5 Computer Algebra in Mathematics Education

5 Computer algebra in mathematics education 5.1 Introduction In this chapter we investigate the possible contributions of computer algebra use to the learning of algebra. First, as a background we consider the use of IT tools in mathematics education in general. We note that for the learning of algebra, most information technology (IT) tools provide limited algebraic facilities and notations (Section 5.2). Then, we focus on computer algebra systems (CAS). We specify what computer algebra is and re- view some didactic issues that arose in early research projects into computer algebra use in mathematics education. In this way we sketch the state of the art on which this study elaborates (5.3). Next we draw up an inventory of the opportunities and pitfalls that computer algebra use presents to mathematics education in general (5.4), and to learning the concept of parameter in particular (5.5). These opportunities and pitfalls are related to the theoretical framework of the study and to the different parameter roles that were identified in Chapter 4. The choice for using the handheld TI-89 symbolic calculator in the teaching experiments is motivated. Finally, we describe the theory of instrumentation that focuses on students develop- ing utilization schemes and instrumented techniques that integrate technical and conceptual aspects (5.6). This is the main theoretical framework we use in Chapter 10 for analysing the interaction between student and computer algebra environment. 5.2 Information technology and the learning of algebra In this section we briefly address the general issue of the learning of mathematics, and particularly algebra, in a technological environment. It offers a background to our focus on the use of computer algebra for the learning of algebra. First, we present Pea’s view of the use of cognitive technology as amplifier and organizer. Second, we discuss the opportunities that computer use offers for visualization and for the integration of multiple representations. Finally, we focus on the use of technology in the algebra learning process in general and in gaining an understanding of the concept of variable in particular. Computer algebra technology is addressed in Section 5.3. Amplifier and organizer Pea suggested an interesting approach to the use of technology in education (Pea, 1985, 1987). He considered cognitive technologies as media that help ‘transcend the limitations of the mind... in thinking, learning, and problem-solving activities’ (Pea, 1987, p. 91). He distinguished two functions of IT tools: the amplifier function and the organizer function. The former refers to the amplification of possibilities, for ex- 77 Chapter 5 ample by investigating many cases of similar situations at high speed. The complex- ity of the examples can be amplified as well, thus allowing for a wider range of examples with an increasing variation. In this way, IT makes accessible insights that otherwise would be hard for students to gain. We agree with Berger that the use of IT thus enlarges the possible ‘zone of proximal development’, to use the Vygotskian terminology (Berger, 1998; Vygotsky, 1978). The amplifier function of an IT tool may also be used to invite algebraic generalization (Mason, 1996). As Heid (2002) indicated, the amplifier function does not affect essentially the curriculum. In its second role, the IT tool can also function as organizer or reorganizer affecting the organization and the character of the curriculum. For example, Hillel et al. (1992) describe how the possibility to simultaneously display graphical and symbolical windows changed the instructional approach to functions. Heid’s study, which is addressed below, shows similar effects (Heid, 1988). The IT function of generic organizer that is described by Tall (Tall, 1989; Tall & Thomas, 1991) is, surprisingly, close to the amplifier function. Technology, accord- ing to Tall, can help to generate a wide scale of examples and non-examples that provide the opportunity for the student to gather data and to develop a cognitive struc- ture on the basis of these experiences. The amplifier-organizer distinction can be helpful in explaining the role of technology in a specific educational setting. In this study computer algebra primarily functions as amplifier. For example, the computer algebra tool is used to generate examples that are a source for generalization. In the meantime, the computer algebra tool acts as an organizer as well: available options, such as the slider for dynamically changing graphs, affect the traditional instructional sequence for the learning of the concept of parameter. Visualization and multiple representations As soon as technology became available for integration into mathematics education, its main opportunities were surveyed by many researchers (Hillel et al., 1992; O’Callaghan, 1998). These perceived opportunities are called affordances. For example, the affordances of the use of graphing calculators from the perspective of RME have been identified (Doorman et al., 1994; Drijvers & Doorman, 1997). The graphing calculator was supposed to allow for: • the use of realistic contexts; • an exploration approach to problem situations; • visualization and the integration of different representations; • the experience of dynamics within a problem situation; • a flexible way of doing mathematics. In similar lists that also concern other technological tools, the issues of visualization and the integration of multiple representations, and exploration are often mentioned. 78 Computer algebra in mathematics education Visualization within functional algebra involves drawing and manipulating graphs. In a technological environment this can be done in a fast, flexible and dynamic way. This offers opportunities for linking multiple representations of a relation, and in particular for linking graphical and algebraic representations (e.g. Doerr & Zangor, 2000; Janvier, 1987; Ruthven, 1990; van Streun, 2000; van Streun et al., 2000; Yerushalmy, 1991, 1997). The ‘algebra of graphs’ example in Fig. 5.1 illustrates how the graphing calculator can be used to explore how changes in the formula affect the graphs (Doorman et al., 1994; Drijvers, 1994). It is an example of an explor- atory approach that aims at combining the graphical and the algebraic representa- tion. Figure 5.1 The algebra of graphs example The function list in the left-hand screen of Fig. 5.1 shows two linear functions. The rule is that the product function Y3 is locked and cannot be changed. The ‘game’ is now to change the formulas of Y1 and Y2 to generate a required visual effect on the graph of the product function Y3, shown in the right-hand screen. For example, the graph of Y3 is required to be a ‘valley parabola’ rather than a ‘mountain parabola’, or it is supposed to touch the horizontal axis, or we want its vertex to be exactly above the intersection point of the two lines. This game could be played with graphical software on a desktop PC as well as with the graphing calculator. Examples such as the algebra of graphs illustrate how working in the IT environment can stimulate the perception of different representations as different views of the same mathematical object, and the linking of the visual and algebraic properties of the functions involved. Furthermore, it shows that an IT environment allows for exploration and variation. Dynamics in the sense of sliders and gradually moving graphs are not included in the algebra of graphs example. Note that the effect of tasks such as the algebra of graphs does not depend only on the affordances of the technological tool, but rather on the exploitation of these affordances embedded in the educational context and managed by the teacher (Doerr & Zangor, 2000). In this study, we intend to use technology for visualization, dynamics and for combining graphical features with algebraic properties. 79 Chapter 5 Learning the concept of variable with IT tools Technology can be used to foster the understanding of the concept of variable. This is relevant to our study, as the concept of parameter builds on the concept of variable. We discuss two examples: the work of Graham and Thomas with the graphing calculator and the Freudenthal Institute’s computer applet ‘number strips’. Graham and Thomas (1999, 2000) propose the following assignment to address the relation between a variable and its numerical value: Store the value 2.5 in A and 0.1 in B. Now predict the results of the ten sequences listed below. Then press the sequences to check your predictions. A+B, B+A, A − 5B, 2A+10B, A/B, AB, BA, 2A+2B, 2(A+B), 4(A+5B) (Graham and Thomas, 2000, p. 270) Figure 5.2 Screens for the Graham and Thomas task The graphing calculator screens that students obtain while working on this task are shown in Fig. 5.2. In the left-hand screen, values are attributed to the variables. After that, the student is supposed to predict the values of the expressions in the right-hand screen. This task addresses the variable as placeholder, as a ‘box’ in which numerical values can be stored, as was described in Section 4.4 (Heck, 2001; Kemme, 1990). When a student shows only the right-hand screen to his neighbour, the task for his colleague is to reconstruct the values of A and B. In that case the variables play the role of unknowns. The authors report that the graphing calculator proved to be an instrument for achieving a significant improvement in student understanding of the concept of variable, and of the roles of placeholder and unknown in particular. The approach in this study is similar to the one described by Tall and Thomas (1991). The Graham and Thomas example focuses on technology use for enhancing the concept of a variable as a placeholder for numerical values, and as unknown.

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