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Article Excerpt
This article reports on results of an exploratory study on undergraduate pre-service teachers' understanding of graphical representations of motion functions. The study described pre-service teachers' explorations using a CBR device. Pre-service teachers' growth was studied in two dimensions: (a) in their learning of the mathematics involved and (b) in their learning of the pedagogy related to the mathematics and the technology used. Through their interaction with the device, pre-service teachers were able to overcome common misconceptions with respect to the mathematics and also to develop pedagogical insights regarding the teaching of the concepts.
MATH IN MOTION: USING CBRS TO ENACT FUNCTIONS
Pre-service teachers enrolled in mathematics education courses bear a dual role. As Bowers and Doerr (2001) note, pre-service teachers are "simultaneously learners and teachers in transition" (p. 115). This is especially true when pre-service teachers are introduced to seemingly "non-traditional" topics such as the development of algebraic reasoning at the elementary school level. Elementary school pre-service teachers are expected to construct knowledge of new content that they may not have had the opportunity to learn or of which they have little or fragmented knowledge (Kilpatrick & Swafford, 2001). Furthermore, pre-service teachers are expected to develop pedagogical insights and models for teaching these unfamiliar concepts and skills to their own future students while they, themselves, have had little (if any) experience observing other teachers teaching these topics.
Teacher preparation institutions have traditionally drawn a line between content and pedagogy. This often has resulted in pre-service teachers being required to enroll first in mathematics courses (often taught separately in the mathematics department) and subsequently in "methods" courses that focus primarily on the development of pedagogical knowledge. However, research has shown that this scheme is not necessarily successful in helping preservice teachers construct interconnected knowledge of mathematics and pedagogy (e.g., Brown & Borko, 1992; Lappan & Theule-Lubienski, 1994). Drawing from this research, the Conference Board of the Mathematical Sciences [CBMS], in a report on its vision of the education of mathematics teachers, argues for a close coordination of content and pedagogical courses in order to facilitate the development of strong knowledge of the mathematics that pre-service teachers will teach (CBMS, 2001). Furthermore, pre-service teachers themselves have indicated that they would be more willing to take additional mathematics courses if they found more connections between the content of the coursework and the mathematics that they would be expected to teach (Smith, 2000).
One way to address this call for coordination of content and pedagogy is to embrace the pre-service teachers' dual role as learners and teachers. This can be accomplished by accommodating their needs in education courses that aim to teach both the new mathematical content and the pedagogy related to this content. A growing number of teacher educators advocate this approach to mathematics education courses. Teacher educators in this group believe that pre-service teachers' understanding of mathematics will be more directly strengthened when they learn by engaging in mathematical tasks similar to those their students will be asked to complete as well as in ways that reflect the style of teaching and learning that they are expected to practice and promote (e.g., Ball and Cohen, 1999; Simon and Schifter, 1991). Smith (2001) argues that the education and professional development of teachers should be "situated in practice." In this view, the everyday tasks and tools that are central to the work of teaching (including the curricular materials that are used in classrooms) should provide the focus and object of teacher education. Allowing pre-service teachers to engage in mathematics learning while reflecting on the practice of teaching affords them the opportunity to build strong networks of interconnected knowledge.
The mathematics of motion, and particularly graphical representations of motion functions, is one of the topics of which pre-service teachers have limited or fragmented knowledge. While the development of algebraic thinking, including the mathematics of motion, is one of the major goals in the K-12 mathematics (NCTM, 2000), the majority of elementary school teachers report that they feel inadequately prepared or unqualified to teach algebra (e.g., Weiss, 1995) and that they spend little or no time teaching the topic (Grouws & Smith, 2000).
In this article we describe a study designed to understand how engaging pre-service teachers in mathematical tasks related to mathematics in motion may impact their understanding of the mathematics involved in the task as well as how they might teach this topic to their own students. Given the limited or fragmented understanding of functions and, in particular, motion functions, of the pre-service teachers involved in this study, their participation allowed us to study them in their dual role of learners of mathematics and teachers in transition. We chose to advance our pre-service teachers' understanding of the mathematics of motion and its pedagogy by engaging them in technology-based mathematical tasks. It was hypothesized that novel technology environments would provide a variety of opportunities for advancement of these pre-service teachers' mathematical understanding of the topic and for their reflection on pedagogical issues related to the mathematics of motion. However, little is known about how technology can be used by pre-service teachers in their dual roles. Our goal was to investigate the use of specific types of technology in each of these two aspects of pre-service teachers' education.
Graphical Representations of Functions--A Brief Review of Education Research
Students' understanding of the various representations of functions, and, in particular, their understanding of graphical representations of functions has been the object of several research studies (for a detailed review of this literature see Leinhardt, Zaslavsky, & Stein, 1990). This literature focuses largely on the misconceptions and misunderstandings that characterize students' responses to the visual qualities of graphs. Inexperienced users of graphs often interpret graphs "iconically." With respect to graphs of motion (both distance versus time and velocity versus time graphs), inexperienced users often interpret such graphs as actual pictures of the path traveled by an object making an overtly direct connection between the visual features of the graph and the situation it represents (e.g., Janvier, 1978; Kaput, 1987). In addition, the literature points to student difficulties in making connections between graphical and other representations of a function--primarily symbolic and numerical (Leinhardt et al., 1990). Our collective experiences with student understanding of graphical representations suggest that graphs do not necessarily aid students to further their understanding of a novel (to them) concept. This difficulty of understanding graphical representations found in novice learners is contrasted by evidence that individuals who are skilled in mathematical problemsolving tend to rely on visual representations (including graphs and diagrams) as tools that add information in the problem-solving process (e.g., Leikin, Stylianou & Silver, in press; Stylianou, 2002). Similar results are described in the science education literature (e.g., Ochs, Jacoby & Gonzales, 1994).
The topic of graphical representations and, particularly, the issues surrounding the learning and teaching of graphical representations gained further importance with the advancement of technology and its new role in classrooms. Modern computer technology and software offered students of mathematics and professionals engaged in mathematical work unprecedented access to easily generated and manipulated types of representations. Educators were quick to realize that these new technological learning environments and tools could lift some of the obstacles interfering with the understanding of graphical representation. Hence, calls for instructional reform in mathematics have explicitly focused on the importance of multiple representations (including numeric, graphic, and symbolic) in the learning of core concepts (e.g., Kaput, 1986). In particular, the Principles and Standards for School Mathematics (NCTM, 2000) includes a representation standard and attests to the importance of multiple representations in mathematics teaching and learning in pre-K-12 classrooms.
The affordability and convenience of graphing calculators, in particular, opened new possibilities for using and creating links among numeric, graphic, and symbolic representations. Most middle and secondary school curricula encourage students to use graphing calculators on a regular basis in order to explore mathematical concepts in their various representations. Indeed, the use of multiple representations appears to have a positive impact on student learning. A number of studies (e.g. Demana, Shoen, & Waits, 1993; Ellington, 2003; Kieran, 2001; Moreno, Rojano, Bonilla, & Rerrusquia, 1999) demonstrate that calculator environments induce students to conceive of algebra as a language for representing general phenomena and relations. Computer environments with graphing capabilities are also being used successfully as tools for representing the relationships of problem situations and as visual support for understanding symbolic expressions (e.g., Stylianou & Shapiro, 2002).
However, at the same time some studies have brought new insights related to students' understanding of multiple representations to...
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