Improving QED-Tutrix by Automating the Generation of Proofs

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.Comment: In Proceedings ThEdu'17, arXiv:1803.0072

Computation of protein geometry and its applications: Packing and function prediction

This chapter discusses geometric models of biomolecules and geometric constructs, including the union of ball model, the weigthed Voronoi diagram, the weighted Delaunay triangulation, and the alpha shapes. These geometric constructs enable fast and analytical computaton of shapes of biomoleculres (including features such as voids and pockets) and metric properties (such as area and volume). The algorithms of Delaunay triangulation, computation of voids and pockets, as well volume/area computation are also described. In addition, applications in packing analysis of protein structures and protein function prediction are also discussed.Comment: 32 pages, 9 figure

Proceedings of the ECCS 2005 satellite workshop: embracing complexity in design - Paris 17 November 2005

Embracing complexity in design is one of the critical issues and challenges of the 21st century. As the realization grows that design activities and artefacts display properties associated with complex adaptive systems, so grows the need to use complexity concepts and methods to understand these properties and inform the design of better artifacts. It is a great challenge because complexity science represents an epistemological and methodological swift that promises a holistic approach in the understanding and operational support of design. But design is also a major contributor in complexity research. Design science is concerned with problems that are fundamental in the sciences in general and complexity sciences in particular. For instance, design has been perceived and studied as a ubiquitous activity inherent in every human activity, as the art of generating hypotheses, as a type of experiment, or as a creative co-evolutionary process. Design science and its established approaches and practices can be a great source for advancement and innovation in complexity science. These proceedings are the result of a workshop organized as part of the activities of a UK government AHRB/EPSRC funded research cluster called Embracing Complexity in Design (www.complexityanddesign.net) and the European Conference in Complex Systems (complexsystems.lri.fr). Embracing complexity in design is one of the critical issues and challenges of the 21st century. As the realization grows that design activities and artefacts display properties associated with complex adaptive systems, so grows the need to use complexity concepts and methods to understand these properties and inform the design of better artifacts. It is a great challenge because complexity science represents an epistemological and methodological swift that promises a holistic approach in the understanding and operational support of design. But design is also a major contributor in complexity research. Design science is concerned with problems that are fundamental in the sciences in general and complexity sciences in particular. For instance, design has been perceived and studied as a ubiquitous activity inherent in every human activity, as the art of generating hypotheses, as a type of experiment, or as a creative co-evolutionary process. Design science and its established approaches and practices can be a great source for advancement and innovation in complexity science. These proceedings are the result of a workshop organized as part of the activities of a UK government AHRB/EPSRC funded research cluster called Embracing Complexity in Design (www.complexityanddesign.net) and the European Conference in Complex Systems (complexsystems.lri.fr)

MGOS: A library for molecular geometry and its operating system

The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present "Molecular Geometry (MG)'' as a theoretical framework accompanied by "MG Operating System (MGOS)'' which consists of callable functions implementing the MG theory. MG allows researchers to model complicated molecular structure problems in terms of elementary yet standard notions of volume, area, etc. and MGOS frees them from the hard and tedious task of developing/implementing geometric algorithms so that they can focus more on their primary research issues. MG facilitates simpler modeling of molecular structure problems; MGOS functions can be conveniently embedded in application programs for the efficient and accurate solution of geometric queries involving atomic arrangements. The use of MGOS in problems involving spherical entities is akin to the use of math libraries in general purpose programming languages in science and engineering. (C) 2019 The Author(s). Published by Elsevier B.V

INCREASING SENIOR HIGH SCHOOL STUDENTS\u27 MATHEMATICAL PROBLEM-SOLVING SKILLS THROUGH IMPROVING THEIR SPATIAL VISUALIZATION SKILLS BY LEARNING AND PRACTICING THREE-DIMENSIONAL DYNAMIC GEOMETRY WITH CABRI 3D

Problem-solving is one of the standards of mathematics processes developed by the National Council of Teachers of Mathematics. Many research studies indicated that there was a relationship between mathematics achievement and spatial visualization skills. But the research results still didn’t indicate whether or not there was a causal mediation relationship between the achievement of learning-and-practicing three-dimensional geometry and mathematical problem-solving skills mediated by spatial visualization skills. Therefore, it’s still necessary for us to specifically investigate whether or not spatial visualization skills can be a medial factor between the increase of mathematical problem-solving skills and the effects of learning and practicing three-dimensional geometry. For effectively increasing mathematical problem-solving skills and enhancing spatial visualization skills through learning and practicing three-dimensional geometry, we need an effective tool. And for investigating which tool was effective for learning and practicing three-dimensional geometry, experimental research was conducted at one of the prestigious public senior high schools in Indonesia. This experimentation applied Cabri 3D (dynamic geometry software) for the treatment group and traditional tools (non-dynamic geometry software) for the control group. The results of this experimental research indicated that both kinds of tools were significantly able to develop and enhance students’ spatial visualization skills, but with different effect sizes. The effect size in the treatment group was bigger than the effect size in the control group. The students in the treatment group outperformed the students in the control group in spatial visualization skills and mathematical problem-solving skills. Further investigation was to investigate the causal mediation relationship between achievement in learning-and-practicing three-dimensional geometry with Cabri 3D and mathematical problem-solving skills mediated by spatial visualization skills. Mediation analysis in this research indicated that there was a significant indirect effect between achievement in learning-and-practicing three-dimensional geometry with Cabri 3D and the increase of mathematical problem-solving skills mediated by the enhanced spatial visualization skills. This experimental research was then completed with questionnaires (students’ feedback). The conclusions of this experimental research are Cabri 3D (dynamic geometry software) is an effective tool for learning and practicing three-dimensional geometry, developing and enhancing spatial visualization skills, and indirectly increasing mathematical problem-solving skills

Aerospace Medicine and Biology: a Continuing Bibliography with Indexes (Supplement 328)

This bibliography lists 104 reports, articles and other documents introduced into the NASA Scientific and Technical Information System during September, 1989. Subject coverage includes: aerospace medicine and psychology, life support systems and controlled environments, safety equipment, exobiology and extraterrestrial life, and flight crew behavior and performance

Tarefas para promover a argumentação na classe de matemática baseada no Software de Geometria Dinâmica

Purpose: The tasks proposed to students have an impact on the cognitive activity they develop and on the construction of concepts and joint meanings. In this article we propose two tasks to promote interactivity in class mediated by argumentation processes with the use of GeoGebra. Description: Dynamic geometry environments allow students to experiment with different types of semiotic representation to mediate the cognitive activity they develop. Although some studies suggest that these environments decrease the cognitive load, it is recognized in the literature the impact that this type of systems generate in school learning, such as their use promotes the development of strategies to extend the problem or explore cases that allow students to generalize on a hypothesis or conjecture. In this sense, the importance of entrainment and of the concepts, definitions and fundamental topics of Euclidean geometry that are necessary for the constructions of the models that are tested is highlighted. Point of view: The teacher\u27s mediation and the presentation of significantly rich and challenging tasks allow generating moments of interactivity where the teacher\u27s actions are articulated with those of the students around the task. Conclusions: it is important for teachers to design activities that promote argumentation spaces in class as an opportunity to learn mathematics, since it allows confronting different points of view and mediating in the construction of meanings.Propósito: Las tareas que se proponen a los estudiantes repercuten en la actividad cognitiva que desarrollan y en la construcción de conceptos y significados conjuntos. En este artículo proponemos dos tareas para promover la interactividad en clase mediadas por procesos de argumentación con el uso de GeoGebra. Descripción: los entornos de geometría dinámica permiten que los estudiantes experimenten con diversos tipos de representación semiótica para mediar la actividad cognitiva que desarrollan. Si bien, algunos estudios sugieren que estos entornos disminuyen la carga cognitiva, se reconoce en la literatura el impacto que este tipo de sistemas generan en el aprendizaje escolar, su uso promueve el desarrollo de estrategias para ampliar el problema o explorar casos particulares que les permita a los estudiantes generalizar sobre una hipótesis o conjetura. En ese sentido, se resalta la importancia del arrastre y de los conceptos, definiciones y tópicos fundamentales de la geometría euclidiana que son necesarios para las construcciones de los modelos que se someten a prueba. Punto de vista: la mediación del profesor y la presentación de tareas significativamente ricas y desafiantes permiten generar momentos de interactividad donde se articulan las acciones del profesor con las de los estudiantes en torno a la tarea. Conclusiones: es importante que los profesores diseñen actividades que promuevan espacios de argumentación en clase como una oportunidad para aprender matemáticas dado que permite confrontar diversos puntos de vista y mediar en la construcción de significadosObjetivo: As tarefas propostas aos alunos têm um impacto na atividade cognitiva que desenvolvem e na construção de conceitos e significados comuns, neste artigo propomos duas tarefas para promover a interatividade na aula mediada por processos de argumentação com a utilização do GeoGebra. Descrição: os ambientes de geometria dinâmica permitem aos estudantes experimentar diferentes tipos de representação semiótica para mediar a atividade cognitiva que desenvolvem. Embora alguns estudos sugiram que estes ambientes diminuem a carga cognitiva, é reconhecido na literatura o impacto que este tipo de sistemas gera na aprendizagem escolar, a sua utilização promove o desenvolvimento de estratégias para expandir o problema ou explorar casos particulares que permitem aos estudantes generalizar sobre uma hipótese ou conjectura. Nesse sentido, destaca a importância do arrasto e dos conceitos, definições e tópicos fundamentais da geometria euclidiana que são necessários para as construções dos modelos que são testados. Ponto de vista: a mediação do professor e a apresentação de tarefas significativamente ricas e desafiantes permitem gerar momentos de interatividade onde as ações do professor são articuladas com as dos alunos em torno da tarefa. Conclusões: é importante que os professores concebam atividades que promovam espaços de argumentação na aula como uma oportunidade para aprender matemática, dado que lhes permite confrontar diferentes pontos de vista e mediar na construção de significados

Représentations dynamiques et tangibles dans l'enseignement mathématique

International audienceDynamic geometry environments offer a new kind of representation of mathematical objects that are variable and behave "mathematically" when one of the elements of the construction is dragged. The chapter addresses three dimensions about the transformations brought by this new kind of representation in mathematics and mathematics education: an epistemological dimension, a cognitive dimension and a didactic dimension. As so often stated since the time of ancient Greece, the nature of mathematical objects is by essence abstract. Mathematical objects are only indirectly accessible through representations (D'Amore 2003, pp. 39-43) and this contributes to the paradoxical character of mathematical knowledge: "The only way of gaining access to them is using signs, words or symbols, expressions or drawings. But at the same time, mathematical objects must not be confused with the used semiotic representations" (Duval 2000, p. 60). Other researchers have stressed the importance of these semiotic systems under various names. Duval calls them registers. Bosch and Chevallard (1999) introduce the distinction between ostensive and non ostensive objects and argue that mathematicians have always considered their work as dealing with non-ostensive objects and that the treatment of ostensive objects (expressions, diagrams, formulas, graphical representations) plays just an auxiliary role for them. Moreno Armella (1999) claims that every cognitive activity is an action mediated by material or symbolic tools. Through digital technologies, new representational systems were introduced with increased capabilities in manipulation and processing. The dragging facility in dynamic geometry environments (DGE) illustrates very well the transformation technology can bring in the kind of representations offered for mathematical activity and consequently for the meaning of mathematical objects. A diagram in a DGE is no longer a static diagram, representing an instance of a geometricalLes environnements de géométrie dynamique offrent un nouveau type d'objets mathématiques variables qui se modifient quand l'un des éléments de la construction est déplacé.Le chapitre aborde trois dimensions relatives aux transformations apportées par ce nouveau type de représentations en mathématiques et dans l'enseignement des mathématiques : une dimension épistémologique, une dimension cognitive et une dimension didactique