subject in the textbook. Tasks seldom invite the students to formulate own problems and questions ADDIN EN.CITE Mellin-Olsen199187Mellin-Olsen, S1991The double bind as a didactical trapBishop, A JMellin -Olsen, Svon Dormolen, JMathematical knowledge: Its growth through teachingDordrechtKluwer39-59(Mellin-Olsen, 1991). Tasks can function like a Russian-doll where a solved task includes solving sub-tasks. In the following paragraph from the second interview in year 7 Kristina describes features of so called text tasks in a test.
Text tasks I found fairly easy, it was just some subtraction and addition, but otherwise you have to read the task very carefully and then sometimes it might come like several times that you have to calculate the first part this and this and then you have to subtract this from that and so on. And then of course you have to write the answer down there with a unit because it is a text task and you have to know what it is all about. (Int2)
According to Stieg Mellin-Olsen (1990, 1991) the influence of the didactical use of different types of tasks is so deeply rooted in school mathematics practice, and in the teachers images of school mathematics, that the school mathematical teaching tradition can be conceptualized as a task discourse. Transformation of participation in school mathematical practice can thus be seen as an issue of re-negotiating the meta-rules of the task discourse.
In the five interviews with Kristina I found significant traces of her participation in a school mathematical practice that continued to be very much dominated by a similar type of task-discourse. In fact, substantial parts of all interviews, and especially of the second to fifth interviews, expose aspects of Kristinas experiences in task related actions and activities. This comes as no surprise when one considers the more than thousand textbook tasks that was part of the school mathematical practice each school year. Generally Kristina starts to talk about tasks in her descriptions of activities that occur within the frame of ordinary lesson design, but also in descriptions of homework, project work and of written evaluations as individual tests and problem solving activities.
An indication of the dominance of task is that Kristina in all interviews introduced task-related issues into the conversation without being prompted. Out of 65 paragraphs with task-related utterances from Kristina, only 15 are answers to task-related questions. In the following extract from the first interview Kristina describes monitoring of affordance to cognitive challenge as a task related issue. With the help of appropriate tasks the teacher can set the cognitive challenge on a level that Kristina finds comfortable.
I: Do you think that you get enough demanding questions and tasks?
Kristina: Yes. Or, I think that they are not too difficult, but you kind of have to start with something easy so you can manage the difficult things later when you get those tasks. And not start directly with the difficult.
An interesting aspect in the first interview is that Kristina experiences such a high confidence in her ability to do tasks that she usually ignores the answer key ADDIN EN.CITE Erlwanger19736see e.g. , on the impact on answer key on students' knowing0Erlwanger, S H1973Benny's conception of rules and answers in IPI mathematicsJournal of Children's Mathematical Behavior12(see e.g. Erlwanger, 1973, on the impact on answer key on students' knowing). She explains that she is used to managing without an answer key because of the lack of textbook in primary school. Nevertheless, having a textbook is the most positive experience next to learning mathematics that she can think of in the beginning of lower secondary school.
I guess it is what you learn... Yes! And then that we have got a mathematics textbook. That is good. I didnt like it when we got papers before. It is much better to have a textbook.
However, Kristinas positive identification with the textbook might not primarily be related to the answer key, but more because the book is an essential sign of order for her in a school mathematical practice so heavily dependent on tasks for learning rules and systems. In such a practice by default the book has epistemic authority: teachers explain assignments to pupils by saying this is what they want you to do here, and the right answers are found in the answer key ADDIN EN.CITE Boaler200022, 1817Boaler, JGreeno, J G2000Identity, agency and knowing in mathematics worldsBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing171-200(Boaler & Greeno, 2000, 181)
The influence of textbook on classroom interaction is explained by Kristina when she relates the necessity of individual work in the mathematics classroom to the nature of the textbook tasks. As the solution procedures of these tasks are already set in advance there is no need for any real collaboration with others. In other types of tasks, in project work and in problems set by the teacher, she sees the potential for learning from other students and to adopt solutions developed by others for her own use. But, she explains, if you have ordinary textbook tasks you must try to manage them by yourself (Int3).
To sum up, I conclude from Kristinas task related utterances that she expects the school mathematical content to emerge from tasks, to be practiced within set tasks and to be assessed with tasks. Taken together, there are indications in the five interviews with Kristina that she continued to see
tasks as a significant feature of the ordinary lesson design,
tasks as tools for regulating the tempo of lessons, the social interaction and the cognitive challenge,
tasks as a significant feature of homework. She should do a certain amount of tasks before she meets the next field of mathematical knowledge,
tasks as a significant feature of project work characterized as school mathematical, i.e. you know directly when you see them that they are school work (Int2),
tasks as a significant feature of tests,
tasks as containers for mathematical procedures. By following the steps in a solution she learns the rules for doing a particular task. Kristina can retrieve forgotten procedures from memory with the help of repetition with tasks set by the teacher in plenary sessions and more individual practice in doing tasks,
tasks as a rationale for written solutions. A written solution displays the cognitive processes of the solver. If she copy solutions from the board she has caught a written trajectory of the cognitive process of the teacher.
tasks as catalysts for personal satisfaction and as indicator of under standing. The quicker she manages to solve a task the better she under stands.
Invisiblity of school mathematics
At several occasions during the interviews Kristina spontaneosly in her utterances compared her experiences of school mathematical practices with her experiences of practices in other school subjects. In the following utterance she compares mathematics with textile work. The statement was given in the third interview at a stage when Kristina expressed a negative emotional relation to mathematics. The only possible way out from the boredom and dullness she experienced in mathematics as well as in physics and chemistry was, as she saw it, to change her own behaviour: to motivate herself into being more alert, to accept that feeling bad about mathematics and that lack of understanding is part of the game I was at the time quite astonished by her very strong negative relation to school mathematics and asked her in one of the follow up questions to look back and reflect on whether there had been some changes during lower secondary school. No, she said
mathematics has never been any fun subject ... like textile work is fun, you can do a lot and you kind of see what you are doing, but mathematics is dull you cannot do anything about it. And there are those that do like mathematics, you have to have those subjects anyhow. (Int3)
The difference between the practices of textile work and the practices of school mathematics is a matter of visibility. In textile work she has experienced participation in doing things that are visible. This has given her a sense of satisfaction that she has not gained from participating in school mathematics. Mathematics at school is from Kristinas point of view closed into a rules and systems practice where she is occupied with learning things to remember, different formulas and ways of calculating (Int5), cognitive and internal activities that indeed may be very far from externalised activities where you can see what you are doing.
Coda
On the basis of considerations that are included in the theoretical frame for this paper I argue that the development of Kristinas experiences is situated in and cannot be separated from the development of the school mathematical practice. Even though I in this paper have chosen to focus on Kristina and her story it is strictly a research based decision. I separated the experiences of Kristina but nevertheless I see her experiences as integrated in and emerging out of the particular school mathematical practice where she was a member. I thus argue that her story can be seen as traces of a trajectory of participation in this practice. In relation to school mathematics her expectations and attitudes have developed precisely because she has participated in a particular school mathematical practice. Not only did she learn mathematics by participating in this practice, she also developed her image of what it takes to learn mathematics in this particular practice. Taken together her school mathematical knowing can be described as a kind of school mathematical disposition. Participating and engagement in school mathematical practice can thus be seen as a process that both produced and transformed her identity as learner and school mathematician ADDIN EN.CITE Wenger1998371Wenger, E1998Communities of practice. Learning, meaning and identityCambridgeCambridge University PressBoaler2000227Boaler, JGreeno, J G2000Identity, agency and knowing in mathematics worldsBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing171-200Lerman2000487Lerman, S2000The social turn in mathematics education researchBoaler, JMultiple perspectives on mathematics teaching and learningLondonAblex Publishing19-44(Boaler & Greeno, 2000; Lerman, 2000).
This opens up for the questions how participation in school mathematical practice affected her identity as learner of mathematics and why participation in school mathematical practice did not boost Kristinas confidence in her ability to learn mathematics even though she was considered to be a good student.
One of my hypothesis is that the her rules and systems - experiences coalesced into a very limited view on the purpose of mathematical activities as finding a precise procedure for each task or type of tasks. Tasks were indeed, as she saw it, used over and over again in school to check her ability to quickly remember and whether she understood correctly which procedure to apply. Along with the expansion of the mathematical content she, in short, became totally over whel med by the load on her memory and presumably, as a result, her feeling of being alienated from the whole business of doing mathematics was growing cons tantly. She experienced the practice as closed and the practice closed her out.
The statement below about mathematics given by Kristina as a grown-up strengthens my hypothesis. In the conversation I had with her about her school mathematical experiences in lower secondary school she argued against my suggestion that mathematical activities and problems perhaps can be seen as having a quite open character. On the contrary, she said, there are a lot of similar problems, the difficulty is in to define what type of problem it is, and then to pick the right procedure. Kristina argued that
there are many similar problems in mathematics, you have to be able define the type of problem and which procedure to use to solve it. Every problem belongs to a certain group of problems. It is difficult to know which procedure to combine with a particular problem if you cannot define the type of problem. And when the amount of procedures increases this dilemma grows, and is worsened by the high pace of teaching. You have to know the types you have gone through before and then they just grow and grow and it gets into some sort of chaos. It becomes too much.
In one of her last statements in the fifth interview Kristina referred to her decision to enter upper secondary school where she reasoned she would meet a tough time with mathematics. This expectation of hers turned out to be partly fulfilled. When I met Kristina again she told me how she had struggled with mathematics the first year in upper secondary school. But then a decision to study biology at the university level completely changed the scenario. Now she became motivated to study mathematics in school. Voluntarily she even repeated some courses, and she had no problem to pass the examination with a high grade in mathematics. However, after secondary school she eventually chose pedagogy as her field of study, because, as she said, in pedagogy there are no right answers; a statement that brings with it a clear, albeit implicit, question: why is there such a strong focus on right answers in school mathematics?
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ADDIN
In this paper the concept reform denotes the intentions by the teachers to change their teaching practices and not necessarily the outcomes. Kristinas story and the traditional school mathematical practice as experienced by her that this paper revel can however be interpreted as an outcome. It is nevertheless not my intention nor is it possible to relate Kristinas story to a discussion of success or failure of the reform.
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