In this chapter, the focus is on arithmetic which for the Netherlands as a trading nation is a crucial part of the mathematics curriculum. The chapter goes back to the roots of arithmetic education in the sixteenth century and compares it with the current approach to teaching arithmetic. In the sixteenth century, in the Netherlands, the traditional arithmetic method using coins on a counting board was replaced by written arithmetic with Hindu-Arabic numbers. Many manuscripts and books written in the vernacular teach this new method to future merchants, moneychangers, bankers, bookkeepers, etcetera. These students wanted to learn recipes to solve the arithmetical problems of their future profession. The books offer standard algorithms and many practical exercises. Much attention was paid to memorising rules and recipes, tables of multiplication and other number relations. It seems likely that the sixteenth century craftsmen became skilful reckoners within their profession and that was sufficient. They did not need mathematical insight to solve new problems. Five centuries later we want to teach our students mathematical skills to survive in a computerised and globalised society. They also need knowledge about number relations and arithmetical rules, but they have to learn to apply this knowledge flexibly and meaningfully to solve new problems, to mathematise situations, and to evaluate, interpret and check output of computers and calculators. The twenty-first century needs problem solvers, but to acquire the skills of a good problem solver a firm knowledge base-comparable with that of the sixteenth century reckoner-is still necessary. 7.1 Introduction Over many centuries teaching arithmetic has played an important part in Dutch education. Interest in this subject started to grow in the sixteenth century when the Netherlands began to develop into an important trade nation and arithmetic finally got

To reach for abstraction is a major but challenging goal in mathematics education: teachers struggle with finding ways how to foster abstraction in their classes. To shed light on this issue for the case of geometry education, we align theoretical perspectives on embodied learning and abstraction with practical perspectives from in-service teachers. We focus on the teaching and learning of realistic geometry, not only because this domain is apt for sensori-motor action investigations, but also because abstraction in realistic geometry is under-researched in relation to other domains of mathematics, and teachers’ knowledge of geometry and confidence in teaching it lag behind. The following research question will be addressed: how can a theoretical embodied perspective on abstraction in geometry education in the higher grades of primary school inform current teacher practices? To answer this question, we carried out a literature study and an interview study with in-service teachers (n = 6). As a result of the literature study, we consider embodied abstraction in geometry as a process of reflecting on, describing, explaining, and structuring of sensory-motor actions in the experienced world through developing and using mathematical artifacts. The results from the interview study show that teachers are potentially prepared for using aspects of embodied learning (e.g., manipulatives), but are not aware of the different aspects of enactment that may invite students’ abstraction. We conclude that theories on embodiment and abstraction do not suffice to foster students’ abstraction process in geometry. Instead, teachers’ knowledge of embodied abstraction in geometry and how to foster this grows with experience in enactment, and with the discovery that cognition emerges to serve action.

No abstract