3—237 Function and Graph in DGS Environment

International Group For the Psychology of Mathematics Education, 2003

Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking Vygotskian perspective of semiotic mediation, a teaching experiment has been designed with the aim of introducing pupils to the idea of function. First data coming from the observations in Italian and French classrooms are presented.

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The study of the concept of the graphic representation in Moroccan first year Baccalaureate, section experimental sciences, finds its origin in the mathematics curriculum that gives an important place to the study of functions. This research aims at exploring how this concept is presented in the Moroccan mathematics curriculum using the theoretical framework of didactic transposition, as it was developed by Y.Chevallard (1985) and that of R. Duval (1993) on the semiotic representations, specifically on the graphic register, and on the conversion of the algebraic register to the graphic register. Our article relies more precisely on the ability of students to solve various tasks involving the use of graphic representation of a function.

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.

2012

In our research and practice, to solve an equation and to represent a graph are placed as tools for problem solving in the curriculum. We set the new integrated unit called "Function and Equation" which is reorganized conventional units in Japan, intended to cultivate students' ability and attitude to apply those in effectively in each school grade. To realize this, we edited the experimental textbooks along the program that we reconstructed. These are verified through lesson studies.

Educational Studies in Mathematics, 2007

Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking the Vygotskian perspective of semiotic mediation, a teaching experiment was designed with the aim of introducing students to the idea of function. This paper focuses on the use of the Trace tool and its potentialities for constructing the meaning of function. In particular, starting from a dynamic approach aimed at grounding the meaning of function in the experience of covariation, the Trace tool can be used to introduce the twofold meaning of trajectory, at the same time global and pointwise, and leads students to grasp the notion of function.

Technology, Knowledge and Learning

In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students' perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students' knowledge develops in specific social and cultural contexts, students' construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students' personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students' engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students' meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher.

ZDM, 2013

Since their appearance new technologies have raised many expectations about their potential for innovating teaching and learning practices; in particular any didactical software, such as a Dynamic Geometry System (DGS) or a Computer Algebra System (CAS), has been considered an innovative element suited to enhance mathematical learning and support teachers' classroom practice. This paper shows how the teacher can exploit the potential of a DGS to overcome crucial difficulties in moving from an intuitive to a deductive approach to geometry. A specific intervention will be presented and discussed through examples drawn from a long-term teaching experiment carried out in the 9th and 10th grades of a scientific high school. Focusing on an episode through the lens of a semiotic analysis we will see how the teacher's intervention develops, exploiting the semiotic potential offered by the DGS Cabri-Géomètre. The semiotic lens highlights specific patterns in the teacher's action that make students' personal meanings evolve towards the mathematical meanings that are the objective of the intervention.

International Group for the Psychology of Mathematics Education, 2011

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. Antecedents Despite its importance, there are not many published articles in the mathematics education research literature that deal with the particularities of functions of two variables. The first published article we found that explicitly treats functions of two variables is by Yerushalmy (1997). In it he insisted on the importance of the interplay between different representations to generalize key aspects of these functions and to identify changes in what seemed to be fixed properties of each type of function or representation. Kabael (2009) studied the effect that using the "function machine" might have on student understanding of functions of two variables, and concluded that it had a positive impact in their learning. In other work, Montiel, Wilhelmi, Vidakovic, & Elstak (2009) considered student understanding of the relationship between rectangular, cylindrical, and spherical coordinates in a multivariable calculus course. They found that the focus on conversion among representation registers and on individual processes of objectification, conceptualization and meaning contributes to a coherent view of mathematical knowledge. Martínez-Planell and Trigueros (2009) investigated formal aspects of students' understanding of functions of two variables and identified many specific difficulties students have in the transition from one variable to two variable functions. Using APOS theory, they related these difficulties to specific coordinations that students need to construct among the set, one variable function, and R 3 schemata. Finally, in a study about geometric aspects of two variable functions, Trigueros and Martinez-Planell (2010) concluded that students' understanding can be related to the structure of their schema for R 3 and to their flexibility in the use of different representations. They gave evidence that the understanding of graphs of functions of two variables is not easy for students, that it can be related to the structure of students' schema for R 3 , and in particular, that intersecting surfaces with planes, and predicting the result of this intersection, plays a fundamental role in understanding graphs of two variable functions and was particularly difficult for students. The way students are taught, and the way mathematical topics are introduced in the textbooks used by students plays an important role on what they learn. In this study we analyze the way graphs of functions of two variables are presented in a widely used textbook, and in standard university classrooms. Our research questions for the part of the study we present here are:  How is the topic "graphs of two-variable functions" introduced in a widely used textbook?

Mathematics, 2021

In this paper, we elaborate on theoretical and methodological considerations for designing a sequence of tasks for introducing middle and high school students to functions and their graphs. In particular, we present didactical activities with an artifact realized within a dynamic interactive environment and having the semiotic potential for embedding mathematical meanings of covariation of independent and dependent variables. After laying down the theoretical grounds, we formulate the design principles that emerged as the result of bringing the theory into a dialogue with the didactical aims. Finally, we present a teaching sequence, designed and implemented on the basis of the design principles and we show how students’ efforts in describing and manipulating the different graphs of functions can promote their production of specific signs that can progressively evolve towards mathematical meanings.