Student Understanding of the Relation between Tangent Plane and the Total Differential of two-Variable Functions

  • Maria Trigueros Gaisman
  • Rafael Martínez-Planell
  • Daniel McGee


Action-Process-Object-Schema (APOS) theory and tools resulting from dialogue with the Anthropological Theory of the Didactic (ATD) were used to analyse data from semi-structured interviews and teaching materials to study students’ understanding of the relationship between tangent planes and the total differential. Results of the study show students’ difficulties relating these ideas and suggest a refinement of the initial genetic decomposition. They also underline aspects of the teaching materials that need to be considered to promote those constructions and development of a complete praxeology for the total differential. This study exemplifies how the dialogue between a cognitive theory and one that focuses on institutional aspects of mathematics education, can provide tools to deeply analyse the teaching and learning of a mathematical topic.


APOS Anthropological theory of the didactic Calculus Differential Moments of study Tangent plane 


La théorie Action-Processus-Objet-Schéma (APOS), ainsi que les résultats d'’un dialogue avec la Théorie Anthropologique du Didactique (ATD), ont été appliqués à l’analyse d'’entretiens semi-structurés et de manuels, afin d'’étudier la compréhension, par les étudiants, de la relation entre plans tangents et différentielle totale. Les résultats de l’étude montrent que les étudiants éprouvent des difficultés à relier ces idées et suggèrent des manières d’affiner la décomposition génétique initiale. Ils permettent également d'’identifier des aspects des supports d’enseignement qu’il s'’agit de prendre en compte pour faciliter la construction de ces notions et le développement d’une praxéologie complète pour la différentielle totale. Cette étude illustre la fertilité d’un dialogue entre une théorie cognitive et une théorie centrée sur les aspects institutionnels en didactique des mathématiques, pour s’outiller en vue d’analyser l’enseignement-apprentissage d’un sujet donné.


APOS Théorie Anthropologique du Didactique Analyse Différentielle Moments d’étude Plan tangent 



study was supporteed by ITAM and Asociación Mexicana de Cultura A.C.

Compliance with Ethical Standards

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Río Hondo # 1 Colonia Progreso Tizapán, Delegación Álvaro ObregónIntituto Tecnológico Autónomo de MéxicoBENITO JUAREZMexico
  2. 2.Universidad de Puerto Rico en MayagüezMayaguezUSA
  3. 3.Kentucky Center for MathematicsNorthern Kentucky UniversityHighland HeightsUSA

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