Effective mathematics learning environments : primary teachers' beliefs and practices

A first priority for changing teaching practice is to make problematic for teachers aspects of their current practice. As part of a project on improving mathematics and science teaching in the middle years of schooling, teachers were asked to rate their practice against components of effective teaching and learning and to rate each of these according to their perceived importance. Findings suggest that primary teachers endorsed the components as representing effective practice, scoring most components higher than their actual practice. Gaps were particularly evident for items relating to challenging students conceptually and higher-order thinking, with these becoming the basis for some of the action planning for change.

Conceptual complexity and apparent contradictions in mathematics language

The article explores the conceptual complexity and apparent contradictions in the language of mathematics and suggests ways to handle these problems. Mathematics, as described the author, is a formal, artificially constructed language. One problem arises from the changes of gradient and rates of change of gradient, continuous functions. Conceptual conflicts also arise when least upperbounds and greatest lower-bounds in discussing sequences and series, convergence and limits are considered.

Non-smooth optimization methods for computation of the conditional value-at-risk and portfolio optimization

We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21-41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters.

Mathematics as a second language : looking for a bridge between mathematical language and the language of thought

Our thoughts are in one language, and mathematical results are expressed in a language foreign to the way we think. Mathematics is a unique foreign language with all the components of a language; it has its own grammar, vocabulary, conventions, synonyms, sentence structure, and paragraph structure. Students need to learn these components to partake in a thorough discussion of how to read, write, speak and think mathematics. Beginning with the students natural language and expanding that language to include symbolism and logic is the key. Providing lessons in concrete, pictorial, written and verbal terms allows the instructor to create a translation bridge between the grammar of the mother language and the grammar of mathematics. This papers presents methods to create the translation bridge for students so that they become articulate members of the mathematics community. The students "mother" language, expanded to include the symbols of mathematics and logic, is the the key to both the learning of mathematics and its effective application to problem situations. The use of appropriate language is the key to making mathematics understandable.

Computers for secondary mathematics: who uses them and how?

In the first stage of a three year ARC-funded study in which the effects of using computers for the teaching and learning of mathematics are being explored, a questionnaire has been developed and is being administered to a large number of students in grades 7-10 in a representative sample of co-educational post-primary schools in Victoria. Using open and closed response formats, the information sought includes: background and biographical details; data on computer ownership and use =96 frequency and mode/s =96 for mathematics learning at home and in mathematics classes; and beliefs about mathematics and the use of computers for learning mathematics. In this paper, the focus will be on some of the preliminary findings related to ownership and use of computers for learning mathematics at home and at school. In previous research, factors such as gender, socio-economic background, and ethnicity have been identified as contributors to inequitable mathematics learning outcomes. The questionnaire data will be analysed to examine if these and other factors are related to any inequities identified in student's access to computers and in how they use computers in their learning of mathematics. The results will be presented and pertinent implications discussed.

The presence of mathematics anxiety in future primary school teachers and factors affecting abatement

This study reflects on the implementation of various teaching initiatives for reducing anxiety toward mathematics in students studying to become primary school teachers. We highlight similarities between these practices and those promoted by the 'Whole Teacher' approach - in particular, the aim to develop attitudes along with knowledge and skills. Here, the negative past associations with mathematics and anxiety toward mathematics that students bring with them have been a key consideration when designing the subject content and delivery. Given the important role these students will have in shaping mathematics education in the future, we suggest frameworks such as that of the 'Whole Teacher' could be extended to the university setting. We investigate four years of student feedback pertaining to a first year undergraduate mathematics unit, contending that the teaching initiatives introduced over time have helped students develop a positive attitude toward mathematics. We note, however, that the student-teacher relationship was still the most prominent factor directly identified by students who previously had a fear or negative attitude toward mathematics.

Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems

Abstract
This paper presents new discrete inequalities for single summation and double summation. These inequalities are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. An application of these discrete inequalities to analyse stability of linear discrete systems with an interval time-varying delay is studied and a less conservative stability condition is obtained. Three numerical examples are given to show the effectiveness of the obtained stability condition.

Equity, social justice and ethics in mathematics education

The performativity policy mindset driving national and international testing highlights issues of equity in access and success according to socio-economic status, geographic location, ethnicity, gender and combinations of these factors. Researchers seek explanations for these inequities in terms encompassing engagement, participation and achievement to identify socially just and ethical practices at system, school and classroom level. The emergence of a theoretical perspective involving redistribution, recognition and participation (Fraser, 2013) is evident in a range of studies concerning leadership, professional learning, pre-service teacher education, and pedagogies that focus on equity and social justice in mathematics education. The challenge of ethical and socially just practices at all levels and social groups is in providing access to deep learning in mathematics and success in “knowledge making” (Jorgensen, 2014).

Experiential learning in social science theory: an investigation of the relationship between student enjoyment and learning

This paper provides an analysis of student experiences of an approach to teaching theory that integrates the teaching of theory and data analysis. The argument that supports this approach is that theory is most effectively taught by using empirical data in order to generate and test propositions and hypotheses, thereby emphasising the dialectic relationship between theory and data through experiential learning. Bachelor of Commerce students in two second-year substantive organisational theory subjects were introduced to this method of learning at a large, multi-campus Australian university. In this paper, we present a model that posits a relationship between students' perceptions of their learning, the enjoyment of the experience and expected future outcomes. The results of our evaluation reveal that a majority of students:

•enjoyed this way of learning;
•believed that the exercise assisted their learning of substantive theory, computing applications and the nature of survey data; and
•felt that what they have learned could be applied elsewhere.

We argue that this approach presents the potential to improve the way theory is taught by integrating theory, theory testing and theory development; moving away from teaching theory and analysis in discrete subjects; and, introducing iterative experiences in substantive subjects.

The discovery of generalized causal models with mixed variables using MML criterion

One major difficulty frustrating the application of linear causal models is that they are not easily adapted to cope with discrete data. This is unfortunate since most real problems involve both continuous and discrete variables. In this paper, we consider a class of graphical models which allow both continuous and discrete variables, and propose the parameter estimation method and a structure discovery algorithm based on Minimum Message Length and parameter estimation. Experimental results are given to demonstrate the potential for the application of this method.

Reason for hope : primary students' mathematics learning environment

This paper reports the results of a student survey of perceptions of classroom learning environment conducted as part of a major investigation into effective pedagogical practices in mathematics and science. All primary school project participants were surveyed, The 36 Likert-type items were subjected to a Partial Credit Model analysis, and response categories subjected to statistical requirements. The results are presented graphically, their meanings examined, and the implications of the findings for both researchers and practitioners are discussed.

Generic versus subject specific pedagogy - should mathematics be in the centre?

Research into pedagogy and school change is a high priority in Australia and many other countries. This paper, which includes some preliminary findings from the Improving Middle Years Mathematics and Science: The role of subject cultures in school and teacher change (IMYMS) project, argues that, while there are key features that are common to quality learning environments across all subject areas, generic formulations of pedagogy fail to take account of the extent to which the disciplines being taught shape pedagogy or the contribution of Pedagogical Content Knowledge (PCK) to effective teaching - i.e. that there really is a need to put "mathematics in the centre ".

Future considerations for the training and development of Australian quality managers

Adequate, appropriate training of quality managers is essential to the development and implementation of effective quality management systems. This study reports on a survey of 235 Australian quality managers to determine their perceptions of their training and development needs, the extent to which these were being met, and their views on the future of their discipline. The study found that there was a general lack of systematization, most programs were short-term and delivered by a fragmented set of providers. Many managers had received no training in the past 5 years. While the respondents were generally satisfied with the training they had received, there was evidence that their insight into their own needs was imperfect. The respondents were divided between those who thought quality management would remain a discrete field and those who saw it being absorbed by other management systems. The challenges of developing an appropriate training and development regime in this environment are discussed.

A matrix-based model for web page community construction and more

The rapid development of network technologies has made the web a huge information source with its own characteristics. In most cases, traditional database-based technologies are no longer suitable for web information processing and management. For effectively processing and managing web information, it is necessary to reveal intrinsic relationships/structures among concerned web information objects such as web pages. In this work, a set of web pages that have their intrinsic relationships is called a web page community. This paper proposes a matrix-based model to describe relationships among concerned web pages. Based on this model, intrinsic relationships among pages could be revealed, and in turn a web page community could be constructed. The issues that are related to the application of the model are deeply investigated and studied. The concepts of community and intrinsic relationships, as well as the proposed matrix-based model, are then extended to other application areas such as biological data processing. Some application cases of the model in a broad range of areas are presented, demonstrating the potentials of this matrix-based model.

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.