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

Abstract

IntroductionFrom looking at research literature is possible to see that research on design, implementation and analysis of mathematical tasks is an actual theme: there is special issue of Journal of Mathematics Teacher Education (2007) with Anne Watson, John Mason and Orit Zaslavsky as editors (Watson and Mason, 2007), book published by Clarke, Grevholm and Millman (2009) concerning Tasks in primary mathematics teacher education and under ICME 11 in Mexico (2008) title of one of Topic Study Groups was Research and development in task design and analysis. In addition several substantial research projects conducted in United States focus on this issue. For example QUASAR project (Quantitative Understanding: Amplifying Student Achievement and Reasoning), involving group of researchers (Stein, Smith, Henningsen & Silver, 2000), aimed at improving mathematics instruction for students by emphasising thinking, reasoning, problem solving and communication of mathematical ideas. One of central aspects of their research was to focus on use of instructional tasks in project classroom and they proposed elaboration of the mathematical tasks framework where kinds of thinking needed to solve tasks were referred to as demands. They reported on observations concerning change of cognitive demands during lesson where a task that starts out challenging ... might not induce high-level thinking and reasoning that was intended as students actually go about working on it (Stein et al., 2009, p.xviii). This aspect is also address by Artigue (1994) arguing that might be tempting to implement too quickly development products arising from research into products for teaching. She characterises processes related to transmission of products from didactic engineering in terms of distortions and she emphasises distinction between 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 distortions of specific mathematical task (the T-shirt task) from its design by group of didacticians at University of Adger (UiA) to its implementation by two different teachers. This research is situated in larger research project conducted at (UiA), Teaching Better Mathematics project (TBM).The structure of article is as follows: First I present central aspects of TBM project and emphasise theoretical constructs of didactical aim and pedagogical means. I also introduce methodological approach adopted in project. Then I turn to an example and explain how T-shirt task was designed by didacticians at UiA and how was implemented by teacher from primary school and by teacher from lower secondary school. Finally, I discuss results and present implications for further collaboration between didacticians and researchers.Central aspects of TBMprojectCo-leaning agreement with teachersThe aims of TBM project are reflected in title: Teaching Better Mathematics. First 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 means to achieve first aim. The nature of 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 didactician get opportunity to observe nature of impact project had on participating schools. In 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 opportunity to develop further understandings of world of other and of his/her own world. …