Papers by Peter Galbraith
Mathematics Education Research Group of Australasia, 2014
As a contribution to honour the foresight of Ken Clements and John Foyster in founding MERGA so m... more As a contribution to honour the foresight of Ken Clements and John Foyster in founding MERGA so many years ago this paper is not a research paper in the usual sense. Rather it sets out to sample the context of Mathematics Education in Australasia and beyond (then and now) and to highlight some challenges as seen by this author. In this personal view I do not intend to expand in detail upon particular strands of research in which I have been involved, although for purposes of illustration examples will be drawn from time to time from this and other work. MERGA is about both people and scholarly activity, and so this paper will make reference to both -for history, culture, and challenge are essential components of the development of any organisation. In considering how it has evolved within and from the traditions established by its history it is useful to view MERGA as a community of practice , whose defining characteristics are summarised as follows: Communities of practice are groups of people who share a concern or a passion for something they do, and learn how to do it better as they interact regularly. (Wenger, 2006, p. 1). In his terms three characteristics are central to a community's structure and purpose. The Domain: Identity is defined by a shared domain of interest, and therefore commitment to a shared competence that is a distinguishing feature of the group. The Community: To pursue interests in their domain, members engage in joint activities and discussions, help each other, share information, and build relationships that enable them to learn from each other. The Practice: Members of a community of practice are practitioners -they develop a shared repertoire of resources, experiences, stories, insights, and ways of addressing characteristic problems that arise in their domain. The community is constituted by these components in combination, and is cultivated by developing them in parallel. This paper draws from all these elements in developing its theme. Mathematics Education had been through a decade and more of at times turbulent activity. The relatively calm seas of mathematics teaching in Australia were ruffled from overseas in the early 1960s and beyond by influences emanating from the UK, the USA, and continental Europe. The return of Bob McMullen (NSW) from a period working on a US based project in Chicago foreshadowed the introduction of elements of New Mathematics thinking into school curricula through the On Course Mathematics series, and the following years saw a proliferation of activity in all states.
Annual Meeting of the American Educational Research Association, 2001
Undergraduate mathematics courses in Australia, as elsewhere, have for some time been integrating... more Undergraduate mathematics courses in Australia, as elsewhere, have for some time been integrating software into their teaching programs. This international trend is stimulated by the increase in technological resources in general and the impact of symbolic manipulator packages. It has been recommended that mathematics departments re-design courses to make the best use of the increased computer power becoming available. This paper examines some of the issues that are emerging as this process unfolds. It focuses on the computer-based undergraduate courses' attitudes towards mathematics and technology. Studies designing attitude scales for use in programs with computer technology, classifying the range of student-generated questions that emerge when learning of mathematical content interacts with a symbolic manipulator environment, and identifying structural properties associated with the Maple environment that can be identified as linking task demand and student success are discussed. (Contains 24 references.) (ASK) Reproductions supplied by EDRS are the best that can be made from the original document.
Woodhead Publishing Limited eBooks, 2007
In a broad sense, computing is an area of knowledge from which a popular and effective technology... more In a broad sense, computing is an area of knowledge from which a popular and effective technology emerged long before a solid, specific, scientific methodology, let alone formal foundations, had been put forward. This might explain some of the weaknesses in the software industry, on the one hand, as well as an excessively technology-oriented view which dominates computer science training at pre-university and even undergraduate teaching, on the other. Modelling, understood as the ability to choose the right abstractions for a problem domain, is consensually recognised as essential for the development of true engineering skills in this area, as it is in all other engineering disciplines. But, how can the basic problemsolving strategy, one gets used to from school physics: understand the problem, build a mathematical model, reason within the model, calculate a solution, be taken (and taught) as the standard way of dealing with software design problems? This paper addresses this question, illustrating and discussing the interplay between modelling and reasoning.
International Conference on Multimodal Interfaces, 2008
Reflecting on disciplinary mathematics in relation to school mathematics leads inevitably to a co... more Reflecting on disciplinary mathematics in relation to school mathematics leads inevitably to a consideration of the mathematical education provided for those (teachers and instructors) who ultimately will become mediators between these different expressions of mathematics. What kind of approaches to disciplinary mathematics should inform their education, and how might its treatment differ from that designed for intending chemists, engineers, economists etc? In this paper suggestions are offered from two perspectives. 1. In terms of the kinds of engagement with content we might encourage for those whose future will involve diagnosis and correction of misunderstandings, ability to unpack and explain difficult concepts, and encourage through example the development of skeptical enquiring minds. 2.A teacher cannot be inspiring if they are only ever a purveyor of the ideas of others. How can we foster the ability to develop original teaching examples from the everyday environment?
The paper commences by reviewing some of the issues currently being raised with respect to the us... more The paper commences by reviewing some of the issues currently being raised with respect to the use of technology in undergraduate mathematics teaching and learning. Selected material from three research projects is used to address a series of questions. The questions relate to the use of symbolic manipulators in tertiary mathematics, to undergraduate student attitudes toward the use of computers in learning mathematics, and to outcomes of using technology in collaborative student activity in preuniversity classrooms. Results suggest that teaching demands are increased rather than decreased by the use of technology, that attitudes to mathematics and to computers occupy different dimensions, and that students adopt different preferences in the way they utilize available resources. These outcomes are reflected back on the literature, and implications for teaching, learning, and research are discussed. (Contains 49 references.) (Author) Reproductions supplied by EDRS are the best that can be made from the original document.
This paper examines the role of systems thinking in higher education, explaining that university ... more This paper examines the role of systems thinking in higher education, explaining that university examples provide a sense of what systems thinking entails when applied within large, complex organizations. It shares insights provided by the system dynamics approach for approaches to organizational leadership in education. A system dynamics approach aims to identify how streams of decisions and resources interact to produce behaviors recognized as problematic for an organization (for the purpose of intervention and performance improvement). The system structure is based on two types of feedback loops: positive (self-reinforcing) and negative (self-correcting or balancing). Archetypes are systems thinking tools that can help construct dynamic pictures of the operation of various systems, providing assistance in diagnosing, predicting, and addressing problems in organizational behavior by identifying structures responsible for such behaviors. The paper presents a sample of archetypes and examples of their occurrence in educational settings (e.g., attempts to introduce and spread technology use throughout the curriculum in all schools in a district). It describes systems thinking and university structures, highlighting publicly funded institutions that have recently faced more stringent operating environments. It concludes by illustrating how a systems approach helps at a. wider institutional level. (Contains 15 references.) (SM) Reproductions supplied by EDRS are the best that can be made from the original document.
Teacher education programs frequently advocate reflection as a way of learning, without always sp... more Teacher education programs frequently advocate reflection as a way of learning, without always specifying how reflective analyses of oneís own teaching should be carried out. This paper examines the role of a mentor as a reflective agent in eliciting and structuring student teachersí verbal reflections on particular lessons they had taught, and traces in detail the effect of the mentorís assistance on one studentís cognitions and practice. Relationships between the mentorís actions, studentsí perceptions of their developing teaching skills, and constraints within the practice teaching context are also explored so as to identify factors which influenced the extent to which students benefited from the mentorís intervention.
Evaluierte Lernumgebungen zum Modellieren, 2018
While modelling initiatives have been noted for many years, it has also been observed that they h... more While modelling initiatives have been noted for many years, it has also been observed that they have not achieved a sustained presence in curricula to the extent hoped for. This chapter draws from two extended modelling programs (one junior secondary and one senior secondary) in identifying and illustrating elements deemed significant for the successful mounting of sustained programs. Stresses that exist when, for example, modelling as real world problem solving is attempted in an environment that sees modelling as a vehicle for teaching other content were necessarily confronted in developing and implementing the initiatives. Such factors are considered as they existed separately within the respective school contexts, and impacted on the structure of the programs. Finally reference is made to a contemporary program to indicate how priorities underpinning similar values are currently manifested.
In terms of achieving educational goals, technology impacts on the nature of mathematical accompl... more In terms of achieving educational goals, technology impacts on the nature of mathematical accomplishment with respect to both scope and purpose. We review the use of technology, actual and potential, within mathematical modelling viewed as real-world problem solving. We consider its role within the total modelling process, as well as its manner of use within individual problem contexts, illustrating ways in which inappropriate uses of technology create problems within modelling activity, as well as how discerning use can increase the power and accessibility of models to new audiences. We then demonstrate how technology provides access to models unavailable to those equipped only with hand methods of solution. Here non-linearity and simultaneity among model relationships means that model equations need to be first developed, parameterised, and then solved by simulation. Methods provided by System Dynamics are illustrated by considering the problem of providing potable water for a pop...
Mathematics Education Research Group of Australasia, 2014
As a contribution to honour the foresight of Ken Clements and John Foyster in founding MERGA so m... more As a contribution to honour the foresight of Ken Clements and John Foyster in founding MERGA so many years ago this paper is not a research paper in the usual sense. Rather it sets out to sample the context of Mathematics Education in Australasia and beyond (then and now) and to highlight some challenges as seen by this author. In this personal view I do not intend to expand in detail upon particular strands of research in which I have been involved, although for purposes of illustration examples will be drawn from time to time from this and other work. MERGA is about both people and scholarly activity, and so this paper will make reference to both – for history, culture, and challenge are essential components of the development of any organisation.
New ICMI Study Series, 2007
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection wit... more except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
Educational Studies in Mathematics, 1998
Gaining insights into students' attitudes and beliefs is a crucial step in understanding how ... more Gaining insights into students' attitudes and beliefs is a crucial step in understanding how the learning environment for mathematics is affected by the introduction of computers and other technology. We review research into the impact of technology on the teaching and learning of mathematics, although systematic evaluations are sparse. We discuss the relationship between affective variables and performance and having
ZDM, 2006
In this article we present, illustrate, test and refine a framework developed by for identifying ... more In this article we present, illustrate, test and refine a framework developed by for identifying student blockages whilst undertaking modelling tasks during transitions in the modelling process. The framework was developed with 14-15 year old students who were engaging in their first experiences of modelling at the secondary level.
The Journal of Mathematical Behavior, 2003
The introduction of technology resources into mathematics classrooms promises to create opportuni... more The introduction of technology resources into mathematics classrooms promises to create opportunities for enhancing students' learning through active engagement with mathematical ideas; however, little consideration has been given to the pedagogical implications of technology as a mediator of mathematics learning. This paper draws on data from a three year longitudinal study of senior secondary school classrooms to examine pedagogical issues in using technology in mathematics teaching -where "technology" includes not only computers and graphics calculators but also projection devices that allow screen output to be viewed by the whole class. We theorise and illustrate four roles for technology in relation to such teaching and learning interactions -master, servant, partner, and extension of self. Our research shows how technology can facilitate collaborative inquiry, during both small group interactions and whole class discussions where students use the computer or calculator and screen projection to share and test their mathematical understanding.
Reflecting on disciplinary mathematics in relation to school mathematics leads inevitably to a co... more Reflecting on disciplinary mathematics in relation to school mathematics leads inevitably to a consideration of the mathematical education provided for those (teachers and instructors) who ultimately will become mediators between these different expressions of mathematics. What kind of approaches to disciplinary mathematics should inform their education, and how might its treatment differ from that designed for intending chemists, engineers, economists etc? In this paper suggestions are offered from two perspectives. 1. In terms of the kinds of engagement with content we might encourage for those whose future will involve diagnosis and correction of misunderstandings, ability to unpack and explain difficult concepts, and encourage through example the development of skeptical enquiring minds. 2.A teacher cannot be inspiring if they are only ever a purveyor of the ideas of others. How can we foster the ability to develop original teaching examples from the everyday environment?
The number of papers and research reports addressing the theory and/or practice of mathematical m... more The number of papers and research reports addressing the theory and/or practice of mathematical modelling with some form of connection to education is growing astronomically. Small wonder then that educational publications featuring articles emerging from this field, present such a plethora of views that even those experienced in the field can become disoriented, let alone those feeling their way in a new area. This paper joins a conversation that concerns itself with meanings, approaches, priorities, and intentions associated with the use of the term 'mathematical modelling' as it occurs in education. For example it will be argued that there are essentially two generic approaches to modelling within education: modelling that acts primarily as a 'vehicle' for the attainment of other curricular priorities, and modelling as 'content' that seeks first to nurture and enhance the ability of students to solve authentic real world or life-like problems. Within these approaches there are particular purposes and perspectives, but the latter are just that -they are not (as sometimes suggested) additional modelling genres. The paper visits areas of relevance to its theme: such as stated priorities of educational authorities in curriculum statements; types of activity that make up the two modelling genres; a selection of writings that canvass a rich array of issues, challenges, and research foci that are currently engaging interest and activity within the field; and the implications of criticisms of modelling, both appropriate and misplaced.
Undergraduate mathematics courses in Australia, as elsewhere, have for some time been integrating... more Undergraduate mathematics courses in Australia, as elsewhere, have for some time been integrating software into their teaching programs. This international trend is stimulated by the increase in technological resources in general and the impact of symbolic manipulator packages. It has been recommended that mathematics departments re-design courses to make the best use of the increased computer power becoming available. This paper examines some of the issues that are emerging as this process unfolds. It focuses on the computer-based undergraduate courses' attitudes towards mathematics and technology. Studies designing attitude scales for use in programs with computer technology, classifying the range of student-generated questions that emerge when learning of mathematical content interacts with a symbolic manipulator environment, and identifying structural properties associated with the Maple environment that can be identified as linking task demand and student success are discussed. (Contains 24 references.) (ASK) Reproductions supplied by EDRS are the best that can be made from the original document.
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Papers by Peter Galbraith