Designing Learning Environments for Developing Understanding of Geometry and Space

may be a sizable stumbling block for the introduction of prevector activities into the lower grades and must be attended to carefully in any instructional design.

Third, the type of material used in building objects relates significantly to children's perceptions of force and stability. The use of flexible foam (with marked cross sections) was effective in drawing students' attention to the fact that tension and compression coexist in most simple situations. (Note: Some arches are entirely compression systems.) However, it also introduced torsion and dead spots that drew children's attention away from the direction of the forces they were applying. Moreover, because we introduced the flexible foam prior to the rigid bamboo, the children tended to apply their thinking in the foam situations to the situations involving the bamboo.

Fourth, students' use of drawings seemed to be far more advanced than were their verbal abilities to describe the stability of the figures they were manipulating.

These preliminary findings suggest that curriculum developers pay careful attention to the ways in which realistic engineering activities are designed if they intend that children abstract relationships between geometry and structure. Variation in the complexity of figures, the types and amount of materials used, and the introduced notations all influenced children's conceptions of the geometry of structure. Nevertheless, engineering and design activities that build on children's intuitions about stability appear promising for helping to expand children's connections between structure and space.

Aguirre, J., & Erickson, G. ( 1984). "Students' conceptions about the vector characteristics of three physics concepts". Journal of Research in Science Teaching, 21( 5), 439-457.

Aguirre, J. M., & Rankin, G. ( 1989). College students' conceptions about vector kinematics. Physics Education, 24, 290-294.

Battista, M. T. ( 1994). "On Greeno's environmental/model view of conceptual domains: A spatial/geometric perspective". Journal for Research in Mathematics Education, 25( 1), 86-89.

Bohm, D., & Peat, R D. ( 1987). Science, order and creativity. New York: Bantam.

Clements, D. H., & Battista, M. T. ( 1992). "Geometry and spatial reasoning". In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464), New York: Macmillan.

de E. Moor ( 1991). "Geometry instruction in the Netherlands (ages 4-14)--The realistic approach". In L. Streefland (Ed.), Realistic mathematics education in primary school: On the occasion of the opening of the Feudenthal Institute (pp. 119-138). Utrecht, the Netherlands: CD-ß Press.

Freudenthal, H. ( 1983). Didactical phenomenology of mathematical structures. Dordrecht, the Netherlands: Reidel.

Gravemeijer, K. ( 1994). "Educational development and developmental research in mathematics education". Journal for Research in Mathematics Education, 25( 5), 443-471.

Hoffer, A. ( 1983). "Van Hiele-based research". In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 205-227). New York: Academic Press.

-264-