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.

Probabilistic reasoning of inconsistent belief in protocol analysis

Security protocols have been recently found with subtle flaws due to incomplete or ambiguous specification. Although formal methods have remarkably assisted in protocol analysis, they ignores the effect of hostile/uncertain environment, which might lead to inconsistent belief that can be held by principals in delivered messages. This discrepant belief may prevent us from representing the insecurity and uncertainty in a real trading situation. Unfortunately, the current approaches lack the ability to handle the inconsistent belief. This article presents a probabilistic method, which intuitively measures the belief from different principals that can be put on the goal of the protocol. The experiments demonstrate our method is useful to enhance the protocol analysis.

Information sharing between heterogeneous uncertain reasoning models in a multi-agent environment : a case study

The issue of information sharing and exchanging is one of the most important issues in the areas of artificial intelligence and knowledge-based systems (KBSs), or even in the broader areas of computer and information technology. This paper deals with a special case of this issue by carrying out a case study of information sharing between two well-known heterogeneous uncertain reasoning models: the certainty factor model and the subjective Bayesian method. More precisely, this paper discovers a family of exactly isomorphic transformations between these two uncertain reasoning models. More interestingly, among isomorphic transformation functions in this family, different ones can handle different degrees to which a domain expert is positive or negative when performing such a transformation task. The direct motivation of the investigation lies in a realistic consideration. In the past, expert systems exploited mainly these two models to deal with uncertainties. In other words, a lot of stand-alone expert systems which use the two uncertain reasoning models are available. If there is a reasonable transformation mechanism between these two uncertain reasoning models, we can use the Internet to couple these pre-existing expert systems together so that the integrated systems are able to exchange and share useful information with each other, thereby improving their performance through cooperation. Also, the issue of transformation between heterogeneous uncertain reasoning models is significant in the research area of multi-agent systems because different agents in a multi-agent system could employ different expert systems with heterogeneous uncertain reasonings for their action selections and the information sharing and exchanging is unavoidable between different agents. In addition, we make clear the relationship between the certainty factor model and probability theory.

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.

The effects of accounting students` ethical reasoning and personal factors on their ethical sensitivity

Purpose – Rest posited that to behave morally, an individual must have performed at least four basic psychological processes: moral sensitivity; moral judgment; moral motivation; and moral character. Though much ethics research in accounting has been focused on component two, ethical judgment, less research has been undertaken on the other three components. The purpose of this study is to focus on component one, ethical sensitivity, of Rest's four-component model.
Design/methodology/approach – A sample of 156 accounting undergraduates was employed to investigate the ethical sensitivity of accounting students and the effects of their ethical reasoning and personal factors on their ethical sensitivity.
Findings – Results of this study show that accounting students vary in their ability to detect the presence of ethical issues in a professional scenario. There is no significant relationship between accounting students' ethical sensitivity and their ethical reasoning (P-score). Accounting students characterized as “internals” are more likely to show an ability to recognize ethical issues than those characterized as “externals.” The results also indicate that an accounting ethics intervention may have positive effect on accounting students' ethical sensitivity development. Hence, an individual who possesses the ability to determine what is ethically right or wrong (high ethical reasoning) may fail to behave ethically due to a deficiency in identifying ethical issues (low ethical sensitivity) in a situation.
Originality/value – Whilst much research has concentrated on ethical reasoning and ethics education to enhance the ethical conduct of accountants, it is important that the profession and researchers also direct their attention and efforts to cultivating the ethical sensitivity of accountants. The findings of this study provide additional evidence to support Rest's theory of a more comprehensive cognitive model of ethical decision-making and suggest a more balanced research effort in evaluating the ethical development of individuals.

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.