Tacit Knowledge and Embodied Insight in Mathematical Modeling

ABSTRACT

Mathematical modeling is a challenging and creative process. If one considers only interim or final solutions to modeling problems or interviews modelers afterward, often only their explicit models are accessible – those expressed in work products or evinced in verbal and written reflections. The inner world of tacit knowledge and its impacts on mathematical modeling remain largely inaccessible to such approaches. To understand when and how tacit knowledge can emerge onto the explicit plane as insight during modeling, we present an exploratory and instrumental single-case study. We analyze the embodied interactions of a pair of ninth grade (age 14–15) students as they collaboratively solved a modeling problem. We introduce the phenomenon of embodied insight as a pattern of interaction, in which tacit knowledge becomes explicit and shapes modeling work, and we analyze three such episodes from this pair’s modeling. This work contributes to the field by illustrating the potential in studying embodied social discourse in collaboration to reveal the operation of tacit knowledge in mathematical modeling and its expression on the explicit plane.

KEYWORDS:

• Tacit knowledge
• embodied insight
• implicit models
• modeling
• modeling cycle

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

¹ The extracts presented here utilize Jeffersonian conventions of transcription (Jefferson, 2004). Conventions used in the extracts of this paper are as follows. Underlining indicates increased volume, emphasis. Numbers enclosed in parentheses indicate pauses in seconds; a period in parentheses (.) indicates a momentary pause. Phrases enclosed in double parentheses (()) are used to describe non-verbal actions. Phrases enclosed in >inward-pointing angles< indicate a sped-up utterance. Double colons,::, indicate an elongated sound. An upward arrow, ↑, indicates a rise in intonation. And square brackets, [], indicate overlapping speech.

² In the member-checking session, Diego expressed concern that “frustration” was too strong a description for Henry’s feelings. However, Henry smiled and said that he was frustrated at that point.

³ Diego’s phrasing in the member-checking session as he described this YouTube video illuminated the texture of his memory of this sudden moment of insight. He said that, for this “video about a mansion,” he remembered that “in the thumbnail … that it shows you before clicking, there was like part of a just like a car, like in an elevator pretty much, and I, when I was like thinking about how can I get rid of that ramp, I just somehow remembered that image in my head, and I was like ‘oh, yeah! We can just have an elevator that just carries the cars all up!’”