Didactic Contract Research Articles

Didactic Situations and Didactical Engineering in university mathematics: cases from the study of Calculus and proof

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic.

Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus

In this paper we outline the main tenets of the commognitive approach and we exemplify its application in studies that investigate the learning and teaching of mathematics at university level. Following an overview of such applications, we focus on three studies that explore fundamental discursive shifts often occurring in the early stages of studying Calculus. These shifts concern the lecturers' and students' communicative practices, routines of constructing mathematical objects and ways of resolving commognitive conflicts. We then propose that commognitive constructs such as subjectification can be deployed towards ‘scaling-up’ the hitherto fine-grained focus of commognitive analyses. Finally, we conclude with observing how the commognitive approach relates to constructs from other sociocultural approaches to research in university mathematics education, such as “legitimate peripheral participation” from the theory of Communities of Practice and “didactic contract” from the Theory of Didactic Situations.

Students individual engagement in GIS

This paper develops two sets of concepts to theorize why students engage differently in Geographical Information Systems (GIS). These theoretical concepts are used as an analytical lens to explore empirical data on the experiences and engagement of students enrolled in an undergraduate GIS course in planning and management. The analysis shows that both the theoretical perspectives and the custom and didactical contract are important to understand students' engagement in GIS. However, it is the personal desiderata that are the key to understanding the students' different engagement. Further, a temporal dimension and contextual awareness are important in understanding students' engagement in a broader perspective.

La enseñanza y su relación con el saber en los estudiantes universitarios colombianos

Se exponen los resultados de una investigación sobre las respuestas de los estudiantes a las exigencias de las formas de los saberes enseñados en la universidad desde la perspectiva de su relación con el saber. Estudio exploratorio, cualitativo, descriptivo y descriptivo en una muestra cualitativa con representatividad teórica de dieciséis estudiantes de seis cursos universitarios, que se desarrollaron en el segundo semestre de 2012 en la Universidad Tecnológica de Pereira, Colombia. Se adoptaron cinco dimensiones de la relación con el saber: (1) sentido; (2) dimensión de lo que es importante en el saber; (3) contrato didáctico; (4) relación de identidad y afectiva con el saber; y (5) actitudes de estudio. Tres rasgos generales surgen de la investigación: (1) si bien los estudiantes, aceptan en su relación con el saber la exigencia de la significación, esperan también los momentos de la designación; (2) obstáculo importante en la relación constituye la distancia entre lo que el curso propone y la representación que el estudiante puede tener de la práctica fuente y la práctica objetivo; (3) rasgo común para los cursos es la distancia entre la percepción de las exigencias del profesor antes y después de la evaluación. Se confirma la hipótesis comprensiva: la respuesta que dan los estudiantes a las exigencias de las formas de los saberes enseñados en la universidad dependen de su relación con el saber y de sus actitudes y tratativas o enfoques que esta relación implica.

Using the Construct of the Didactic Contract to Understand Student Transition into University Mathematics Education

In this article the concept of the Didactic Contract is used to investigate student ‘transition’ from upper secondary into university mathematics education. The findings are anchored in data from the TransMaths project, more particularly the case of an ethnic minority student's journey from his school to a university mathematics course taught at a large inner-city university in England. Results show that there is a transformation (or rupture) of the Didactic Contract from school to university mathematics, and that this is likely to have serious consequences for students' success, or failure, at this crossroads of students' mathematical development, in particular if students are left to ‘bridge the gap’ from one contract to the other. Further, the author argues that in order to help to smooth the passage, a refinement of the Didactic Contract is helpful, a re-conceptualisation in terms of Normative Didactic Contract and Personal Didactic Contract, and each needs to be considered at each level of development. This is likely to raise awareness about the necessary conditions for success at both sides of the transition junction, and providing appropriate support for students at both levels is likely to provide access for more students to successfully stay in higher education mathematics.

Can peer assisted learning be effective in undergraduate mathematics?

We report the implementation and evaluation of a ‘peer assisted learning’ (PAL) scheme designed to reduce the so-called ‘cooling off’ phenomenon in undergraduate mathematics. ‘Cooling off’ occurs when mathematics undergraduates lose motivation and interest in their studies, despite having previously actively chosen to study it at higher levels. We found that, despite concerns about the novel didactic contract inherent in PAL schemes, a majority of students chose to engage with the scheme, and that the student leaders of the PAL sessions were generally capable of implementing a student-centred pedagogy. Furthermore, we found that students who attended the PAL sessions had higher achievement in their final examinations, even after controlling for their lecture attendance and prior attainment. We conclude by arguing that PAL may provide a useful mechanism for reducing the prevalence of the ‘cooling off’ phenomenon in some – but not all – groups of mathematics students.

Some Specific Features of the Educational Learning Context – An Outlook on Secondary Mathematics

Educational learning seen as an individual process which is carried out, though, in a social context, is determined by the multitude of variables specific to the didactic situation.Starting with the results obtained by pupils in National Assessments in Mathematics, which clearly aren’t encouraging, we decided to study the existing relationships between different dimensions of the didactic situation and the performance levels reached by pupils in mathematics learning. The goal of the inquiry was to identify the factors with considerable impact on the learning process such as, specific features of the didactic contract, the mathematics teacher's attitude towards the learning errors and the didactic strategy elements. For this purpose, to collect data, we resorted to the school documents’ analysis (inspection reports) and the inquiry based on the questionnaire applied to the pupils.The results have shown that there are some didactic practices of the mathematics teacher with negatively impact on the learning process. More than the others, the mathematics teacher is more oriented to the subject taught than to the learning process itself.Recognizing these situations, which can be expressed in an explicit or an implicit manner, is the first step in making the mathematics learning process more efficient.

A multidimensional approach to explore the understanding of the notion of absolute value

The study aimed to investigative students’ conceptions on the notion of absolute value and their abilities in applying the specific notion in routine and non-routine situations. A questionnaire was constructed and administered to 17-year-old students. Data were analysed using the hierarchical clustering of variables and the implicative method, while the qualitative approach was used to identify and analyse students’ errors. The results revealed students’ strong tendency to use algorithmic processes even in situations in which this kind of reasoning was not suited. This tendency and students’ errors in the tasks were assumed to occur primarily due to obstacles of didactic origin concerning the didactic contract and subsequently due to epistemological obstacles grounded in the history of mathematics.

Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics

One of the manifestations of learning is the student’s ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9–10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity.

Students’ reported justifications for their representational choices in linear function problems: an interview study

Thirty-six secondary school students aged 14–16 were interviewed while they chose between a table, a graph or a formula to solve three linear function problems. The justifications for their choices were classified as (1) task-related if they explicitly mentioned the to-be-solved problem, (2) subject-related if students mentioned their own characteristics as representational users, (3) context-related if contextual features surrounding the choice were mentioned and (4) representation-related if formal characteristics of the representations were pointed out. Justifications were mostly task- and subject-related, although contextual and representational features also played an important role. Some students (reportedly) tried to reconcile different (task-, subject-, context- and representation-related) factors before selecting a representation, which was interpreted as an attempt to use their meta-representational competence to make appropriate representational choices. The influence of the didactical contract and the experimental contract on students’ representational choices, as well as the tensions between them, are also discussed.

Entre el autodidactismo, la solidaridad y la certificación Procesos de estudio de las matemáticas en cuatro plazas comunitarias del INEA

By way of an empirical study, the author investigated the study procedures of the Mathematics courses in four community halls coordinated by the Instituto Nacional para la Educacion de los Adultos (National Institute for Adult Education-INEA). The Education for Life and Work model is the guiding framework for the actions taken, though in reality, the mathematics activities are guided by an ‘institutional didactic contract,’ providing certification for the said activities and guiding the course content towards the objective of examinations and accreditation. The study process leading to the certification is based on self-directed learning and social solidarity. The investigation shows that certification —together with personal goals— reorient the formal institutional objectives, thus affecting the study of mathematics, already impoverished by the two pillars that constitute their livelihood.

About the Joint Action Theory in Didactics

Didactics describes the teaching-learning process which can be conceptualised as a system comprising three subsystems: the teacher, the student and the knowledge at stake, definable as the power of acting. Within this framework, joint action is understood as a unit of the joint attention of teacher and students, joint affordance, and common ground. An analysis of the grammar of didactic action identifies the specific cooperation of the teacher and students. Under this condition, a didactic contract identifies a system of (largely implicit) expectations, and the didactic milieu identifies the structure of the problem at stake, which can be viewed as antagonistic elements producing resistance to the students’ action, particularly in some designed cases.

Teaching with new technology: four ‘early majority’ teachers

This paper explores how four good teachers, who do not have a special interest in technology, meet the challenge of introducing the rapidly developing mathematics analysis software (e.g. spreadsheets, function graphers, symbolic algebra manipulation and dynamic geometry) into their classrooms. These teachers’ practice is viewed through the lens of Roger’s framework for the diffusion of innovation and Pierce and Stacey’s pedagogical opportunities map. Data on teachers, views and practices were collected over 2 years. ‘Pedagogical Maps’ give a picture of the teachers’ perception and uptake of pedagogical opportunities. New practices have been added slowly to each teacher’s repertoire and their increasing fluency in practical ability to teach with the technology resulted in some changes to the classroom didactic contract. Overall, new technology seemed to have been absorbed into current practice, more than changing practice. At this stage of their development, these teachers do not identify the distinctive new mathematical capabilities as contributing to the major relative advantage of the innovation. Instead, they see the relative advantage mostly in the incremental improvements to capabilities of earlier calculators, and meeting the need for students to be up to date. One of the current challenges is that significant changes in both software and hardware design have been happening so rapidly that these early majority teachers felt almost constantly hampered by the need to learn and teach new technical skills and so continue to make limited progress in taking advantage of opportunities to approach mathematics concepts in new ways.

Designing representations of trigonometry instruction to study the rationality of community college teaching

We describe the process followed to design representations of mathematics teaching in a community college. The end product sought are animated videos to be used in investigating the practical rationality that community college instructors use to justify norms of the didactical contract or possible departures from those norms. We have chosen to work within the trigonometry course, in the context of an instructional situation, “finding the values of trigonometric functions,” and specifically on a case of this situation that occurs as instructors and students are working on examples on the board. We describe the design of the material needed to produce the animations: (1) identifying an instructional situation, (2) identifying norms of the contract that are key in that situation, (3) selecting or creating a scenario that illustrates those norms, (4) proposing alternative scenarios that instantiate breaches of those norms, and (5) anticipating justifications or rebuttals for the breaches that could be found in instructors’ reactions. We illustrate the interplay of contextual and theoretical elements as we make decisions and state hypothesis about the situation that will be prototyped.

Examining the didactic contract when handheld technology is permitted in the mathematics classroom

The use of mathematics analysis software (MAS) including handheld scientific and graphics calculators offers a range of pedagogical opportunities. Its use can support change in the didactic contract. MAS may become an alternative source of authority in the classroom empowering students to explore variation and regularity, manipulate simulations and link representations. Strategic use may support students to direct their own learning and explore mathematics, equipping them to share their findings with the teacher and the class with more confidence. This paper offers a framework for examining the impact of the use of MAS on the didactic contract. Lessons were observed in 12 grade 10 classes, with 12 different teachers new to MAS. MAS technology was used with a variety of didactic contracts, mostly traditional. The framework drew attention to many ways in which the teaching differed. Analysis of the didactic contract must consider both the teaching of mathematics and of technology skills, because these have different characteristics. In all classes, both teachers and students saw the teacher as having a responsibility to teach technology skills. Students saw technology skills as the main point of the lesson, but the teachers saw the lesson as primarily teaching mathematics—one of the mismatches which may need negotiation to adapt didactic contracts to teaching with MAS.

Pupils’ visual representations in standard and problematic problem solving in mathematics: their role in the breach of the didactical contract

ABSTRACT This study investigated the modes of representations generated by kindergarteners and first graders while solving standard and problematic problems in mathematics. Furthermore, it examined the influence of pupils’ visual representations on the breach of the didactical contract rules in problem solving. The sample of the study consisted of 38 kindergarteners (age 5–6) and 34 first graders (age 6–7). Two standard problems (addition and subtraction) and two problematic problems were given to the participants. The majority of kindergarteners used a variety of spontaneous visual representations in order to solve both types of problems. In contrast, first graders mainly used symbolic representations corresponding to the numbers involved in the text of the problems. Results also suggested that the visual representations prevented kindergarteners from obeying the didactical contract rules. In fact, many kindergarteners were inclined to draw descriptive pictures about the meaning of the problems, without ...

GEOMETRIC AND ALGEBRAIC APPROACHES IN THE CONCEPT OF “LIMIT” AND THE IMPACT OF THE “DIDACTIC CONTRACT”

The present study explores students’ abilities in conversions between geometric and algebraic representations, in problem- solving situations involving the concept of “limit” and the interrelation of these abilities with students’ constructed understanding of this concept. An attempt is also made to examine the impact of the “didactic contract” on students’ performance through the processes they employ in tackling specific tasks on the concept of limit. Data were collected from 222 12th-grade high school students in Greece. The results indicated that students who had constructed a conceptual understanding of limit were the ones most probable to accomplish the conversions of limits from the algebraic to the geometric representations and the reverse. The findings revealed the compartmentalized way of students’ thinking in non-routine problems by means of their performance in simpler conversion tasks. Students who did not perform under the conditions of the didactic contract were found to be more consistent in their responses for various conversion tasks and complex problems on limits, compared to students who, as a consequence of the didactic contract, used only algorithmic processes.

Liens faibles, méfiance et apprentissage collectif

The paper focuses on the relevance of the social structures involved in the development of a collaborative learning process. It analyses the issue concerning the relationship between the necessary trust in the didactic contract, the devolution to collective rules, and the care of self. Three models will be considered in order to raise questions on face-to-face strategies: the network trust, the institution trust, and the calculation trust.

中学校の図形領域における証明の教授・学習に関する考察 : 教授学的契約と委譲の概念を視座として

中学校の図形領域における証明の教授・学習に関する考察 : 教授学的契約と委譲の概念を視座として

The gendered construction of physical education content as the result of the differentiated didactic contract

Background: Research on gender in physical education has pointed out the social construction of gendered bodies and minds through the curriculum. It has been shown that girls do not benefit from equal opportunities to participate in physical activities. Developed within the theoretical framework of the didactique tradition, this paper tries to understand how teacher and students (girls and boys) jointly construct gendered content knowledge during formal episodes of teaching and learning in physical education. Purpose: To focus on student learning with special attention to the gendered content taught and learned with the aim of understanding the differentiated process which is as the root of gender issues in physical education. Setting and participants: Sixteen students (six girls and ten boys) and two teachers during coeducational volleyball units in two high schools (Lycée) in France. Research design and data collection: The study investigates how the attack is learned in volleyball according to gender through a design experiment within a didactique research methodology. The protocol included collecting data regarding student intentions, actions and interpretations of content through the use of lesson observation and pre- and post-lesson participant interviews. Data analysis consisted of the identification and evolution of the differentiated didactic contract (DDC) across the pedagogical interactions which occurred during two learning tasks concerning ‘how to perform an efficient attack’. Findings: Students do not develop the same nature and degree of understanding and performance of the attack in volleyball due to the differentiated ways in which the didactical interactions evolve during formal lessons in physical education. Five factors influence the process: (1) the gendered nature of knowledge; (2) the ways in which the knowledge is developed by the teacher; (3) the gendered perspectives individuals bring into and invoke during the didactical interactions; (4) the evolution of the enacted content knowledge as a consequence of the ongoing process through (5) the tacit and implicit negotiations which originate the differentiated didactic contract. Conclusions: The study supports a research agenda which gives attention to the domain-specific content taught and learned during classrooms interactions. It supports the theoretical idea that curriculum is enacted through micro-social interactions around particular content and within specific contexts.