University student understanding of the gradient of a function of two variables: an approach from the perspective of the
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University student understanding of the gradient of a function of two variables: an approach from the perspective of the theory of semiotic representation registers Oihana Moreno-Arotzena 1 & Ion Pombar-Hospitaler 1 & José Ignacio Barragués 1 Accepted: 2 October 2020/ # Springer Nature B.V. 2020
Abstract
The aim of this research is to study how students recognise the concept of gradient of a function of two variables in its different representations and how they make conversions between these representations. Three groups of university students with different profiles took part in this research, and Duval’s theory of semiotic representations was taken as a framework to analyse the resulting data. The paper describes recognition and conversion difficulties and demonstrates that conversion direction is an important factor when determining students’ success in conversions between representations. We discuss the implications of this study on teaching and learning the concept of gradient at undergraduate level. Keywords Gradient . Function of two variables . Semiotic representation . Recognition . Conversion
1 Introduction and purpose of the study As they learn about the functions of several variables, students’ mathematical experience begins to broaden, opening up highly useful science and engineering concepts: vector-valued function, curvilinear integral, multiple integral, differential equations, etc. However, the relevance of the function of several variables contrasts against scarce research into how it is taught and learned (Trigueros & Martínez-Planell, 2010; Martínez-Planell, Trigueros, & McGee, 2015a; Suárez-Aguilar, 2015). The little research that is available has shown, firstly, that it is important that students should be able to use multiple representations of the concept and understand the relations between them (Kabael, 2011) and, secondly, that the concepts of
* José Ignacio Barragués [email protected]
1
University of the Basque Country, Donostia-San Sebastián, Spain
Moreno-Arotzena O. et al.
multivariable analysis are abstract and difficult for students (Martínez-Planell et al., 2015a). Presenting these functions as a simple generalisation of the functions of a variable is not enough. As Yerushalmy (1997) points out, this generalisation requires forms of representation and operation to be designed, and the concept of rate of change to be rethought. For real functions of a real variable, the change in the dependent variable takes place when the independent variable moves along its axis. However, when talking about functions of two variables, a specific direction (vector) must be chosen initially to evaluate the rate of change. This leads to mathematical problems that must be solved, such as seeking the direction that gives the maximum rate of change (gradient vector), the value of this maximum (gradient vector modulus), its relationship with the rates of change obtained when the displacement directions are taken as parallel to the coordinate axes (partial derivatives), plus how all t
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