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Relating the different derivatives in the development of the Geometric Differential Calculus Schema for two-variable functions.

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Title: Relating the different derivatives in the development of the Geometric Differential Calculus Schema for two-variable functions.
Authors: Trigueros, María1 (AUTHOR) mtriguerosg@gmail.com, Martínez-Planell, Rafael2 (AUTHOR), Borji, Vahid3 (AUTHOR)
Source: ZDM - Mathematics Education. Dec2025, Vol. 57 Issue 7, p1357-1370. 14p.
Subject Terms: *Cognitive learning theory, *Teaching methods, Derivatives (Mathematics), Multivariable calculus, Schemas (Psychology), Differential geometry
Abstract: This study examines students' construction of relations among the different concepts related to the rate of change of two-variable functions. It uses the notions of Schema and Schema development from APOS theory to analyze students' constructed relations between different components of the Geometric Differential Calculus Schema after they conclude a multivariable calculus course using APOS theory's didactic methodology. The course emphasized the tangent plane at a point on a surface to give meaning to the different concepts associated with function change. Results obtained contribute to research on the teaching and learning of the differential calculus of two-variable functions. This study contributes to the state of knowledge by showing how students construct relations between the tangent plane and the different derivatives defined for these functions. It also contributes to the state of knowledge by showing how using different representations helps students understand the various approaches to rates of change in this type of function. Findings inform about relations between Schema components that students can establish, those they have difficulty constructing, and about didactic strategies to help university students construct a deeper understanding of change in multivariable calculus functions. [ABSTRACT FROM AUTHOR]
Copyright of ZDM - Mathematics Education is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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  Data: <searchLink fieldCode="JN" term="%22ZDM+-+Mathematics+Education%22">ZDM - Mathematics Education</searchLink>. Dec2025, Vol. 57 Issue 7, p1357-1370. 14p.
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– Name: Abstract
  Label: Abstract
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  Data: This study examines students' construction of relations among the different concepts related to the rate of change of two-variable functions. It uses the notions of Schema and Schema development from APOS theory to analyze students' constructed relations between different components of the Geometric Differential Calculus Schema after they conclude a multivariable calculus course using APOS theory's didactic methodology. The course emphasized the tangent plane at a point on a surface to give meaning to the different concepts associated with function change. Results obtained contribute to research on the teaching and learning of the differential calculus of two-variable functions. This study contributes to the state of knowledge by showing how students construct relations between the tangent plane and the different derivatives defined for these functions. It also contributes to the state of knowledge by showing how using different representations helps students understand the various approaches to rates of change in this type of function. Findings inform about relations between Schema components that students can establish, those they have difficulty constructing, and about didactic strategies to help university students construct a deeper understanding of change in multivariable calculus functions. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of ZDM - Mathematics Education is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s11858-025-01728-6
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      – SubjectFull: Derivatives (Mathematics)
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      – SubjectFull: Multivariable calculus
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              Text: Dec2025
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