Material representations in mathematical research practice

Abstract

Mathematicians' use of external representations constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians' use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process.

Mathematical cognition and enculturation: introduction to the Synthese special issue

“…In their article, Johansen and Misfeldt (2018) analyze the embodied physical experiences in practices of mathematics based on an interview study of active research mathematicians. Their analysis lends support to the position that it is crucial to include social and cultural contexts when analyzing the cognitive use of material representations such as diagrams by mathematicians.…”

Dynamic attentional mechanisms of creative cognition

“…Observations of real-life creative performance-and not just tasks confined to laboratory settings-also point to the importance of dynamic interactions between external and internal attention. For example, when working on non-trivial proofs and problems, mathematicians tend to not only focus on the internal contents of the mind, but they do so while constantly interacting with the environment by creating and manipulating diagrams and inscriptions using chalk and blackboard or pen and paper (Johansen & Misfeldt, 2020;Marghetis et al, 2019). We see this facilitation as akin to an extension of working memory, enabling greater creativity.…”

Models as Epistemic Artifacts for Scientific Reasoning in Science Education Research

“…But on closer inspection, such a distinction between concrete and inconcrete models tends to lead astray. For instance, there is accumulating evidence that the perceptual and sensorimotor engagement with external mathematical representations is crucial for mathematical reasoning over and above them functioning as mere scaffolds for mnemonic and communicative tasks [98,99]. On the other hand, the Phillips-Newlyn model, a hydraulic model of a macroeconomy in which colored water flows and accumulates in a system of tanks and channels, does not reduce it to its material embodiment.…”