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

IntroductionFrom 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).The structure of the article is as follows: First I present central aspects of the TBM project and emphasise the theoretical constructs of didactical aim and pedagogical means. I also introduce the methodological approach adopted in the project. Then I turn to an example and explain how the T-shirt task was designed by didacticians at UiA and how it was implemented by a teacher from primary school and by a teacher from lower secondary school. Finally, I discuss the results and present implications for further collaboration between didacticians and researchers.Central aspects of the TBMprojectCo-leaning agreement with teachersThe aims of the TBM project are reflected in the title: Teaching Better Mathematics. First it promotes to develop better understanding and competency in mathematics for pupils in schools (Better Mathematics), and second to explore and develop better teaching approaches (Teaching Better) as a means to achieve the first aim. The nature of the project is developmental research and we collaborate with in-service teachers from 4 kindergarten, 6 primary and lower secondary schools and 3 upper secondary schools. Our collaboration with teachers is organised around workshops, approximately 4-5 per year, and school visits during which a didactician get the opportunity to observe the nature of the impact the project had on the participating schools. In the project we see teachers and didacticians as working together as co-learners (Wagner, 1997). This implies that both teachers and didacticians are engaged in action and reflection, and by working together, each has the opportunity to develop further understandings of the world of the other and of his/her own world. …