Hur kan dubbelt så långt bli fyra gånger större?

Abstract

Previous research has shown that a large majority of 12- to 16-year old students have a tendency to improperly apply the linear model when solving non-linear problems about the relation between lengths and area of enlarged and reduced plane geometric figures (De Bock et al., 1998). A major deficiency associated with the passage from one-dimensional to two-dimensional units is linked to ‘the illusion of linearity’, an explicit belief in a linear relation between lengths and areas of similarly enlarged or reduced figures. The main aim of this study was to investigate whether it could contribute to students’ deeper understanding of scaling of two-dimensional geometric figures when they were given the opportunity to simultaneously discern aspects related to linear and quadratical scaling. A Learning Study approach was adopted to improve instruction and the students’ learning outcomes. The interest was to, based on a perspective of Variation Theory, study how mathematical content was treated regarding the offered aspects of the object of learning and to what extent the instruction was powerful in helping the students to overcome the “illusion of linearity” and develop their understanding of the object of learning. In total 46 students and three teachers participated. The results of the study describe in what way the differences in treatment of the content handled in the classroom affect the students’ learning. The analysis is based on the videotaped lessons, the transcription of the conversations between the teacher and the students, and the students’ results at pre and post-tests. The major qualitative difference in student learning is to what extent the students managed to discern the differences in change of length and area in the same figure simultaneously when scaling two-dimensional geometric figures. The results show how the students’ learning outcome increased during the third cycle, which suggests that the handling of the content in this cycle was more powerful. The activities that took place during the third cycle was more intentionally systematic, both regarding in what order the critical aspects were highlighted, and how the relationship between them was explicitly problematized in several occasions. Another interesting finding is that the teachers encouraged the students to communicate about the content and by doing so the teachers and students were given an opportunity to jointly enact patterns of variation where students’ misconceptions, or way of looking at the content, could be, not only visualized, but also problematized and the critical aspects could explicitly be discerned. The students also got the opportunity to more explicitly express what they had discerned and were able to justify their answers more distinctly.