Professional Mathematicians' Level of Understanding: An Investigation of Pseudo-Objectification

This research study investigated how professional mathematicians understand and operate with highly-abstract, advanced mathematical concepts in their own work. In particular, this study examined how professional mathematicians operated with mathematical concepts at different levels of understanding. Moreover, this study aimed to capture what factors influence professional mathematicians' level of understanding for particular mathematical concepts.

To frame these research goals, three theoretical levels of understanding were proposed, process-level, pseudo-object-level, object-level, leveraging two ways that Piaget (1964) described what it meant to know or understand a mathematical concept. Specifically, he described understanding an object as being able to "act on it," and also as being able to "understand the process of this transformation" (p. 176). Process-level understanding corresponds to only understanding the underlying processes of the concept. Pseudo-object-level understanding corresponds to only being able to act on the concept as a form of object. Object-level understanding corresponds to when an individual has both of these types of understanding. This study is most especially concerned with how professional mathematicians operate with a pseudo-object-level understanding, which is called pseudo-objectification.

For this study, six professional mathematicians with research specializing in a subfield of algebra were each interviewed three times. During the first interview, the participants were given two mathematical tasks, utilizing concepts in category theory which were unfamiliar to the participants, to investigate how they operate with mathematical concepts. The second interview utilized specific journal publications from each participant to generate discussion about influences on their level of understanding for the concepts in that journal article. The third interview utilized stimulated recall to triangulate and support the findings from the first two interviews.

The findings and analysis revealed that professional mathematicians do engage in pseudo-objectification with mathematical concepts. This demonstrates that pseudo-objectification can be productively leveraged by professional mathematicians. Moreover, depending on their level of understanding for a given concept, they may operate differently with the concept. For example, when participants utilized pseudo-objects, they tended to rely on figurative material, such as commutative diagrams, to operate on the concepts. Regarding influences on understanding, various factors were shown to influence professional mathematicians' level of understanding for the concepts they use in their own work. These included factors pertaining to the mathematical concept itself, as well as other sociocultural or personal factors.