Opening up open-ended mathematics tasks

This discussion has evolved from ideas in Peter Sullivan and Pat Lilburn's Open-Ended Maths Activities; Using 'Good' Questions to Enhance Learning (1997). This is a compilation of open-ended questions, and methods for generating open-ended questions, along with a brief rationale for using such questions. This classroom-oriented teaching resource developed from earlier research and teaching development work by Peter Sullivan and David Clarke, and others. Two methods are offered for constructing a 'good question'.

Introductory computer programming as a means for extending spatial and temporal understanding

Should computer programming be taught within schools of architecture?

Incorporating even low-level computer programming within architectural education curricula is a matter of debate but we have found it useful to do so for two reasons: as an introduction or at least a consolidation of the realm of descriptive geometry and in providing an environment for experimenting in morphological time-based change.

Mathematics and descriptive geometry formed a significant proportion of architectural education until the end of the 19th century. This proportion has declined in contemporary curricula, possibly at some cost for despite major advances in automated manufacture, Cartesian measurement is still the principal ‘language’ with which to describe building for construction purposes. When computer programming is used as a platform for instruction in logic and spatial representation, the waning interest in mathematics as a basis for spatial description can be readdressed using a left-field approach. Students gain insights into topology, Cartesian space and morphology through programmatic form finding, as opposed to through direct manipulation.

In this context, it matters to the architect-programmer how the program operates more than what it does. This paper describes an assignment where students are given a figurative conceptual space comprising the three Cartesian axes with a cube at its centre. Six Phileban solids mark the Cartesian axial limits to the space. Any point in this space represents a hybrid of one, two or three transformations from the central cube towards the various Phileban solids. Students are asked to predict the topological and morphological outcomes of the operations. Through programming, they become aware of morphogenesis and hybridisation. Here we articulate the hypothesis above and report on the outcome from a student group, whose work reveals wider learning opportunities for architecture students in computer programming than conventionally assumed.

Productive on-line student support : developing a system for primary mathematics trainee teachers

This paper outlines the development project for the 'Productive on-line student support system', a student "self-help" system, at Deakin University. The aim of this project was to provide Deakin primary teacher education students with a web-based learning tool that allowed them to assess and diagnose their strengths and weaknesses in mathematics, and supports students in their mathematics learning, and in so doing produce mathematically competent graduates. This project was, like similar programs, a development of peer or cross-age tutoring common in primary and secondary schools. A grant under the Deakin University Strategic Teaching and Learning Grant Scheme enabled a staff team from the mathematics education group, to develop a sophisticated and well-designed system that catered for a wide range of student needs, provided useful feedback, and was engaging and easy to use. The under-pinning software for the system was WebCT, available to staff through the Deakin Studies On-line system, to which students are connected also. The 'Productive on-line student support system' enabled students to determine their own mathematical needs, and have these addressed whenever they wished, as often as they wished, and allowed self-monitoring of progress. An outline of the system and examples of the assessment materials will be presented.

Problematising the mathematics curriculum : some examples from Australia and Japan

Problem solving is often seen as being the core of mathematics. While there are many examples of teaching for and about problem solving, there are relatively few examples of teaching mathematical content through problem solving. This paper uses data from three, apparently quite different, mathematics lessons from Australia and Japan to explore different ways in which mathematics can be taught successfully through problem solving and to analyse some of the characteristics of such lessons. It also attempts to identify some of the supports and constraints for adopting a problem solving approach to the teaching of mathematics that exist in the quite different contexts of Japan and Australia.

Analyzing Australian middle years students' preferred mathematics classroom practices

The Middle Years of schooling have become an issue for mathematics teachers and educators, and calls for the reform of this period of schooling are frequent. However, the suggested reforms appear to be divided in their views of what would be best for these students. This paper provides a brief overview of what some Middle Years students say about their needs. Some 1500 students in the Middle Years of schooling (Years 5 to 9) were surveyed about their preferred mathematics classroom activities: the methods and results of the survey may be extremely relevant to those contemplating Middle Years reform.