Analysis of two-variable function graphing activities

This is a report of an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic ideas of two-variable functions. Each of the research cycles used semi-structured interviews with students to test an initial conjecture of needed mental constructions, develop supporting classroom activities, and improve the conjecture. The article summarizes findings from each of the research cycles and shows the improvement in students' understanding of functions of two variables.

Journal of Mathematical Behavior, 2019

This article reports on an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a two-variable functions and its graphical representation. Each of the three research cycles used semi-structured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student learning of function of two variables.

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.

This is a study about how graphs of functions of two-variables are taught. We are interested in particular in the techniques introduced to draw and analyze these graphs. This continues previous work dedicated to students' understanding of topics of twovariable functions in multivariable calculus courses. The model of the "moments of study" from the Anthropological Theory of the Didactic (ATD) is used to analyze the didactical organization of the topic of interest in a popular calculus textbook, and in a typical classroom presentation. In so doing we obtain information about the institutional dependence of findings in previous studies.

Abstract: The founder of the anthropological theory of the didactical, Yves Chevallard, was recently awarded the Hans Freudenthal medal, given in recognition of “a major cumulative program of research” in mathematics education. The aim of this paper is to present, to a Nordic audience, an outline of this theoretical programme, and to highlight some of the features which we believe to be of particular relevance to research on mathematics education.

Africa Education Review, 2017

This article reports on a study of six Grade 11 learners" levels of understanding of concepts related to the function definition and representation. Task-based clinical interviews were used to elicit the learners" interpretations and reasoning when working with these function-related concepts. Indicators for Action-Process-Object-Schema (APOS) theory conception levels were formulated and used to categorise learners" written and interview responses into conception levels of understanding of the function concept. According to the South African Curriculum Assessment Policy Statement (CAPS) curriculum, Grade 11 learners are expected to be operating at the object and schema levels after instruction. However, the results indicated that learners were operating at the action and process levels as their understanding was characterised by vague definitions of function-related concepts and memorisation of procedures for translating between symbolic and graphical representations.

Eurasia Journal of Mathematics, Science and Technology Education

The goal of this paper is the combined use of two theories, APOS and OSA, for the analysis of the university students' understanding on the graph of the function and its derivative. For this, we study the students' understanding to solve one graphing problem in relation to the first derivative and characterize their schemas in terms of levels (intra, inter and trans) of development of the schema for sketching ′ when given the graph. We present a multiple case study in which 14 students of the first course of Calculus in one university of Iran participated voluntarily. Results show that most of the students in our study had major problems in developing mental constructions and doing the practical work needed to solve the problem, particularly those mental constructions that have to be made to calculate the derivative at the critical points and to determine the speed of the variation of the inclination of the tangent lines to , which is why most of them have constructed a schema at the intra level of development of the schema for sketching ′ when given the graph. We finish with some final conclusions.

Educational Studies in Mathematics, 2005

The Anthropological Theory of Didactics describes mathematical activity in terms of mathematical organisations or praxeologies and considers the teacher as the director of the didactic process the students carry out, a process that is structured along six dimensions or didactic moments. This paper begins with an outline of this epistemological and didactic model, which appears as a useful tool for the analysis of mathematical and teaching practices. It is used to identify the main characteristics of the mathematical organisation around the limits of functions as it is proposed to be taught at high school level. The observation of an empirical didactic process will finally show how the internal dynamics of the didactic process is affected by certain mathematical and didactic constraints that significantly determine the teacher’s practice and ultimately the mathematical organisation actually taught.

HAL (Le Centre pour la Communication Scientifique Directe), 2022

We used Action-Process-Object-Schema (APOS) theory to analyze the possibility of replication of results obtained in our previous research on student understanding of two-variable functions when using a similar teaching approach in a different institutional context. The experience was conducted at a university in a different country from those in previous studies. The experience consisted in comparing two groups of the same course, one taught through lectures and the other using collaborative work and activities designed with APOS theory. In this study, we show a summary of results obtained through a comparison of students' performance in both groups. Findings show the generalizability of results obtained in previous studies and the possible replication of didactic aspects across institutions. In particular, it was found that using APOS theory's didactical approach favors a deeper understanding of functions of two variables.

Educational Research and Reviews, 2018

The purpose of this study is to analyse the content knowledge on graphs of pre-service elementary mathematics teachers from an anthropological perspective. 112 pre-service elementary mathematics teachers participated in the study. Concentric mixed pattern research method has been used in the study. The data was collected through Graphic Content Knowledge Scale and interview method in order to examine the content knowledge of primary school math teacher candidates related to graphics anthropologically. This scale which was developed within the frame of determined institutional recognitions includes graphic concept and its usage conditions, graphic types and different display forms, making proper transformations between graphics related to a given context, graphic reading, graphic interpreting and graphic drawing skills. In addition, for the purpose of having a more detailed review of the individual recognitions of the 10 participants, a semi-structured interview scale has been developed and used during the interviews. Data acquired have been analysed by using ecologic and praxiologic approach suggested within the framework of Anthropological Theory of the Didactic. The knowledge scale and interview data have been subjected to content analysis and descriptive analysis to interpret the individual recognitions of the pre-service teachers within the scope of institutional recognitions. As a result, it has been observed that the individual recognitions of pre-service teachers related to column, circle and line graphs conform to the institutional recognitions, however in institutional recognitions the techniques specified for column graphs are being used for histogram, hence they have difficulties in comprehending the differences between histogram and column graph. It has also been concluded that they were not aware of a theory based on graph knowledge.