Introducing students to geometric theorems: how the teacher can exploit the semiotic potential of a DGS

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.     

Learning about multiplication by comparing algorithms: “One times one,but actually they are ten times ten”

Multiplication algorithms in primary school are still frequently introduced with little attention to meaning. We present a case study focusing on a third grade class that engaged in comparing two algorithms and discussing “why they both work”. The objectives of the didactical intervention were to foster students' development of mathematical meanings concerning multiplication algorithms, and their development of an attitude to judge and compare the value and efficiency of different algorithms. Underlying hypotheses were that it is possible to promote the simultaneous unfolding of the semiotic potential of two algorithms, considered as cultural artifacts, with respect to the objectives of the didactical intervention, and to establish a fruitful synergy between the two algorithms. As results, this study sheds light onto the new theoretical construct of “bridging sign”, illuminating students’ meaning-making processes involving more than one artifact; and it provides important insight into the actual unfolding of the hypothesized potential of the algorithms.     

Designing a duo of material and digital artifacts: the pascaline and Cabri Elem e-books in primary school mathematics

This paper focuses on a duo of artifacts, constituted by a physical artifact and its digital counterpart. It deals with the theoretically and empirically underpinned design process of the digital artifact, the e-pascaline developed with Cabri Elem technology, in reference to a physical artifact, the pascaline. The theoretical frameworks of the instrumental approach and the theory of semiotic mediation together with the analysis of teaching experiments with the pascaline support the design in terms of continuity and discontinuity between the two artifacts. The components of the digital artifact were chosen from among the components of the physical artifact that are the object of instrumental genesis by the students and that are analyzed as having a semiotic potential that contributes to didactical aims. Components instrumented by students which had inadequate semiotic potential were eliminated. With the resulting duo, each artifact adds value to the use of the other artifact for mathematical learning.     

Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spaces

The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct events, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own pre-analytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and didactical foundations and then by presenting empirical findings from a design-based research project. Twenty-eight students aged 9?C11 participated in tutorial, task-based clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal event space as a semiotic means of objectifying these presymbolic notions. I elaborate on the thesis via micro-ethnographic analysis of key episodes from a paradigmatic case study.     

Students’ understanding of algebraic notation: A semiotic systems perspective

The ideas of equivalence and variable are two of the most fundamental concepts in algebra. Most studies of students’ understanding of these concepts have posited a gap between the students’ conceptions and the institutional meanings for the symbols. In contrast, this study develops a theoretical framework for describing the ways undergraduate students use personal meanings for symbols as they appropriate institutional meanings. To do this, we introduce the idea of semiotic systems as a framework for understanding the ways students use collections of signs to engage in mathematical activity and how the students use these signs in meaningful ways. The analysis of students’ work during task-based interviews suggests that this framework allows us to identify the ways in which seemingly idiosyncratic uses of the symbols are evidence of meaning-making and, in many cases, how the symbol use enables the student to engage productively in the mathematical activity.     

Artifacts and signs after a Vygotskian perspective: the role of the teacher

The notion of mediation, widely used in the current mathematics education literature, has been elaborated into a pedagogical model describing the contribution of integrating tools to the human activity, and to teaching and learning mathematics in particular. Following the seminal idea of Vygotsky, and elaborating on it, we postulate that an artifact can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical signs, that are detached from the use of the artifact, but that nevertheless maintain with it a deep semiotic link. The teaching organization proposed in this paper is modeled by what we have called the didactical cycle. Starting from assuming the centrality of semiotic activities, collective mathematical discussion plays a crucial role: during a mathematical discussion the intentional action of the teacher is focused on guiding the process of semiotic mediation leading to the expected evolution of signs. The focus of the paper is on the role of the teacher in the teaching–learning process centered on the use of artifacts and in particular a dynamic geometry environment. Some examples will be discussed, drawn from a long-term teaching experiment, carried out over the past years as part of a National project. The analysis is accomplished through a Vygotskian perspective, and it mainly focuses on the process of semiotic mediation centered on the use of artifacts and on the role of the teacher in this process.     

Drawing inferences from learners’ examples and questions to inform task design and develop learners’ spatial knowledge

Examples that learners generate, and questions they ask while generating examples, are both sources for inferring about learners’ thinking. We investigated how inferences derived from each of these sources relate, and how these inferences can inform task design aimed at advancing students’ knowledge of scale factor enlargement (i.e. scaling). The study involved students in two secondary schools in England who were individually tasked to generate examples of scale factor enlargements in relation to specifically designed prompts. Students were encouraged to raise questions while generating their examples. We drew inferences about students’ thinking from their examples and, where available, from their questions. These inferences informed our design and implementation of a set of follow-up tasks for all students, and an additional personalised task for each student who raised any questions. Students showed increased knowledge of, and confidence with, scale factor enlargement independently of whether they asked questions during the exemplification task.     

Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

Definitions and examples in elementary calculus: the case of monotonicity of functions

This paper is devoted to the investigation of students’ understanding and handling of examples in the framework of an example-based introductory mathematics undergraduate course. The plan of the course included a wide use of graphs in standard lectures, tutoring sessions as well as in examinations. This study deals with the notion of increasing function, which has been introduced by means of both the standard definition and a range of examples and non-examples, most often conveyed through graphs. We have analysed students’ interpretations of the notion of increasing function as they applied them in a set of written examination tests. The data gathered have been completed by a number of interviews of students whose answers were difficult to interpret. The outcomes underline the importance of linguistic and semiotic competence and suggest that the design of innovative teaching paths should take care of the linguistic and semiotic skills needed to handle the representations involved.     

Transparent Rule Based CAS to Support Formalization of Knowledge

The complexity of computer algebra systems hinders many students to develop an adequate mental model of such a system. As a result, they are often suspicious about the results and the didactical benefit is limited. The paper suggests that it is possible to design a transparent version of a computer algebra system that gives students a transparent access to the inner working of such a system. Didactical uses of such a system are discussed.     

Declining participation in post-compulsory secondary school mathematics: students’ views of and solutions to the problem

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.     

On a research programme concerning the teaching and learning of linear algebra in the first-year of a French science university

The paper discusses the current approach to the teaching of linear algebra in the first year at a French science university and the main difficulties that students have with this material. A brief account is given of the first steps towards the design of a teaching experiment. From a joint didactical and historical survey a first hypothesis is drawn: epistemological specificity, the use of ‘meta-lever’, the use of changes of settings and points of view, and the importance of the concept of rank. The main aspects and objectives of the teaching design with which we experimented over a whole teaching semester for five years with around 200 students are presented. Finally, the type of evaluations that were set up and the difficulties encountered are explained. The conclusion deals with issues on the teaching and learning of linear algebra as well as issues on methodological and theoretical points in relation to the original didactical framework.     

“No! He starts walking backwards!”: interpreting motion graphs and the question of space, place and distance

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs. The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization. Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”     

Polish teachers’ conceptions of and approaches to the teaching of linear equations to grade six students: an exploratory case study

In this article we present an exploratory case study of six Polish teachers’ perspectives on the teaching of linear equations to grade six students. Data, which derived from semi-structured interviews, were analysed against an extant framework and yielded a number of commonly held beliefs about what teachers aimed to achieve and how they would achieve them. In general, teachers’ aims were procedural fluency founded on students understanding the equals sign as a relational rather than an operational entity and the balance scale as a representation supportive of students’ understanding of an equation as the equivalence of two expressions. The analyses also indicated that the ways teachers proposed to conduct their lessons, whereby they pose single problems for individual work before inviting whole class sharing of solutions, resonates with the didactical traditions found in other East and Central European countries previously influenced by the Soviet Union.     

From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation

Research on the use of artifacts such as textbooks and digital technologies has shown that their use is not a straight forward process but an activity characterized by mutual participation between artifact and user. Taking a socio-cultural perspective, we analyze the role of artifacts in the teaching and learning of mathematics and argue that artifacts influence the didactical situation in a fundamental way. Therefore, we believe that understanding the role of artifacts within the didactical situation is crucial in order to become aware of and work on the relationships between the teacher, their students and the mathematics and, therefore, are worthwhile to be considered as an additional fundamental aspect in the didactical situation. Thus, by expanding the didactical triangle, first to a didactical tetrahedron, and finally to a ??socio-didactical tetrahedron??, a more comprehensive model is provided in order to understand the teaching and learning of mathematics.     

Some didactical and Epistemological Considerations in the Design of Educational Software: The Cabri-euclide Example

We propose to use didactical theory for the design of educational software. Here we present a set of didactical conditions, and explain how they shape the software design of Cabri-Euclide, a microworld used to learn “mathematical proof ” in a geometry setting. The aim is to design software that does not include a predefined knowledge of problem solution. Key features of the system are its ability to verify local coherence, and not to apply any global and automatic deduction.     

Functions: a modelling tool in mathematics and science

It is difficult for the students to transfer concepts, ideas and procedures learned in mathematics to a new and unanticipated situation in science. An alternative to this traditional transfer method stresses the importance of modelling activities in an interdisciplinary context between mathematics and science. In the paper we introduce a modelling approach to the concept of function in upper secondary school is introduced. We discuss pedagogical and didactical issues concerning the interplay between mathematics and science. The discussion is crystallized into a didactical model for interdisciplinary instruction in mathematics and science. The model is considered as an iterative movement with two phases: (1) the horizontal linking of the subjects: Situations from science are embedded in contexts which are mathematized by the students, (2) the vertical structuring in the subjects: The conceptual anchoring of the students' constructs from the horizontal linking in the systematic and framework of mathematics and science respectively.     

A didactical situation for the enhancement of meta-analogical awareness

The aim of this study was to propose a didactical situation for the confrontation of the epistemological obstacle of linearity (routine proportionality) and consequently for the enhancement of meta-analogical awareness. Errors caused by students’ spontaneous tendency to apply linear functions in various situations are strong, persistent and do not disappear with traditional instruction. The effects of a didactical situation on the way students perceive and handle proportional and non-proportional relations were examined. The situation consisted of four parts which referred to the situations of action, formulation, validation and institutionalisation and was presented as a game to four twelve-year students of different abilities. The results showed the potential of the application of a didactical situation towards enhancing students’ meta-analogical awareness and therefore their ability to discern and handle linear and non-proportional relations.     

The Case of Gaël

The Theory of Didactical Situations has had a central position in French mathematics education research since the early 1970s. A major component of this theory is the didactical contract, a completely implicit but highly powerful aspect of the relationship between teacher and student. In this article we relate the series of tutorial sessions which provoked the original formulation of that theory, and in which the theory was validated by its first application.Gaël was an intelligent child who was failing exclusively in mathematics. He was one of nine cases studied between 1980 and 1985 (at the Bordeaux COREM³). After observing him in class and offering him various learning situations, both didactical and adidactical, we arrived at the hypothesis that Gaël was implementing a strategy of avoidance of the “conflict of knowing,” which we characterized as “hysteroid type avoidance,” whereas the others exhibited “obsessional type avoidance” (note that these behaviors should not be confused with the psychiatric categories of the same name, which are serious personality disorders). It was possible to offer psychological explanations for this behavior, but they did not provide the means for correcting the avoidance, and they focused the interest of the researchers on a characteristic of the child or on his competencies, rather than remaining at the level of his behavior and the conditions which provoked it or which might modify it. This behavior demonstrated the refusal, conscious or not, of the child to accept his share of the decision-making responsibilities in a didactical situation and hence to learn while working with an adult.Studying Gaël's behavior enabled the experimenters to explore and understand the constraints of the didactical situation, interpreted as a “didactical contract.” It is the simulacrum of a contract, an illusion, intangible and necessarily broken, but a fiction that is necessary in order for the two protagonists, the teacher and the learner, to engage in and carry out the didactical dialectic. The didactical means to get a student to enter into such a contract is devolution. It is not a pedagogical device, because it depends in an essential way on the content. It consists of putting the student into a relationship with a milieu from which the teacher is able to exclude herself, at least partially (adidactical situation). The mechanism implemented was devised to engage Gaël progressively but explicitly in a challenge in which the teacher could be “on the student's side.”The mathematical aspects of this situation subsequently proved to be one of the fundamental didactical situations of subtraction.