Algebraic Insight Underpins the Use of CAS for Modelling

Computer Algebra Systems (CAS) performs algorithmic processes quickly and correctly. Concern is commonly expressed that students using CAS will merely be pushing buttons but this paper indicates that, while CAS may assist students, this facility impacts on only one section of the mathematical modeling process: CAS may be used to help find mathematical solutions to mathematically formulated problems. Controlling and monitoring the use of CAS to perform the necessary routine processes requires the mathematical thinking referred to as algebraic insight. This paper sets out a framework of the aspects, and elements of algebraic insight and illustrates the importance of students developing each of the two key aspects: algebraic expectation and ability to link representations. This framework may be used for both planning teaching and monitoring students’ progress. CAS Support Mathematical Analysis in the Modeling Process Mathematical analysis tools are now not only increasingly powerful but affordable and available. In particular, Computer Algebra Systems (CAS), available for PC’s and hand held calculators, offer students support to allow them to work successfully through more complicated or time consuming mathematical manipulations and calculations. Heid (2003) describes clearly three key ways in which CAS can function as a cognitive technology: • Students can use CAS for the repeated execution of routine symbolic procedures in rapid succession, without diminished accuracy and increased fatigue usually associated with the repetitive execution of byhand routines... • Students can assign rote symbolic tasks to the CAS so that they can concentrate on making ‘executive’ decisions... • Students can use the CAS to apply routine symbolic algorithms to complicated algebraic expressions, without the confusion students sometimes experience when trying to apply a routine procedure to a complicated expression. (pp. 34-35). Pierce TMME,vol2,no.2,p.108 This capacity for CAS to be used by students to share cognitive load has obvious advantages for mathematical modeling. A CAS allows the user to work in numeric, graphic or symbolic modes and to move between these with mathematical precision and relative ease. For example, in modeling real world situations in order to solve estimation or optimization problems, it is common to begin by collecting and entering numeric data into a software package. CAS allows us to use the graphic mode to examine any pattern in this data; make use of the CAS’s statistical capabilities to perform an appropriate regression on the data; and store the result in the graphic function editor ready for graphing or transfer to the symbolic mode. The model which has been created can be examined and refined for the particular case then the impact of changing the various parameters may be explored until a general model is developed or that notion discarded. Monitoring CAS Work Requires Algebraic Insight It must be clear though that CAS does not reduce the need for students to develop their skills in mathematical thinking. Figure 1, below, illustrates the typical process for mathematical modeling. Starting with a real world situation (top left) which must be formulated as a mathematical problem, the mathematician typically collects numeric data or moves immediately to a symbolic representation of the situation ( top right). Using symbolic, graphic, numeric or geometric methods the mathematician works on the abstract version of the problem in order to progress towards some particular or general solution. Once a mathematical solution has been developed (bottom right) this abstract solution must be interpreted in terms of the real world (bottom left) and checked for applicability in the situation where this process began. If the solution is not adequate then the process must be repeated. This diagram highlights the fact that, currently, technologies like CAS only impact on one section of the modeling cycle, that is, the process of moving from the mathematically formulated problem to a symbolically formulated particular or general solution. CAS assists with routines but does not take over the role of mathematical thinking. This is illustrated by Pierce and Stacey, (2001a) who report the following extract from a group interview conducted with first year undergraduate mathematics students working with CAS available for all aspects of learning and assessment: Interviewer: One of the other things that people argue about is whether or not people are really doing mathematics when working with a computer-algebra system. Are you doing it or is the machine doing it? Who’s doing the maths? Student A: I reckon that we are actually doing it. The computer only spits out an answer to what you type into it Student B: It’s just like with a calculator...it’s just going a bit further, we’re not just doing multiplication and division quickly, we’re doing simple differentiations and stuff quickly. Student C: Also, you still have to interpret the answer or for that matter interpret the question so you can convert it into what the Pierce TMME,vol2,no.2,p.109 computer wants ...you’re still doing a lot of mathematics. (pp. 153154). Figure 1. A model of problem solving showing the places of symbol sense and algebraic insight (Pierce & Stacey, 2002) The processes of formulating and solving the mathematical problem then interpreting the solution all require what Fey (1990) and Arcavi (1994) call symbol sense. As Fey (1990) pointed out: Even if machines take over the bulk of computation, it remains important for users of those machines to plan correct operations and to interpret results intelligently. Planning calculations requires sound understanding of the meaning of operations – of the characteristics of actions that corresponds to various arithmetic operations. Interpretation of results requires judgement about the likelihood that the machine output is correct or that an error may have been made in data entry, choice of operations, or machine performance. (p.79) Symbol sense is a broad concept encompassing a feel for the power of symbols; an ability to use symbols to express relationships; a sense of when to use symbols and when to use another approach; a sense for which symbolic manipulations will aid progress towards solution of a problem; an ability to recognise equivalent symbolic expressions; an ability to interpret the meaning of symbols in a given context and much more. In this paper we concentrate on the part of symbol sense required to monitor progress towards the solution of a mathematically formulated problem. This is the phase of the modelling process where a CAS may be able to perform the algorithmic tasks involved accurately and quickly. However, in order to direct and monitor this work the user needs the part of symbol sense we call algebraic insight. Pierce TMME,vol2,no.2,p.110 Technology to date does not impact on the processes of formulation and interpretation; it does however offer alternative methods to progress between the mathematically formulated problem and a mathematical solution. Methods which were, in the past, considered too time consuming or tedious are now accessible. For mathematics teachers and students, limited by the constraints of class timetables and a crowded curriculum, CAS can offer the possibility of tackling interesting real problems which could not previously have been tackled in the time available. The support of CAS to correctly execute the algorithmic routines and manipulation required in a solution process may allow students to test their conjectures and develop their higher level mathematical thinking instead of setting their focus at the micro level of the steps involved in these routines. However, studying the value of the output from such a process of shared cognition will be dependent on correct input and the execution of appropriate commands. Checking that mathematical expressions have been correctly entered in to CAS and that the output at each stage makes sense certainly requires symbol sense. As stated above, to draw specific attention to this part of symbol sense we refer to it as algebraic insight. Its place in the broader scheme of thinking required to work within and between the three mathematical representations typically afforded by CAS is illustrated in Figure 2 and the key aspects, elements and some common instances of this concept are outlined in Figure 3. Figure 2 indicates that algebraic insight has two key aspects: first the thinking which allows us to monitor working within the symbolic mode of operating, that is algebraic expectation; and second the ability to link representations, in this case to link the symbolic with graphical or numeric representations. These two elements of algebraic insight will be discussed and illustrated in the following section. Figure 2. The place of algebraic insight and its components within the senses needed when working with CAS. (Pierce and Stacey, 2004) Pierce TMME,vol2,no.2,p.111