suggested enrichment program using cinderella (dgs) in developing geometric creativity

Suggested Enrichment Program Using Cinderella (DGS) in

Developing Geometric CreativityMohamed El-Demerdash

The University of Education - Schwaebisch Gmuend

Nov. 27th 2008

Working Title

The Effectiveness of an Enrichment Program Using Interactive Dynamic Geometry

Software in Developing Mathematically Gifted Students' Geometric Creativity in

the High Schools

Purposes

lDeveloping an enrichment program in Euclidean geometry to enhance the geometric creativity of the mathematically gifted students in the high schools using dynamic geometry software.

lMeasuring the effectiveness of the enrichment program using the interactive dynamic geometry software in developing mathematically gifted students' geometric creativity in the high schools.

Talk Structure

l Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric Creativity.

lGeometric Creativity Test (GCT).

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.

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.

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.

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.

Aims1. 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.

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.

Content

lStudent’s HandoutslTeacher’s GuidelCD ROM

Enrichment Activities

1. Problem Solving Activities2. Redefinition Activities3. Construction Activities4. Problem Posing Activities

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).

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).

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).

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).

Talk Structure

l Suggested Enrichment Program Using Cinderella (DGS) in Developing Geometric Creativity.

lGeometric Creativity Test (GCT).

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.

Creativity Components of the Test1. Fluency: the student’s ability to pose or come up with many

geometric ideas or configurations related to a geometricproblem.

2. Flexibility: the student’s ability to vary the approach or suggesta variety of different methods toward a geometric problem.

3. Originality/Novelty: the student’s ability to try novel orunusual approaches toward a geometric problem.

4. Elaboration: the student’s ability to redefine a single geometricproblem to create others, which are not the geometricproblem, situation itself, or even its solutions but rather thecareful thinking upon the particular aspects that govern thegeometric problem or situation changing, one or more ofthese aspects by substituting, combining, adapting, altering,expanding, eliminating, rearranging, or reversing.

Preliminary Form of the Test

Grading Method of the Test

1. Fluency: The number of relevant responses. Each relevant responseis given one point.

2. Flexibility: The number of different categories of relevant responses:answers, methods, or questions. Each flexibility category is givenone point.

3. Originality/Novelty: It is the statistical infrequency of responses inrelation to peer group. The more statistical infrequency the responsehas, the more originality it manifests.

4. Elaboration: It is graded by the number of follow-up questions orproblems 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 creativitythinking ability in geometry.

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.

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.

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.

Item-internal Consistency

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.

The Suitable Time Range for the Test

The suitable time-range for the test iscalculated as the mean of the time firststudent took (60 minutes) and the last onetook (143 minutes), so the suitable time ofthe test was calculated as 100 minutes.

Questions

Thank you very much!