From designing to implementing mathematical tasks: investigating the changes in the nature of the T-shirt task

From looking at research literature it is possible to see that research on design, implementation and analysis of mathematical tasks is an actual theme: there is a special issue of the Journal of Mathematics Teacher Education (2007) with Anne Watson, John Mason and Orit Zaslavsky as editors (Watson and Mason, 2007), a book published by Clarke, Grevholm and Millman (2009) concerning “Tasks in primary mathematics teacher education” and under ICME 11 in Mexico (2008) the title of one of the Topic Study Groups was “Research and development in task design and analysis”. In addition several substantial research projects conducted in the United States focus on this issue. For example the QUASAR project (Quantitative Understanding: Amplifying Student Achievement and Reasoning), involving a group of researchers (Stein, Smith, Henningsen & Silver, 2000), aimed at improving mathematics instruction for students by emphasising thinking, reasoning, problem solving and the communication of mathematical ideas. One of the central aspects of their research was to focus on the use of instructional tasks in project classroom and they proposed the elaboration of “the mathematical tasks framework” where the kinds of thinking needed to solve tasks were referred to as “cognitive demands”. They reported on observations concerning the change of cognitive demands during a lesson where “a task that starts out challenging … might not induce the high-level thinking and reasoning that was intended as the students actually go about working on it” (Stein et al., 2009, p.xviii). This aspect is also address by Artigue (1994) arguing that it might be tempting to implement too quickly development products arising from research into products for teaching. She characterises the processes related to the transmission of products from didactic engineering in terms of distortions and she emphasises the distinction between the activities of conducting research and of engaging in teaching. My aim, in this article, is to follow Artigue’s argumentation and to investigate, trace and characterise the distortions of a specific mathematical task (the T-shirt task) from its design by a group of didacticians at University of Adger (UiA) to its implementation by two different teachers. This research is situated in a larger research project conducted at (UiA), the Teaching Better Mathematics project (TBM)

Developing Algebraic Thinking in a Community of Inquiry : Collaboration between Three Teachers and a Didactician

In this thesis I report from a study of the development of algebraic thinking of three teachers, from lower secondary school, and a didactician from a university in Norway (myself). The thesis offers an account of the relationship between the participants’ development of algebraic thinking and the processes related to the creation and development of a community of inquiry. In addition, the thesis presents elements of the relationship between the teachers’ development of algebraic thinking and their thinking in relation to their teaching practice. My theoretical framework was elaborated according to the criteria of relevance and coherence. In order to conceptualise the participants’ development of algebraic thinking within the community of inquiry, I started from Wenger’s theory of community of practice and expanded it in order to include both the dimension of inquiry and Karpov’s ideas of cognitive and metacognitive mediation. Methodologically, I understand my study as a case study, within a developmental research paradigm, addressing the development of algebraic thinking within a community of inquiry consisting of three teachers and a didactician. The collaboration between the teachers and the didactician was organised through regular mathematical workshops, and interviews with each teacher both before and after classroom observations. During the workshops, the participants engaged with some mathematical tasks which were offered by the didactician. The results of this study indicate that the participants’ development of algebraic thinking is deeply interwoven with the processes related to the creation and development of the community of inquiry. It seems that the participants’ confidence in the community was developing gradually while the confidence in the subject-matter was related to the nature of the mathematical tasks with which the participants engaged. In addition, the study shows how the teachers engaged in a process of both looking critically into their own teaching practice as a consequence of their collaborative engagement within the community of inquiry, and of envisaging possible implications for their future teaching practice. Furthermore, I offer insights into my own development both as a didactician and as a researcher and how these relate to research outcomes. Overall, the thesis contributes to a better understanding of issues related to collaboration between in-service teachers and a didactician from a university, while focusing on the development of algebraic thinking. Implications are also suggested concerning the way algebra could be addressed in schools

Developing Algebraic Thinking in a Community of Inquiry : Collaboration between Three Teachers and a Didactician

In this thesis I report from a study of the development of algebraic thinking of three teachers, from lower secondary school, and a didactician from a university in Norway (myself). The thesis offers an account of the relationship between the participants’ development of algebraic thinking and the processes related to the creation and development of a community of inquiry. In addition, the thesis presents elements of the relationship between the teachers’ development of algebraic thinking and their thinking in relation to their teaching practice. My theoretical framework was elaborated according to the criteria of relevance and coherence. In order to conceptualise the participants’ development of algebraic thinking within the community of inquiry, I started from Wenger’s theory of community of practice and expanded it in order to include both the dimension of inquiry and Karpov’s ideas of cognitive and metacognitive mediation. Methodologically, I understand my study as a case study, within a developmental research paradigm, addressing the development of algebraic thinking within a community of inquiry consisting of three teachers and a didactician. The collaboration between the teachers and the didactician was organised through regular mathematical workshops, and interviews with each teacher both before and after classroom observations. During the workshops, the participants engaged with some mathematical tasks which were offered by the didactician. The results of this study indicate that the participants’ development of algebraic thinking is deeply interwoven with the processes related to the creation and development of the community of inquiry. It seems that the participants’ confidence in the community was developing gradually while the confidence in the subject-matter was related to the nature of the mathematical tasks with which the participants engaged. In addition, the study shows how the teachers engaged in a process of both looking critically into their own teaching practice as a consequence of their collaborative engagement within the community of inquiry, and of envisaging possible implications for their future teaching practice. Furthermore, I offer insights into my own development both as a didactician and as a researcher and how these relate to research outcomes. Overall, the thesis contributes to a better understanding of issues related to collaboration between in-service teachers and a didactician from a university, while focusing on the development of algebraic thinking. Implications are also suggested concerning the way algebra could be addressed in schools

Mediated action in teachers’ discussions about mathematics tasks

This paper presents analyses of teachers’ discussions within mathematics teaching developmental research projects, taking mediation as the central construct. The relations in the so-called ‘didactic triangle’ form the basic framework for the analysis of two episodes in which upper secondary school teachers discuss and prepare tasks for classroom use. The analysis leads to the suggestion that the focus on tasks places an emphasis on the task as object and its resolution as goal; mathematics has the role of a mediating artefact. Subject content in the didactic triangle is thus displaced by the task and learning mathematics may be relegated to a subordinate position

Mediated action in teachers’ discussions about mathematics tasks

Published version of an article in the journal: ZDM. Also available from the publisher at: http://dx.doi.org/10.1007/s11858-012-0423-0This paper presents analyses of teachers’ discussions within mathematics teaching developmental research projects, taking mediation as the central construct. The relations in the so-called ‘didactic triangle’ form the basic framework for the analysis of two episodes in which upper secondary school teachers discuss and prepare tasks for classroom use. The analysis leads to the suggestion that the focus on tasks places an emphasis on the task as object and its resolution as goal; mathematics has the role of a mediating artefact. Subject content in the didactic triangle is thus displaced by the task and learning mathematics may be relegated to a subordinate position

Developing Algebraic Thinking in a Community of Inquiry : Collaboration between Three Teachers and a Didactician

In this thesis I report from a study of the development of algebraic thinking of three teachers, from lower secondary school, and a didactician from a university in Norway (myself). The thesis offers an account of the relationship between the participants’ development of algebraic thinking and the processes related to the creation and development of a community of inquiry. In addition, the thesis presents elements of the relationship between the teachers’ development of algebraic thinking and their thinking in relation to their teaching practice. My theoretical framework was elaborated according to the criteria of relevance and coherence. In order to conceptualise the participants’ development of algebraic thinking within the community of inquiry, I started from Wenger’s theory of community of practice and expanded it in order to include both the dimension of inquiry and Karpov’s ideas of cognitive and metacognitive mediation. Methodologically, I understand my study as a case study, within a developmental research paradigm, addressing the development of algebraic thinking within a community of inquiry consisting of three teachers and a didactician. The collaboration between the teachers and the didactician was organised through regular mathematical workshops, and interviews with each teacher both before and after classroom observations. During the workshops, the participants engaged with some mathematical tasks which were offered by the didactician. The results of this study indicate that the participants’ development of algebraic thinking is deeply interwoven with the processes related to the creation and development of the community of inquiry. It seems that the participants’ confidence in the community was developing gradually while the confidence in the subject-matter was related to the nature of the mathematical tasks with which the participants engaged. In addition, the study shows how the teachers engaged in a process of both looking critically into their own teaching practice as a consequence of their collaborative engagement within the community of inquiry, and of envisaging possible implications for their future teaching practice. Furthermore, I offer insights into my own development both as a didactician and as a researcher and how these relate to research outcomes. Overall, the thesis contributes to a better understanding of issues related to collaboration between in-service teachers and a didactician from a university, while focusing on the development of algebraic thinking. Implications are also suggested concerning the way algebra could be addressed in schools

Évaluations nationales et internationales en mathématiques : quelles analyses didactiques? - Compte-rendu du Projet Spécial n°4.

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