System Description: GCLCprover + GeoThms

The power of The Geometer's Sketchpad® (GSP) for teaching and learning lies in its potential for enabling students to explore geometric concepts and recognize relationships among these concepts as illustrated through the use of multiple representations. This article explores two problems and examines how the use of tables along with GSP sketches may assist students in understanding these concepts. The first activity investigates the relationship of the length of a trapezoid's median (or midsegment) to that of its bases. The second involves showing how the area of a regular polygon approaches that of a circle when the number of sides increases. In both activities, we will start with the geometric object, look at the measurements, tabulate these, and make a graph to illuminate the situation.

The research question addressed by this study is whether or not students made effective use of dynamic geometry software to explore the validity of a geometrical statement. Particular attention was paid to the use of definitions and the types of justifications students employed. Interviews conducted with six secondary mathematics preservice teachers were video and audio taped and video output of the interviewee's Geometer's Sketchpad (GSP) work was recorded on video tape. The interviews were transcribed along with a synchronized record of the GSP work. The transcripts and videotapes were analysed and coded. Results indicated that reasoning based on prototypically constructed categories and difficulty correctly parsing mathematical statements, especially the ‘for all’ quantifier, interfered with the effective use of dynamic geometry software in justifying mathematical assertions. It may be that explicit instruction in how to make use of software programs to explore the validity of an assertion is needed.

We present an application of automatic theorem proving (ATP) in the verification of constructions made with dynamic geometry software (DGS). Given a specification language for geometric constructions, we can use its processor to deal with syntactic errors. The processor can also detect semantic errors - situations when, for a given concrete set of geometrical objects, a construction is not possible. However, dynamic geometry tools do not test if, for a given set of geometrical objects, a construction is geometrically sound, i.e., if it is possible in a general case. Using ATP, we can do this last step by verifying the geometric constructions deductively. We have developed a system for the automatic verification of regular constructions (made within DGSs GCLC and Eukleides), using our ATP system, GCLCprover. This gives a real-world application of ATP in dynamic geometry tools.

Contents: Preface. Part I: Why Teach Geometry? E.P. Goldenberg, A.A. Cuoco, J. Mark, A Role for Geometry in General Education. K. Gravemeijer, From a Different Perspective: Building on Students' Informal Knowledge. D. Chazan, M. Yerushalmy, Charting a Course for Secondary Geometry. R.L. Devaney, Chaos in the Classroom. Part II: Studies of Conceptual Development. J. Pegg, G. Davey, Interpreting Student Understanding in Geometry: A Synthesis of Two Models. R. Lehrer, M. Jenkins, H. Osana, Longitudinal Study of Children's Reasoning About Space and Geometry. R. Lehrer, C. Jacobson, G. Thoyre, V. Kemeny, D. Danneker, J. Horvath, S. Gance, M. Koehler, Developing Understanding of Space and Geometry in the Primary Grades. D.H. Clements, M.T. Battista, J. Sarama, Development of Geometric and Measurement Ideas. M.T. Battista, D.H. Clements, Students' Understanding of Three-Dimensional Cube Arrays: Findings From a Research and Curriculum Development Project. J.A. Middleton, R. Corbett, Sixth Grade Students' Conceptions of Stability in Engineering Contexts. K. Raghavan, M.L. Sartoris, R. Glaser, Interconnecting Science and Mathematics Concepts: Area and Volume. D. Dennis, J. Confrey, Geometric Curve-Drawing Devices as an Alternative Approach to Analytic Geometry: An Analysis of the Methods, Voice, and Epistemology of a High School Senior. K.R. Koedinger, Conjecturing and Argumentation in High School Geometry Students. Part III: Defining a New Semantics of Space: Computers, Software, and the Electronic World. E.P. Goldenberg, A.A. Cuoco, What Is Dynamic Geometry? M. de Villiers, An Alternative Approach to Proof in Dynamic Geometry. J. Olive, Opportunities to Explore and Integrate Mathematics With the Geometer's Sketchpad. D.L. Watt, Mapping the Classroom Using CAD Program: Geometry as Applied Mathematics. L. Zech, N.J. Vye, J.D. Bransford, S.R. Goldman, B.J. Barron, D.L. Schwartz, R. Kisst-Hackett, C. Mayfield-Stewart, The Cognition and Technology Group at Vanderbilt, An Introduction to Geometry Through Anchored Instruction. K.A. Renninger, S.A. Weimar, E.A. Klotz, Teachers and Students Investigating and Communicating About Geometry: The Math Forum.

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.

GLI (Geometric Locus Identifier), an open web-based tool to determine equations of geometric loci specified using Cabri Geometryand The Geometer's Sketchpad, is described. A geometric construction of a locus is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, following the Groebner basis method, compute the locus equation and its graph. Moreover, an OpenMath description of the geometric construction is given. GLI can be efficiently used in mathematics education, as a supplement of the locus functions of the standard dynamic geometry systems. The system is located at http://nash.sip.ucm.es/GLI/GLI.html .

Have you ever thought of exploring geometric relationships on the computer? The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) challenges us to pay increased attention to “developing an understanding of geometric objects and relationships” and “using geometry in solving problems” (p.70). The document also advises placing decreased emphasis on memorizing geometric vocabulary, facts, and relationships. This article describes several geometric investigations that were developed in the spirit of the curriculum standards and are appropriate for use at advanced elementary and middle school levels, as well as with older students. They encourage exploration, creativity, and discovery.

Geometry learning at the junior secondary level should focus on the connection between students’ intuitive, spatial thinking and formal, deductive reasoning. It is crucial for students at this stage to develop abstract concepts based on knowledge of and reasoning with geometric properties. Dynamic geometry tools are promising resources in classroom teaching for enriching students’ experience in geometry.
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\nStudies about students’ learning with dynamic geometry tools usually focus on the long-term development of tool use in the contexts of problem solving and explorations. Data are mostly obtained from interviews and observations with students working in small groups or individually.
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\nThis study begins with an original approach. It aims at understanding students’ perceptions of geometric properties in simple dynamic figures without assuming any prior experience of students with dynamic geometry tools. A special web-based platform was developed to capture students’ results of dragging in dynamic figures. Quantitative and qualitative data were obtained for analysis from this platform and task-based interviews respectively.
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\nIn a set of 8 tasks designed in this study, students were asked to drag free points in pre-constructed figures to create examples of geometric configurations with parallel lines. Results of the tasks were collected through the web-based platform from 1,589 secondary 1 to 4 Hong Kong students in 11 schools of different backgrounds. In the next stage, 24 secondary 2 students from another 6 schools took part in task-based interviews, enabling detailed observations and analysis of their reasoning.
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\nThe basic assumption of this design is that students working on the tasks will generate examples of geometric figures that reflect their understanding of relevant geometric properties. Their results could then be analyzed in the form of collective and personal example spaces with critical dimensions of variation to be identified (Watson & Mason, 2005).
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\nThe findings of the study indicate how participating students vary in their ways of discerning critical geometric properties in dynamic figures. In particular, the results reflect students’ general limited awareness of basic geometric properties, such as equal opposite sides or angles, while manipulating and interpreting pre-constructed figures. Graphical representations of collective example spaces, developed in this study, provide useful means for revealing dimensions of variation in examples generated. It is hoped that the findings and method of the study can inform classroom teaching with dynamic geometry tools that capitalizes on variation in students’ perceptions.

Multi-agent deep reinforcement learning (MADRL) has been used for disaster rescue and other robotic applications. In most MADRL cases, agents are trained only to deal with specific tasks, so once there are unpredictable changes in target tasks, the earlier trained-models might no longer work, and agents have to get trained again from scratch in limited time, which means traditional MADRL is unsuited to unpredictable post-disaster environments. In order to promote the scalability and flexibility of the multi-agent system, meta-learning, which has achieved few shot learning successfully in supervised deep learning field, could reuse the earlier models as the prior knowledge to guide and speed up the new task learning process. In this work, we propose a framework to apply meta learning methods to multi-agent deep deterministic policy gradient (MADDPG), and achieve efficient adaption to new tasks with less time and fewer samples. Some common experiences in the previous tasks learning will be refined as the meta knowledge. Once the tasks get changed with new scenarios, agents will get retrained with good initial network parameters based on this meta knowledge. Although the new scenarios might be different from the previous scenarios, agents could quickly adjust their policy and get better performance within only few episodes. In two experiments, our method performs better learning efficiency than others, getting higher rewards in the first several episodes of new tasks learning.

As a kind of mathematical geometry education software, dynamic geometry software is an application software that can interactively create and manipulate geometric objects. Its vivid image changes and colorful geometric objects provide teachers with an efficient Geometry teaching tool. Comparing and analyzing the existing geometry software, we found that there is a curve smoothness problem in its trajectory drawing. In this article, we propose the uniform point selection method based on the principle of the trajectory generation geometric method. Under the premise of ensuring rendering efficiency, extensive case studies have shown that the smoothness of the trajectory curve generated by this method has been improved.

Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra {mathcal{A}} we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying {mathcal{A}}—that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category {mathsf{C}} can be applied directly to these geometric objects to automatically yield an extension {tilde{F}} acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant {tilde{K}_{ f}} of the extension {tilde{K}} of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.

Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy characteristics is proved and relations to the equivalence problem, theory of geometrical objects and connection theory are briefly mentioned.

CalGeo is a three-year project supported by EU programme Comenius 2.1. Amongst the objectives of this project is the design of an in-service teacher education programme which employs dynamic geometry tools for teaching Calculus in upper secondary education. In this paper we present the project, its main objectives and the produced material; an example of a learning environment/activity designed for the introduction to the notion of derivative at Year 12; and, some results of the application of this activity in a real classroom situation. In this activity we use the tangent line and the property of local straightness to introduce the formal definition of derivative. Several cases of differentiable and non-differentiable functions are discussed through their geometrical and symbolic representations.

We propose a new approach to building appearance models of 3D objects which is based on Canonical Correlation Analysis (CCA). In appearance based modeling, instead of building an explicit object model (e.g., 3D geometrical object model), a low dimensional object representation is obtained from a set of images. In standard appearance models typically Principal Component Analysis (PCA) is used for feature extraction. In our experiments we compare the performance of standard appearance models based on PCA and models based on CCA for 3D pose estimation. Results indicate that, while getting by with a smaller number linear features, CCA-based models perform consistently better.

This article proposes a possible path devoted to upper secondary school and early university students, as well as training teachers, with the aim to build a conscious approach to the learning/teaching of the conics, which uses, for an educational purpose, the close relationship between conics as loci of points of a plane and conics as sections of a cone. In this path, we will refer to some elements taken from the history of mathematics relating to a particular conic: the parabola. These elements could help students to discover and realize the transition from a parabola considered as a curve in a plane to the same parabola considered on a cone of which it is a section, as well as the inverse passage, and to grasp the profound link between two presentations of the same geometric object. Both steps will be carried out through constructions made with the use of the GeoGebra dynamic geometry software. In addition, it will be highlighted how the construction of conics by points has allowed the creation of lenses and mirrors, which represents a practical application of geometry very relevant to physics and astronomy. Such a practical approach could help students to overcome the difficulty in understanding conics by making the argument less abstract. Moreover, this path could build up an environment in which teachers and students could explore some semiotic registers and their changes, through which Mathematics expresses itself. In the final part, an educational experiment of the path that was proposed will be shown to the students of the Master’s degree course of “mathematics education” at the Department of Mathematics and Computer Science of the University of Calabria. The results of this experiment are described in detail and seem to confirm that the twofold view of the parabola as a section of a cone and as locus of points of a plane helps the students in understanding its meaning in both theoretical and applicative fields.