---

Mathematical Education Cognition in Mathematics / Enseignement des mathématiques Cognition et mathématiques
(Florence Glanfield, Organizer)

---

COLIN CAMPBELL, Information Systems and Technology, Pure Mathematics; and Waterloo
The expanding role of Mathcad in mathematical courses at Waterloo

Instructors seeking to add dynamic mathematical explorations to their courses have turned to Mathcad in record numbers at the University of Waterloo. From Physics and Chemistry to Engineering and Mathematics dozens of instructors have adopted Mathcad mainly because of it's ease of use for faculty and students alike (compared to Maple and MATLAB).

The challenge with all technology is to use it effectively. The presenter will describe what has been done so far, particularly in Linear Algebra courses with support from UW's Teaching and Learning Through Technology Centre (LT3), and seek feedback from participants on ways to improve.

JAMIE CAMPBELL, University of Saskatchewan, Saskatoon, Saskatchewan
Cognitive architecture and numerical skills

Educated adults acquire an elaborate system of numerical skills that has its roots in basic perceptual processes in infancy and that ultimately involves the functional integration of a diverse set of complex cognitive procedures. These component skills include the ability to comprehend and produce written and spoken numbers, to count by various increments, as well as to retrieve, calculate, or estimate the results of both simple and complex arithmetic problems. Numerical competence emerges through the functional and conceptual integration of these various numerical activities. The question of how to conceptualize the cognitive architecture underlying numerical skills has been the focus of much theoretical debate over the last few years. I will review research that supports an encoding-complex view of basic numerical cognition. This approach emphasizes representational concreteness (i.e., number processing is based on perceptual and linguistic codes, rather than on abstract codes), hyperspecificity (i.e., skilled performance depends on implicit memory processes that are specialized for code-specific input/output pathways), and interactivity (i.e., competition and resolution of competition among numerical cognitive operations).

ALAN COOPER AND ERIC WOOLGAR, Department of Mathematics and Statistics, Langara College, Vancouver, British Columbia  V5Y 2Z6 and Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta  T6G 2G1
Web-based resources for teaching and learning: building a location and retrieval service for the Canadian mathematical community

There is a large body of web-based material intended to support the teaching and learning of mathematics, but its quality is uneven and it is still not easy to quickly locate the best available resources related to any particular topic. Both users and authors will benefit from improved indexing and review of such materials. A number of current projects are directed towards providing such a service, and the speakers have received an Endowment Fund Grant to support work towards the development of a Canadian version, but it is not yet clear what will best meet the needs of the mathematical community.

Our intent is both to review the current state of affairs, and to initiate a discussion of what features of such a service would be most useful for both authors and users of the material. A report and hands-on lab regarding work to date will therefore be followed by a working group of those who wish to participate in further development.

IMPLICATIONS FOR TEACHING I
discussion of Orzech/Orzech talk

IMPLICATIONS FOR TEACHING II
discussion of Campbell/May/Rabinowitz

IMPLICATIONS FOR TEACHING III
discussion of Sierpinska talk

SHERRY MAY AND MICHAEL RABINOWITZ, Memorial
To be announced

GRACE ORZECH AND MORRIS ORZECH, Queen's
Dealing with dimension & teaching and learning issues

The two of us teach courses for students with quite different mathematical backgrounds. It was therefore interesting to find that students in both classes exhibit similar (and to us surprising) misconceptions about dimension, a notion that appears as an issue in both our courses. Observing this phenomenon has not led us to dramatic success in overcoming student difficulties, but it has informed our own ideas about how to promote mathematical understanding and has influenced the conversations we have with our students. Our reflection on what our students seem to think has influenced not only the way we teach, but the mathematical material we select and how we present it.

ANNA SIERPINSKA, Concordia
Is it enough to `understand' when learning mathematics? Is it necessary?

Motto: `In mathematics, nothing is necessary, or sufficient'.-a well known saying.

This talk will look at the relations between understanding in mathematics and other mental processes and acts such as doing mathematics, memorising, taking another person's point of view, taking another point of view, constructing mental schemata, theoretical thinking, ingenuity. Underlying the discussion of these relations will be, mainly, the question posed in the title of the talk. Two other questions will be asked to provoke further discussion: `Can understanding in mathematics be taught?'; `Is it possible to prevent our teaching from becoming a complete impediment to understanding?'