Martin Carlsen

Reasoning with Paper and Pencil: The Role of Inscriptions in Student Learning of Geometric Series

The purpose of this article is to analyse how students use inscriptions as tools for thinking and learning in mathematical problem-solving activities. The empirical context is that of learning about geometric series in a small group setting. What has been analysed is how students made use of inscriptions, self-made as well as those provided by text books and teachers, and the role these inscriptions played in the coordination of students’ learning/communication. Through the use of inscriptions (made on the chalkboard and with paper and pencil), the students externalised their thinking while engaging in mathematical reasoning on the topic of geometric series. The inscriptions were significant as anchor points for arguments in the ongoing discussions. Three main issues are highlighted: (a) how the inscriptions used contribute to the process of appropriation, (b) how the students use inscriptions to externalise and clarify their ideas and attempts at meaning-making, and (c) how the inscriptions are conducive to closing the gap between the original problem as given in the text book and the mathematisation necessary. It is argued that inscriptions, through their material nature, play a decisive role in learning mathematical reasoning.

Engaging with mathematics in the kindergarten. Orchestrating a fairy tale through questioning and use of tools

ABSTRACT The aim of this study is to analyse how a kindergarten teacher orchestrated a mathematical activity involving a fairy tale. Taking a sociocultural perspective on learning and development, naturally occurring talk-in-interaction has been analysed in order to scrutinise the subtleties of the orchestration. The fairy tale Goldilocks and the Three Bears was used both as context for the activity, but also as a means through which mathematical concepts were engaged with. The analysis shows that the orchestration bears characteristics of questioning and use of tools such as voice, facial expressions, and concrete materials. The orchestration is also characterised by sincere planning in advance of the activity and by addressing pre-formulated learning objectives.

Children’s Engagement with Mathematics in Kindergarten Mediated by the Use of Digital Tools

The purpose of this chapter is to study children’s engagement with mathematics in kindergarten mediated by digital tools within the context of interactive whiteboard (IWB). Our study reports from a research project aiming at analysing in what ways digital tools may nurture children’s appropriation processes relative to mathematics. In adopting a sociocultural perspective on learning and development, observations together with video recordings have been analysed. Our study shows that the children make sense of the digital tools and are able to apply the tools purposefully due to the interaction with an adult within the zone of proximal development (ZPD). The children externalise their reasoning through multimodal interaction, such as drag-and-drop actions, verbal responses, and non-verbal pointing gestures. Their engagement and interaction with the digital tool made the children become involved in an initial process of appropriating mathematical tools such as pair of contrasts, comparison of weights, and the transitive ordering relation.

Designing Playful Inquiry-Based Mathematical Learning Activities for Kindergarten

This study addresses issues related to the process of designing mathematical activities for kindergarten based on the constructs of playful learning and inquiry. The mathematical activities were designed for 5-year-old children in a Norwegian kindergarten setting. In order to address this design process, we focus at issues of mediating design principles and ideas to kindergarten teachers, who are the ones to orchestrate and implement the mathematical activities with children. The study is situated within a design research methodology in which observations, written and personal communication have been used as sources of data. Our analysis shows that the kindergarten teacher closely followed the written instructions given, that she only occasionally orchestrated an inquiry approach to the learning of mathematics, and a limited implementation of playfulness in the activities. These results were due to issues with too detailed written instructions, a possible experienced expectation to carry out all parts of the activities, a possible experienced power relationship between researchers and the kindergarten teacher as well as limited experience in orchestrating mathematical activities and lack of time to adopt playfulness and inquiry as a way of being.