This book review explores Equity in Discourse for Mathematics Education: Theories, Practices, and Policies, edited by Beth Herbel-Eisenmann, Jeffrey Choppin, David Wagner, and David D. Pimm

In this paper, the distinction between Bernstein's horizontal and vertical discourse is used to show how two children are restricted in their possibilities to learn mathematics. The social relationships set up within contexts, both of the problems being solved, and between participants, contributed to the horizontal or vertical discourse being employed. In a circular motion, these discourses then reinforced the social relationships that could come into play. It is argued that mathematical exclusion can occur when social relationships, not only within problem contexts but also within interactions, miscue the kind of discourse which is foregrounded. Children can become confused over the sort of discourse that contributes to mathematics learning.

In this chapter, an adapted version of Dowling's distributive strategies is used to show how two children, Klara and Teo, are provided with differential enhancement in the mathematical learning opportunities that they are offered. The analysis shows that the use of everyday settings of mathematics problem, including expectations about the social relationship in those settings, can cause children to collude in the kind of enhancement that they experience. Expectations about the social relationships, within the problems being solved and between the participants, contributed to the two children using strategies which channeled them towards operating in different domains.

The informal character of preschool mathematics, engaged in during children’s play, places complex requirements on preschool teachers. It also leads to challenges in developing appropriate analytical tools for researching teacher work. In this paper a framework, the ”didaktic space”, is described and used to analyse interactions between preschool teachers and children in relationship to mathematical learning situations. An interaction between a preschool teacher and a group of children about how to compare their temperatures is analysed, using this framework. The analysis focuses on how the teacher’s contributions as well as those of the children changed as the role of the mathematics changed. The paper discusses how the didaktic space offers a nuanced understanding of preschool mathematical situations, both to researchers and to teachers.

In order to better understand the mathematics that preschool children engage in, we categorised the activities made available in one Swedish preschool. Although using Bishop’s 6 categories of mathematical activities and Walkerdine’s distinction between instrumental and pedagogical activities overcame some of our categorisation issues, they did not solve all of them. Some activities could be included in several categories which brought us back to a discussion about who was making what kinds of distinctions and for what purposes

This paper investigates how three children provided mathematical explanations whilst playing with a set of glass jars in a Swedish preschool. Using the idea of semiotic bundles combined with the work on multimodal interactions, the different semiotic resources used individually and in combinations by the children are described. Given that the children were developing their verbal fluency, it was not surprising to find that they also included physical arrangements of the jars and actions to support their explanations. Hence, to produce their explanations of different attributes such as thin and sameness, the children drew on each other’s gestures and actions with the jars. This research has implications for how the relationship between verbal language and gestures can be viewed in regard to young children’s explanations.

Teachers of mathematics have adopted willingly a variety of digital technologies. Yet for this to be done successfully, it has been suggested that teachers need to integrate knowledge of content, pedagogy and technologies (Mishra & Koehler, 2006). In this paper, one teacher’s story of planning and implementing a lesson on patterns using a digital resource on an Interactive White Board is analysed using Bower’s (2008) analysis of affordances of e-learning tools. We find that this captures better the dynamic decision making process of the teacher than Mishra and Koehler’s (2006) TPCK model of teachers’ knowledge. Nevertheless for this experienced teacher it was the mathematical and pedagogical considerations, which drove his desire to incorporate digital technologies in the lesson.

Mathematics homework is an activity done by large numbers of students across the world. However, it is not without controversy, with concerns being raised about its academic value and whether parents have the appropriate resources to actively support or teach their children. In this article, we use the narratives of two 10-year-old girls to consider how emotional and mathematical trauma can arise from doing mathematics homework with family help. This is often the undiscussed outcome of homework interactions, but one that can have profound implications for relationships between children, their parents, the school and mathematics as a discipline. The way that the children described their and other participants’ actions in the narratives provided information about the children’s agency whilst doing school mathematics in the home. We discuss the opportunities and constraints on children doing homework as a consequence of the social and institutional relations that they operate within. The constraining influence of schooling over the opportunities provided within the home situations was the main determiner of the emotional and mathematical trauma experienced by the children.

The importance of mathematics or its alter ego 'numeracy' is being cemented in the public's mind with the instigation of national, high-stakes testing in Australia. Discussions about national testing in press releases, online news articles and online public comments tacitly attribute importance to mathematics. In these discussions, children are positioned as commodities, with mathematics achievement being the value that can be added to them. Deficit language identified some children as being less valuable commodities and less likely to gain value from schooling. In the same public discourse, the value of the sort of mathematics that can be assessed in these tests appeared to be so accepted that it did not need to be mentioned. This has social justice implications.

Although preservice primary teachers’ limited mathematical knowledge has been well documented, little research has been conducted on programs to improve it. We report on first-year, teacher education students’ use of recommended internet resources on different mathematics topics. Our findings suggest that many of our preservice teachers had not previously used internet resources for learning, except to do research. They also saw mathematics learning as occurring only when they are taught by a teacher and so internet resources were of limited value. Ultimately these beliefs, if left unchanged, will have an impact on their teaching of mathematics to primary school students.

Often professional development is seen as something that is provided pre-packaged to teachers who adopt or reject it depending upon their previous beliefs and knowledge. This view does not take into account the influence of context on the professional development. Yet, there is little research from the perspective of the professional development facilitators and so the complexity in which they operate tends to be ignored. In this paper, we explore the constraints and opportunities in the setting up a mathematics professional development project in one school whose students came from a low socioeconomic area. We unpack some of the main decision making points using Kemmis and Grootenboer’s (2008) ideas on the connection between practice architectures and dispositions, as described by Aristotle. By identifying how the circumstances and context shaped the setting up of the professional development, we were better able to understand how we influenced the situation and thus how we affected the later decision making of participants. Such unpacking has potential for better understanding how the rhetoric surrounding Education for All can become a reality.

Australia introduced national testing in literacy and numeracy in 2008 in order to ensure that all children reached basic benchmarks. In January 2010, school results were published online, making these tests high-stakes, especially for schools likely to have poor results. In 2009, a research project was conducted on what supported and hindered teachers in a school in a low socio-economic area to take up mathematics professional development opportunities. This paper explores the impact of national testing on perceptions of teachers’ agency, particularly the constraints it imposed on taking up these opportunities.

Currently there is much discussion internationally about what and how mathematics should be integrated into preschool programs. In Swedish preschools, there is a strong tradition of children’s play. Using video data, we identify how one teacher in a Swedish preschool recognises and builds on mathematical teaching moments that arise from children’s play. The role of respectful listening and asking challenging questions is important in the development of children’s mathematical curiosity. We use this data to explore whether using teaching moments is an appropriate teaching practice in preschools for ensuring that children have good mathematical knowledge to begin school with.

Internationally there is much discussion about what and how mathematics should be integrated into preschool programmes. In Swedish preschools, there is a strong tradition of valorising children’s play. Using video data, we identify how one teacher in a Swedish preschool recognises and builds on mathematical teaching moments that arise from children’s play. We use these data to discuss how the guided play context facilitates some teacher actions while constraining others. Respectful listening and asking challenging questions seem to stimulate the development of children’s mathematical curiosity. However, the teacher is not able to insist that her suggestions for activities are taken up by the children.

Recently, attention has been focussed on the mathematics learnt in preschools and how this contributes to children’s subsequent learning in schools. This paper explores the dilemma of trying to increase preschool children’s learning of mathematics, whilst allowing their play to guide that learning. In Sweden, the revised curriculum for preschools specifies more mathematics to be covered. Yet, like other countries, Sweden traditionally has seen preschools as places where learning arises from children’s play. We suggest two avenues for further research that would contribute information to increasing the likelihood of play supporting mathematics learning.

In the Swedish preschool curriculum, it is stated that children’s learning should happen through play. There is also an expectation that children will engage with ICT and be provided with situations that require them to engage with mathematical concepts. Consequently, the object of this research project is to evaluate mathematical games designed for interactive tables in regard to children’s needs and intersts. Four mathematical games are analysed to determine whether they utilise the affordances of the interactive tables in ways that were likely to support preschool children’s possibilities for learning mathematical concepts and language as well as how to interact with each other. This investigation provides information about how the affordances of interactive tables can be utlised to best meet the needs of young children for future design projects.

New norms and values introduced into preschools through curricula changes require teachers to adapt their teaching. However, in making adjustments in order to increase children’s agency, teachers can loose sight of how they wield their own power. In this project, a professional development facilitator worked with five Swedish teachers. Although the preschool teachers considered their primary role to be carers who resisted interfering in children’s explorations, the initial analysis suggests that they were wielding considerable power in their interpretations of what was occurring. Consequently, it has become important for the professional development facilitator to unpack with teachers the power that they wield in order to develop their teaching. This project description discusses initial ideas for doing this.

In this paper, a methodology is proposed for gaining insights into preservice teachers’ understandings about young children’s mathematics learning. Using data from a Swedish and Norwegian pilot study, it is possible to see how a set of questions about a stimulus photo of children playing with some glass jars provided insights into the preservice teachers’ mathematical and pedagogical understandings. Although the preservice teachers seemed to be able to recognise a range of mathematical activities and respond to children engaging in them, they often gave only implicit, general information. This raises questions about teacher educators’ expectations about whether preservice teachers, at the end of their courses, should be able to provide more explicit descriptions of what children are doing and suggestions for how to develop their mathematical understandings. Information of this kind can inform teacher educators about what could be improved in future mathematics education courses in early years programmes.

This article investigates a map-drawing activity in a Swedish preschool from the perspective of how children are positioned as being or becoming mathematicians by the children themselves, the teacher and the researcher. The children positioned themselves as being mathematicians, who were capable of expressing and using their own experiences and skills. The researcher also focused on the children’s expertise and so positioned them as being mathematicians. Nevertheless, there were times when the children acknowledged their need for more skills and knowledge to solve a problem and thus positioned themselves as becoming mathematicians. Similarly, some of the teacher’s questions also focussed on developing children’s mathematics skills, which thus emphasized children’s incompleteness or state as becoming mathematicians. By contrasting the teacher’s role with those of the children and the researcher, it is possible to identify how the teacher affects children’s socialisation and learning through her questioning.

In many countries around the world, young children use different kinds of information and communication technologies (ICT) on a daily basis. In this chapter, the use of games or apps on these technologies is explored in relationship to children’s learning of mathematical thinking. The work of Biesta on education and socialisation is combined with that of Radford on subjectification and objectification to theorise young children’s learning of mathematical thinking. Two Swedish preschool children’s interactions with a balance game on an Interactive Table are analysed to consider the value of this theory and what it says about the affordances of the game.

In Sweden, where almost all children attend preschool, preschools are significant sites of socialisation. Therefore, there is a need to investigate how this socialisation may be affected by implementation of a revised curriculum. The changes in preschool curriculum are an indication of how schoolification is influencing Early Years education. This is because preschool teachers and work teams use the curriculum to plan activities for children who will be socialised by participating in these activities. This article investigates the goals and guidelines in the revised preschool curriculum and considers how an increased emphasis in those related to mathematics may affect the kind of socialisation children could gain. The goals and guidelines support teachers’ pedagogical practices and hence are worth investigating. The concepts of being and becoming are used to consider how the goals and guidelines position children as having or needing to gain norms and values, skills and knowledge. Consequently, they are considered to need to acquire the skills to perform as members of their society or as knowledgeable participants when constructing their everyday lives in preschool. The goals and guidelines related to mathematics emphasise children’s becoming, and thus their incompleteness. This results in less opportunity for teachers to perceive children as having relevant experience and skills to contribute to activities and to produce creative cultural understanding. Consequently, the schoolification of the preschool curriculum through the increased emphasis in the goals and guidelines for school subjects is likely to affect the kinds of activities that preschool teachers plan and provide to children, and thus the kind of socialisation they receive in preschool.

This paper examines the development of two iterations of mathematics curricula over a 15-year period for classrooms teaching in te reo Māori, the endangered Indigenous language of Aotearoa New Zealand. Similarities and differences between the two iterations are identified. Although parameters set by the New Zealand Ministry of Education about what the curricula would look like and how they would be developed were not always commensurate with Māori aspirations, analysis suggests that Māori were able to use opportunities to ensure that their agendas for language development and revitalisation were achieved. Spaces were made available because of the government’s ideological assumptions, but were used by Māori to achieve their ideological aims. However, neither iteration was smooth, with Māori having to determine how to operate within these contested spaces. The result of Māori requirements to have language recognised as an important issue was that both process and product of curriculum development were affected.

This paper examines the development of mathematics curricula for classrooms teaching in te reo Māori, the endangered Indigenous language of New Zealand. Although the parameters set by the New Zealand Ministry of Education about what the curricula would look like and how they would be developed were not always commensurate with Maori aspirations, the analysis suggests that Māori were able to identify spaces to ensure that their agendas for language development and revitalisation were achieved. These contested spaces were made available because of the government’s ideological assumptions, but then were used by Māori to achieve their ideological aims.

The present call for structured mathematics programmes in preschools continues a trend from the nineteenth century, in which young children's lack of mathematical knowledge was considered to have a detrimental effect on their individual futures and those of the wider society. In this paper, an investigation of the philosophies behind several early childhood programmes shows that there is a long-standing acceptance that those not living in poverty should make decisions about the education, including the mathematics education, that children who are living in poverty should engage in. Consequently, the philosophies behind these programmes, and with them the advocated mathematics education, contribute to a homogenised view of the child. This fails to recognise the attributes that children and their communities have and situates those living in poverty as being deviant. The strong promotion in this century of structured mathematics education programmes is solidifying this homogenising process in a manner not seen in previous early childhood programmes.

Children bring a wealth of mathematical knowledge from home to school. This paper describes one six/seven year old girl’s home interactions over 20 weeks about measurable quantities. Almost all measurement discussions occurred when actions were not straight-forward. Consequently the measurement attributes usually used to introduce measurement at school, such as length, were rarely discussed because actions associated with them could be done without discussion. Time was discussed more often than any other attributes. Although time is considered difficult to learn because of its abstract nature, it may in fact be an easier concept for introducing ideas of iteration.

In mathematics education research, English has become the lingua franca in many situations. There are many advantages of having a lingua franca within such a diverse community. However in this paper, it is argued that the practices that are most affected by the need for a lingua franca, such as conference attendance and the writing of journal articles, also contribute to mathematics education research becoming monocultural, both in what is researched and how it is reported. Fictional dialogues are used to explore the construction of this monocultural nature of mathematics education research. In considering the collective praxis of researchers in this field, there is a need to identify the constraints on our possibilities for participating in those practices, along with the ways that those possibilities are affected by our participation. In this way, we not only begin a dialogue on these issues but have the possibility to locate other ways of participating.

Traditionally, multicultural education is not seen as something that those from the mainstream need to concern themselves with unless it is as a holiday fill-in activity. Intercultural education on the other hand explores the responsibility for the construction of culture by mainstream society. In this article, I explore my role in the delivering of mathematics education courses in teacher education programs in three countries: New Zealand, Australia, and Sweden. I focus on my attempts to raise mainstream as well as minority students’ awareness of intercultural understandings. Different approaches, including the use of storytelling, were used to match local circumstances but all had the same aim of making preservice teachers reappraise the role that mathematics education has in marginalising some children's cultural backgrounds.

Although refuted many times, the commonly accepted story about Indigenous communities in Australia is that they had few counting word and thus were lacking in ways to quantify amounts. In this paper, we use the case of quantifying to discuss how Indigenous mathematics can be used, not just to help Indigenous students transition into the classroom but also back into their home communities. We argue that mathematics education must take seriously its responsibility to support Indigenous students to gain school mathematics and also to help maintain the use of traditional mathematical ideas. If this does not occur, mathematics educators will contribute, intentionally or unintentionally to the loss of Indigenous knowledge that present and future generations of Indigenous people will hold them responsible for.

The limited mathematical knowledge of preservice primary teachers is an international concern. The areas of mathematical difficulties have been well documented, which has led to many universities instituting testing regimes to ensure that preservice teachers have appropriate knowledge of primary school mathematics. In our study, the preservice teachers identify some benefits for being tested, but these were often related to having sufficient knowledge so that they did not lose face in front of students. We suggest that this emphasis on performance could exacerbate preservice teachers’ reliance on procedural rather than conceptual understanding. Thus, there needs to be consideration of the type of testing that is instituted so that it connects to preservice teachers’ professional identities as teachers.

In this chapter, we explore, from a social justice perspective, conceptions of learners in transition between contexts and evaluate pedagogical practices that have been advocated for such learners. Learning occurs as learners reflect on their transition between contexts, particularly when there are differences in what content knowledge is valued, the relationships between participants and how activities are undertaken. From this perspective, productive pedagogical practices for learners in transition are those that build and sustain relationships between learners and mathematics and between learners and others, including those outside the classroom. We look specifically at examples of pedagogical practices that draw on ethnomathematics and critical mathematics education for their inspiration.

Undervisning om mätning behöver handla om att utveckla barns förståelse för de matematiska begreppen egenskap, enhet och skala. Detta är ett ändrat fokus från att först se på längden, sedan på area, volym och massa, och slutligen på tid. Författarna diskuterar i artikeln hur problemlösning kan vara ett användbart sätt att underlätta och utveckla förståelse för olika mätningsbegrepp

Language can be simultaneously both a support and a hindrance to students’ learning of mathematics. When students have sufficient fluency in the mathematics register so that they can discuss their ideas, they become chiefs who are able to think mathematically. However, learning the mathematics register of an Indigenous language is not a simple exercise and involves many challenges not only for students, but also for their teachers and the wider community. Collaborating to Meet Language Challenges in Indigenous Mathematics Classrooms identifies some of the challenges—political, mathematical, community based, and pedagogical— to the mathematics register, faced by an Indigenous school, in this case a Mäori immersion school. It also details the solutions created by the collaboration of teachers, researchers and community members.

In this article, we explore how a school in Aotearoa [New Zealand] infuses the identity of Indigenous students into the school-based curriculum through the promotion of their language and culture in mathematics lessons. Bernstein’s pedagogic device illustrates how teachers’ practices were influenced by being able to think the “unthinkable.” This came from the contestation that arose when competing bodies of knowledge had to be integrated both at the school level and at the classroom level.For equity to be achieved regarding students’ mathematics learning, parents’ and the community’s aspirations for students’ education need to be infused into debates about the knowledge that teachers are expected to include in their teaching. This enables the local context to make a positive contribution to students’ learning. It also implies that programs for improvement should not be imposed on schools unless there are opportunities for them to be adapted to the needs of individual schools.

For a large number of reasons, most of the teachers with whom we work do not have the language of instruction, te reo Māori, as their first language. Teaching in a language, which is an additional language, is not an unusual situation for many teachers around the world. Using data from our ongoing, longitudinal research, we explore the sort of language knowledge that could be included within mathematical pedagogical content knowledge. This has implications for teacher education programs designed for teachers who do not teach in their first languages.

Many mathematics teachers around the world teach in a language different from the one in which they studied or completed their teacher education. Often these teachers must learn both the registers of mathematics and of mathematics education to teach in the additional language. This paper examines the factors that help teachers to learn these registers in Maori, the Indigenous language of New Zealand. Many of these teachers are second-language learners of the Maori language and attended Englishmedium schools and teacher-education programmes. After a brief discussion about the key role of language in teaching mathematics, this paper examines data from teachers at two Maori-immersion schools and a professional development facilitator. The analysis provides initial understanding of the factors that support or hinder their learning of the mathematics registers. Finally, a research agenda is suggested for further investigation of this issue.