Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction

Theorem acquisition and deductive proof have always been core elements in the study and teaching of Euclidean geometry. The introduction of dynamic geometry environments,DGE (e.g., Cabri-Géomètre, Geometer's Sketchpad), into classrooms in the past decade has posed a challenge to this praxis. Student scan experiment through different dragging modalities on geometrical objects that they construct, and consequently infer properties(generalities, theorems) about the geometrical artefacts. Because of the inductive nature of the DGE, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern. In this paper, we will describe and study a ‘Cabri proof by contradiction’ of a theorem on cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. We will discuss how their construction motivates a visual-cognitive scheme on `seeing' proof in DGE, and how this scheme could fit into the theoretical construct of cognitive unity of theorems proposed by Boero, Garuti and Mariotti(1996). The issue of a cognitive duality and its relation to visualization will be raised and discussed. Finally, we propose a possible perspective to bridge the experimental-theoretical gap in DGE by introducing the idea of a dynamic template as a visualizer to geometrical theorem justification and acquisition. This revised version was published online in July 2006 with corrections to the Cover Date.     

CONJECTURING IN OPEN GEOMETRIC SITUATIONS USING DYNAMIC GEOMETRY: AN EXPLORATORY CLASSROOM EXPERIMENT

In this paper I describe a classroom teaching experiment carried out with a class of Year 10 students. This experiment was twofold. On the one hand it was aimed at developing and trying out a new mode of working in the classroom, taking into account the possibilities offered by dynamic geometry software as support in the conjecturing and proving process in geometry. On the other hand, from the research point of view, it provided the possibility of testing out in the classroom a theoretical model describing and interpreting students’ use of Cabri-Géomètre in open geometric problems, with a particular focus on dragging. The main findings relate to the evolution in the use of dragging in Cabri and the production of rich conjectures, which can provide the basis for development and evolution towards the proving process.     

A dynamic approach to ‘simple’ algebraic curves

Despite its low relevance for teaching in schools the desideratum “Algebraic Curves’ has always been an issue within mathematical didactics. This article aims to continue this tradition with respect to a project oriented mathematics instruction by applying current dynamic geometry software (Cabri II+).     

Design decisions in interactive geometry software

Software design is increasingly being recognized as an important factor in student learning when interacting with interactive geometry software (IGS). A categorization of the operations possible within an IGS is used to identify and analyse design decisions made in a number of current IGS programs: Cabri II Plus, Cabri 3D, Cinderella, GeoGebra and Geometer’s Sketchpad. The analysis, in the context of exploring the area of a circle, is focused on construction, dragging, and alternative spatial and semantic views. A wide diversity of both design issues and individual design decisions was identified, illustrating both the scarcity of research in this area and a number of inevitable tensions, such as between functionality and complexity, and between static and dynamic geometry, related to which future research questions might be posed.     

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.     

CABRI AS A COGNITIVE TOOL FOR THE EVOLUTION OF A SENSE OF PROVING

Cognitive technologies have been described in the literature as reorganisers of thinking processes, especially where problem solving is concerned. This paper aims to analyse the possible use of Cabri-Géomètre as a cognitive tool in the elaboration of mathematical justifications in the context of problem-based mathematics. Some empirical examples are given to illustrate the significance of the specific learning situation. The complexity of learning environments incorporating computer-based activities is stressed as a condition for them to be effective in the introduction of the idea of mathematical justification and its evolution towards a sense of proving.     

The Aim of Justification and Epistemic Difference-Making Principles

The idea that truth is the aim of justification is one that is often defended by theorists who uphold different views about the nature of epistemic justification. Despite its prevalence, however, it is not quite clear how one is to cash out the metaphor that justification aims at truth. Some theorists, for example, have objected that the thesis would leave no room for justified false beliefs and unjustified true beliefs. In this paper, I offer an account of what it is for justification to aim at truth using the recently revived idea of difference-making according to which facts often make a difference to other facts. It will be argued that, thus understood, the thesis can illuminate a number of controversial debates in epistemology and that, given its explanatory power, it has a lot to recommend it.     

Archimedes with Cabri: Visualization and Experimental Verification of Mathematical Ideas

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. This snapshot explores a classic geometry theorem of Archimedes. The author shows how the use of a dynamic geometry environment can deepen and guide the process of proof. The Cabri environment enabled exploration in a new way that connected steps in the proof to geometric properties and intuitions that served as a foundation for deductive reasoning. Computer Math Snapshots Editor: Uri Wilensky , Center for Connected Learning and Computer-Based Modeling Northwestern University, USA E-mail: uri@northwestern.edu     

A cognitive analysis of dragging practises in Cabri environments

Dragging in Dynamical Geometry Software (DGS) is described by introducing a hierarchy of its functions. This is suitable for classifying different attitudes and aims of students who investigate a geometric problem, such as exploring, conjecturing, validating and justifying. Moreover the hierarchy has cognitive features and can be used to describe the twofold modulities namely ascending and descending in which students interact with external representations (e.g. Cabri drawings). Switching from one modality to the other through dragging often allows them to produce fruitful conjectures and to pass from the empirical to the theoretical side of the question. The genesis of such different functions in students does not happen automatically but is the consequence of specific didactical interventions of the teacher in the pupils' apprenticeship of Cabri practises. A worked-out example illustrates the theoretical concepts introduced in the paper.     

Prefixed tableaus and nested sequents

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that Gentzen sequent calculi and tableaus are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.     

On the strategic origin of Brownian motion in finance

This paper is concerned with the strategic use of a private information on the stock market. A repeated auction model is used to analyze the evolution of the price system on a market with asymmetric information.  The model turns out to be a zero-sum repeated game with one-sided information, as introduced by Aumann and Maschler.  The stochastic evolution of the price system can be explicitly computed in the n times repeated case. As n grows to ∞, this process tends to a continuous time martingale related to a Brownian Motion.  This paper provides in this way an endogenous justification for the appearance of Brownian Motion in Finance theory. Received: February 2002     

The construction of meanings for geometric construction: Two contrasting cases

Dynamic geometry software seems to offer new approaches to the teaching and learning of geometry. Interest has been particularly intense in Britain where geometry has almost disappeared from the curriculum. In this paper we draw on our experiences of using Logo with children and adult students as a way of thinking about the design of geometrical activities for young children who lack cultural support in this domain and who are at early stages of conceptual development. We explore how the nature of Cabri Geometry activities, in conjunction with previous connected experiences, may influence the construction of meaning for geometric construction.     

Third-Person Internalism: A Critical Examination of Externalism and a Foundation-Oriented Alternative

This paper starts with an examination of the major problems of foundation-oriented epistemology in Sect. 2. Then, in Sects. 3–4, it is argued that the externalistic re-definition of knowledge deprives this concept from useful applications to human’s epistemic practice. From the viewpoint of cultural evolution, the condition of justification is the most important ingredient of knowledge. An alternative foundation-oriented conception of knowledge called third-person internalism is developed in Sect. 2 and Sect. 5. It combines insights of externalism with the requirement of second-order justification. The application of third-person internalism to contextualistic positions leads to an important constraint on contextualism (Sect. 6). The final section (Sect. 7) sketches new prospects for a foundation-oriented epistemology which are based on epistemic optimality arguments.     

Challenging Students to View the Concept of Area in Triangles in a Broad Context: Exploiting the Features of Cabri-II

This study focuses on the constructions in terms of area and perimeter in equivalent triangles developed by students aged 12–15 years-old, using the tools provided by Cabri-Geometry II [Labore (1990). Cabri-Geometry (software), Université de Grenoble]. Twenty-five students participated in a learning experiment where they were asked to construct: (a) pairs of equivalent triangles “in as many ways as possible” and to study their area and their perimeter using any of the tools provided and (b) “any possible sequence of modifications of an original triangle into other equivalent ones”. As regards the concept of area and in contrast to a paper and pencil environment, Cabri provided students with different and potential opportunities in terms of: (a) means of construction, (b) control, (c) variety of representations and (d) linking representations, by exploiting its capability for continuous modifications. By exploiting these opportunities in the context of the given open tasks, students were helped by the tools provided to develop a broader view of the concept of area than the typical view they would construct in a typical paper and pencil environment.     

Justifying the methods of OR

Operational research practitioners use mathematical, statistical, scientific, and other methods to structure and analyse issues in order to advise and assist their clients. In doing so they apply values, follow rules and use methodologies. The paper examines the justification of these methods, values and methodologies. Starting with a conceptual model drawn from the philosophy of science, a justification framework is developed for operational research (OR). Making a distinction between OR academic research and OR practice helps to clarify the issues. OR research is similar to scientific, mathematical and social science research; OR practice, as technology, is closer to engineering. While OR academic researchers will seek justification in the academic discipline within which they choose to work, it is argued that the justification of OR practice lies in its usefulness. For academic OR, justification lies in the justification of mathematics, statistics, science and social science; for practice, it is practitioners who decide what usefulness means in their context.     

A few pedagogical designs in linear algebra with Cabri and Maple

Since the 90’s, with the creation of new electronic environments for learning and teaching, several research groups in Mathematics Education have been emerging and developing. This article elaborates few pedagogical designs in Linear Algebra supported by both the geometrical micro-world Cabri and the computer algebra system Maple. Stumbling blocks in the learning of Linear Algebra are examined, more exactly linear transformations, eigenvectors, quadratic forms, conics with changes of bases and finally singular values. Encountering a special group of students very eager to explore the world of linear algebra, we initiated a classification of linear transformations of the Euclidean plane R² via ellipses.     

Information technology and systems justification: A review for research and applications

Information technology (IT) such as Electronic Data Interchange (EDI), Radio Frequency Identification Technology (RFID), wireless, the Internet and World Wide Web (WWW), and Information Systems (IS) such as Electronic Commerce (E-Commerce) systems and Enterprise Resource Planning (ERP) systems have had tremendous impact in education, healthcare, manufacturing, transportation, retailing, pure services, and even war. Many organizations turned to IT/IS to help them achieve their goals; however, many failed to achieve the full potential of IT/IS. These failures can be attributed at least in part to a weak link in the planning process. That weak link is the IT/IS justification process. The decision-making process has only grown more difficult in recent years with the increased complexity of business brought about by the rapid growth of supply chain management, the virtual enterprise and E-business. These are but three of the many changes in the business environment over the past 10–12 years. The complexities of this dynamic new business environment should be taken into account in IT/IS justification. We conducted a review of the current literature on IT/IS justification. The purpose of the literature review was to assemble meaningful information for the development of a framework for IT/IS evaluation that better reflects the new business environment. A suitable classification scheme has been proposed for organizing the literature reviewed. Directions for future research are indicated.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Research on the Evolution of Thinking Dynamics Attractor

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