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.     

Determinism versus nondeterminism

This paper is the third item of the sequence of articles, which has been devoted to investigate nondeterminism within automata theory from didactic point of view. The first two articles discussed the teaching of notions and proofs, while this paper deals with the didactic approach of teaching problem solving. xab]de|KurzreferatDieser Aufsatz ist der dritte in eimer Folge von Artikeln, die der Untersuchung von Nichtdeterminiertheit in der Automatentheorie aus didaktischer Perspektive gewidmet sind. Die ersten beiden Aufsätze beschreiben die Behandlung von Begriffen und Beweisen, während sich dieser Artikel mit didaktischen Strategien zum Problemlösen befasst.     

Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school

This paper presents a study aimed at investigating the didactic potentiality of the combined use of two different kinds of artefacts for the purpose of constructing and conceptualizing mathematical meanings related to the notion of axial symmetry. In our view, the process of meanings construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that a potential synergy may occur between the use of different artefacts, synergy that can foster the integration of different and complementary meanings providing a rich support to the development of the expected mathematical meaning. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. Specifically, the construct of semiotic potential provides the tool for describing the potentialities of the two artefacts, while that of didactic cycle offers a model for the organization of the different activities. The paper reports on a teaching sequence and its implementation in a teaching experiment, involving pupils at fourth grade level. We describe them, within the chosen theoretical framework, and provide the analysis of key episodes of the teaching sequence. We show evidence supporting our main hypothesis about the combined use of an artefact that can be manipulated (paper and pin), and a digital artefact (Dynamic Geometry Environment) in the development of the notion of axial symmetry and its properties: the combined, intentional and controlled use of the two artefacts may develop a synergy, so that each activity enhances the potential of the other.     

Structural stability and dynamic geometry: Some ideas on situated proofs

In this paper we survey the historical and contemporary connections in mathematics between classical “conceptual” tools versus modern computing tools. In this process we highlight the interplay between the inductive and deductive, experimental and theoretical, and propose the notion of Situated proofs as a didactic tool for the teaching of geometry in the 21^st century.     

Didactic approach for teaching nondeterminism in automata theory

Nondeterminism plays a central role in almost all fields of computer science. It has been incorporated naturally as well as in the theory of automata as a generalization of determinism. Although nondeterministic finite automata do not have more recognition power than deterministic ones their importance and usefulness is of no doubt.Unfortunately, the operation of the mathematical model of nondeterministic automata is difficult to understand which means a didactic challenge for every teacher and lecturer. This paper gives a didactic approach to introduce the notion of finite (deterministic and) nondeterministic automata.As a teaching tool we make use of the automata theory package developed in Maple. We put emphasis on the process character of learning and work out a method that promotes the repetitive experiments.     

Mathematics teaching development as a human practice: identifying and drawing the threads

The didactic triangle links mathematics, teachers and students in a consideration of teaching?Clearning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers?? personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle.     

Problem solving in France: didactic and curricular perspectives

In this paper, we address the issue of problem solving in classrooms in France through two different and complementary approaches: didactic research and curricular choices. These two approaches correspond to two different, but not independent perspectives on problem solving and we investigate the existing links between them. We show that in France, the solving of problems is given a central role both in didactic research and curricular choices and that problem solving, as generally understood, is an object taking controversial positions, and we try to elucidate the rationale behind such positions. This paper is structured into two main parts: the first part devoted to didactic research, the second part to curricula, and one last part, discussion, wherein we question didactic research and curricula as regards their potential for influencing the reality of problem solving in classroom practices.     

Mediated action in teachers’ discussions about mathematics tasks

This paper presents analyses of teachers?? discussions within mathematics teaching developmental research projects, taking mediation as the central construct. The relations in the so-called ??didactic triangle?? form the basic framework for the analysis of two episodes in which upper secondary school teachers discuss and prepare tasks for classroom use. The analysis leads to the suggestion that the focus on tasks places an emphasis on the task as object and its resolution as goal; mathematics has the role of a mediating artefact. Subject content in the didactic triangle is thus displaced by the task and learning mathematics may be relegated to a subordinate position.     

The intangible task – a revelatory case of teaching mathematical thinking in Japanese elementary schools

This paper deals with the nature of teaching mathematical thinking and presents a case study of a single Japanese lesson where the characteristics of mathematical thinking and the teaching thereof are identified in relation to multiplication. The raison d’être for this teaching is questioned and investigated by looking at how multiplication is described in the curriculum and representative textbook material. It is seen how Japanese teachers are institutionally conditioned to incorporate mathematical thinking in the context of multiplication, something which may appear in contrast to other countries. The lesson is analysed using the notion of praxeologies and didactic co-determination conceptualised in the Anthropological Theory of the Didactic.     

Kaizen teaching and the learning habits of engineering students in a freshman mathematics course

Which are the teaching methods that actually contribute to the learning of mathematics? The answer to this certainly is the holy grail of didactic and pedagogy, and should be supported by large scale statistical evidence. Our article aims at providing an initial step into this direction by first illustrating a teaching paradigm that is suited for the generation of large scale data sets: based on industry best practice quality assurance standards we introduce the Kaizen teaching paradigm which enforces Kolb’s reflective learning cycle on the students’ side. Second, we present and analyze the data we obtained through our pilot implementation at a engineering freshman mathematics course in the Sultanate of Oman. These emphasize the effectiveness of Kaizen teaching and once again show the necessity of continuous learning. A practice that seems to be forgotten in traditional university engineering courses due to the mere size of the audience. In particular it seems that a Markovian estimator for students’ performance may have to be considered.     

Revisiting the didactic triangle: from the particular to the general

The basic notion of a didactic triangle is explained with historical annotations on its origins and subsequent theorization in the literature. Instances of its application to classroom environments to demonstrate its representational capabilities are presented. Generalizations of the triangle are proposed that integrate the role of technology, the researcher in mathematics teaching developmental research, and mediating complexes in the student?Cteacher?Ccontent interfaces. Further, the use of the didactic triangle as a heuristic device is also discussed.     

Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.     

Reductive-holistic cycle: A model for the study of the didactic procedure

The concept ofreality level may be useful as a catalyst among several systems in the area of knowledge. This concept is leading us to ask, if we can make a reduction from a reality level to another, that is to the problem ofreductionism. Relative to it is the problem ofholism. At the end these concepts are connected to the category theory and adjoint functors. Within the framework of this aspect we set up a model for the study of the didactic procedure. This model is a feedback system between two reality levels or categories, these of the teacher and of the student. So, the article seeks to enhance and improve the teaching of mathematics by its attempt to understand both student's and teacher's knowledge in the same terms.     

How to connect school mathematics with students' out-of-school knowledge

In this paper we present an explorative study for which special cultural artifacts have been used, i.e. supermarket receipts, to try to construct with 9-year old pupils (fourth class of primary school) a new mathematical knowledge, i.e. the algorithm for multiplication of decimal numbers. Furthermore also estimation and approximation processes have been introduced, procedures that are not commonly used in ordinary teaching activity. In our study the receipts, through some modifications, have become more explicitly tools of mediation and integration between in and out-of school knowledge, so they can be utilized to create new mathematical goals, thus becoming real mathematizing tools and constituting a didactic interface between in and out-of-school mathematics. In agreement with ethnomathematical perspective we deem that it is a task for the teacher to know, in order to be able to profitably take account of the teaching, the life experienced by the pupil. Future mathematics teachers should be prepared a) to see mathematics incorporated into real world, b) to investigate mathematical ideas and practices of their pupils, and c) to look for ways to incorporate into the curriculum elements belonging to the sociocultural environment of the pupils, as a starting point for mathematical activities in the classroom. In this way the motivation, interest and curiosity of the pupils will be increased and the attitude towards mathematics of both pupils and teachers will be changed.     

Truth versus validity in mathematical proof

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between truth and validity in mathematical proof for mathematics education.     

Using E-Exercise Bases in Mathematics: Case Studies at University

E-Exercise Bases (EEB) are now used in the teaching of mathematics, especially at university. We discuss here the consequences of their use on the students’ activity during computer lab sessions. Results stem from observations of several teaching designs organised in different French universities with three e-exercise bases. The analysis focuses on new tasks and on specific solving strategies, which appear using these resources. Moreover, specific didactic contract clauses are studied.     

A model of professional competences in mathematics to update mathematical and didactic knowledge of teachers

This paper describes part of a research and development project carried out in public elementary schools. Its objective was to update the mathematical and didactic knowledge of teachers in two consecutive levels in urban and rural public schools of Region de Los Lagos and Region de Los Rios of southern Chile. To that effect, and by means of an advanced training project based on a professional competences model, didactic interventions based on types of problems and types of mathematical competences with analysis of contents and learning assessment were designed. The teachers’ competence regarding the didactic strategy used and its results, as well as the students’ learning achievements are specified. The project made possible to validate a strategy of lifelong improvement in mathematics, based on the professional competences of teachers and their didactic transposition in the classroom, as an alternative to consolidate learning in areas considered vulnerable in two regions of the country.     

Using Video‐Based Pedagogy in an Elementary Mathematics Methods Course

The study explored the impact of using video‐based pedagogy on preservice teachers' cognitions about teaching mathematics. The use of video‐based pedagogy was integrated into the course, Methods for Teaching Elementary School Mathematics. A variety of written and interview data were collected during the course and, in the following semester, during student teaching. Evidence from case studies of three preservice teachers indicates that they engaged in reflection and reconstruction of their beliefs about how children learn mathematics and moved from a more didactic perspective of teaching mathematics toward a student‐centered perspective. Such movement appears to have been influenced by the use of video‐based pedagogy.     

Mathematical modelling as a tool for the connection of school mathematics

We start introducing some aspects of the theoretical framework: the Anthropological Theory of Didactics (ATD). Then, we consider on the research domain commonly known as “modelling and applications” and briefly describe its evolution using the ATD as an analytical tool. We propose a reformulation of the modelling processes from the point of view of the ATD, which is useful to identify new educational phenomena and to propose and tackle new research problems. Finally, we focus on the problem of the connection of school mathematics. The reformulation of the modelling processes emerges as a didactic tool to tackle this research problem. We work on the problem of the articulation of the study of functional relationships in Secondary Education and present a teaching proposal designed to reduce the disconnection in the study of functional relationships in Spanish Secondary Education.     

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.