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.

Introducing Engineering Education in the Middle School

This paper first describes a new three-year, longitudinal project that is implementing engineering education in three middle schools in Australia (grade levels 7-9). This important domain is untapped in Australia. Hence, as a starting point, we conducted a context analysis to help situate engineering education in a school system. We report on this analysis with respect to findings from one of two literature-based surveys that gathered middle-school student responses in mathematics (n=172) and science (n=166) towards understanding their dispositions for engineering education. ANOVA indicated gender differences for 3 out of 23 items in both mathematics and science. In addition, the majority of students agreed or strongly agreed with 17 of the 23 survey items, however, there were some differences between mathematics and science. We conclude the paper with some recommendations for establishing engineering education in schools, including the development of partnerships among engineering and education faculties, school systems, and industry to develop contemporary engineering resources to support school-level mathematics, science, and technology.

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).

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.

Current evidence on integrated treatment for serious mental disorder and substance misuse

Substance misuse in people with serious mental disorders is common and has a wideranging negative impact. The multiplicity of problems suggests that this comorbidity is better conceptualized as a type of complex disorder than by ‘dual diagnosis’. Problems with sequential and parallel treatments have led to the development of integrated approaches, with one practitioner or team addressing both the substance use and mental disorder. These treatments are typically characterized by motivation enhancement, minimizing treatment-related stress, emphasizing harm reduction as well as abstinence, and assertive outreach. A review of published randomized trials demonstrates that superior effects to controls are rarely consistent across treatment foci and over time. While motivational interventions assist engagement, more intervention is usually required for integrated treatment programs to improve long-term outcomes more than control conditions. More intensive case management does not consistently improve impact, but extended cognitive-behavioral therapies have promise. Suggestions for maximizing treatment effects and improving research evidence are provided.

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.

Integrating engineering education within the elementary and middle school mathematics curriculum

Many nations are experiencing a decline in the number of graduating engineers, an overall poor preparedness for engineering studies in tertiary institutions, and a lack of diversity in the field. Given the increasing importance of mathematics, science, engineering, and technology in our world, it is imperative that we foster an interest and drive to participate in engineering from an early age. This discussion paper argues for the integration of engineering education within the elementary and middle school mathematics curricula. In doing so, we offer a definition of engineering education and address its core goals; consider some perceptions of engineering and engineering education held by teachers and students; and offer one approach to promoting engineering education within the elementary and middle school mathematics curriculum, namely through mathematical modeling.

Using picture books in the Secondary English classroom

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

The arobase in the libr@ry : new political economies of children’s literatures and literacies

This essay--part of a special issue on the work of Gunther Kress--uses the idea of affordances and constraints to explore the (im)possibilities of new environments for engaging with literature written for children (see Kress, 2003). In particular, it examines a festival of children's literature from an Australian education context that occurs online. The festival is part of a technologically mediated library space designated by the term libr@ry (Kapitzke & Bruce, 2006). The @ symbol (French word "arobase") inserted into the word library indicates that technological mediation has a history, an established set of social practices, and a political economy, which even chatrooms with "real" authors may alter but not fully supplant.

Using Media Technologies in English

English has long been the subject where print text has reigned supreme. Increasingly in our networked and electronically connected world, however, we can be using digital technologies to create and respond to texts studied in English classrooms. The current approach to English includes the concept of ‘multiliteracies,’ which suggests that print texts alone are necessary but not sufficient’ (E.Q, 2000) and that literacy includes the flexible and sustainable mastery of a repertoire of practices. This also includes the decoding and deployment of media technologies (E.Q, 2000). This has become more possible in Australia as secondary students have increasing access to computers and online platforms at home and at school. With the advent of web 2.0., with its interactive platforms and free media making software, teachers and students can use this software to access information and emerging online literature in English covering a range of text types and new forms for authentic audiences and contexts. This chapter is concerned with responding to literary and mediated texts through the use of technologies. If we remain open to trying out new textual forms and see our digital ‘native students’ (Prensky, 2007) as our best resource, we can move beyond technophobia, become digital travellers’ ourselves and embrace new digital forms in our classrooms.

Wrapped fragments : Drapery, the eighteenth century portrait bust and the male subject

The depiction of drapery (generalised cloth as opposed to clothing) is a well-established convention of Neo-Classical sculpture and is often downplayed by art historians as of purely rhetorical value. It can be argued however that sculpted drapery has served a spectrum of expressive ends, the variety and complexity of which are well illustrated by a study of its use in portrait sculpture. For the Neo-Classical portrait bust, drapery had substantial iconographic and political meaning, signifying the new Enlightenment notions of masculine authority. Within the portrait bust, drapery also served highly strategic aesthetic purposes, alleviating the abruptness of the truncated format and the compromising visual consequences of the “cropped” body. With reference to Joseph Nollekens’ portraits of English statesman Charles James Fox and the author’s own sculptural practice, this paper analyses the Neo-Classical use of drapery to propose that rendered fabric, far from mere stylistic flourish, is a highly charged visual signifier with much scope for exploration in contemporary sculptural practice.

Re-thinking pedagogy for middle school students with little, no or severely interrupted schooling

The arrival of substantial cohorts of English language learners from Africa with little, no or severely interrupted schooling is requiring new pedagogic responses from teachers in Australia and other Western countries of refugee re-settlement. If the students are to have optimal educational and life chances, it is crucial for them to acquire resources for conceptually deep and critical literacy tasks while still learning basic reading and writing skills. This requires teachers to extend their pedagogic repertoires: subject area teachers must teach language and literacy alongside content; high school teachers must teach what has been thought of as primary school curriculum. The aim of this article is to describe some teacher responses to these challenges. Data are drawn from a study involving an intensive language school and three high schools, and also from the author’s experience as a homework tutor for refugees. Stand-alone basic skills programs are described, as are modifications of long-established ESL programs. It is also argued that teachers need to find ways of linking with the conceptual knowledge of students who arrive with content area backgrounds different from others in their class. Everyday life experiences prior to, and after re-settlement in the West, are rich with potential in this regard.

Developing the descriptions of landmark teaching styles : a spectrum inventory

Mosston & Ashworth‟s Spectrum of Teaching styles was first published in 1966 and is potentially the longest surviving model of teaching within the field of physical education. Its longevity and influence is surely testament to its value and influence. Many tools have also been developed through the years based on The Spectrum of Teaching Styles. In 2005 as part of a doctoral study, this tool was developed by the author, Dr Edwards and Dr Ashworth for researchers and teachers to identify which teaching styles were being utilised from The Spectrum when teaching physical education. It could also be utilised for self-assessment of the teaching styles and individual uses, or those who work with Physical Education Teacher Education courses. The development of this tool took approximately 4 months, numerous emails and meetings. This presentation will outline this process, along with the reasons why such a tool was developed and the differences between it and others like it.