An aspect of mathematical understanding: the notion of 'connected knowing'

Much advice about teaching for understanding implies that teacher should help children to develop connections between aspects of their experience, knowledge, and skills. This paper outlines points from the literature about different types of connections and describes relevant points from four case study teachers said and did.

Using the results of diagnostic testing to promote mathematical pedagogical knowledge

Teachers require a range of knowledge bases, including both content knowledge and pedagogical knowledge (Shulman, 1986). In recent times there have been calls from a variety of sources for teacher preparation courses to improve the mathematical knowledge of teachers, particularly primary teachers. These calls have been underlined by the recent formation of bodies such as the Institutes of Teachers in Victoria and NSW, as well as the development of teaching standards by professional bodies including the Australian Association of Mathematics Teachers. Rather than simply adopt a "back-to-basics" approach, work is required that uses the results of educational research to design courses that help pre-service students to understand how and why errors are made (by themselves and by children in their own classrooms). Diagnostic testing of preservice students is the first step in the process. However, it is not enough to simply test students and to remediate their misconceptions. Instead, the aim is to use the results of the testing to improve students' pedagogical knowledge as well as their subject content knowledge. This paper outlines one approach to the use of diagnostic testing with pre-service students and how the results can be used to assist in the development of pedagogical knowledge.

Mathematical knowledge of some entrants to a pre-service education course

Standards for teaching emphasise the need for teachers to have deep content knowledge. To assess the mathematical knowledge of students enrolling in its B.Ed. program, the University of New England has introduced a mathematics diagnostic test. This work is the first stage of an ongoing research project into the numeracy needs of students entering the B.Ed. program. The test is a pen-andpaper test that replaces previous on-line, multiple-choice tests. This paper reports on the test results, discusses some common errors made by students and outlines the future direction of the research.

Associations between student pursuit of novel mathematical ideas and resilience

As part of a broader study of characteristics of situations that promote or inhibit spontaneous student exploration of novel mathematical ideas, student inclination to display these spontaneous behaviours was studied in conjunction with resilience. Resilience was operationalised using the dimensions of optimism (Seligman, 1995). Indicators of optimism were identified in post-lesson video-stimulated reconstructive interviews with year eight students. Students who demonstrated the pursuit of novel mathematical ideas were found to also display indicators of resilience.

Re-thinking multiculturalism: performing the Cronulla beach riot

Since 11 September 2001, Australia’s race relations have been an issue of significant cultural concern, particularly relations between Anglo-Celtic and Middle-Eastern Australians. Riots on Cronulla Beach, Sydney, in December 2005 heightened this concern. This paper looks at the events at Cronulla and the debates they catalysed about race relations in Australia, and examines how these discourses have been shaped by arguments from both the Right and the Left. Informed by the discourse of critical multiculturalism, we examine several performance-based arts activities that made the riots their subject matter and argue that these arts practices reflect a larger cultural concern about the currency of traditional forms of multiculturalism, and promote instead an emphasis on understanding racial conflict as a critical negotiation over shared territories and values.

Diversions

The article focuses on the mathematical games developed based on the polyomino shapes created by American mathematician Solomon Golomb. It details the mechanics of the pentomino game created by Golomb. The spatial thinking strategy board games Cathedral and Blokus were also based on the families of shapes and the idea of Golomb's game. Information on how to play the Blokus games is given.

The 'V' factor: thinking about values as the epicentre of leadership, learning and life

This paper places the ‘V’ factor at the epicentre of meaningful and productive living, learning and leadership endeavour. It explores the authors’ reflections upon their own professional experiences, research studies and review of the literature as they work towards establishing ‘working with’ leadership practices appropriate for a knowledge era. This paper also reports on Nanschild’s preliminary PhD research findings of “A Values-Based Approach to Leadership Development: Implications for Organisational Change”.

The answer is correct but can I explain it? Student explanations in mathematics

This paper explores an aspect of mathematics that has been overlooked often - student explanation, or communication. The data used is derived from a large-scale project in Victoria. Australia, the Improving Middle Years Mathematics and Science (IMYMS) project. Primary student explanations on two open-response items are examined to illustrate the bene tits of incorporating this type of item in mathematical assessments. Techniques for analysing the data and displaying the results of the analysis are presented. The analysis shows clearly how more sophisticated thinking and explanations develop - information of benefit to teachers of these students.

Thinking teaching: seeing mathematics teachers as active decision makers

Many teachers are now incorporating a broader range of strategies into their teaching, including problem solving, investigations and open-ended questions. Among other things, such teaching requires teachers to talk less but to make more decisions. The acknowledgment of this complexity and the centrality of active decision making have implications for teacher education and teacher development, and for the strategies and resources used.