Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Characterizing Interaction with Visual Mathematical Representations

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.     

Row-Strict Quasisymmetric Schur Functions

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.     

Interaction analysis of mathematical communication in primary teaching: The epistemological perspective

Communication between students and teacher in the mathematics classroom is a form of social interaction which focuses on a specific topic:mathematical knowledge. This knowledge cannot be introduced into classroom interaction “from the outside”, but grows through the communicative process, in the course of interactive exchanges between the participants of discussion. Although mathematical communication must be seen and analysed in the same way as any other form of communication, the particularity of interactive constructions of mathematical knowledge and its specificsocial epistemology within the context of teaching processes has to be taken into consideration. Also, the institutional influences of school institutions and those of teaching (analysed in the frame of general socio-interactive research approaches) must be considered. An epistemology-oriented interaction research approaches the specificity of amathematical classroom and communication culture in its analyses.     

Students’ Conceptions of Congruency Through the Use of Dynamic Geometry Software

This paper describes students’ interactions with dynamic diagrams in the context of an American geometry class. Students used the dragging tool and the measuring tool in Cabri Geometry to make mathematical conjectures. The analysis, using the cK￠ model of conceptions, suggests that incorporating technology in mathematics classrooms enabled a measure-preserving conception of congruency with which students’ could shift focus from shapes to properties. Students also interacted with dynamic diagrams in a novel way, which we call the functional mode of interaction with diagrams, relating outputs and inputs that result when dragging a figure. Students’ participation in classroom interactions through discourse and through actions on diagrams provided evidence of learning using tools within dynamic geometry software.     

Triple-loop networks with arbitrarily many minimum distance diagrams

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form (“L-shape”).In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.     

On a class of linked diagrams,I. Enumeration

Recurrence rules are derived for enumerating the linked diagrams first encountered by Touchard in 1952. These may be characterized as the subclass of irreducible diagrams in the full set of (2n ? 1)!! complete pairings on 2n points. In addition, a recurrence rule is given for the number of symmetric irreducible diagrams. This enables one to calculate the number of symmetry-reduced irreducible diagrams.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Interactions with diagrams and the making of reasoned conjectures in geometry

Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions on the work of a teacher as she manages students’ mathematical work is illustrated.     

Une ambition éditoriale “universelle et confraternelle” : le bulletin bibliographique de L'Enseignement mathématique (1899–1920)

In 1899 Henri Fehr and Charles Laisant founded L'Enseignement mathématique (EM) with the ambition to involve teachers in the then-growing internationalization movement of mathematics. To this purpose, their editorial project gave an important place to a bibliographical bulletin reviewing periodicals which could be of interest for the world of mathematical education. This article is dedicated to the study of this bulletin, from its creation to the 1920s, and to the initiatives and choices that Laisant and Fehr made to carry out this internationalist editorial ambition, as well as to the limits and constraints of their project. During that time, many bibliographical initiatives for periodicals developed in the mathematical press, which can be considered as a first form of interaction between journals. Our study will concern initially the year 1899 and this interaction in which EM took part, dealing at first with the bulletin of EM, then, secondly, with the confrontation between bibliographical sections of other journals. Lastly, considering the first thirty years of the 20th century, we will study the different dynamics at work in the world of mathematical periodicals which the EM serves to depict.     

Developing a theory of mathematical growth

In this paper I formulate a basic theoretical framework for the ways in which mathematical thinking grows as the child develops and matures into an adult. There is an essential need to focus on important phenomena, to name them and reflect on them to build rich concepts that are both powerful in use and yet simple to connect to other concepts. The child begins with human perception and action, linking them together in a coherent way. Symbols are introduced to denote mathematical processes (such as addition) that can be compressed as mathematical concepts (such as sum) to give symbols that operate flexibly as process and concept (procept). Knowledge becomes more sophisticated through building on experiences met before, focussing on relationships between properties, leading eventually to the advanced mathematics of concept definition and deduction. This gives a theoretical framework in which three modes of operation develop and grow in sophistication from conceptual-embodiment using thought experiments, to proceptual-symbolism using computation and symbol manipulation, then on to axiomatic-formalism based on concept definitions and formal proof.     

Use of words and visuals in modelling context of annual plant

This study looks at the various verbal and non-verbal representations used in a process of modelling the number of annual plants over time. Analysis focuses on how various representations such as words, diagrams, letters and mathematical equations evolve in the mathematization process of the modelling context. Our results show that (1) visual representations such as flowcharts are used not only in the process to symbolization, but also used in the justification of symbols, (2) some of the visual representations serve as a bridge between the words in the problem context and the symbols that represent the mathematical equations of the number of annual plants and (3) words and context help to introduce visual representations and symbols. Also, once students come up with the visual representations and symbols, they show better understanding about words used in the problem context. These observations imply that the modelling and mathematization process is not just one-directional and linear from words describing real-life situations to the symbols in mathematical equations and expressions. Rather, the mathematization can be promoted through using other visuals that help make this transition smooth by organizing the given information in a way that can be used towards mathematization.     

Mathematical tables in Ptolemy's Almagest

This paper is a discussion of Ptolemy's use of mathematical tables in the Almagest. By focusing on Ptolemy's mathematical practice and terminology, I argue that Ptolemy used tables as part of an organized group of units of text, which I call the table nexus. In the context of this deductive structure, tables function in the Almagest in much the same way as theorems in a canonical work, such as the Elements, both as means of presenting acquired knowledge and as tools for producing further knowledge.     

Two problems in the 筭數書 Suanshu shu (Book of Mathematics): Geometric relations between circles and squares and methods for determining their mutual relations

Much research has been devoted to two problems, Yi yuancai fang (From a circular timber [find] a square) and Yi fangcai yuan (From a square timber [find] a circle), both of which appear in the Suanshu shu, an early Han dynasty mathematical work written on bamboo slips, excavated from tomb 247 at Zhangjiashan in Hubei Province, China. In this article, the geometric relations between circles and squares and the methods for determining their mutual relations in these two problems are interpreted in a different way, and an alternative approach is offered for reconciling these two problems.     

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.     

Sur la conception des objets et des méthodes mathématiques dans les textes philosophiques de d'Alembert

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (41 and 2). Then we come to the ideas of definitions, primitive ideas, simple ideas, and their generation as well as their generalization (43 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (45), we approach the question of the reality of mathematical objects with the example of the irrational (46). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (47); principles of geometry (48); the notion of limit as the basis of infinitesimal calculus (49). His reflections, even if unfinished, were not without posterity (410).     

Technological Aids in the Teaching of Mathematics∗

In learning mathematics a relationship needs to be known as a detail and this also needs to be understood relative to the over‐all pattern and structure of mathematics. For some users of mathematics, techniques can be developed for presenting formulae sequentially so that they are available for term‐by‐term substitution, but the needs of creative mathematical thinkers are not met in this way. Present human resources are one mathematics graduate per 500 secondary school pupils and one mathematics graduate per 1300 students in further education. There is a pressing need to supplement these teaching resources by aids made available by educational technology, and also for research into suitable student terminals. Sequentially presented material is already available but there is a great need for visual material which can be presented synoptically. It is suggested that a steering team could mobilize many units where there are mathematicians and audio‐visual facilities to provide a large library of linking sequences to be available at computer‐controlled student terminals, relating each formula or relationship to other aspects of mathematics so that its place in the whole structure of mathematics is presented. Examples are given. There is also a need to devote some resources to the study of the effectiveness of particular diagrams and the order of presentation of visual materials since some current researches indicate these may be critical factors.