Yip-Cheung Chan

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.

Implementation of Tool-Based Mathematics Lesson: A Duo of Material and Digital Tools

This paper reports a case study on the implementation of a tool-based mathematics lesson. The design of a tool-based task and the implementation and evaluation of mathematics lessons were analyzed in the lens of a theoretical framework underpinning a duo interplay of material tools and digital tools serving as teaching and learning aids. It was found that material tools and digital tools serve different functions in the manipulation processes. The tools played different roles in the tool-based learning environment where students develop mathematics senses with interplay of the tools.

Feedback and Discrepancies of a Physical Toolkit and a Digital Toolkit: Opportunities and Pitfalls for Mediating the Concept of Rotational Symmetry

In this chapter, excerpts of lessons on using tool-based tasks to teach the concept of ‘rotational symmetry’ were analyzed. Both the instrumental approach and the theory of semiotic mediation were adopted as theoretical frameworks. We compare a lesson carried out with a tailor-made physical tool and one carried out with the software PowerPoint (a digital tool). The analysis focuses on the opportunities and pitfalls that these two tools offer and on how the tasks could (or could not) exploit the semiotic potential of the tool used. In particular, the notions of feedback and discrepancy are theorized. Hypotheses on these notions in the context of designing and implementing tool-based mathematics tasks are raised. We propose that the critical features of the object of exploration, the discrepancy opportunity and pitfall of the tool and the task as well as the teachers’ sensitivity and insights into the discrepancy are important considerations for tool-based mathematical task design. They provide a useful guiding framework for investigating the pedagogical affordances of different mathematical tools. We hope that this chapter can provide insights into how the choice of the tools and the design of tool-based tasks may enhance exploitation of the semiotic potential of the tools.

A multiple case study of teachers referring to their own religious beliefs in mathematics teaching

Abstract There is a need to integrate religious education and spiritual education across school curriculum. This paper reports one of the few empirical studies on bridging the intention-practice gap in classrooms. Six school teachers deliberately designed and implemented mathematics lessons which referred to their own religious beliefs in teaching. It unfolds teachers’ intention to enact their religious beliefs in mathematics classroom teaching. Different modes were identified. Implications to religious education in schools are offered.

Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simson line

Dynamic geometry environment (DGE) is a powerful tool for exploration and discovering geometric properties because it allows users to (virtually) manipulate geometric objects. There are two possible components in the process of exploration in DGE, viz. experimentation and theoretical consideration. In most cases, there is interplay between these two components. Different people may use DGE differently. Depending on the specific mathematical tasks and the background of individual users, some approaches of interplay are more experimental whereas some other approaches of interplay are more theoretical. In this paper, different approaches of exploring a geometric task using Sketchpad (a DGE) by three individual participants will be discussed. They represent three different approaches of interplay between experimentation and theoretical consideration. An understanding of these approaches may contribute to an understanding on the mechanism of exploration in DGE.