Theory and its role in mathematics education

The increased recognition of the theory in mathematics education is evident in numerous handbooks, journal articles, and other publications. For example, Silver and Herbst (2007) examined ―Theory in Mathematics Education Scholarship‖ in the Second Handbook of Research on Mathematics Teaching and Learning (Lester, 2007) while Cobb (2007) addressed ―Putting Philosophy to Work: Coping with Multiple Theoretical Perspectives‖ in the same handbook. And a central component of both the first and second editions of the Handbook of International Research in Mathematics Education (English, 2002; 2008) was ―advances in theory development.‖ Needless to say, the comprehensive second edition of the Handbook of Educational Psychology (Alexander & Winne, 2006) abounds with analyses of theoretical developments across a variety of disciplines and contexts. Numerous definitions of ―theory‖ appear in the literature (e.g., see Silver & Herbst, in Lester, 2007). It is not our intention to provide a ―one-size-fits-all‖ definition of theory per se as applied to our discipline; rather we consider multiple perspectives on theory and its many roles in improving the teaching and learning of mathematics in varied contexts.

Future directions and perspectives for problem solving research and curriculum development

Since the 1960s, numerous studies on problem solving have revealed the complexity of the domain and the difficulty in translating research findings into practice. The literature suggests that the impact of problem solving research on the mathematics curriculum has been limited. Furthermore, our accumulation of knowledge on the teaching of problem solving is lagging. In this first discussion paper we initially present a sketch of 50 years of research on mathematical problem solving. We then consider some factors that have held back problem solving research over the past decades and offer some directions for how we might advance the field. We stress the urgent need to take into account the nature of problem solving in various arenas of today’s world and to accordingly modernize our perspectives on the teaching and learning of problem solving and of mathematical content through problem solving. Substantive theory development is also long overdue—we show how new perspectives on the development of problem solving expertise can contribute to theory development in guiding the design of worthwhile learning activities. In particular, we explore a models and modeling perspective as an alternative to existing views on problem solving.

Methodologies for investigating relationships between concept development and the development of problem solving abilities

This paper is the second in a pair that Lesh, English, and Fennewald will be presenting at ICME TSG 19 on Problem Solving in Mathematics Education. The first paper describes three shortcomings of past research on mathematical problem solving. The first shortcoming can be seen in the fact that knowledge has not accumulated – in fact it has atrophied significantly during the past decade. Unsuccessful theories continue to be recycled and embellished. One reason for this is that researchers generally have failed to develop research tools needed to reliably observe, document, and assess the development of concepts and abilities that they claim to be important. The second shortcoming is that existing theories and research have failed to make it clear how concept development (or the development of basic skills) is related to the development of problem solving abilities – especially when attention is shifted beyond word problems found in school to the kind of problems found outside of school, where the requisite skills and even the questions to be asked might not be known in advance. The third shortcoming has to do with inherent weaknesses in observational studies and teaching experiments – and the assumption that a single grand theory should be able to describe all of the conceptual systems, instructional systems, and assessment systems that strongly molded and shaped by the same theoretical perspectives that are being used to develop them. Therefore, this paper will describe theoretical perspectives and methodological tools that are proving to be effective to combat the preceding kinds or shortcomings. We refer to our theoretical framework as models & modeling perspectives (MMP) on problem solving (Lesh & Doerr, 2003), learning, and teaching. One of the main methodologies of MMP is called multi-tier design studies (MTD).

Cooperative learning in Thailand : professional development to enhance primary education

The overall purpose of this study was to develop a model to inform the design of professional development programs and the implementation of cooperative learning within Thai primary school mathematics classrooms. Action research design, with interviews, surveys and observations, was used for this study. Survey questionnaires and classroom observations investigated the factors that influence the implementation of cooperative learning strategies and academic achievement in Thai primary school mathematics classrooms. The teachers’ interviews and classroom observation also examined the factors that need to be addressed in teacher professional development programs in order to facilitate cooperative learning in Thai mathematics classrooms. The outcome of this study was a model consisting of two sets of criteria to inform the successful implementation of cooperative learning in Thai primary schools. The first set of criteria was for proposers and developers of professional development programs. This set consists of macro- and micro-level criteria. The macro-level criteria focus on the overall structure of professional development programs and how and when the professional development programs should be implemented. The micro-level criteria focused on the specific topics that need to be included in professional development programs. The second set of criteria was for Thai principals and teachers to facilitate the introduction of cooperative learning in their classrooms. The research outcome also indicated that the attainment of these cooperative learning strategies and skills had a positive impact on the students’ learning of mathematics.

Increasing stochastic perturbations enhances acquisition and learning of complex sport movements

Traditionally, the aquisition of skills and sport movement has been characterised by numerous repetitions of presumed model movement pattern to be acquired by learners. This approach has been questioned by research identifying the presence of individualised movement patterns and the low probability of occurrence of two identical movements within and between individuals. In contrast, the differential learning approach claims advantage for incurring variability in the learning process by adding stochastic perturbations during practice. These ideas are exemplified by data from a high jump experiment which compared the effectiveness of classical and a differential training approach with pre-post test design. Results showed clear advantages for the group with additional stochastic perturbation during the aquisition phase in comparison to classically trained athletes. Analogies to similar phenomenological effects in the neurobiological literature are discussed.

Recognising and respecting Torres Strait Islander mathematics knowledges and home languages through mainstream mathematics education [Abstract]

This abstract provides a preliminary discussion of the importance of recognising Torres Strait Islander knowledges and home languages of mathematics education. It stems from a project involving Torres Strait Islander Teachers and Teacher Aides and university based researchers who are working together to enhance the mathematics learning of students from Years 4-9. A key focus of the project is that mathematics is relevant and provides students with opportunities for further education, training and employment. Veronica Arbon (2008) questions the assumptions underpinning Western mainstream education as beneficial for Aboriginal and Torres Strait Islander people which assumes that it enables them to better participate in Australian society. She asks “how de we best achieve outcomes for and with Indigenous people conducive to our cultural, physical and economic sustainability as defined by us from Indigenous knowledge positions?” (p. 118). How does a mainstream education written to English conventions provide students with the knowledge and skills to participate in daily life, if it does not recognise the cultural identity of Indigenous students as it should (Priest, 2005; cf. Schnukal, 2003)? Arbon (2008) states that this view is now brought into question with calls for both ways education where mainstream knowledge and practices is blended with Indigenous cultural knowledges of learning. This project considers as crucial that cultural knowledges and experiences of Indigenous people to be valued and respected and given the currency in the same way that non Indigenous knowledge is (Taylor, 2003) for both ways education to work.

The Brisbane media map : participatory design and authentic learning to link students and industry

The Brisbane Media Map is both an online resource and a tertiary-level authentic learning project. The Brisbane Media Map is an online database which provides a detailed overview of about 600 media industry organisations in Brisbane, Australia. In addition to providing contact details and synopses for each organisation’s profile, the Brisbane Media Map also includes supplementary information on current issues, trends, and individuals in the media and communication industry sectors. This resource is produced and updated annually by final-year undergraduate Media and Communication students. This article introduces the Brisbane Media Map, its functionality and systems design approach, as well as its alignment with key learning infrastructures. It examines authentic learning as the pedagogical framework underpinning the ongoing development work of the resource and highlights some synergies of this framework with participatory design principles. The Brisbane Media Map is a useful example of an authentic learning approach that successfully engages students of non-traditional and non-design areas of study in human-computer interaction, usability, and participatory design activities.

Assessment in education : principles, policy and practice

The English examination system provides a market in which a limited number of providers are accredited to offer curriculum-based examinations in many subject areas and at several levels. The most significant are the General Certificate of Secondary Education (GCSE) and the General Certificate of Education, Advanced (A level). Because these examinations are used for high-stakes purposes, including higher education and employment selection for individuals and programme evaluation for institutions, it is desired that scores from various exams be ‘comparable’ in several respects: across syllabuses and examination boards within a subject area, across years, and even across subject areas. Just how to accomplish this goal has been a topic of continual research and debate for over 50 years, through many changes of examination and institutional structures. But ever year, tens of thousands of scores must be reported, and every year, users expect them to ‘be comparable’ and use them as if they are.

Deepening our understanding about school leadership : learning from educators and non-educators

Increasingly, leadership is argued as a way forward to improve performance and practice in a variety of contexts. School leadership is no different. There is little doubt that in the current globalised world characterized by change and complexity, effective school leadership is a key requirement. The contribution of this chapter is framed around a synthesis of current research, writing and theoretical insights regarding leadership. It draws upon three bodies of writing, Firstly, it begins by distilling several key themes and trends regarding educational leadership from the current research and writing. Secondly, it reports on the findings of a current research project carried out by the authors that explored the leadership stories of ten outstanding leaders from non-educational settings in Australia. Finally, it concludes by referring to some of the paradoxes and tensions inherent in the work of school leaders. It is argued that understanding and endeavouring to reconcile these dilemmas is a pre-requisite for school leaders as they continue to operate in an environment fraught with change and complexity.

Third space in cyberspace : Indigenous youth, new technologies and literacies

This chapter analyses the affordances and constraints of an online literacy program designed for Indigenous Australian youth through a partnership between the Indigenous community, university staff and local schools. The after-school program sought to build on the cultural resources and experiences of the young people through a dialogic process of planning, negotiating, implementing, reflecting, and renegotiating the program with participants and a range of stakeholders. In the majority of cases, students presented themselves as part of pervasive global popular cultures, often hot-linking their webpages to pop icons and local sports stars. Elders regarded their competency as a potential cultural tool and community resource.

Using picture books in the Secondary English classroom

This Chapter explores how teachers can use children's picture books in the Secondary English classroom.

Assessment for learning revisited : an Asia-Pacific perspective

In March of this year, 2009, the Third International Conference on Assessment for Learning was held in Dunedin, New Zealand. Colleagues from Australia (4), Canada (6), Europe (5), New Zealand (7), United Kingdom (5) and the United States of America (4) met to advance the understanding and practices of Assessment for Learning at all levels of education. An important outcome of this meeting was a position paper on Assessment for Learning (AfL) that has been reproduced with permission in this editorial because of its significance to a recurrent theme of the majority of the articles published in this special Asia-Pacific issue of the journal. The text of this statement is set out in italics below.

Learning to drive with cognitive impairment : the experience of young ADHD diagnosed drivers and their parent supervisors

A subset of novice drivers exhibit executive function impairments which may adversely impact on the learn-to-drive period and subsequent driving experience, potentially explaining their overrepresentation in traffic offences and crashes. This paper presents the results of a qualitative analysis of a small series of in-depth semi-structured interviews undertaken individually with affected young drivers (n = 7) and each of their parent supervisors (n = 6). Young drivers were selected on the basis of their ADHD diagnosis, as a sample particularly affected by executive function impairments. Standardised rating scale measures confirmed the currency of the young drivers’ ADHD symptoms and executive function impairment. Results are discussed in terms of common experiences of the young affected drivers and those of their parents as supervising drivers of the learn-to-drive process and subsequent driving behaviour. Key themes included difficulties that were related to core executive function impairments symptomatic of ADHD. Themes also included common emotions that the young drivers associated with driving, with particular types of impact on their driving behaviour. Common strategies that were used by both the young driver and their parent during this learning process and their perceived effectiveness are also discussed. Those that were perceived to be most effective tended to focus on reducing the cognitive load for the young driver when introducing new information and skills.

Toward a fuzzy domain ontology extraction method for adaptive e-learning

With the widespread applications of electronic learning (e-Learning) technologies to education at all levels, increasing number of online educational resources and messages are generated from the corresponding e-Learning environments. Nevertheless, it is quite difficult, if not totally impossible, for instructors to read through and analyze the online messages to predict the progress of their students on the fly. The main contribution of this paper is the illustration of a novel concept map generation mechanism which is underpinned by a fuzzy domain ontology extraction algorithm. The proposed mechanism can automatically construct concept maps based on the messages posted to online discussion forums. By browsing the concept maps, instructors can quickly identify the progress of their students and adjust the pedagogical sequence on the fly. Our initial experimental results reveal that the accuracy and the quality of the automatically generated concept maps are promising. Our research work opens the door to the development and application of intelligent software tools to enhance e-Learning.

Levelling expectations across different years in an undergraduate degree to support work-related learning : evidence for a multi-pronged approach.

One of the ways in which university departments and faculties can enhance the quality of learning and assessment is to develop a ‘well thought out criterion‐referenced assessment system’ (Biggs, 2003, p. 271). In designing undergraduate degrees (courses) this entails making decisions about the levelling of expectations across different years through devising objectives and their corresponding criteria and standards: a process of alignment analogous to what happens in unit (subject) design. These decisions about levelling have important repercussions in terms of supporting students’ work‐related learning, especially in relation to their ability to cope with the increasing cognitive and skill demands made on them as they progress through their studies. They also affect the accountability of teacher judgments of students’ responses to assessment tasks, achievement of unit objectives and, ultimately, whether students are awarded their degrees and are sufficiently prepared for the world of work. Research reveals that this decision‐making process is rarely underpinned by an explicit educational rationale (Morgan et al, 2002). The decision to implement criterion referenced assessment in an undergraduate microbiology degree was the impetus for developing such a rationale because of the implications for alignment, and therefore ‘levelling’ of expectations across different years of the degree. This paper provides supporting evidence for a multi‐pronged approach to levelling, through backward mapping of two revised units (foundation and exit year). This approach adheres to the principles of alignment while combining a work‐related approach (via industry input) with the blended disciplinary and learner‐centred approaches proposed by Morgan et al. (2002). It is suggested that this multi‐pronged approach has the potential for making expectations, especially work‐related ones across different year levels of degrees, more explicit to students and future employers.