In this survey paper, we describe the state of the field of research on teaching mathematics with technology with an emphasis on the secondary school phase. We synthesize themes, questions, results and perspectives emphasized in the articles that appear in this issue alongside the relevant foundations of these ideas within the key journal articles, handbooks and conference papers. Our aim is to give an overview of the field that provides opportunities for readers to gain deeper insights into theoretical, methodological, practical and societal challenges that concern teaching mathematics with technology in its broadest sense. Although this collection of articles was developed prior to the global coronavirus pandemic, we have taken the opportunity to survey the contributing authors to provide some country perspectives on the impact the pandemic has had on mathematics teaching with technology in the period January–July 2020. We conclude the survey paper by identifying some areas for future research in this increasingly relevant topic.

Teacher education is central to the development of the professional knowledge of pre-service teachers. The main goal of this paper is to reflect on the development that the analysis (done by a group of pre-service secondary teachers) of a set of tasks, based on elements related to domains of KTMT—Knowledge for Teaching Mathematics with Technology—can bring to the knowledge of pre-service teachers of mathematics. Specifically, the goal was to investigate the following questions: (1) What are the factors that guide the pre-service teachers’ task discussion? (2) Which KTMT domains are emphasized by pre-service teachers during task discussion? The elements taken into account are the characteristics of the tasks (focus on cognitive level, structuring level and technology role), the use of representations (focus on balance and articulation of representations), and the equilibrium between experimentation (focus on digital technology affordances) and justification (focus on argumentation and proof). The methodology of this case study involves a qualitative approach. The main conclusions suggest that influences in the pre-service teachers’ discussion of tasks fell into the following categories: the potentialities of technology, the type of tasks, and the prospective teachers’ experience with a set of tasks, and analysis of some real students’ reports. With regard to KTMT, although it was possible to identify some global development, Teaching and Learning and Technology Knowledge was the domain in which stronger development took place.

Innovative methods can change the paradigm of teaching mathematics and inspire teachers to espouse new ideas and gain new experiences. The flipped classroom (FC) is currently an innovative pedagogical approach that has high potential to transform the teaching of mathematics. In the case study described in this paper, we investigated one mathematics teacher’s transformation of teaching in two mathematics classrooms through implementing interventions based on FC methods; furthermore, we identified several key points of FC design as well as challenges and opportunities afforded by teaching mathematics in FCs. The results of the study showed that the tasks posed by the teacher, the implemented discourse, teacher feedback and scaffolding, and the teaching–learning environment were changed in FCs, although the approaches used by the teacher to analyze the tasks and students’ learning were similar to those used in non-FCs, which points out the strengths of traditional teaching approaches. The study indicates that although teaching mathematics in FCs created some difficulties for teaching, well-designed FCs offered a great opportunity to promote students’ mathematical thinking and understanding. Overall, the results highlight that through FC, teachers can develop students’ mathematical potential with FCs.

The purpose of this study was to investigate preservice teachers’ (PSTs') beliefs about their intentions to integrate information and communication technologies (ICTs) in their future mathematics classrooms. The main research objective was to examine the extent to which PSTs’ beliefs predicted their intentions to integrate ICTs in their classrooms. We adopted a sequential mixed-methods design in which a survey questionnaire was the main data collection method followed by focus group and individual interviews. 147 secondary mathematics PST majors at two South African universities responded to the questionnaire. We purposefully selected eighteen of the preservice teachers for individual and focus group interviews. Quantitative results partly affirmed usefulness beliefs as the strongest predictor of attitude, which in turn was the strongest predictor of intentions to integrate mathematics teaching and learning ICTs. Although survey data showed that the influence of superiors and peers had the strongest influence on subjective norms, interview data contrastingly revealed that learners’ needs and societal expectations in the digital age pressured PSTs more to shift their professional identities. While quantitative results showed that self-efficacy beliefs had the strongest influence on PSTs’ control on ICT integration, the most recurrent self-efficacy theme from interviews was PST’s plea for training in the use of ICT tools. A wide range of ICT tools were identified as applicable to mathematics classrooms. This signified varying conceptions of which ICT tools were appropriate for mathematics teaching and learning. A recommendation is that mathematics teaching and learning ICT’s should be integrated as early as possible in initial teacher education curricula.

Dynamic representational technologies (DRTs), influenced by the seminal work of James Kaput and colleagues, have been in use in mathematics classrooms for decades. In this paper, we analyze 24 classrooms in the United States where teachers support students’ conceptual learning with technologies that support explorations of dynamic connections both within and across mathematical representations. These DRTs, built in alignment with Kaput’s principles, form part of a curricular activity system that embeds a central pedagogical routine. Yet despite the use of common DRTs, lessons, and professional development, classroom teaching practices varied widely. We characterize and analyze levels of technology use, which vary from using the technology as a static resource to taking advantage of dynamism to support students’ emerging explanations of mathematical concepts. There are important implications for further research into classroom use of DRTs and, more broadly, for curriculum developers and teacher educators.

Educators often argue that students should engage in activities such as conjecturing, proving, and deductive reasoning. The underlying principle is that learning mathematics means doing as mathematicians do. However, “mathematician” implicitly refers to a pure mathematician at university. The aim of our paper is to critically question the logic model underpinning these premises in order to suggest which aspects of mathematicians’ practice could be salutary for students in schools and university and which are not. We argue that aligning learning with these practices might not meet the broader educational goals of pragmatism and vocationalism. We show that activities attributed to pure mathematicians are largely ignored by biologists, engineers, and physicists and in workplace settings. In contrast the practices of professional modellers are highly valued. We argue that such practices are desirable for learning to use and apply mathematics. Next, we illustrate the suitability of practices from studies of professional modellers and applied mathematicians for classroom learning using empirical data. We conclude that the interpretation of mathematicians’ practice to be emulated must be broadened to include professional modellers’ practices to better serve meeting educational goals for more students.

The introduction of digital technologies into mathematics classrooms requires an adaptation by teachers. To develop an understanding of technology integration in classrooms, this research focused on the use of dynamic geometry by a mathematics teacher who was a novice in technology-use, in the context of Turkish secondary school mathematics. Professional instrumental genesis and hiccup constructs were chosen as theoretical constructs for the study. Following these constructs, the classroom practices of the teacher were analysed to shed light on the character of the professional instrumentation and the process through which it developed, by highlighting the lesson hiccups. This case study analysis illustrated a number of potential problems and ways in which the teacher adapted her classroom practices, and provided some indications of the growth of teaching schemes in the course of appropriating dynamic geometry.

As technologies that put the body at the center of mathematics learning enter formal and informal learning spaces, we still know little about the teaching methods educators can use to support students’ learning with these specialized systems. Drawing on ethnomethodology and conversation analysis (EMCA) and the Co-Operative Action framework, we present three multimodal ways that educators can be responsive to learners’ embodied ideas and help them transform sensorimotor patterns into mathematically significant perceptions. These techniques include (1) encouraging learners to use gesture to express and reflect on their ideas, (2) presenting multimodal candidate understandings to check comprehension of learners’ embodied ideas, and (3) co-constructing multimodally expressed embodied ideas with learners. We demonstrate how these techniques create opportunities for learning and discuss implications for a multimodal, embodied practice of responsive teaching.

Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for a measure of collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate.