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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 each of these cognitive and non-cognitive competencies in the Chinese curriculum since 1902, and a comprehensive set of snapshots of the current performance of Grade 8 students on each. The assessment tools provide powerful base-line data for monitoring students’ mathematical competencies into the future. The detail in the studies will assist international researchers to more deeply understand some of the paradoxes in PISA results, such as Chinese students reporting low classroom exposure to “applied problems” while also demonstrating outstanding performance on items emphasizing PISA’s formulate process. The book gives insight into the strong Chinese tradition of mathematics education, changing over time in response to dramatic social and economic forces but retaining unique features and also becoming increasingly well integrated with international thought.

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 chapters demonstrate that teaching with any model requires care and attention to achieve conceptual clarity. Teachers need to act entirely within in the world of the model, only later adding other models to round out the mathematical picture. The teaching of integers is a key point for maintaining students' confidence that mathematics is a subject that makes sense. These chapters demonstrate how use of models can assist in this goal.

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 teachers with an automated diagnosis of their own students’ stages of development in specific topics and to report on an individual’s errors and misconceptions, in order to inform teaching. Our claim is that teachers’ diagnostic competence increases when they have easy access to information about their own students’ thinking. In turn, this can further improve teaching, and hence learning. By drawing together evaluative data from four sources, we highlight aspects of teachers’ initial responses to formative assessment and the effect of using this system on their knowledge for teaching and the subsequent changes to teaching practice. Overall, teachers report that using the smart-tests has improved their knowledge of the thinking of individual students as well as of students in general (i.e., their pedagogical content knowledge), and that they can use this information in several ways to adjust their teaching. Paradoxically, using smart-tests reduces the demand for teachers to have specific knowledge for diagnosis and at the same time increases this knowledge and so improves their diagnostic competence.

This chapter is based on the Plenary Panel on International Comparative Studies we delivered at the 13th International Congress on Mathematical Education (ICME-13) in 2016. In the past a few decades, international comparative studies have transformed the way we see mathematics education and provide insight for improving student learning in many ways. Out of several possibilities, we selected four lessons we have learned from international comparative studies: (1) examining the dispositions and experiences of mathematically literate students, (2) documenting variation in students’ thinking in different cultures, (3) appreciating the varying meanings and functions of common lesson events, and (4) the importance of making global research locally meaningful. Throughout the paper, we point out future directions for research to expand our understanding and build up capacity in international comparative studies.

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 knowledge in concert with other types of knowledge for teaching. In this article, we explore the idea that teachers of mathematics act as applied mathematicians in applying mathematical knowledge to the resolution of teaching problems. This task involves the complex interplay of mathematical and teaching knowledge and processes of problem solving with success judged according to how well students learn. The article discusses these ideas through an examination of seven scenarios.

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 like a mathematician’ to teachers so that they can develop sensitivity to their own and to their students’ problem solving experiences. Additionallyteachers’ appreciation of teaching for, about and through problem solving can be stimulated when they study students’ problem solving attempts in depth. Three studies on classroom implementation of problem solving all give attention to making the initiatives an ongoing part of normal teaching practice. This section illustrates that over the three decades since problem solving first rose to international attention, mathematical problem solving has become an important goal in many counties. There are now long-term design research projects working with sizeable groups of ‘ordinary’ teachers to establish practical guidelines for sustainable problem solving initiatives.Problem solving remains an elusive goal, with no silver bullets, but this new phase of research will widen its reach into classrooms.

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 Roger’s framework for the diffusion of innovation and Pierce and Stacey’s pedagogical opportunities map. Data on teachers, views and practices were collected over 2 years. ‘Pedagogical Maps’ give a picture of the teachers’ perception and uptake of pedagogical opportunities. New practices have been added slowly to each teacher’s repertoire and their increasing fluency in practical ability to teach with the technology resulted in some changes to the classroom didactic contract. Overall, new technology seemed to have been absorbed into current practice, more than changing practice. At this stage of their development, these teachers do not identify the distinctive new mathematical capabilities as contributing to the major relative advantage of the innovation. Instead, they see the relative advantage mostly in the incremental improvements to capabilities of earlier calculators, and meeting the need for students to be up to date. One of the current challenges is that significant changes in both software and hardware design have been happening so rapidly that these early majority teachers felt almost constantly hampered by the need to learn and teach new technical skills and so continue to make limited progress in taking advantage of opportunities to approach mathematics concepts in new ways.

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 aligned with mathematical literacy for all students. Australia was selected as the comparison country because of its relatively high international achievement in mathematical literacy. Three comparable chapters in each book were analysed for these characteristics. When considering the percentages of problems showing each characteristic, there was considerable similarity between the textbooks. However, the Australian mathematics textbooks contain many more problems than the Iranian textbook and each chapter includes problems in a diverse range of contexts and the mathematical tasks range from simple applications of formulas to real world modelling although of a constrained nature. With many fewer problems, the Iranian textbook has considerably less diversity of context and very little opportunity for students to engage in mathematical modelling, the key process of mathematical literacy.

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 multimedia stimulus material to every interviewee. Provision of a personal copy of the interview material, including all the stimulus resources, to the interviewees provided for a richer interview than would have been possible with a face-to-face interview in the more limited timeframe that would have been imposed. The opportunity for interviewees to reflect on questions and edit responses resulted in high quality data and interviewees found the electronic interview convenient and satisfying. Further advantages of the electronic format were the savings in organisation, travel and transcription. We believe that electronic interviews offer great potential to other researchers.

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 research conducted with nine teachers in nine classes across two schools. The lesson came at the end of their study of quadratic functions (Year 10). Technology offered the ability to link representations and the teaching challenge was to use this ability effectively. Four issues emerged: variable naming; reducing cognitive load; maintaining motivation; and retaining a clear learning focus.

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. Two main misconceptions leading to incorrect placement of negative decimals on a number line are identified. One relates to having separate number ‘rays’ for positive and negative numbers, which are aligned according to context. The other (with several variations) results from creating the negative part of the number line by amalgamating translated positive intervals. These misconceptions explain a large percentage of wrong answers. The most important implication for education at school, as well as in teacher education, is that the teaching of negative numbers and of the number line must not be confined to integers, as is frequently the case, but must also include negative fractions and decimals.

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 link representations. Strategic use may support students to direct their own learning and explore mathematics, equipping them to share their findings with the teacher and the class with more confidence. This paper offers a framework for examining the impact of the use of MAS on the didactic contract. Lessons were observed in 12 grade 10 classes, with 12 different teachers new to MAS. MAS technology was used with a variety of didactic contracts, mostly traditional. The framework drew attention to many ways in which the teaching differed. Analysis of the didactic contract must consider both the teaching of mathematics and of technology skills, because these have different characteristics. In all classes, both teachers and students saw the teacher as having a responsibility to teach technology skills. Students saw technology skills as the main point of the lesson, but the teachers saw the lesson as primarily teaching mathematics—one of the mismatches which may need negotiation to adapt didactic contracts to teaching with MAS.

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 availability in classrooms also provides opportunities for positive changes to teaching and learning. Very many examples are documented in the professional and research literature, and in this paper we organize them into 10 types. These are displayed in the form of a ‘pedagogical map’, which further classifies them according to whether the opportunity arises from new opportunities for the mathematical tasks used, change to interpersonal aspects of the classroom or change to the point of view on mathematics as a subject. The map can be used in teacher professional development to draw attention to possibilities for lessons or as a catalyst for professional discussion. For research on teaching, it can be used to map current practice, or to track professional growth. The intention of the map is to summarise the potential benefits of teaching with technology in a form that may be useful for both teachers and researchers.

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 quadratic functions. Following observation of nine different classes and teachers, four key principles emerged: focus on the main goal for that lesson (despite the possibilities offered by having many representations available); identify different purposes for using different representations to maintain engagement; establish naming protocols for variables that are treated differently when working with pen and paper and within a machine; and reduce any sources of cognitive load that are not essential.

The use of Computer Algebra Systems (CAS) in years 9 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 experienced secondary teachers who were working with years 9 and 10 students. The literature review shows that CAS is valued for calculation and manipulation capabilities, the option of alternative representations, the opportunity for systematic exploration and for prompting rich discussion. However, the technical overhead, initial workload for the teacher and unresolved questions about the perceived relative contribution of machine and by-hand work to learning currently pose obstacles to teaching with CAS in the middle secondary years. The teachers who contributed data to this study perceived that using CAS in their teaching is, on balance, worth the effort. However, they believed that CAS is of most benefit to their high ability students and may present an obstacle to their low ability students’ learning of mathematics.

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 explanations according to the mode of reasoning used. Seven modes were identified, making a classification scheme which both refined and extended previous schemes. Most textbooks provided explanations for most topics rather than presenting “rules without reasons” but the main purpose appeared to be rule derivation or justification in preparation for practise exercises, rather than using explanations as thinking tools. Textbooks generally did not distinguish between the legitimacies of deductive and other modes of reasoning.

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 three topics in nine eighth-grade textbooks from four Australian states for procedural complexity, type of solving processes, degree of repetition, proportion of ‘application’ problems and proportion of problems requiring deductive reasoning. Overall, there was broad similarity between the characteristics of problems in the textbooks and in the Australian Video Study lessons. There were, however, considerable differences between textbooks and between topics within textbooks. In some books, including the best-selling textbooks in several states, the balance is too far towards repetitive problems of low procedural complexity.

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 learning mathematics. The paper presents a model of how technology use can enhance mathematics achievement, a review of other instruments and a psychometric analysis of the MTAS. It also reports the responses of 350 students from 6 schools to demonstrate the power of the MTAS to provide useful insights for teachers and researchers. ‘Attitude to learning mathematics with technology’ had a wider range of scores than other variables studied. For boys, this attitude is correlated only with confidence in using technology, but for girls the only relationship found was a negative correlation with mathematics confidence. These differences need to be taken into account when planning instruction.

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 mathematical literacy as it is likely to be encountered in modern workplaces; and new questionnaire items targeting mathematics. Procedures for quality assurance that arise in the preparation of an assessment for use in many countries around the world are illustrated with some items and results from the 2011 international field trial. The paper will provide background for the interpretation of the results of the PISA 2012 survey, which are to be published in December 2013.

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 was based in his expertise in the teaching of mathematical modelling, and very concrete and practical guidance to the complex and wide-ranging work of the group. This article documents and celebrates Werner Blum’s contribution, from the perspective of MEG colleagues and collaborators.

The basic theory of Rasch measurement applies to situations where a person has a certain level of a trait being investigated, and this level of ability is what determines (to within a measurement error) how well the person does on each item in a test. This paper responds to frequent suggestions from colleagues that the use of Rasch measurement would be profitable in analysing a set of data on students’ understanding of decimal notation. We demonstrate misfit to the Rasch model by showing that item difficulty estimates show important variation by year level, that there is significant deviation from expected score curves, and that success on certain splitter items does not imply a student is more likely to score well on other items. The explanation given is that conceptual learning may not always be able to be measured on a scale, which is an essential feature of the Rasch approach. Instead, students move between categories of interpretations, which do not necessarily provide more correct answers even when they are based on an improved understanding of fundamental principles. In this way, the paper serves to highlight the assumptions built into the Rasch model and to discuss its applicability to describing the progress of learning with various characteristics.

Those who market people or products choose their images very carefully. They create positive associations in the public's mind by photographing their clients with sporting heroes or national icons. In this paper we present a variety of evidence to show that a major and overlooked reason for teachers' use and choice of real world problems is to take advantage of this ‘halo effect’ to improve studients' attitude towards learning mathematics. Analysis of interviews, reports, and results of a brief survey from teachers of middle secondary school classes indicate that they place a very high priority on positive attitudes and hence both choose and enhance real world problems to promote studients' affective engagement through simple pleasures. Pleasant sensory stimuli, generally non-cognitive and peripheral to the situation to be modelled, are used to promote a positive view of mathematics. This is a good strategy for creating enjoyable and memorable lessons, but there is a danger that it may override more substantive learning goals.

This paper reviews the presentation of problem solving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problem solving in the documents is reviewed and contrasted, and illustrative problems from teachers’ support materials are used to demonstrate how problem solving is now more often treated as a teaching method, rather than a goal in itself. The paper also analyses how the curriculum documents describe the growth of students’ abilities in the process areas of mathematics, and assesses the guidance that this provides for teachers. At each stage, the paper suggests directions for research that would be useful in assisting curriculum documents to promote the fundamental but elusive goal of making students better problem solvers.

This study analyses some of the consequences of adopting a functional/modelling approach to the teaching of algebra. The teaching of one class of 17 students was observed over five weeks, with 15 students undertaking both pre- and post-tests and 6 students and the teacher being interviewed individually. Use of graphics calculators made the predominantly graphical approach feasible. Students made considerable progress in describing linear relationships algebraically. They commented favourably on several aspects of learning concepts through problems in real contexts and were able to set up equations to solve contextualised problems. Three features of the program exerted a ‘triple influence’ on students’ use and understanding of algebraic symbols. Students’ concern to express features of the context was evident in some responses, as was the influence of particular contexts selected. Use of graphics calculators affected some students’ choice of letters. The functional approach was evident in the meanings ascribed to letters and rules. Students were very positively disposed to the calculators, and interesting differences were observed between the confidence with which they asked questions about the technology and the diffidence with which they asked mathematical questions.

This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.

The support of technology for working with multiple representations of functions has substantial potential for teaching calculus. For teaching differentiation, these representations relate to finding difference quotients, finding gradients of curves and tangents, and using symbolic differentiation rules. For students to use them all and link them together requires a wide range of skills, which have been organised into aDifferentiation Competency Framework. This paper also describes a balancedDifferentiation Competency Test that was created from the Framework and used in two Year 11 classes to monitor students’ understanding of introductory differentiation. The Framework helps identify both student achievement and teaching focus.