Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simon line Yip-Cheung CHAN Faculty of Education The University of Hong Kong Hong Kong The author acknowledges the financial supports of HKU CRCG Conference Grants for Research Students & Sik Sik Yuen Education Research Fund.

How do experimentation (realized by visual feedback of DGE software) and theoretical consideration (based on user’s mathematical knowledge) interplay in the process of dynamic geometry exploration?

An empirical study This is part of the author’s PhD research project. Here, I will report the exploration processes of 2 participants individually working on a mathematics task using Sketchpad (a DGE). They represent 2 different approaches of interplay between experimentation and theoretical consideration in the exploration process.

Mathematics task Use Sketchpad to construct the following geometrical configuration: ABC is a triangle on a plane. P is an arbitrary point on that plane. Let X, Y, Z be the feet of the perpendicular lines drawn from P to the sides AB, BC and AC respectively. (Sides AB, BC and AC can be extended if necessary.) Use Sketchpad, find all possible positions of P such that X, Y, Z are collinear (i.e. X, Y, Z lie on the same straight line). You may use any exploration techniques and dragging strategies learnt in earlier sessions. Write down what you have found.

Simon line Consider triangle ABC. Let P be a point and X, Y, Z be the feet of perpendicular lines drawn from P to the (extended) sides AB, BC and AC respectively. The locus of P such that X, Y, Z are collinear is the circumscribed circle of triangle ABC. Posamentier, A. S. (2002). Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students. Emeryville, California: Key College Publishing.

All participants started the exploration in more-or- less same ways. They first constructed the figures by Sketchpad. Some participants used colour lines as visual aids. Then, they randomly dragged P aimed at finding the locations of P where X, Y, Z were visually collinear. In other words, they tried to get some feelings to the question and search a possible direction of exploration.

Case 1: Tracy (All names reported in this presentation are pseudonyms.) Female Secondary school mathematics teacher Obtained a B.Sc. in mathematics Studying (part-time) M.Sc. in mathematics Knew basic commands of Sketchpad before participating this research project but seldom uses this software in her teaching

A line joining X & Z was constructed. This served as a “guide” for locating suitable positions of P.

P was randomly dragged until Y lied on that line (drag-to-fit strategy)

Trying to “mark” the suitable positions by “trace”

The “trace” method seems not satisfactory. A new method of marking the found positions was invented.

Repeated this method to locate and mark suitable positions of P.

Based on her intuition, she guessed that the locus is a circle. But what kind of circle is it?

She constructed the circumscribed circle by locating the intersection of perpendicular bisectors (circumcenter) and verified that all marked points lied on this circle.

P was merged onto the circumscribed circle. Animation was used to show that the required locus is the circumscribed circle, as expected.

Another line joining X, Y was constructed to ensure X, Y, Z were collinear.

Critical features observed in Tracy’s exploration She failed to use “trace” to mark the suitable positions  led to inventing a new method Wandering dragging of P  P was accidentally positioned at vertex C  led to making the conjecture (the locus is the circumscribed circle) She was unsatisfied to visual determination of collinearity of the points  Constructed 2 lines and checked whether they were overlapped.

Interplay between experimentation and theoretical consideration in Tracy’s exploration Experimentation took the “prominent role” throughout the exploration process. Exploration depended mainly on visual feedback of Sketchpad Math knowledge took the “subordinate role” mainly for the purpose of development of utilization skills and techniques

Case 2: Kelvin (All names reported in this presentation are pseudonyms.) Male Secondary school mathematics teacher Obtained B.Sc. & M.Phil. in mathematics Studying (part-time) M.Ed. Familiar with Sketchpad before participating this research project Sometimes use this software as an auxiliary tool in his teaching

A line joining X & Y was constructed. This served as a “guide” for locating suitable positions of P. Then, P was randomly dragged until Z lied on that line. This method was repeated to locate other positions of P.

Guesses were made by intuition based on the figure shown on screen.

Guess 1: XP is parallel to BC

Guess 2: XPYB is concyclic

Working on paper: Use coordinate geometry

After tedious computations by using coordinate geometry, he found that the locus of P is the circle represented by From this equation, x-coordinate of center = ½. But what does this mean?

“The center lies on the perpendicular bisector of one of the sides of the triangle [in this case, side AB]. But, by symmetry, it should…. Let’s try side BC as well.” “The center should be the intersection point of perpendicular bisector of AB and perpendicular bisector of BC.” “But, what is the radius?”

Trying to find the radius. Finally, he concluded that the locus should be the circumscribed circle of triangle ABC.

Critical features observed in Kelvin’s exploration He failed to mark the suitable positions of P (“put a nail on the suitable position”) (“trace could not help”)  This led him to switch the path of exploration (coordinate geometry) He succeed to find the equation of the locus (indeed, a “rigorous” proof was found), but he did not satisfy to this answer because it could not give a geometric meaning. Instead, Sketchpad experimentation gave him geometric insight and led to mathematical understanding.

Interplay between experimentation and theoretical consideration in Kelvin’s exploration Experimentation & theoretical consideration were partner, i.e. they complemented to each other. Experimentation & theoretical consideration “worked hand-in-hand” in the exploration process. Both experimentation and theoretical consideration significantly contributed to the discovery of geometric property.

Discussion Different approaches of interplay between experimentation and theoretical considerations are observed. How does it relate to the user’s background, e.g. perception of DGE, perception of mathematics, experience of using DG software and mathematics background? How does it relate to the specific mathematics task?

Further discussion Different people perceived and used Sketchpad differently. Tracy’s Sketchpad, Kelvin’s Sketchpad,…. A central idea of instrumentation theory is the development of utilization scheme. This study may contribute to an understanding on how to develop utilization schemes in DGE. Implication to future development in interactive software: As different people use interactive software differently, flexibility for catering individual differences should be one of the major concerns in future development of interactive software. (e.g. “put a nail on the position”)

Further questions What is the best way to describe the interplay between experimentation and theoretical consideration in DG exploration? How does the description of interplay help us understand the user’s utilization schemes (in the terminology of instrumentation theory)? How does the interplay relate to other theoretical perspectives, e.g. variational dragging scheme (Leung, 2003, Leung & Chan, 2006, Leung, Chan & Lopez-Real, 2006)?

Questions, comments and suggestions are welcome! Thank you!