Francis Lopez-Real

Peer Assessment of a Group Project in a Primary Mathematics Education Course

ABSTRACT This paper discusses the problem of discriminating between individual contributions in the assessment of group projects. It is argued that peer assessment is a necessary element of any proposed structure but that the use of any ‘weighting‐factor’ system is contrary to the philosophy underpinning collaborative groupwork. A structure that separates the process and product elements of the assessment is described and a set of generic criteria for the process peer assessment identified. This structure was implemented with a group of students at Hong Kong University and evaluated using questionnaires and in‐depth interviews. The paper discusses the results of this evaluation and in particular identifies the need for ‘benchmarking’ when using the criteria and also the problematic nature of the ‘insider‐outsider’ syndrome as an important cultural factor.

Dragging as a conceptual tool in dynamic geometry environments

Some ‘drag-to-fit’ solutions given by student teachers to three geometric construction problems in a dynamic geometry environment (DGE) are analysed. The responses of a group of experienced mathematics teachers to the question whether or not such solutions can be considered ‘legitimate’ are then discussed. This raises fundamental questions concerning the concept of legitimacy, the relationship between DGEs and Formal Axiomatic Euclidean Geometry, the nature of ‘conceptual tools’ in different geometric environments, and the functions of dragging in DGEs. The authors argue that, if dragging is viewed as a conceptual tool, then certain drag-to-fit solutions, although soft constructions, may still be considered as conceptually legitimate and therefore valid. Finally, some important questions are raised concerning the impact that teachers’ different attitudes towards legitimacy might have on students’ learning through DGEs.

Properties of tangential and cyclic polygons: an application of circulant matrices

In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon P n with n even is slippable (or conformable) and the sum of a set of non-adjacent sides (or interior angles) of P n satisfies certain equalities. On the other hand, for a tangential (or cyclic) polygon P n with n odd, it is rigid and the sum of a set of non-adjacent sides (or interior angles) of P n satisfies certain inequalities. These inequalities give a definite answer to the question raised by Lopez-Real concerning the alternating sum of interior angles of a cyclic polygon.

Collaborative Reflection through Sharing Significant Incidents

This chapter looks at the use of significant incidents in serving teachers’ own classrooms as a means of encouraging and developing reflective practice. The nature of the incidents is structured according to classifications suggested by the teachers. These include successful and unsuccessful teaching strategies, unusual pupil suggestions, pupil misconceptions, and other unanticipated events. After describing and analysing the incidents the teachers’ reflections were shared in group sessions with other teachers and with a university tutor. This then becomes a second level of reflection where the teacher’s analysis and own reflections of an incident are themselves subjected to discussion with all the members of the group. This leads to a form of collaborative reflection during which not only the individual gains further insights but all members of the group gain a mutual benefit from each other’s experiences. Great mutual trust is required for this kind of sharing but it is argued that this form of collaborative reflection can be a more powerful means of learning than more individualised and introspective modes and can lead to more refined and critical reflection on the part of the teachers. The chapter provides examples of significant incidents described by the teachers, their own analyses and reflections, subsequent group discussions and the insights arising from them, and finally the thoughts of some of the teachers on the whole process.