Dynamic Geometry Environments (DGE)動(dòng)態(tài)幾何環(huán)境（DGE）

1、.The dragging process in 3D environments and gender differences by using the drag modeMathias HattermannInstitute for mathematics educationJustus Liebig-University, Gießen, Germany Working group 3: Information technologies in teaching and learningShort description of dynamic geometry environmen

2、ts (DGE)In the last three decades technology offered new possibilities to improve teaching and learning of geometry in a special way. Mathematics educators see dynamic geometry environments as a powerful tool to enrich and further the learning process in regular mathematics classrooms. The first dyn

3、amic geometry environments (DGEs) appeared in the eighties with the goal to provide a family of diagrams as representing a set of geometrical objects and relations instead of a single static diagram. DGEs try to imitate Euklids geometry and are characterized by three properties. The drag-mode allows

4、 the user to drag e.g. the points of a given drawing and the diagram will change according to the new position of the point, while all the geometrical relations used in the construction are preserved, e.g. the mid-perpendicular of a segment will stay the midperpendicular after a point of the segment

5、 has been dragged. The second property of DGEs is the possibility of macro-constructions, which allow the user to group several construction steps under one command. For example, with the help of a macro-construction it is possible to construct the tangents of a given circle through a given point, w

6、hich is located outside the circle with the help of one command. The third property of DGEs is the command “trace or locus of points”. With the help of this command, the user is able to investigate the locus of a special point, by dragging a basic point. For example, the user can figure out what cur

7、ve is described by the intersection point of two heights of a given triangle when the given point C is dragged on a straight line.The most popular DGEs for plane geometry are Cabri II Plus, Geometers Sketchpad, Cinderella 2.0. and others. In German schools, “Euklid DynaGeo” is used very often.DGE, o

8、ne of the best-researched type of software within didactics of mathematicsThe developers of DGEs achieved the aim to develop programs which follow the laws of geometry while the user is manipulating points, e.g. by using the drag mode, just like material objects react by following the laws of physic

9、s. For mathematicians, teachers and developers these programs are powerful tools which act in a logical way. However, for students the situation is quite different.When DGEs are used in mathematical classrooms, problems occur which are not relevant in a paper and pencil environment. Investigations o

10、n different issues were made by researchers to prevent mistakes by using DGE and to improve lessons. Topics of investigation concerning dynamic geometry environments were the move from the spatial to the theoretical, the use of the drag mode, proving and justifying processes, relationships between g

11、eometric and functional aspects in a study of transformations (locus and trace) to mention only the main themes (cf. Arzarello et al., Jahn, Kadunz, Laborde et al.).· Dynamic Geometry Environments and the move from the spatial to the theoreticalStudents have problems by interpreting diagrams, w

12、hich are resistant under the drag mode. The students often do not relate this behaviour to geometrical independence from the dragged points. In students minds, there exist two separate “worlds: the mechanical world of the computer diagram (the spatial) and the theoretical (or geometrical; see Soury-

13、Lavergne 1998). As a consequence, big problems in understanding the dependency relationship in a DGE occur for students not seeing a relation between the spatial-graphical level and the theoretical level (Hoyles, 1998; Jones, 1996). · The drag modeThe drag mode is a key element in DGEs and the

14、most important tool provided by the software. It could be shown that students do not use the drag mode from the beginning, they are careful by using the drag mode because they do not want to destroy their drawing (Rolet, 1996). The importance of the teachers role in a DGE by using the drag mode is m

15、entioned by Straesser (1992).Numerous investigations about the way dragging is used by students in solving problems have been undertaken. Arzarello and Olivero distinguish different kinds of dragging, in particular- wandering dragging is moving the points on the screen randomly in order to discover

16、configurations, - guided dragging is done with the intention to obtain a particular shape,- lieu muet dragging is moving a point with the constraint of keeping a particular property satisfied at the initial state, the variable point follows a hidden path even without being aware of this.(Arzarello,

17、Olivero 2002)To sum up: Several schemes of utilization of the drag mode are constructed by students and the using of the drag mode sometimes differs from developers, researchers and teachers intention. · Proving and justifying processesA highly disputed point in discussions between researchers

18、is the role of the computer in proving processes. On the one hand the “authority” of the computer could lead to a greater resistance to proving, because students do not see the need of it. On the other hand, an adequately chosen and presented proof could further students to see sense for the need fo

19、r proofs. The role of measurement, a tool which is provided by most DGEs, in proving processes was scrutinized in several studies ( Hollebrands, 2002; Vadcard, 1999). There exist novel ways offered by DGS enhance understanding and the need for proofs.o Students must give explanations for the fact th

20、at a drawing remains inflexible by using the drag mode (Jones, 2000)o A teaching experiment is designed in which students produce deductive arguments to justify the correctness of their constructions. The critical role of the teacher in guiding the discussion is part of the study (Marrades&Gutie

21、rrez, 2000)o With the help of an adequate sequence of tasks in a DGE researchers achieve the aim to generate an inner conflict in students minds about why an unexpected property is true and further the need for proofs with this course of action (Hadas, Hershkowitz & Schwarz, 2000).Recent develop

22、ment concerning DGEsIn the last two years researchers developed three-dimensional geometry software, which fulfils the definition of a DGE more or less. Cabri 3D and Archimedes 3D support the drag mode, while to my knowledge - Archimedes 3D is the only software supporting the macro construction and

23、the “trace function of points”, which is expanded in this environment to “trace of surfaces”.As far as I know, there is nearly no research in the use of three dimensional DGEs, but developers claim that students do not use the drag mode as in two dimensional environments. First they do not use it at

24、 all and try only to investigate the drawing from different perspectives without using the drag mode. After encouraging the students to use the drag mode by the teacher it is remarkable that students can drag points only on one surface without interrupting the process of dragging in three dimensiona

25、l environments. It is interesting that dragging in 3D-environments is something different from the use in 2D-environments, where the drag mode is easier to handle. In fact, the control of dragging in 2D-environments is far more direct than in 3D-environments. Research interestI am interested in the

26、use of DGE in (German) schools and would like to scrutinize students behaviour by using DGE, especially the drag mode in three-dimensional environments in an explorative study. An explorative study of students use of drag mode within 3D-environments could even throw some light on results for 2D-drag

27、ging. In this study and as a second research interest, I would like to analyse differences (if existent?) in gendered use of the drag mode. Some research studies suggest that there are differences between visualization, spatial relations and spatial orientation according to gender. I want to investi

28、gate if it is possible to empirically find differences in using the drag mode in a three dimensional environment between men and women concerning spatial relations or spatial orientation. If existent, an interpretation of these differences would be valued.As for methodology, I would like to scrutini

29、ze students behaviour in an empirical study trying to adapt the methods of Arzarello and his group. Consequently, a comparison with results from Arzarello et al. (for the two dimensional, planar case) with results from a 3D-study is in place. References:Arzarello, F., F. Olivero, et al. (2002). A co

30、gnitive analysis of dragging practises in Cabri environments. Zentralblatt für Didaktik der Mathematik 34(3): 66-72. Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studi

31、es in Mathematics, 44 (1-3), pp.127-150.Hollebrands K. (2002). The role of a dynamic software program for geometry in high school students developing understandings of geometric transformations. In D. Mewborn (Ed.), Proceedings of the 24th PME-NA Annual Meeting, pp.695-706.Hoyles.C. (1998). A cultur

32、e of proving in school mathematics. In D.Tinsley & D.Johnson (Eds.), Information and communication technologies in school mathematics (pp.169-181). London, UK: Chapman and Hall.Jahn, A. P. (2002). "Locus" and "Trace" in Cabri-géomètre: relationships between geometri

33、c and functional aspects in a study of transformations. Zentralblatt für Didaktik der Mathematik 34(3): 78-84. Jones, K. (1996). Coming to know about dependency within a dynamic geometry environment. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th PME International Conference, 3, pp. 145-151.Kadunz, G. (2002). Macros and Modules in Geometry. Zentralblatt für Didaktik der Mathematik 34(3): 73-77. Marrades, R.&Gutierrez, A. (2000). Proofs produced by secondary school students