This application was created by Danh Nam Nguyen aimed to provide dynamic exercises in the form of GeoGebra Applets so that mathematics students can manipulate on the figures in an active and creative way. Multiple success stories are being reported by the educators who see advantage of using GeoGebra as the means for visualizing mathematical concepts, representing mathematical concepts and relationships dynamically, linking different branches and topics in mathematics, supporting modeling practices, building virtual manipulatives, developing conceptual understanding, and teaching conjecturing and proving.

Embacher et al. (2006) used different kinds of instructional materials (e.g., ‘traditional’ worksheets on paper, interactive applets, quizzes). Students were guided towards discovering the concepts of derivative and/or integral. Students found that the dynamic and interactive material helpful to understand and visualize underlying mathematical concepts. Karadag (2009) found that visual representation systems and linked multi-representational systems encourage students to interact with mathematical understanding. Rather than dealing with the grammar of algebra only, students may benefit from direct interaction with the visually represented mathematical concepts.

Dynamic mathematics software (DMS) provides new opportunities for using both computer algebra system (CAS) and dynamic geometry software (DGS). GeoGebra is a DMS that freely available on-line, and supplemented with a variety of dynamic worksheets. This software also allows students to make and test assertions and prepare for more formal proof writing. Students can build a geometric construction and simultaneously observe how changes in a formula in the algebra window are affected by the manipulation of the construction and vice versa. Teachers can use this software to construct interactive applets on the internet to improve students’ proving abilities. By participating in these explorative tasks, the student will engage in realizing geometric invariants and formulating conjectures activities. As a result, students take produced arguments for granted that support to construct formal proofs. This software can be downloaded freely at the following website: http://www.geogebra.org. In short, the development of DMS provides students with many opportunities to explore and discover mathematics concepts according to their own individual needs and pace. This dynamic environment could motivate students to explain their empirical conjectures using formal proofs and provide an opportunity to link empirical and deductive reasoning together. A DMS can also be utilized to gain insight into a deductive argument, support experimentation, and thus lead to conviction.

Learning mathematics within a dynamic geometry software involves transitions in the learning process between figures and concepts, between perceptual activity and mathematical knowledge. Typically, a geometrical problem cannot be solved while remaining only at the perceptual level of figures on the screen. Conceptual control is needed and this requires explicit knowledge. The use of the dragging function validates procedures and is the crucial instrument of mediation between figure and concepts, perception and knowledge. In particular, Arzarello et al presented some features of such transitions in the move from conjecturing to proofs in geometry when using a dynamic geometry software. He also reports on students’ elevating the use of the language of mathematical argumentation, particularly when justifying constructions, and when students were working with the dragging mode.

## This application was created by Danh Nam Nguyen aimed to provide dynamic exercises in the form of GeoGebra Applets so that mathematics students can manipulate on the figures in an active and creative way. Multiple success stories are being reported by the educators who see advantage of using GeoGebra as the means for visualizing mathematical concepts, representing mathematical concepts and relationships dynamically, linking different branches and topics in mathematics, supporting modeling practices, building virtual manipulatives, developing conceptual understanding, and teaching conjecturing and proving.

## Embacher et al. (2006) used different kinds of instructional materials (e.g., ‘traditional’ worksheets on paper, interactive applets, quizzes). Students were guided towards discovering the concepts of derivative and/or integral. Students found that the dynamic and interactive material helpful to understand and visualize underlying mathematical concepts. Karadag (2009) found that visual representation systems and linked multi-representational systems encourage students to interact with mathematical understanding. Rather than dealing with the grammar of algebra only, students may benefit from direct interaction with the visually represented mathematical concepts.

## Dynamic mathematics software (DMS) provides new opportunities for using both computer algebra system (CAS) and dynamic geometry software (DGS). GeoGebra is a DMS that freely available on-line, and supplemented with a variety of dynamic worksheets. This software also allows students to make and test assertions and prepare for more formal proof writing. Students can build a geometric construction and simultaneously observe how changes in a formula in the algebra window are affected by the manipulation of the construction and vice versa. Teachers can use this software to construct interactive applets on the internet to improve students’ proving abilities. By participating in these explorative tasks, the student will engage in realizing geometric invariants and formulating conjectures activities. As a result, students take produced arguments for granted that support to construct formal proofs. This software can be downloaded freely at the following website: http://www.geogebra.org. In short, the development of DMS provides students with many opportunities to explore and discover mathematics concepts according to their own individual needs and pace. This dynamic environment could motivate students to explain their empirical conjectures using formal proofs and provide an opportunity to link empirical and deductive reasoning together. A DMS can also be utilized to gain insight into a deductive argument, support experimentation, and thus lead to conviction.

Learning mathematics within a dynamic geometry software involves transitions in the learning process between figures and concepts, between perceptual activity and mathematical knowledge. Typically, a geometrical problem cannot be solved while remaining only at the perceptual level of figures on the screen. Conceptual control is needed and this requires explicit knowledge. The use of the dragging function validates procedures and is the crucial instrument of mediation between figure and concepts, perception and knowledge. In particular, Arzarello et al presented some features of such transitions in the move from conjecturing to proofs in geometry when using a dynamic geometry software. He also reports on students’ elevating the use of the language of mathematical argumentation, particularly when justifying constructions, and when students were working with the dragging mode.