## Abstract

Developing deep conceptual understanding of what Ma (1999) calls fundamental mathematics is a well-accepted goal of teacher education. This paper presents a microanalysis of an intriguing episode within a course designed to encourage such understanding. An adaptation of Krummheuer’s (1995) elaboration of Toulmin’s (1958/2003) diagrams is used to examine video recordings and transcripts of a group of graduate students in secondary mathematics education grappling with the idea of a three-dimensional line having negative slope. The graduate students’ understandings of slope are examined using an expansion of Stump’s (1999, 2001b) categories of conceptions of slope. The episode ends in an interesting impasse, in which the graduate students agree to pursue the idea no further, purposely ignoring the question of negative slope, despite the clear intention of the task. The analysis explores the argumentation, factors of the learning environment, and conceptions of slope that may have contributed to this impasse.

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## Notes

- 1.
Note that defining is an activity in which meaning is assigned to a concept. When done collaboratively, the process often involves argumentation, the negotiation and discussion between collaborators with certain, preexisting ideas as they attempt to reach mutually acceptable conclusions.

- 2.
Pseudonyms are used.

- 3.
References to color indicate work that was done in the analysis process.

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Moore-Russo, D., Conner, A. & Rugg, K.I. Can slope be negative in 3-space? Studying concept image of slope through collective definition construction.
*Educ Stud Math* **76, **3–21 (2011). https://doi.org/10.1007/s10649-010-9277-y

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### Keywords

- Secondary mathematics teachers
- Slope
- Teacher education
- Concept image
- Argumentation