## Invited Speakers

## Zooming in and zooming out: towards a realistic view on tool use in mathematics education

Since the origin of mankind, humans have been developing tools to extend their scope and to carry out tasks more easily and more efficiently. Also in the field of mathematics and mathematics education, tools have been widely used. Nowadays, the omnipresence of powerful digital mathematical tools raises important questions about their impact on mathematics teaching and learning, as is witnessed by CERME working groups on the topic over the years, and by recent publications (Ball et al., 2018; Monaghan, Trouche, & Borwein, 2016; Trgalová, Clark-Wilson, & Weigand, 2018). These questions not only concern the curriculum, but the relationship between tool use and learning in particular. To address the latter question, I will first zoom in on some specific examples of using digital tools in mathematics learning. Next, I will zoom out and present some overall findings on the effects this use is having. After a theoretical view on didactical functionalities of digital tools and on instrumental approaches to their use, I will address two relationships in some more detail: (1) the alignment of the use of digital tools with principles from the theory of Realistic Mathematics Education (RME, the domain-specific instruction theory for mathematics, developed in the Netherlands by Freudenthal, Treffers and colleagues), and (2) the relationship between tool use and embodied and extended views on cognition. As a conclusion, I will claim that both an RME lens and an embodied view on tool use may contribute to identifying criteria for a meaningful integration of tool use in mathematics education.

Ball, L., Drijvers, P., Ladel, S., Siller, H.-S., Tabach, M., & Vale, C. (Eds.) (2018). *Uses of Technology in Primary and Secondary Mathematics Education; Tools, Topics and Trends*. Cham, Switzerland: Springer International Publishing.

Monaghan, J., Trouche, L., & Borwein, J. (2016). *Tools and Mathematics*. Cham, Switzerland: Springer International Publishing.

Trgalová, J., Clark-Wilson, A., & Weigand, H.-G. (2018). Technology and resources in mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven, *Developing Research in Mathematics Education. Twenty Years of Communication and Collaboration in Europe* (pp. 142–161). New York: Routledge.

**History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future**

The field of history of mathematics in mathematics education – often referred to as the history and pedagogy of mathematics domain (or, HPM domain) – can be characterized by an interesting and rich past and a vibrant and promising future. In this plenary, I will describe the development of the field, and in doing so, I will highlight the ways in which research in the field of history of mathematics in mathematics education offers important connections to more mainstream aspects of research in mathematics education, with a particular emphasis on student learning. To begin, I will succinctly situate the HPM domain within mathematics education, with careful attention to the development leading up to establishing the International Study Group on the Relations between the History and Pedagogy of Mathematics in 1976. Precipitated by the creation of the HPM Group, research in the HPM domain has continued to grow in last 40-plus years, and includes all levels of learners and teachers. Part of this growth has been marked by the creation of a thematic working group on history in mathematics education, beginning with CERME 6 in 2009. Next, I will provide an overview of different approaches and frameworks that are useful in empirical research in the HPM domain. In doing so, I will highlight specific examples in which research on the use of history of mathematics contributes to the broader landscape of research in mathematics education. Recent examples include the application of Sfard’s (2008) thinking as communicating framework in research within the *Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources* (or, TRIUMPHS) project in the United States, as well as the work of colleagues in Denmark and Brazil. These examples also address the role of history of mathematics in the learning of a wide variety of mathematical concepts, ranging from the function concept to determinants of matrices, as well as topics in analysis and abstract algebra. Furthermore, efforts to extend knowledge about beliefs research via detection and inspection of domain-specific beliefs, and the contributions of working with primary historical sources on pre- and in-service teachers’ mathematical knowledge for teaching hold great potential for both strengthening theoretical and empirical connections to research in mathematics education. Finally, after a brief analysis of ongoing discussions for calls to strengthen empirical work in the HPM domain in light of certain pitfalls and dilemmas facing the field, I will propose directions for research in the coming years, and the ways in which Thematic Working Group 12 can contribute to bridging research in this important field with the broader mathematics education research community.

**Extensions of number systems: continuities and discontinuities revisited**

The extension of number systems from natural to rational and real numbers and related arithmetic is a prominent theme in mathematics from primary to upper secondary education. Students’ difficulties in mastering the transition from one number system to another are well documented related to integers and rational numbers. Several theories of different scope have been proposed to explain these difficulties (e.g. natural number bias, conceptual change). These theories mostly focus on the students’ conceptual development and not so much on the opportunities that students were exposed to in order to develop or extend their number concept(s). Using the transition from natural numbers to integers as an exemplary case, the plenary talk focuses on transitions from one number system to the other with a particular interest in continuities and discontinuities in the teaching of these number systems. Since its foundation at CERME7 in 2011 TWG02 “Arithmetic and Number Systems” gives a flourishing account of the multifaceted issues related to the teaching and learning of different number systems. Drawing on this work, I will first give an overview of the main issues and aims in the teaching of both number systems, natural numbers and integers respectively. Among these, number sense and conceptual development, mental calculation and flexibility as well as algebraic thinking will be the most prominent ones. With the latter aspect, I will also relate to the work of TWG03 “Algebraic Thinking”. Comparing the issues that are focused in research on different number systems, I will argue that discontinuities in the teaching of the two number systems are likely. These will be substantiated with results from a textbook analysis. Second, I will elaborate on different understandings of the notion of “transition” related to the teaching of number systems. Based on this and the analysis in the first part, I will thirdly analyze, how the transition from natural numbers to integers can be characterized in the analyzed textbooks and if it is consistent with one of the notions of “transition”. The guiding question is if the transition from one number system to the next is consistent with one of the notions of “transition”. Third, I will argue for more continuity on the level of opportunities to learn in the teaching of number systems and related arithmetic in order to support a more coherent construction of the number concept(s).