The document describes a suggested enrichment program that uses dynamic geometry software to develop geometric creativity in mathematically gifted high school students. It includes four types of enrichment activities: problem solving, redefinition, construction, and problem posing. It also describes a geometric creativity test to assess students' creativity before and after the program. The test measures fluency, flexibility, originality, and elaboration. It was validated by experts and pilots found it reliably measures creativity with high validity. The suitable time for the test is 100 minutes.

1. Suggested Enrichment Program
Using Cinderella (DGS) in
Developing Geometric Creativity
Mohamed El-Demerdash
The University of Education - Schwaebisch Gmuend
Nov. 27th 2008

2. Working Title
The Effectiveness of an Enrichment Program
Using Interactive Dynamic Geometry
Software in Developing Mathematically
Gifted Students' Geometric Creativity in
the High Schools

3. Purposes
l Developing an enrichment program in
Euclidean geometry to enhance the geometric
creativity of the mathematically gifted students
in the high schools using dynamic geometry
software.
l Measuring the effectiveness of the enrichment
program using the interactive dynamic
geometry software in developing
mathematically gifted students' geometric
creativity in the high schools.

4. Talk Structure
l Suggested Enrichment Program Using
Cinderella (DGS) in Developing Geometric
Creativity.
l Geometric Creativity Test (GCT).

5. Talk Structure
l Suggested Enrichment Program Using
Cinderella (DGS) in Developing Geometric
Creativity.
l Geometric Creativity Test (GCT).

6. Bases
l The characteristics of the mathematically gifted
students and the nature of mathematical giftedness.
l The nature of creativity and geometric creativity.
l General principles of developing enrichment
programs for the mathematically gifted students.
l The contemporary trends in planning and organizing
enrichment programs for the mathematically gifted
students.
l The characteristics of interactive dynamic geometry
software.

7. Principles
l The program should provide opportunities for the
mathematically gifted to explore some mathematical
ideas using the IDGS in a creative fashion.
l 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.
l 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.

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

9. Principles
l 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.
l The suggested enrichment program activities should
address standards for school mathematics, for example
the ones recommended by the National Council of
Teachers of Mathematics (NCTM) as it is one of the most
popular standards in the field of teaching and learning
mathematics.

10. Aims
1. Construct dynamic figures.
2. Come up with many construction methods to construct
dynamic configurations for an assigned figure.
3. Come up with many various and different construction
methods to construct dynamic configurations for an assigned
figure.
4. Come up with novel and unusual methods to construct
dynamic configurations to an assigned figure.
5. Produce many relevant responses (ideas, solutions, proofs,
conjectures, new formulated problems) toward a geometric
problem or situation.
6. Produce many various and different categories of relevant
responses (ideas, solutions, proofs, conjectures, new
formulated problems) toward a geometric problem or
situation.

11. Aims
7. Generate many unusual ("way-out"), unique, clever responses or
products toward a geometric problem or situation.
8. Make new conjectures and relationships by recognizing their
experience toward the aspects of the given problem or situation.
9. Investigate the made conjectures by different methods in different
situations.
10. Generate many different and varied proofs using the formal logical
and deductive reasoning toward a geometric problem or situation.
11. Generate many follow-up problems by redefining (modifying,
adapting, expanding, or altering) a given geometric problem or
situation.
12. Apply different learning aspects of geometry (concepts,
generalizations, and skills) in solving a geometric problem or
situation.

12. Content
lStudent’s Handouts
lTeacher’s Guide
lCD ROM

13. Enrichment Activities
1. Problem Solving Activities
2. Redefinition Activities
3. Construction Activities
4. Problem Posing Activities

14. Enrichment Activities
1. Problem Solving Activities
2. Redefinition Activities
3. Construction Activities
4. Problem Posing Activities

15. Problem Solving Activities
… 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 follow-up
problems related to the original problem
(e.g. activities 1, 5, and 6).

16. Enrichment Activities
1. Problem Solving Activities
2. Redefinition Activities
3. Construction Activities
4. Problem Posing Activities

17. Redefinition Activities
… 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).

18. Enrichment Activities
1. Problem Solving Activities
2. Redefinition Activities
3. Construction Activities
4. Problem Posing Activities

19. Construction Activities
… 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 constructing facility of
Cinderella application (e.g. activities 7, 8,
9, and 10).

20. Enrichment Activities
1. Problem Solving Activities
2. Redefinition Activities
3. Construction Activities
4. Problem Posing Activities

21. Problem Posing Activities
… 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).

22. Talk Structure
l Suggested Enrichment Program Using
Cinderella (DGS) in Developing Geometric
Creativity.
l Geometric Creativity Test (GCT).

23. Specifying the Aim of the Test
The aim of the geometric creativity test is to
assess the geometric creativity of the
mathematically gifted students in terms of
creativity components before and after
administering the suggested enrichment
program.

24. Creativity Components of the Test
1. Fluency: the student’s ability to pose or come up with many
geometric ideas or configurations related to a geometric
problem.
2. Flexibility: the student’s ability to vary the approach or suggest
a variety of different methods toward a geometric problem.
3. Originality/Novelty: the student’s ability to try novel or
unusual approaches toward a geometric problem.
4. Elaboration: the student’s ability to redefine a single geometric
problem to create others, which are not the geometric
problem, situation itself, or even its solutions but rather the
careful thinking upon the particular aspects that govern the
geometric problem or situation changing, one or more of
these aspects by substituting, combining, adapting, altering,
expanding, eliminating, rearranging, or reversing.

25. Preliminary Form of the Test

26. Grading Method of the Test
1. Fluency: The number of relevant responses. Each relevant response
is given one point.
2. Flexibility: The number of different categories of relevant responses:
answers, methods, or questions. Each flexibility category is given
one point.
3. Originality/Novelty: It is the statistical infrequency of responses in
relation to peer group. The more statistical infrequency the response
has, the more originality it manifests.
4. Elaboration: It is graded by the number of follow-up questions or
problems that are posed by redefining – substituting, combining,
adapting, altering, expanding, eliminating, rearranging, or reversing
– one or more aspects of the given geometric problem or situation.
Each correct response is given one point.
5. Overall Geometric Creativity It is the sum of fluency, flexibility,
originality, and elaboration scores that represents the creativity
thinking ability in geometry.

27. Content Validity of the Test
For validating the GCT, the test was
presented, in its preliminary form, to a
group of judges specialized in teaching and
learning mathematics in China, Egypt, and
Germany. For reviewing the items, in their
initial form, for clarity, readability, and
appropriateness to measure what it is
designed to measure, and the level of
mathematically gifted students in the high
schools.

28. Piloting of the Test
1. The reliability coefficient for the test.
2. Item-internal consistency reliability for the
test items.
3. Experimental validity for the test.
4. The suitable time-range for the test.

29. Reliability Coefficient for the Test
The reliability coefficient (Cronbach's α
(alpha)) for all test items as they measure
geometric creativity was calculated using
SPSS. It was 0.83, a high reliability
coefficient. Consequently, the GCT
prepared was proven reliable to measure
the geometric creativity ability as a whole.

30. Item-internal Consistency

31. The Experimental Validity of the Test
The experimental validity of the test was
calculated as the square root of the test
reliability coefficient. It was 0.922 and that
shows the geometric creativity test has a
high experimental validity.

32. The Suitable Time Range for the Test
The suitable time-range for the test is
calculated as the mean of the time first
student took (60 minutes) and the last one
took (143 minutes), so the suitable time of
the test was calculated as 100 minutes.

33. Questions

34. Thank you very much!