"Applications GeoGebra into Teaching Some Topics of Mathematics at the College Level" by Ljubica Diković

This paper presents results from a didactic experiment in teaching calculus at a business-technical school in Uzice, Serbia. The objective of this study was to determine whether there were any significant differences in Students' math achievement before and after GeoGebra-based instruction in important concepts in a calculus class. With a sample size of 31 students (19 females and 12 males) and Cronbach Alpha of 0.784 for the instrument, GeoGebra applets were designed to provide alternative representations to already existing representations of calculus concepts. In addition to traditional lectures and problem-solving sessions, these participants also participated in mathematical experimentations with GeoGebra, i.e. working in a computer lab to carry out investigations, individual research, and group work. Pre- and post-tests were administered at the beginning and end of the course. A paired-samples t-test showed that the mean scores were statistically significantly higher at the end of course.

It is interesting to note that the author fails to mention whether the distribution of the data satisfied all the underlying assumptions of this test (eg. normality, variances, independent, etc.) Solely based on the small sample and the instrument, the author goes on to suggest that the use of GeoGebra applets had a "positive effect on the understanding and knowledge of students" when teaching differential calculus (slope of tangent lines, connection between slope of the tangent line and graph of the gradient function, continuity/discontinuity of functions, connection between continuity and differentiability, etc).

On a positive note, citing other research findings, the author suggests several advantages of using GeoGebra for teaching and learning mathematics: user-friendly interface, guided discovery learning through multiple presentations and experimentations, manipulate mathematical objects, cooperative learning environment through task-oriented interactive situations, etc. Some of the disadvantages are that the uses GeoGebra require basic skills in algebraic commands, minimal training, and that independent explorations and experimentations may be inappropriate for some students.

Diković, L. (2009). Applications GeoGebra into teaching some topics of mathematics at the college level. Computer Science and Information Systems,6(2), 191-203.

"Teaching and Learning Calculus with Free Dynamic Mathematics Software GeoGebra" by Markus Hohenwarter

This article talks about some of the advantages of embedding technology in the classrooms for teaching and learning mathematics by using GeoGebra. The author of this article, Markus Hohenwarter, is the creator of GeoGebra. As part of his graduate studies in mathematics education and computer science at the University of Salzburg in Austria, during 2001/2002, he developed GeoGebra with the support of Austrian Academy of Sciences. From 2006 onwards, development of GeoGebra continues at the Florida Atlantic University.

GeoGebra is an open-source dynamic software that is designed for teaching and learning mathematics for all levels. It is based on the principles of a constructivist learning environment, which connect geometry, algebra, calculus, and some other areas of mathematics into an easy-to-use software environment. Since this software is free of charge, students and teachers can use it in the classrooms and at home to learn and teach calculus, for instance.

This article briefly looks at other powerful dynamic learning environments such as Geometer's Sketchpad and Cabri Geometry. These two software packages are not only proprietary, but also require considerable amount of time commitment for mastering, which renders these suitable for only higher education. And also, these packages are geared towards addressing one area of mathematics (geometry, algebra, calculus, or statistics). In contrast to these commercial packages, teachers, educators, and researchers develop GeoGebra worksheets and methods. For instance, fully functional dynamic calculus worksheets can be created on the fly in the classrooms or customize existing worksheets, created by others, for classroom use. That is, dynamic and interactive figures, functions, constructions, or objects can be manipulated by moving points, changing graphs, or manipulating parameters with sliders.

The author goes on to suggest that by providing such a dynamic teaching and learning environment, the process of students' learning becomes mathematical experiments, i.e. a student centered approach to learning mathematics.

Hohenwarter, M., Hohenwarter, J., Kreis, Y., & Lavicza, Z. (2008). Teaching and learning calculus with free dynamic mathematics software GeoGebra. In 11th International Congress on Mathematical Education. Monterrey, Nuevo Leon, Mexico.

"Proof, Explanation and Exploration: An Overview of Mathematics learning and Teaching" by Gila Hanna

In addition to looking at the role of proofs in mathematics education, this article also talks about the importance of dynamic geometry software in teaching mathematical proofs. According to Hanna, as educators in mathematics, a key task for us is to first understand the role of proofs in teaching mathematical concepts and ideas so that we are in a better position to promote conceptual understanding of mathematics through proofs.

For instance, the use of dynamic geometry software can provide more effective ways of challenging students and also teaching sound and logical mathematical reasoning to produce valid proofs for geometrical propositions and objects. The author goes on to suggest that dynamic software tools can not only provide an avenue towards heuristics, exploration, and visualization of mathematical concepts and ideas, but these tools can also pose challenges to teaching proofs.

The author notes that the term 'mathematical understanding' is somewhat elusive. By this, I think that the author is wondering what it really means to gain an understanding of mathematics, here, through proofs. For instance, expert mathematicians view proofs to be most valuable when it leads to understanding. That is, understanding proofs assists them to clearly verify, explain, systematize, discover, construct, communicate, and explore mathematical concepts and ideas. However, students and novices view proofs as alien, unnatural, not useful, or simply too boring as a human activity. That is, proofs are not appealing to the learners. One reason that novices feel this way about proofs is because, according to Hanna, teaching of proofs in traditional classrooms amounts to rote learning of mathematical proofs.

In order to move away from learning mathematical proofs through lower order thinking skills, dynamic software tools are available in order to explore, discover, and construct mathematical proofs, properties, and conjectures through dynamic investigations, explorations, demonstrations, constructions, problems, arts, or puzzles. For example, these tools can provide students to "check a large number of cases or even ... an infinite number of cases" through dynamic manipulation of mathematical objects.

It will be interesting to see if students' views of proofs as intuitive collection of properties change to that of viewing proofs as a system of theoretical ones through dynamic geometry software.

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational studies in mathematics, 44(1-2), 5-23.