Hops, steps and jumps : mathematical progress in the early years

While curriculum frameworks ale major influences on learning, teachers know that children progress at different rates. Sometimes this is evident within a particular topic. and at other times more obvious across different topics. In this paper. we present the hops, steps, and jumps of numeracy learning of some 3000 Australian children. All were assessed using I Can Do Maths, and their achievements mapped to provide a detailed picture of how children hop, step and jump on their numeracy journey. This mapping provides teachers with infonnation about key hurdles to numeracy learning for Australian children.

Mathematical and pedagogical understanding as situated cognition

One mathematics lesson was planned by two Grade 2 teachers together. TheirSeparate teaching of it was videotaped. and each teacher was interviewed before and after her lesson. The "same" lesson resulted in different sets of worthwhile learning outcomes. In this research report. the notion of situated cognition is used as a tool for analysis of how this divergence seemed to happen. It is agued that the teachers' development and uses of mathematical concepts were mediated by the social situations.

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.

Increasing access to mathematical thinking

This article is a contribution to understanding of teacher actions that can contribute to a successful mathematics learning experience, defined as one that engages all students, especially those who may sometimes feel alienated from mathematics and schooling, in productive and successful mathematical thinking and learning. We offer an example of a task that can form the basis of such a learning experience. The key elements are that the task is open-ended, that the teacher offers specific pedagogical prompts to support student leaning, that the teacher builds a sense of community by ensuring that there are some common experiences, and the teacher prepare prompts that can be used to support students who are experiencing difficulty, or to extend those students who complete the
basic task readily.

Children's mathematical thinking in different classroom cultures

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and 1 conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and  expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.

Book review : The active modeler : mathematical modeling with Microsoft® Excel by E. Neuwirth and D. Arganbright

A recently-published work in by two well-known authors in the field of spreadsheet modeling is reviewed.