Students' images of two-variable functions and their graphs

Eric Weber, Patrick Thompson

Research output: Contribution to journalArticlepeer-review

33 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)67-85
Number of pages19
JournalEducational Studies in Mathematics
Issue number1
StatePublished - Sep 2014


  • Calculus
  • Covariational reasoning
  • Quantitative reasoning
  • Three dimensions
  • Two-variable functions

ASJC Scopus subject areas

  • Education
  • General Mathematics


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