Suggested Enrichment Program Using Cinderella (DGS)
© 20072008 Mohamed ElDemerdash
The suggested enrichment program, which is presented here, is a main part a Ph. D. thesis that is registered in the university of education in Schwäbisch Gmünd since January 2007 under supervising of Professor Dr. Ulrich Kortenkamp. The working title for this thesis is: "The effectiveness of an enrichment program using dynamic geometry software in developing mathematically gifted students’ geometric creativity in the high schools".
One of the thesis purposes is to develop a suggested enrichment program in Euclidean geometry to enhance the geometric creativity of the mathematically gifted students in high schools using dynamic geometry software, Cinderella application. In the coming sections, much details about the suggested enrichment program will be presented in terms of: why has Cinderella been used in developing the suggested enrichment program? principles, aims, content, and appropriateness of the suggested enrichment program as well as the role of both the student and the teacher in the suggested enrichment program.
Why Cinderella?
Even though there are many computer software applications for doing mathematics in general and for doing geometry in particular, the present researcher chose Cinderella to use in developing the suggested enrichment program because Cinderella may provide the mathematically gifted students with a suitable dynamic environment in which they can engage in exploring, conjecturing, testing, and confirming geometric relationships. Moreover, Cinderella has special features that could be useful in teaching and learning geometry among the mathematically gifted students such as:

Cinderella provides an environment, with a range of geometric objects and tools, in which the mathematically gifted students can not only simply construct geometric constructions with extreme accuracy and ease but also interactively explore the dynamic behavior of the constructed figures through the dynamic tools, which are embedded in Cinderella.

Cinderella special facilities of constructing, dragging, measuring, calculating, animating, and tracing loci provide many opportunities to the mathematically gifted students to dynamically visualize a variety of geometric situations and problems.

Cinderella has builtin automatic proving facility. This feature enables the mathematically gifted students to automatically check the correctness of their mathematical ideas and conjectures.

Cinderella has the possibility to directly publish interactive figures onto the Web. This enables the mathematically gifted students not only to immediately publish their work on interactive web pages but also to communicate mathematically with others through the World Wide Web, the electronic mail, and the web pages.

Cinderella has powerful modes for geometric transformations (e.g. reflection, translation, rotation, similarity, among others). These modes of geometric transformations could be very useful in simplifying constructions for the mathematically gifted students.

Cinderella has the possibility to design interactive exercises as web pages that mathematically gifted students can solve on their own.
Principles of Developing the Suggested Enrichment Program
Based on reviewing the related literature and prior studies, the present researcher was able to determine the following principles for designing and developing the suggested enrichment program using dynamic geometry software for developing mathematically gifted pupils' geometric creativity in the high schools:

The suggested enrichment program should provide the mathematically gifted students with opportunities to explore some mathematical ideas using the DGS in a creative fashion.

Activities within the suggested enrichment program should provide the mathematically gifted students with opportunities to reinvent the mathematical ideas through both exploration and the refining of earlier ideas.

The enrichment activities should be designed and presented in a constructivist way that encourage the mathematically gifted students to make new connections to their prior experiences and construct their own understanding.

Teaching the instructional activities, within the suggested enrichment program, should follow van Hiele phases of learning geometric concepts: Information, guided orientation, explicitation, free orientation, and integration.

The suggested enrichment program activities should correspond to the students’ skills, since they should experience success in order to motivate to continue in the program.

The suggested enrichment activities should challenge students’ thinking, enhance students’ achievement, and develop students’ geometric creativity.

The instructional activities, within the suggested enrichment program, should be designed to be effective in revealing geometric creativity and in distinguishing between the mathematically gifted students in terms of the geometric creativity and their responses.

The suggested enrichment program activities should address standards for school mathematics recommended by the National Council of Teachers of Mathematics (NCTM) as one of the most popular standard in the field of teaching and learning mathematics.
Aims of the Suggested Enrichment Program
The overall aim of the suggested enrichment program is to cultivate and develop keen creative abilities in geometry among the mathematically gifted students in the high schools using Cinderella application as a mediation environment. This overall aim can be interpreted in the light of the definition of geometric creativity, and the main characteristics of Cinderella. It branches out into the following specific aims:
Through the suggested enrichment program, the mathematically gifted students will hopefully be able to use Cinderella to:

Construct dynamic figures.

Come up with many construction methods to construct dynamic configurations for an assigned figure.

Come up with many various and different construction methods to construct dynamic configurations for an assigned figure.

Come up with novel and unusual methods to construct dynamic configurations to an assigned figure.

Produce many relevant responses (ideas, solutions, proofs, conjectures, new formulated problems) toward a geometric problem or situation.

Produce many various and different categories of relevant responses (ideas, solutions, proofs, conjectures, new formulated problems) toward a geometric problem or situation.

Generate many unusual ("wayout"), unique, clever responses or products toward a geometric problem or situation.

Make new conjectures and relationships by recognizing their experience toward the aspects of the given problem or situation.

Investigate the made conjectures by different methods in different situations.

Generate many different and varied proofs using the formal logical and deductive reasoning toward a geometric problem or situation.

Generate many followup problems by redefining (modifying, adapting, expanding, or altering) a given geometric problem or situation.

Apply different learning aspects of geometry (concepts, generalizations, and skills) in solving a geometric problem or situation.
Content of the Suggested Enrichment Program
The suggested enrichment program consists of three interrelated parts: student’s handouts, a teacher’s guide, and a CDROM. They (student’s handouts, the teacher’s guide, and the CDROM) cover 12 enrichment activities. These enrichment activities are openended and divergentproduction geometric situations and problems that require many various and different responses. These enrichment activities are designed in four categories to develop the geometric creativity components based upon specific teaching and learning strategies using different facilities of Cinderella application, which are:

Problem solving activities, where the student is given a geometric problem with a specific question and then invited not only to find many various and different solutions but also to pose many followup problems related to the original problem (e.g. activities 1, 5, and 6).

Problem posing activities, where the student is given a geometric situation and asked to make up as many various and different questions or conjectures as he can, that can be answered in direct or indirect ways using the given information (e.g. activities 11 and 12).

Construction activities, where the student is asked to come up with as many various and different methods as he can to construct a geometric figure (e.g., parallelogram) using the constructing facility of Cinderella application (e.g. activities 7, 8, 9, and 10).

Redefinition activities, where the student is given a geometric problem or situation and invited to pose as many problems as possible by redefining – substituting, adapting, altering, expanding, eliminating, rearranging or reversing – the aspects that govern the given problem (e.g. activities 2 and 4).
These are not presented as hardandfast categories, but as a framework of designing activities that might help the mathematically gifted students to develop their geometric creativity.
Student’s Handouts
The first main part of the suggested enrichment program is represented in the student’s handouts. There are 15 handouts prepared to guide the student throughout the suggested enrichment program. For each activity, a student’s handout is prepared to assist students to smoothly go through the activity and promote discussions between the teacher and the students as well. There are two versions of these handouts: one in English and the other in German.
Teacher’s Guide
The second main part of the suggested enrichment program is the teacher’s guide. It is designed to make the teacher’s work and progress in the course easier and more effective. The guide does not restrict the teacher’s work, but is flexible enough for any creative additions. The teacher’s guide includes the following:
Introduction
An introduction for the teacher to the suggested enrichment program is provided in the beginning of the teacher’s guide. It aims at introducing the suggested enrichment program, its purposes, its content, and its teaching/learning strategies, as well as a brief explanation of the concept of geometric creativity.
Aims of the Suggested Enrichment Program
The overall aim and the specific aims of the suggested enrichment program are objectively formulated to guide both the teacher and the student toward achieving them through the course. They precisely describe what could be expected from the mathematically gifted students by the end of the course.
Cinderella Getting Started
In this part the researcher introduces some teaching hints to the teacher to help him manage the introductory sessions, which is about Cinderella getting started through the designed handouts (Handout 2 and Handout 3).
Moreover, the teacher’s guide covers 12 enrichment activities, each of them includes the following elements:
The title of the activity
The researcher writes the title for each activity in the teacher’s guide that consists with the title written in the student’s handouts.
The activity problem
In each activity the problem statement of the activity is presented.
Activity content analysis
The researcher presents a mathematical content analysis for each activity into three main categories of content: concepts, generalizations, and skills to inform the teacher of the learning aspects that might be covered in the activity.
Objectives
Instructional objectives are objectively formulated for each activity to guide both the teacher and the students during teaching and learning processes. They precisely describe what could be expected from the mathematically gifted students by the end of the activity.
Materials
A list of proposed materials is suggested for each activity, which includes: computers with Cinderella application installed on them, LCD projector, and student’s handout of the activity.
Vocabulary
A List of new mathematical vocabularies is presented for each activity.
Prerequisites
A list of Cinderella prerequisite skills is presented for each activity.
Teaching and learning strategies
Specific teaching and learning strategies using different facilities of Cinderella application are presented for each activity including the warm up.
CDROM
The third part of the suggested enrichment program is the accompanying CDROM that contains all dynamic configurations prepared for the suggested enrichment program in two formats cdy and html. In addition, it contains html index, in both languages English and German, for all activities of the program and their dynamic configurations that can be used to easily access any activity and any configuration within it.
Appropriateness of the Suggested Enrichment Program
For judging the appropriateness of the suggested enrichment program, the researcher presented it, in its initial form, to a group of experts, who are experienced in teaching and learning mathematics. These experts were asked to decide on the appropriateness of the suggested enrichment program (the student’s handouts, the teacher’s guide, and the CDROM) and suggest any changes to modify it within the framework of the following criteria:

To what extent the suggested enrichment program is appropriate to the level of the mathematically gifted students in the high schools.

To what extent the enrichment activities included in the suggested enrichment program are appropriate to develop the geometric creativity among the mathematically gifted students in the high schools.

To what extent the use of Cinderella application is appropriate to the learning actions adopted in the suggested enrichment program.

To what extent the teacher’s guide is appropriate to guide the teaching and learning processes during the program.

To what extent the directions set in the teacher’s guide are clear.

Further comments for development.
The experts stated that the suggested enrichment program is appropriate to the level of the mathematically gifted students in high schools confirmed that the enrichment activities are also appropriate to develop the geometric creativity among the students. The experts also remarked that the use of Cinderella as proposed in the suggested enrichment program is appropriate to access and manipulate the program’s mathematical content. As for the teacher’s guide, the experts asserted that the teacher’s guide is appropriate to guide the teacher during the course of the program and the directions set in the teacher’s guide are clear. Finally and more importantly, it is recommended that it would be much appropriate to administer the suggested enrichment program to students in grades 912 as the students in German high schools study geometry in grades 79, so that they would have enough experience and background in geometry that allows the suggested enrichment.
Timerange for the Program Activities
After finishing the preparation of the suggested enrichment program and deciding on its appropriateness, the researcher attempts a pilot experimentation for the program that aims at determining the suitable timerange for each activity and ensuring the experimental appropriateness of the instructional treatment using Cinderella. In this respect, the student’s handouts were translated into German and one of the preservice student teachers was trained to be able to administer the program in German. Afterwards, the suggested enrichment program was administrated to a sample of 11 preservice student teachers of mathematics, 7 male and 4 female, in the university in Schwaebisch Gmuend at the beginning of the summer semester of the academic year 2008. In the light of the pilot experimentation of the suggested enrichment program, some minor modification were made and the timerange for each activity of the program was determined as follows:
Table 1
Timerange for the Program Activities
The Activity  No. of Sessions  Time 
Introductory Session: Cinderella Getting Started  1  90 min. 
Activity 1: Dragging and Measuring Facilities of Cinderella  1  90 min. 
Activity 2: Automatic Proving Facilities of Cinderella  1  90 min. 
Activity 3: Developing Macroconstructions  1  90 min. 
Activity 4: Animating and Tracing Loci Facilities of Cinderella  1  90 min. 
Activity 5: Midpoints of the Sides of a Quadrilateral  1  90 min. 
Activity 6: Angular Bisectors of a Parallelogram  1  90 min. 
Activity 7: Constructing a Parallelogram  1  90 min. 
Activity 8: Constructing a Rhombus  1  90 min. 
Activity
9: Constructing a Rectangle Activity 10: Constructing a Square 
1  90 min. 
Activity 11: Posing Geometric Problems  1  90 min. 
Activity 12: Finding Geometric Relationships  1  90 min. 
The Student’s Role
The student in the suggested enrichment program has several roles, which are varying according to the learning actions the student will move through during the enrichment activities of the program. There are 7 main learning actions the student will move through most of them during the enrichment activities and then there are 7 different roles of the student corresponding to them. Here are the learning actions and the student’s roles that correspond to each of them.
Constructing
The first learning action in most of the enrichment activities is to construct dynamic configurations using the constructing facility of Cinderella. In construction activities, the student’s role is to use the constructing facility of Cinderella to come up with various and different dynamic configurations for the assigned figure. In other enrichment activities, the student’s role is to come up with only one dynamic configuration for the assigned problem or situation.
Observing
The observing action includes using the dragging facility of Cinderella to alter the constructed configuration and visually observe the ripple effect. The student’s role is to drag free points or semi free points and visually observe what is invariant under dragging. Knowing what is invariant when a configuration is dragged is not always obvious. However, very often the student is guided through the observation process using the problem statement to see what is invariant.
Conjecturing
Here the observation should be reported in the form of mathematical theorems; so the student’s role is to think up mathematical conjectures based upon his/her own observation and formulate them in the form of mathematical theorems.
Investigating
This action is concerned with looking into the correctness of the formulated conjectures using different facilities of Cinderella by different methods in different situations. The student’s role is to use different facilities of Cinderella to design endeavors to visually and dynamically examine the correctness of his/her conjectures.
Proving
Deductive reasoning and producing formal proofs are vital learning actions to decide on the correctness of the formulated conjecture as well as logically convince students of the validity of the conjectures. The student’s role is to build various and different mathematical proofs for the formulated conjectures using the formal logical and deductive reasoning.
Elaborating
The elaboration action includes redefining the original situation by redefining – substituting, combining, adapting, altering, expanding, eliminating, rearranging, or reversing – one or more of the situations’ aspects then speculating on how this single change would have a ripple effect on other aspects of the situation at hand. The student’s role is to think carefully to pick one or more of the situations’ aspects and redefine it/them and then again use different facilities of Cinderella to design an endeavor to explore the ripple effect on the situation, and then formulate new conjectures or situations.
Posing
The posing action in the suggested enrichment program includes using the information that is available in the problem or the situation with different facilities of Cinderella to make up and find new problems that could be answered or deduced, in direct or indirect ways, from the given information in the situation. The student’s role is to use the given information with different facilities of Cinderella to find and pose new problems. In this action, the student’s role is extended to be not only problem solver but also problem finder and problem poser.
Consequently, the roles of the student during the suggested enrichment program could be described as: dynamicconfiguration constructor, observer, conjecture maker, endeavor designer, investigator, explorer, elaborator, problem solver, problem finder, and problem poser. All of these roles are supposed to be played in a collaborative context in which the student actively interacts with the teacher and other students using the proposed materials.
The Teacher’s Role
The role of teacher is important to facilitate the learning actions that the students move through during the enrichment activities of the program. So, it is expected that the teacher manage the enrichment session by:

Presenting the activity problem to the students.

Encouraging the students to use Cinderella to construct a dynamic configuration for the problem or the situation.

Helping the students through the observation process to see what is invariant in the problem configuration under dragging.

Encouraging the students to express their geometric ideas in their own words and in different forms.

Assisting the students to use the different facilities of Cinderella to investigate the correctness of their ideas in different methods.

Encouraging the students to produce a mathematical proof for their investigated conjectures using the formal logical steps. Here also the teacher might help by giving the students some hints in the form of questions to guide/direct their attention toward the proving process.

Directing the students' attention towards the problem aspects that could be redefined in order to find or pose new related problems.
Resources Used in Developing the Suggested Enrichment Program
Christou, C., Mousoulides, N., Pittalis, M., & PittaPantazi, D. (2005). Problem solving and problem posing in a dynamic geometry environment. The Montana Mathematics Enthusiast, Vol.2, No. 2, pp. 125143
Contreras, J. (2003). Using dynamic geometry software as a springboard for making conjectures, solving problems and posing problems. Retrieved February 20, 2006, from http://www.usm.edu/pt3/pa/jc01.html ElRayashy, H. A. M. & Ibrahim AlBaz Mohamed, A. (2000). A proposed strategy on group mastery learning approach in developing geometric creativity and reducing problemsolving anxiety among preparatory stage students. Journal of Mathematics Education in Faculty of Education  Banha, Zagazig University, Vol. 3, July 2000, pp. 65207 Friedrich, H. (1999). DGS in schools. Retrieved April 17, 2008, from: http://mathwww.unipaderborn.de/~hugo/artikel/pdf Haja, S. (2005). Investigating the problemsolving competency of preservice teachers in dynamic geometry environment. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 8187. Melbourne, Australia, July 1015 Haylock, D. (1997). Recognizing mathematical creativity in schoolchildren. Zentralblatt für Didaktiek der Mathematik, Vol. 29, No. 3, pp. 6874 Keyton, M. A search for an allencompassing problem in elementary geometry. Eightysomething! Winter 1997, pp. (1213) Olkun, S., Sinoplu, N. B. & Derzakulu, D. (2005). Geometric explorations with dynamic geometry applications based on van Hiele levels. International Journal for Mathematics Teaching and Learning. Retrieved February 20, 2006, from http://www.cimt.plymouth.ac.uk/journal/default.htm RichterGebert, J. & Kortenkamp, U. (2006). Cinderella.2 documentation. Retrieved February 8, 2007, from http://doc.cinderella.de RichterGebert, J. & Kortenkamp, U. (1999). The interactive geometry software Cinderella. SpringerVerlag, Heidelberg Weth, T. (1998). Kreative Zugänge zum Kurvenbegriff. Mathematikunterricht, Vol. 44, No. 45, pp. (3860) Weth, T. (2007). Kegelschnitte und höhere Kurven als Ortslinien in Dreiecken. Retrieved September 28, 2007, from: http://www.didmath.ewf.unierlangen.de/kegel_weth/index.htmlPlease use Safari web browser to view these pages!