Kaye Stacey

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• BSc (Hons) MSc, DPhil, DipEd FAustMS
• Professor Emeritus at University of Melbourne

This paper provides an overview of the ideas in the Routledge book “Learning and Teaching for Mathematical Literacy: Making Mathematics Useful for Everyone” (Burkhardt, Pead and Stacey 2024). Mathematical literacy is interpreted broadly as the ability to use mathematics in meaningful ways across many situations for personal life and life as an info...

Abstract The digitization of society impacts not only professional and private life, but also education. For the case of mathematics education, the question is how curricula may benefit from digital technology, and how digitization affects curricu- lum policies. To address these questions, we distinguish the intended, the implemented, and the attain...

The paper presents an analysis of elementary grade mathematics teaching in India focusing on mathematical reasoning. Mathematical reasoning is an important tool to make sense of mathematical ideas, which in turn makes mathematics more accessible to learners and promotes social equality. The paper analyzes 112 lessons from Grades 1 to 8 by 21 teache...

The paper begins with a discussion of the characteristics of digitization that influence school mathematics, its goals and practices: the digitization of everything, the invisibility of computation, the era of big data and information on everything, and changes in the social role of mathematics due to increased global competitiveness. This leads di...

The data mining method of Statistical Implicative Analysis is used to reveal details of the inner structure of algebraic competence as seen in data collected from beginning algebra students as part of the development of the SMART online diagnostic tests. The implicative analysis of the data shows the logical relation between algebraic sub-competenc...

This study investigated the mathematical connections found in the solutions provided by 22 preservice secondary mathematics teachers to a set of algebra problems. Interest in, and research on, mathematical connections has gained prominence of the past decade. Here, we use the Extended Theory of Mathematical Connections or ETMC to explore the types...

Assessing students' (mis)conceptions is a challenging task for teachers as well as for researchers. While individual assessment , for example through interviews, can provide deep insights into students' thinking, this is very time-consuming and therefore not feasible for whole classes or even larger settings. For those settings, automatically evalu...

Chapter 3: Seeing algebra through the eyes of a learner Chapter 3: Seeing algebra through the eyes of a learner 45 3.1 Introduction – putting on teachers’ bifocal spectacles 45 3.2 What do algebraic letters represent? 46 3.2.1 Fruit salad algebra – misleading teaching with long term consequences 47 3.3 The process-object duality 49 3.4 The meaning...

This volume—Research into Mathematics Education in Australasia 11—reviewing research from Australasia published since the last review of 2018, provides perspectives on many hundreds of research papers presenting findings from a diversity of projects and varied settings. Some reports are from large, funded projects while others tell the story of tar...

NOTE: THE PUBLISHED PAPER (OCT 2024) IS AVAILABLE THROUGH RESEARCHGATE. THIS IS PREPRINT. The paper presents an analysis of elementary grade mathematics teaching in India focusing on mathematical reasoning. The paper analyzes 112 lessons to Grades 1 to 8 by 21 teachers in 5 schools in a big city of India. Because classrooms were strongly teacher ce...

This article is a personal reflection on Pólya's work on problem solving, supported by a re-reading of some of his books and viewing his film Let Us Teach Guessing. Pólya's work has had lasting impact on the goals of school mathematics, especially in establishing solving problems (including non-routine problems) as a major goal and in establishing...

One reason why mathematics is a compulsory educational subject globally is the fact that mathematical knowledge is applied in multiple and different areas outside mathematics. The PISA mathematics frameworks see mathematics as an essential tool for students beyond school as they face issues and challenges in the diverse contexts where mathematical...

This article reports an in-depth analysis of 3010 student responses to a short online diagnostic constructed-response assessment on solving linear equations. The aim of the paper is to share some insights into students’ equation solving, and also to demonstrate the process that we have developed to analyse, summarize, visualize and grasp meaning fr...

The Victorian Curriculum and Assessment Authority (VCAA) introduced the use of Computer Algebra System (CAS) technology (calculator and software) into the senior secondary mathematics curriculum and examination assessment in three phases, starting with a research-based pilot from 2000, followed by parallel implementation of CAS and non-CAS subjects...

Individual, diagnosis-guided support for learners is one of the most important factors in understanding mathematics and learning efficiently. Especially in the field of algebra, many students often still lack basic competencies to handle variables, algebraic expressions and equations in a proper way. Digitally supported diagnostic systems offer the...

This chapter identifies aspects of the study of mathematical competencies of Chinese students that are likely to be of special interest to international readers. Perhaps the most striking feature is the scale of the work. Within an overarching framework for conceptualizing mathematical competencies, there are detailed reviews of the treatment of ea...

This chapter provides an overview of some themes which have emerged over two decades of Bärbel Barzel’s work related to the teaching and learning of school mathematics with technology. The themes which are discussed include technology supporting mathematical communication, technology supporting cognitive activities and technology supporting an open...

The thesis of RL Hayes describes an attempt to improve the teaching and learning of negative number concepts and operations. A teaching method based on the neutralisation model and using counters in the form of 'integer tiles' as the manipulative materials was developed and evaluated. The thesis begins with a review of the history and development o...

Proceedings of the ICMI Study Group conference on the teaching and learning of algebra. Supporting the full review book "The Future of the Teaching and Learning of Algebra" published by Springer 2004. Also available from The University of Melbourne https://minerva-access.unimelb.edu.au/handle/11343/35000

Proceedings of the ICMI Study Group conference on the teaching and learning of algebra. Supporting the full review book "The Future of the Teaching and Learning of Algebra" published by Springer 2004. Also available from The University of Melbourne https://minerva-access.unimelb.edu.au/handle/11343/35000

The research project discussed in this paper reopens the issue of teaching negative number concepts and operations. The experimental work in schools has only recently commenced. In conjunction with a detailed evaluation of current teaching strategies, an experimental card/counter teaching strategy will be carefully 'classroom tested'. Short and lon...

These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student’s learning. Together the chapt...

The National Statement on Mathematics for Australian Schools advises that algebra learning begins with the study of sequences and patterns leading to their description as algebraic rules relating dependent and independent variables. This study assessed the success of 512 students in seven schools at year levels 7 to 10 in recognizing and describing...

In this chapter, we describe the design of an online system for the formative assessment of students’ understanding of mathematics and discuss how it develops diagnostic competence and influences teaching. The smart-test system covers many mathematics topics studied by students between about 10 and 16 years of age. It is programmed to provide teach...

Comparing is one of the most basic intellectual activities. We consciously make comparisons to understand where we stand, both in relation to others as well as to our own past experiences. International comparative studies have completely transformed the way we see mathematics education. The focus of this ICME-13 Topical Survey is to discuss the wa...

This book addresses the problématique of the teaching and learning of school algebra. The problématique addressed by this book can be posed in general terms as follows. On the one hand, algebra is considered a central subject to be studied in junior high and high schools in almost all educational systems around the world. The reasons for this centr...

This book addresses the problématique of the teaching and learning of school algebra.‘ Problématique’ is a word borrowed from French and it describes not just problems in isolation, but comprehensive challenges posed by a certain large-scale theme. In other words, it consists of articulating questions and dilemmas arising in a certain area. The pro...

This chapter identifies common themes among six research papers from around the globe concerning how teachers can support their students’ mathematical problem solving. Three of the studies indicate how teachers’ understanding and awareness of students’ problem solving can be deepened.One approach is to offer deeper practical experience of ‘working...

It argues that the main purpose of educational research is to improve student learning, and that international comparative studies are no exception.

This chapter describes the purpose of the Framework for the PISA surveys of mathematical literacy and its evolution from 2000 to 2012. It also describes some of the analysis and scholarship on which the key constructs of the Framework are based, and links to kindred concepts in the wider mathematics education literature. The chapter does not intend...

The OECD PISA international survey of mathematical literacy for 2012 is based on a new Framework and has several new constructs. New features include an improved definition of mathematical literacy; the separate reporting of mathematical processes involved in using mathematics to solve real world problems; a computer-based component to assess mathe...

Werner Blum served as a member of the PISA Mathematics Expert Group from late 2000 until after the completion of the PISA 2012 survey. He was also a member of the Questionnaire Expert Group for the duration of the PISA 2012 survey preparation period. He provided an important theoretical perspective to the thinking and work of the MEG, much of which...

This chapter contains short descriptions from contributors in ten countries (Chile, Denmark, France, Indonesia, Iran, Israel, Korea, Singapore, Spain and USA) about some ways in which the PISA Framework and results have influenced thinking and action about mathematics education. In many countries, the PISA results have been a call to action, and ha...

This chapter describes the way in which PISA theorises and operationalises the links between the real world and the mathematical world that are essential to mathematical literacy. Mathematical modelling is described and illustrated and the chapter shows why it is used as the cornerstone to mathematical literacy. It discusses how this concept has de...

This paper explores how four good teachers, who do not have a special interest in technology, meet the challenge of introducing the rapidly developing mathematics analysis software (e.g. spreadsheets, function graphers, symbolic algebra manipulation and dynamic geometry) into their classrooms. These teachers’ practice is viewed through the lens of...

Changes to students’ understanding of mathematical notation may be brought about by using technology within mathematics. Taking equality as a case study, the paper provides brief epistemological, historical, didactical, and computational reviews of its symbolic representation in pen-and-paper and technology-assisted mathematics, most especially in...

In this chapter the authors describe a case study of an electronic interview with accompanying multimedia stimulus resources that was used to obtain in-depth data from a specific group of professionals. Set in the context of mathematics education, the case study sought data related to the interviewees' beliefs, professional decision-making, reflect...

Some of mathematics teaching is routine, like an exercise from a textbook for which you have received instruction and already know what to do. On other occasions, however, teaching mathematics is challenging, involving problems of teaching for which the solutions may not be readily apparent. These situations require the application of mathematical...

Most successful Bayesian network (BN) applications to datehave been built through knowledge elicitation from experts.This is difficult and time consuming, which has lead to recentinterest in automated methods for learning BNs from data. We present a case study in the construction of a BN in anintelligent tutoring application, specifically decimal m...

This chapter reviews the way that the decreasing cost and increasing availability of powerful technology changes how mathematics is assessed, but at the same time raises profound issues about the mathematics that students should be learning. A number of approaches to the design of new item types, authentic assessment and automated scoring of constr...

In this chapter the authors describe a case study of an electronic interview with accompanying multimedia stimulus resources that was used to obtain in-depth data from a specific group of professionals. Set in the context of mathematics education, the case study sought data related to the interviewees’ beliefs, professional decision-making, reflect...

This study presents a content analysis of the new Iranian Grade 9 mathematics textbook and two Australian Year 9 mathematics textbooks, examining the extent to which the problems show characteristics associated in the literature with promoting mathematical literacy. The new Iranian book was produced to meet a range of needs including several well a...

We illustrate three basic types of reasoning used in mathematics by showing how they operate in practical and mathematical situations. The importance and function of the different types of reasoning in each situation is outlined. As a consequence we note that while introducing new techniques by example is good from a pedagogical viewpoint, several...

In a study of quality mathematics teaching, an electronic interview with multimedia stimulus resources was used with relevant professionals to obtain in-depth data relating to their own beliefs, and their reflective observations and evaluations of mathematics teaching practice. The electronic format enabled presentation of the same questions and mu...

PISA, the OECD’s international program of assessment of reading,scientific and mathematical literacy (www.oecd.org/pisa), aims to assess the ability of 15 year olds to use the knowledge and skills that have acquired at school in real world tasks and challenges. It also uses questionnaires to gather data on students’ attitudes to learning and the co...

This paper explores misconceptions of the number line which are revealed when pre-service primary teachers locate negative decimals on a number line. Written test responses from 94 pre-service primary teachers provide an initial data source which is supplemented by group responses to worksheets completed during a lesson and individual interviews. T...

Current technologies incorporating sophisticated mathematical analysis software (calculation, graphing, dynamic geometry, tables, and more) provide easy access to multiple representations of mathematical problems. Realising the affordances of such technology for students’ learning requires carefully designed lessons. This paper reports on design re...

This chapter reports on the use of dynamic geometry to support the use of real world contexts to enhance the learning of mathematics in the middle secondary years. Dynamic geometry, either linked to real world images or used to create dynamic simulations, not only can provide opportunities for students to collect real or simulated data to make conj...

The use of mathematics analysis software (MAS) including handheld scientific and graphics calculators offers a range of pedagogical opportunities. Its use can support change in the didactic contract. MAS may become an alternative source of authority in the classroom empowering students to explore variation and regularity, manipulate simulations and...

Understanding linear functions is key to students developing a sound foundation for algebra. The common linear function rule y = mx + c presents students with the need to develop a conceptual understanding of both variables (x and y) and parameters (m and c) and to comprehend the different roles they play in the rule. Our Australian study explored...

This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its avai...

This session is also available as an audiorecording.

The use of Computer Algebra Systems (CAS) in years9 and 10 classrooms as a tool to support learning or in preparation for senior secondary mathematics is controversial. This paper presents an analysis of the positive and negative aspects of using CAS identified in the literature related to these year levels, along with the perceptions of 12 experie...

Understanding that mathematics is founded on reasoning and is not just a collection of rules to apply is an important message to convey to students. Here we examined the reasoning presented in seven topics in nine Australian eighth-grade textbooks. Focusing on explanatory text that introduced new mathematical rules or relationships, we classified e...

Taking advantage of pedagogical opportunities afforded by new technology requires appropriately designed lessons. This article reports on the use of ‘lesson study’ to research principles for the design of a lesson aiming to take advantage of multiple representations. The lesson, for year 10 students who had personal access to TI-Nspire, focused on...

This study presents a content analysis of the new Iranian grade 9 mathematics textbook, examining the extent to which it includes aspects of mathematical literacy. Two chapters were analyzed for the extent of use of real world contexts, and the extent of emphasis on the mathematical modelling processes of formulation and interpretation. The new tex...

Australian eighth-grade mathematics lessons were shown by the 1999 TIMSS Video Study to use a high proportion of problems of low procedural complexity, with considerable repetition, and an absence of deductive reasoning. Using definitions from the Video Study, this study re-investigated this ‘shallow teaching syndrome’ by examining the problems on...

This paper reports on the effectiveness of an intervention designed to improve nursing students' conceptual understanding of decimal numbers. Results of recent intervention studies have indicated some success at improving nursing students' numeracy through practice in applying procedural rules for calculation and working in real or simulated practi...

Extensive studies have documented various difficulties with, and misconceptions about, decimal numeration across different levels of education. This paper reports on pre-service teachers' misconceptions about the density of decimals. Written test data from 140 Indonesian pre-service teachers, observation of group and classroom discussions provided...

Understanding that mathematics is not just an arbitrary collection of rules to follow is basic to good mathematics learning, but studies show that many classrooms exhibit little mathematical reasoning. In order to better understand the nature of reasoning in schools, this study examined the modes of explicit reasoning in the explanations, justifica...

This is a Schooling Issues Digest, published by the Australian Government Department of Education, Employment and Workplace Relations to provide status reports on complex research data in a non-technical format for a wide audience. It summarizes the performance of Australian students in mathematics in PISA 2000, 2003 and 2006 and TIMMS 1995, 1999 a...

The Mathematics and Technology Attitudes Scale (MTAS) is a simple scale for middle secondary years students that monitors five affective variables relevant to learning mathematics with technology. The subscales measure mathematics confidence, confidence with technology, attitude to learning mathematics with technology and two aspects of engagement...

In today's world, where the volume of knowledge everyone must deal with is increasing exponentially, many educators agree that schools must focus on developing skills for life-long learning. But what does that mean for an area such as algebra? Teachers' goal in school algebra should be to guide students to "work smarter" with algebraic symbols and...

This study examines the procedural complexity and mathematical solving processes required by problems on two topics in seven Year 8 textbooks from four Australian states. The study used definitions from the 1999 TIMSS Video Study. Although variation existed between textbooks, the majority of problems were of low procedural complexity, requiring onl...

Permitting the use of computational technologies in school mathematics makes a significant impact on teaching and learning processes, on the curriculum itself and on the assessment of mathematics. When examinations permit the use of technology (e.g. scientific, graphics or symbolic calculators), teachers must decide how to balance the development a...

Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns. Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names. The principal explanation given in the literature has been a general link to l...

This chapter presents three principles which assist in designing and evaluating assessment, and offers some examples of their use in secondary school mathematics. These principles can be used to guide teaching with a range of technology, although the examples used here are drawn from experiences with graphics calculators, including those with a sym...