The online interactive sketches that you are invited to investigate have been designed for use in a small study with grade 7 students. They involve reflections, rotations, and translations and have been prepared with The Geometer’s Sketchpad, a dynamic geometry program licensed for use by the Ontario Ministry of Education, then saved as html files. The resulting JavaSketches can be viewed with a browser.

Dynamic geometry is the exploration of geometric relationships by observing geometric configurations in motion. Although these configurations are usually constructed by the students themselves, web-based sketches can also be used. My thesis research focused on examining the use of pre-constructed, web-based dynamic geometry sketches in a small case study in two Ontario secondary schools. This limited study indicated that web-based sketches can provide opportunities for students of different ability levels to investigate geometric ideas, and to develop visual, analytic, and deductive reasoning skills. Since many geometric concepts are formed before students enter secondary school, my present research is now focused on the use of these sketches by students in late elementary –specifically grade 7.

I have included a brief overview of the relationship to the literature for those who are interested. Otherwise you may go directly to the sketches at: http://www.yorku.ca/sinclair
From that page please Follow the link under Fields Online Math Conference.

Background

The growing importance of visual and spatial understanding in our world has led to recommendations for an increased emphasis on a visual approach to the teaching of mathematics (cf. Eisenberg & Dreyfus, 1991; Barwise & Etchemendy, 1998). In their summary analysis of research into geometry and spatial reasoning, Clements and Battista (1992) concluded that students can also gain greater understanding of spatial relationships by constructing and exploring a model created with dynamic geometry software.

Extensive studies of Cabri, a dynamic software program, have been undertaken by researchers such as Balacheff, Mariotti, Laborde and Straesser and have led to an understanding of some components of Cabri's "epistemological domain of validity" (Arzarello et al, 1998). For example, we know that a geometry problem cannot be solved simply by perceiving the images on the Cabri screen, even if these are animated. The student must bring some explicit mathematical knowledge to the process. A similar systematic analysis is needed to reach an understanding of the underlying epistemology of pre-constructed web-based dynamic diagrams.

Much early research with geometry software focused on elementary students and the use of Logo to develop concepts of shape and measurement (Papert, 1980; Clements and Battista, 1989; Cohen and Geva, 1987). However, since the advent of dynamic geometry software the emphasis has been on studies related to proof in secondary classrooms (Olive, 1998; de Villiers, 1998; Hadas and Hershkowitz, 1999; Mariotti, 2000; Marrades and Gutierrez, 2001). Currently, as noted by Jones (2000) there is a need for research on the use of dynamic software at the elementary level where the foundational geometric understanding is developed.

There is also a need for research to inform the continuing discussion about whether it is better to give students powerful general-purpose programming and construction tools or to have them interact with pre-constructed, interactive models. Wilensky (1990) strongly supports student constructions because he believes that students develop a deeper understanding of the object by explicitly connecting the parts. LOGO researchers found that the activity of constructing shapes increased elementary students' ability to describe geometric relationships (cf. Cohen and Geva, 1987; Kieran and Hillel, 1990; Hoyles, 1991). Support for constructing also comes from researchers who have examined the dynamic geometry environment. Hoyles and Noss (1994) maintain that when an improperly constructed figure falls apart under dragging, the student is forced to notice relationships among the geometric objects. Hadas and Hershkowitz (1999) point out that the experience of not being able to construct an object that seems intuitively possible to make, is a powerful incentive for students to investigate geometric ideas.

While not minimising the importance of constructions, some researchers believe that pre-constructed sketches can also play an important role in the development of mathematical understanding. Whiteley (personal communication, 2000) contends that pre-constructed dynamic geometry diagrams are valuable as learning tools because ability to recognise the connections between geometric objects is necessary before students can effectively carry out many constructions. Schumann and Green (1994), one of the few research teams to focus on the use of ready-made sketches, found that pre-constructed sketches are especially helpful for weaker students.

In their research, Schumann and Green used pre-constructed diagrams of the type that accompany Cabri or The Geometer's Sketchpad. The difference between these sketches and web-based sketches is that additional construction details can be added by the student. Since most web-based sketches do not permit the addition of constructed elements, research on interpreting mathematical images is also pertinent to this project. Goldenberg, Cuoco and Mark (1998) report that mathematical pictures and diagrams are difficult to interpret because they contain a great deal of information, represented in a concise but "nonsequential" format. Wheatley (1998) contends that the meaning we take from an image depends in part on what we know about what we are looking
at. These two conclusions have important implications for the use of pre-constructed diagrams. If we are the 'builder' of a mathematical configuration we are aware, at some level, of underlying relationships in the diagram. If we are interpreting someone else's construction there will be gaps in our knowledge. Research is needed to determine whether this hampers our ability to read the information and, if it does, how we can produce diagrams that help the viewer overcome the lack of background knowledge.

The project I am presently involved in will begin to address these identified research needs by preparing and testing sketches and accompanying materials in preparation for a long-term study to extend our understanding of the use of pre-constructed web-based dynamic geometry sketches in elementary classrooms. Such an extended study is needed to determine if results of dynamic-geometry research, pertaining to student-constructed dynamic geometry diagrams, is applicable to pre-constructed web-based dynamic diagrams. For example, in their study of the transition from exploring with Cabri, to conjecturing and proving, Arzarello, Micheletti, Olivero, Robutti, Paola, and Gallino (1998) classified modalities of dragging and identified those that were linked with productive student reasoning. One modality, "dragging test," is associated with constructing. When a figure has been constructed correctly it will pass the dragging test, that is, it "will not be messed up by dragging" (p. 33). The other two forms, "wandering dragging" and "lieu muet" (dummy locus), are associated with student explorations of already constructed models. It appears that the two modalities not linked to construction might apply to web-based diagrams. During the piloting phase of this project, we will be focusing on identifying and eliminating technical and design problems, to produce sketches that will help determine to what extent prior research results with dynamic sketches apply to pre-constructed web-based diagrams.

Recent research has also examined how we take meaning from an image that is moving--a topic which has important implications for the design of web-based dynamic diagrams. Rensink (2000) in a study of change blindness reports that we are only able to grab four to six visual objects at once and that focused attention is needed to notice change. In earlier research, I reported that colour and motion helped students notice mathematical details, but that measurements went largely unnoticed despite the fact that they were continually updated under dragging.

Developing the mathematical tasks around the sketches is as important as designing the sketches themselves.. Hadas and Hershkowitz (1999) have done extensive research on the activities that students undertake using regular dynamic software, but research is needed to establish guidelines for tasks that are designed around web-based dynamic sketches.

Bibliography

Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in Cabri and modalities of transition from conjectures to proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 32-39). Bellville, South Africa: Kwik Kopy Printing.

Barwise, J. & Etchemendy, J. (1998). Computers, visualization, and the nature of reasoning. In T. W. Bynum and J. H. Moor, (Eds.), The digital phoenix: How computers are changing philosophy. London: Blackwell. 93-116.

Clements, D. H., & Battista, M. T. (1989). Learning of geometric concepts in a Logo environment. Journal for Research in Mathematics Education, 20(5), 450-467.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 420-464). New York: MacMillan Publishing Co.

Cohen, R., & Geva, E. (1987). The effects on young children of learning turtle geometry programming through the use of Logo Microworlds. Toronto: Ontario Institute for Studies in Education.

de Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 369-394). Mahwah, NJ: Lawrence Erlbaum Associates.

Goldenberg, E. P., Cuoco, A. A., & Mark, J. (1998). A role for geometry in general education. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 3-44). Mahwah, NJ: Lawrence Erlbaum Associates.

Hadas, N., & Hershkowitz, R. (1999). The role of uncertainty in constructing and proving in computerized environments. In O. Zaslavsky (Ed.), Proceedings of the 23rd PME Conference (Vol. 3, pp. 57-64). Haifa, Israel.

Hoyles, C. (1991). Microworld/schoolworlds: The transformation of an innovation. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 1-17). New York: Springer-Verlag.

Hoyles, C., & Noss, R. (1994). Dynamic geometry environments: What's the point? Mathematics Teacher, 87(9), 716-717.

Jones, Keith. (2000). Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics. 44: 55-85.

Kieran, C., & Hillel, J. (1990). It's tough when you have to make the triangles angle: Insights from a computer-based geometry environment. Journal of Mathematical Behavior, 9, 99-127.

Mariotti, Maria Alessandra. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics. 44: 25-53.

Marrades, Ramon & Gutierrez, Angel. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics. 44: 87-125.

Olive, J. (1998). Opportunities to explore and integrate mathematics with the Geometer's Sketchpad. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 395-418). Mahwah, NJ: Lawrence Erlbaum Associates.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books, Inc.

Rensink, R. A. (2000). The dynamic representation of scenes. Visual Cognition, 7, 17-42.

Schumann, H., & Green, D. (1994). Discovering geometry with a computer - using Cabri Géomètre. Sweden: Studentlitteratur, Lund.

Wheatley, G. H. (1998). Imagery and mathematics learning. In M. Sharma, J. Schmittau, & V. Schell (Eds.), Focus on Learning Problems in Mathematics, 20 (2 & 3), 65-77.

Wilensky, U. (1990). Abstract meditations on the concrete and concrete implications for mathematics education [Online]. Available: www.tufts.edu/~wilensk/papers/concrete.htm