Number 1, November 1996.
Number 1, November 1996.
Jean-Baptiste Lagrange
DIDIREM research group
Institut universitaire de formation des maîtres
153, rue de St Malo 35043 Rennes Cedex FRANCE
Tel: (+33 2) 99 54 64 44 Fax: (+33 2) 99 54 64 00
Email : lagrange@univ-rennes1.fr
Learning Mathematics is potentially a major application of Symbolic and Algebraic Computation. Secondary students have now easily access to systems of Computer Algebra on computers as well as on hand held calculators. In the recent years, investigations in the field of the "didactics of computer algebra" indicated improved understanding as well as learning problems occurring in the classroom use of symbolic systems (see Monaghan, 1994). The author is involved with the DIDIREM team in a large scale examination of the French DERIVE and TI92 experiment. He claims that, for a successful use of Computer Algebra, we need more reflection on the complex links between technical skills and conceptual understanding in the learning of mathematics.
In 1993, at the beginning of the French DERIVE experiment, many papers in the field argued that computer algebra potentially relieves students of the technical part of Mathematics, and extends their heuristical and conceptual activity. A group of French teachers, supported by the ministry of Education (see Hirliman, to appear) worked to actualise those assumptions. They experimented the use of Computer Algebra in the everyday practice of Mathematics by their students. They assumed that the students could focus on the conceptual part in this practice. In addition, they created a lot of new classroom situations. In those situations, Computer Algebra was supposed to play a key role in students' understanding of a given mathematical topic. The DIDIREM team of researchers in "didactique" was in charge of a survey of this experience (see Artigue, to appear). Questioning teachers and students, examining classroom situations, we noticed that the change in the relationship between technical skills and understanding seemed not so clear, as compared with the expectations of the teachers.
First, the technical work did not vanish when doing mathematics using Computer Algebra. Not all students did actually welcome to be relieved of the usual pen and paper skills : some of them considered those skills as something useful to succeed in Mathematics. It appeared also that using Computer Algebra requires specifical skills and knowledge. For instance, when a student obtains an output using the system, this output is not always the usual expression generally accepted in the pen and paper context. In this situation, few students could transform the system's output to obtain the usual writing. A consequence is that, although most students thought of Computer Algebra as a helpful tool for "double checking", they generally miss the techniques to perform effectively this double check.
Understanding mathematics with the help of Computer Algebra is not a view that students generally considered much. Even when they liked the new classroom situations that they experienced using Computer Algebra, they did not generally recognise that those situations could bring them a better comprehension of a mathematical concept. If a better understanding of a topic is reported by students, it happened generally when they had an easy practice of the symbolic capabilities related with the topic. Therefore, one cannot describe the change in the relationship between the technical and conceptual part of the mathematical activity as "less techniques, more understanding". Symbolic computation brings new skills as well as new understandings, interacting with the usual pen and paper practices. To analyse this interaction, and develop a successful use of Computer Algebra, a key point is the "computational" conversion of a mathematical subject (see Balacheff , 1994) when it is instanced inside a Computer Algebra package. For instance, factoring a polynomial appeared very different to the students (and to the teachers) with the DERIVE package, as compared with the usual practice. First, the user of a computer algebra system has to select a "level" of factoring which is actually a class of possible outputs and the associated algorithm. In addition, factoring is systematically complete, when, in the usual practice, factoring partially is possible and often useful. Then factoring with DERIVE is not just reading an output on a computer screen. Students must develop specific techniques for using DERIVE's factoring in problems and therefore, enlarge their understanding of this topic.
With this "computational conversion" in mind, a question for research could be what knowledge students should have of fundamentals of symbolic computing, to use wisely a Computer Algebra package. On this question, specialists in symbolic computation and didacticians could fruitfully collaborate.
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