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Abstracted Quantitative Structures: Using Quantitative Reasoning to Define Concept Construction

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Quantitative Reasoning in Mathematics and Science Education

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 21))

Abstract

Over the most recent several decades, researchers have argued the importance of quantitative and covariational reasoning for students’ learning. These same researchers have illustrated the importance of these reasoning processes with respect to local and longitudinal development. In both grain sizes, researchers are detailed in their descriptions of the intended topics or reasoning processes. There is, however, a lack of specificity of generalized criteria for concept construction from a quantitative reasoning perspective. In this chapter, we introduce such criteria through the construct of an abstracted quantitative structure, which has its roots in quantitative reasoning, covariational reasoning, and various Piagetian notions. In introducing the construct, we focus on ideas informing its development and its criteria, and we use it to characterize examples of student actions. We close with comments regarding implications for both teaching and research.

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Notes

  1. 1.

    As we elaborate below, no conceptual structure is truly representation free.

  2. 2.

    We underscore that we do not consider an AQS to be an exhaustive description of the meanings that can be associated with some concept. An AQS is a construct that can be used to describe a meaning for a concept that is rooted in quantitative and covariational reasoning, and we limit our discussion to such meanings except when contrasting them with alternative meanings to clarify AQS defining criteria.

  3. 3.

    We acknowledge that we can identify creative ways to partition the symbol, but it is not used for such purposes. A stronger example of the distinction between a symbol and figurative material that permits quantitative operations is provided in the following section.

  4. 4.

    We thank a reviewer for pointing out the relationship between an educator seeking to engender learning and their having constructed an AQS.

  5. 5.

    We note that we do not consider Caleb’s activity as an indication of C3 as he had previously experienced both circle and segment contexts repeatedly during the teaching experiment. His engagement suggested they were not novel relative to his perceived goal and activity.

  6. 6.

    This task is a modification of the task Saldanha and Thompson (1998) presented.

  7. 7.

    We note that there are several other key aspects to understanding the sine relationship, including measuring quantities in radius lengths, proportionality, and periodicity (Bressoud, 2010; Moore, 2014; Thompson, 2008).

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grants No. DRL-1350342, No. DRL-1419973, and No. DUE-1920538. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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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: Karagöz Akar, G., Zembat, İ.Ö., Arslan, S., Thompson, P.W. (eds) Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Era, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_3

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