De cultuur en de praktijken van wiskundeklassen in het zesde leerjaar. Een poging om de relaties tussen sociale en individuele aspecten in vraagstuklessen te ontrafelen

摘要

During the last decades, numerous initiatives have been taken in many co untries to reform and innovate mathematics classroom practices. First, t his reform movement is characterized by a shift in focus from rules, mec hanic procedures, and memorization to students’ acquisition of higher-or der problem-solving skills. Second, an important role is ascribed to the use of reality in mathematics classrooms. Third, the reform movement pa ys attention to the establishment of an innovative classroom culture inv olving a more active role for students. Inspired by these three characteristics of the reform movement in mathem atics education, scholars have designed powerful learning environments a imed at eliciting in students appropriate mathematics-related beliefs an d problem-solving skills. In Flanders, for instance, Verschaffel et al. (1999) designed, implemented, and evaluated an innovative learning envir onment for teaching and learning how to solve mathematical word problems in upper primary school. The learning environment aimed at students’ ac quisition of an overall metacognitive strategy for solving mathematical application problems, and of positive mathematics-related beliefs. The b asic principles of this innovative learning environment were threefold a nd clearly reflected the main ideas of the reform movement in mathematic s education: (1) using a set of realistic and complex problems; (2) appl ying a variety of highly interactive instructional techniques and classr oom organization forms; and (3) establishing appropriate classroom norms concerning (learning) to solve mathematical problems. Meanwhile, the reform-based ideas have led to a new generation of mathem atical textbooks. However, the question remains to what extent these inn ovative ideas also effectively impact today’s regular mathematics classr ooms. This dissertation aims at a dual goal that has guided the scope an d design of two studies that will be reported. The first goal relates to an in-depth analysis of the way in which problem-solving lessons based on an innovative textbook are implemented in regular mathematics classro oms. The second goal focuses on the analysis of the relationship between teachers’ instructional approaches to the teaching of mathematical prob lem solving, and teachers’ and students’ beliefs, on the one hand, and s tudents’ problem-solving skills, on the other hand. In the first, exploratory study we investigated the way in which ten six th-grade teachers implemented the same two problem-solving lessons from an innovative textbook. The analysis was based on a coding scheme which covers the three basic principles of Verschaffel et al.’s (1999) learnin g environment (i.e., the nature of the tasks, the instructional techniqu es and classroom organization forms, and the classroom norms) that are a ssumed to influence students’ mathematics-related beliefs and problem-so lving skills. Additionally, the students of all participating classrooms were administered a questionnaire and test to measure respectively thei r mathematics-related beliefs and problem-solving skills.Two teachers wh o revealed in the pilot study a considerably different approach towards learning to solve mathematical word problems volunteered to participate in the second, more extended and more in-depth investigation. We observe d both teachers’ approaches to word problem solving over a seven-month p eriod (from September until the Easter holidays). The complex and multif arious data were studied from different perspectives, relating to three themes that are characteristic for the aforementioned reform movement in mathematics education: (1) the way in which reality is dealt with in ma thematics lessons, (2) the focus on heuristic and metacognitive skills i n the teaching and learning of problem solving, and (3) the establishmen t of an innovative classroom culture in which students are actively invo lved in making mathematical contributions (i.e., a broad distribution of authority). At the beginning and at the end of the seven-month observat ion period, we investigated students’ beliefs, the way in which they app roached realistic word problems, and their general mathematics achieveme nt level. Moreover, we conducted in-depth interviews with both teachers and all students about their beliefs about the installed classroom cultu re.