Learning mathematics through programming : An instrumental approach to potentials and pitfalls

ion and concept formation: APOS theory The idea that programming could be helpful in mathematics education in the late 1980s also developed in the context of teaching mathematics in high school and college. Here the geometric and artistically framed LOGO program was less popular. On the contrary, teachers often utilized common programming languages such as BASIC, COMAL and PASCAL to support learning. One of the outspoken hopes was to create a process-oriented approach to abstract mathematics, basing abstract constructions in concrete numerical computations. The arguments for this approach were often based in constructivism and radical constructivism, which claims that all abstract learning has a concrete starting point, as well as in the and in the discussions of process-object duality (Sfard, 1991). Ed Dubinsky’s work is probably the clearest description of the learning potential of programming (see Breidenbach et al., 1992). His theory is often referred to as APOS theory and it is situated in a radical constructivist framework (Glasersfeld, 1995). APOS is an acronym for action, process, object, and scheme. The theory describes mathematical concept formation as beginning with performing actions on well-understood mathematical objects; these actions can be organized in processes and encapsulated into objects. These objects can be related to one another in schemas. The encapsulation stage is, as famously described by Sfard (1991), crucial and hard. And the Learning mathematics through programming: An instrumental approach to potentials and pitfalls (Morten Misfeldt and Stine Ejsing-Duun) 2526 schematic aspects of concept formation is similar to Skemp’s relational understanding (1971). This rather general learning theory of mathematical concept formation relates to the use of computers because they can significantly empower and enrich the concrete numerical calculations that are – in this conception – the necessary foundation for concept formation. Process approach to mathematics: Algorithmic thinking The ability to think in algorithms and procedures is promoted as an important learning goal in mathematics. Algorithmic thinking describes students’ ability to work with algorithms understood as systematic descriptions of problem-solving and construction strategies, cause-effect relationships, and events. A recipe is a good example of an algorithm: (1) Add all dry ingredients together. (2) Stir. (3) Add 2/3 cup of the water and stir. (4) If the dough is steady, then stir for 2 minutes. Otherwise, go to step (3) and add more water. Algorithmic thinking is about being able to develop, execute, and make machines to perform such algorithms. Donald Knuth (1985) views algorithms as a crucial phenomenon constituting the intersection between computer science and mathematics. He traces the study of algorithms to the mathematical masterpiece Al Kwarizm from the 9th century (Katz, 1993). Knuth defines algorithms as follows (Knuth, 1985, p. 170): I tend to think of algorithms as encompassing the whole range of concepts dealing with well-defined processes, including the structure of data that is being acted upon as well as the structure of the sequence of operations being performed; some other people think of algorithms merely as miscellaneous methods for the solution of particular problems, analogous to individual theorems in mathematics. Hence algorithms, according to Knuth, consist of both a recipe and the actual objects dealt with by the recipe. Knuth analyzes the difference between mathematical thinking and algorithmic thinking. He finds that a first approximation algorithmic thinking relates to (1) representation, (2) reduction, (3) abstract reasoning, (4) information structures, and (5) algorithms. Mathematical thinking can, according to Knuth, relate to all of these, however other aspects are also present such as (a) formula manipulation, (b) behavior of functions, (c) dealing with infinity, and (d) generalization. Hence algorithmic thinking is strongly related to mathematical thinking but emphasizes specific and slightly different aspects than other types of mathematical thinking. Before we introduce classroom examples exemplifying these learning potentials, we will introduce the instrumental approach that we use as a general theoretical framework for the use of ICT for mathematics teaching. This framework will be used to analyze the cases and create a connected description of the different learning potentials. THE INSTRUMENTAL APPROACH The instrumental approach (Guin, Ruthven, & Trouche, 2005) addresses students’ use of technology when learning mathematics from the perspective of appropriating digital tools for solving mathematical tasks. It builds on (Verillon & Rabardel, 1995), and views computational artifacts as mediating between user and goal (Rabardel & Bourmaud, 2003). The approach presupposes a continuation and dialectic between design and use, in the sense that a pupil’s goal-directed activity is shaped by his use of a tool (this process is often referred to as instrumentation), and simultaneously the goal-directed activity of the pupil reshapes the tool (this process is often referred to as instrumentalization) (Rabardel & Bourmaud, 2003, p. 673). In students’ work with technology the distinction between epistemic mediations and pragmatic mediations (Guin et al., 2005; Rabardel & Bourmaud, 2003) operationalize the difference between learning with technology and just using technology to solve tasks. Epistemic mediations relate to goals internal to the user – affecting his or her conception of, overview of, or knowledge about something Rabardel & Bourmaud (2003) use the example of a microscope, and Lagrange (in Guin et al., 2005, ch. 5) refers to experimental uses of computers) and pragmatic mediations related to goals outside of the user – making a change in the world (Rabardel & Bourmaud use the example of a hammer, Lagrange (in Guin et al., 2005, ch. 5) refers to the mathematical technique of “pushing buttons”). Finally, Rabardel & Bourmaud (p. 669) introduce sensitivity to a broader conception of the orientation of the mediation. Instrumented mediations can be directed towards (a combination of ) the objects of an activity (the solution of a task), other subjects (classmates, the teacher), and oneself (as a reflective or heuristic process). Hence the theoretical framework consists of Learning mathematics through programming: An instrumental approach to potentials and pitfalls (Morten Misfeldt and Stine Ejsing-Duun) 2527 the concepts: instrumental genesis, as consisting of instrumentation and instrumentalization, the concepts epistemic and pragmatic mediations, as well as a sensitivity towards the orientation of an instrumented mediation. The orientation of the mediation can be towards oneself, external objects, and other subjects. EXAMPLES OF LEARNING MATHEMATICS