Barry Kissane

In recent years, calculators have become common objects in primary school classrooms, as they also have in other places, including homes and shops. Being accessible to children is, however, no guarantee that calculators will be educationally beneficial, and some have even suggested that they might be harmful. An important part of the legacy of Derek Hurrell, reflected throughout this book, is that children might learn about mathematics if given appropriate tasks and opportunities to engage productively with them in a supportive environment...

Mathematics is often interpreted as a ‘useful’ activity, with limited attention paid to its potential to be significant for other reasons. While utility is of course important, it is of diminished significance if students are not engaged with, interested in or attracted to mathematics. Many mathematicians and others over time have drawn attention to the beauty of mathematics and its deep aesthetic qualities, and mathematics is connected richly to our collective cultural heritage. One of the broad aims of mathematics education is to develop positive attitudes towards mathematics; while syllabuses make reference to this broad aim in different ways, it is often hard to see how it is addressed explicitly in official documents. In this chapter, we first consider some ways in which mathematics is connected to a wider world beyond its practical applications. We address in particular the significance of some of the aesthetic aspects of mathematics and its cultural heritage, both its history and contemporary views of its place in the world.

Mathematics is often interpreted as a ‘useful’ activity, with limited attention paid to its potential to be significant for other reasons. While utility is of course important, it is of diminished significance if students are not engaged with, interested in or attracted to mathematics. Many mathematicians and others over time have drawn attention to the beauty of mathematics and its deep aesthetic qualities, and mathematics is connected richly to our collective cultural heritage. One of the broad aims of mathematics education is to develop positive attitudes towards mathematics; while syllabuses make reference to this broad aim in different ways, it is often hard to see how it is addressed explicitly in official documents. In this chapter, we first consider some ways in which mathematics is connected to a wider world beyond its practical applications. We address in particular the significance of some of the aesthetic aspects of mathematics and its cultural heritage, both its history and contemporary views of its place in the world.

Although officially approved for use in the Singapore secondary curriculum, including examinations, scientific calculators are still often misunderstood as devices solely for arithmetic computation and even at times regarded as unhelpful for students learning mathematics. Yet the development of calculators in recent decades has been focused on their use as educational devices, and the design of calculators has been heavily influenced by the needs of secondary school students. In this paper, we first consider the educational potential of calculators in education, drawing on a model developed for this purpose. We then focus on the educational design of tasks and activities using calculators for various educational purposes, including the development of mathematical concepts and important processes such as reasoning and modelling. Attention focuses on the design of experiences that incorporate modern calculator capabilities and their affordances for important learning goals, as well as the implementation of these in classrooms. Examples and analysis of some appropriate tasks are provided.

School mathematics in the 21st century seems likely to be characterized in part by new expectations regarding the complex lives of students after they leave school. In this chapter, an analysis of four domains of 21st century life is provided: working, shopping, participating in society and personal satisfaction. A productive worker will need to use the mathematics they have learned at school in the increasingly rapidly changing world of work. A careful consumer will need to develop expertise in understanding the many choices they need to make regularly over the course of their adult lives. The informed citizen will need to engage with, understand and even contribute to the rapidly changing world of information in the 21st century for both personal and national benefit. The balanced person will need to become aware of and appreciate mathematical perspectives on everyday events and on mathematics itself. Some examples are offered in each of these domains, recognizing that an important part of the craft of mathematics teaching is to interpret and implement the curriculum to achieve the best outcomes for students. While some of these outcomes are determined by short-term transition from school to later study, others require a longer perspective.

Information and communications technology (ICT), increasingly available at both home and school, offers new kinds of learning experiences for primary children. This chapter explores the distinctive contribution of ICT to learning experiences of students, focusing on those that are not readily available through traditional media of pencil and paper and organised practice of skills. According to a Becta report in the UK, arguments and evidence for using ICT in the number curriculum focus on the role of feedback, observation of patterns and relationships and the development of visual imagery. The chapter highlights examples that use virtual manipulatives on the Internet, other web-based experiences, and hand-held devices such as Apple's iPad® and iPod Touch® and modern scientific calculators. The focus of the examples is on the development and understanding of important ideas in the number curriculum, such as those related to number patterns, place value, decimals, fractions, factors and ratios, rather than to the refinement of arithmetical skills. New learning experiences using ICT demand a new kind of attention to the role of the teacher, in a variety of contexts, including individual ICT use, small-group use, whole-class use (e.g. through interactive whiteboards) and home use.

Recent Australian Institute of Teaching and School Leadership (AITSL) standards for primary mathematics teacher education highlight the significance of mathematical content knowledge for graduating teachers. Online learning systems have become popular for school mathematics in recent years. As the development and maintenance of such systems requires more resources than a teacher education faculty is likely to have available, this study identifies ways of adapting a commercial resource intended for schools to one that is appropriate for teacher education. The study reported is based on the experiences of the authors in building Cambridge HOTmaths into the design of two early undergraduate units for pre-service primary mathematics teachers at Murdoch University over 2011 and 2012. Students are provided with a licence for personal use of the online learning system, and a small mark was allocated to reflect individual student’s engagement with the online materials, partly to recognise the significant learning of mathematics involved, but also to provide an inducement to use it. The software is used periodically in plenary lectures and in workshops, according to individual teacher preferences and practices; specific advice for HOTmaths use is provided in weekly learning guides accessed online by all students. Differences between school use and teacher education use are described in the paper. Data on the extent to which students have made use of HOTmaths are presented, together with student feedback regarding the value of the online experience. Different approaches taken by teaching staff to the use of HOTmaths are also reported. Together, the evidence suggests that there is considerable merit in adapting this resource for use in primary mathematics teacher education.

Nurturing reflective learning in the school depends critically on teachers being appropriately reflective themselves. In this chapter, the author describes some pre-service primary teachers reflecting on their own experiences of learning mathematics. The pre-service teachers were required to reflect on how they had learned mathematics and on how technology was used to support their learning. The chapter draws on a sample of the resulting reflections to illustrate the value of such an activity for the pre-service teachers themselves, while the results of the reflections also provided learning opportunities for their teacher.

This chapter discusses the nature of numeracy, which is concerned with the use of mathematics in context, reflecting some recent Australian work concerned with numeracy across the curriculum, and some Australian initiatives to highlight mathematics in wider contexts than the mathematics classroom. Australian national curriculum developments, in which numeracy is identified as a General Capability to be addressed in all school curricula, not only in mathematics curricula, are described. In the primary years, situations that demand some level of mathematical thinking arise in all aspects of the curriculum, sometimes unexpectedly, and thus provide opportunities for numeracy to be developed providing we are alert to the possibilities, while in the secondary years, mathematics is important for learning in other subjects, although this is not always well recognized. Students need to develop a capacity to use mathematics in their everyday lives, their school lives and eventually in the world of work. The chapter explores examples of numeracy demands and opportunities across the curriculum, and considers the roles of teachers to develop the necessary mathematical, contextual and strategic expertise to address these.

Apart from easing some of the computational burden, an attraction of ICT for mathematics education is that we may have new opportunities for pupils to learn and teachers to teach. This chapter describes some recent examples of technologies that seem likely to be appropriate for use in Singapore, and analyses some of their potential for mathematical applications and modelling, especially in probability and statistics. Particular attention is paid to hand-held calculators because they are the most likely to be affordable and available, especially when the constraints of examinations are taken into account. Additionally, some computer software with particular strengths is acknowledged: spreadsheets, which are likely to be widely available where computers are available; Fathom and Tinkerplots, innovative commercial software for statistics. The Internet offers considerable potential for mathematics education, despite relatively high costs of obtaining access to it, and seems likely to grow in significance in Singapore in the next several years. Some ways in which the Internet might support increased attention to applications and modelling are identified. The chapter concludes with recognition of the pervasive significance of professional development of teachers.