Conceptual understanding of the dot product

The aim of this study is to investigate whether it is possible to illuminate the development of conceptual understanding of the dot product through analyses of smallgroup dialogues. In the study we will focus on language, i.e. on the nature of the argumentation that develops. The article presents a rationale for conceptual learning and collaborative learning from a socio-cultural perspective. The article focuses on four sequences that make intelligible how the students use mathematical language and show a highly coordinated thinking-together mode. In spite of inaccurate mathematical formulations, the problem-solving process evolves and the students understand each other. The sequences also show how the students’ argumentation evolved, how it changed because of the listeners’ contributions, and in which way definitions are understood, used and applied.