Sebastian Rezat

Sebastian Rezat, Paderborn University, Paderborn, Germany

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).