The teacher, of course, plays an important role in the students’ learning of how to integrate computer algebra into mathematical work. It appears that teachers may be very influential in privileging certain ways of using (or not using) a computer algebra device. How do teachers do that? What professional development is needed for teachers in order to be able to integrate the use of the technology in an effective way in their teaching?
(Revised versions of these papers published in IJCAME vol 9, no 2, 2002)
We had a varied and interesting time in this group. Even with a topic as focused as 'CAS and teachers' it was clear that the different interests of the group members created different foci and questions. For example, members who were involved in in-service work with teachers wanted to know how best to train teachers whilst curriculum developers focused on how teachers might help students. At the core, however, everyone was united in trying to understand the phenomena of teachers using CAS.
The following notes are based on the group's presentation to the Symposium final session. They do not report in detail on the discussions we had but summarise the areas and questions we covered.
Members of the group:
John Berry (University of Plymouth, UK),
Roza Leikin (Israel Institute of Technology, Haifa, Israel),
Jean-Baptiste Lagrange (Université de Rennes 1, France),
Matija Lokar (University of Ljubljana, Slovenia),
John Monaghan (University of Leeds, UK) [CHAIR],
Adrian Oldknow (Institute of Education, University of London, UK),
Pieter Schadron (Texas Instruments, The Netherlands),
Kaye Stacey (University of Melbourne, Australia),
Carel Van de Giessen (Almende College, The Netherlands),
Rose Zbiek (University of Iowa, USA)
PUPILS:
TEACHERS:
PARENTS:
TRAINERS:
CURRICULUM DEVELOPERS:
We don't know the answer but Kaye Stacey et al's Symposium paper addresses this indirectly. It shows how different teachers will 'privilege' different aspects of mathematics and CAS tools according to their developmental paths as teachers. It would be somewhat naive to assume that a simple cause and effect analysis can be applied to this question.
This is an urgent question for our Australian colleagues. It concerns students but does it also concern teachers? Of course it does. Teachers in innovative CAS projects are deeply concerned about such issues. They live with the students and are responsible for their mathematical development. We need to do more than just understand the issues facing teachers, we need to help them.
It was noted that some students (and teachers - see Stacey et al's paper) write keystrokes. What does this say about such teachers? Is this useful for the students (their development? their marks in exams?) We thought it would depend on the nature of the task.
Not surprisingly we thought it was important and something that teachers should address whether they use CAS or not. Whether it is something that become more important in CAS-based mathematics is an interesting question for further thought and research.
Adrian Oldknow offered the acronym STAR: strategist, technician, accountant, reporter This is his way of thinking of students doing mathematics with CAS (or any technology). It wasn't developed a great deal by us, but several people came back to it as an interesting idea.
It was felt that wee need a set of outcomes for CAS mathematics. Not so much to reify learning objectives but more so that participants in CAS projects (all participants: students, teachers, parents, curriculum developers) can report/feedback on how students achieve with respect to these outcomes. What outcomes do participants want? Are there intrinsic outcomes to CAS-based teaching and learning which are distinct to desirable outcomes for teaching and learning using any medium?
Michèle Artigue, in her presentation, writes: "To analyse the life of a mathematical object in an institution, to understand the meaning in the institution of "knowing/understanding an object" is to identify the praxéologies which bring it into play."
From the point of view of teachers it is crucially important to understand their positioning in their institutions.
Kathy Heid, in her presentation, mentions Pea's amplifiers and organisers. From the point of view of working with teachers we should not try to specify what these 'are' but start from what teachers in CAS projects see them as being.
The subject of discursive knowledge (knowing that) vs procedural knowledge (knowing how) appears very important. These are not necessarily distinct categories but we should be aware that what teachers may talk about during in-service courses may not easily translate into what they are able to do when they are back in class with 30 students.