Gestures have been shown to reflect speakers’ embodied thinking about mathematical concepts and play a role in conveying understandings in teaching/learning interactions. However little research has been done to consider the similarities and differences in the functions that a particular gesture might have in mathematics classrooms in different parts of the world. In this paper, the occurrence of a metaphorical gesture to do with addition in a bilingual mathematics classroom in the UK and two Spanish-speaking classrooms in Chile is investigated. To do this, we elaborate on Systemic Functional Linguistics to consider how the gesture is integrated with other modes of communication to reflect the immediate context of situation as well as a wider mathematics education context of culture. An analysis of the interactions in three classrooms illustrates how the gesture seemed to convey extra meanings, sometimes complementary and sometimes contradictory, to what was expressed through other modes. Besides adding meaning to the mathematical ideas conveyed verbally, the gesture could potentially convey meanings to participants in the interactions about interpersonal relationships which were not as evident in the verbal communication.

Using Engeström's third generation of activity theory, we explore three master's students' views of their completed research projects on children's argumentation in number stories as potentially contributing to their forthcoming teaching of Grade 1 students. Activity theory was chosen because it provided opportunities to consider how two different activity systems, research as part of teacher education and mathematics teaching in Grade 1, might overlap around the shared artefact of mathematical argumentation through number stories. The three interviews are analysed using Engeström's description of four levels of contradictions identified in a matrix of principles and questions. The analysis raised some issues with the use of Activity Theory to understand the master's students' learning from the contradictions between the two activity systems they were in-between.

Med gjeninnføring av lek i skolens læreplan er det behov for å vurdere hvordan lek har vært knyttet til matematikk i barnehagen. Vi bruker en modell som kobler sammen barns lek med det å stille og å løse problemer, til å analysere barnehagebarns engasjement med digitale apper og 2. trinns elevers skriving av regnefortellinger. Analysen viser hvordan lek kan oppstå, når barna stiller problemer, og når de prøver ut ulike løsningsstrategier. Vi anser at resultatet av forskningen kan gi innsikt til matematikklærere i de første skoleårene om potensialet for lek i arbeid med problemstilling og problemløsning.

In this study, we aim to explain choices that groups of preservice teachers made to implement modelling tasks in primary school. We found that the preservice teachers decided to facilitate everyday-inspired contexts for the tasks and that this could be explained by their reasoning that modelling is supposed to be an activity close to reality. Essentially, the groups preferred two different approaches to introducing students to the tasks: (1) by inviting students to physically participate in situations related to their local communities that were familiar to them (e.g. trash picking), and (2) through pre-structured problems that were recognisable to students (e.g. estimation of the number of beads in an irregular box). We found evidence that the preservice teachers tended towards avoiding focusing on possible mathematical answers to the tasks. This could be explained by lacking experience with how to find approximate solutions, or that the preservice teachers were engaged in observing students’ solution strategies and mathematical behaviour.

In our action research projects as teacher educators, we focus on broadening preservice teachers’ understandings about language diversity in mathematics classrooms, away from just improving students’ fluency in the language of instruction. To undertake these projects, we developed a flexible analytical tool, for identifying those pedagogical practices which could be improved and so broaden preservice teachers’ understandings about language diversity. By documenting the development of the analytical tool, we provide insights into how action research projects can contribute to aims for social justice. Using data from a wider project, the analytical tool is discussed in regard to its possibilities for identifying practices that could be improved, its validity and reliability, and its potential for use in other action research projects.

In this article, we investigate how the introduction of the pedagogical method, "mathematics in three acts" to preservice teachers influenced their mathematical modelling, while on practicum. We analyzed documentation, of groups of preservice teachers, on their experience of teaching modelling lessons. One finding was that the groups were able to use "mathematics in three acts" to produce modelling problems that fulfilled certain criteria in our selected framework, but they had difficulty finding modelling problems for the lower grades (grades 1-3). We also found that the preservice teachers appeared to emphasize the subprocess of working mathematically and devalue the subprocesses simplifying/structuring, mathematising, and validating, when guiding the pupils through a modelling process in a "mathematics in three acts" lesson.

Recently modelling and applications were included in the revised mathematics curriculum for the Norwegian grade levels 1–10. The focus on mathematical modelling in primary grade education is a challenge, because of limited experience with modelling at this educational level. This chapter is based on the study of documents written by primary grade pre-service teachers, containing their reflections on modelling activities they had implemented during a practice period. From this content, we studied what procedural choices and assessments the pre-service teachers let the pupils make and how they facilitated their critical thinking. We found that pre-service teachers often emphasised mathematical exploration, but that they tended to offer specific tasks to assist pupils with this. Pupils were not often given the opportunity to narrow the modelling problem and decide how to collect and represent data.

Recently modelling and applications were included in the revised mathematics curriculum for the Norwegian grade levels 1–10. The focus on mathematical modelling in primary grade education is a challenge, because of limited experience with modelling at this educational level. This chapter is based on the study of documents written by primary grade pre-service teachers, containing their reflections on modelling activities they had implemented during a practice period. From this content, we studied what procedural choices and assessments the pre-service teachers let the pupils make and how they facilitated their critical thinking. We found that pre-service teachers often emphasised mathematical exploration, but that they tended to offer specific tasks to assist pupils with this. Pupils were not often given the opportunity to narrow the modelling problem and decide how to collect and represent data.

Keywords: Independent modelling processes, Critical thinking, Pupils’ inquiries, Pre-service teachers’ scaffoldings, Primary grade, Document analysis

The purpose of this study is to get insight into the aspects of mathematical modelling (MM) that 16 primary preservice teachers (PTs) focus on when planning teaching for school practicum. The analyzed planning session was part of the PTs mathematics education course in their second year. All the PTs have the concern to choose a topic that interests the students for the modelling activity, and to let students make choices by themselves during modelling. The unpredictability of the MM is experienced as uncomfortable. Three categories are identified in the data: student perspective, teacher perspective and activity perspective. The results are relevant in exploring the gap between the potential that MM holds for the students’ learning and the teachers’ reluctance to implement it.

Keywords: mathematical modelling, preservice teachers, primary school.

In this paper, I investigate practices in teacher education for language-responsive teaching of mathematics. I use action-research in a mathematics education course for primary school (grades 1–7), to systematically investigate my practices to support pre-service teachers in identifying language demands of modelling activities. Two sets of practices were identified regarding teaching about supporting communication and supporting multimodality. These practices were associated with pre-service teacher actions of talking, noticing, planning, and applying language-responsiveness. The paper concludes with recommendations for further changes and improvements in these practices. The study contributes to insights on improving teacher education practice for preparing language-responsive mathematics teachers.

Although critical mathematics education has been promoted as supporting learners to engage with social justice issues, there has been little research on how learners anticipate using mathematics to make a case for a change in behaviour by an audience. The aim of this study was to identify how small groups of preservice teachers discussed the kinds of mathematical representations needed to convince specific audiences to change their behaviour. It was found that the preservice teachers took into consideration their chosen audience’s background experiences and mathematical knowledge. As well as considering the kinds of representations which were most likely to be convincing for these audiences, the preservice teachers discussed whether their calculations should be available for scrutiny in their presentations.

Preservice teachers begin their mathematics teacher education with a set of understandings about different aspects of mathematics education which will affect their engagement with mathematics teacher education courses. However, very little is known about these understandings. In this paper, the design of a survey to find out about specific aspects, emphasised in the new Norwegian curriculum, is described along with initial results from 96 preservice teachers. The results suggest that using scenarios can provide relevant information about PTs’ understanding about different aspects of teaching mathematics. These results may provide teacher educators with potential starting points for planning their own teaching.
Keywords: Preservice teachers, argumentation, multilingual classrooms, digital tools, modelling

In this chapter, we investigate one group of preservice teachers’ descriptions of their practicum experience at the beginning of their second year of their teacher education. The five preservice teachers were placed in a class where 19 of the 24 students used more than one language in their daily encounters. The preservice teachers discussed their experiences both in writing and in an oral presentation, particularly in relationship to the teaching of a unit of work on mathematical modelling. They were also interviewed about these experiences. From what the teachers seemed to notice, we identify how students’ use of multimodalities disturbed or confirmed preservice teachers’ expectations about multilingual students learning of mathematics. For teacher educators, the outcomes of this research provides an understanding of how practicum teaching involving an innovative project can support preservice teachers to improve their understandings about teaching in multilingual mathematics classrooms as well as what may still be needed to increase their professional competence.

Preservice teachers begin their teacher education with experiences that affect their possibilities for accessing and integrating new learning into their teaching practices. Yet often mathematics teacher education courses treat preservice teachers as a homogenous group. Responses to an electronic survey from the beginning of two compulsory mathematics education courses showed that preservice teachers could be divided into two clusters. The preservice teachers in each cluster give similar responses to different aspects of mathematics teaching, suggesting they share similar sets of views. These differences should be recognised in their future mathematics teacher education courses.
Keywords: Preservice teachers, cluster analysis, argumentation, modelling. 

This paper reports on a case study following four preservice teachers (PTs) who implemented modelling in their practice teaching at first grade. During the modelling activity the students made, and responded to, different forms of mathematical representations. We have applied a categorical framework to classify the responses into four broad stages of structural development. The aim was to investigate how the PTs supported the students at various stages. One finding was that the PTs had tendency to overlook responses that could be classified into the pre-structural stage. When the PTs themselves offered representations, these were aiming towards a more sophisticated level.

Keywords: Modelling, first grade, preservice teachers, representations, structural development

Digital technology has become a key part of the development of society and influences people’s ways of being, thinking, and communicating. Young people’s digital cultures are continuously developing, and this article investigates how two students’ productive struggle when programming in mathematics can be analyzed and understood from a cultural perspective. The data consists of two seventh-grade students who share a computer and program a pentagon. The task proves to be challenging and the students face several kinds of struggles. However, the students are persistent, apply communication qualities, and make continuous refinements in ways that create an interesting interweaving of mathematics and programming. The students’ communication qualities are used to describe characteristics of a productive struggle in the mathematics classroom and how such a culture can be supported.

Programming is being included in many educational policies, also in Norway. A study involving first-year pre-service teachers and year four students is undertaken to address the increased emphasis on programming. The focus is on links between ScratchJr functions and students’ mathematical explanations and justifications. The results indicate that some functions in ScratchJr have the potential to foster such mathematical argumentation, but it requires appropriate mathematical tasks and teacher awareness about how to support the students’ work.

Preservice teachers’ evaluations on school students’ mathematical argumentation are rarely the focus of mathematics education research. Yet, understanding how they evaluate the quality of their future students’ mathematical argumentation is important. In this paper, a discussion by a group of multilingual preservice teachers is analysed to determine the properties they considered to be connected to high-quality mathematical argumentation. The preservice teachers, who had two different dominant languages, discussed whether examples of Grade Four students’ work, written in those two languages, displayed the qualities of being clearly written, mathematically correct and complete. From an analysis of this discussion, we identify how the preservice teachers’ multilingual backgrounds provided potential and realised learning opportunities about students’ mathematical argumentation.

Keywords: Multilingual preservice teachers, mathematical argumentation, multimodal representations

In this paper, we apply the ideas of Lord (1994) about critical colleagueship to understand how
mathematics teacher educators (MTEs) can work together to become more critical in their teaching
practices. There is relatively little research on MTEs’ learning and development from a critical
perspective. Our study examines a group of MTEs working together to develop novel teaching and do research about initiating critical discussions. During two meetings, the MTEs discussed their
different perspectives after using indices such as the Body Mass Index (BMI) in teaching. Identified examples of Lord’s elements were a willingness to seek and try out promising ideas, and being open to share perspectives and ask for arguments. Such collaboration supports reflections for developing teaching and research.

As part of a large research project about supporting preservice teachers to learn about teaching argumentation for critical mathematics education in multilingual classrooms, we outline a framework for considering the knowledge, skills and practices that we, as teacher educators, consider mathematics teachers need. Our objective for describing such a framework is to provide a discussion document for teacher educators, primarily at our institution, but also for others who aim to improve their mathematics teacher education practices and want to determine theoretically how to navigate the complexity of changing our practices.

Dette kapittelet handler om tilrettelegging for elevers argumenter og agens som del av demokratisk opplæring i matematikkundervisning. Agens handler om å påvirke, og delt agens studeres ut fra elevers og lærers initiativer til samtale og hvordan det får virkning for fortsettelsen. Søkelyset rettes mot lærerens grep for at elever skal kunne og ville ta agens, mot å identifisere åpninger som skapes for å ta agens, og hvordan lærer møter elevens evne til å ta agens. Vi undersøker hvordan argumenter utvikles ut fra lærers og elevers initiativ gjennom Toulmins argumentstruktur. Empirien er hentet fra et prosjekt der læreren la til rette for elevers arbeid med problemstillinger knyttet til trafikksikkerhet på elevenes skolevei. Vi ser på lærerens ytringer og grep i en plenumsdiskusjon der elevpresentasjoner blir diskutert. Det blir identifisert hvordan demokratisk opplæring i matematikk kan knyttes til lokalt engasjement, deltakelse i å utvikle argumenter og ved å erfare matematikk som del av det levde liv.

In this paper, we investigate how preservice teachers express their awareness about language diversity in mathematics classrooms. We use “language as resource” and “language as a problem” as theoretical constructs. The data are from PT´s group assignments, including transcriptions of dialogues between the PTs and multilingual students, and from PTs´ written reflections about teaching in multilingual classrooms. We found that in the dialogues, PTs and multilingual students produced meanings together through negotiating a shared repertoire from out-of-school discourse and school-mathematics discourses. Nevertheless, the students´ home languages were not evident as resources in PT´s dialogues and reflections. A deficit perspective dominates in their reflections. Next step will be to arrange a co-learners study to explore how languages as resource can be present in teacher education and schools.

There has been little attention in mathematics education research about how to include issues to do with language diversity in teacher education. This paper describes the process used by two teacher educators to examine their own practices of linking multilingual perspectives to mathematics education in their work with preservice teachers. By systematically analysing their discussion about a three-hour, mathematics-teacher-education workshop on proportional thinking, the teacher educators were able to identify a series of Discourses. They considered that these Discourses underlay their decision making about how language diversity could be raised with preservice teachers. The results highlight the complexity connected to raising language diversity issues in mathematics teacher education. For example, deciding what challenging content should be provided to preservice teachers is affected by the need to develop relationships with them as well as managing their learning. Joint reflection by the teacher educators was needed to ensure that the aim of challenging preservice teachers about how to deal with language diversity issues in mathematics classrooms could be achieved.

I denne artikkelen undersøker vi potensielle sammenhenger mellom skriving og forståelse i matematikkfaget. Materialet består av 28 elevtekster der elever fra 7. og 10. trinn løser en regneoppgave og skriver om hvordan de tenker og går frem for å løse den. Ved å analysere elevtekster har vi søkt innsikt i hva elever vektlegger når de individuelt løser en oppgave skriftlig. Artikkelen har utspring i erfaringer fra satsingen på Ungdomstrinn i utvikling, særlig med kunnskapsområdet skriving som grunnleggende ferdighet. Der viste det seg at et fåtall av matematikklærerne ser på seg selv som skrivelærere, noe som også støttes i forskningslitteraturen. Analysene av elevtekstene viser hvordan elever posisjonerer seg som skrivere i matematikk gjennom språk, forklaringsmåte og utregning, og hvordan de anvender ulike representasjoner. Avslutningsvis diskuteres forholdet mellom å forklare og å tenke i elevtekstene. Vi finner at en kombinasjon av utregning og forklarende tekst gir mer informasjon om elevens matematikkforståelse og holdning til faget enn utregningen med tall alene. Skrivingen kan dessuten utvikle fagspråk og bidra til læring i matematikk. Vi finner også at selve oppgavedesignet gir føringer for hvordan elevene løser oppgaven. I tillegg kan elevenes skriving gi læreren et bedre utgangspunkt for å veilede elevene i matematikk.