
06356584
j
2014f.00754
Weber, Eric
Thompson, Patrick W.
Students' images of twovariable functions and their graphs.
Educ. Stud. Math. 87, No. 1, 6785 (2014).
2014
Springer Netherlands, Dordrecht
EN
I60
twovariable functions
three dimensions
calculus
quantitative reasoning
covariational reasoning
conceptualization
doi:10.1007/s1064901495480
Summary: This paper presents a conceptual analysis for students' images of graphs and their extension to graphs of twovariable 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 onevariable functions to graphs of twovariable 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 twovariable functions, and discuss implications for instruction.