Proceedings of the 14th International Congress on Mathematical Education

The following sections are included:

• Editor’s Notes
• Table of Contents

Embodied design is a proactive educational research program that promotes and investigates humans’ universal capacity to understand STEM concepts. The program’s empirical work is centered on design-based research projects that contribute theory to the Learning Sciences through the practice of building, implementing, and evaluating experimental pedagogical architectures that inform instructional practice. Using both historical and emerging technologies, embodied-design activities are typically two-stepped: (1) draw on students’ evolutionary inclination for purposeful sensorimotor engagement with the natural environment; and only then (2) introduce heritage symbolic artifacts that students initially adopt to enhance the enactment, evaluation, or explanation of their intuitive judgments and actions, yet, in so doing, find themselves adopting normative disciplinary forms, language, representations, and solution procedures. Embodied-design researchers apply mixed methods — from ethnomethodological conversation analysis through to multimodal learning analytics and cross-Recurrent Quantification Analysis — in analyzing empirical data of learning process, including records of students’ motor actions, sensory behavior, and multimodal utterance in conversation with peers and instructors. Several decades of projects across numerous mathematical content domains have increasingly implicated perception — a hypothetical Psychology construct believed to govern sensorimotor and cognitive behavior — as pivotal in explaining students’ capacity to first solve challenging motor-control coordination problems and then bridge through to discursive articulation of their movement strategy. As they attempt to operate the educational technology according to an unknown interaction regimen, new information patterns, e.g., an imaginary line connecting their hands, come forth spontaneously into students’ perceptual experience as their cognitive means of managing the enactment of the activity’s targeted movement forms. These emergent, proto-mathematical, multimodal, dynamical ontologies are then languaged and entified into consciousness, grounding the meaning of conceptual terminology and procedural routines. The embodied-design framework has been applied in building technologies for students of intersectional diversity, including populations of minoritized epistemic — linguistic practices and atypical neural, cognitive, and sensorial capacity.

Mathematics is one of the oldest disciplines in the world. Bishop (1991) expressed its value regarding human relationships and social institutions as Openness — that is, mathematical constructs such as propositions and ideas are open to human deliberation. Even before such systematization, many problems were solved and simultaneously created since earliest civilizations. This effort became the foundation for further endeavors.

What is the “Problem” in problem-solving? It has various types. Especially, the open-ended approach (Shimada, 1977) has been developed in Japan as a method to evaluate and develop mathematical thinking. Furthermore, problem posing can be an extension of problem solving. While posing various problems we may notice the patterns among those problem variations. In this sense, problem posing itself can be a problem. What is “Solving” in the problem-solving? It is dependent on the type and characteristic of problem. For example, the open-ended problems provide more than one solution. Socially open-ended problems provide solutions together with values. Problem posing requires developing problems and such development itself can be a solution. Therefore, importantly, the meaning of solving a problem is extended beyond traditional problem solving.

This paper explores the idea of problem-solving in mathematics to appreciate the value of openness under the Open Science movement (OSF, 2021). Open science is a movement accommodating experts and non-experts to have access to the outputs of scientific research and can participate in the research activities. This is essential for future citizens and is related to the ethical dimension of mathematics education (Ernest, 2012).

This lecture offers a reflection on the challenge posed by the current trend of curricula and standards to recommend starting the learning of proof from the very beginning of the compulsory school. This trend pushes on the fore the notion of argumentation, it is here discussed as well as its relations to proof as a convincing and an explaining legitimate means to support the truth of a statement in the mathematics classroom. Eventually, a didactical concept of mathematical argumentation is discussed and elements of its characterization are proposed.

I examine the question of why language diversity matters in mathematics education, offering four responses, illustrated with examples drawn from my research. The four responses look at the nature of language diversity, its role in learning and teaching mathematics, its connection with social stratification, and its connection with the ecological crises faced by our planet.

This session article unpacks mathematics teaching and learning focused on racial equity and social justice. Specifically, the session will explore the intersection of mathematics teaching and learning with racial equity and social justice across four critical reasons: a) Building an informed society; b) Connecting mathematics to cultural and community histories as valuable resources; c) Confronting and solving real-world mathematics as a tool to confront inequitable and unjust contexts; d) Use mathematics as a tool for democracy and creating a more just society.

Challenging tasks are essential in developing and demonstrating mathematical understanding. They provide opportunities to learn and the motivation for student to engage with learning. This chapter highlights how the real-world and digital technologies provide many opportunities to design and implement challenging tasks for all learners. The affordances of technology-rich teaching and learning environments need more attention if teachers and their students are to be better enabled to maximise opportunities for learning mathematics. A range of tasks are presented and discussed. Planning by teachers for varied student responses is critical in enabling ‘as needed’ in-the-moment scaffolding to keep students engaged with mathematical thinking.

Curriculum reform is a fundamental factor in pushing forward educational development and reform. The aim of this study is to present the current situation of mathematics curriculum reform for compulsory education in Chinese mainland since 2000. In this study, we examine the development and implementation of Chinese mathematics curriculum standards for compulsory education. Based on mathematics curriculum standards, this study introduces the reform of mathematics textbooks, classroom instruction and mathematical achievement assessment.

It is a great challenge for the teachers to practice differentiated instruction in the heterogeneous mathematics classroom because there is a great demand for a valid and reliable diagnostic assessment. To address this issue, this study sought to develop and validate an online Cognitive Diagnostic Assessment (CDA) with Ordered Multiple-Choice (OMC) items for Grade Four Topic of Time. However, this paper only focuses on the results of six cognitive models for conversion between time units. Each cognitive model was measured by an assessment comprising seven OMC items. The quality of the online CDA with OMC items was assured with robust psychometric properties, convincing reliability, and satisfactory model-data fit. Perhaps this instrument could support the teachers in diagnosing pupils’ cognitive strengths and weaknesses, followed by practicing differentiated instruction in the mathematics classroom.

The present lecture engages a speculative reading of “The Masters”, a science-fiction novel written by Ursula K. Le Guin to narrate a state where citizens are governed by the law of negating mathematics education. In this oppressive context, Le Guin crafts a collective whose desire to practice mathematics subverts the fear for death used as punishment for mathematical heresy. This allows to ponder into thinking as “negation” and “affirmation” and, consequently, to speculate two interweaved assemblages of mathematics education; first, the assemblage where negating mathematics enforces masculine knowledge enclosures and second, the assemblage of affirming the practice of mathematics as knowledge commons. The chapter contributes by rethinking of mathematics education as/for/with the commons and by discussing about speculation as an act of thinking.

Digital technologies have been evident in the field of mathematics education since the late 1970s, a time of great optimism and enthusiasm for how emerging technologies would impact on school mathematics as a subject — and how mathematics would be taught and learned. Some fifty years on, whilst the pace of educational technology design accelerates, the parallel global transformation of school mathematics curricular — and associated high stakes assessment systems — lag noticeably behind. Within the mathematics education research field, there is general agreement about the barriers to systemic change: an underestimation of the professional needs of the teaching workforce; insufficient and inequitable access to suitable technologies; an unrealistic or ill-defined vision for students’ digitally-enhanced mathematics learning experiences; challenges in the design and “at-scale” uses of (mathematical) technologies in classrooms and its role within high-stakes assessments (Clark-Wilson, Robutti, and Thomas, 2020; Hoyles, 2018). The (re)emergence of computer programming, which was commonplace in UK mathematics classrooms of the 1980s has prompted some rethinking but, to date there are no widely accepted definitions of what a student’s school mathematics educational experience in the digital age should comprise. The coronavirus pandemic prompted a global upskilling of students’, parents’ and teachers’ digital skills within all phases of education and put technology, in its most general sense, on the map. In this invited lecture, I will offer a vision for how students’ experiences of learning school mathematics with and through (mathematical) technologies might be reconceived. Alongside this, how the parallel assessment processes might be designed to enable a more student-centric approach that takes account of multiple sources of evidence. Most crucial to this is the role of teachers, whose expertise is more vital than ever as they support students to actively engage with substantive dynamic mathematical tools that make core mathematical ideas more tangible. The lecture concludes by highlighting how a deeper understanding of the theoretical construct of the “hiccup” (Clark-Wilson, 2010; Clark-Wilson and Noss, 2015) might underpin wider understanding of the process of teachers’ classroom-based learning concerning the adoption of mathematical technologies towards this vision.

In this paper, I will focus on how teachers can use typical problems to develop both conceptual fluency as well as procedural skills. These typical problems will be considered as mathematical tasks that teachers can opt to select, reformulate or create new ones for use in class. Teachers may find it easier to make sense of the mathematics and pedagogical considerations in the implementation of these tasks, as they are readily available in textbooks and past examination papers as compared to rich tasks. I will demonstrate that typical problems do have affordances for developing conceptual fluency and in that sense are equally good, if not better, better than the so-called rich tasks. With limited time at their disposal, teachers have to be strategic in noticing the affordances of typical problems and in optimally using their available time for selecting and using relevant tasks for their lessons. As such, the gist of the paper, connecting to my previous work, will be on how teachers can use typical problems in their day-to-day practice to enhance the learning of their students, against a backdrop of teacher noticing.

In this article, I use the term “textbook transformation” to refer to the development of a textbook or a series of textbooks based on another selected pre-existing textbook(s), which leads to the formation of a new textbook(s). By mainly drawing on my own mathematics textbook research and development experiences, particularly in transforming a popular Chinese mathematics learning resources series, One Lesson One Exercise, or Yi Ke Yi Lian in Chinese, to the English learning resource series the Shanghai Mathematics Project Practice Books and developing the Zhejiang Secondary Mathematics Project Textbooks over the last two decades, I argue that the means of textbook transformation can be classified into five types: translation, adaptation, revision, rewriting and a combination of them, based on the selected pre-existing textbook(s). Following this classification, the article analyzes and discusses the approaches, issues and challenges in textbook transformation by using concrete examples from available textbooks, and illustrates in particular how social and cultural factors play an essential role in textbook transformation and its significance in international exchange and collaboration in the development of school mathematics textbooks.

This paper discusses the critical role of a learning trajectory in teaching mathematics using realistic mathematics education (RME) approach. The first part will present a brief history of RME and how RME was adopted into the Indonesian context. It is followed by a discussion of some of the principles and characteristics of RME. Furthermore, it is explained the learning trajectory, hypothetical learning trajectory (HLT), and how the principles and characteristics of RME are integrated inti HLT in learning mathematics. In the next section, the development of HLT through design research is discussed, and in the last section, some examples of HLT are given and their impact on students’ mathematical abilities.

“Art and Mathematics” has been considered in mathematics education primarily for the possibility of teaching and learning mathematics through art. Many reasons are implied in this: to give meaning to mathematics; to motivate or contextualise teaching; to broaden mathematical visualization, among others. However, neither colonizing art by mathematics nor instrumentalizing mathematics by art, we have been considering this pair for the experimentations that can happen in the exercise of thinking. Taking this into account, in this presentation, first, I introduce the idea of visuality in differentiation with the concept of visualization in mathematics education to point out some theoretical concepts of the research. Then, I present some research works I have been developing, especially those that have been effects of the production of a methodological stance that occurs at the interface between paintings, visuality and mathematics education. Finally, I draw some conclusions, outlining an ethical, aesthetic, and political stance for teaching mathematics with arts.

This paper purposed to examine the process employed by two lower secondary school teachers in designing a lesson for students through discussion with three university researchers. The teachers conducted practical research at a professional development program in mathematics and collaboratively designed a lesson for Grade 7 on the geometrical transformation of figures. The results revealed the incorporation of considerable changes to the lesson objectives and development, indicating the emergence of different perspectives. The paper employs the study results to discuss the significance of designing lessons that enhance student problem-solving to the professional learning for teachers.

This chapter aims to provide a holistic portrayal of the features of Chinese lesson study (LS), the mechanisms of Chinese LS, and its recent development. Recommendations for further improvement of Chinese LS are provided and implications of Chinese LS on the practice of LS internationally are discussed.

The diffusion of technology in the teaching and learning is more complex than other fields. In order to understand the complexity of the factors and processes affecting teachers’ integration of technology, in mathematics education in particular, we need to use many complementary lenses. For that end, in this study, we have used three theories: the technological pedagogical content knowledge (TPACK), innovation diffusion theory (IDT), and the zone theory. TPACK describes the types of knowledge that teachers need to integrate technology effectively in their teaching practices. IDT describes the developmental processes that individuals go through as they adopt/reject a technological innovation. While the zone theory identifies the limiting and assisting factors teachers face when they decide to integrate technology in their teaching. The result of this study was a new framework named the Ladder and Slider framework to introduce the three theories together using the networking theory. The purpose of the Ladder and Slide framework is to visualize easily the complexity of technology integration, consequently that will influence a better design of professional development. A pilot phase done with four in-service secondary mathematics teachers using GeoGebra in their teaching is presented with the new framework followed by some conclusions and recommendations.

We will explore the role of history as a resource through which students can gain experience with authentic mathematical modeling in scientific contexts i.e., when mathematical modeling is used as a research tool, a practice, to gain knowledge in other areas. Three modeling episodes from the 20^th century will be presented and analyzed with respect to modeling strategies, practices, items used in the modeling construction and cross-disciplinary epistemic issues — and an analytical framework for analyzing modeling episodes in scientific context will be presented. The framework will be discussed with respect to the modeling cycle in mathematics education, highlighting issues in the framework which are not featured explicitly in the modeling cycle. It will be illustrated how and in what sense history makes it possible to invite students into the work place of scientists that used and experimented with mathematical modeling as a research practice, i.e. its significance in creating such teaching and learning environments. Finally, the value of developing students’ historical awareness for preparing them for tertiary studies where mathematical modeling might play a role will be discussed.

This article deals with maths education in the middle and high school in Kharkiv City (Ukraine) and in Academic Gymnasium No. 45 in particular. It shows the whole structure of education and the ways of motivation for learning maths at the high level by students. It also shows the obvious success of the strategy of complex maths teaching and analyzes its positive results for the last 25 years.

Enabling students to achieve a deep and connected understanding of mathematical concepts is an important aim in Singapore mathematics education. While current forms of instruction in the mathematics classroom can engender detailed expositions of a concept and links between targeted concepts and earlier concepts, much of this information is structured by the teacher and neglects the role of students’ perspectives of the information that is transmitted to them. With the demonstrated efficacy of constructivist learning designs that build upon students’ prior knowledge structures, one of such designs was implemented in Singapore’s mathematics classrooms to not only afford deeper learning but also transform mathematics teaching and practice. In this paper, this constructivist learning design that was introduced to Singapore’s secondary mathematics classroom is described and its rationale, efficacy, and the measures that were taken to ensure its sustainability discussed. The paper concludes with reflections of how to sustain such constructivist designs beyond research, and suggestions on proliferating their use among the Singapore teaching fraternity.

Fostering student agency means developing students’ willingness and ability to engage in their own learning. This chapter presents the views of 21 expert elementary school mathematics teachers in Shanghai on fostering student agency. Interview data show that all of the expert teachers value the importance of getting students take ownership of learning and they believe that teachers can contribute significantly in developing student agency. When described the essential features of a classroom where students exercise agency, in addition to focusing on creating an environment that supports student to take up space and actively engage in learning, the expert teachers in Shanghai placed special emphasis on achieving satisfactory learning outcomes. When sharing their strategies for fostering student agency, they commonly mentioned the importance of teachers as role models for their students, which has been less addressed in the literature.

Based on the Flanders Interaction Analysis System (FIAS) and the Information Technology-Based Interactive Analysis Coding System (ITIAS), nine high school math lessons from the National and Local Public Service Platform for Educational Resources were selected as the research objects and were analyzed to investigate the characteristics of teacher-student interaction in mathematics classroom of Chinese senior high schools in the information technology environment.

There have been reviews or meta-analysis showing that using manipulatives is an effective intervention for learning mathematics for students with disabilities, including autism spectrum disorders (ASD), without concentrating on the effects on generalization and maintenance. We conducted a meta-analysis to evaluate the effect of manipulatives on generalizing and/or maintaining mathematical skills for individuals with ASD and whether the effect varies with different participant characteristics, study design, intervention characteristics and mathematical content, focusing on the single-case studies. After application of the What Works Clearinghouse design standards, a total of 11 studies were included in the review: three studies collected data points during generalization phases, five studies collected data points during maintenance phases, the other three studies collected both generalization and maintenance data. Aggregate Tau-U and non-overlap of all pairs effect sizes (NAP) were calculated for each study and conducted moderator analyses. Overall, effect size scores ranged from small to significant effects across all comparisons. On average, most comparisons from the baseline to generalization and maintenance produced medium to large effects. Whereas, minor effects were found in most of the intervention of generalization and maintenance comparisons. Further moderator analysis regard to generalization and maintenance revealed that out of seven variables analyzed, only manipulatives types served as a moderator for maintenance. The findings suggest that manipulatives interventions were likely to result in mixed effects on mathematical skill generalization and maintenance within children with ASD, especially virtual manipulatives. Limitations and implications for future research and practice are discussed.

It is believed that a knowledge of the history of mathematics could improve or expand an individual’s understanding of the nature of mathematics, and hence may challenge teachers’ epistemological beliefs of mathematics and, as a result, cause teachers to reconstruct their beliefs. Wilder reminds us that mathematics is a part of, and is influenced by, the culture in which it is found. As such, the culture dominates its elements, and in particular its mathematics. For instance, a Chinese mathematician living about the year 1200 C.E. would have mainly focused on computing with numbers and solving equations without paying attention to geometry as the ancient Greeks understood it. In contrast, a Greek mathematician of 200 B.C.E. would have focused more on geometrical proofs than on algebra and numerical computation as the Chinese practiced it. This paper aims to question the conventional view that treats mathematics as a significant instrument for developing one’s personal career, instead advocating that we should regard mathematics as a cultural discipline of human endeavor in our teaching. I will interpret the history of mathematics in terms of a sociological macro-view and investigate the rise and fall of mathematics in the European and Chinese cultures to shed more light on the intellectual value of mathematics in education.

E-learning has become popular these years, and the advantages of flipping the classroom are also widely depicted in literature. However, the widely used element in video instruction, overlaying of a small video of the instructor over lecture slides, is understudied. A new technology called Learning Glass, which can be used for recording lectures and allowing instructors to write lecture notes while maintaining face-to-face contact with students, was used to record instructional videos. The effect of the presence of instructors in instructional videos for university students in two metropolitan universities (Los Angeles and Hong Kong) was studied. Participants were randomly assigned to watch a video with and without the presence of the instructor. The extent to which the participants have grasped the video materials was assessed via pre- and post-tests. Participants’ satisfaction towards the video was also evaluated via a survey near the end of the experiment. The effect of the instructor’s presence and where the participants come from was studied. It was found that the instructor’s presence did not impose a statistically significant difference towards participants’ acquisition of the video contents. One possible reason is that individual learning preference is more important than instructing all learners with one approach. It was, on the other hand, found that participants from Los Angeles were more willing to recommend videos to the others and to watch more for learning. This may be related to the fact that e-learning is more popular in Los Angeles. Results of this study may help us recognize the implication of the presence of the instructor in videos as well as providing a better learning environment in the future.

This contribution analyzes the origin of the competence construct, its evolution and how it is conceptualized by different authors in different fields. The objective is to reveal the complexity of the idea that the construct is meant to capture; in fact, only by bringing out this complexity can we hope to make the construct truly operational and useful for practice and educational research. In particular, I discuss the multidimensional artefact-like character of the construct of competence trying to reveal the several distinct related dimensions which contribute to form this single theoretical concept.

The need to improve teachers’ preparation to teach mathematics is shared by many countries. E-learning professional development (PD) programs appear as an attractive option due to their flexibility and availability. Suma y Sigue is an e-learning PD program for Chilean teachers that focuses on the development of Mathematical Knowledge for Teaching (MKT). The program is characterized based on a constructivist perspective of learning by using a contextualized problem-based approach. This article describes the instructional design of the program learning activities that demonstrate how mathematical tasks centered on the construction of MKT are articulated and implemented. The learning performance of the participants in a specific course within the program is analyzed. The findings show empirical evidence of improvement in teachers’ knowledge. The detailed description of the course and participants’ performance can aid PD developers to design principles and the use of different instructional strategies, especially when the course focuses on MKT development.

Attitudes towards mathematics has a long history in mathematics education research. Over the time, research on attitudes and, more in general, on affective aspects developed a wide range of methodologies and perspectives in mathematics education, playing a growing role in the field. In this chapter, I will describe the development of the research about attitude in mathematics education, discussing the main issues emerged in this field. In particular, I will discuss the definition problem, that is the emergence of the need for a clear definition of the construct, and the ground for the development of our (TMA) three-dimensional model of attitude (Di Martino and Zan, 2010). In the last part of the chapter, some fields of application of the TMA model will also be discussed.

In my work, I seek to understand how interactions between instructors, students, and resources — both inside and outside of the classroom, create opportunities for mathematics learning in post-secondary settings. Various methodological decisions have advanced this work. I showcase the evolution of two inter-dependent research strands that together have helped me understand the centrality of resource use by instructors and students and its implications for student learning.

Studies have indicated that the development of Mathematical Knowledge for Teaching (MKT), is rooted in teaching experience occasioned in teachers’ daily work. To determine the role of teaching experience in the development of MKT, a special tool was required to capture all the MKT tenets and their combinations for analysis of mathematics teacher’s proficiency. In this article the effectiveness of a tool developed purposely to examine the relationship between years of teaching experience and the development of Mathematical Knowledge for Teaching (MKT) is shared. This article has been drawn from a larger study on MKT proficiency status carried out in Kenya involving 117 trained secondary school mathematics teachers with varying years of teaching experience and academic backgrounds. Both descriptive and inferential statistics were found to be interpreted accurately using this tool. Using this tool, this study found a very weak positive relationship (β = 0.171) between teaching experience and MKT proficiency. The study established that MKT proficiency is not progressive, it is non directional and can regress in spite of teaching experience. From this finding it is my proposition that this pedagogical tool can sufficiently be used to discuss exhaustively teachers’ MKT proficiency.

This invited lecture summarized my work on language and learning mathematics. I described a theoretical framework for academic literacy in mathematics (Moschkovich, 2015a, 2015b) that can be used to analyze student contributions and design lessons. The presentation included a classroom example and recommendations for instruction that integrates attention to language. Although the example is from a bilingual classroom, the theoretical framing and the recommendations are relevant to all mathematics learners, including monolingual students learning to communicate mathematically.

For decades, reformers have emphasized discussion over recitation and lecture. Yet, traditional communication patterns are still dominant in mathematics classrooms internationally. In an effort to better understand this challenge, the present study investigates patterns and contributions of research on discussion in mathematics teaching. Based on systematic search in the Eric database, and in selected journals of mathematics education, 72 studies were reviewed. Based on analysis and discussion of the reviewed studies, it is suggested to develop conceptual clarity and include definitions of core terms like discussion, to consider alternative methods for studying discussion in teaching, and to consider shifting the focus from teacher actions to the entailments of the work of leading mathematical discussions.

Language as resource is a challenging research approach in mathematics education because it examines how language can function to support mathematics learning and teaching. The approach originally started to develop in response to discourses of non-mainstream languages and cultures as problems or obstacles to mathematics teaching and learning. In this text, I revisit and bring together four empirical studies in order to discuss four major findings that are arguments to explain the complexity and importance of the language as resource approach. These four arguments are: 1) the huge potential of all languages to make mathematical meaning; 2) the critical realization of some languages in the mathematics classroom; 3) the critical communication of mathematical meaning in classroom teaching talk; and 4) the huge potential of teaching talk to support mathematics learning for understanding.

The didactic perspective on mathematics and language focuses on topic-specific instructional approaches for integrating language learning opportunities into mathematics instruction. From a didactic perspective, a sound and research-based specification of language demands is crucial for providing well-focused learning opportunities. For this, the paper (1) presents the topic-specific specification grid as a useful practical tool for specifying mathematically relevant language demands and (2) explains its underlying theoretical framework by making explicit the four incorporated lenses: epistemic, conceptual, functional, and discursive. The theoretical framework for specifying topic-specific language demands combines various linguistic theory elements and is empirically grounded in findings on typical language demands while mathematics learning.

I focus here on several aspects of cultures related to digital technologies and mathematics education. One first aspect is that any integration of digital technologies for mathematical (or other) teaching and learning creates and transforms the classroom culture. On the other hand, in order for learning to be meaningful through the use of digital technologies, these may need to be embedded in a certain “culture” that empowers students to engage pro-actively with those technologies. I present different types of teaching and classroom “cultures” that have been found when using digital technologies, how these impact mathematical learning, as well as different conditions and teacher-training opportunities for the use of digital technologies found in different countries, illustrating all of these with examples from my own experience and from literature. I discuss how the different conditions and access opportunities in different regions and cultures create digital gaps. Finally, I discuss what could be done to support teachers to create meaningful contexts and classroom cultures when integrating digital technologies within established school systems (but at the same time transforming these), so that these can empower learners (e.g., to “do mathematics”) and promote the construction of knowledge.

Teacher professional development is important in order for teachers to effectively address changing contextual realities. Effective professional development builds on teachers’ experience and relates to their practice. The paper presents guiding ideas and lessons learnt from teacher development component of a research project that aimed at improving numeracy performance of pupils by focusing on teachers’ assessment practices. Based on conclusions, recommendations are made for possible approaches to future PD especially in similar contexts.

We refer to four general theories of argumentation that provide insights on innovative current approaches in mathematics education. Through several examples of tasks, we show the richness of argumentative practices in the learning and teaching of mathematics that have some bonds with these general theories of argumentation. We show, however, that these theories do not capture the specific processes and the complexities of argumentation in the learning and teaching of mathematics according to innovative pedagogies. We pledge for new advances in mathematics education based on design-based research that fosters deliberative, epistemological, rhetorical, and structural aspects of argumentation.

The article focuses two areas related to learning opportunities of future mathematics teachers within their university studies, namely the elementary mathematics from a higher standpoint and practical activities as part of university studies. Thereby the article refers to several comparative studies on mathematics teachers’ professional competence which are shortly summarized in the beginning. Afterwards along with conceptual considerations about elementary mathematics from a higher standpoint examples for the integration of the concept in studies on teachers’ professional competence as well as practical experiences from university courses are described. Subsequently empirical results from a study evaluating the professional development of future teachers in longer practical activities are depicted.

In this talk, I will introduce mathematics as a code for interdisciplinary dialogues through a story on infinity.

A variety of tangible manipulatives and digital environments are commonly used in mathematics education. Instead of comparing and opposing the two types of artefacts, we propose to study their combination, with a simplified model of just two artefacts. We define duo of artefacts as a specific combination of complementarities, redundancies and antagonisms between a tangible artefact and a digital one in a didactic situation (Soury-Lavergne, 2021). A duo is designed to provoke a joint instrumental genesis regarding both artefacts, and to control some of the schemes and mathematical conceptualizations developed by pupils during its use. Learning is described in terms of evolution of conceptions in the sense of Balacheff (2013). This lecture illustrates the model with two examples of duo of artefacts for primary school, one in arithmetic and one in geometry. We argue that in addition to be a research tool, duos of artefacts are also a way to support the integration of technology into teachers’ practices.

This lecture is about the intertwining of digital and tangible artefacts when manipulatives are introduced in the situations with technologies. My work has its origins in a collaboration with my esteemed Italian colleague, Michela Maschietto, who is in charge of the laboratory of mathematic machines in Modena. In 2010, I was working on technologies, especially dynamic geometry, and she introduced me to some mechanical machines for problem solving in geometry and arithmetic. We began to work as a duo of persons before elaborating together the idea of duo of artefacts. My research question is about how to design and to provide students and teachers with digital technologies and didactical situations using these technologies, that would generate meaningful uses regarding the learning of mathematics and also that could be appropriated by the teachers and be integrated into their practices. Since the eighties and the emergence of personal computers in education, the problem has not been solved. The idea of “duo of artefacts” is a proposal to tackle the question.

This paper present first some assumptions that ground my thinking about technology and the learning of mathematics, which explain how I came to the idea of focusing on the articulation of tangible and digital technologies. Then, I will present in detail the combination of digital and tangible artifacts that constitute a duo of artifacts, which is a model to study systems of instruments. I will illustrate it by two examples, one in geometry, and one in arithmetic. The conclusion raises the main characteristics of the model of duo of artefacts that may be used both for research purposes as well as for providing teachers and students enhanced learning environments. The whole lecture is available on video at the following url: https://videos.univ-grenoble-alpes.fr/video/20249-icme14-invited-lecture-ssoury-lavergne/

In this article, we elaborate on the following goals that we have for developing caring and socio-politically aware beginning teachers of mathematics and the strategies that we use to reach them: 1) Understand what it means to achieve equity, access, and empowerment in a mathematics classroom; 2) Develop equitable pedagogical strategies, 3) Examine and overcome barriers related to student engagement and achievement, 4) Confront negative beliefs about students from different race/ethnicity, socio-economic status, gender, ability; and sociolinguistics groups and move forward in a positive manner, 5) Develop an advocacy stance.

A great mathematical education should build mathematical virtues, not just mathematical skills. Virtues are what make mathematical experiences enriching and they serve one well no matter what one does in life. They enable human beings to flourish.

In this chapter I present some theoretical and methodological approaches on language in mathematics education. Language can be mainly viewed as the means to represent mathematical meanings or as constructing mathematical meanings itself. These approaches in turn lead to different methodologies, which in turn lead to different types of results. I argue that researchers should be cautious before adopting a particular theoretical framework, since sometimes frameworks are based on neighbouring concepts. I provide examples of such concepts, namely positioning and norm. The complexity of language and interaction in mathematics classrooms calls for more holistic and less dichotomised approaches. The combination of approaches is also an effective way to conduct research on language in mathematics education; examples of such combinations are provided.

According to UNESCO (2015), the equity gap in education is exacerbated by the shortage and uneven distribution of professionally trained teachers, especially in disadvantaged areas. Target 4.c (MOI) of the SDG 4 is therefore aimed at substantially increasing the supply of qualified teachers, including through international cooperation for teacher training in developing countries, especially least developed countries and Small Island developing States by 2030. It further states that teachers are one of the fundamental conditions for guaranteeing quality education and therefore there is need to empower and adequately recruit, remunerate and motivate professionally qualified teachers and educators, and support them within a well-resourced, efficient and effectively governed systems (UNESCO, 2015). The knowledge of teachers in the last three decades was mainly influenced by a well-known scholar Lee Shulman who categorized teacher knowledge into seven categories among which content knowledge is included. However, much research on in-service teachers focused on the pedagogical content knowledge hypothesizing the mastery of content as much as they are graduated from recognized training institutions. Based on this categorization, the present paper presents an analysis of Rwandan mathematics school subject leaders’ Content Knowledge (CK). The presentation is based on partnership established between governmental and nongovernmental institutions led to development and implementation of certified Continuous Professional Development (CPD) programs for primary and secondary mathematics teachers. Findings reveal a lack of teachers’ preparedness to adopt the new curriculum teaching approaches, there is also lack of appropriate physical facilities in schools to accommodate every leaner’s individual needs among other hindering factors. Recommendations include systematic CPD programs for in service teachers to complement preservice training so that they can adapt various reforms and in-service teachers to establish their individual professional development plans.

In this paper, I detail the ways in which a South African initiative focused on mental mathematics in the early grades (the Mental Starters Assessment Project — MSAP) can be considered as an intervention aligned with the idea of ambitious instructional practice. In building this argument, I take note of the fact that the materials associated with the MSAP initiative are relatively prescriptive in their format, a feature that has sometimes been argued to work against the goals of ambitious instructional practice. The reasons for considering the MSAP an ambitious instructional practice initiative is linked in the paper with the attention given in the materials to working across the strands of mathematical proficiency, with local conditions and cultures driving the relatively prescriptive format of materials provision.

Dominant discourses in teacher development often posit teachers as being lacking in knowledge, beliefs, or skills, thus justifying the “need” for further development and for educational reforms. This perspective shaped the analysis of Filipino teachers’ explanations of fraction concepts using the constructs of content knowledge and pedagogical content knowledge, leading to an interpretation that reinforced deficit narratives about teachers. However, there are increasing contestations of these deficit research narratives (Adiredja, 2019) that neither acknowledge the larger context that contributes to the ways teachers perform nor highlight the productive resources that teachers may draw upon in their teaching. This paper aims to illustrate a reconceptualization of the research away from focusing on what teachers lack towards identifying the ways by which teachers’ fractional explanations reflect their constructed perception of ideal mathematics teaching as shaped by the broader system where education takes place. This is my attempt to acknowledge my own participation in the deficit perspective and challenge the narrative about education in a developing country.

This lecture presents a didactic proposal that combines mathematical modeling and digital technologies in the framework of a teacher education program for future mathematics teachers in a public university of Argentina. After presenting the theoretical assumptions underpinning this proposal, the characteristics of the program and an annual mathematics education course that forms part of its curriculum are described. This course covers topics related to mathematical modeling and the use of digital technologies, among others. Details are given of the characteristics of the modeling scenario created within the framework of this course, for preservice teachers to experience the development of open modeling projects. A synthesis of the modeling experiences developed in 2020 during the COVID-19 pandemic is shown. These experiences were carried out in groups of preservice teachers, allowing them to choose freely a real-life topic of their own interest and the use of various digital technologies. The topics chosen by each group, the role of technologies, the learnings recognized by the preservice teachers and the difficulties and limitations detected are detailed. The text concludes with some reflections on the relevance of this type of experience in teacher education.

The development of Chinese mathematics competition activities can be divided into the following stages: Stage I (1956-1964): Considered as the birth of China’s competition mathematics, Stage II (1978-1984): A working committee was set up by the Chinese Mathematical Society (CMS) in order to standardize and institutionalize the development of mathematics competition activities in China, Stage III (1985-present): This period sees a flourish of mathematics competition activities. China began to participate in the International Mathematical Olympiad (IMO) and in 1990, the 31^st IMO was successfully held in Beijing on an unprecedented scale.

After more than three decades’ exploration and practice, an ever-enriching, relatively stable system has evolved in the Chinese high school mathematical Olympiad practice.

Competition mathematics education is beneficial to the development of gifted students’ mathematical ability in various aspects.

Competition mathematics was introduced to China in 1956 (cf. Hua, 1956a, 1956b). In the same year, mathematics competitions for senior high school students were held in Beijing, Tianjin, Shanghai and Wuhan respectively. Luogeng Hua (known as Loo-keng Hua in the west) personally served as chairman of the Beijing Competition Committee and engaged in the preparation of test materials. Many famous senior mathematicians, including Luogeng Hua, Zhongsun Fu, Jiangong Chen, Buqing Su, Xuefu Duan, and Zehan Jiang also made lectures during the competitions (Sun and Hu, 1994).

The development of Chinese mathematics competition activities can be divided into the following stages (cf. Chen and Zhang, 2013):

Stage I (1956-1964): Considered as the birth of China’s competition mathematics, competitions were mainly advocated and personally directed by senior mathematicians and held in a few key cities of China.

Stage II (1978-1984): After the ten-year political turmoil came to an end, mathematics competitions were resumed. A working committee was set up by the Chinese Mathematical Society (CMS) in order to standardize and institutionalize the development of mathematics competition activities in China.

Stage III (1985-present): This period sees a flourish of mathematics competition activities. China began to participate in the International Mathematical Olympiad (IMO) and in 1990, the 31st IMO was successfully held in Beijing on an unprecedented scale. The level of mathematics competitions in China quickly caught up with the international standard and continued to maintain its leading position afterwards. Meanwhile, various kinds of competitions at all levels were launched and a wide and diverse range of learning materials was readily accessible. The competition-oriented training became the “second classroom” for a proportion of students, or even the “second school” for a few top students.

When students are asked to examine their understanding individually or in small groups, information can become part of a feedback process that supports students’ learning. As designers of technology to support learning, we are interested in supporting such feedback processes in the context of guided inquiry instruction. This paper explores the potential of automatically associating mathematical descriptions with student submissions created with interactive diagrams. The paper focuses on the feedback processes that occur when students use the descriptions provided by the technology as resources for reflection and learning. We discuss the design of personal feedback processes where students reflect on and communicate their own learning, utilizing individually-reported multi-dimensional automatic analysis of their submissions in response to example-eliciting tasks. While there is much research and development work to be done, we consider mathematical descriptions of student work as an important contribution to broader developments in learning analytics.

This paper shares several realistic mathematics education (RME) projects designed and implemented at the Department of Mathematics Education, Universitas Sriwijaya, Indonesia. These projects are developed using the design research method. They are based on the foundational work of Freudenthal and his successors. The development of the PMRI approach (the Indonesian version of RME) started at an ICMI Regional Conference in Shanghai in 1994, where Robert Sembiring met Jan de Lange. We will also briefly reflect on how RME was adapted in Indonesia, inspiring research and development in mathematics education. The focus of this paper will be on (1) designing a learning environment within the PMRI approach to support students’ learning mathematics literacy; (2) designing an international journal on mathematics education; (3) creating PISA-like tasks on mathematics using the Indonesian context; and (4) our way of surviving the current COVID-19 context. I will discuss these issues and illustrates their examples from PMRI practices.

This Appendix lists all participants who actually gave invited lectures in the ICME-14 (on-line or on-site), according to the conference video recording, and all lecturers who contributed to this volume (indicated with asterisks), together with his/her co-authors, if any…

The following section is included:

• Index of Authors