Theme 1 focused on essential questions regarding the relationship between paper-and-pencil techniques, CAS techniques, and conceptual understanding. The plenary talk, given by Michèle Artigue, began by presenting the theoretical framework developed and used by the French researchers in mathematical didactics. She discussed some key points of Chevallard's anthropological approach in didactics. The French research globally relies on: mathematical objects that arise from institutional practices - Praxeologies; praxeologies can be seen as complexes of the elements tasks-techniques-technology-theory. Techniques have both pragmatic and epistemic value; the advance of knowledge goes with making tasks and techniques routine. Then she explained the need the group felt to complement this theoretical framework by an ergonomic approach, more precisely that developed by Verillon and Rabardel around instrumental genesis. Michèle analyzed research findings indicating the unexpected complexity of instrumentation. These useful examples and their analysis led to the issue of instrumented techniques and their relationship with conceptualization. Using the examples, she emphasized that technology changes the epistemic value of techniques, thus the epistemic value of instrumented techniques relies on adequate didactical situations, some of which do not have immediate counterparts in a paper-and-pencil environment.
Ken Ruthven reacted to the plenary talk, whose aim he summarised as finding a systematic way of thinking about the learning of mathematics in a technological environment. He introduced the term virtual mathematical experience: the idea that our sense of interacting with mathematical objects is shaped by, and expressed through, the tasks and techniques that give them form and substance, and the technical and theoretical discourses surrounding them. Thus, students develop cognitive schemes adapted to the tasks, techniques, and discourses they encounter in their cultural system. Ken introduced an additional aspect, the communicative interface where cognitive schemes meet cultural techniques. Where tasks or techniques are not standard, it becomes necessary to account for the solutions explicating personal schemes in relation to the cultural systems. Using examples from studies in France he discussed the situated nature of the concept of simplification. His conclusion was that effective use of computational technologies requires their instumentalization to create mathematical tools, and that instrumentation involves the reorganization of technical and conceptual aspects of mathematical thinking.
Revised versions of these papers are due for publication in the International Journal of Computers for Mathematical Learning in 2003
Two short papers were given by members of the group.
(Revised version published in IJCAME vol 9, no 2, 2002)
This paper describes the setup of an elementary calculus and linear algebra course for first-year university students in social sciences. In this course a computer algebra environment was incorporated into a more traditional course, with special attention to the connection between them. The teaching strategy integrated paper-and-pencil and CAS techniques. The links between technical work using CAS and conceptual understanding have been explored.
[Word file of the van Herwaarden and Gielden presentation] (104 KBytes)
(Related paper published in IJCAME vol 9, no 3, 2002: Kidron & Zehavi, The Role of Animation in Teaching the Limit Concept)
This paper focuses on how animations may affect the teaching and learning of the concepts of differentiation and limit. The examples demonstrate the advantages of CAS-created animation to visualize dynamic process on one hand, and misconceptions reinforced by the animation on the other hand. In conclusion, students need to be exposed to well-chosen examples in order to help them interpret and control the dynamic graphics.
[Word file of the Kidron presentation] (147 KBytes)
Chevallard's theory provides a possible explanation for the key problem of changing the existing practice to one based on the integration of CAS technology into the teaching of mathematics; namely, that we are not even sure about the types of tasks that would characterize the new praxeology. As Michèle and Ken emphasized, the challenge is to design didactical situations that highlight the epistemic value of the instrumented techniques. The participants agreed that the way in which the technological analysis interweaves with the mathematical analysis is difficult but essential part of the task design. Therefore, the group discussion was devoted to the question: What are the implications of the complexity of instrumentation for task design and research in CAS environments? Here are some of the participants' suggestions for designing tasks and the associated research (based on the notes taken by Lynda Ball).
| TASKS | RESEARCH |
|
Tasks designed to address known difficulties of students |
Are the difficulties encountered in a paper and pencil environment actually present in a CAS environment? |
|
Diverse tasks designed for conceptual development |
Examine the relationships between techniques and concepts when using CAS and other technologies. Encourage the implementation of tasks under different conditions (to obtain an international perspective). |
|
Tasks designed to integrate the different types of representations available in a CAS environment |
What role does visualization play in different environments? Under what circumstances is it appropriate to work with two or more representations? |
|
Tasks that encourage a good classroom discussion and for which the resulting discussion opens 'webs of meaning' (a term used by Noss & Hoyles) that go beyond that which occurs within the regular curriculum. |
Exploring the opportunities to extend the curriculum. Carry out classroom experiments with rich tasks that would not really be feasible without the CAS. |
|
Tasks that motivate socio-mathematical norms that arise in the new mathematical environment |
Examine the types of communication and accounting in a CAS classroom. Examine the tension between the cognitive and cultural aspects. |
|
Tasks for assessment of the impact of integrating CAS into teaching |
Identify those difficulties created by the complexity of instrumentation when students are using the CAS. Compare students' achievement when implementing alternative approaches to the use of CAS. |
|
Tasks specially designed for teachers to experience doing mathematics with CAS |
Examine the process of teachers' development in introducing CAS. Examine the transition of teachers from graphic calculators to the CAS culture. Classify and analyse tasks developed by teachers who are experienced users of CAS. |
The discussion concluded by emphasizing the difficulties in developing criteria for CAS-based tasks, and by inviting the participants to share their 'successful' tasks. It was also suggested to consider the design of tasks as a theme of the next CAME Symposium (scheduled for 2003).