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Article Excerpt
This article introduces technology-based teaching ideas that facilitate the development of qualitative reasoning techniques in the context of quadratic equations with parameters. It reflects on activities designed for and used with prospective secondary mathematics teachers in accord with standards for teaching and recommendations for teachers in North America. The main educational implications of the proposed didactics include an emphasis on using geometric constructions in the context of algebra, emergence of residual mental power that can be used in the absence of technology, and development of skills in problem posing.
INTRODUCTION
For over two decades, emerging advances in the computerization of mathematics education have opened exciting opportunities to revisit, enhance, and extend secondary school mathematics curricula. As mentioned by many authors (Stephens & Hartman, 2004; Kersaint, Horton, Stohl, & Garafalo, 2003; Mistretta, 2002; Chamblee & Slough, 2002; Browning & Klespis, 2000), these opportunities can only be realized if teachers of mathematics are well prepared to appropriately incorporate computers into their teaching. Appropriate use of technology in this context may include the joint use of computing activities and cognitive follow-up tasks that are true reflections of these activities. This requires a deep understanding of mathematics--an appreciation of the subject matter as a rich interplay of concepts, ideas, and techniques, including those traditionally taught at the secondary school level. The Conference Board of the Mathematical Sciences (2001) recommends that prospective teachers of secondary school mathematics (referred to below as teachers) become familiar with the ways of exploring fundamental mathematical concepts taught at this level from an advanced standpoint. Furthermore, the Board suggests that effective mathematics teacher education programs should take full advantage of technological tools, which enable informal entries into grade-appropriate advanced mathematical ideas and facilitate movement from informal to formal reasoning. In other words, the teachers should become familiar with technology-enabled approaches to revealing hidden domains of secondary school mathematics curricula (Abramovich & Brouwer, 2003).
This article explores one such domain associated with topics dealing with quadratic functions and corresponding equations. As an appropriate software tool, less generally described in the literature, the authors introduce the Graphing Calculator 3.2 (Avitzur, Gooding, Herrmann, Piovanelli, Robbins, Wales, & Zadrozny, 2002), referred to below as the GC. Its ability to graph a relation from any two-variable equation without the need to convert the latter into a form suitable for traditional "function grapher" software or a hand-held graphing calculator is an important advantage of the GC over other computer graphics software. This paper will show how mathematics teacher educators can draw on this powerful feature of the GC to develop effective techniques for the study of families of quadratic equations depending on parameters. Those techniques can also be extended to include the exploration of more complex algebraic structures.
The main up-to-date document of mathematics education reform in North America, Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), in which technology is elevated to the status of being a principle, suggests that all students in grades 9-12 should be proficient in using a variety of representations of functional relations including those depending on parameters. The ability to recognize and analyze change in a variety of contexts, and to judge the meaning of such a change in the manipulation of a parameter, are among the most useful mathematical skills that one needs to live and function successfully in the 21st century. Within secondary school algebra, such skills can be developed through making connections between external, representational characteristics of functions and equations, and their internal, structural properties. Teachers are unlikely to facilitate such connections for their students unless they have been involved in exploring and solving problems with parameters. Furthermore, the appropriate use of graphing technology in this exploratory context provides a support system for teachers to learn new problem-solving strategies and develops a strong foundation for the study of mathematics from an applied perspective.
This article introduces pedagogical ideas for developing qualitative reasoning techniques using the GC in the context of pre-service secondary mathematics teacher education. It reflects the first author's work with teachers (mathematics majors) over the last decade. It focuses on the exploratory nature of technology-enhanced learning of algebra and shows how an alternative look at the familiar can create a highly inquiry-oriented learning environment suitable to current standards and recommendations of mathematics education reform. These recommendations include the importance of courses within which the teachers "could examine ... [the] use of computer tools in exploring algebraic ideas" (Conference Board of the Mathematical Sciences, 2001, p. 41), the need for the teachers to understand "the ways that basic ideas of ... algebraic structures underlie rules for operations on expressions, equations, and inequalities" (Conference Board of the Mathematical Sciences, p. 40), and the development of rich problems for the teachers that "convey important aspects of mathematical thinking" (Steen, 2004, p. 869).
Through a number of illustrations (all of which have been tested successfully through appropriate educational applications at the tertiary level), the authors will demonstrate how re-conceptualization of graphing strategies can help teachers develop useful skills in using a combination of computer-enhanced visualization and qualitative reasoning for exploring advanced mathematical ideas. It will be argued that this approach provides occasion for teachers to gain mathematical power that eventually can partially replace the use of technology. The authors will also show how this approach can be put to work to develop teachers' research-like experience in mathematics through technology-enabled problem posing.
ATTEMPTS TO CHANGE TRADITIONAL APPROACHES TO TEACHING ALGEBRAIC EQUATIONS
Solving quadratic equations in one variable is a traditional topic in secondary school algebra. Students do not encounter insurmountable difficulties in this topic in which all their cognitive efforts are concentrated on either factoring trinomials or using quadratic formulas. Such treatment of the topic nurtures procedural skills alone and pays little attention to conceptual development. The reformed vision of secondary school algebra, however, goes beyond the need for students to remember formulas and master factorization techniques. This vision presents algebra as a dynamic, engaging, and conceptually-oriented subject matter. It defines algebraic activities as the exploration of patterns leading to conjecturing and, ultimately, to formal demonstration of mathematical propositions discovered. Such activities may be based on the blend of informal and formal reasoning and can be amplified by the use of technology (Fey, 1989; Dugdale, Thompson, Harvey, Demana, Waits, Kieran, McConnell, & Christmas, 1995; Kaput, 1995; Akst, 1998; Yerushalmy & Chazan, 2002). As demonstrated below, topics that deal with functions and equations that are dependent on parameters offer a perfect milieu for the activities.
Ursini and Trigueros (2004) reported that students, both at the secondary and tertiary levels, have difficulties working with parameters because parameters are meaningless to them. These authors suggested using geometric contexts to help learners comprehend the meaning of parameters. However, whereas their approach does create context in which parameters arise, it seems to be a traditional one from an algebraic perspective (i.e., focusing on the study of discriminants in the case of quadratic equations). As will be shown below, geometric representations of algebraic equations, made possible by the use of the GC when a parameter becomes one of the coordinates in the Cartesian plane, has the potential of creating context in which parameters can be both meaningfully interpreted and easily visualized.
The use of equations and functions depending on parameters in the context of in-service teacher education was reported by Zaslavsky (1995) who suggested utilizing...
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