Operations have to operate on something and that something is the figurative material contained in the operations, figurative material that has its origin in the construction of the operations.

- Steffe, personal communication

## Inspiration

3+4

You probably thought of the word "seven" or imagined the symbol "7" without even thinking (at least in a conscious way...we're always thinking and anything brought to mind is due to some form of brain activity). If so, you didn't do mathematics. Said more precisely, you didn't engage in reasoning unique to mathematics.

Piaget was interested in the mental operations that were particular to mathematical reasoning, as he viewed such reasoning as unique due to its logico/operational structure. Both Steffe and Thompson have carried forward this tradition, which I work to build upon in ways truthful to that view.

## Description

The primary focus of my research is students' quantitative reasoning, and specifically their construction and enactment of quantitative operations and relationships. Returning to Steffe's quote above, quantitative operations and relationships necessitate figurative material for their enactment; one cannot enact quantitative operations and construct quantitative relationships without such material, whether imagined or "physically" present. It follows that actions in the context of inscriptions, glyphs, and other symbolic forms are not indicative of quantitative operations. Such actions might symbolize or point to quantitative operations and relationships, but they do not afford them. For instance, the symbol "3" was socially negotiated as a way to signify those operations involved in measuring some magnitude as three of some unit. The symbol “3” does not afford the enactment of quantitative operations. On the other hand, a segment (more naturally) provides figurative material to assimilate via quantitative operations as having a measure of “3”. Formulas (or tables) and graphs/phenomenons follow the same distinction.

- Moore et al., 2022, p. 40

To be clear, actions involving inscriptions, glyphs, and other symbolic forms are critically important for the practice of mathematics. But such actions are hardly unique to the mathematical world; we regularly engage with inscriptions, glyphs, and other symbolic forms that symbolize entire systems of images and conceptual structures. A most salient experience of this occurs when you read your favorite book or travel to a new city. We could not function without the symbolization of complex ideas...well, we could, but life and progress would look a lot different. Regardless, my research interest is the construction and enactment of quantitative operations and relationships, and thus you will find me immersing participants in playgrounds that afford such actions. There are times I'm interested in their symbolizing those operations, but bear in mind a participant acting within those symbolized forms gives little insight into their quantitative reasoning. They do, however, give all sorts of useful insights into how they perceive those symbolized forms as re-presenting quantitative operations and relationships (or something entirely else).

- Moore et al., 2022, p.41

## Further Reading

Moore, K. C., Liang, B., Stevens, I. E., Tasova, H. I., & Paoletti, T. (2022). Abstracted quantitative structures: Using quantitative reasoning to define concept construction. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative Reasoning in Mathematics and Science Education (pp. 35-69). Springer International Publishing. https://doi.org/10.1007/978-3-031-14553-7_3

Stevens, I. E. (2022). “A=2πrh is the surface area for a cylinder”: Figurative and operative thought with formulas. In Karunakaran, S. S., & Higgins, A. (Eds.) Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education (pp. 613–621). Boston, MA. READ

Liang, B. & Moore, K. C. (2021). Figurative and operative partitioning activity: Students’ meanings for amounts of change in covarying quantities. Mathematical Thinking and Learning, 23(4), 291-317. https://doi.org/10.1080/10986065.2020.1789930

Stevens, I. E. (2019). Pre-Service Teachers’ Construction of Formulas through Covariational Reasoning with Dynamic Objects. [Unpublished doctoral dissertation]. University of Georgia. READ

Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L. F., & Liang, B. (2019). Pre-service teachers’ figurative and operative graphing actions. The Journal of Mathematical Behavior, 56. https://doi.org/10.1016/j.jmathb.2019.01.008