This study concerns the teaching of Calculus in secondary school, with a focus on the introduction of the limit notion (sequences and functions). Prior research has highlighted recurrent difficulties of first-year university students with this notion. We try to show that the secondary school curriculum in French-speaking Belgium offers many continuities between high school and university. However, the analysis of a textbook shows discrepancies with the potential identified in the curriculum.
The concept of function traverses the mathematics studied at school and university and is an important transitional link between the two. We examine how an interval perspective on function may help with constructing graphical antiderivative functions. In doing so a number of important constructs needed to build such a graphical understanding are considered, along with how the students in the study linked them and built with them. In addition, some of the difficulties they faced and possible reasons for them are explained. The evidence presented shows that an interval perspective on function was important in being able to construct antiderivative functions graphically. Hence, we propose that this interval perspective on function may prove useful in helping students in transition to construct a local perspective on function. In turn we suggest what a potential path for thinking about graphical antiderivatives could look like and the kind of activities that could assist transition students along it.
In France, current curricular choices tend to make the role of order in analysis invisible. They implicitly favour a topological point of view related to metric spaces, which we find in the notion of absolute value, generalised to that of distance. These choices lead us to leave aside a complementary perspective that emphasises ordered sets, in which the connected parts of ℝ, which are intervals, play a central role. In our epistemological study and using two examples from the secondary-tertiary transition, we show that considering the relationship between connectedness and completeness in the set of real numbers encourages an explicit consideration of the role of order in analysis, especially in proofs. Finally, we suggest some avenues of research opened up by this work to help take into account the dual nature of this transition.
This introductory paper presents the aims and structure of the special issue. The seven contributions gathered in the issue explore secondary-tertiary calculus transition through three thematic clusters: cognitive development and conceptual foundations, epistemological and instructional continuity, and curriculum structure and pedagogical proposals. This paper situates these studies within current international research trends, emphasizing the need for epistemological unity, representational fluency, and teacher empowerment to foster coherent and meaningful calculus learning trajectories.
Integral calculus presents persistent challenges for students in general and for those transitioning from secondary to tertiary education in particular. This study examines the development of "Accumulative Thinking" as a foundation for understanding integration. In a purposely designed learning activity, pairs of Grade 11 students explored accumulation concepts using the context of water flowing into a pool. Using the Abstraction in Context framework, we analysed students’ processes of constructing knowledge during this activity. Our findings indicate that most students constructed elements of Accumulative Thinking, preparing them for future studies; the findings also demonstrate how real-world contexts can facilitate the development of Accumulative Thinking.
The triad order, continuity, and reconceptualization that appears in the title of this paper refers to a juxtaposition of three aspects of calculus education. Order refers to differentiation followed by integration (DI approach) versus integration followed by differentiation (ID approach) versus Thompson’s integrated approach (TI approach) which view differentiation and integration inseparable. Continuity refers to the impact of these approaches on student learning as they transition from high school to university. Reconceptualization refers to the effort to reform calculus learning and teaching by reeducating future secondary teachers relearn calculus concepts and ideas through the lens of quantitative reasoning. This is an analytic paper. It begins with an analysis of the cognitive and pedagogical features of the three approaches, DI, ID, and TI, and continues with a discussion of the continuity problem concerning the transition from school mathematics to university mathematics, focusing on the difficulty to reform calculus education in the U.S. To advance this reform, it is necessary to examine in depth the current approaches to calculus education in the U.S., as well as alternative approaches advocated by mathematicians and mathematics education scholars. The analysis of the three approaches, DI, ID, and TI, aims at contributing to this essential examination. As part of this examination, the paper offers a calculus module for prospective secondary teachers who have already taken the “mainstream” calculus sequence. The module, while akin to the TI approach, its development and implementation rest on a separate theoretical framework.
Our research team has been working on the development of aid sessions aimed at students with difficulties in mathematics, that take place prior to a classroom lesson and that aim at preparing these students to fully engage into classroom activity. One of the main issues during the development of each aid session is the necessity to prescisely articulate the content of the aid session with the classroom task. In order to clarify under which conditions the articulation of the aid session and the classroom task optimizes the functions of the aid session, we elaborated a classification of possible articulations. We conducted a new classroom experimentation and based our analysis of the articulations between aid sessions and classroom tasks on concepts from the anthropological theory of the didactic and the joint action theory in didactics.
In previous research, we developed a framework for a didactic analysis of mathematics textbooks in the field of geometry at primary school. As a continuation of this work, the present study investigates the possibility of establishing a link between an analysis of a textbook’s “intrinsic quality” and the effects observed on its use by a teacher when conducting learning activities focused on the concept of parallelism in Year 4 (students aged 9–10). We reveal the numerous adaptations of the textbook's suggestions made by the teacher of the class studied. Our analyses also show that the expertise of a teacher using a textbook of "low didactic quality" may not be sufficient to detect and compensate for the failings of this textbook.
This article offers a didactic analysis of an activity derived from scientific outreach, at the interface between mathematics and computer science. We analyse an activity based on a particular combinatorial game and a machine which learns to win at this game, conceived within the framework of unplugged computer science. Using the theoretical framework of the theory of didactical situations and a didactic engineering methodology, we analyse learning potentials, and organize a didactic situation, experimented with 14-15 year-old pupils. A priori and a posteriori analyses enable us to show the potential for mathematical learning, AI-related learning, and the way in which these two learning stakes are articulated in the students' work. The results highlight the relevance of the theory of didactical situations for scientific outreach contexts, as well as for the didactics of computer science and the interactions between mathematics and computer science.
In the learning of mathematics, solving exercises is an fundamental component for practice and deepening knowledge. This study examines the characteristics of exercises presented in ten textbooks used in the final year of primary education in France (Grade 5) within a teaching sequence focused on the ordering of decimal numbers. Drawing on activity theory in mathematics didactics, we developed a systematic analytical framework. Our analysis reveals significant variability among the exercises proposed in the ten textbooks on this content. However, we also identified certain general trends: exercises mainly aimed at the direct application of knowledge with little contextualization, few changes in register, and mainly offered during the lesson and before assessment. These results highlight a predominant focus in textbook exercises toward practicing procedural skills rather than fostering deeper conceptual understanding.
In this article we present the analysis grid we designed to analyze classroom sessions integrating programmable floor robots, as part of a research project aimed at studying the potential of educational robotics in the development of spatial knowledge. We based our tool on three levels. The first is based on classroom observations of different aspects of the integration of programmable robots: the robot itself, spatial knowledge and the programming used to move the robot. The second level focuses on the spatial tasks at stake in the lessons, while the last level focuses on the conditions under which the robot is integrated into the classroom. We present the conception and theoretical grounds for this grid and implement it on a case study from our research.
In our modern society, individuals are often faced with situations in which they have to make decisions spontaneously in an environment involving chance, in which the use of probability remains an adequate tool occupying an important status. In this article, we present an empirical study exploring the manifestation of certain biases and heuristics among moroccan learners during such decision-making. We also propose comparisons between learners from different school cycles within the moroccan educational system. In order to carry out this project, we selected a sample of 410 individuals, to whom we administered a questionnaire composed of six questions, involving four heuristics and one bias. The results we obtained showed that "classical" teaching does not develop, in the learners, the erroneous spontaneous intuitions that often remain next to the taught models and retain their areas of action.
The ways teachers deal with their students' errors reveal their practices, but also their relationship to the knowledge they teach and to the students to whom they address. Between improvisation regulated in situation and components of professional culture, these modes of doing have to do both with the multi-temporality and with the multi-addressing of teaching activity. Promoting the reflexivity of students and teachers on errors and their methods of processing requires thinking about and setting up space-times where debates on norms can develop between students and teachers, between students, between teachers, and even between researchers and teachers.
In this article, we propose to shed light on the successive confrontations between a sociological and a didactic point of view during the analysis of a corpus of data consisting of sessions devoted to the teaching of numbers, in which two female teachers from the same kindergarten class took charge of pupils' errors. These confrontations gradually brought to light the theoretical and epistemological presuppositions underlying these points of view on knowledge in school teaching and learning situations. This led not only to new avenues of analysis in this area, but also to the question of how sociological and didactic analyses can complement each other in understanding the process of constructing inequalities at school, over and above the simple juxtaposition or even renewed use of concepts specific to each of these disciplinary research orientations.
In this paper, we examine the practices of two French primary school teachers: Camille (with pupils aged 4-6) and Anne (with pupils aged 9-10). Assuming that these teaching practices could help to limit or even reduce inequalities at school, we analyse them through the prism of various 'features' relating to the handling of the error(s) we observed in their respective classes.
Two studies have been carried out on mathematics teaching practices and the question of how error is taken into account in this teaching. The first is a large-scale survey of over 1,300 teachers of 5th grade pupils; the second is a collaborative research project carried out in a LéA on numerical problem solving at this same school level. The quantitative research divided the respondents into five groups, characterized by their teaching practices and, in particular, the way in which errors are taken into account in teaching. As the teachers involved in the LéA completed the questionnaire of the large-scale survey, a comparison of the two studies was carried out. It is this comparison that is the subject of this article.
Based on observations made in six kindergarten classes in contrasting social contexts, this article develops two parts. The first looks at the different forms of interpretation to which the same task can give rise on the part of the pupils, and at how close they are to school expectations. The second looks at the ways in which teachers take up students' responses when they don't correspond to what they expect, and what this produces (or not) in terms of learning. In so doing, the article shows how the way in which “mistakes” are apprehended by teachers, and the moments of reworking they offer pupils, are key moments in the construction of both learning and inequalities at school.
This article addresses questions relative to teachers of mathematics and sciences. We first establish the notion of the specificity of both disciplines as an epistemological background to underpin our ways of tackling questions about teachers’ practices, knowledge and training initiatives. Addressing issues about practices (of teachers and of teacher education) in relation to each disciplines’ specificity invites taking into account the nature of scientific and mathematical activity in itself. These orientations lead to consider the possibilities, in contrast to prescriptions, for adressing issues of teachers’ knowledge and practices, and their implications for teacher education practices.
