The Development of Students Geometrical Thinking through Transformational Processes and Interaction Techniques in a Dynamic Geometry Environment

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Introduction

Applying Comenius' rationale that "learning has been becoming more and more an activity", Freudenthal (1971, p. 415) adds that he dismisses the question "whether people learn better by active building up the subject than by passive reception of a ready made matter." This idea accords with the constructivist hypothesis as Mariotti (2002) declares "that learning results from a process of active adaptation of the learner to his /her environment, rather than a passive reception of information or instruction". On the other hand Nardi (1996, p.35) writes that "it is not possible to fully understand how people learn or work, if the unit of study is the unaided individual with no access to other people or to artefacts for accomplishing the task at hand". From this perspective, we consider the computer to constitute a fundamental artefact with a crucial role to play in learning processes. Computers, according to Pea (1985, p.167), are "reorganizers of mental functioning". The technological environment of the computer provides cognitive tools through which the user's communicative expression can be improved. Papert (1980) writes that children need tools to think with; a fundamental question concerns the ways in, and procedures through, which computers could be used effectively in education as cognitive tools to promote and reinforce cognitive processes, and act catalytically upon the quality of knowledge. According to Roschelle, Pea, Hoadley, Gordin, & Means (2000, p.79), research into cognitive processes has shown that learning is most effective when four core conditions hold: (1) active engagement, (2) participation in groups, (3) frequent interaction and feedback, and (4) connections to real world contexts"; all supported through effective uses of technology.

Dynamic geometry systems have been described as computational environments that embody some subdomain of mathematics or science, generally using linked symbolic and graphical representations. Through computers' environment and dynamic geometry environments especially can allow students to explore the various solution paths individually and in small groups in which they make decisions and receive feedback on their ideas and strategies.

Mental schemes are developed by the students during problematic situations. Cobb, Yackel, and Wood (1992, p.4) suggest that "students construct mental representations that correctly or accurately mirror mathematical relationships located outside the mind in instructional representations".

Consequently: Which is the role of the computer in students constructing mental schemes? How can the computer contribute to the configuration of cognitive units, and thus operate as a reference point for organizing, pursuing, and retrieving information, and thus facilitating the reusing and handling of the schemes in a wide range of situations? How can the construction of rigorous proofs be affected by dynamic geometry systems? What impact can dynamic geometry systems have a student's van Hiele level?

Introducing new representational infrastructures (Kaput, Noss, & Hoyles, 2002, p.2) such as dynamic geometry systems in the teaching and learning process makes it necessary to investigate the way in which students create mathematics and support reasoning. The focal point of interest, and subject under analysis, are the students' answers and the way in which they represent and verbally formulate concrete and abstract situations in problems.The present study a) focuses on the insightful data provided by a comparison of the experimental and control groups during the research process b) reports and describes a study undertaken to investigate the benefits of using (semi) pre-designed sketches relating to rigorous proof at the secondary school level; van Hiele levels are used as descriptor for the analysis.

The Role of the Dynamic Geometry Enviroment in Problem Posing

Writing in "Crossroads in mathematics: Standards for introductory college mathematics before calculus" Daniel Alexander offers the following reflections on the value of students learning geometry (quoted in Larew, 1999): "Geometry is a vehicle that provides much of the basic core of knowledge that the student of mathematics should possess . …