### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 67-85 |

Number of pages | 19 |

Journal | Educational Studies in Mathematics |

Volume | 87 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Social Sciences(all)

### Cite this

**Students' images of two-variable functions and their graphs.** / Weber, Eric; Thompson, Patrick.

Research output: Contribution to journal › Article

*Educational Studies in Mathematics*, vol. 87, no. 1, pp. 67-85. https://doi.org/10.1007/s10649-014-9548-0

}

TY - JOUR

T1 - Students' images of two-variable functions and their graphs

AU - Weber, Eric

AU - Thompson, Patrick

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Calculus

KW - Covariational reasoning

KW - Quantitative reasoning

KW - Three dimensions

KW - Two-variable functions

UR - http://www.scopus.com/inward/record.url?scp=84905197445&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905197445&partnerID=8YFLogxK

U2 - 10.1007/s10649-014-9548-0

DO - 10.1007/s10649-014-9548-0

M3 - Article

VL - 87

SP - 67

EP - 85

JO - Educational Studies in Mathematics

JF - Educational Studies in Mathematics

SN - 0013-1954

IS - 1

ER -