The materiality of mathematics: Presenting mathematics at the blackboard

Abstract

Sociology has been accused of neglecting the importance of material things in human life and the material aspects of social practices. Efforts to correct this have recently been made, with a growing concern to demonstrate the materiality of social organisation, not least through attention to objects and the body. As a result, there have been a plethora of studies reporting the social construction and effects of a variety of material objects as well as studies that have explored the material dimensions of a diversity of practices. In different ways these studies have questioned the Cartesian dualism of a strict separation of 'mind' and 'body'. However, it could be argued that the idea of the mind as immaterial has not been entirely banished and lingers when it comes to discussing abstract thinking and reasoning. The aim of this article is to extend the material turn to abstract thought, using mathematics as a paradigmatic example. This paper explores how writing mathematics (on paper, blackboards, or even in the air) is indispensable for doing and thinking mathematics. The paper is based on video recordings of lectures in formal logic and investigates how mathematics is presented at the blackboard. The paper discusses the iconic character of blackboards in mathematics and describes in detail a number of inscription practices of presenting mathematics at the blackboard (such as the use of lines and boxes, the designation of particular regions for specific mathematical purposes, as well as creating an 'architecture' visualising the overall structure of the proof). The paper argues that doing mathematics really is "thinking with eyes and hands" (Latour, 1986). Thinking in mathematics is inextricably interwoven with writing mathematics.

Closing seminars and lectures: The work that lecturers and students do

“…Towards the scheduled end time of a lecture, the lecturer has just finished proving one direction of an equivalence proof (cf. Greiffenhagen, 2014). He now has to decide whether proving the other direction will fit into the remaining time.…”

I See What You Are Doing: Student Views on Lecturer Use of Tablet PCs in the Engineering Mathematics Classroom

“…Although writing boards (blackboards or whiteboards) have remained a preferred delivery technology for many university teachers of mathematics disciplines (Greiffenhagen, 2014), the centralisation of responsibility for the management of teaching rooms in many institutions has encouraged the development of standardised technologies in generic rooms. The data projector, connected to a standard desktop computer, has become a standard configuration in many university teaching spaces (AETM, 2015), including in the study institution, with less emphasis being placed on the provision of large-scale traditional writing boards.…”

Writing-in-interaction

“…Public writing is well illustrated by the use of the blackboard in classroom. Greiffenhagen (2014) has explained in detail how the lecturer uses writing and inscription of mathematical symbols in conjunction with talk during a public lecture in logic. Pitsch (2006), on the other hand, has studied how the public inscriptions on the blackboard can also be produced more collaboratively, as a result of discussion and negotiation.…”

“…This is mainly the case of blackboard writing, typically in school settings (Pitsch 2006(Pitsch , 2007Greiffenhagen 2014) but also in meetings (Schmitt 2001;Mondada 2011). In these settings, the issue of visibility matters for the entire group of participants and for the very organization of their activity.…”

“…In all of these cases, the participants are involved in a form of multiactivity (Haddington et al 2014) (although these previous studies do not use this term) in which a) talking and writing are happening 'at the same time' and b) writing is related, and even dependent on talking. In terms of multiactivity it is precisely the detailed analysis of when and how writing takes place that allow to specify the position of writing relatively to other actions: as shown by studies of multiactivity, it is fundamental to situate the exact temporality of both activities in their emergent details in order to understand multiactivity as occurring in parallel or in succession.…”