Is proof dead in the age of computers?

Some knowledge of what it means to construct a proof is an extremely important part of mathematics. All mathematics teachers and students should have some exposure to the ideas of proof and proving. This paper deals with the issue of creating proofs in mathematics problems.

The contexts of mathematics tasks and the context of the classroom: are we including all students?

Mathematics teachers are encouraged to use realistic contexts in order to make mathematics more meaningful and accessible for all students. However, the focus group research reported in this article shows that decisions on the suitability of contexts are complex and multidimensional. Similarly, the way the task contexts are presented, and the way the tasks are incorporated into classroom routines have potential to alienate some groups of students. We suggest that teachers and researchers should be sensitive to difficulties that students might experience as a result of both the task and classroom contexts, and take specific steps to avoid or overcome the difficulties.

Progressive discourse in mathematics classes - the task of the teacher

This paper uses data from two mathematics lessons to explore the nature of progressive discourse and examine critical features of teacher actions that contribute to mathematics classrooms functioning as communities of inquiry. Features found to promote progressive discourse include a focus on the conceptual elements of the curriculum and the use of complex, challenging tasks that problematised the curriculum; the orchestration of student reporting to allow all students to contribute to progress towards the community's solution to the problem; and a focus on seeking, recognizing, and drawing attention to mathematical reasoning and justification, and using this as a basis for learning.

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 ".

Fun or profit : primary students' perceptions of mathematics and science

As the number of students pursuing mathematics and science in higher education decline, it becomes imperative· that we look for the causes of the decline. As part of the Australian Improving Middle Years Mathematics and Science (IMYMS) project, students were asked to rate their perceptions of classroom practice in mathematics and science and their attitudes to these subjects. Results of this survey reveal little difference in perceptions of classroom practice, but significant differences in students' attitudes between mathematics and science. Differences were particularly evident for items relating to the usefulness of mathematics and science (mathematics was more useful) and enjoyment of the subjects (science is more fun). If teachers are aware of such perspectives, it may be possible to change students' attitudes.

Effective student engagement depends on students enjoying their studies in mathematics and science, being confident in their ability and recognising the relevance of these subjects to everyday life, now and in the future.
(Education Training Committee, 2006, p. xvii)

Science and technology are the widely acknowledged foundation of Australia's future development. Underpinning these are the key learning areas of mathematics and science. However, Australia is experiencing a decline in numbers of mathematics and science students in higher education. Moreover, studies over the last two decades have shown a general decline in Australian students' interest and enjoyment of science across the compulsory secondary school years, with a particularly sharp decline across the primary to secondary school transition (e.g. Adams, Doig, & Rosier 1991; Goodrum, Hackling, & Rennie, 200 I) and a decline in the numbers of students studying' advanced mathematical courses in upper secondary school (Thomas, 2000).

Improving teaching and learning in the middle years of schooling (Years 5 to 9) is receiving particular attention because of the coincidence of the disengagement of students with the significance of these years for the preparation of students for their future role in society. Thus the Improving Middle Years Mathematics and Science: The role of subject cultures in school and teacher change (IMYMS) project, which is the source of data for this paper, is investigating the role of mathematics and science' knowledge and subject cultures in mediating change processes in the middle years of schooling.

Mathematics and science are sometimes seen as "love-hate" subjects, rating highest for subjects disliked, but also rating relatively highly among preferred subjects (Hendley & Stables, 1996). Students, even primary aged students, can often shed light on what constitutes good practice (see, for example, 'van den Heuvel-Panhuizen, 2005). Students' attitudes towards mathematics and science and their perceptions of what they regard as positive aspects of classroom practice have been shown to decline from the primary years to junior secondary (Race, 2000). The decline in interest in science in the early years of secondary school is of particular concern, since it is in these years that attitudes to the pursuit of science subjects and careers are formed (Speering & Rennie, 1996). Students' negative attitude towards the relevance of science ,content for their lives was a strong theme in the report by Goodrum, Hackling, & Rennie (2001) on the status and quality of teaching and learning of science.

As part of the IMYMS project, the IMYMS Student Survey was administered to all students in 2004 and 2005. The survey included a 36 item section on students' perceptions of classroom practice and attitudes towards mathematics and science, and a 24 item section on students' learning preferences. Students completed separate, parallel surveys for mathematics and science.

This paper focuses on students' perceptions and attitudes. It explores the differences in 700 Year 5 and 6 students' perceptions of their learning environment and their attitudes to mathematics and science during 2005, the second (and final) year of schools , involvement in the IMYMS project.

Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms

The basic unit of school based mathematics teaching is the lesson. This article is a contribution to understanding teacher actions that facilitate successful lessons, defined as those that engage all students, especially those who may sometimes feel alienated from mathematics and schooling, in productive and successful mathematical thinking and learning. An underlying assumption is that lessons can seek to build a sense in the students that their experience has elements in common with the rest of the class and that this can be done through attention to particular aspects of the mathematical and socio-mathematical goals. We examine three teacher actions that address the mathematical goals: using open-ended tasks, preparing prompts to support students experiencing difficulty, and posing extension tasks to students who finish the set tasks quickly; as well as actions that address the socio-mathematical goals by making classroom processes explicit. To illustrate and elaborate these actions, we describe a particular lesson taught to a heterogeneous upper primary (age 11–12) class.

Describing elements of mathematics lessons that accommodate diversity in student background

We are researching actions that teachers can take to improve mathematics learning for all students. Structural elements of the lessons being trialled include making aspects of pedagogy explicit to seek to overcome differences in familiarity with schooling processes, and sequencing tasks with the potential to engage students. This article reports research on teachers building learning communities by preparing variations to set tasks in order to address differences in students’ backgrounds.

Planning and teaching mathematics lessons as a dynamic, interactive process

We are researching actions that teachers can take to improve mathematics learning for all students, with particular attention to specific groups of students who might experience difficulty. After identifying possible barriers to learning, we offered teachers mathematics lessons structured in a particular way. Teachers’ use of the model outlined in this paper seemed productive and their resulting planning and teaching proved to be dynamic and interactive. This paper uses excerpts from a conversation between two teachers to illustrate specific aspects of the model.

Sociocultural research on mathematics education : an international perspective

This volume--the first to bring together research on sociocultural aspects of mathematics education--presents contemporary and international perspectives on social justice and equity issues that impact mathematics education. In particular, it highlights the importance of three interacting and powerful factors--gender, social, and cultural dimensions. Sociocultural Research on Mathematics Education: An International Perspective is distinguished in several ways:

* It is research based. Chapters report on significant research projects; present a comprehensive and critical summary of the research findings; and offer a critical discussion of research methods and theoretical perspectives undertaken in the area.
* It is future oriented, presenting recommendations for practice and policy and identifying areas for further research.
* It deals with all aspects of formal and informal mathematics education and applications and all levels of formal schooling.

As the context of mathematics education rapidly changes-- with an increased demand for mathematically literate citizenship; an increased awareness of issues of equity, inclusivity, and accountability; and increased efforts for globalization of curriculum development and research-- questions are being raised more than ever before about the problems of teaching and learning mathematics from a non-cognitive science perspective. This book contributes significantly to addressing such issues and answering such questions. It is especially relevant for researchers, graduate students, and policymakers in the field of mathematics education.

Teacher professional growth : a study of primary teachers involved in mathematics professional development

The overall purpose of this study was to examine whether professional development programs can act as appropriate vehicles for the professional growth of teachers of primary mathematics. A longitudinal study was conducted of primary teachers involved in a Victorian mathematics professional development program — Exploring Mathematics In Classrooms (EMIC). The professional growth of six teacher participants in one EMIC course was examined over a period of 18 months. The teachers selected were from four different schools located in the southern metropolitan region of Melbourne. The central interest of this study was in teacher professional growth and accordingly the perspective sought was predominantly that of the teacher. A case study research approach was adopted and data were gathered through observations, interviews, questionnaire, and the collection of teacher work documents. A theoretical model of teacher professional growth was used to represent the teachers' growth. The study generated data on the nature of teacher professional growth and the features of professional development programs likely to influence teacher professional growth. All of the teachers reported and demonstrated growth with respect to their mathematics teaching, in areas associated with their: Classroom Practice, Knowledge and Beliefs, and Professional Attributes. The teachers' growth was highly individualistic, with no two teachers demonstrating exactly the same professional growth outcomes, or the same growth processes. The data provided evidence to confirm that teacher growth is a complex and gradual learning process. For each of the teachers several different routes to change and growth were evident, drawing attention to the non-linear nature of growth. The teachers' responses to the professional development program were influenced by various contextual and personal factors. The data provided evidence of a strong link between the content and outcomes of professional development programs — the outcomes reported and demonstrated by the teachers reflected the content of the EMIC program. Key factors associated with mathematics professional development programs perceived as influencing growth were: program content; program structure; and program presentation. A significant finding was the strong influence on teacher growth of the presenters of professional development programs—some data suggested that the 'quality' of the program presenter is fundamental to the success of any professional development program. The study provided insight into the processes involved in teacher professional growth and factors associated with the way in which professional development programs influence growth. The theoretical model of teacher professional growth used in this study has been elaborated and recommendations which might inform the design and implementation of future professional development programs have been made.

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.

Keeping all students on the learning path

A powerful notion to guide thinking about whole-class mathematics teaching is Vygotsky’s zone of proximal development (ZPD). Our research with primary and secondary teachers over the last six years has identified roles of teachers in relation to the ZPD, and ways of overcoming some typical barriers to students’ movement through their zones. Methods have included focus groups of experts, video analysis of classroom interactions, classroom observation, and analysis of lesson plans and teachers’ reflections teaching processes their outcomes. The research has involved the gradual development, trailing, evaluation, and adjustment of a six-component model for planning and teaching mathematics. The focus of this paper is on the use of one of its components, “differentiated learning trajectories”.

Geocaching : a worldwide treasure hunt enhancing the mathematics classroom

Geocaching is a global treasure hunt that invites people of all ages to discover actively the beauty of their environment through the assistance of a Global Positioning Systems (GPS), mathematical know-how, and a bit of foraging. If you are seeking a new way to engage your students in a motivating and exciting real-life task, then geocaching might be the answer. The purpose of this article is to describe the experience of our shcool-based geocaching project undertaken with children in Prep (5-6) and the senior primary Grades (ages 10-12). We will share the potential for mathematical learning and engagement. It is argued that geocaching provides the opportunity for rich engagement with key mathematical concepts that goes beyond what can be acheived during a typical lesson.