How are graphs of two-variable functions taught?

Educational Studies in Mathematics

In this study we analyze students' understanding of two-variable function; in particular we consider their understanding of domain, possible arbitrary nature of function assignment, uniqueness of function image, and range. We use APOS theory and semiotic representation theory as a theoretical framework to analyze data obtained from interviews with thirteen students who had taken a multivariable calculus course. Results show that few students were able to construct an object conception of function of two variables.

HAL (Le Centre pour la Communication Scientifique Directe), 2020

In this article we present a short review of our research on student understanding of function of two variables. We describe results dealing with basic aspects such as geometrical understanding and understanding of the definition, then we consider results on student understanding of some notions of the differential calculus: plane, tangent plane, partial derivatives, directional derivatives, and the total differential. Keywords: Teaching and learning of analysis and calculus, teaching and learning of specific topics in university mathematics, functions of two variables, APOS.

International Journal of Mathematical Education in Science and Technology, 2013

In a series of previous studies, the authors have described specific mental constructions that students need to develop, and which help explain widely observed difficulties in their graphical analysis of functions of two variables. This new study, which applies Action-Process-Object-Schema theory and Semiotic Representation Theory, is based on semi-structured interviews with 15 students. It results in new observations on student graphical understanding of two-variable functions. The effect of research findings in designing a set of activities to help students carry out the specific constructions found to be needed is briefly discussed.

epiDEMES

This article presents a review of our research on students’ understanding of the calculus of bivariatefunctions. It summarizes findings from studies conducted during 15 years of research on the topic with the aim ofdisseminating our overall results in an accessible format. The results discussed underscore the new challenges thatstudents face when dealing with this new type of function and suggest that the belief that students can easily generalizefrom their knowledge of one-variable functions is not sustained by research, so the different foundational notionsnecessary for the context of bivariate functions need to be considered explicitly during instruction. We include researchbased suggestions that have practical value for teaching these functions, and update the state of research in thisimportant area of the didactics of mathematics, which makes the need for further research apparent. . Une revue de nos recherches sur la compréhension par les étudiants du calcul des fonctions biva...

HAL (Le Centre pour la Communication Scientifique Directe), 2021

A review of our research about student understanding of the calculus of bivariate functions is presented. It summarizes findings from studies conducted throughout 15 years of research activities on the topic. Its aim is to communicate our research results as a whole in an accessible way. The results we discuss underscore the new challenges that students meet when dealing with this new type of function and suggest that the belief that students can easily generalize from their knowledge of one-variable functions is not sustained by research, so different foundational notions that are necessary for the context of bivariate functions, need to be explicitly considered during instruction. We include the presentation of research-based suggestions that have practical value for teaching these functions and bring to light the status of research in this important area of the didactics of mathematics, making apparent the need for further research.

International Journal of Science and Mathematics Education, 2019

In this paper, we will examine the mathematical knowledge that prospective mathematics teachers draw upon when graphing function graphs and curves, with a special focus on the occurrence of asymptotes. Three tasks which involved a graph of a rational and exponential function and a hyperbola as a conic section were designed and administered to students. We performed this study within the framework of Anthropological Theory of the Didactic to examine the relationship of prospective mathematics teachers' available knowledge with the knowledge to be taught in upper secondary schools and scholarly knowledge relevant for teaching. By studying prospective mathematics teachers' knowledge, we aim to understand the feasibility of our proposed reference epistemological model for graphing functions and curves in the upper secondary school. Our findings reveal students' shortcomings with respect to the choice of the appropriate graphing praxeology for given tasks. Students' graphing strategies relied mostly on plotting points obtained by evaluating a formula, which is a dominant approach in the textbooks we analysed. Plotting points did not lead students to examine asymptotic behaviour, along with the observed monotonicity of a function. Their graphing strategies were found to be predominantly dependent on the particular setting in which the task was presented. Additionally, in our study, the idea of an asymptote as a tangent line at infinity in the geometric setting was questioned.

Mathematics Education Research Journal, 2019

The aim of this research was to explore the mathematical connections that pre-university students make when they sketch the graph of a derivative function and an antiderivative function. Also, we tried to explain the origin of the mathematical connections identified. We assume mathematical connections as a cognitive process through which a person makes a true relationship between two or more mathematical ideas, concepts, definitions, theorems, or meanings with each other. Task-based interviews were used to collect data which included two graphical tasks that involved the derivative function and the antiderivative function. Through thematic analysis, we identified five types of intramathematical connections: procedural, different representations, part-whole, feature, and reversibility, which can serve as a preliminary theoretical framework to study mathematical connections in Calculus in future research; this is a contribution of this research. In addition, results indicated that Mexican students seldom used visualization to solve graphical tasks, so in future research, classroom intervention proposals should be developed to promote the use of visualization including the development of the ability to make mathematical connections, in order to improve their mathematical understanding.

This is a study about the didactical organization of a research based group of activities designed using APOS theory to help university students make constructions needed to understand and graph two-variable functions, but found to be lacking in previous studies. The model of the "moments of study" of the Anthropological Theory of Didactics is applied to analyze the activities in terms of their institutional viability.

The Journal of Mathematical Behavior, 2015

APOS Theory is applied to study student understanding of the differential calculus of functions of two variables, meaning by that, the concepts of partial derivative, tangent plane, the differential, directional derivative, and their interrelationship. A genetic decomposition largely based on the idea of a directional slope in three dimensions is proposed and tested by conducting semi-structured interviews with 26 students who had just taken a course in multivariable calculus. The interviews explored the mental constructions of the genetic decomposition they can do or have difficulty doing. Results give evidence of those mental constructions that seem to play an important role in the understanding of these important concepts.

Proceedings of the 25th Annual Conference on Research in Undergraduate Mathematics Education, 2023

Given the challenge of visualizing the main constructs of two-variable functions and their differential and integral calculus, it is important to consider the use and perceived potential of resources to contribute to students' understanding of multivariable calculus. This case study considers how four instructors attempt to utilize resources in their multivariable calculus teaching and their motivations to do so. We study how these instructors think about the digital and non-digital resources that they use to foment students' understanding. We also look at their reporting of the ways that instructors, students, and resources interact in multivariable calculus to determine if resource use is meant to facilitate visualization, reasoning, or communication. With this, the study proposes to contribute to the discussion of resource use in multivariable calculus.