Reasoning and proof (R&P) are key elements in current reform efforts, but notorious for the problems they create for teachers. We present results from a pilot to an intervention study that seeks to alleviate these problems for prospective primary and lower secondary teachers in Denmark. The study introduces R&P in contexts that are “sufficiently close” to both academic mathematics and to instruction in school. The pilot asks if this is a feasible approach. The part of the pilot presented here consists of responses by 57 prospective teachers to a qualitative questionnaire. The results show that many feel strongly about R&P, one way or another, but also that they have considerable problems with these processes to some extent irrespectively of their affective commitment. The results of the pilot confirm our approach for the main study.
Research on teachers’ knowledge and beliefs has grown big in recent years. Thelarger parts of these fields are built on acquisitionist interpretations of humanfunctioning. We explore the potentials of a participationist framework forunderstanding the role of the teacher for emerging classroom practices. Theframework is built on social practice theory and symbolic interactionism and adoptsa processual approach to understanding the role of the teacher. We use theframework in a qualitative study of two teachers with different prior experiences. Thestudy suggests that the framework has some potential and sheds light on the dynamicrelationships between the teacher’s engagement in the practices of the mathematicsclassroom and other, personally significant, past and present ones.
Mathematical reasoning and proving (R&P) are notoriously difficult for students at all school levels as well as for many prospective teachers for elementary school. The latter is particularly worrying as current recommendations for school mathematics emphasise R&P as significant in their own right and as ways of supporting students’ learning of important mathematical contents. I outline the background and rationale to an intervention study that seeks to develop prospective elementary teachers’ own proficiency with reasoning and proving as well as their ways of working with these processes when teaching. The pilot to the intervention suggests that the research participants faced even greater difficulties than anticipated. We suggest balancing proving that and proving why in mathematics teacher education to address these difficulties, using tasks that arise or may be developed from school classrooms.
Paul Cobb fik i sommer overrakt ICMI’s Freudenthalmedalje for sin matematikdidaktiske forskning. Hensigten med denne artikel er at introducere Cobbs forfatterskab. Hovedvægten er på den socialkonstruktivistiske forståelse af undervisning og læring fra 1990’erne, men der knyttes an til tidligere og senere arbejder. Pointen er at nok er der teoretiske skift i Cobbs arbejder over de seneste tre årtier, men der er også en rød tråd i form af en dobbeltinteresse i at teoretisere over praksis og bidrage til dens videreudvikling. Sammen med en teoretisk pragmatisme peger det mere på gradvis, kontinuert udvikling i Cobbs tilgang end på fundamentale brud i den.
Ræsonnement og bevis har en stærk position reformforslag for matematikundervisning. Imidlertid giver området problemer for elever på alle nieauer såvel som for en del studerende, fx i læreruddannelsen. Begge grupper har ofte svært at skelne mellem empiriske og matematiske argumenter; de ser ikke matematiske ræsonnementer som en måde at udvikle bedre forståelser af det indhold, der ræsonneres om; og generelt synes det svært at bryde med den tradition, at ræsonnement og bevis introduceres sent i skoleforløbet og mest i relation til geometri.
For at imødegå problemerne er det blevet anbefalet i skolen at fokusere på ræsonnementer som del af undersøgende og kommunikative aktiviteter. Ræsonnement og bevis skal da ikke primært verificere på forhånd givne resultater, men ses som en helhed af at undersøge matematiske sammenhænge, formulere hypoteser, og be- eller afkræfte foreløbige resultater. Det indebærer et skift i fokus fra proving that til proving why. Da et andet af tidens forslag er at koble læreruddannelsen tæt til matematik i skolen, er det nærliggende at foreslå et tilsvarende skift i uddannelsen af matematiklærere.
Jeg præsenterer baggrunden for og resultater fra piloten til et interventionsstudie, der har fokus på ræsonnement og bevis i uddannelsen af matematiklærere til grundskolen. I studiet (RaPiTE - Reasoning and Proving in Teacher Education) arbejder vi på den ene side med proving why, men på den anden argumenterer vi, at skiftet fra proving that til proving why ikke må overdrives. Hvis lærere skal forholde sig matematisk til uforudsigelige forslag og ideer fra elever, som arbejder undersøgende, må de have erfaringer med forskellige måder at afgøre sandhedsværdien af matematiske udsagn. Argumentet er, at matematiklæreruddannelsen skal balancere proving why med proving that og i praksis arbejde på måder, der ligger ”tilstrækkeligt tæt” på såvel matematik som fag som på skolens undervisning. Jeg giver eksempler på, hvad det kan betyde i praksis.
Studies of professional identity are generally conducted using participatory frameworks and from the perspective of a particular development initiative. They provide understandings of teachers’ move towards more comprehensive participation in the practices the initiative promotes. Studies in line with this main trend, however, leave questions of teacher identity unanswered when teachers are not enrolled in long-term development programmes. I argue that to address such questions a different framework is needed, one that maintains the participatory stance, but focuses on the individual teacher rather than a development initiative. It is the intention of the Patterns-of-Participation framework (PoP) that I introduce to re-centre the individual in this sense. To make my point, I discuss how research frameworks may be conceptualized and compared and use the resulting “frameworks framework” to contrast studies of the main trend with the intentions of PoP.
Studies of teacher identity generally conceive identity in participatory and processual terms and seek to understand how cultural and social contexts inform and transform teachers’ tales of themselves as professionals as well as their contributions to classroom practice. Also, they often view identity changes from the perspective of a particular development initiative and provide understandings of if and how teachers move from the periphery to more comprehensive modes of participation in the practices it promotes. Foregrounding particular initiatives, however, these studies leave questions of teacher identity unanswered in the majority of cases in which teachers are not enrolled in long-term development programmes. Referring to a longitudinal study of a novice teacher, Anna (cf. Skott, 2013), I address questions of the latter type. I follow Anna at her school, Northgate, for periods of time over the first three years after her graduation. Somewhat in line with other studies of identity, I draw on social practice theory in my interpretation of her contributions to classroom interaction, most notably on the notions of practice (e.g. Wenger, 1998) and figured worlds (e.g. Holland et al., 1998). Rather than focusing on the role of one particular set of practices (e.g. as promoted by a development initiative), I ask: (1) what prior and present practices and figured worlds does Anna re-engage and participate in, as she interacts with her students? (2) What changes, if any, occur among the practices and figured worlds that dominate her contributions to classroom interaction over the first few years of her career? I use a conceptual framework called Patterns of Participation (PoP). PoP combines social practice theory with an interpretation of symbolic interactionism to redefine teacher learning and to re-centre the individual (rather than a development initiative) in what is still a participatory account of professional identity. The results suggest that among practices and figured worlds that are important for Anna’s approach to instruction, some relate to her pre-service education; others to collaboration with her colleagues and the leadership at Northgate; still others to contexts with little apparent connection to her education and profession as a teacher. Also, there are significant shifts in the relative importance of these practices and figured worlds over time and the ones related to the reformist intentions of her teacher education programme are generally subdued by others based at the school. I argue that PoP is helpful for understanding changes in Anna’s professional identity and shedding light on significant shifts in the practices and discourses that inform her contributions to classroom practice.
Studies of professional identity are generally conducted using participatory frameworks and from the perspective of a particular development initiative. They provide understandings of teachers’ move towards more comprehensive participation in the practices the initiative promotes. Studies in line with this main trend, however, leave questions of teacher identity unanswered when teachers are not enrolled in long-term development programmes. I argue that to address such questions a different framework is needed, one that maintains the participatory stance, but focuses on the individual teacher rather than a development initiative. It is the intention of the Patterns-of-Participation framework (PoP) that I introduce to re-centre the individual in this sense. To make my point, I discuss how research frameworks may be conceptualized and compared and use the resulting "frameworks framework" to contrast studies of the main trend with the intentions of PoP.
Current developments in mathematics education require teachers to play a different and more profound role than few years ago. The first half of this article discusses the theoretical background of these developments in terms of their epistemological and meta-mathematical orientations. Based on this theoretical analysis, the teacher's new role is summed up as one of forced autonomy. Next, the article summarises two different responses on the part of the research community to the situation of forced autonomy. These are the ones of focussing primarily on teachers' meta-mathematical and mathematical qualifications. Subsequently, the results of an empirical study are outlined. These results are used both to develop an empirically grounded elaboration of the notion of forced autonomy and to challenge the above-mentioned responses on the part of the research community. The concluding section calls for the adoption of broad perspectives on teacher qualifications, integrating pedagogical, mathematical and meta-mathematical perspectives.