The role of DGS in developing mathematically gifted students' geometric creativity

Status Quo Theses Enrichment Program GCT Experimentation Conclusion
The Role of DGS in Developing Mathematically
Gifted Students’ Geometric Creativity
Mohamed El-Demerdash
Mathematics and Computer Science Institute
The University of Education, Schw¨abisch Gm¨und
Dissertation Disputation
May 6, 2010
Mohamed El-Demerdash Doctoral Dissertation Disputation

Dissertation Working Title and Timeline

The Eﬀectiveness of an Enrichment Program Using Dynamic
Geometry Software in Developing Mathematically Gifted
Students’ Geometric Creativity in High Schools

The Eﬀectiveness of an Enrichment Program Using Dynamic
Geometry Software in Developing Mathematically Gifted
Students’ Geometric Creativity in High Schools
Dissertation Development Timeline

Disputation Structure
The Status Quo
Theses (Solution Claims)
The Enrichment Program
The Geometric Creativity Test (GCT)
Study Experimentation Procedures
Conclusion

The Status Quo

The Status Quo
“. . . The student most neglected, in terms of realizing full poten-
tial, is the gifted student of mathematics. Outstanding mathe-
matical ability is a precious societal resource, sorely needed to
maintain leadership in a technological world” (Sheﬁeld, 2000)

The Status Quo
“. . . The student most neglected, in terms of realizing full poten-
tial, is the gifted student of mathematics. Outstanding mathe-
matical ability is a precious societal resource, sorely needed to
maintain leadership in a technological world” (Sheﬁeld, 2000)
“. . . many programs for the mathematically gifted students are
inadequate and poorly designed; leaving the classroom teachers
behind, struggling in an attempt to eﬀectively meet the edu-
cational needs of the mathematically gifted” (Rotigel & Fello,
2004)

The Status Quo
“. . . even though geometric thinking is very important in many
scientiﬁc, technical, and occupational ﬁelds as well as in study-
ing mathematics, geometry is often neglected in school mathe-
matics especially in most high schools” (Olkun et al., 2005)

The Status Quo
“. . . even though geometric thinking is very important in many
scientiﬁc, technical, and occupational ﬁelds as well as in study-
ing mathematics, geometry is often neglected in school mathe-
matics especially in most high schools” (Olkun et al., 2005)
“. . . some possible reasons for this negligence would be the lack
of resources, such as concrete materials, computer applications,
and the lack of knowledge and expertise about how to use com-
puters and other materials for instructional purposes” (Olkun
et al., 2005)

The Status Quo
The negligence of realizing full potential of the mathemat-
ically gifted.

The Status Quo
ically gifted.
The inadequate of the existing text books and programs.

The Status Quo
ically gifted.
The elimination of teaching geometry in high schools.

The Status Quo
ically gifted.
Teachers’ lack of knowledge and expertise in:

The Status Quo
ically gifted.
Enriching the existing text books and programs.

The Status Quo
ically gifted.
Enriching the existing text books and programs.
Using the integrated environments with technology.

Theses

Theses
Thesis #1
The school mathematics textbooks should include open-ended and
divergent-production questions.

Theses
Thesis #1
The school mathematics textbooks should include open-ended and
divergent-production questions.
Thesis #2
The mathematically gifted students should be given the opportunity
to think like mathematicians do and re-invent mathematics by going
through the process which leads them to the product and not merely
learn the product (theorems and axioms).

Theses
Thesis #3
Learning and developing creativity, its operations and its skills,
should be taken into consideration and associated with mathemat-
ics curricula while they are designed. Furthermore, it should be a
part of daily lesson plans that mathematics teachers prepare for their
classrooms.

Theses
Thesis #3
Learning and developing creativity, its operations and its skills,
should be taken into consideration and associated with mathemat-
ics curricula while they are designed. Furthermore, it should be a
part of daily lesson plans that mathematics teachers prepare for their
classrooms.
Thesis #4
Special schools or classes should be provided for the mathematically
gifted students at diﬀerent educational levels and stages.

Theses
Thesis #5
The use of integrated learning environments with technology, for
instance using Computer Algebra Systems (CAS) and Dynamic
Geometry Software (DGS) is strongly recommended for better learn-
ing outcomes.

Theses
Thesis #5
The use of integrated learning environments with technology, for
instance using Computer Algebra Systems (CAS) and Dynamic
Geometry Software (DGS) is strongly recommended for better learn-
ing outcomes.
Thesis #6
In-service and pre-service teachers of mathematics should be trained
on the ways and techniques that can be used to enrich the math-
ematical content for mathematically gifted students and the tech-
niques of developing creative thinking in the ﬁeld of mathematics.

The Enrichment Program
The Principles of Developing the Enrichment Program
The Content of the Enrichment Program
The Types of Enrichment Activities
The Appropriateness of the Enrichment Program
The Pilot Experimentation of the the Enrichment Program

Principles

Principles
1 The activities should provide the mathematically gifted students
with opportunities to explore some mathematical ideas using
the DGS in a creative fashion.

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

Principles
2 Activities within the program should provide the mathematically
gifted students with opportunities to reinvent the mathematical
ideas through both exploration and the reﬁning of earlier ideas.
3 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
4 Teaching the instructional activities, within the program, should
follow van Hiele phases of learning geometric concepts: Infor-
mation, guided orientation, explicitation, free orientation, and
integration.

Principles
integration.
5 The program activities should correspond to the students’ skills,
since they should experience success in order to stay motivated.

Principles
integration.
5 The program activities should correspond to the students’ skills,
since they should experience success in order to stay motivated.
6 The enrichment activities should challenge students’ thinking,
enhance students’ achievement, and develop students’ geomet-
ric creativity.

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

Principles
7 The instructional activities, within the enrichment program,
should be designed to be eﬀective in revealing geometric cre-
ativity and in distinguishing between the mathematically gifted
students in terms of the geometric creativity and their responses.
8 The enrichment program activities should address standards for
school mathematics, for example the ones recommended by
(NCTM) as it is one of the most popular standards in the ﬁeld
of teaching and learning mathematics.

Content
Student’s Handouts
English Version
German Version
CD-ROM
English Version
German Version
Teacher’s Guide

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

1 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.
Activity 1
Activity 5
Activity 6

2 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.
Activity 2
Activity 4

3 Construction Activities
. . . the student is asked to come up with as many various and diﬀer-
ent methods as he can to construct a geometric ﬁgure (e.g., parallel-
ogram) using constructing facility of Dynamic Geometry Software.
Activity 7
Activity 8
Activity 9
Activity 10

4 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.
Activity 11
Activity 12

Appropriateness
Subjects

Appropriateness
Subjects
. . . were 8 experts in teaching and learning mathematics.

Appropriateness
Subjects
1 . . . of the program to the level of the mathematically gifted
students in high schools.

Appropriateness
Subjects
2 . . . of the enrichment activities to develop the mathematically
gifted students’ geometric creativity.

Appropriateness
Subjects
3 . . . of Cinderella to the program’s learning actions.

Appropriateness
Subjects
3 . . . of Cinderella to the program’s learning actions.
4 . . . of the teacher’s guide to mentor the program’s teaching
and learning processes.

Pilot Experimentation
Voluntary Sessions
In-seminar Sessions

Voluntary Sessions
Subjects were 11 pre-service student teachers of mathematics, 7
male and 4 female, in the PHSG at the beginning of the summer
semester 2008.
In-seminar Sessions

Voluntary Sessions
Subjects were 11 pre-service student teachers of mathematics, 7
male and 4 female, in the PHSG at the beginning of the summer
semester 2008.
In-seminar Sessions
Subjects were 30 pre-service student teachers, 15 male and 15 fe-
male, in the PHSG during the seminar of “Computer im Mathe-
matikunterricht” summer semester 2008.

The Geometric Creativity Test
Test Design

Test Design
6-step process

Test Design
6-step process
1 Speciﬁcation of the Aim of the Test

Test Design
6-step process
. . . is to obtain a validated and reliable instrument to measure
geometric creativity.

Test Design
6-step process
2 Speciﬁcation of the Components that the Test Measures

Test Design
6-step process
Fluency

Test Design
6-step process
Fluency
Flexibility

Test Design
6-step process
Fluency
Flexibility
Originality/Novelty

Test Design
6-step process
Fluency
Flexibility
Originality/Novelty
Elaboration

Test Design
6-step process
Fluency
Flexibility
Originality/Novelty
Elaboration
3 Creation of a Preliminary Version of the Test

Test Design
4 Setting-up of a Grading Method for the Test

Test Design
5 Content Validity Check

Test Design
Subjects
. . . 13 judges from China, Egypt, and Germany.

Test Design
Subjects
. . . clarity and readability of test items.

Test Design
Subjects
. . . test items are appropriate to measure what they are
designed to measure.

Test Design
Subjects
. . . test items are appropriate to measure what they are
designed to measure.
. . . test items are appropriate to the level of the mathematically
gifted students in high schools.

Test Design
6 Test Piloting

Test Design
6 Test Piloting
Subjects
. . . were 30 pre-sevice student teachers, 15 male and 15 female,
in the PHSG during the summer semester 2008.

Test Design
6 Test Piloting
Subjects
The Reliability Coeﬃcient

Test Design
6 Test Piloting
Subjects
Item-internal Consistency Reliability

Test Design
6 Test Piloting
Subjects
Experimental Validity

Test Design
6 Test Piloting
Subjects
Experimental Validity
Determining a Suitable Time-range

Procedures of the Experimental Study
Subjects
. . . were 7 mathematically gifted students in Landesgymnasium
f¨ur Hochbegabte (LGH).

Subjects
. . . were 5 eleventh graders, 1 tenth grader, and 1 ninth grader
(2 male, 5 female).

Subjects
(2 male, 5 female).
. . . have IQ > 130, 14/15, and high interest in solving
mathematical problems.

Subjects
(2 male, 5 female).
Experimental Design

Subjects
(2 male, 5 female).
Experimental Design
The pre-experimental one-group, pretest – intervention – posttest
design was used to investigate the eﬀectiveness of the enrichment
program in developing the mathematically gifted students’ geometric
creativity.

Administering the Pretest

Administering the Enrichment Program

Administering the Posttest

Experimental Results

There are statistically signiﬁcant diﬀerences at the level 0.05
between the mean ranks of the subjects’ scores on the pre-
post measurements of the geometric creativity and its four sub-
components in favor of the post measurement.

There are statistically significant differences at the level 0.05
between the mean ranks of the subjects’ scores on the pre-
post measurements of the geometric creativity and its four sub-
components in favor of the post measurement.
The enrichment program using DGS has a suitable level of
effectiveness in developing the mathematically gifted students’
geometric creative potential and its sub-components.

Conclusion
The positive impact of the enrichment program can be traced
back to its content and designed enrichment activities.

Conclusion
The eﬀectiveness of the enrichment program can be also traced
back to the use of DGS and its diﬀerent facilities throughout
the teaching course.

Conclusion
The results can be also ascribed to the characteristics of the
mathematically gifted.

Conclusion
Recommendations

Conclusion
Recommendations
. . . are given in the six suggested theses presented in the beginning
of the disputation.

Theses
1 Reforming school mathematics textbooks to include open-
ended and divergent-production questions.
2 Giving opportunities to students to re-invent mathematics.
3 Including creativity skills and operations in every day mathe-
matics lesson plan.
4 Providing special schools and classes for the mathematically
gifted students.
5 Using the integrated learning environment with technology is
strongly recommended for better learning outcomes.
6 Training in-service and pre-service teachers on the ways and
techniques of enriching the current content and the use of tech-
nology.

Thank you very much!

The role of DGS in developing mathematically gifted students' geometric creativity

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