José Ricardo Mafra; Maria Cecilia Fantinato (Brazil)

Perceived techniques and processes of craftswomen in Santarém/PA.

María del Carmen Bonilla (Peru)

Tools of history of mathematics and dynamic geometry in the pre-service training in intercultural bilingual education.

Ramesh Neupane (Nepal)

Teaching and learning mathematics in a cultural context: ping as a project.

Sudhakar Agarkar (India)

Understanding the units of length measurement used by tribal people in India. André Gerstberger; Ieda Maria Giongo (Brazil)

Ethnomathematics look at mobile usage regarding teaching mathematics processes in elementary education final years.

The presentations focused on and discussed ethnomathematics as a line of study and research of mathematics education from the many diverse perspectives and points of view brought to the group from all over the world. Together, presenters are investigating the roots of mathematical ideas and practices, starting from the way individuals see and use mathematical thinking in different cultural groups. In other words, ethnomathematics studies seek to identify mathematical practices that begin with the knowledge of the others in their own terms and rationality.

Because ethnomathematics studies the cultural roots of mathematical knowledge beginning with the various ways in which different cultural groups mathematize including academic mathematics, our study of ethnomathematics considers the historical evolution of mathematical knowledge with the acknowledgment of all social and cultural factors that form this ongoing and dynamic development.

## Discussions and Reflections

The various presentations recognized how members of distinct cultural groups develop unique techniques, methods, and explanations that allow them alternative understandings, comprehensions, new actions, and a transformation of societal norms. Such historical research field has been a foundational area in ethnomathematics and continues to build a database of examples of mathematical thinking in distinct cultural groups.

It is evident from the discussion from TSG35 that the theoretical basis of an ethnomathematics program offers a valid alternative to traditional studies of history, philosophy, cognition, and pedagogical aspects of mathematics. Therefore, there is a growing sensitivity to the understanding and comprehension of mathematical ideas, procedures, and practices developed by the members of distinct cultural groups. This is due primarily to the expansion of studies related to culture, history, anthropology, linguistics and ethnomathematics.

Because ethnomathematics offers a broader view of mathematics, including its ideas, notions, procedures, processes, methods, and practices rooted in distinct and diverse cultural environments, this aspect leads to increased evidence of cognitive processes, learning capabilities, and attitudes that influence the learning processes occurring in classrooms. In addition to reflecting on social and political dimensions of ethnomathematics, another important aspect of this program is the possibility for the development of innovative approaches for a dynamic and glocalized society as outlined by D'Ambrosio.

The results of our discussions within the TSG35 show that it is important to understand the diverse sociocultural representations and concepts of ethno developed from distinct ideas, procedures, practices, and dimensions of space and time through the relationships between members of cultural groups. This aspect shows that, currently, a more sensitive understanding of diverse mathematical ideas, procedures, and practices developed by members of diverse cultural groups has become increasingly available through the growth of the fields of ethnology, culture, history, anthropology, linguistics, and ethnomathematics.

The insights gleaned from the presentations and from our discussions demonstrated the breadth and depth of how ethnomathematics influences the teaching and learning of mathematics, how it broadens our understanding of the nature of mathematical knowledge, and how it helps to create a just and inclusive society. Our discussions at the TSG 35 shows that it is necessary to pursue the current agenda of the ethnomathematics program in order to continue its progressive trajectory that contributes to the achievement of social justice and peace with dignity for all.

## Final Considerations

From the presentations and discussions conducted during the regular and oral communication sessions of TSG35, it is possible to identify three unique characteristics that are interrelated.

From the discussions provided in this topic study group it is possible to conclude that mathematical knowledge is constructed by developing ideas, procedures, and practices that are common to the members of distinct cultural groups. This mathematical knowledge enables these members to elaborate and use their abilities that include the universal processes of counting, locating, measuring, drawing, representing, playing, understanding, comprehending, explaining, and modeling to solve problems they face daily.

Ethnomathematics provides mathematics educators an important framework to enable the transformation of mathematics so that it can better contribute in realizing the dream of a just and humane society. In this regard, mathematics is considered a powerful tool to help people build a civilization with dignity for all, in which inequity, arrogance, violence, and bigotry have no place, and in which threatening life, in any form, is rejected.

Presenters in this group shared the necessity for further discussions of issues related to mathematics education, classroom practices, and valuing the mathematical knowledge developed in specific cultural groups, and which helps to clarify the nature of mathematical knowledge.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 36: Task Design, Analysis and Learning Environments Programme Summary

Jere Confrey, Jiansheng Bao, Anne Watson, Jonei Barbosa and Helmut Linneweber-Lammerskitten

## The Programme

Our TSG36 program included contributions from continents around the world: Africa, Asia, Europe, North and South America, and Oceania. Plenaries were presented by Anne Watson (UK), Koeno Gravemeijer (NED), Kazuhiko Nunokawa (JP), Berta Barquero, Ioannis Papadopoulos, Mario Barajas, and Chronis Kynigos (SP, GR) Angelika Kulberg (SWE), and Celia Hoyles and Richard Noss (UK). Twenty six papers and thirteen posters were presented. Confrey (USA) authored a closing summary. The presentations are listed below (posters omitted due to space constraints); the text references these contributions.

Topic Study Group 36's presentation began with the framework proposed by a prior ICME Topic Study Group (Watson) organized into the categories (or parameters) of Theory, Intentions, Likely Activity and Implementation. Over the course of the conference, additional components of a framework for TSG 36 emerged around (a) tasks, (b) learning environments, and (c) theory. Within the component of tasks, the topics addressed included how tasks are sequenced and structured (Gravemeijer, Brady et al., Goa et al.) what representations and tools were used in tasks (Thiel-Schneider, Johnson), what kinds of activities and actions

Co-chairs: Jere Confrey, Jiansheng Bao.

Team members: Anne Watson, Jonei Barbosa, Helmut Linneweber-Lammerskitten.

J. Confrey (&)

North Carolina State University, Raleigh, NC, USA e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
J. Bao

East China Normal University, Shanghai, China e-mail<EMAIL_ADDRESS>
were taken (Palatnik, Schäfer & Linneweber-Lammerskitten), the meaning of cases and classifications (Kulberg) and their effects on instruction and the type of feedback provided. Within the topic of learning environments, topics included student-to-student and student-to-teacher interactions (Mok et al.), curricular approaches (Goa et al.), assessment (Sharma et al.) and teacher knowledge and teacher roles in instruction (Dietiker et al., Lee) A variety of views of the student and learning were addressed. The use of theories in the presentations ranged from "grand theories" such as Realistic Maths (Gravemeijer), Socio-Epistemological Theory (Montiel et al.), Socio-Constructivism and Constructionism (Hoyles & Noss, Barquero et al.), and intermediate or "bridging theories" such as local instructional theories (Gravemeijer), task variations (Kulberg) and exemplifying (Kim & Park), problem solving approaches (Nunokawa), project-based instruction (Barquero et al.), learning trajectories (Confrey), and place-based design (Zender & Ludwig). Discussions indicated that task design work is evolving towards ever more careful description of local instructional theories to ensure that grand theories are held accountable to their impact on the design and research. Attention to addressing how tasks reside within sequences (Brady et al.), hierarchies (Tan et al.), frameworks of mathematical relations (Gravemeijer), and cases (Kulberg) arose across presentations.

The TSG also had a strong and consistent focus on both the design and research processes. This focus drew attention to iterative and agile design (Confrey), emergent modeling (Gravemeijer), shifting communities of interest (Barquero et al.), mind maps (Barquero et al.), progress maps (Isidro Camac et al.), navigation (Zander et al.), pre- and post-design (Kulberg), studies of classroom practices and implementation (Johnson), clinical interviewing (Tan et al.), and crowd-sourcing (Zander et al.).

From the talks and presentations as a whole. four themes emerged that bear further examination and development. The TSG36 members varied in the extent to which they acknowledged an important role for (1) re-envisioning the purpose and characterizations of doing mathematics, (2) considering how to design from the perspective of the student, (3) articulating, sharing, and strengthening the foundation of professional knowledge, and (4) articulating a role for learning systems in providing feedback, diagnostic/formative assessment, and supporting iterative design.

Attempts to re-envision the purpose and characterization of doing math came from a variety of presentations. Hoyles and Noss stated "in technology-enhanced mathematics classrooms, the use of digital tools can disrupt routine practices in a transformative sense, and ensuing breakdowns can promote further reflection and thinking again." Olsher et al. described how "The design principles of this game setting provide a unique and innovative implementation to the use of online dynamic figures in a game setting that is strongly rooted in meaningful mathematical work of students." Barquero et al. also emphasized in their joint enterprise of building an e-book that "… also, mathematical modelling was considered as a crucial approach to cope with most of the extra-mathematical questions, and progressively build up mathematical models to study the complexity of the questions posed along the unit, linking and articulating the models appearing at each step."

Consideration of how to design from the perspective of the student arose repeatedly in presentations emphasizing the need to connect to student experiences (Johnson, Nunokawa) and as relevant to their current and evolving identities. For instance, Nunokawa addressed how students learn to appreciate what kind of advantages the new ideas have over the previous ones. The importance of leveraging multiple representations (Johnson) arose along with articulating different student models (Kraemer et al., Rojas et al.). Wilkes described how sophisticated mathematics could be recognized in elementary students' activities.

Ways to strengthen the foundation of professional knowledge around the implementation of tasks was a topic that for which numerous scholars offered, as virtually everyone rejected the notion of materials that are "teacher proofed." Kulberg and Montiel et al. discussed how teaching can be improved through the way that teachers understand and approach cases. Stories were used to reach teachers and strengthen their involvement in task activities (Noruzi et al., Dietiker et al.). Joubert & Mostert and Lee involved teachers as co-designers and partners. Viewing teachers as central to the process of identifying and fostering emergent thinking and supporting discourse (Olsher et al.) was critical to many participants (learning to listen and support discourse and argumentation) There was also an emphasis on teacher community in this process including comparing novice and experienced teachers (Gao & Zhang) and studying collaborative teacher tasks (Lee et al.).

The final theme emerged around the idea of learning systems, systems in which tasks are embedded but allow students to proceed in personalized ways and at times, include assessments and feedback (Basila, Confrey). Sometimes this resided in Microworlds and dynamic software environments such as a similarity game (Olsher et al.) and for others the learning system actually included the learning systems that permitted a multi-site, multi-theory construction of an e-textbook. (Barquero et al.) with two geographically disparate communities designed an e-book. MathCity Math (Zender & Ludwig) even described a system that leveraged GPS and permitted mathematics activities to be constructed in real time at different locations around a city.

Overall the TSG 36 showed evidence that the field of task design, analysis and learning environments is continuing to grow and offer new forms of insights.

• Watson, Anne: Parameters for practice and research in task design in mathematics education

• Gravemeijer, Koeno: A personal take on instructional design

#### Tuesday 26 July 2016

#### Tuesday 26 July 2016

#### Wednesday 27 July 2016

#### Friday 29 July 2016

#### Friday 29 July 2016

• Kim, Dong-Won; Park, JinHyeong: Building mathematical statements through exemplifying

• Montiel, Gisela; López-Acosta, Luis; Cantoral, Ricardo; Scholz, Olivia: Design-based socioepistemological research

<sup>•</sup> Dietiker, Leslie; Brakoniecki, Aaron; Miller, Elyssa R.; Richman, Andrew S.: Enacted task design: tasks as written in the classroom

#### Friday 29 July 2016

#### Friday 29 July 2016

#### Saturday 30 July 2016

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 37: Mathematics Curriculum Development

Anita Rampal, Zalman Usiskin, Andreas Büchter, Jeremy Hodgen and Iman Osta

## The Programme

Co-chairs: Anita Rampal, Zalman Usiskin.

Team members: Andreas Büchter, Jeremy Hodgen, Iman Osta.

A. Rampal

Delhi University, New Delhi, Delhi, India e-mail<EMAIL_ADDRESS>
Z. Usiskin (&) University of Chicago, Winnetka, Chicago, IL, USA e-mail<EMAIL_ADDRESS>
## Summary

Curriculum concerns what we want students to learn, and when we want them to learn what we want them to learn; it is the heart of mathematics education. Curriculum development involves the processes that influence policy, the development of curriculum materials and pedagogical practices, and planning and implementation.

Other TSGs at ICME-13 discussed curriculum development in specific topic areas or with specific populations or age groups. TSG 37 sought papers of broader concern: the status of curriculum development in countries or regions; policies and influences shaping the development process; implementation, learnings, and future visions for curriculum development.

Usiskin identified five kinds of curriculum: ideal (intended), textbook (materials), implemented (taught), tested, and learned. The first four of these are created by people called policy makers, curriculum developers, teachers, and item writers, respectively. He offered three broad paradigms of curriculum development: traditional, test-influenced, and innovative. In the traditional paradigm, these five kinds are developed in the order shown above. In the test-influenced paradigm, tested curriculum specifications occur before materials are written. Discussions during TSG-37 sessions indicated that test-influenced curricula are exceedingly common throughout the world, with PISA and TIMSS results having particular impact in some countries.

For the innovative paradigm, Usiskin offered a sequence of events: (1) work by pioneers, the individuals who design the curriculum and assume a role comparable to policy makers in traditional curricula; (2) proselytizing of and by first users, who become apostles for the curriculum; (3) use by those disenchanted with the old curriculum; (4) acceptance by government agencies and teacher trainers if there is evidence of success; (5) if accepted; a forcing of the enchanted, those people who were content with the old curriculum and need persuasion or coersion to change. At this point, test creators take notice and, with enough support, the tests change to reflect the new ideas. But simultaneously, there can be pushback by those who question the need for the changes that the new curriculum embodies.

Rampal and Lipka each emphasized the need for curriculum to be developed from the cultures and lives of all students, rather than developing content from the hierarchy of mathematics. Rampal discussed efforts in the national primary mathematics curriculum of India to rethink mathematics, for and from life, for all children, through pedagogies of empathy that enable democratic participation. She noted the daunting challenges of curriculum development for public education in the 'gatekeeping' subjects of science and math in socially and culturally diverse poor countries, in the light of national pressures to reserve professional opportunities for the 'talented' privileged, and increasing global pressures for curriculum homogenization and standardization, tied to aggressive formulations of learning outcomes, with increasing transnational corporate interests in promoting 'low fee' private schools for the poor.

The curricular materials and textbooks developed by Rampal et al. follow a constructivist approach and also involve thematic issues of work, entrepreneurship, heritage, craft knowledge, history of monuments and pre-historic cave paintings, etc. using contexts that invoke and integrate concepts already learnt. In a similar vein, Lipka engaged the TSG participants in activities involving the "center of everything" from a curriculum he and individuals from the Yupiaq community in Alaska designed together. He noted that indigenous knowledge is rarely part of a core academic subject and that in the USA and elsewhere, indigenous people and their knowledge have been suppressed. His work uses indigenous knowledge to create an accessible curriculum for teaching the foundations of mathematical thinking for elementary school students. This work has included scholars and knowledge holders at Saami University, at the University of Greenland, and in Micronesia with Carolinian knowledge holders from Yap State and Chuuk who are associated with their respective Departments of Education. This work fits squarely into the role of ethnomathematics in mathematics curriculum.

Rojano presented a study to evaluate, from an international perspective, the current mathematics curriculum for compulsory education in Mexico. The Mexican curriculum was compared with those of the United Kingdom, Chile, and South Korea. Four influences on conceptions of school mathematics were noted: the international context of mathematics education; mathematics education research; the role of mathematics in the broader school curriculum; and distinctive traits in the quality of curriculum design. Differences were found concerning how school mathematics is conceived in the four curricula. Differences also exist in the presentation of the curriculum—e.g., how much detail, the presence or absence of discussions of relevance and of problem-solving. Notable in all four curricula was the lack of explicit references to research literature, as well as their inclusion of innovations in terms of content or teaching approaches (use of ICT or early algebra, for instance). However, all four programmes showed signs of the influence of education research. Suggestions are given for a redesign of the manner in which the Mexican curriculum is presented.

In England in 2014 a new national mathematics curriculum was introduced. This curriculum introduces some content in earlier grades than before and is accompanied by rhetoric identifying "mastery" as an overriding goal for all students—not just a few. Mastery is specified as mathematical fluency and conceptual understanding, and the ability to reason mathematically and solve problems. Townsend presented a case study of a primary school teacher (whose job entails teaching all subjects) who struggles with bringing all students to mastery, against longstanding beliefs that mastery is possible only for some students. Townsend concluded that teachers like this one "will be left to mediate tensions between adopting new ways of working alongside ingrained practices resulting from historical authoritative discourses."

Prendergast reported on his study of the knowledge of tertiary mathematics lecturers about Project Maths, a recent change in the secondary mathematics course content, teaching, and assessment in Ireland. Most lecturers were aware that there was more emphasis on understanding, more mathematics in context, and more exploration and activities. Those aware of changes in assessment pointed out the fact that it is now more difficult to predict the kinds of questions. Yet in all these domains, the majority of the lecturers indicated they had made no change and did not plan to make any change.

Hirsch described the development of a new course, Transition to College Mathematics and Statistics designed for the last year of secondary school, primarily for college-intending students whose planned programs of study do not require calculus. The course is notable for the United States for including of concepts from algebra and functions, statistics and probability, discrete mathematics, and geometric visualization. These branches of mathematics are connected by the central themes of the course, modeling of our world and the nurturing of mathematical habits of mind. A suite of curriculum-embedded technological tools has been developed for the course. The course was developed with the aid of extensive field-testing and has been tested in a variety of settings in the past few years, and the testing suggests that students are able to solve more complex and more open-ended problems. In so doing, they seem to be more able to work in teams and assess their own work and the work of the team.

Teuscher compared the ideal curriculum as found in the U.S. Common Core State Standards in Mathematics (CCSSM) with the approach to isometries and congruence in six current middle school textbook series. Geometric transformations have only been universally found in books at this level since 2010. The researchers found that three series defined congruence without regard to isometries even though the CCSSM ask for a definition in terms of isometries. Very little work was done on properties of isometries, and errors were found in some of the materials with regard to the discussion of orientation. The researchers concluded that teachers need to be aware of these difficulties, but this is unlikely because so many teachers themselves have never previously encountered this content.

Guorui examined how two teachers in each of two schools in Beijing and Hong Kong utilized mathematics textbooks to develop the taught curriculum. The Pythagorean Theorem was chosen so as to have uniform content, though the teachers were purposely selected so that in each school one teacher had at least 18 years and the other had less than 5 years experience. The experienced teachers tended to go beyond the textbook more than the inexperienced. The researchers pointed out the variety of factors that influence textbook use beyond the textbook itself, including the education system, school policies and practices, and teacher personal traits, themselves all influenced by socio-cultural factors.

In all, the papers demonstrate the complexity of developing mathematics curriculum reflecting the broad range of desires and needs of mathematics in our world while simultaneously taking into account the correspondingly broad socio-cultural backgrounds of students and the variety of teachers and schools in any system. Animated discussion in each of the sessions and participants enriched the deliberations by relating some if these issues to participants' own specific contexts.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 38: Research on Resources (Textbooks, Learning Materials etc.)

Lianghuo Fan, Luc Trouche, Chunxia Qi, Sebastian Rezat and Jana Visnovska

TSG38 focuses on issues related to mathematics teaching and learning resources, which mainly refer to school mathematics textbooks but also include other resources such as teacher manuals, student learning and assessment materials, and online resources. It is the hope of the organising team that TSG 38 would bring to foreground and examine various theoretical and methodological approaches used to study teaching and learning resources.

In the pre-congress call for contribution, it was stated that TSG38 sought contributions addressing broadly the areas of resources, teachers, and students, as outlined in the list of possible questions below, with a particular interest in analyses of the evolution of interactions between resources, teachers and students in a time of transition. The following three aspects were particularly highlighted in the call for contribution.

About the resources themselves: Among the learning materials available in mathematics classrooms in different countries, what role do textbooks and other curricular or learning resources play in mathematics teaching, learning, and assessment? How does the digitalization of information and communication affect this role? Is it possible to have a common definition of e-textbooks, and how could we characterize the differences between the traditional textbooks and e-textbooks?

Co-chairs: Lianghuo Fan, Luc Trouche.

Team members: Chunxia Qi, Sebastian Rezat, Jana Visnovska.

L. Fan (&)

University of Southampton, Southampton, UK e-mail<EMAIL_ADDRESS>
L. Trouche École Normale Supérieure de Lyon, Lyon, France e-mail<EMAIL_ADDRESS>
About the teachers: What are the main features of the teacher resource systems in different countries, their crucial resources, stability, flexibility and evolutions? What are the relationships between their individual and collective resources, and how could we model such relationships? What about the relationships between resource designers and users? What are the consequences of evolutions at stake for the teaching of mathematics, and for teacher knowledge and professional development?

About the students: what are the main features of the student resource systems in different countries? What is the effect of modern ICT (particularly internet) on their use and the design of resources? How do these evolutions affect their behavior, learning and relationships concerning the subject of mathematics?

TSG38 received a larger-than-expected number of submissions. According to the statistics released, it was one of the five largest TSGs in terms of the number of submissions received. After the process of review, 67 contributions were accepted, 61 for oral presentation and 6 for poster presentation.

Given the large number of contributions, the programme of TSG38 was organised into four 90-min regular sessions, plus 10 concurrent parallel sessions which took place in 4 time slots. As Sebastian Rezat was unable to attend the congress due to some unforeseen reason, these sessions were chaired by the two chairs, the two team members and the IPC liaison person Birgit Pepin. The four regular sessions are as follows:

### Session 1 (26 July 2016). Setting the scene: What role do textbooks and other curricular or learning resources play in mathematics teaching, learning, and assessment? (Chair: Birgit Pepin)

#### Session 2 (27 July 2016). How does the digitalization of information and communication affect the role of resources? (Chair: Jana Visnovska)

### Session 3 (29 July 2016). Teachers' collective work through resources (Chair: Chunxia Qi)

### Session 4 (30 July 2016). Teachers' and students' interactions through resources (Chair: Lianghuo Fan and Luc Trouche)

The 10 parallel sessions in 4 time slots (TS), the first two on 26 July and the last two 29 July, were organised with focuses on different areas concerning the research of mathematics textbooks and resources.

On 26 July, TS1 (60 min) consisted of 3 parallel sessions: the first parallel session focusing on "Textbook analysis and comparison" (Chair: C. Qi; presenters: X. Yang, M. S. Aguilar, S. Y. Jeong, and K. Oh), the second on "Text evaluation and digital tools" (Chair: L. Trouche; presenters: H.-D. Janetzko, A. Pu, and A. M. Bijura), and the third on "Teachers' work and learning with tools" (Chair: J. Visnovska; presenters: R. Lucena, R. Yap, and R. M. A. Filho). TS2 (90 min) consisted of 2 parallel sessions: the first parallel session focusing on "Problem solving" (Chair: L. Fan; presenters: E. Bingobali\*, E. Santaolalla, S. Walter, and C. A. Fuentes) and the second on "Representation and illustration in textbooks" (Chair: B. Pepin; presenters: X. Liu, P. Pausigere, V. Sarveswary, R. E. Borba, V. C. Lianos, and G. Glasnovic).

On 29 July 2016, TS3 (60 min) consisted of 3 parallel sessions with all focusing on "Students and teachers": the first parallel session had three presentations (Chair: J. Visnovska; presenters: N. Podevano, X. Jia and L. Zhao), the second parallel sessions had four presentations (Chair: L. Trouche; presenters: V. Gitirana, I. Ercan, E. Benitez, and X. Shao), and the third also four presentations (Chair: L. Fan; presenters: Z. Zhu, C. C. Assis, M. Ribeiro, and M. A. Huntley). TS4 (90 min) consisted of 2 parallel sessions: the first parallel session focusing on "Collective work" (Chair: C. Qi; presenters: C. Qi, A. Bapat, S. Basturk, J. Slisko, and D. Wijayanti) and the second on "Teacher use and interaction with resources" (Chair: L. Trouche; presenters: H. Siedel, F. Bifano, D. Paez, L. Jaber, L. Ahl, and E. Aydin\*).

The two presentations marked with \* by E. Bingobali and E. Aydin were made through video presentations due to the fact that they were unable to attend the congress due to unforeseen reasons taking place in Turkey.

To conclude, it is worth noting that ICME-10 was the first congress in the history of the ICMEs that a group with specific focus on mathematics textbooks (including learning and teaching materials), Discussion Group 14, was programmed. According to the organisers, DG14 received a much smaller number of submissions and accepted only 9 of them for presentation (Fan, Turnau, Dole, Gelfman, & Li, 2008). Compared with about nearly 70 presentations, it signals to us a clear and rapid growth of the interest of the international mathematics education community in the area of mathematics textbooks and resources (also see Fan, Zhu, & Miao, 2013), taking into account digital evolutions (Pepin, Gueudet, Yerushalmy, Trouche, & Chazan, 2015). On the other hand, it should be also noted that among all the submissions, a great majority of studies reported are on textbooks (compared to other resources), on teaching resources (compared to learning resources) and on printed text (compared to digital form), while methodologically, most are on textbook analysis, textbook comparison and teachers' use and interaction with textbooks and resources.

Overall, we are encouraged by the fact that there were so many contributions to TSG38 and wish to thanks all the contributors, presenters, reviewers, and the participants to help TSG38 a success. A post-congress monograph, based on contributions and presentations at TSG38, will be published by Springer.

## References

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 39: Large Scale Assessment and Testing in Mathematics Education

Rae Young Kim, Christine Suurtamm, Edward Silver, Stefan Ufer and Pauline Vos

## Introduction

Topic Study Group 39 aimed to address issues related to large-scale assessment, evaluation and testing in mathematics at all levels. Sound large-scale assessment (LSA) has the potential to provide important feedback about students' mathematical thinking, about classroom mathematical culture, or about a country's curriculum emphasis. Furthermore, LSA can have a strong influence in mathematics education as it often defines the mathematics that is mediated, valued and worth knowing.

Our TSG sought contributions of research in and new perspectives on LSA in mathematics education. We saw these issues as falling into three main strands: purposes and use, design and development, and teacher-related issues. Prospective contributors were requested to address one or more of the following topics:

Purposes and Use

Co-chairs: Rae Young Kim, Christine Suurtamm.

Team members: Edward Silver, Stefan Ufer, Pauline Vos.

R.Y. Kim (&)

Ewha Womans University, Seoul, Republic of South Korea e-mail<EMAIL_ADDRESS>
C. Suurtamm University of Ottawa, Ottawa, Canada e-mail<EMAIL_ADDRESS>

Design and Development

Teacher-related issues

We initially received over 40 papers for the TSG covering a wide range of areas of interest from all over the world. We discussed how to organize the sessions and participated in reviewing the papers. Each paper was evaluated by two reviewers including co-chairs, team members, and the authors of the papers submitted to TSG 39. Based on the reviews of these papers, 12 of these contributions were chosen for extended papers, 14 were chosen for oral communication, and 12 were recommended for poster presentations. Considering the topics and issues of the papers, we categorized the papers into three extended paper sessions, three oral communication sessions, and one poster session (at general exhibition) facilitated by co-chairs and team members as chair. In addition, we had a joint session with Topic Study Group 40: Classroom Assessment for Mathematics Learning to share mutually interesting issues, ideas, and practices around assessment through intensive discussion. We collaboratively produced a pre-conference publication with the classroom assessment group as well. Since some papers were withdrawn, 11 papers were presented in extended paper sessions, 1 paper was presented in the joint session (along with 2 from TSG 40), 11 were presented in oral communication sessions, and 8 were shown in the poster session in the end.

All the sessions of TSG 39 were organized to create a sense of community among all the presenters and participants who share common interests and ideas about large-scale assessment to improve mathematics education. The participants contributed greatly to the sessions and brought in perspectives from a wide range of knowledge, experiences, and practices. They were asked to read all of the papers before coming to the TSG 39 sessions and to bring some questions and comments on the papers. We also generated online space to facilitate further discussion out of sessions. The following are the leading questions in the discussion:

Topic Study Group No. 39: Large Scale Assessment and Testing … 567

## Main Ideas and Discussions in Each Session

Each session consisted of three or four 15-min presentations, short questions and comments after each presentation, and a 20-min whole group discussion at the end. Although each session was originally organized by the main themes, various issues and questions related to several themes came up together in the sessions. Thus, we summarized what was presented and discussed by the main themes shown above: Purposes and use, design and development, and teacher-related issues.

## Purposes and Use

More than 17 papers were presented regarding this main theme with various perspectives throughout the sessions. The presentations showed that large-scale assessments have been implemented for multiple purposes and uses in mathematics education. One group of papers focused on the use of large-scale assessments to evaluate systems and to make student placements. For instance, there are analyzing issues in specific regions such as gender and socioeconomic status (SES) in Brazil (e.g., Chagas and Kleinke) and the case of bonus points in Ireland (e.g., Treacy). Some papers presented the use of assessments to make student placements (e.g., Reddy) or to predict student performance by finding some factors or determinants (e.g., Alagoz and Ekici; Seifert, Eilerts, and Rinkens; Weitz and Venkat).

Another group of presentations showed that large-scale assessments could be used to reveal the features of student achievement and affective characteristics in certain contexts or across national contexts. Many papers focused on the analysis of student achievement in specific regions such as Taiwan (e.g., Tam and Leung), Belgium (e.g., Deprez, Nijlen, Ameel, and Janssen), and Thailand (e.g., Jaikla, Changsri, and Inprasitha) or across countries in terms of cognitive domains or levels (e.g., Kanageswari). While discussing several issues and concerns in each context, we also found commonalities across contexts.

The results from large-scale assessments contribute to analysis of factors related to student achievement. For instance, the relationship between self-efficacy and student achievement by their cognitive levels (e.g., Zhou, Liu, Q., and Liu, J.), the effects of socioeconomic status (SES) and opportunity to learn (OTL) at classroom and country levels on student achievement (e.g., Bokhove), the relationship between the use of ICT and mathematics achievement (e.g., Kanoh), didactic contract (e.g., Ferretti, Gambini, and Giorgio), and the factors influencing affective characteristics (e.g., Hwang, Kim, H., and Kim, W.). Some papers suggested a natural model of analysis of student abilities (e.g., Dimitric) or items measuring students' geometric intuition (e.g., Bai, Huang, and Zhang). We discussed pedagogical and political issues from the results of studies as well as methodological concerns around data analysis and interpretation.

## Design and Development

Many presentations brought up methodological issues around the design and development of tasks in large-scale assessments. For instance, the validity of the assessment (e.g., Bansilal; Grapin; Kasoka, Jakobsen, and Kazima), cross-cultural adaptations of measures (e.g., Marcinek and Patrová), cultural sensitivity and validity (e.g., Philpot), perceived task difficulty different from empirical one (e.g., Beitlich, Lehner, Strohmaier, and Reiss), and equivalent assessment design (e.g., Inekwe). In addition, many studies showed that individual or cultural differences in solving problems, especially word problem (e.g., Strohmaier, Beitlich, Lehner, and Reiss) or problems with realistic situations (e.g., Chen, Liu, Zhao, Song, and Li), could influence the reliability and validity of large-scale assessment.

Another group of presentations pointed out that large-scale assessments have often measured low level of cognitive demands (e.g., Dogbey and Dogbey; Drüke-Noe and Kühn), which could not reflect current goals in mathematics education. In order to enhance student learning through large-scale assessment, some presentations suggested new ways of evaluating student abilities by developing new items to measure geometric intuition (e.g., Bai, Huang, and Zhang) or providing a new guideline and prescription for interpreting problem situations with multicultural values (e.g., Djepaxhija, Vos, and Fuglestad).

## Teacher-Related Issues

Although a relatively small number of papers focused on this theme, we discussed how the results from large-scale assessments could be used for improving teaching practice and teacher knowledge. Since teaching is a cultural activity in a situated context, we also discussed cross-cultural adaptation issues of using measures of Mathematical Knowledge for Teaching (MKT) from a certain context to another (e.g., Marcinek and Patrová) and considered qualitative approaches such as using video clips (e.g., Bruckmaier and Krauss).

We learned from the joint session with the classroom assessment group that large-scale assessment and classroom assessment could complement each other to improve mathematics teaching and learning. In particular, Burkhardt argued that high-stakes assessment could be "a tool for improvement" by playing the roles not only in assuring accountability of systems but also in "measuring student performance", "defin(ing) performance goals for teaching and learning", and "largely determin(ing) the balance of classroom activities in most classrooms." This implies that large-scale assessment and classroom assessment can inform each other and enhance student learning in constructive ways.

## Concluding Remarks

All the participants actively participated in the sessions and brought up interesting and important issues around large-scale assessments. We finally found that there were both decontextualized commonalities and contextualized differences across different contexts. In this sense, it was productive to collaborate with TSG 40, the classroom assessment group, to elaborate our discussions around assessments and improve assessments for student learning. We also came to the conclusion that further discussion needs to be continued to develop the emerging ideas from this topic study group.

Acknowledgements The contribution of all the authors and participants in TSG 39 are deeply acknowledged.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 40: Classroom Assessment for Mathematics Learning

Denisse R. Thompson, Karin Brodie, Leonora Diaz Moreno, Nathalie Sayac and Stanislaw Schukajlow

## The Programme

Prior to the conference, members of the two topic study groups on assessment (TSG 39 on Large Scale Assessment and TSG 40 on Classroom Assessment) collaborated to develop a topical survey on Assessment in Mathematics Education (Suurtamm et al., 2016). The survey addressed five main issues related to the current state of assessment:

Twelve papers were presented in the four main sessions of TSG 40, with the third session held jointly with TSG 39: Large-Scale Assessment and Testing in Mathematics Education. In addition, twelve papers were presented in the oral communication sessions and fourteen posters were also presented. Papers addressed topics related to all but the first issue in the pre-conference topical survey. The focus of the main topic study group sessions is indicated below.

Co-chairs: Denisse R. Thompson, Karin Brodie.

Team members: Leonora Diaz Moreno, Nathalie Sayac, Stanislaw Schukajlow.

D.R. Thompson (&) University of South Florida, Tampa, USA e-mail<EMAIL_ADDRESS>
K. Brodie University of Witswatersrand, Johannesburg, South Africa e-mail<EMAIL_ADDRESS>
Session 1 included three papers focusing on assessment tools used by teachers, with three different models to assess student learning. Swan and Foster focused on design approaches to formative assessment with tasks developed that incorporate pre-assessment activities, formative feedback questions, and sample work for students to critique. Sia and Lim approached assessment for learning by developing a cognitive model for the concept of time and identifying attributes describing the knowledge and skills needed to answer tasks related to that concept; by associating the attributes with items and then finding students' patterns of success with the attributes, teachers have detailed feedback to make inferences to enhance student progress. Krieger, Platz, Winter, and Niehaus focused on the use of the open source web application, Internet Mathematics Assessment System (IMathAS), to help students in writing proofs; building the system for a proof requires identifying segments or phrases that would be appropriate as well as an acceptable sequence so that grading can be semi-automatic.

Session 2 included four papers focusing on teacher judgments and teacher learnings that occur as part of their assessment process. Sayac collected assessment tasks from primary teachers of varying experience levels and then analyzed the tasks for the level of complexity and nature of the competencies; she found most tasks were low level on both complexity and competency level. Hardie researched the sources that New Zealand teachers reference in making overall teacher judgments about student progress for parents or to document teaching and learning; teachers tended to use formal and informal methods in assessing their students' progress and had to learn to be skilled users of evidence from various assessment methods. Marynowski provided results from a case study of two secondary teachers in which she investigated both their beliefs about assessment as well as their actual practices, including students' perceptions of their teachers' practices; she found that neither teacher fully recognized how various practices related to their beliefs. In the final paper from this group, Pai described the beginning analysis of a phenomenological study investigating three phases of an ephemeral assessment cycle, namely eliciting, interpreting, and acting; he also explored factors that influence these phases.

Session 3 was held jointly with TSG 39, with a focus on psychometric models and other issues at the boundary of classroom and large-scale assessment. Two papers from TSG 40 were presented in this session. Bostic and Sondergeld described a process by which they developed and validated measures for assessing problem-solving ability of middle grades students, including linking of performance between the seventh and eighth-grade measures using Rasch IRT analysis. Ariza-Hernández, Rodríguez-Vásquez, and Arciga-Alejandre focused on assessing undergraduates' understanding of real functions of real variables using Sierpinska's four categories of understanding, namely identification, discrimination, generalization, and synthesis; students' achievement was analyzed using Bayesian IRT analysis to determine which categories of understanding students had achieved.

The three papers presented in the final session focused in some way on aspects of self-assessment within the mathematics classroom. O'Shea described assessment for learning through student autonomy, describing actions by an expert teacher to develop an environment in which grade 5 children engaged in critical dialogue to take control of their own learning. Veldhuis, van den Heuvel-Panhuizen, and Zhao compared the effects in the Netherlands and China of professional development support for teachers around issues of assessment on their students' achievement; teachers in the two countries varied in the length of time they participated in the professional development, with resultant differential effects on students' achievement. Straumberger described the use of self-diagnosis sheets as a means to help students assess confidence at using various competencies, and relating achievement to the accuracy of their self-diagnosis.

Three short papers were presented in each of the four oral communication sessions, with the focus of the sessions providing an opportunity to expand on issues raised during the regular topic study group sessions. Most papers focused on issues related to formative assessment, including the design of tasks, use of rubrics, specific formative assessment strategies, or the use of digital or electronic environments to create tasks. In addition, various papers provided cultural perspectives on the use of formative assessment in different countries.

The papers in the topic study group sessions and oral communications, as well as the poster presentations, indicate that assessment at the classroom level is a pressing issue around the globe. Teachers and researchers are engaged in collaborative efforts investigating assessment for learning as well as assessment as learning, all in the service of enhancing mathematics learning of all students. The various papers also indicate that much work remains to be done within the overall assessment sphere as the mathematics education community attempts to better understand the interplay of teachers' assessment practices and students' learning.

Authors of several of the papers, oral communications, and posters will be expanding their contributions as part of a post-monograph publication related to the work of the TSG.

## Reference

Suurtamm, C., Thompson, D. R., Kim, R. Y., Moreno, L. D., Sayac, N., Schukajlow, S., et al. (2016). Assessment in mathematics education: Large-scale assessment and classroom assessment (ICME-13 Topical Surveys). SpringerOpen.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 41: Uses of Technology in Primary Mathematics Education (Up to Age 10)

Sophie Soury-Lavergne, Colleen Vale, Francesca Ferrara, Krongthong Khairiree and Silke Ladel

TSG 41 at ICME-13 will explore these issues:

Many types of digital technology and environments are available for primary education since before the turn of the century. Yet, individual drill and practice software and interactive tools for exposition still appear to dominate practice in primary classrooms where technology is used. Around the world today, young children bring their experience with hand-held and other technology into the classroom. Moreover, teachers are normally more comfortable using digital tools in the classroom that they use in their personal life. In recent years, these have included tools to communicate in the cloud. Are primary teachers keeping up with digital natives? And which types of technology use are emerging to enrich and foster mathematics learning at primary school?

Co-chairs: Sophie Soury-Lavergne, Colleen Vale.

Deakin University, Burwood, Australia e-mail<EMAIL_ADDRESS>
Team members: Francesca Ferrara, Krongthong Khairiree, Silke Ladel.

S. Soury-Lavergne

École Normale Supérieure de Lyon, Lyon, France e-mail<EMAIL_ADDRESS>
C. Vale (&)

Regarding the first two issues, we want to learn more about factors and practices that enable teachers to efficiently embed technology use in the classroom, including contexts of differently advantaged communities. These factors might involve the design of the technology, curriculum innovation, instructional leadership, collaborative teacher inquiry, or other interventions. Contributions concerning this theme will need to identify specific technology for primary mathematics teaching and learning and the context. Regarding the third question for this TSG we want to know about innovations in the design of digital technology and tasks. Contributions for the last issue will focus on the impact of innovative technology and environments on children's mathematical inquiry, problem solving and reasoning.

The sessions chaired by Chairs Colleen Vale and Sophie Soury-Lavergne the following contributions were presented: Kevin Larkin, "Enhancing student learning using geometry apps: utilising the homogeneity and heterogeneity of clusters of apps"; Annie Savard, "Robotic tasks: affordances for mathematics learning?"; Anne Voltolini, "Duo of digital and material artifacts dedicated to the learning of geometry at primary school".

The Wednesday session chaired by Silke Ladel the following scholars presented the titles: Patricia Moyer-Packenham, "Using virtual manipulatives on iPads: how app alignment promotes young children's mathematics learning"; Sophie Soury-Lavergne, "Duos of artefacts to enhance mathematical learning"; Sean Chorney, "Exploring the social dimension of using TouchCounts".

On Friday Krongthong Khairiree chaired the session with: Catherine Attard, "Is current research assisting the implementation of contemporary ICT in the primary mathematics classroom?"; Nigel Calder, "Reshaping the learning experience through apps: affordances"; Shannon Larsen, "Using 1-1 mobile technology to support student discourse".

And on Saturday Krongthong Khairiree, "Enhancing students' visualize skills in solving word problems using bar model and the Geometer's Sketchpad"; Stéphane Cyr, "Impact of a video game on fractions concept learning in elementary school students"; Piata Allen, "He puawaitanga harakeke—using technology to accelerate learning in indigenous language schools".

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 42: Uses of Technology in Lower Secondary Mathematics Education (Age 10–14)

Lynda Ball, Paul Drijvers, Bärbel Barzel, Yiming Cao and Michela Maschietto

There is no doubt that digital technology nowadays has a tremendous impact on society. As a consequence, the question arises as to what impact this has on the teaching and learning of mathematics. The ICME-13 Topic Study Group «Uses of technology in lower secondary mathematics education» addressed this topic for education to 10–14 year olds and aimed to:

This focus is related to the topic of other ICME-13 TSGs. TSGs 41 and 43, respectively, focused on primary or upper secondary; TSG 44 addressed e-learning and blended learning. As there was one general TSG on in-service and professional development of secondary mathematics teachers (TSG 50), the TSG 42 described here included both a learner's and a teacher's perspective on digital technology in lower secondary mathematics education.

The following themes were core in the work of TSG 42.

Co-Chairs: Lynda Ball, Paul Drijvers.

Team members: Bärbel Barzel, Yiming Cao, Michela Maschietto.

L. Ball

P. Drijvers (&) Freudenthal Institute Utrecht University, Utrecht, The Netherlands e-mail<EMAIL_ADDRESS>
The University of Melbourne, Melbourne, Australia e-mail<EMAIL_ADDRESS>
• Evidence for effect:

What are the research findings about the benefits for student learning of the integration of digital tools in lower secondary mathematics education?

How can digital technology be used to promote communication and collaborative work between students, between teachers, and between students and teachers? What are the potential professional development needs of teachers integrating digital tools into their teaching, and how can technology act as a vehicle for such professional development activities?

Even if these themes are not exhaustive for the topics of the TSG, most contributions in these proceedings focus on one of them, and as such offer an excellent overview of the advancements in this field.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 43: Uses of Technology in Upper Secondary Education (Age 14–19)

Stephen Hegedus, Colette Laborde, Luis Moreno Armella, Hans-Stefan Siller and Michal Tabach

## The Programme

TSG 43 addressed the use of technology in upper secondary mathematics education from four points of view:

The group received 42 submissions for a presentation and 5 posters coming from 23 different countries. From these submissions, 12 were selected for a long presentation during the sessions of the Topic Study Group. The other submissions gave rise to short oral communications in slots external to the sessions of the Topic Group. In order to stimulate and structure the discussion, four additional presentations were planned by the Topic Group for introducing the four main themes of the group: they were done by L. Moreno Armella and C. Brady, S. Hegedus and

Co-chairs: Stephen Hegedus, Colette Laborde.

Team members: Luis Moreno Armella, Hans-Stefan Siller, Michal Tabach.

S. Hegedus

C. Laborde (&) Cabrilog, Fontaine, France e-mail<EMAIL_ADDRESS>
Southern Connecticut State University, New Haven, CT, USA e-mail<EMAIL_ADDRESS>
S. Dalton, by H.-S. Siller, by M. Tabach and J. Trgalova. These presentations gave rise to an ICME 13—Topical Survey (Hegedus et al., 2016).

The programme of the TSG was organized with the intention of meeting three aims: bringing information, supporting discussion and formulating critical questions for the future.

The first session was devoted to these four presentations providing a state of art presentation of the four main issues mentioned above and formulating critical questions related to those issues. The long presentations took place in the second and third sessions in parallel. The fourth session was a collective session divided into three parts:

A large diversity of themes was addressed by the presentations and during the discussion slots. The following section summarizes the main issues and questions of the group sessions.

## Main Issues and Questions

#### Technologies at use

Although the group called for presentations about new emerging technologies, the presentations mainly dealt with "classical" technology like Dynamic Geometry Environments or CAS, but with a stronger focus on Dynamic Geometry Environments in the long presentations as well as in short oral communications.

Three presentations only reported about the use of emerging technologies: multimodality involving various sensory modalities (like sight, touch, sound), 3D augmented realities, 3D pen, Wii graph a software application modeling the movement of two controllers of the game console Nintendo Wii. It is interesting to note that the two studies about 3D pen and Wii console shared a common theoretical framework in which diagrams and gestures are strongly linked and in which the meaning of mathematical objects is to be found in the interactions between diagrams and gestures. This framework can also be linked to the notion of co-action (Moreno-Armella & Hegedus, 2009): the learner is co-acting with the representational systems of mathematics offered by dynamic technologies.

Topics and questions

The most mentioned technological environments were Dynamic Geometry Environments or Dynamic Mathematics Environments. In the presented studies, they were used

Some participants mentioned that it would be nice to see more applications/investigations in other areas than geometry, algebra, in particular. There is an innate appeal to the visual nature of DGE, so that CAS teaching questions are challenging. Of course, it must be kept in mind that algebra is not only accessible through CAS.

However, some presentations addressed issues related to the use of CAS. A focus was made in particular on justification and proof. There may be changes in the type of justification brought in textbooks when using a CAS. The example of Denmark was given. In this country, more than one textbook make use of CAS assisted proof, i.e. a proof of a theorem or a statement when steps are outsourced to a CAS. The number of steps may vary from one or few steps to the whole proof, the CAS playing the role of an authority. A connected issue was addressed about the norms for evaluating the work of students solving problems by means of a CAS. In absence of verbalization, it may be difficult to reconstruct the work of the students.

This discussion about CAS and justification raised the questions of the emergence of new norms when using technology and not only for CAS. A study devoted to the use of graphing calculators showed that students using calculators were asking less critical and less why questions than those not using calculators. There is presently an absence of policy that may make the situation difficult for teachers.

Problems as well as contribution brought by the use of technologies for teachers in their teaching and teacher education were addressed by several presentations and gave rise to discussion. As shown in some presentations, new opportunities of interaction between the teacher and the students may emerge due to the presence of technology providing feedback to students' actions. The nature of these opportunities and the role of the teacher within these new forms of interactions constitute a wide field to investigate. For example, what are the new ways of interaction among multiple students or between students and their teacher that can occur as a result of these digital technology tasks?

It seems that there is a need of investigating the impact of research on teacher development: in what ways can research provide or has research provided teachers with knowledge, recommendations, and professional development around an optimal use of student work with digital technologies to take advantage of what the digital tools have to offer?

Teachers' choices, views and norms seem to play a crucial role for a successful integration. A study reported how new socio and socio-mathematical norms emerge in technology enhanced lessons done by preservice teachers. In particular, more justifications are requested from students by the teacher and discussion among students about the "reasons" was encouraged. This led to a more general question: Are there any interesting ways of taking advantage of technology for working with proofs in math classrooms?

Teachers' beliefs regarding discovering learning and time constraints are most influential when determining frequency of technology use. Teachers supporting discovery learning are more inclined to resort to the use of technology whereas paying much attention to time constraints may prevent them from integrating often technology often into their teaching.

## Conclusion

From the whole set of presentations and discussions emerged the complexity of the questions of the use of technology in the teaching and learning of mathematics. The studies shed light on many interactions between the mathematical structures and their representations in technology environments, between the learners, the environments and the task they are faced with, between the teacher interventions and the students' work on technology, between teachers' beliefs and the frequency and type of use of technology.

Research developed several theoretical tools for analyzing these interactions but the question of the dissemination of these studies and their possible use by teachers is still remaining.

How to disseminate research-based tasks with technology (designed by researchers or teacher educators) in a way to help teachers use them "properly"? In the case of collaborative design of tasks (by groups of teachers and researchers/teacher educators), how should researchers/teacher educators act in a way not to impose their research view but leading towards tasks with required educational quality?

"How to motivate teachers to adopt technology?" was among the final questions of the group …

## References

Hegedus, S., Laborde, C., Brady, C., Dalton, S., Siller, H.-S., Tabach, M., et al. (2016). Uses of technology in upper secondary mathematics education. ICME-13 topical surveys. Cham: Springer Open.

Moreno-Armella, L., & Hegedus, S. J. (2009). Coaction with digital technologies. ZDM Mathematics Education 41(4), 505–519.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 44: Distance Learning, e-Learning, and Blended Learning

Rúbia Barcelos Amaral, Veronica Hoyos, Els de Geest, Jason Silverman and Rose Vogel

In Topic Study Group 44 we built on current and emerging research in distance learning, e-learning, and blended learning. Specifically, we pushed the boundaries of what is known through an examination and discussion of recent research and development in teaching and learning through these modalities, with a focus on primary, secondary, and higher education. Some of the subtopics considered were utilization of both Web 2.0 and Web 3.0 resources in e-learning, blended learning, and distance education modalities (for example, how are OER utilized as a resource by users); MOOC (what are the affordances and constraints of this approach through specific cases); emerging work on the usage of mobile technologies (such as cell phones and tablets) for distance learning; transitioning traditional classroom practices to use online affordances and constructing bi-learning environments; enabling mathematical collaboration in online mathematics education; online distance education and blended learning in the professional development of mathematics teachers; e-portfolio for reflected mathematics teaching and learning; orchestrating productive mathematical conversations in an online or blended learning setting; the role of the faculty/moderator in online mathematics education; emergence and sustainability of communities of practice in online environments of collaboration and co-construction of resources; research methodologies and paradigms for studying online and blended mathematics education; and evaluation and effectiveness of distance education, e-learning, and bi-learning.

Co-chairs: Rúbia Barcelos Amaral, Veronica Hoyos.

Team members: Els de Geest, Jason Silverman, Rose Vogel.

R.B. Amaral

São Paulo State University, São Paulo, Brazil e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
V. Hoyos (&) National Pedagogical University, Mexico City, Mexico e-mail<EMAIL_ADDRESS>
© The Author(s) 2017 G. Kaiser (ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13 Monographs, DOI 10.1007/978-3-319-62597-3\_71

The TSG served to disseminate significant contributions as seen from international perspectives by providing an overview of the current state-of-the-art research, sharing and discussing emerging work (trends, ideas, methodologies, and results), and a calling for the development of a canon of research for online, blended, and distance math education. As a part of TSG 44 activities, research presentations by distinguished people in the field, posters, and sessions of collective discussion and reflection around previously accepted research contributions were considered.

On the first day during the conference, two sessions of work were accomplished, one on the topic of cases and perspectives of distance learning, e-learning, and blended learning and the other on online student learning, both chaired by Rose Vogel. The scholars who presented the talks in the first session were Fabian Mundt and Mutfried Hartmann; Kar Fu Yeung, Rachel Ka Wai Lui, William Man Yin Cheung, Eddy Kwok Fai Lam, and Nam Kiu Tsing; Karin Landenfeld, Martin Göbbels, and Antonia Hintze (invited paper); and Tatjana Hrubik-Vulanovic. The scholars in the second session were Jonathan T. Lee; Bijeta Kumar and Bibhya Sharma; and Yasuyuki Nakamura, Tetsuya Taniguchi, Kentaro Yoshitomi, Shizuka Shirai, Tetsuo Fukui, and Takahiro Nakahara.

There were also two sessions on the second day covering the topics of online collaborative learning and teacher PD through online tools and learning, both chaired by Rubia Barcelos. The first topic's presentations were given by Arthur Powell (invited talk); Kadian M. Callahan and Anne-Marie S. Marshall; and Mandy Lo, Julie-Ann Edwards, Christian Bokhove, and Hugh Davis. The talks on the second topic in the afternoon were given by Maman Fathurroman, Hepsi Nindiasari, Nurul Anriani, and Aan Subhan; Tatyana A. Oleinik; Andrey I. Prokopenko and Sergeevich S. Zub; Maria E. Navarro, Veronica Hoyos, Victor Raggi, and Sergio Vazquez; and Eugenia Taranto, Virginia Alberti, and Sara Labasin.

The third session, was chaired by Jason Silverman and covered the topic of mathematics teacher education at a distance and mediated by ICT. The talks were by Elizabeth Fleming, Daniel Chazan, Pat Herbst, and Dana Grosser-Clarkson (Invited Paper); Cosette Crisan; Tamar Avineri, Hollylynne S. Lee, Dung Tran, Jennifer N. Lovett, and Theresa Gibson; and Yaniv Biton and Osnat Fellus. During the afternoon, a third session was chaired by Veronica Hoyos on the topic of distance learning and quantitative assessment, and talks were given by Tajana Hrubik-Vulanovic, Ferlisa Bundalian Lavador, Mary J. Castilla, and Richard Vinluan and Maxima Joyosa Acelajado.

The last session on the topic of communities of learning at a distance and enhanced by ICT, was chaired by Veronica Hoyos. The following scholars presented their work on that topic: Marcelo Borba (Invited talk); Giovannina Albano, Maria Polo, and Pier Luigi Ferrari; and Angela María Restrepo.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 45: Knowledge in/for Teaching Mathematics at Primary Level

### Carolyn A. Maher, Peter Sullivan, Hedwig Gasteiger and Soo Jin Lee

As the teaching of primary level mathematics (ages 5–13) is complex, it requires teachers to master a variety of types of knowledge. These are outlined as follows:

Understanding of important mathematical concepts that underpin meaningful student learning of the main strands of the mathematics curriculum;

Co-chairs: Carolyn A. Maher, Peter Sullivan. Team members: Hedwig Gasteiger, Soo Jin Lee.

C.A. Maher Rutgers University, New Brunswick, USA e-mail<EMAIL_ADDRESS>
P. Sullivan (&) Monash University, Melbourne, Australia e-mail<EMAIL_ADDRESS>

TSG 45 participants explored the types of knowledge represented by these various challenges, and how teachers can be supported in their learning.

The major presentations for the group were:

The presentations covered a broad range of topics for supporting the improvement in knowledge for teachers of mathematics. In particular topics addressed included knowledge for:

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 46: Knowledge in/for Teaching Mathematics at the Secondary Level

Ruhama Even, Xinrong Yang, Nils Buchholtz, Charalambos Charalambous and Tim Rowland

The program of TSG 46 focused on three themes:

The first three sessions centred on the three themes, while the fourth was devoted to summary, discussion and reflections. Below we describe the activities that took place during the four sessions.

Session 1: Conceptualization and Theorization of Knowledge in/for Teaching Mathematics at the Secondary Level (Chairs: Nils Buchholtz and Tim Rowland)

The problem: A number of international studies investigate the professional knowledge for teaching mathematics at the secondary level and for this purpose draw back on various different theoretical conceptualizations. Within the session the similarities and differences of different conceptualizations were analyzed and discussed, but also the current challenges of these conceptualizations were faced, especially with regard to the interaction between theoretically-assumed knowledge

Co-chairs: Ruhama Even, Xinrong Yang.

Team members: Nils Buchholtz, Charalambos Charalambous, Tim Rowland.

R. Even (&)

© The Author(s) 2017

589

Weizmann Institute of Science, Rehovot, Israel e-mail<EMAIL_ADDRESS>
X. Yang

Southwest University, Chongqing, China and University of Hamburg, Hamburg, Germany e-mail<EMAIL_ADDRESS>
G. Kaiser (ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13 Monographs, DOI 10.1007/978-3-319-62597-3\_73

facets and their visible manifestation in the practice of teaching. The current challenge seems to be to differentiate rather better current conceptualizations for teaching mathematics at the secondary level according to the theoretically sound and empirically-based integration of action-based knowledge facets. Orienting theoretical conceptualizations more to practical school-based contexts offers a basis for empirical research that is oriented more to the realities of mathematics teaching in school. The invited contributors of the session presented promising perspectives in this field, with a valuable overview from the first presenter:

Presentation 1: Conceptualization and Theorization of knowledge in/for teaching mathematics at secondary level, by Michael Neubrand from the University of Oldenburg in Germany.

Presentation 2: Academic mathematics or school mathematics? What kind of content knowledge do mathematics teachers need?, by Aiso Heinze and Anika Dreher from IPN—Leibniz Institute for Science and Mathematics Education in Germany (together with Anke Lindmeier, IPN, Germany).

Presentation 3: Analysing secondary mathematics teaching with the knowledge quartet, by Tim Rowland from the Universities of Cambridge and East Anglia in the UK (together with Anne Thwaites and Libby Jared from the University of Cambridge, UK).

Session 2: Methods for Measuring, Assessing, Evaluating and Comparing Knowledge in/for Teaching Mathematics at the Secondary Level (Chairs: Charalambos Y. Charalambous and Xinrong Yang)

The problem: The last two decades have seen considerable work not only in theorizing the knowledge needed for the work of teaching mathematics, but also in operationalizing and measuring this knowledge. These last two facets pose significant challenges to scholars working on exploring teacher knowledge and its effects on instructional quality and student learning, since at least two critical questions need to be addressed when it comes to considering these issues: (a) what (aspects of teacher knowledge) to measure—especially given the multifaceted nature of (recent) teacher knowledge conceptualizations—and (b) how best to measure them to ensure that valid and reliable data are collected, and legitimate inferences are drawn. Although these questions have attracted significant scholarly interest for elementary school grades, the field of measuring teacher knowledge at the secondary school is still developing. These issues were taken up by both Session-2 presentations. The first of these pointed to the importance of focusing on the knowledge entailments of key mathematical teaching tasks as opposed to simply attending to different types of knowledge; further capitalizing on videos to measure teacher knowledge as embedded in practice; and measuring teacher knowledge in cost-efficient ways. The second presentation made a case about the importance of measuring both generic teaching tasks as well as content-specific tasks, and the knowledge entailments associated with them.

Presentation 1: Measuring Secondary Teachers' Knowledge of Teaching Mathematics: Developing a Field, by Heather C. Hill of the Harvard Graduate School of Education, the USA.

Presentation 2: Measuring Instructional Quality in Mathematics Education, by Lena Schlesinger and Armin Jentsch of the University of Hamburg, Germany.

### Session 3: Connections Between Knowledge and Practice of Teaching Mathematics at the Secondary Level (Chair: Ruhama Even)

The problem: That expertise in mathematics teaching requires adequate mathematical knowledge is a trivial statement, but what "adequate" means is not clear. In many countries, the education of secondary school mathematics teachers traditionally includes a strong emphasis on advanced mathematics courses at the college or university level, taught by mathematicians, assuming that it would contribute to the quality of classroom instruction. This tradition, however, has been reconsidered in recent years, and the relevance of advanced mathematics courses to the quality of secondary school mathematics teaching is being debated. Is there a need for advanced mathematics studies in the professional education and development of secondary school mathematics teachers? What might be the relevance of advanced mathematics courses taught by research mathematicians to teaching secondary school mathematics? This issue was the focus of the three presentations in session 3, all of which reported on studies that addressed the overarching question: What are the relevance and the contribution of advanced mathematics studies to secondary school mathematics teaching?

Presentation 1: Accommodation of teachers' knowledge of inverse functions with the group of invertible functions, by Nicholas H. Wasserman from Teachers College in the USA.

Presentation 2: Senior secondary school teachers' advanced mathematics knowledge and their teaching in china, by Haode Zuo and Frederick K.S. Leung from the University of Hong Kong in China.

Presentation 3: Teachers' views on the relevance of advanced mathematics studies to secondary school teaching, by Ruhama Even from the Weizmann Institute of Science in Israel.

Session 4: Summary, Discussion and Reflections (Chairs: Xinrong Yang and Ruhama Even)

In this session, Xinrong Yang from Southwest University in China, Nils Buchholtz from the University of Hamburg in Germany, and Charalambos Charalambous from the University of Cyprus in Cyprus reflected on the first three sessions. Below is a summary of their reflections.

Current theoretical conceptualizations of knowledge in/for teaching mathematics at the secondary level primarily focus on knowledge as a personal disposition that can be tapped for empirical surveys. At the theoretical level, drawing on the seminal work of Shulman (1986), various dimensions of knowledge are often distinguished and segregated depending on assumed content-related aspects, or on aspects of practical teaching. When such knowledge is operationalized in empirical studies, it becomes more possible to separate these different facets empirically. Michael Neubrand and Tim Rowland pointed out that the more context-oriented knowledge gets analyzed in such studies, the harder it gets to empirically differentiate the knowledge in actu from other factors such as the teacher's personality or the affective level, which leads us to look more at the performance of mathematics teachers and at classifications of situations in which mathematical knowledge surfaces in teaching.

In retrospect, thirty years after Shulman's (1986) pioneering work, we can now claim that much has been accomplished on different fronts. Reflecting on this rapidly accumulating work, Charalambos Charalambous argued that the polyphony in the different theoretical frameworks and conceptualizations advanced thus far seems to be productive; he nevertheless voiced concern as to whether this polyphony will eventually be turned into cacophony, in the sense that we might run the risk of creating a Tower of Babel when it comes to talking about, studying, and measuring teacher knowledge. He thus suggested that scholars invest more in exploring synergies between different conceptualizations. Given that a shift seems to be observed from studying components of teacher knowledge to investigating tasks of teaching and the knowledge requirements these tasks impose on teachers (cf. Gitomer & Zisk, 2015), the need to develop a comprehensive framework encompassing such tasks and detailing their knowledge requirements for teachers was also underlined. Finally, the merit of employing the different approaches pursued so far to study teacher knowledge was highlighted. At the same time, a series of open issues was also outlined. For example, at what level of granularity should teacher knowledge be measured to ensure both its predictive validity and generalizability? To what extent might certain measures be culturally specific, and what might the implications of this specificity be? To what extent might certain items used in teacher-knowledge measures function differently when used in different contexts? To what extent does the knowledge measured actually impact teachers' teaching practice and students' mathematics achievement? This indicative list of questions indicates that there remains significant uncharted terrain to explore when working on studying teacher knowledge, and its effects on instruction and student learning.

## References

Gitomer, D. H., & Zisk, R. C. (2015). Knowing what teachers know. Review of Research in Education, 39(1), 1–53.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 47: Pre-service Mathematics Education of Primary Teachers

Keiko Hino, Gabriel J. Stylianides, Katja Eilerts, Caroline Lajoie and David Pugalee

## The Programme

Topic Study Group 47 (TSG 47) included paper presentations on significant new trends and developments in research, theory, and practice about all different aspects that relate to the mathematics education of pre-service primary teachers. The phrase "different aspects" was interpreted broadly to include (among others) the following:

TSG 47 offered a forum for an overview of the current state-of-the-art, invited contributions from experts in the field (Fou-Lai Lin and Skip Fennel), presentation of high-quality research reports from TSG participants, and discussion of directions

Co-chairs: Keiko Hino, Gabriel J. Stylianides.

Team members: Katja Eilerts, Caroline Lajoie, David Pugalee.

K. Hino (&)

Utsunomiya University, Utsunomiya, Japan e-mail<EMAIL_ADDRESS>
G.J. Stylianides University of Oxford, Oxford, UK e-mail<EMAIL_ADDRESS>for future research. In discussing the findings of research studies that took place in different countries, the TSG participants also had an opportunity to learn about practices used around the world in relation to the mathematics education of pre-service primary teachers such as similarities and differences in the formal mathematics education of teachers, types and routes of teacher education, and factors that can influence similarities or differences.

Associated with the TSG there were in total 19 regular presentations (8-page papers), 29 oral communications (4-page papers), and 18 posters. The regular presentations (8-page papers) were organized around five themes as described below. Although several presentations (and associated papers) addressed issues that spanned several themes, practical considerations related to the organization of the TSG sessions during the conference necessitated a best-fit approach.

## Theme 1: Mathematics-Content and Mathematics-Specific Pedagogical Preparation

This theme is about the mathematical and pedagogical aspects of teachers' preparation in teacher education. The following presentations were offered under this theme:

Papers emphasized the importance of pedagogy focused tasks to promote professional growth (Lin and Hsu). Papers focusing on content explored math-specific domains (Pilous et al.), specific areas such as proportional reasoning (Fox), and the role of computer technology in geometry (Lo).

## Theme 2: Activities and Assessment Tools Used in Mathematics Teacher Education Programs

This theme is about activities and tools for assessing prospective teachers' knowledge or skills used in mathematics teacher education programs. The following presentations were offered under this theme:

Fennell presented current and emerging challenges related to elementary education programs in the United States. Effective characteristics of the learning environments were found through prospective teachers' activities of designing non-routine mathematical problems (Kool & Keijzer). Through various assessment tools, certain aspects of teachers' content knowledge or teaching skills were found to be in need of more stimuli or to be built in teacher preparation programs (Vula & Kingji-Kastrati, and Shaughnessy & Boerst).

## Theme 3: Mathematical Knowledge for Teaching and Beliefs

This theme is about the mathematical knowledge that teachers need for their work and about teachers' beliefs and how the might affect teaching practice. The following presentations were offered under this theme:

The sessions provided multiple perspectives on the mathematical knowledge and beliefs for teaching at the primary level. Participants explored self-efficacy as it relates to pre-service teachers' mathematics background. Measures of knowledge and beliefs were the focus of several sessions including the use of image vignettes and teachers' beliefs for topic specific knowledge. Mathematical knowledge for teaching and evaluation was explored in terms of argumentation.

## Theme 4: Experiences in Mathematics Classrooms/Teacher Educators' Knowledge for Teaching

This theme is about prospective teachers' experiences in mathematics classrooms or opinions about the learning opportunities for teaching to diverse students in the teacher education programs, and the work of mathematics teacher educators. The following presentations were offered under this theme:

Faculty cooperation and near peers were shown as playing vital roles to make rich field experience (Doğan & Kılıç, and Eriksen et al.). The prospective teachers' learning opportunities for teaching to diverse students were found not homogenous even within a country (Çelik et al.). Four practices by the teacher educators were identified as they connect preservice teachers' learning to the practice of teaching mathematics to students (Li & Superfine).

## Theme 5: Developing Ability to Notice

This theme is about prospective teachers' developing their ability to notice. The following presentations were offered under this theme:

The sessions provided multiple perspectives on the ability to notice and the development of that ability amongst prospective teachers. The analysis of a role-play with pre-service primary school teachers involving the use of a calculator has been used to illustrate the complexity of learning to notice and learning to act in-the-moment (Lajoie). Writing narratives have been used as a successful way to help pre-service teachers develop their skill of noticing pupils' mathematical thinking (Ivars and Fernández). Pre-service teachers' skills to recognize, identify and make instructional decisions have been examined in a context in which they were provided with opportunities to engage in noticing practices (Gupta et al.).

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 48: Pre-service Mathematics Education of Secondary Teachers

Marilyn Strutchens, Rongjin Huang, Leticia Losano, Despina Potari and Björn Schwarz

## The Programme

During Topic Study Group 48 regular sessions, significant new trends and developments in research and practice on the mathematics education of prospective secondary teachers were discussed. An overview of the current state-of-the-art and recent research reports from an international perspective were provided. In keeping with the call for papers, presentations focused on similarities and differences related to the development of mathematics content and pedagogical content knowledge of teachers; models and routes of teacher education, curricula of mathematics teacher education; the development of professional identities as prospective mathematics teachers and a variety of factors that influence these different aspects; field experiences and their impact on prospective secondary mathematics teachers' development of the craft of teaching; the impact of the increasing availability of various technological devices and resources on preparing prospective secondary mathematics teachers; and others.

We received fifty-four 4-page submissions to TSG 48. From the submissions, 20 papers were selected for presentations during the regular meetings. Each of these 20 papers was scheduled in one of the following TSG sessions based on the topic of the paper: (1) Field Experiences (two sessions); (2) Prospective Teachers

Team members: Leticia Losano, Despina Potari, Björn Schwarz.

R. Huang Middle Tennessee State University, Murfreesboro, USA e-mail<EMAIL_ADDRESS>
Co-chairs: Marilyn Strutchens, Rongjin Huang.

M. Strutchens (&) Auburn University, Auburn, USA e-mail<EMAIL_ADDRESS>
Knowledge (two sessions); (3) Technologies, Tools and Resources (one session); and (4) Prospective Teachers Professional Identities (one session). Also, we invited four speakers to submit an extended article, one for each of the major themes: Blake E. Peterson and Keith R. Leatham (field experiences), João Pedro da Ponte (teachers' knowledge), Rose Zbiek (tools, technologies, and resources), and Márcia Cristina de Costa Trindade Cyrino (teachers' identities). Thus, 24 papers were scheduled for the TSG 48 regular meetings. We had 3 cancellations. Below is a list of the regular session presenters and the titles of their presentations:

## Field Experiences

### Teachers knowledge

#### Technologies, tools and resources

#### Teachers professional identities

Next highlights from the regular TSG 48 sessions are presented. A major theme discussed in the field experiences sessions is the notion of creating opportunities for prospective secondary mathematics teachers (PSMTs) to learn effective strategies for teaching students and simultaneously ensure that their students are developing the mathematical skills and knowledge that they need. Three presentations (Peterson & Leatham; Martin & Strutchens; Losano & Villarreal) focused on structuring field experiences in such a manner that more than one teacher candidate is placed with the same mentor teacher to capitalize on collective planning, teaching, and monitoring of student growth, as well as fostering student-focused teaching strategies for the teacher candidates. Other papers (Potari & Psycharis; Heinrich) focused on teacher candidates noticing students' actions and asking the appropriate questions or redirecting instruction to better meet students' needs. Jackson and Mohr-Schroeder helped participants to think about field experiences that go beyond the regular mathematics classroom and that help teacher candidates to connect mathematics to other STEM areas. Akcay and Boston presented a study that determined pre-service teachers' ability to integrate technological tools into instructional activities and showcase portfolios in mathematics in ways that support students' high-level thinking and reasoning. Participants were also happy to support Hulya Kilic who had to present her talk virtually, since academic travels abroad were forbidden by the Turkish government during the time of the conference.

Within the PSMTs' teacher knowledge sessions, 'structures in mathematics teacher education that support the development of PSMTs' knowledge' was a major theme. Ponte reviewed studies which focused on lesson study in teacher education and addressed emerging challenges. He described strategies that have been used in teacher education, such as microteaching, reflection in oral and written-form, and face-to-face and digital context for planning and reflecting. He also discussed the constrains that exist in using lesson study in initial teacher education. Arnal-Bailera et al. discussed how grading mathematics exams can be embedded in PSMTs' programs in ways that promote reflection and provide opportunities for professional learning. Park, Oh, and Kwon addressed the analysis of textbooks as an approach in teacher education to bring PSMTs closer to the curriculum and to their future work as teachers. Lin, Yang, and Chang discussed a course in which PSMTs had opportunities to understand students' mathematical thinking, cultivate the competencies of exploration and practice, and develop positive beliefs towards mathematics teaching and learning.

In the second session on PSMTs' knowledge, four presentations focused on teacher education practices related to specific mathematical content areas and processes. Manouchechri's study prepared PSMTs to become familiar with mathematical modelling. The study shows that teachers experienced difficulty enacting the mathematical practice of mathematical modelling. Lin and his colleagues examined PSMTs' teaching of mathematical induction in the context of their field experiences. Juhász et al. focused on the development of PSMTs content knowledge and pedagogical content knowledge in a teacher education course based on problem solving. Finally, Ölmez, Izsák, and Beckmann pointed out that the quantitative definition for multiplication is linked to PSMTs capacity to visualize the relationships between the multiplier and multiplicand in strip diagrams. Coincidentally, the research threads discussed in PSMTs' knowledge reflected the same emphases as some of the studies found in the topical survey that the group published (Strutchens et al., 2016).

Three presentations were given in the technology focused session. Zbiek discussed the conceptual tools for framing secondary mathematics teacher preparation and technology use. She argued that TPACK (Technological, Pedagogical, And Content Knowledge), long used as a framework for knowledge and recently proposed as an orientation towards technology use, is productively enriched by elaboration. Within this frame, she further illustrated conceptual tools for framing technology, content, and pedagogy. She suggested that PSMTs should encounter multiple forms of technology in all venues of their preparation: content courses, pedagogy course, and practical experiences. Wu examined how PSMTs learn to teach mathematical concepts via a video-based approach. The participants' reflection reports documented how they learned from an expert teacher, their peers, and self-reflections. Moreno and Llinares shared PSMTs' perspectives on the use of technology for supporting mathematics learning. They found that PSMTs perspectives on the use of technology were defined by the way in which technological resources were used and the nature of the mathematical activity.

Two papers were presented at the session devoted to PSMTs' professional identity. They revolved around two notions related to identity. The first one is PSMTs' self-perceptions and was employed by Hine in his analysis of how PSMTs understand and perceive their "readiness" to teach mathematics based on their pre-service education. The second one, employed by Durandt and Jacobs, is PSMTs' attitudes. The authors investigated PSMTs' attitudes towards modeling based on their initial exposure to a model-eliciting task. Although invited speaker, Márcia Cyrino, was unable to attend the congress due to force majeure circumstances, her article<sup>1</sup> introduces new aspects for the notion of identity. Particularly, her work highlights that identity is related to agency and vulnerability. One of the issues discussed during the session was the following: What are the theoretical links between the notions of belief, attitude, conception, emotion, agency and identity? Furthermore, participants agreed on the importance of reflecting on methodological tools for collecting data—interviews, narratives, surveys, field notes, etc.—best suited for capturing not only the PSMTs' self-perceptions, attitudes and professional identity but also its development.

Overall, TSG 48 sessions were well received and attended. Participants were intrigued that the countries had so many issues in common around prospective secondary mathematics teacher education.

## Reference

Strutchens, M., Huang, R., Locano, L., Potari, D., Ponte, J. P., Cyrino, M. C., et al. (2016). The mathematics education of prospective secondary teachers around the world. ICME-13 Topical Surveys. Berlin: Springer.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

<sup>1</sup> Cyrino, M. C. T. Teacher professional identity construction in pre-service mathematics teacher education: analysing a multimedia case.

## Topic Study Group No. 49: In-Service Education and Professional Development of Primary Mathematics Teachers

Akihiko Takahashi, Leonor Varas, Toshiakira Fujii, Kim Ramatlapana and Christoph Selter

## The Program

Professional development is a never-ending pursuit for a teacher. This is why teacher and teacher educators are involved in the learning process throughout their entire professional lives. The complexity of mathematical teaching practices means in-service teachers face many challenges, such as the demands of new curricula, the introduction of new technologies in the classroom, and the adaptation of teaching practices for students with different abilities and in varying contexts. To address these challenges in each country's agenda for 21st century education, teachers' professional development should link the intended curriculum with students' success.

Topic Study Group (TSG) 49 discussed not only the experience and approaches of effective in-service teacher education and professional development of primary school mathematics teachers, but also contributed to building up a comprehensive overview of the state of the art, the impact of extended and lasting policies (e.g., accountability systems, standardized testing of educational outcomes, etc.), new categories and emphasis introduced by educational researchers (e.g., pedagogical

Co-chairs: Akihiko Takahashi, Leonor Varas.

Team members: Toshiakira Fujii, Kim Ramatlapana, Christoph Selter.

A. Takahashi (&) DePaul University, Chicago, USA e-mail<EMAIL_ADDRESS>
L. Varas Universidad de Chile, Santiago, Chile e-mail<EMAIL_ADDRESS>
© The Author(s) 2017 G. Kaiser (ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13 Monographs, DOI 10.1007/978-3-319-62597-3\_76

605

content knowledge, mathematical knowledge for teaching, mathematical competencies, etc.), and current discussions on the available evidence for effective professional development programs and the need for related specific research.

To achieve this goal, researchers were invited to contribute to TSG 49. 50 papers were kindly submitted. The authors of 20 papers were invited to present at TSG 49s themed sessions. The authors of 16 papers were invited to present at the oral communication sessions. In addition, the authors of 14 papers were invited to present at the poster sessions. The following is a list of the papers presented at TSG 49s themed sessions and oral communication sessions:

## TSG Session 1: Lesson Studies

## TSG Session 2: Developing Models of Efficient In-Service Teacher Training

## TSG Session 3: New Relations and Partnership Experiences in Teachers' Professional Development

• Ader, E. Investigating Classroom Teachers' Development of Quality of Implementation of Mathematical Tasks.

## TSG Session 4: Scaling Up Sustainable Interventions

## Oral Communication Session 1

## Oral Communication Session 2

## Oral Communication Session 3

## Oral Communication Session 4

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 50: In-Service Education, and Professional Development of Secondary Mathematics Teachers

Jill Adler, Yudong Yang, Hilda Borko, Konrad Krainer and Sitti Patahuddin

## The Programme

The aim of TSG 50 at ICME-13 was to share, discuss and advance knowledge and understanding of key aspects of research, policy and practice in the in-service education and professional development of secondary mathematics teachers. In TSG 50, there were a total of 97 contributions (19 Paper presentations, 49 Oral presentations and 29 Poster presentations) and well over 100 participants given this total does not include the TSG organizing team, and some paper co-authors.

During ICME-13 conference, the work of TSG 50 was organized into four main TSG Sessions, and these were supplemented by oral communication and poster sessions. The four main sessions each focused on key questions. Session 1 reported and reflected on A Survey of the Field of Research on Mathematics Professional Development. The session was chaired by Jill Adler, and focused on two key questions: Q1. What key research questions, theories and methods are used to study in-service education and professional development of secondary mathematics teachers? Q2. What are the accumulating results of research? What do we still not know or understand? What are the implications for policy and practice and across contexts?

Co-chairs: Jill Adler, Yudong Yang. Team members: Hilda Borko, Konrad Krainer, Sitti Patahuddin.

J. Adler (&)

University of the Witwatersrand, Johannesburg, South Africa e-mail<EMAIL_ADDRESS>
Y. Yang Shanghai Academy of Educational Sciences, Shanghai, China e-mail<EMAIL_ADDRESS>
Hilda Borko, Stanford University, USA, presented a "stimulus paper" entitled Research on Mathematics Professional Development. This paper is a review of research in the field, and Hilda's presentation was followed by reflective comments from each of the remaining TSG 50 Team Members. Konrad Krainer, reflected on the format of the study, in particular focusing on the research process, its strengths and absences. Yudong Yang, drew attention to the Asian perspective and specifically to important work that has been done on Lesson Study, an aspect of the field not dealt with directly in the review. Sitti Patahuddin emphasized the ICT perspective of professional development for secondary mathematics teachers. Jill Adler pointed out that research in the field of education development was not incorporated in the review, and was important to consider as there was evidence of failure in many developing contexts of professional development initiatives focused on reform, and as Chair of the session concluded with a short summary of TSG 50's first session, and introduction to the remaining sessions of the TSG for all participants.

The second main session focused on the question: Q3. What have we learned about secondary mathematics teachers' learning through participation in PD, related to beliefs, knowledge and practice? There were two parallel sessions, one chaired by Sitti Patahuddin, focused on teacher learning in relation to their knowledge, beliefs and practice; the other was chaired by Yudong Yang, and focused on teacher educator/facilitator/coach learning. The presentations' titles and their authors are the following, the first four in the teacher learning session and the remaining four in the teacher educator learning session:

The third main session focused on the question: Q4. What innovative professional development programs with/without ICT for in-service mathematics teachers at secondary level have been developed and implemented in different cultural contexts? What is the role of professional learning communities in these programs, including online communities? Here too there were two parallel sessions, chaired respectively by Jill Adler and Hilda Borko. One focused on a range of Professional Development programs including a specific focus on Learning and Lesson Study initiatives; and the other focused on those with a specific ICT focus. The presentations' titles and their authors are as following, with the last three on ICT:

The fourth main session (chaired by Konrad Krainer and concluded by Jill Adler) focused on the question: Q5. What do we know about the impact, sustainability and scalability of PD programs? What are the various ways in which sustainability, scalability and effectiveness are defined and then assessed? The presentations' titles and their authors are as following:

Supplementing these main TSG sessions, were six parallel Oral Communications sessions where 49 papers were presented and discussed; and two sessions for 29 Poster papers (due to the pages' limitation, the titles and presenters of Oral Communication and Poster will not be listed here). It is also not possible to communicate the richness of research and ideas presented and discussed. We nevertheless mention some exemplary aspects that were highlighted.

In the review of the field presented in session 1 most studies examined primarily utilized qualitative methods, which can offer detailed accounts of PD models and rich descriptions of teachers' experiences in the PD while providing proof of concept that the PD design can effectively support teacher learning. Opportunities to analyze excerpts of classroom video or mathematical tasks were provided as examples of contexts and practices promoting learning in professional communities. Knowledge of student mathematical thinking was drawn attention to as critical for teacher learning. In addition, PD facilitators' learning was identified as a new focus of research in the field. These studies demonstrated the continued value of foundational, exploratory, development, and design research for growing the knowledge base on mathematics PD. Still, very few studies investigated causal links between participation in PD and teacher or student outcomes. In addition research that examines what it takes to bring these programs to scale on the one hand, and that compares the effectiveness of different mathematics professional development programs on the other, needs further development.

Secondary mathematics teachers' professional development programs in specific contexts were introduced and discussed in TSG 50. The analysis of sustainable effects is crucial for short-term and long-term programs. The way in which sustainable impacts were researched, the theoretical models and empirical findings were discussed. More evaluation and analysis of sustainable impacts are needed in the future, along with sustainability in teacher education.

The use of technology in mathematics professional development was also a popular topic in TSG 50. Here foci were on the use of technology as interaction tools in and for mathematics teachers' professional development, as it provides resources and platforms for teachers at scale; and on digital technology as an interactive way to support mathematics teachers' learning that has great potential prospect. However, research on the effectiveness of using technology as a mode of interactions is limited. Further research is also needed to understand the possible challenges of the transformation in technology-based programs, such as the design of digital PD to match with the context and needs of teachers, the roles of the facilitators, the specialized pedagogy to conduct digital mathematics professional development, and the utilization of big data in understanding digital PD effectiveness.

In conclusion, TSG 50 was a learning space where mathematics professional development programs were overviewed theoretically, using different analytical frameworks, as well as empirically. New programs for secondary mathematics teachers' professional development in specific context were shared, including some attention to Lesson study, though less so in the Asian context; and all of these with or without information and communication technology. Presentations explored teacher educators' role and their knowledge and skills promotion within diverse backgrounds; and the impact of sustainability of professional development. Others introduced technology-based professional development programs. Some presenters put forward more research questions, including the prospective of using big data to better understand the effectiveness of teachers' PD. However, we are still short of evidence-based studies and of tools for measuring and evaluating teachers' learning. These areas remain a challenge for researchers in this field.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 51: Diversity of Theories in Mathematics Education

Tommy Dreyfus, Anna Sierpinska, Stefan Halverscheid, Steve Lerman and Takeshi Miyakawa

## The Programme

## Session I

### Session II

#### Session III

• Presentation 6: Yasuhiro Sekiguchi: Theories and traditions: Tensions between mathematics teaching practices and a recent school reform in Japan.

Co-chairs: Tommy Dreyfus, Anna Sierpinska.

Team members: Stefan Halverscheid, Stephen Lerman, Takeshi Miyakawa.

T. Dreyfus (&) Tel Aviv University, Tel Aviv, Israel e-mail<EMAIL_ADDRESS>
A. Sierpinska Concordia University, Montreal, Canada e-mail<EMAIL_ADDRESS>

Session IV

## Summary

In the closing discussion we have asked what were the implied or explicit ideas about some of the fundamental questions about mathematics, its teaching and its learning, mentioned in the abstract of TSG51? We look at some of these questions in turn.

#### What is Mathematics?

Anna Sfard's response was that mathematics is a collection of collections of stories about different things (shapes, numbers, sets, functions, …) told using specialized discourses characterized by (1) keywords and their uses (e.g., number, function, limit, derivative, …); (2) visual mediators (sign systems) intended to clarify what the particular story is about (e.g., positional number systems, functional notation, graphs, …); (3) routine actions of the storytellers (abstraction, generalization, deduction, induction, reasoning by contradiction to prove the impossibility of some hypothetical story, testing stories for internal consistency and coherence with other stories, …), and meta-rules (explicit definitions of technical terms, laws of logic, …) (Sfard, 2007).

Ricardo Cantoral said that this is a philosophical question and therefore best left to philosophers. A question for mathematics educators is what are the differences between mathematics as a body of theoretical knowledge and mathematics as a school subject?

In relation to the nature of mathematics, both Anna Sfard and Ricardo Cantoral addressed a question that was not explicitly posed in the abstract of our group: Where do mathematical concepts come from? For Anna Sfard, mathematical concepts come from a feedback loop between practical activities and discourses; for Ricardo Cantoral—from cultural practices, techniques, traditions. It is an important question in mathematics education for, whenever we plan to teach a mathematical concept, we seek the sources of its meaning so that we can construct instructional situations that will help students to construct these meanings for themselves. This question underlies Davydov's concept of "object sources" and Brousseau's concept of "fundamental situations" for particular mathematical notions.

#### What is Mathematics as a School Subject?

Regarding mathematics as a school subject, there were differences of opinion between Anna Sfard and Ricardo Cantoral. Anna Sfard described school mathematics as a discourse obtained from mathematicians' mathematics by "customization" to the needs and capacities of young learners. She said that it differs from mathematicians' mathematics mostly by the meta-rules, which are less strict and also different in nature.

Ricardo Cantoral, on the other hand, proposed that school mathematics is not only a transposition of scholarly mathematical knowledge. For him, school mathematics is part of the culture of a given society and place; it contains cultural traditions, riddles and games known from popular culture, technology, and other things used in that culture to construct mathematical knowledge in the classroom. School mathematics is culturally situated. There is popular mathematical knowledge, technical mathematical knowledge and scientific mathematical knowledge, and all have to be taken into account when building theories of teaching and learning mathematics at school. There is no hierarchy among the three kinds of knowledge. They are all part of human wisdom (Cantoral, 2013).

#### What Does it Mean to Teach Mathematics in General or a Particular Mathematical Concept or Process? What Does it Mean to Teach it Well?

According to Ricardo Cantoral, to teach a fundamental mathematical concept (e.g., derivative), we need to find a "cultural basis" for it and help students anchor their understanding of the concept in this cultural basis (e.g., the idea of taste—sweetness, salinity—can be used as a cultural basis for the concept of rate and hence of slope of a linear function).

Higinio Dominguez looked at a more social-interactive aspect of teaching and proposed that to teach well it is necessary to engage in "reciprocal noticing" with students: "In reciprocal noticing, what is noticed is not individual reasoning but rather the emerging and continuous influence of people's reasoning WITH (not for) one another …."

Cristina Frade's perspective was focused on the person of the teacher, but it was not psychological: the question was how a person constructs her or his identity as a teacher. This construction is an important element of becoming a teacher—and therefore, a condition of "teaching well". It is social in nature, by means of language: the individual develops a vocabulary with which to justify their actions, compare their past and present behaviors, and generally narrate their live stories.

A theory of teaching well was the main concern of Yasuhiro Sekiguchi's presentation. The theory of "School as Learning Community" (SLC) was proposed in the 1980s by Manabu Sato and has become the foundation of a school reform in Japan (Saito et al., 2015). Preparation for teaching well is sought through a well-developed system of "lesson study" conferences and developing collaboration between children, teachers, parents, and people in the region.

### What Does it Mean to Know Something or a Given Particular Thing in Mathematics (e.g., Fractions)?

According to Anna Sfard, to know something in mathematics means to "extend one's discursive repertoire by individualizing" the discourse in which a particular mathematical story or a collection of such stories is told in school mathematics. "To individualize a discourse means to be able to communicate according to its rules" with others and with oneself. Carolina Tamayo Osorio pointed to a subtle transitional aspect of the process of individualization of a discourse: based on her observations of the Gunadule indigenous Community of Alto Caiman (Colombia) and Wittgenstein's notion of grammar, she proposed that to learn something in mathematics is to extend one's repertoire of language games by constructing a third "border" grammar between one's native grammar and the grammar of school mathematics.

Ricardo Cantoral's take on "knowing something in mathematics" can be summarized as follows: To be able to participate in a certain social (cultural) practice, not only to be able to perform a certain individual intellectual act. This practice may take place in school, but also out of school (e.g., building a log cottage requires knowledge that is technical mathematical knowledge).

Yasuhiro Sekiguchi stressed that in the SLC theory, learning is a collaborative endeavor. For an individual to know something, a whole community must know it; school must be a learning community.

The question, What does it mean to know something well? did not raise much discussion, but it is worth mentioning Anna Sfard's thesis that success and failure in mathematics are elements of a discourse. An individual who participates in that discourse constructs his or her identity as a "success" or a "failure" in mathematics.

#### Meta-Questions

In her presentation in the last session, Michèle Artigue criticized the questions asked in the abstract from the perspective of the French approach to mathematics education. Some of the questions contained the verb "should" and thus suggested value judgements and a normative point of view that the French didactic culture strongly rejects. Such questions are not scientific: answers to them are not verifiable by scientific means. A scientific question would be: what are the consequences (for the practice of teaching and learning, for example) of such and such normative perspective? Other participants pointed out that in other cultures the construction of mathematics education research as "science" is not necessarily a priority, and that choices made in posing research questions and selecting aspects to consider are inevitably value-laden and guided by more or less explicit answers to the fundamental questions asked in the abstract for our group. So it is useful, for understanding the motives and aims of research, to know the researchers' position on these questions.

The last question debated in the group was,

#### Why Do We Have So Many Theories in Mathematics Education?

Several hypotheses appeared.

A theory is created or borrowed because it is useful. But usefulness is relative to values and needs (problems?). Since the latter are diverse in mathematics education, a diversity of theories is needed to respond to them. (Anna Sfard)

Teaching and learning of mathematics is culturally situated. Cultures of mathematics education in different populations must be studied empirically. To explain differences between these cultures, existing theories may not be enough. (Ricardo Cantoral)

In Japan, "theory" is always a theory of some practice. A practice develops, somebody notices it, reflects upon it and constructs a theory of this practice. Practices evolve, change; new practices emerge. Hence many theories. (Takeshi Miyakawa)

## References

Cantoral, R. (2013). Teoria Socioepistemológica de la Matemática Educativa. Estudios sobre construccion social del conocimiento. Barcelona, España: Gedisa.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group 52: Empirical Methods and Methodologies

David Clarke, Alan Schoenfeld, Bagele Chilisa, Paul Cobb and Christine Knipping

TSG 52, "Empirical methods and methodologies," was devoted to explorations of common themes and underlying issues in the use of empirical methods and methodologies. The challenge for the group was defined as follows:

Research in mathematics education employs an extensive range of Methods, Methodologies, and Paradigms (M/M/Ps) in the service of key goals. But which M/M/P combinations help us understand which phenomena, in robust and reliable ways?

The group met for four sessions at ICME-13. The first three sessions were devoted to parallel sessions in which the assembled groups explored methodological issues related to the following six goals central to ongoing research in mathematics education:

Co-chairs: David Clarke, Alan Schoenfeld.

Team members: Bagele Chilisa, Paul Cobb, Christine Knipping.

D. Clarke (&) University of Melbourne, Melbourne, Australia e-mail<EMAIL_ADDRESS>
A. Schoenfeld University of California, Berkeley, CA, USA e-mail<EMAIL_ADDRESS>
6. Understanding the Role of Culture and Language in Shaping the Teaching and Learning of Mathematics.

These are diverse goals, which might be addressed using research designs that integrate different M/M/P combinations. For each goal, one might be ask: "Suppose you have an hypothesis about this goal. How do you set about evaluating it?" Alternatively, "Suppose you are trying to explain some aspect of individual or group behavior relevant to that goal. How would you characterize and then theorize that behavior?" Or, "How might cultural, historical and political perspectives shape one's understandings of the contingencies related to realizing this particular goal?" One could imagine radically different approaches not only across the six goals, but within each goal. The question, then, was whether the group could, in the final collective discussion, find some degree of coherence in the varied approaches, and produce a larger frame within which the discussions of the six goals could be situated.

That frame emerged as the group considered a broad framing of empirical work in education, as an act of modelling—the idea being that empirical methods and methodologies are, in effect, ways to characterize "real world" phenomena. The challenge, then, was to problematize the enterprise. The challenge was taken up with enthusiasm, which led to some discussions that pushed the boundaries of previous framings.

The group rejected the standard version of the modelling process,

for a more complex one,

and began to problematize this representation by noting all the ways that empirical work could go wrong—e.g., (arrow 1) for decades of medical research, the "human population" was represented by male experimental subjects, and the results of many studies later turned out not to apply to women. Similarly, (arrow 2) what one decides to consider important in learning makes a big difference—we noted that nations rank differently on TIMSS and PISA, because the two mathematics tests capture different aspects of what might be considered mathematically important. Group members similarly (arrow 3) discussed alternative interpretations of statistical findings—looked at one way, certain results seemed significant, but from another perspective, they were not. Nor were challenges limited to the quantitative realm; questions of how to be confident about interpretations of discourse analyses received attention, as did issues of triangulation and the use of multiple methods. Likewise mapping back from analyses to the "real world" was an issue: the fact that many "significant" laboratory studies did not translate to meaningful learning gains in real classrooms was raised as an issue of concern.

Perhaps the most interesting, and passionate, part of the conversation dealt with what has been called the "political turn" or "socio-political turn" in mathematics education. Members of the group noted that the conceptual analytic models in the second figure often were shaped by tacit social biases—that the "clean" analytic descriptions in the figures could obscure various forms of racism, sexism, gender bias, and more. There was a general concern among the group that the field needs to attend in more explicit ways to the possibility of such bias, and address it in our work. At the same time, there was appreciation for the fact that, although there is a lot more to be done, the field has made tremendous progress since the first ICME in 1969.

Open Access Except where otherwise noted, this chapter is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/.

## Topic Study Group No. 53: Philosophy of Mathematics Education

Paul Ernest, Ladislav Kvasz, Maria Bicudo, Regina Möller and Ole Skovsmose

What is the philosophy of mathematics education? It can be an explicit position that is formulated, reformulated, criticized, refined, etc. But it can also refer to implicit assumptions and priorities, including paradigmatic assumptions that one need not be aware of, but which might be identified through, let us call it, a philosophical archaeology.

The philosophy of any activity comprises its aims or rationale. Thus we ask: what is the purpose of teaching and learning mathematics? An answer explains why we engage in these practices and what we hope will be achieved. But just considering such purposes quickly leads to seeing the divergence in aims and values of different groups.

A broader view of the philosophy to mathematics education looks at the applications of topics such as epistemology, philosophy of mathematics, ethics and aesthetics. It applies philosophical methods to a critical examination of the assumptions, reasoning and conclusions of mathematics education, systematically enquiring into fundamental questions:

Co-chairs: Paul Ernest, Ladislav Kvasz.

Team members: Maria Bicudo, Regina Möller, Ole Skovsmose.

P. Ernest (&) University of Exeter, Exeter, UK e-mail<EMAIL_ADDRESS>
L. Kvasz Charles University Prague, Prague, Czech Republic e-mail<EMAIL_ADDRESS>

The philosophy of mathematics education matters because it gives people new 'glasses' through which to see the world. It enables people to see beyond official stories about the society, mathematics, and education. It provides thinking tools for questioning the status quo, for seeing 'what is' is not what 'has to be'; enabling us to imagine alternatives possibilities. A preconference overview of the field was published: Ernest et al. (2016), available free from Springer Open.

At the conference, itself the sessions included expert presentations on key questions and issues of the field and was time made available for questions, discussion and participation.

The Tuesday session began with a welcome and an introduction from Paul Ernest and Ladislav Kvasz. Chaired by Ladislav Kvasz and Ole Skovsmose two main presentations were given:

Paul Ernest An overview philosophy of mathematics education. Due to personal circumstances the second speaker was unable to attend so the second presentation was:

Paul Ernest The collateral damage of learning mathematics.

The Wednesday session was chaired by Paul Ernest and Ladislav Kvasz and consisted of two presentations:

Ole Skovsmose The politics of meaning in mathematics education

Maria Bicudo Developments in philosophy in/of mathematical education: ontological questions posed by the presence of computers etc.

On the Friday, there were two parallel sessions chaired by Paul Ernest and Ladislav Kvasz. The presentations were as follows:

On Saturday, the session was chaired by Paul Ernest and Maria Bicudo, and the following talks were given.

There was a closing panel discussion with the main presenters Paul Ernest, Ladislav Kvasz, Maria Bicudo, Regina Möller, Ole Skovsmose and the audience. Lastly, the publication strategy of the group was discussed, with two elements. First, there is the publication of all papers presented at TSG 53 including poster presentations, in Philosophy of Mathematics Education Journal (2016) number 31, a special issue dedicated to TSG 53 papers. Second, the publication of the best papers rewritten and expanded, plus invited outside expert contributions in a special monograph (Ernest, forthcoming).

## References