# Students’ images of two-variable functions and their graphs

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## Abstract

This paper presents a conceptual analysis for students’ images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of one-variable functions to graphs of two-variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.

## Keywords

Two-variable functions Three dimensions Calculus Quantitative reasoning Covariational reasoning## Notes

### Acknowledgments

This research was supported by National Science Foundation (NSF) Grant No. MSP-1050595. Any recommendations or conclusions stated here are of the authors and do not necessarily reflect official positions of the NSF. Any recommendations or conclusions stated here are of the authors and do not necessarily reflect official positions of the NSF.

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