Abstracts - Earli

the paper is to explore possible ways to capture this ongoing assessment. The lessons with handson work, described in this study, are part of a project in which the teachers and researchers workedcollaboratively to explore possible meanings of qualities of knowledge the pupils are expected todevelop according to the national mathematics curriculum. In part of the video-recordings from theproject (two out of four classrooms), we have analysed the classroom interaction. For the analysis,we have used two approaches that appear slightly different. The first analysis employs the conceptmultimodality (Kress et al, 2001). For the second analysis, Goodwin’s (2002) concept"participatory framework" has been used. In the excerpts we used for the analyses, the focuses areon the dialogues between the teachers and their respective group of students, specifically on howthe communication contributes to forming the mathematical knowledge. When studying theclassroom communication in these situations, we find many incidents of formative assessment –that is communication that can be expected to contribute to the forming of the students’mathematical knowledge. Our conclusion is that these kinds of studies can contribute tounderstanding the importance of the various kinds of feedback-interaction between teacher andstudents, especially with regards to what content is established in the classroom.Learning and instruction crossroads: What paths lead to equations?Ricardo Machado, University of Lisbon, PortugalMargarida Cesar, University of Lisbon, PortugalMathematics is stated by many students as problematic. One of the most rejected and fearedcontents are algebra. But it assumes an important role in compulsory education curricula.Therefore, teachers need to change students’ social representations about Mathematics, namelyequations, implementing effective ways to promote their performances. One of the decisions thatPortuguese teachers make is about the nature of the tasks. When confronted to open tasks studentsuse different solving strategies. Some studies illuminated that adopting collaborative work as adaily practice and associating it to open and significant tasks related to elementary algebraimproves students’ performances (César, 1994; Céêsar, Perret-Clermont, & Benavente, 2000;César & Torres, 1998). This work is based in two research projects (César, 1994; César & Torres,1998). The first one is a quasi experimental study and it explored the characteristics of dyad workvs. individual work (equations, 7th grade). The second one is the development of this first project.It studies and promotes collaborative work in school setting to improve students’ academicachievement (5th to 12th grade) and it follows an ethnographic approach based in action-researchstudies. The data collecting instruments were the participant observation (including audio tapedpeer interactions), questionnaires, interviews and documents. Students’ protocols illuminate theexistence of five solving strategies: graphic representation (drawings and schemes), asimplification strategy (variable manipulation), by trials, arithmetic and algebraic (equations).These strategies were shaped by students’ knowledge and ways of reasoning, but also by the levelof complexity of the problems. Students who used a subtractive or algebraic strategy in the easiestproblems were the ones who were able to change their strategies into effective ones when theproblems were more difficult, i.e., the ones showing a most effective performance had a differentpattern of solving strategies.– 213 –