Studies of Conceptual Development
From the questioning of basic assumptions of how we teach traditional geometric concepts to the increased interaction of reasearchers and teachers in creating research-based, teacher-designed activities, researchers with their colleagues, classroom teachers, have begun to rethink what happens in mathematics (in particular, geometry) classrooms.
In this section, we sample that range of work, looking at models of student thinking; studies of interactive, hands-on and computer- enhanced methods of exploring geometric concepts; and the emergence and growth of students' reasoning from the primary grades through high school. In chapter 5, Pegg and Davey graft the SOLO taxonomy of modes and levels of thinking ( Biggs & Collis, 1991) onto the main trunk of the van Hiele cognitive model. They examine what is perhaps the predominant theory in K-12 geometry education--the model of "levels" of teaching and learning proposed by the van Hieles--and take the position that although the van Hiele theory offers a broad framework in which to view cognitive growth in geometry, the theory is not sufficiently fine grained to account for individual differences. The synthesis they propose suggests a way to fine-tune how we look at cognitive reasoning and conceptual development.
Lehrer, Jenkins, and Osana, in chapter 6, report on a 3-year longitudinal study of elementary-grade students that questions (as do Pegg and Davey in chap. 5) the adequacy of the van Hiele model as a description of the progression of children's thinking. Their broad-scale portrait of children's emergent spatial reasoning skills suggests that current curricular practices in elementary school promote little conceptual development and that, for many children, the opportunities to develop and integrate spatial reasoning skills may substantially diminish before they leave elementary school.
In the next chapter, Lehrer, Jacobson, et al. suggest both that geometry education begins in students' informal knowledge and grows in a classroom culture that depends on skilled teachers, teacher-developed models of student cognition, and a comfortable interface between research (and researchers) and teachable moments (and teachers). This particular collaboration between researchers and elementary-grades teachers worked to design activities that stimulated children's development of spatial reasoning and their ability to conjecture. The authors trace the growth and development of children's conceptions of area and its measure as teachers structure instruction, based on children's informal knowl