Apprehensions in the Graphic Register of Two-variable Functions

Katia Vigo Ingar,Maria José Ferreira Da Silva

doi:10.13189/ms.2017.050201

Abstract

The objective of this article is to present part of a doctoral thesis, which deals with an extension of Duval's study in relation to apprehensions in the graphic register of a two-variable function. Its relevance is extensive in teaching and learning Differential Calculus of two variables since the information the graph of this type of functions may provide is important to build knowledge on two-variable functions and for its applications. For graphic representation and knowledge building, we rely on CAS Mathematica, given that its dynamism allows performing operations in the graphic register. Because of this, we ask ourselves, how do apprehensions take place in the CAS graphic register of two-variable functions? Our research is qualitative and exploratory since the proposed object of study has not been studied a lot. We believe the interaction of apprehensions in the CAS graphic register allows students to conjecture properties of the two-variable functions when, for instance, a student applies those notions to optimization problems.

Highlights

In the last two decades, the study of two-variable functions has developed progressively, as shown by the works of Yerushalmy [12], Kabel [7], Montiel et al [9] and Trigueros and Martínez-Plannel [10], [11] since the theoretical frameworks of the APOS theory (Action, Process, Object, Schema), the Onto-semiotic approach and the Anthropological Theory of the Didactic (ATD), even though the studies on the visualization process are scarce in the area of calculus in two variables
In the graphic register of a two-variable function, Ingar [5] states that the perceptual apprehension of the CAS graph, as the example shown in figure 1, allows identifying a paraboloid, fulfilling the epistemological function to identify objects, as stated by Duval [1]
We show the section of the graph in the horizontal plane z=0, gotten when we wrote the command ContourPlot3D[{z==20}, {x, -3, 3}, {y, -3, 3}, {z, 0, 25}, AxesLabel→{“X”, “Y”, “Z”}, AxesOrigin→{0,0,0}, Boxed→False] and the command Show to show the sections. We believe these apprehensions are necessary to understand the possible variations in the graphic register of two-variable functions, due to the fact that we are interested in studying the cognitive activities students mobilize to develop visualization in such register, since the apprehensions in this register allow to organize relations among units of representation, i.e. among visual variables

Summary

Introduction

In the last two decades, the study of two-variable functions has developed progressively, as shown by the works of Yerushalmy [12], Kabel [7], Montiel et al [9] and Trigueros and Martínez-Plannel [10], [11] since the theoretical frameworks of the APOS theory (Action, Process, Object, Schema), the Onto-semiotic approach and the Anthropological Theory of the Didactic (ATD), even though the studies on the visualization process are scarce in the area of calculus in two variables This is demonstrated with the works of Zimmermann and Cunningham [13], Guzmán [4] and Duval [2]. In the graphic register of a two-variable function, Ingar [5] states that the perceptual apprehension of the CAS graph (graphic register using Mathematica), as the example shown in figure 1, allows identifying a paraboloid, fulfilling the epistemological function to identify objects, as stated by Duval [1]

Visualization in the Graphic Register

Apprehension in the Graphic Register of Two-variable Functions

Final Considerations