On the Importance of Visuo-Spatial Thinking for Research Mathematicians

While in society mathematics is often thought of as formal and rigid, mathematicians themselves frequently consider the discipline creative and visual. To challenge stereotypes, we focus on visuo-spatial thinking by research mathematicians (n = 232). Via theObject-Spatial Imagery andVerbalQuestionnaire (Blazhenkova& Kozhevnikov, 2009), together with open questions, we ask the following: (1) Are mathematicians visuo-spatial thinkers? (2) Is the degree of visual thinking correlated with mathematical subdiscipline? (3)Which role does visual thinking play in mathematical research? The Object-Spatial Imagery and Verbal Questionnaire results indicate that mathematicians are more strongly visuo-spatial thinkers than scientists, humanities researchers or visual artists. The degree of visuo-spatial thinking does not correlate to how ‘visual’ the mathematical subdiscipline is as measured by average figure environment per article, obtained through text mining 3,799 arXiv articles. In open questions, two thirds of respondents (n = 222) report using visual mental imagery during mathematical research. Some mathematicians mention metaphors for research that refer to spatial movement, such as rock climbing, moving through a jungle or attacking the problem like an insect. Our study contributes to the research agenda set by Alcock et al. (2016), which aims to improve our understanding of mathematical cognition for the purpose of elucidating the nature of mathematical thinking and inform policymakers to address challenges in mathematics education. We conclude that visualisation plays an important part in the practice of mathematics, contrary to common belief. As Hadamard wrote in 1945: ‘deductions in the realm of numbers may be, at least in several mathematical minds, most generally accompanied by images’.Public Significance Statement We asked more than 200 mathematicians about their thinking processes. Through a variety of measures and questionnaires, we found a pervasive role for visualisation in the work of most research mathematicians. This suggests that visuo-spatial skills may deserve attention when designing educational mathematics curricula and in future research in mathematical cognition, in particular on how experts do advanced mathematics.