1. ORDER OF CONSTRUCTION in DYNAMIC GEOMETRY ENVIRONMENT Varda Talmon The University of Haifa, Israel CET- The Center for Educational Technology, Israel Under supervision of Prof. Michal Yerushalmy ATCM 2004, Singapore

2. This presentation is a part of a larger study (My PhD dissertation) that examines various instruments (Artigue, 2002; Mariotti, 2002; Rabardel & Bourmaud, 2003) developed by users while using dragging in DGEs[1]. It addresses one aspect of dragging: the connection between dynamic behavior and the sequential order of construction. Introduction [1] This presentation deals with two DGEs: The Geometer’s Sketchpad (Jackiw, 1995), and The Geometric Supposer (Schwartz, Yerushalmy, & Shternberg (1998). Talmon V. ATCM 2004, Singapore

3. Dynamic Behavior Dragging produces a Dynamic Behavior (DB) for each element in the construction. DB refers to the degree of freedom of the dragged element (is it possible to drag the element, and if so, along what path?) and to the response of related elements (changes and invariance during the dragging). Talmon V. ATCM 2004, Singapore

4. Decisions based on the axiomatic system and concept of Euclidean Geometry Decisions based on the axiomatic system and concept of Euclidean Geometry Design decisions User's decisions Decisions based on the axiomatic system and concept of Euclidean Geometry Decisions based on the axiomatic system and concept of Euclidean Geometry Design decisions User's decisions background Three sets of decisions determine the Dynamic Behavior of a figure within DGE: Talmon V. ATCM 2004, Singapore

5. Decisions based on the axiomatic system and concept of Euclidean Geometry Example: The DB of a figure reflects the definition on which its construction procedure is based. The construction of the first parallelogram is based on the proposition that if the two diagonals of a quadrangle intersect one another in the middle, than the quadrangle is a parallelogram. The second construction is based on the proposition that if a quadrangle has two pairs of parallel sides, than the quadrangle is a parallelogram. Two diagonals that intersect one another in the middle Two pairs of parallel sides For Example, these two parallelograms: Talmon V. ATCM 2004, Singapore

6. Design Decisions For Example: The Supposer main menu includes a Shapes Menu that provides pre-constructed well known geometric objects. This allows students to inquire a figure even if they do not know how to construct it. Let us assume that a student is asked to inquire the properties of the rectangle. Using the Shapes Menu, the student browses through the parallelograms family and gets a pre-constructed rectangle. The student can also use the Quad Menu and construct his own quadrangle by determining the dimensions of its diagonals. Talmon V. ATCM 2004, Singapore

7. Design Decisions The DB: The first rectangle is dynamic. All its vertices and sides are draggable. The second rectangle is fixed and obviously cannot be changed by dragging. Browsing through the hierarchy of the parallelograms family. Talmon V. ATCM 2004, Singapore

8. Design Decisions On the other hand, the option Commands in the Construct menu of the Sketchpad is enabled only when the appropriate prerequisites have been selected (the objects used to define the construction). This requires pre-knowledge: User must know a procedure to construct the figure based on the geometric properties of the object. Talmon V. ATCM 2004, Singapore

9. Design Decisions Talmon V. ATCM 2004, Singapore As we just have seen, the DB depends on the designers decisions about the modes of the construction.

10. User’s Decisions To construct a figure within DGE, the student has to create a geometrical procedure of the construction. One of his main decisions concerns the sequential organization of the procedure (order of construction). The connection between Dynamic Behavior and the Order of Construction will be our focus in the time left for this presentation. Talmon V. ATCM 2004, Singapore

11. The sequential organization of a procedure in DGE produces a hierarchy of dependencies, as each part of the construction depends on something created earlier (Jones, 2000). This hierarchy of dependencies is one of the main factors that determine DB within DGE (Jackiw & Finzer, 1993; Laborde, 1993). This order is often explicitly organized by terms such as parent and child. hierarchy of dependencies & Dynamic Behavior An element that is related to a previous one The previous element Talmon V. ATCM 2004, Singapore

12. Order on paper & Order in DGE On paper the order of construction is central in complex or advanced constructions, but it is not critical in many basic constructions. For example, a point on a line or a line through a point are geometrically identical in a paper sketch, but these two constructions have different DBs. On paper: A ∋ a In DGE: C on ABIn DGE: Line through A Talmon V. ATCM 2004, Singapore

13. Goldenberg et al. (In preparation) raise the following question: what do users expect would be the effect of dragging point B on point E placed arbitrarily on a line perpendicular to AB. The study: Users expectations Dragging point B using the Sketchpad: Dragging point B using the Supposer and the Cabri: Talmon V. ATCM 2004, Singapore

14. Users expectations & Order of construction We took this question further and conducted a study on the issue of understanding the DB of geometrical objects in a Dynamic Environment. One of the main questions of the study was: what are users expectations of DB and interpretations of the connection between the order of construction and the DB of its elements in terms of parent-child relations? Talmon V. ATCM 2004, Singapore

15. The Study The main tool of the study was an interview consisting of three parts: 1. To perform a given procedure within one of the two following environments: the Sketchpad or the Supposer. 2. To predict the dynamic behavior of the constructed geometric object. We used transparencies to demonstrate the predictions. 3. To drag several points of the object and explain the specific behavior they saw on the screen. 15 ninth-grade pupils and 10 M.A. Math-Education students participated in the study. Talmon V. ATCM 2004, Singapore

16. Shoval: A 9 th grade student Varda:Let us look at point C. Can we drag point C? Shoval: Yes, we can. But the whole shape will be preserved. Varda:First of all, tell me if we can drag point C. Shoval: Where to? Varda: Wherever you like. Shoval: We can drag it [point C] but the construction will move too. On the screen: To demonstrate his predictions, I placed a transparency on the computer screen, and Shoval copied the figure he constructed and (while the transparency was still on the screen) made a sketch representing his conjecture about the shape of the figure after dragging point C. After following a given construction procedure using the Supposer, Shoval predicts the DB of point C. Talmon V. ATCM 2004, Singapore

17. Shoval’s transparency page Shoval predicts that point C (a point on a line) can be dragged freely. He predicts that dragging point C would affect other elements of the construction including point C’s parents (it relocates B) Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E On the screen: Shoval copies “a triangle” (an object he didn’t construct) He predicts that dragging point C would translate the “whole triangle” Here is his drawing: Talmon V. ATCM 2004, Singapore

18. Avi’s transparency page: Avi predicts that dragging point B would not relocate Its child (E) but would affect its parents (Break AB). Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E Avi: A 9th grade student Talmon V. ATCM 2004, Singapore

19. We found 7 Categories of Reverse- Order predictions of DB Prediction 2 Prediction 2 – A constrained point (a point on a figure, e.g. on a segment or a circle) can not be dragged. Prediction 1 Prediction 1 – A free point (without any constraints) cannot be dragged Prediction 3 Prediction 3 – A constrained point can be dragged without any constraints. Prediction 4 Prediction 4 – Dragging a child changes the location of its parent. Prediction 5 Prediction 5 – The existence of a child affects the Dynamic Behavior of its parent. Prediction 6 Prediction 6 – Dragging a parent does not change the location of the child. Prediction 7 Prediction 7 – Dragging a point changes the location of a free point. Talmon V. ATCM 2004, Singapore

20. Examples of the 7 Categories of Reverse-Order predictions of DB Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E Procedure A: 1. AB (j) is an arbitrary segment 2. C is a point on AB (j) 3. (k) is a perpendicular to AB (j) through C 4. E is a point on (k) 5. EB (l) is a segment between B and E Prediction 1 A (B) can’t be dragged Prediction 2 C can’t be dragged Prediction 5 C can’t be dragged because of k Prediction 3 C can be dragged Freely Prediction 7 Dragging C relocates A (B) Prediction 6 Dragging A doesn’t relocate C Prediction 4 Dragging E Relocates C Talmon V. ATCM 2004, Singapore

21. Explanations: 1. The extent to which the figure-image held by the interviewees is reflected in their reverse – order predictions. Two possible reasons: 2. The extent to which the dragging – image held by the interviewees is reflected in users' predictions as a mathematical transformation. The action of dragging was sometimes interpreted by users as a mathematical transformation such as Translation, rotation or homothetic. This was reflected in their predictions as they seemed to deduce properties of dragging from the properties of these three transformations. We found that interviewees tend to relate to figures that can be seen on the screen but are not logically recognized by the software. They use these figures to explain their predictions. For example, three segments are treated as a triangle. Talmon V. ATCM 2004, Singapore

22. Why does this matter? Dynamic Geometry Environments (DGEs) have become standard tools for students, teachers, and mathematicians. Moreover, using DGE established new norms of learning geometry and deepen our understanding of tool-situated learning. Dragging geometric constructions is one of the fundamentals of these environments and unquestionably enhances the esthetics and power of DGEs. At the same time it changes the nature of the dependencies declared in the construction. Thus it introduces new complexity to learning using DGE. It seems that Euclidean axioms and theorems only partially influence the DB of constructions. Talmon V. ATCM 2004, Singapore

23. Why does this matter? To understand more fully the stages of instrumental genesis of DGE and software design decisions, questions such as, “What do users observe while dragging one element of the construction?”, “How do users interpret DB, and what are their perceptions of it?”, should be further investigated. The study defines categories and terms that could hopefully contribute to the crystallization and depth of further research questions as well as to curriculum and software design considerations. Therefore… Talmon V. ATCM 2004, Singapore

24. Acknowledgment Special thanks to Michal Yerushalmy, for being a teacher and a friend, and to Beba Shternberg, Nicholas Jackiw and Paul Goldenberg for discussing with me the ideas that appear in this presentation. Talmon V. ATCM 2004, Singapore