book of abstracts - umacome to malaga on time conference on mathematics education technology...
TRANSCRIPT
COME TO MALAGA ON TIMEConference On Mathematics Education
Technology Oriented.
MALAGA Organizes New
TIME 2010Technology and its Integration
into
Mathematics EducationJuly 6th-10th, 2010
E. T. S. I. Telecomunicaciones e Informatica, Malaga, Spain
This symposium combines 2 conferences:
11th ACDCA Summer Academy
9th Intl TI-Nspire & Derive Conference
Book of Abstracts
Editors:
Jose Luis Galan GarcıaGabriel Aguilera VenegasPedro Rodrıguez Cielos
Editors:Jose Luis Galan GarcıaGabriel Aguilera VenegasPedro Rodrıguez Cielos
Printed in SpainJuly 2010
Composition developed by editors using LATEX2ε
Contents
Prologue 11Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chairs & Committees 15Conference and Organizing Committee . . . . . . . . . . . . . . . . . . . . . . 17Program Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Keynote Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Google Maps and buses information 19General view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Malaga City Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Conference Venue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Bus number 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Bus number 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bus Line A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Shuttle Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Conference Venue Maps 29Ground floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31First floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Third floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Schedule 35General Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Wednesday, 7th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Thursday, 8th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Friday, 9th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Saturday, 10th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Abstracts 43
Keynote Lectures 45
K001: The role of CAS when learning algebra and developing functional
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thinkingBarbel Barzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
K002: Using the Real Power of Computer AlgebraMichel Beaudin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
K003: Tangible and dynamic mathematicsColette Laborde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
K004: Some Applications of Algebraic System SolvingEugenio Roanes-Lozano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
ACDCA Strand 51
A001: Teaching and Assessing Polygons Using TechnologyN. Radovic, T. Soucie, R. Svedrec, and I. Kokic . . . . . . . . . . . . . . . . . 53
A002: Learning Math, Doing Math: Fractals and Dynamic GeometryH. Flores and K. Alvarenga . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A003: Enhancing Mathematical Problem Solving through the use of DynamicSoftware Autograph
I. Karnasih and M. Sinaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A004: Arguing and Reasoning supported by Technology
Helmut Heugl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A005: Base Technologies for Tutoring
Walther Neuper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A006: The Positive Aspects of Modeling Process in Teaching Mathematics
Natalija Budinski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A007: Sliders – A Dynamic Support for Teaching Mathematics
Josef Bohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60A008: Physics Through GeoGebra Window
Abdul Sahib Hasani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A009: Web based education and assessment in the Bologna process
E. Lopez, E. Domınguez and G. Ramos . . . . . . . . . . . . . . . . . . . . . . 63A010: Problems and Prospects of Remote Teacher Training in Uniform E-Learning Environment
V. Bogun and E. Smirnov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A011: Mathematics lessons and Classroom examples, inspired by the “Dres-den Morning Post”
Rainer Heinrich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A012: Software GOLUCA: Knowledge representation in Mental Calculation
L. Casas, R. Luengo, V. Godinho and J.L. Carvalho . . . . . . . . . . . . . . . 67A013: WebQuest on Conic Sections as a Learning Tool for UndergraduateStudents
A. Kurtulus and T. Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A014: An analysis of arguments for and against the CAS
Harry Silfverberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A016: Competence, didactic situations and Virtual Environments for Teach-ing and Learning
O. Leon and C. Guzner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A017: The Intergeo Project
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M. Fioravanti and T. Recio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A018: Analogy and dynamic geometry software together in approaching 3dgeometry
M. F. Mammana, B. Micale and M. Pennisi . . . . . . . . . . . . . . . . . . . 74A019: Expanding Room for Tacit Knowledge in Mathematics Education
Jozef Hvorecky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A020: Using Technology to Support Mathematics Teaching
Martin Harrison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A021: Teaching Math With Advanced Learning Blocks
B. Horvat, M. Lokar and P. Luksic . . . . . . . . . . . . . . . . . . . . . . . . 77A022: Flexible Mathematics Content Preparation
B. Horvat, M. Lokar and P. Luksic . . . . . . . . . . . . . . . . . . . . . . . . 78A023: Designing Task for CAS Classrooms
Matija Lokar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A025: The use of notebooks in mathematics instruction. What is man-ageable? What should be avoided? A field report after 10 years of CAS-application
Peter Hofbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A026: The Role of Dynamic Geometry Software in the Process of Learning:GeoGebra Example about Triangles
M. Dogan and R. Icel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A027: Visualization and Interface in Exploration of Functions
Vladimir Nodelman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A029: Integrating Computers into Mathematics classes in a Unique way –Classroom Examples
R. Hoffmann and R. Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A030: Categorizing CAS Use within One Reform-Oriented United StatesMathematics Textbook
Jon Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A031: Critical Issues of Technology Use in Undergraduate Mathematics
Greg Oates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A032: On Assessment of Teaching A Mathematical Topic Using Neural Net-works Models (with a case study)
H. M. Mustafa, A. Al-Hamadi and M. H. Kortam . . . . . . . . . . . . . . . . 88A033: On the Visualization of the Calculus concepts
D. Takaci and J. Maksic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A035: The Impact of Computer Use on the Teaching of Geometry in Grade8
Sirje Pihlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A036: Folding and unfolding cones and cylinders with Cabri 3D: how to doit and how to use it in the classroom
Jean-Jacques Dahan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A037: CAS in the Swedish national tests in upper secondary level
G. Wastle and T. Hellstrom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A038: 3D-Dynamical Geometry in Building Construction
Raul Manuel Falcon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A039: Those Magic Moments when you realise you could not do this withchalk!
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Douglas Butler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97A040: Overcoming difficulties in understanding of the nonlinear program-ming concepts
Alla Stolyarevska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A042: A Computational Measure of Heterogeneity on Mathematical Skills
E. M. Fedriani and R. Moyano . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A043: Using Maxima in the Mathematics Classroom
E. M. Fedriani and R. Moyano . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A044: Applications of Multimedia Technology to study the ordinal thinkingevolution of scholars from 3 to 7 years old
P. Hernandez and J. L. Gonzalez . . . . . . . . . . . . . . . . . . . . . . . . . 101A045: The Past and the Future of Computer Algebra in Mathematics Edu-cation – A personal (& nostalgic) perspective
Bernhard Kutzler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A046: Technology and the Yin&Yang of Mathematics Education
Bernhard Kutzler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A047: An example of learning based on competences: Use of Maxima inLinear Algebra for Engineers
A. Dıaz, A. Garcıa and A. de la Villa . . . . . . . . . . . . . . . . . . . . . . . 105A048: Learning Math in the context of European Space for Higher Education
Salvador Merino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A049: Using intelligent adaptive assessment models for teaching mathematics
J. Galvez, E. Guzman, and R. Conejo . . . . . . . . . . . . . . . . . . . . . . . 108A050: How to assess the quality of interactive dynamic geometry resources?The Intergeo experience
C. Laborde and S. Soury-Lavergne . . . . . . . . . . . . . . . . . . . . . . . . 109A051: Free Dynamic Software for Exploring Multi-Dimensional Relations
Sharon Whitton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A052: Taking advantage of Sherman’s march
P. Guerrero and M. Toril . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A053: GeoGebra Workshop for the Initial Teacher Training in Primary Ed-ucation
Natalia Ruiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A055: Problem solving using Geogebra
Kaja Maricic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A056: Dynamic Applets for Differential Equations and Dynamical Systems
Robert Decker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A057: Lessons learned using Publishers’ Web-based software to assess stu-dent work
June Decker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A058: Interactive Teaching Materials on the Web – Their Diversities andVariety
Vlasta Kokol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A059: On the Visualization of the Function
R. Vukobratovic and D. Takaci . . . . . . . . . . . . . . . . . . . . . . . . . . 118A060: Using Mathematics Journals to Enrich the Methods Course Experi-ences of Prospective Mathematics Teachers
Reda Abu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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A061: Teaching and Learning Geometric Transformations in the Context ofa Dynamic Environment
H. Bahadir Yanik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
TI-Nspire & Derive Strand 121
D001: C++ as a programming language for CASLluıs Parcerisa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
D002: Coding Theory for the ClassroomJosef Bohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D003: 20 Years International DERIVE & CAS-TI User GroupJosef Bohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D008: Teaching Linear Regression in the PC LabKarsten Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D009: Solving Second Order ODEs, Two Non-analytical Methods RevisitedM. Beaudin and G. Picard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D010: Using Rational Arithmetic to Develop a Proof. “What Josef and CarlSaw”
J. Bohm and C. Leinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129D011: Beyond Newton’s Law of Cooling Updated Methods for EstimatingTime Since Death
Carl Leinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130D012: Using DERIVEr’S Graphics Tools For Geographic Profiling of SerialOffenders
Carl Leinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131D013: Using Mathematical Tools In Forensic Investigations
C. Leinbach and P. Leinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132D014: Laplace Transforms, ODEs and CAS
G. Picard and C. Trottier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133D015: Could it be possible to replace DERIVE with MAXIMA?
A. Dıaz, A. Garcıa and A. de la Villa . . . . . . . . . . . . . . . . . . . . . . . 134D016: WIRIS – Derive, enabling compatibility
E. Badia, S. Egido, R. Eixarch and D. Marques . . . . . . . . . . . . . . . . . 136D017: Construction of mathematical knowledge using graphic calculators(TI-84 plus & CAS) in the mathematics classroom
Fernando Hitt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138D018: 3D Analytic geometry in using the link between CAS and Geometryapplications of the TI N’Spire
Jean-Jacques Dahan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140D019: WIRIS, a complete CAS with DGS
Ramon Eixarch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141D020: Experiments with geometric loci
Wolfgang Moldenhauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142D021: Mathematical Models in Biology
Mazen Shahin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143D022: The ElGamal Public-Key Cryptosystem – A Full Implementation Us-ing DERIVE
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Johann Wiesenbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144D023: A virtual laboratory for blended-learning: Numerical Methods usingWIRIS
A. Mora, E. Merida and R. Eixarch . . . . . . . . . . . . . . . . . . . . . . . . 145D024: Integrating a new Derive 6 Video User Guide into Virtual Teaching
J. Bustos, J. L. Galan, Y. Padilla and P. Rodrıguez . . . . . . . . . . . . . . . 147D025: A CAS routine for obtaining eigenfunctions for Bryan’s effect
S. V. Joubert, C. E. Coetzee and M. Y. Shatalov . . . . . . . . . . . . . . . . 149D026: Interactive Explorations of Mathematics with TI-Nspire Technology
K. Stulens and G. Brothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150D027: 11 Years of Master Theses in Engineering using DERIVE in the Uni-versity of Malaga
G. Aguilera, J. L. Galan and P. Rodrıguez . . . . . . . . . . . . . . . . . . . . 151D028: Teaching Differential Equations and its Applications Using Derive 6as a PeCas
G. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla and P. Rodrıguez . . . . . . 152D030: Teaching Calculus and Numerical Analysis using CAS according toBologna Process
J. M. Gonzalez, T. Morales, M. L. Munoz, M. A. Atencia and S. Merino . . . 153D031: E-Learning and Joomla
S. Merino, J. Martınez, J. L. Galan, P. Rodrıguez, M. L. Munoz, J. M. Gonzalez,P. Cordero, Y. Padilla, G. Gutierrez, A. Mora, E. Merida and F. Rodrıguez154
D032: Using Notes with Interactive Math Boxes in TI-Nspire Software Ver-sion 2
Wade Ellis, Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155D033: Exploring Rose Curves with the TI-Nspire Calculator
James Petty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
TICEMUS 2010 (Spanish Strand) 157
T001: La calculadora ClassPad como recurso didactico para la ensenanza delas Matematicas. Interpretacion matematica a traves de la calculadora
Agustın Carrillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159T002: Esquemas de Argumentacion en la Matematica Escolar
Homero Flores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160T003: Matematicas en Entornos Virtuales de Aprendizaje. Caso WIRIS enMoodle
Ramon Eixarch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161T004: Manuales Electronicos para la Ensenanza de la Geometrıa en 2o deBachillerato
G. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla y P. Rodrıguez . . . . . . . 162T005: Utilizacion de vıdeos en las asignaturas de Matematicas de 2o deBachillerato
G. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla y P. Rodrıguez . . . . . . . 163T006: Criterios topologicos en la evaluacion y promocion del alumnado enSecundaria
M. T. Davila, E. M. Fedriani y R. Moyano . . . . . . . . . . . . . . . . . . . . 164
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T007: Explorando la Geometrıa en Educacion Secundaria con los Graficos dela Tortuga
Rosario Vera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165T008: TutorMates: Un nuevo paradigma en la ensenanza de las Matematicas
A. Pascual y A. Limon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167T009: Matematicas 2.0 con Descartes
F. J. Rodrıguez, J. R. Galo y J. A. Salgueiro . . . . . . . . . . . . . . . . . . . 169T010: Descartes en Wikispaces
F. J. Rodrıguez, J. R. Galo y J. A. Salgueiro . . . . . . . . . . . . . . . . . . . 171T011: Programacion Recreativa versus Matematica Recreativa
M. Ruiz y B. C. Ruiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172T012: Los materiales de Descartes como catalizadores de la reflexion meto-dologica
Jose R. Galo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173T013: Nuevas tecnologıas y ensenanza: Introduccion constructiva, geometricay dinamica del concepto de derivada
C. Caballero y J. Bernal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175T014: Matematicas Manipulativas con DESCARTES
J. L. Alcon, J. R. Galo y J. G. Rivera . . . . . . . . . . . . . . . . . . . . . . . 177T015: Un experimento de ensenanza para la construccion del concepto deintegral definida usando un programa de geometrıa dinamica
C. Aranda y M. L. Callejo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178T016: Las Matematicas ante el problema de incomprension de un idioma
Ana Lopez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179T017: Un Acercamiento al Calculo desde la Realidad Virtual con DESCARTES
J. L. Alcon, J. R. Galo y J. G. Rivera . . . . . . . . . . . . . . . . . . . . . . . 181T018: Influencia del empleo de una metodologıa didactica basada en el usode las TIC en el tratamiento efectivo de la diversidad en el aula ordinaria dematematicas
J. J. Larrubia, Y. Padilla, M. Alvarez, J. A. Argote, L. Troughton, N. Puertas,V. Perez, M. D. Domene, M. A. Medina, E. de la Plata, R. Ros, R. Loringy E. Garcıa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
T019: Introduccion geometrica de los numeros reales con GeogebraA. Caro y A. M. Martın . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
T020: Elaboracion de paginas web dinamicas con GeogebraA. Caro y A. M. Martın . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
List of Participants 187List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Authors Index 195
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Prologue
Prologue
Dear colleagues,
It is our pleasure to welcome you to Malaga for TIME 2010 (Technology and itsIntegration into Mathematics Education). This symposium combines 2 conferences:
• 11th ACDCA Summer Academy
• 9th Intl TI-Nspire & Derive Conference
From the very beginning, TIME Conferences have been a very important meetingpoint for professionals in Mathematic Education. In this occasion, TIME 2010 hasjoined 134 delegates from 34 different countries and a total of 154 participants.
Since TIME 2010 has been hold in Spain, a special strand in Spanish has also beenhold: TICEMUS 2010 (TIC en Educacion Matematica en Universidad y Secundaria)specially focused to Spanish High School teachers.
107 contributions have been finally accepted due to their high quality after beingpeer-reviewed by program committee members. Some proposals were rejected since theydid not pass the peer-reviewing process. The distribution of these 107 contributions areas follows:
• 4 Keynote Lectures.
• 55 for the ACDCA Strand.
• 28 for the TI-Nspire & Derive Strand.
• 20 for the TICEMUS Strand.
These 107 contributions can also be classified in 4 Keynote Lectures, 18 Workshopsand 85 Lectures and they have been distributed in up to seven parallel sessions.
The editors deeply thank to:
• Gilles Picard, our Conference Program colleague, not only for all his work withinorganization matters, but also for his continuous support.
• Josef Bohm, Bernhard Kutzler and Vlasta Kokol-Voljc, Program Committee Co-Chairs, for all their work conducting the peer-review process and their continueshints and suggestions regarding organization.
• All the Program Committee Members for their work in the peer-review process.
• Barbel Barzel, Michel Beaudin, Colette Laborde and Eugenio Roanes-Lozano foraccepting the offer of being keynote speakers and providing 4 wonderful and veryinteresting keynote lectures.
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• Salvador Merino, Yolanda Padilla and Ma Angeles Galan, our Organizing Com-mittee colleagues. Without their work, this meeting would have been significantlydifferent. We also want to thank in this point our 12 students from the localorganization, for their invaluable help.
• All the sponsors, in alphabetic order: Applied Mathematic department, E.T.S.I.Informatica, E.T.S.I. Telecomunicacion, Picasso’s Museum, Texas Instruments,Malaga Tourism Bureau, Antequera Town Hall, Malaga Town Hall, Unicaja andUniversity of Malaga, for their generous support.
• And last but not least, all the participants (both delegates and accompanyingpersons) for their interest and contribution to make this meeting to be a veryimportant event.
We wish you a wonderful and productive TIME 2010 Meeting.
Malaga, July 2010
The editors:Jose Luis Galan Garcıa
Gabriel Aguilera VenegasPedro Rodrıguez Cielos
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Chairs & Committees
Conference and OrganizingCommittee
Conference Committee Co-Chairs
Jose Luis Galan Garcıa, SpainGilles Picard, Canada
Pedro Rodrıguez Cielos, SpainGabriel Aguilera Venegas, Spain
Organizing Committee
Jose Luis Galan Garcıa, SpainPedro Rodrıguez Cielos, Spain
Gabriel Aguilera Venegas, SpainSalvador Merino Cordoba, Spain
Yolanda Padilla Domınguez, SpainMa Angeles Galan Garcıa, Spain
Students in Organizing Committee
Jairo Bustos HerediaAna Isabel Butler Lucena
Carlos Canet EspinosaOlimpia Dıaz Fernandez
Silvia Carolina Figueroa ArdilaLydia Flores Martos
Pedro Garrido MartınIsidro Gomez Sanchez
Francisco Gomez SantiagoDaniel Jimenez MazureCarlos Lorente MorenoNuria Sanchez Jimenez
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Program Committee
Program Committee Co-ChairsACDCA Summer Academy TI-Nspire & DERIVE Conference
Bernhard Kutzler Josef BoehmVlasta Kokol-Voljc Jose Luis Galan Garcıa
Program Committee Members
ACDCA Summer Academy TI-Nspire & DERIVE ConferenceBenno Grabinger, Germany Gabriel Aguilera, SpainWilfried Herget, Germany Sarvari Csaba, Hungary
Helmut Heugl, Austria Terence Etchells, United KingdomPeter Jones, Australia Homero Flores, Mexico
Steve Joubert, South Africa Rainer Heinrich, GermanyDjordje Kadijevic, Serbia Guido Herweyers, Belgium
Ed Laughbaum, USA Peter Hofbauer, AustriaZsolt Lavicza, United Kingdom Steve Joubert, South Africa
Walther Neuper, Austria Carl Leinbach, USAKathleen Pineau, Canada Pedro Rodrıguez, Spain
Marlene Torres-Skoumal, Austria Karsten Schmidt, GermanyYuzita Yaacob, Malaysia Eno Tonisson, Estonia
Nurit Zehavi, Israel Agustın de la Villa, Spain
Keynote Speakers
Barbel Barzel, GermanyMichel Beaudin, CanadaColette Laborde, France
Eugenio Roanes-Lozano, Spain
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Abstracts
Keynote Lectures
K001: The role of CAS whenlearning algebra and developing
functional thinking
Barbel BarzelUniversity of EducationFreiburg (im Breisgau)
Germany
Keynote Lecture
Abstract
What are the specific benefits of integrating computer algebra into mathematicsteaching and what are problems which arise when computer algebra is always avail-able. These questions are discussed along certain teaching examples. Results of researchprojects and case studies are included when trying to give an answer to these questions.It is the style of teaching and the type of questions in the classroom situation which haveto change, so that CAS can really lead to an improvement of learning mathematics.
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K002: Using the Real Power ofComputer Algebra
Michel BeaudinEcole de Technologie Superieure
MontrealCanada
Keynote Lecture
Abstract
Almost everyone is convinced that Computer Algebra (CA) is a good tool that shouldbe used for solving complicated problems. The main idea that will be developed in thistalk is the following: we also think that CA can be used to solve problems with differentdegrees of complexity and to explain mathematical concepts. CA can be used to teachmathematics and represents a nice opportunity to do more mathematics instead of less.Isn’t it surprising to observe that too often, despite all this technology, we still continueto follow a classical way, avoiding to link (apparent) different subjects?
Instead of removing items from the curriculum, our point of view is to try to solve aproblem using different approaches – don’t forget that students rapidly forget what youhave said some weeks ago – and also to reinforce paper and pencil techniques with theaid of an appropriate tool – illustrating the real possibilities of teaching mathematicsusing CA. Derive, Nspire CAS software and Voyage 200 symbolic calculator will beused.
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K003: Tangible and dynamicmathematics
Colette LabordeProf. Emerita
University Joseph FourierGrenobleFrance
Keynote Lecture
Abstract
The dynamic mathematics environments offer representations of mathematical ob-jects that can be manipulated and behave mathematically when varying. Variation,variables, co-variation are reified in this kind of environment, they become central andprovide new ways of approaching mathematics focusing on variation. These environ-ments can give rise to new kinds of learning situations in which varying the objects playsvarious roles. The talk will analyze these different roles and bring dragging into focus.It will be illustrated by means of examples in Cabri II Plus and Cabri 3D as well as inCabri Elem for elementary school.
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K004: Some Applications ofAlgebraic System Solving
Eugenio Roanes-LozanoAlgebra Dept., School of EducationUniversidad Complutense de Madrid
MadridSpain
Keynote Lecture
Abstract
Curiously enough, unlike what happens with linear system solving, algebraic (poly-nomial) system solving is not widely known among those mathematicians that are notfamiliar with computer algebra. But, as happens with linear system solving, a problemthat can be expressed or modelled in such a way can therefore be resolved or decided. Asa continuation of my keynote lecture at TIME’2004 (where a geometric interpretationof algebraic systems was given), an overview of some applications of algebraic systemsolving will be now given. They include a wide variety of fields such as graph colouring,mechanical theorem proving in Geometry, logic, decision taking in a railway interlocking,etc.
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ACDCA Strand
A001: Teaching and AssessingPolygons Using Technology
N. Radovic T. SoucieDepartment of Geodesy Matka Laginje Elementary School
University of Zagreb CroatiaCroatia
[email protected] [email protected]
R. Svedrec I. KokicOtok Elementary School Trnsko Elementary School
Croatia [email protected] [email protected]
Lecture for the ACDCA Strand
Abstract
Geometry is an integral component of mathematics learning because it allows stu-dents to analyze and interpret our physical environment. Furthermore, it equips studentswith tools they can apply in other areas of mathematics. It is therefore critical to helpstudents build solid foundations and understanding of geometric concepts as well as gainadequate geometry related skills.
In our presentation, we will show a unit on Polygons in which dynamic geometrysoftware is used for exploring and learning new concepts as well as assessment. Throughvarious age appropriate activities students explore polygons and their properties, makeconjectures, test them, reason about geometric ideas as well as demonstrate understand-ing and ability to apply their knowledge of polygons.
The activities used in our model create a motivating and engaging environment wheretechnology allows students to discover mathematics on their own and construct their ownunderstanding. The dynamic geometry software allows students to work at their ownpace and feel free to take risks. It requires active engagement of students and encourageshigher level thinking.
Keywords
geometry, teaching, assessment, dynamic geometry software, polygons
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A002: Learning Math, Doing Math:Fractals and Dynamic Geometry
H. Flores K. AlvarengaColegio de Ciencias y Humanidades Department of Mathematics
UNAM Federal University of SergipeMexico Brazil
[email protected] [email protected]
Workshop for the ACDCA Strand
Abstract
Learning Mathematics, Doing Mathematics is a teaching model under constructionin the last years. The main feature of this Model is to create a Teacher and Learning En-vironment in which students work co-operatively in the achievement of common learninggoals. One of these learning goals is the fostering of mathematical thinking skills suchas the ability for pattern recognition and generalization.
In this context we find that the use of the Dynamic Mathematics software The Ge-ometer’s Sketchpad is of great help in the design of activities aimed to the achievementof the aforementioned goal. In particular, its capabilities for constructing fractals arevery useful.
In this workshop we are going to construct and explore two fractals from the per-spective of the mathematical knowledge we can promote in the constructions and thequestions we can pose to students. The fractals that we are going to explore are a squarespiral fractal and the Koch curve.
The workshop is intended for Secondary and High School teachers and researchersinterested in the use of Dynamic Geometry software in a learner-centered setting. Thereis not necessary to have experience with the software.
Keywords
Co-operative Teaching Model, Fractals in Teaching, Dynamic Geometry, Pattern Recog-nition and Generalization.
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A003: Enhancing MathematicalProblem Solving through the use of
Dynamic Software Autograph
I. Karnasih and M. SinagaSampoerna School of Education
MedanIndonesia
Lecture for the ACDCA Strand
Abstract
The purpose of this study was to investigate students’ mathematical problem solvingand mathematical connection, in cooperative learning setting using Dynamic SoftwareAutograph, for students in Grade XI, in Medan, Indonesia.
This experimental study was conducted to 34 high school students in the SchoolYear 2009/2010. The collection of the data was done using observation sheets, docu-mentation, attitude scale, and test. Repeated measure test was delivered to students forfour times, in the first class session (first measurement), after 2 class sessions (secondmeasurement), after 4 times class session (third measurement), and after 6 class session(fourth measurement).
The result of the test showed that:
1. Cooperative learning using Dynamic Software Autograph in teaching Statistics inGrade XI improved students’ problem solving ability. This can be shown fromthe results, that there were significant improvement in students’ problem solvingability of the four tests given;
2. Cooperative learning using Dynamic Software Autograph in teaching statistics inGrade XI improved students’ mathematical connection ability. This can be shownfrom the results that there were significant improvement in students’ mathematicalconnection ability of the four tests given, even though there was no significantdifference between the first and the second measurement;
3. Students activity during teaching learning processes improved, shown by the ob-servations conducted during the lessons. Compared to the second measurement,the students’ activeness improved 37% on the third measurement and improved56,3% in the forth measurement.
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The result of analysis of the questionnaire given to the students showed that: 20 stu-dents (58.8%) like using Cooperative Learning methods, 31 students (91.2%) like usingDynamic Software Autograph, and 27 students (79.4%) like using cooperative learningwith Dynamic Software Autograph. Based on this result, it is hoped that mathematicsteachers have to consider the use of Dynamic Software Autograph in teaching and learn-ing secondary mathematics, especially in teaching statistics, and teaching other units ofmathematics such as Algebra, Geometry, Trigonometry, and Calculus.
Keywords
Cooperative learning, Dynamic Software Autograph, Mathematical Problem Solving,Mathematical Connection, Secondary School.
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A004: Arguing and Reasoningsupported by Technology
Helmut HeuglInstitute of Discrete Mathematics and Geometry
Vienna University of TechnologyAustria
Lecture for the ACDCA Strand
Abstract
Standards and Curricula express expected competences at certain steps of the learn-ing process. The task of the teachers is to accompany, to support the development of thecompetences of the learners. I will give some examples of how technology can promotethis development of competence in the field of arguing and reasoning.
The first part of the lecture deals with clarification of terms by discussing pairsof contradictory concepts like
– competence versus skills
– competence oriented versus calculation oriented mathematics education
A characteristic of competence orientation is to look at the learning processfrom a different angle, not from the contents which have to be executed (as most ofthe teachers are doing) but from the angle of mathematical activities like modelling,calculating, interpreting, arguing and reasoning which have to be performed by usingthe contents.
In the second part I will delve into the development of competence in the fieldof arguing and reasoning. The development of arguing does not start with deductiveproofs. As Freudenthal said: “Students at first should learn to suppose before learningto prove.” I will show the role of technology in the heuristic, experimental phaseas well as in the exactifying phase of the learning process. May be “exactifying” is aneologism, but by using this word I wanted to verbalize, that the emphasis of my thesisis the process of becoming more and more exact and not the product which could beentitled “exact phase”. In many cases technology makes the heuristic phase possible inthe first place. In the exactifying phase calculating can be transferred to technology,learners can center their activities on the thinking technology of proving.
Technology tools which I will use in the heuristic phase are “Technology supportedlearning paths” which ACDCA has produced in cooperation with the organizations “Ge-ogebra” and “Mathe online”. The project was called “Diversity of Media in MathematicsEducation”. Especially in the exactifying phase CAS like TI Voyage or TI nSpire arethe main tools.
Keywords
Competence, arguing, supposing, reasoning, learning path, Geogebra, TI Voyage
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A005: Base Technologies forTutoringWalther Neuper
Institute for SoftwaretechnologyGraz University of Technology
Austria
Lecture for the ACDCA Strand
Abstract
Tutoring is concerned with individuals: individual learners on different levels, withdifferent pace in learning, individual teachers with different teaching styles, emphasiz-ing specific examples, and individual programmers: frequently teachers themselves whocreate pieces of software on their own, because they are not satisfied with availableproducts.
This issue of individualization has lead to an unmanageable variety in educationalsoftware, usually small pieces of software with a narrow thematic scope, without freedomfor variants in user input, without user guidance, without making to underlying mathknowledge transparent. The trend is being reinforced by systems supporting applet gen-eration etc – but not supporting the requirements of freedom for input, of user guidanceand of transparent knowledge.
This talk identifies 3 basic technologies powerful enough to promise general accom-plishment of 3 basic requirements in tutoring:
1. The technology of Computer Theorem Proving (CTP) provides most general meansto prove “satisfiability modulo a theory” for user input; this allows to check userinput as generous and liberal as possible.
2. Program interpretation in debug-mode is a general means for user guidance: aprogram describing how to solve a problem is stopped at “break points”, and theinterpreter hands over control to the user (or a dialog module).
3. Transparent knowledge is again provided by CTP technology: all the math knowl-edge is represented in a human readable format (traditional language of math) ona separate language level, which can be inspected from the context of any “breakpoint”.
The technologies mentioned are considered general enough to cover both, algebra andgeometry (while other domains like graph theory and others are not yet considered).
The feasibility of exploiting these technologies for educational math software is demon-strated by experiments at Graz www.ist.tugraz.at/projects/isac, performed with Isabellehttp://isabelle.in.tum.de.
Keywords
tutoring , computer theorem proving , human readable math knowledge , interpretationin step mode , Isabelle
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A006: The Positive Aspects ofModeling Process in Teaching
Mathematics
Natalija BudinskiPrimary and grammar school “Petro Kuzmjak”
Ruski KrsturSerbia
Lecture for the ACDCA Strand
Abstract
This paper proposes modeling based learning as a tool for learning and teachingmathematics. The example of modeling real world problems related to exponentialfunctions and differential equations is described, as well as the statistical data that areobtained by tracking students’ achievements in the modeling process.
The modeling process in mathematical education tends to follow the didactical cycleof activities and reach a desirable level of accomplishments of students’ activities andcompetencies. In this process students were encouraged to use software such as Geogebrain order to develop and foster deeper learning of mathematics, connected with the realworld problems.
Keywords
Modeling, real world problem, exponential function, educational software
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A007: Sliders – A Dynamic Supportfor Teaching Mathematics
Josef BohmACDCA &
International DERIVE & CAS-TI User GroupAustria
Lecture for the ACDCA Strand
Abstract
Technology in Math Education is based on four representation forms:
• numerically
• analytically
• graphically and
• verbally.
Didactics demand that the teacher should at least three of them when teaching andexercising mathematics.
We will compare the sliders offered by many programs which are popular in tech-nology supported mathematics education (MS Excel, DERIVE, TI-Nspire, Autograph,WIRIS, GeoGebra, . . . ).
At one hand sliders can intensify the graphic representation and at the other handthey form a link between analytic and graphic representation by emphasising and inter-preting the meaning and interpretation of one or more parameters.
We will show and discuss a collection of examples from various fields of school mathe-matics with sliders playing an important role connecting analytical, graphical and verbalapproaches to concepts and models.
Keywords
representation forms, didactical use of sliders, comparing mathematical assistants
60
A008: Physics Through GeoGebraWindow
Abdul Sahib HasaniIslamic Azad University, Abadan Branch
Iran
Lecture for the ACDCA Strand
Abstract
We start seeing physics (Nature, the surrounding environment) with naked eye thenwe would resort to use any available device to see better and to get more insight into“physics”. The commonly used devices are paper and pencil; one can draw figures andcurves and do some calculations then draw some conclusions. But because holding aphysical handle is easier than confining ourselves to the paper and pencil, we do experi-ments to better our understanding. Continuing this method we may design experimentsthat need more expenditure of time and effort. Hence we resort to the virtual world inorder to see the real world. In this paper I present some virtual experiments by GeoGebrathat include the electromotive force induced in a loop
(http://www.geogebra.org/en/upload/files/english/ABDUL-SAHIB/InducedEMF RectangularLoop.html)
while entering a uniform magnetic field which first done by MATLAB and locating thefocus of spherical mirrors that first done by graphing calculator CASIO CFX-9850G
(http://www.ictmt9.org/contribution.php?id=24&lang=en)then continued by GeoGebra due to the efficiencies and the simplicity in using thissoftware. Also image formation in spherical mirrors, the relation between refractiveindex and the deviation angle in the prism and the condition for light emergence fromthe prism, inclined plane that acts as a simple brachistron, a euro that rolling aroundstationary euro, and the speed of a slipping ladder are investigated.
(http://www.geogebra.org/en/upload/files/english/ABDUL-SAHIB/circling.html)These experiments are more flexible than their real counterpart. For example the
dimension of the spherical mirror can easily be changed to see its effect on the focusposition or to see when the spherical mirror equation hold. In addition to the easeof changing the refractive index and the prism angle, The curve of relation betweendeviation angle with the incident angle traced simply by changing the incident angle andthe user can check the arrangement that satisfy the mathematical relation that ties thetwo angles to the refractive index. As to the induced electromotive force it visualize themoving loop and the electric wave which inspires a relation between the wave shape andthe shape of the loop - I investigated such relation for circular loop mathematically, andfor trapezoidal and triangular loops only experimentally and due to their characteristicsI called these virtual tools virtual signal generator. The rectangular loop is clearly statedin the textbooks as an example or an exercise.
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Keywords
Physics, virtual experiments, GeoGebra, graphing calculator, Dynamic geometry, MAT-LAB
62
A009: Web based education andassessment in the Bologna process
E. Lopez, E. Domınguez and G. RamosDepartment of Computer Languages and Computer Science
University of MalagaSpain
[email protected], [email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
The creation of the European higher education area by means of the Bologna pro-cess implies a series of changes in teaching and evaluation methodologies. These includeemphasizing the participation of the student and the incorporation of new technologiessuch as web based education. Hence, a full restructuring must be carried out for mostsubjects, which must take into account new tradeoffs between the contents and the aimsof the students. Moreover, teaching strategies and practices must be updated accord-ingly. Here we discuss a case study in full, which is based on our own experience. Inorder to adapt the Automata Theory subject to the new structure of Computer Sciencedegrees, a novel teaching methodology has been developed that belongs to the ECTScredit system implantation experience sponsored by the Computer Engineering Schoolof the University of Mlaga. This methodology emphasizes assessable classroom activi-ties, along with different kinds of projects to be completed during the course. A widerange of optional assignments is provided in order to cover the variability of students’interests. Furthermore, the performance of the students is evaluated through the coursewith the help of web based assessment tools, so that the student is always aware ofhis/her progresses. These tools are centralized in a virtual campus, which also serves asa framework for additional activities such as discussion forums and collaborative tasks.These adaptations are supported by a careful design of the way that the theoretical con-cepts are introduced. In particular, the notation, the presentation style and the relativeemphasis on the different sections of the syllabus are guided by students’ capabilitiesand the overall goals of the degree. These modifications should result in an improvementof the motivation and the integration of the contents of the subject with the ComputerScience body of knowledge that the School tries to convey.
Keywords
Web based education, web based assessment, Bologna process, virtual teaching.
63
A010: Problems and Prospects ofRemote Teacher Training in
Uniform E-Learning Environment
V. Bogun and E. SmirnovDepartment of Mathematics
Yaroslavl State Pedagogical UniversityRussia
Lecture for the ACDCA Strand
Abstract
There is a transition not only from using ICT locally to using ICT within local orglobal networks, but also from using ICT stationary resources to implementation of smallresources of information packed into various handheld devices. In this paper we discussdidactic problems and prospects observed in an educational process of teacher trainingby using a uniform E-learning environment along with small resources of informationprovided by graphic calculators. We identified three main didactic problems generatedby this approach. These were:
a) An absence in networks some dynamic resources for realization of educational set-tlement projects including interconnected activity: design activity is reduced tocreation of presentations and similar documents, that there are no computing andlogic projects that is inadmissible;
b) Monitoring absence of educational student’s activity and intermediate control oneach of mathematics sections;
c) an absence of possibility on implementation of computing and logic projects andintermediate testing on each of mathematics sections.
Among various prospects provided by this approach three are particularly importantfindings. These were:
a) The dynamic system of students testing within of project activity in mathemat-ics the with completely automated processes of values generating of initial data,correctness both obviously erroneous results and checks of answers on tests;
b) The possibility of demos generating of corresponding tests, implementation of high-grade monitoring of students educational activity and taking into account resultsof settlement projects performance and activity thanking completely automatedmechanisms of data processing;
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c) The corresponding applied software based on using of Web-server Apache on thebasis of programming language PHP and control systems by relational databasesMySQL.
Keywords
Dynamic Internet site, remote educational process, graphic calculator, interactive envi-ronment, teacher training.
Observations
We have created original monitoring system of REP with new algorithm programmingimplementation.
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A011: Mathematics lessons andClassroom examples, inspired by the
“Dresden Morning Post”
Rainer HeinrichMinistry of education and sport Saxony,
Germany
Lecture for the ACDCA Strand
Abstract
The use of new tools for mathematics at school wins increasingly importance. Thisoffers consequences as well as on aims and contents of mathematics at school as like onmethods in the lessons. The use of new technology opens a variety of new possibilities.The lecture should show general aims of mathematics education at the example of a newteaching curricula in Saxony.
The examples are taken from the ”Dresden Morning Post”, an ”infamous” newspaperin Saxony. It is an aim of the lecture to illustrate how to find interesting mathematicalexamples in situations of everyday life.
• What is the effect of speed restriction in the traffic?
• Where is the optimal point for a soccer player to hit the goal?
• How often should tickets in a tram be controlled?
With motivating examples the contribution of graphic computers and CAS shall bepresented for the realization of didactic strategies like motivating, discovering, visualizingand experimenting.
Keywords
Curricular standard, Ckassroom Examples using graphic calculators, CAS as pedagogicaltool.
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A012: Software GOLUCA:Knowledge representation in Mental
CalculationL. Casas, R. Luengo, V. Godinho and J. L. Carvalho
Facultad de Ciencias de la EducacinUniversity of Extremadura
Spain
Lecture for the ACDCA Strand
Abstract
In this paper we present the GOLUCA software (Godinho, V. Luengo, R. & Casas,L., 2009) that automatically allows obtain graphical representations in the form ofPathfinder Associative Networks (Schvaneveldt, 1990) of the cognitive structure of asubject in a particular area of expertise. In this case, we studied the strategies used formental calculation. Usually, it appears that many teachers, although considered com-petent in mental calculation, are not consciously aware of the strategies they use andtherefore not use them methodically to improve the teaching for their students.
In the context of a teacher training activity were identified in the first place, what werethe most frequently used strategies. Once identified, subjects were asked to perform a testusing the software GOLUCA. All possible pairs formed between different strategies werepresented to subjects at random, and they were asked to allocate a score of similarity,depending on the relationship they felt that had between one and another strategy.
Using GOLUCA software, networks were obtained representing the relationship be-tween different strategies, visually highlighting what the most important were, the rela-tionship between them, and the consistency of this relationship. We can see one of thesenetworks:
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Among the results of the work can be noted that the technique allows visually identifysignificant differences between teachers who are considered at higher or lower mentalcalculation skills and also identify the most appropriate strategies to become aware ofteach them effectively.
The GOLUCA software, with its ease of use and versatility, is useful for identifyingthe cognitive structure of the subjects in this area and others that are being investigated.
Keywords
Knowledge Representation, Mental Calculation, Pathfinder Associative Networks, Soft-ware.
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A013: WebQuest on Conic Sectionsas a Learning Tool for
Undergraduate Students
A. Kurtulus and T. AdaDepartment of Primary School Mathematics EducationEskisehir Osmangazi University, Anadolu University,
Turkey
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
WebQuests incorporate technology with educational concepts through integratingon-line resources with student-centered, activity-based learning. According to Sunal andHaas (2002), WebQuests are problem-solving activities for students that incorporate theInternet, computer-based materials, and other available resources. In order to effec-tively implement a WebQuest assignment, faculty must understand the various needsof each student involved. Many graduate students in teacher education programs areoften exposed to technology in the classroom and may also develop WebQuests in thesecourses. Specifically, We propose a WebQquest to teach conic sections, their geometricproperties, and algebraic representations to model physical situations and solve appli-cation problems. The goal of this topic is for prospective students to solve applicationproblems involving physical representations of conics sections. They must be able torecognize the conic that will model a situation, write its equation, and solve it eitheralgebraically or graphically for needed information. Finding and using the appropriateconic model is often a matter of recognizing the locus definition in an applied context.In some situations, the solution to the equation must them be interpreted in the contextof the problem situation. The WebQuest is entitled: “Creating a carpet design usingconic sections equations.”
Keywords
WebQquest, conics sections, teacher education, computer-based materials.
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A014: An analysis of arguments forand against the CAS
Harry SilfverbergDepartment of Teacher Education
University of TampereFinland
Lecture for the ACDCA Strand
Abstract
As is well known Finland has been one of the most successful countries in threesuccessive PISA tests in the comprehensive school level mathematics. On the otherhand, the educational policy adopted in upper secondary school mathematics educationhas been rather moderate, perhaps we could even say conservative. Unlike in manyother European countries, CAS has not been introduced into mathematics education inFinland on a large scale.
My presentation is going to deal with an on-going study project aimed at classifyingand understanding various arguments which have been given in order to justify atti-tudes both for and against the use of CAS into upper secondary school mathematicseducation. The study consists of two parts: a qualitative one and a quantitative one.The qualitative part is made up of the interviews which I have made with six influentialexperts of mathematics and mathematical education who have a central role in formingthe educational policy in Finland. Their arguments both for and against the use of CASare examined in this part of my study. The quantitative part handles an inquiry aboutprospective mathematics teachers’ notions of the pros and cons of the use of CAS. Boththe themes handled in the half-structured interviews and the statements of the inquiryare based on the points of view which have emerged from earlier research (Burrill et al.2003, Drijvers 2002, Gardiner, 2001, Guin et al 2005, Thunberg & Lingefjrd 2006), whichallows assessing the degree to which attitudes in Finland, respectively, are either in linewith or deviate from the argumentation presented in other countries. In my presenta-tion I am going to deal with preliminary results of both studies. According to that partof the data which is already analysed prospective teachers evaluated CAS applicationsamong the least important possibilities to apply technology into mathematics teaching.Practically all of the informants did not believe to take CAS in use during the threenearest following years. The estimation of the probability of the taking CAS in use wasalmost systematically lower than the estimation of the importance of CAS.
Keywords
CAS, attitudes, educational policy.
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A016: Competence, didacticsituations and Virtual Environments
for Teaching and Learning
O. Leon and C. GuznerNational Cuyo University
Department of Math Teaching – National Technological UniversityArgentina
Lecture for the ACDCA Strand
Abstract
In the last decade, there has been a notable increase in the use of Information andCommunication Technologies (ICTs) in the development of teaching tools and, conse-quently, their integration in different disciplinary areas of different educational levels.University has not escaped to this reality, and although most modern technologicalmeans are far from being available in every classroom - at least in our country- it isincreasingly common to have them.
Within the arsenal of available technologies, the Virtual Environments for Teachingand Learning (VLEs), or Learning Management Systems (LMS), have enjoyed rapiddevelopment and integration in the context above, although often there is no consensuswhat is “better” because in its assessment are weighted various factors, both institutionaland economic.
We hold that, beyond those mentioned above, the didactical issue should also beconsidered, particularly in relation to VLEs offer the opportunity to deepen not only incontent but mostly in more general procedures of activities in each discipline, favoringthe emergence of different models of learning, competence based. In light of these con-siderations, this study is part of the problem mentioned, with respect to the LMS andVLEs particularly in the teaching of mathematics at university level, taking as a the-oretical framework for the latter the “theory of didactic situations” of Guy Brousseau.The paper tries to examine what possibilities are offered by e-learning environments, toenact the referred theory. This requires taking into account how are affected issues suchas:
a) the organization and planning of the educational process,
b) the structure of knowledge itself (to suit the new context of building knowledge-transfer);
c) the relationships between the relevant actors (institution, teacher, student) andthe medium in which they develop activities;
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d) personal factors (individual characteristics, values, motivations, behaviors, satisfac-tion level, etc.)., institutional and social (impact, supply and demand for training,sustainability, etc.);
e) the design and development activities, and assessment instruments and processes;
f) programming, renewal and format of educational materials and resources.
Keywords
Platforms, didactic, mathematics, learning, competence.
Observations
The work forms part of the master’s thesis Teaching educational tool technologiesweb based design under a didactic perspective and the research project The realization ofCompetency-based approach, applications in the microcurriculum level.
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A017: The Intergeo Project
M. Fioravanti and T. RecioDepartment of Mathematics, Statistics and Computer Science
University of CantabriaSpain
[email protected] and [email protected]
Workshop for the ACDCA Strand
Abstract
Dynamic geometry software allows one to construct geometrical figures where someobjects (free points, lines, etc.) may be dragged with the mouse, while some relations,such as perpendicularity, for instance, are defined to hold. Thus, one can observe anddeduce properties that may be found at each of the diverse placements of the sameconstruction. In this way each construction is not just one figure but a potentiallyinfinite number of figures. Therefore, interactive geometry software is a powerful tool forteaching mathematics, far beyond a mere technological compass and ruler transposition.
Interactive geometry programmes have been available for more than twenty years.In spite of the large and diverse amount of constructions and teaching materials forinteractive geometry available from different sites, the use of interactive geometry in theclassrooms is far from desirable.
Intergeo (http://i2geo.net) is an EU-co-funded econtentplus project, with the par-ticipation of academic institutions and software developers from six European countries,which gives access to more than 2000 existing resources related to Dynamic Geometry,and it helps users to create new ones. Of special relevance in the project is the cre-ation of an Internet portal Intergeo, in ten languages, which collects all the informationrelated to it, and makes it available to the user. Resources are suitably classified andthe portal has a search engine which allows the fast finding of good quality materialrelated to a particular classroom theme. The materials are created using programs suchas Cinderella, Cabri, GeoGebra, Geonext, Geoplane/Geospace, TracenPoche, Wiris, andthe like. All of the explicitly mentioned programs -some of them are open source, someare commercial– are members of the Intergeo consortium. The quality of the resourcesis voluntarily evaluated by the education community. This has a twofold purpose: toprovide teachers with information about the resources, so that they can be used in theirclassrooms with reliability, and to suggest the authors the possible ways to improve theresources.
The aim of the workshop is to present the Intergeo project, comment on its objectives,and to explain the audience how to become active users of the Intergeo web portal, howto submit new resources, how to search for content and how to collaborate in the qualitytesting of the available resources. Those attending the workshop will be able to practicesearching and evaluation of resources with real cases.
Keywords
Interactive geometry, dynamic geometry, teaching resources.
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A018: Analogy and dynamicgeometry software together in
approaching 3d geometryM. F. Mammana, B. Micale and M. PennisiNucleo di Ricerca e Sperimentazione Didattica
Department of Mathematics and Computer SciencesUniversity of Catania
Italy
[email protected], [email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
We present a didactic proposal on Euclidean geometry, both plane and three di-mensional geometry, consisting in a guided research activity that leads the students todiscover unexpected properties, of two apparently distant geometrical objects, that is,quadrilaterals and tetrahedra.
The explicit aim of the activity is to look for the analogies between figures of the planeand figures of the space, but indirectly it aims to retrieve the interest of the students tothe study of three dimensional geometry that, for sure, is more complex than the twodimensional one. In fact, it presents difficulties of conceptual type as well as difficultiesof linguistic type, but above all difficulties in realizing and interpreting the drawing intwo dimensions of three dimensional figures.
A new and catchy approach to three dimensional geometry has been realized by meansof an efficacious conceptual tool, the analogy, and an operative one, a dynamic geometrysoftware. The use of the analogy turns out to be precious because it represents a bridgethat create a significant link between two and three dimensions; while the software, CabriGomtre, permit to pass from observations on quadrilaterals realized with Cabri II Plusto explorations on tetrahedral realized with Cabri 3D.
This kind of activity represents an involving way to perform geometry aiming at:enhancing the rapresentation capacity and exploration ability of geometric sistuations;favouring autonomous creation of conjectures; boosting the consequent need of elabo-rating a rational argomentation of what discovered with its highest completion in thedemonstration.
Keywords
Quadrilaterals, tetrahedra, dynamic geometric software, mathematical laboratory, math-ematical discussion.
Observations
The proposal that we present has been developed within the Progetto Lauree Sci-entifiche - Scientific University Degrees Project - in the subproject for Mathematics ofCatania.
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A019: Expanding Room for TacitKnowledge in Mathematics
Education
Jozef HvoreckyDepartment of Information SciencesSchool of Management, Bratislava
Slovakia
Lecture for the ACDCA Strand
Abstract
Knowledge is present in two forms: explicit and tacit. Explicit knowledge is codified,precisely and formally articulated. Tacit knowledge is exclusively stored in human brains.It is subconsciously understood or applied, developed from direct action and experience,shared through conversation, story-telling etc. As precision and exactness are “obvious”attributes of Mathematics, its education predominantly concentrates on the consolidationof explicit knowledge.
Substantial portions of explicit knowledge are now stored in the tools like ComputerAlgebra Systems and Dynamic Geometry Systems. For their proper application, under-standing “what to do and why” becomes more important than “how”. The presence oftacit knowledge (understanding of context, ability to foresee the consequences, evaluationof validity of outcomes, etc.) becomes critical. Teaching Mathematics must concentrateon additional faces of knowledge transfer. The social context of mathematics must startplaying a substantial role in its education otherwise these advanced tools will be appliedin an incorrect and/or misleading way.
Our proposed lecture is based on the most popular model of knowledge transfer- Nonaka-Takeuchi’s SECI model (Socialization, Externalization, Combination, andInternalization). As Combination is the core of the current courses, three remainingcomponents deserve our special attention. In our lecture, we will indicate how they canbe included into the current content by solving problems that require formal thinkingand reasoning. The method was successfully tested in our database courses, in theircore formal section - query creation. A pedagocical experiment has been executed. Itsstatistically proved results show that our (less formal) approach speeds up the queryformation, preserves the correctness of outcomes and the students feel more comfortablywith their work.
Now, another experiment with financial mathematics is under preparation. In thetime of the conference, its first outcomes should be known and demonstrated.
Keywords
Tacit knowledge, explicit knowledge, SECI model, social aspects of education, less-formalapproach to exact problems.
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A020: Using Technology to SupportMathematics Teaching
Martin HarrisonMathematics Education Centre
Loughborough UniversityUnited Kingdom
Lecture for the ACDCA Strand
Abstract
New technologies aimed at improving the student learning experience have evolvedin recent years. We have used technology to develop a wide range of web-based learningresources to help deliver mathematics teaching and support student learning. Theserange from mathematics workbooks to complete narrated lectures, podcasts to videotutorials and short online quizzes to full-blown computer aided assessments.
One of our main objectives is to investigate the use of emerging technologies formathematics and statistics support and evaluate their effectiveness. Consequently ouruse of technology has significantly increased in recent years and this paper reviewssome early developments before considering in detail how technology is currently in-terwoven into our teaching with reference to teaching a typical engineering mathe-matics module with web-based resources. Some of these resources, e.g. mathcen-tre (http://www.mathcentre.ac.uk/), are openly available online; these help studentsmake the transition from school-level to university-level mathematics. We use others suchas HELM (http://helm.lboro.ac.uk/) to specifically cater for teaching our own stu-dents and encourage student engagement; they are “locally” available online via “Learn”(http://learn.lboro.ac.uk/), the university’s VLE (Virtual Learning Environment).We also use “Learn” to deliver formative and summative CAA (computer aided assess-ments), to deliver “narrated lectures” produced on a Tablet PC, and to deliver podcasts.Regular computer-based testing provides instant feedback and encourages students tolearn and practice their mathematics, narrated lectures provide an alternative to thetraditional lecture-tutorial approach and podcasts help reinforce key concepts and tech-niques. It is difficult to ascertain how beneficial a learning mechanism each is, but allare generally popular with students.
Keywords
Technology, Mathematics Support, Computer-Aided Assessment, VLE (Virtual LearningEnvironment).
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A021: Teaching Math WithAdvanced Learning Blocks
B. Horvat, M. Lokar and P. LuksicInstitute of mathematics, physics and mechanics and
Faculty of mathematics and physicsUniversity of Ljubljana
Slovenia
Lecture for the ACDCA Strand
Abstract
A new role of a teacher for the 21st century is here. As stated in numerous papersthis new role means that teachers should be oriented more towards guiding the learnerthrough the learning process. In this process information and communication technology(ICT) plays a significant role. So it is not surprising that more and more e-resources areavailable to be used in the learning process. But through analyzing existing resourceswe often find that the authors of resources do not use the opportunities offered by thenew technologies. One of the most important drawbacks of the existing resources is thatauthors too often forget (or neglect the fact) that teachers are the ones who will guidethe learner through the resources.
Recent studies have shown that teachers need e-learning content that they can easilyadapt and reuse for their own purposes. This means that lessons should be made out ofsmall learning blocks or, as they are also called, “knowledge objects”. A new conceptof how to create really useful e-learning content has evolved in Slovenia; namely, by“putting the teacher back into the game”. The selection of proper technologies and toolsfor managing e-learning content and the establishment of a user-friendly and easy-to-useenvironment for creating and modifying e-learning content, are essential to ensure basicsupport and popularization of e-learning.
In this talk, we will present new ideas with proofs of concepts of “modular, reallyinteractive e-content” build on the top of the mathematical knowledge using open-sourcesolutions, open standards and some programming. E-learning content, which will be dis-cussed, is not intended to be an electronic teaching book, but the add-on to the standardlearning material. Some preliminary results can be seen at http://www.nauk.si.
Keywords
E-learning content, educational content preparation, modular blocks, ICT in math teach-ing.
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A022: Flexible MathematicsContent Preparation
B. Horvat, M. Lokar and P. LuksicInstitute of mathematics, physics and mechanics and
Faculty of mathematics and physicsUniversity of Ljubljana
Slovenia
Lecture for the ACDCA Strand
Abstract
E-learning materials contribute a lot to the process of learning mathematics, sinceit is possible to use them to interactively present some mathematical structures, to givean explanation that is not found in textbooks or to offer online examinations. Butnevertheless, most of the existing e-materials are static in the sense that teachers cannotadapt them to their needs, but must use them exactly as they were prepared. Recentstudies have shown that teachers need e-learning content that they can easily adapt andreuse for their own purposes. This means that resources should be made out of smalllearning blocks. A new concept of how to create really useful e-learning content hasevolved in Slovenia; namely, by “putting the teacher back into the game”.
The project “Mathematics for Secondary Schools”, which is among the projects ofNAUK.si (NApredne Ucne Kocke - Advanced Learning Blocks), offers teachers the abil-ity to create their own materials or to modify already existing ones. With the knowledgethat is needed to edit articles on Wikipedia it is now possible to create one’s own e-learning content, which will have no shortage of interactivity and is also readily availableto students.
At the presentation, we will use the system for producing e-learning content callede-Sigma and show some examples. We will show how to combine the resources with aninteractive simulation in GeoGebra, build a quiz in which the next question dependson the correctness of the answer to the previous or is selected at random from a givendatabase, create a question where the answers are given as images, offer feedbacks whensolving etc. Some preliminary resources can already be seen at http://www.nauk.si.
Keywords
E-learning content, content creation, modular blocks, ICT in math teaching, high-schoolmathematics.
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A023: Designing Task for CASClassrooms
Matija LokarInstitute of mathematics, physics and mechanics and
Faculty of mathematics and physicsUniversity of Ljubljana
Slovenia
Lecture for the ACDCA Strand
Abstract
When designing CAS tasks we should not look on a CAS as an isolated tool but alsoin connection with other tools. This paper examines how with an appropriate selection oftools several shortcomings of CAS tasks can be avoided. It requires tasks to be adaptableto the needs of their users with respect to procedural and conceptual issues. Finally, itunderlines that designing and evaluating CAS tasks should take care of the whole processof their design, usage and modification.
When a certain task is evaluated, it is too often observed within a closed environment,from the perspective of a specific CAS used. Task designers too often wish to stick withthe same tool at any cost. Several problems could be seen in a different light if anothertool was to be used. Task design should exhibit also the appropriate use of tools.
So instead of arguing about the best way to cope with problems a specific toolintroduces into a certain task, the task itself should be designed in such a way thatstudents are actively encouraged to choose the most suitable tools.
Tasks should be prepared so that they can be adapted according to a particulardidactical situation. If a task designer overcomes his desire “to stay within the sameenvironment”, it is possible to design the task in a more flexible way. In this way theepistemic value of the task would be improved. Namely, instead of speculating theprecise ratio of both the procedural and the conceptual approach that would make thetask suitable for all students, the fact that this ratio is different for each student andeach particular set of circumstances, should be considered. Therefore, the tasks shouldbe designed in such a way that this ratio can easily be adapted to the needs of the user.
In the talk I will try to show how specific examples of tasks can be seen quite differ-ently if different tools and different approaches are combined and used instead of stickingwith only one tool.
Keywords
CAS, task design, procedural/conceptual approach, tools.
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A025: The use of notebooks inmathematics instruction. What is
manageable? What should beavoided? A field report after 10
years of CAS-application
Peter HofbauerCommercial Highschool Horn, AustriaUniversity of Education Lower Austria
University of Applied Sciences bfi ViennaAustria
Lecture for the ACDCA Strand
Abstract
Computer Algebra Systems (CAS) have been changing mathematics instruction re-quirements for many years. Since the tendency of using CAS in mathematics instructionhas been rising for decades and reports have often been positive, the implementation ofnotebook/netbook classes seems to be the consequent next step of mathematics instruc-tion supported by computers. Experiences that have been made with the use of CAS inPC-rooms can be transformed directly into the classroom. Hence the use of CAS is nolonger limited to certain rooms. The permanent availability of the notebook/netbookwith installed CAS offers the chance to realize these concepts that have already beenapproved with the use of CAS so far.
This speech shall show what these concepts could look like and that the use ofnotebooks/netbooks is not only the further development of teaching in PC-classes. Ex-amples from personal experience in teaching will especially show meanders and thought-provoking impulses in order to support teachers finding their way into teaching mathe-matics instruction in notebook/netbook classes successfully.
Keywords
CAS-based curricula and teaching methods, Dispensable and indispensable mathemati-cal, skills and abilities, Classroom examples using CAS, Future of CAS in Education.
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A026: The Role of DynamicGeometry Software in the Process ofLearning: GeoGebra Example about
Triangles
M. Dogan and R. IcelDepartment of Primary Mathematics Education,
University of Selcuk, KonyaTurkey
Lecture for the ACDCA Strand
Abstract
Specific benefits of integrating software into mathematics teaching and learning areappreciated all over the world. It is obvious that this consideration has to be discussedalong with certain teaching examples. Furthermore, classroom situations may also giveopportunities to see possible effects on teaching and learning of mathematics. Thus, itcan be said that computer can really lead to an improvement of teaching and learningmathematics by establishing possible benefits of software.
As dynamic mathematics software, GeoGebra and its use is getting increasingly com-mon all over the world as well. In addition to constructing geometry as a dynamic pro-gram, it also provides, as a key element of learning geometry, visualization, conjecture,creation, discovery, proof and etc. This study is about using GeoGebra for trianglesin eighth grade. The study is conducted in the 2009-2010 academic year. Two eighthgrade classes from a primary school have been selected as experiment and control groups.Before the classroom activities, a pre-test was applied to the both groups to determinethe students’ attainment level. The questions covered seventh grade objectives for thesubject. In the official curriculum, teaching of triangle for eighth grade takes total offifteen hours with eight different objectives. These objectives are mainly concentratedon the construction of triangles with specific properties such as; drawing a triangle witha given measures of sufficient elements, constructing mediator, perpendicular bisector,angle bisector and altitude of a triangle etc. Some of them aim to establish special fea-tures of triangles such as; determining the relationship between the sum or differenceof two sides’ lengths of the triangle and the length of the third side, determining therelationship between the sides’ lengths of a triangle and corresponding angles’ degreesbetween the sides, explaining the equality and similarity terms associated with triangles.etc.
A three weeks course has been planned in accordance with the official course curricu-lum for the experiment group. The course contained GeoGebra activities and practices
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about the stated achievements. The planned and GeoGebra constructed sixteen mainactivities which demand effective use of GeoGebra for this grade shared with the studentsduring the learning and teaching process. Simultaneously, the control group continuedtheir formal teaching and learning procedure. After the three weeks, a post-test wasapplied to both groups simultaneously. The post-test contained questions about all thestated objectives for the eighth grade. The post-test has been used to see possible effectsof GeoGebra on students’ success. This presentation will mainly contain selected samplesfrom the classroom used sixteen GeoGebra activities. It will give a great opportunity todiscuss the issues and outcomes of the real classroom applications with the colleagues.The presentation will also include some basic findings of the tests.
Keywords
Dynamic Geometry, GeoGebra, Students’ Success, Triangles.
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A027: Visualization and Interface inExploration of Functions
Vladimir NodelmanDepartment of Computer Science
Holon Institute of TechnologyIsrael
Lecture for the ACDCA Strand
Abstract
Studying of functions and their graphs is the most popular mathematics subject sup-ported by computer programs. This role “are able to perform” the amateur programs,written by schoolboys, and the commercial products, specially created for studying math-ematics or for application by professional engineers and mathematicians.
This paper presents analysis of typical lacks of existing software and offers few ways toovercome those, including features of visual modelling and interface support of adequatestudying activities with correct models.
Evident examples of such shortage provide incorrect graphing of functions with finiteand/or removable discontinuities, absence or weak support of functional correspondencevisualization in multivariate cases. The fundamental activities of domain element’s imageand range element’s preimage finding, being initial acts of correspondence recognition,need appropriate software maintenance.
Findings of equation’s roots and function’s zeroes are just definite cases of the func-tion’s preimage finding task. We discuss and present different ways of modelling andinterface support of these activities in Calculus, Complex Analysis and Linear Algebrastudies.
Proposed solutions are illustrated by pilot version of the author’s program “VisuMat-ica”.
Keywords
Visualization, interface, function, model, activity, image, preimage.
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A029: Integrating Computers intoMathematics classes in a Unique
way – Classroom Examples
R. Hoffmann and R. KleinKibbutzim College of Education,
Tel Aviv,Israel
ronit [email protected] and ronit [email protected]
Lecture for the ACDCA Strand
Abstract
In our efforts to integrate computers into mathematics classes and expose the studentsto new teaching methods, we developed two technology based courses. These courses aretaught to mathematics B.Ed and M.Ed students in a teacher training college. Our aimis to provide the students with tools to solve various kinds of mathematical problemsassisted by computers and to help them increase their mathematical abilities.
The students are exposed to new and vital subjects which are ordinarily absent fromthe regular school programs (in Israel). They learn how to deal with problems in amodern and different way. The students get acquainted with the mathematical ideasand numerical methods embedded in the computer, calculator and graphic calculator orin other words, understand “the story behind the key”.
In our presentation we will briefly describe the content of the courses. We will startwith computing square and cubic roots using the intuitive “trial and error” methodfollowed by Heron’s method (100 a.d.) and generalize both to first and second order nu-merical methods. This will enable them to solve even equations which have no analyticsolving formulas (exponential, trigonometric or polynomial of a degree greater than 4).We use the Mathematix software (an Israeli CAS) with Excel spreadsheets or GeoGe-bra to obtain the graphs of the appropriate functions in order to define the number ofsolutions (if any) and to find them numerically. We will present additional examples(as time allows) such as xe−x − 0.25 = 0 and the third degree polynomial equationx3 +2x2 +10x− 20 = 0 which was solved by Fibbonacci (1225). The solution presentedis 1.36880810 and nobody knows how it was reached.
We hope that this modern model of teaching will be integrated into the curriculumand our students will be the agents who will incorporate it into schools. We believe thatour task as teacher educators is to raise the students’ curiosity as to how math servesup to date technology and by doing so raise the next hi- tech generation.
Keywords
Mathematix, Excel spreadsheets, GeoGebra, teacher training, Solving problems assistedby a computer, Numerical methods.
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A030: Categorizing CAS Use withinOne Reform-Oriented United States
Mathematics Textbook
Jon DavisDepartment of MathematicsWestern Michigan University
United States
Lecture for the ACDCA Strand
Abstract
Mathematics teachers in the United States frequently use textbooks to guide theirclassroom instruction (Grouws & Smith, 2000; Weiss, Banilower, McMahon, & Smith,2001). Recently, three US high school (age 14 - 18) mathematics textbook programshave been developed which incorporate computer algebra systems (CAS) to varyingdegrees. A framework adapted from Heid and Edwards (2001) was developed and usedto categorize the different roles of CAS as embedded within textbook activities designedfor students within one program’s advanced algebra textbook (Flanders et al., 2010).
Altogether, CAS played six distinct roles within this textbook. Of the 226 tasks thatwere connected to CAS use, 52% were isolated/non-formulaic, 4% were isolated/formulaic,7% were pattern generation, 2% were parameter manipulation, and 3% were method exe-cution, and 32% were a variety of different calculator specific roles such as a spreadsheet.CAS use was also categorized in terms of active or passive student engagement. An ex-ample of active engagement is where the student must use CAS to perform some task orsolve a problem while passive engagement is where the use of CAS is portrayed in thetextbook within an example that students are simply asked to read about. The analysesshowed that 37% of the tasks were passive and 63% of the tasks were coded as activestudent interaction with CAS. CAS instances were also coded in terms of whether thetechnology was a primary method to solve problems or a secondary method such as whenstudents are asked to use CAS to check their solutions to problems that they solved withpaper and pencil. A total of 68% of the tasks were primary uses of CAS and 32% weresecondary.
A framework adapted from Stylianides (2008) was used to examine how CAS wasto promote reasoning and proof. Out of 226 tasks where CAS was used, 30% or 68tasks were devoted to reasoning and proof. The majority of these 68 tasks (65%) askedstudents to notice a symbolic pattern that was generated on CAS. That is, these tasksasked students to use CAS in a certain way (e.g., manipulating the parameters of aquadratic equation and noting how the graphical representation changes) and state withcertainty the patterns that they detected. In fewer instances, (24%) students were asked
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to make conjectures or state with less certainty the patterns that they believed to existafter using CAS. CAS were used by students on only 4 occasions (6%) to test a conjectureto see if it always held and the technology was rarely used in developing proofs (6%).The implications of these results on students’ use of technology and its role in reasoningand proof will also be discussed.
Keywords
proof, reasoning, curriculum, computer algebra systems, reform.
References
• Flanders, J., Lassak, M., Sech, J., Eggerding, M., Karafiol, P. J., McMullin, L., etal. (2010). Advanced algebra. Chicago, IL: Wright Group/McGraw Hill.
• Grouws, D. A., & Smith, M. S. (2000). NAEP findings on the preparation andpractices of mathematics teachers. In E. A. Silver & P. A. Kennedy (Eds.), Re-sults from the seventh mathematics assessment of the National Assessment of Ed-ucational Progress (pp. 107-139). Reston, VA: National Council of Teachers ofMathematics.
• Heid, M. K., & Edwards, M. T. (2001). Computer algebra systems: Revolution orretrofit for today’s mathematics classrooms? Theory Into Practice, 40(2), 128-136.
• Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. Forthe Learning of Mathematics, 28(1), 9-16.
• Weiss, I. R., Banilower, E., McMahon, K., & Smith, P. S. (2001). Report of the2000 national survey of science and mathematics education. Chapel Hill, NC:Horizon Research.
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A031: Critical Issues of TechnologyUse in Undergraduate Mathematics
Greg OatesDepartment of Mathematics
University of AucklandNew Zealand
Lecture for the ACDCA Strand
Abstract
The effective integration of technology into the teaching and learning of mathematicsremains one of the critical challenges facing contemporary tertiary mathematics. Thispresentation reports on several key findings from a PhD study investigating the useof technology in some 72 different undergraduate mathematics courses, representing 31tertiary institutions in 8 different countries. Given that integrated technology is inter-preted in a considerable variety of ways in the literature, and that technology use inundergraduate mathematics also varies widely, the initial phase of the study sought toidentify the features of an Integrated Technology Mathematics Curriculum (ITMC), asa means of characterising and comparing technology use in and between courses. Thisphase of the study resulted in the development of a taxonomy of integrated technologythat identifies six characteristics, each with a number of associated elements, which maybe used to assess the degree of integration in a particular undergraduate mathematicscourse or department. The findings suggest that while the underlying complexity ofthe taxonomy limits a categorical definition of integrated technology, it does provide aneffective means for examining the issues confronting those wishing to implement andsustain integrated technology in undergraduate mathematics. An integrated, holisticapproach, which aims for curricular consistency across all the characteristics describedin the taxonomy, provides the basis for a more effective and sustainable ITMC. The pre-sentation then highlights two critical elements identified in the taxonomy, using evidencegathered from an observational study of technology implementation at The Universityof Auckland. The first of these revisits the issue of changes to the relative value ofcurriculum topics, when using computer algebra systems, as previously considered withrespect to secondary school mathematics (Artigue, 2002; Stacey, 2003). This study sug-gests that issues of curricular value as described by Artigue (2002) and Stacey (2003)are a critical factor in the successful implementation of integrated technology, and thata re-examination of the relative values of fundamental topics remains a significant chal-lenge for undergraduate mathematics in a rapidly evolving technological environment.The second concerns assessment issues, and suggests that aspects of assessment such ascurricular congruency, equity and the advantages and affordances provided by differenttechnologies require continued attention and constant vigilance, if integrated technologyis to be successfully implemented and sustained.
Keywords
Curriculum review, Undergraduate mathematics, Assessment, Policy.
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A032: On Assessment of Teaching AMathematical Topic Using Neural
Networks Models (with a casestudy)
H. M. MustafaEducational Technology Department-Faculty of Specified Education-Banha
University Egypt Currently With Computer Engineering Department,Faculty of Engineering, Albaha University
Kingdom of Saudi Arabia
mustafa [email protected]
A. Al-HamadiInstitute for Electronics, Signal Processing and Communications (IESK)
Otto-von-Guericke-university Magdeburg
M. H. KortamAlSafa Modern Private Schools
Safwa - Eastern ProvinceKingdom of Saudi Arabia
Mohammed [email protected]
Lecture for the ACDCA Strand
Abstract
This piece of research belongs to a rather challenging interdisciplinary approachadopting realistic and fairly assessment of educational processes based on Computer-Assisted Learning (CAL) module(s). This trend is confirmed by the strong connectionsbetween two of the most prosperous research areas of educational psychology, and cog-nitive sciences. Recently, an assessment approach considering output achievement aslearning parameter of a mathematical learning topic [1].
Conversely, comparative and fairly assessments for three different experimental edu-cational methodologies are presented. Herein, those assessment processes are performedexploiting realistic modeling of time response parameter (learning convergence time)rather than output achievement parameter presented at [1]. In more details, two care-fully (CAL) modules are designed for development of a multimedia tutorial packages.Both modules designed for teaching ”How to solve long division problem” by sequentialprocesses as: Divide, Multiply, Subtract, Bring Down, and repeat (if necessary) [2].
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It is worthy to note that designed modules are concerned with effective utilizationof visual and/or auditory tutorial multimedia materials. They are provided for teach-ing adopted mathematical topic (Long Division Problem) to children at the fifth gradeclass level (in a primary school), with average age about 11 years old. Obtained assess-ing results show that optimal teaching methodology reached if both materials (visualand auditory) have been applied simultaneously to reinforce the retention of long di-vision mathematical learned topic. This interesting conclusive remark agrees likewisethe cognitive multimedia theory revealing that simultaneous application -in practicaleducational field- of visual and auditory tutorials is highly recommended [3]. Moreover,obtained practical case study results are supported by recently published simulation re-sults of learning time response by Artificial Neural Networks modeling [4]. Furthermore,suggested tutorial CAL modules motivated by Multi-Sensory associative memories andclassical conditioning theories. In below, three given figures to illustrate output printscreen samples of fairly designed assessment computer program.
Keywords
Associative memories, Computer Assisted Learning, Multimedia Learning Theory, Neu-ral Networks Modeling.
References
[1] H.M. Hassan, Ayoub Al-Hamadi, “On Teaching Quality Improvement of A Math-ematical Topic Using Artificial Neural Networks Modeling (With A Case Study)”,published at 10th (Anniversary!) International Conference Models in DevelopingMathematics Education held in Dresden, Saxony, Germany on September 11-17,2009.
[2] Interactive practice with long division with no decimals: Daisy Maths - LongDivision http://Argyll.epsb.ca/jreed/extras/longdiv/
[3] R. Lindstrom, The Business Week Guide to Multimedia Presentations: CreateDynamic Presentations That Inspire, New York: McGraw-Hill, 1994.
[4] H.M. Hassan, “On Simulation of E-learning Convergence Time Using ArtificialNeural Networks” published at the 7th International Conference on Education andInformation Systems, Technologies and Applications (EISTA) held in Orlando,USA, on July 10-13, 2009.
Fig.1. Basic print screen sample for initial mathematical Long Division process.
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Fig.2. A print screen for fairly assessment Fig.3. A print screen for fairly assessmentprocesses results with no mistake. processes results with two mistakes.
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A033: On the Visualization of theCalculus concepts
D. Takaci and J. MaksicDepartment of Mathematics and InformaticsFaculty of Sciences, University of Novi Sad
Novi Sad,Serbia
Lecture for the ACDCA Strand
Abstract
We analyzed the role of computers in teaching and learning calculus, in particular atdetermining derivatives of functions. In this talk we will present and analyze the studentswork on the visualization of the definition of the first derivative of function, by usingmathematical packages Scientific WorkPlace, and GeoGebra, as a contribution to activelearning. In fact the visualizations of a limit process will be presented. The students,working on their tasks, pointed out quite a few good and bad effects of computer-aidedlearning process caused by using the mentioned packages.
Keywords
CAS, visualization, teaching-learning process, calculus, computers, limit process, func-tions.
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A035: The Impact of Computer Useon the Teaching of Geometry in
Grade 8
Sirje PihlapInstitute of EducationUniversity of Tartu
Estonia
Lecture for the ACDCA Strand
Abstract
The research on the efficiency of computer-assisted learning has provided contradic-tory results. The objective of the current study is to clarify the impact of using computeron learning results and the motivation of the students who learn Geometry in the 8th
grade. The survey was carried out among the students of the 8th grades of four schoolsin Estonia in school year 2004/05. In the present paper, the students’ performance inthe experimental (N=116) and control classes (N=176) is compared. In addition totraditional methods of teaching, computers were used in teaching Geometry in the ex-perimental classes. In the control classes the computers were not used. According to theresearch carried out in the area, the application of computers has not generated betterresults. As for the students’ attitude to the learning process, it has been noted that theapplication of computers has definitely improved the students’ attitude to Mathematics.The students of the experimental classes were posed a question if using computers haschanged their attitude to Mathematics. Thirty two per cent of the students were of theopinion that the use of computers has improved their attitude to Mathematics. The stu-dents find that the use of computers makes studying more interesting, easier and moreunderstandable.
Keywords
Computer-assisted learning, geometry, dynamic geometry system.
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A036: Folding and unfolding conesand cylinders with Cabri 3D: how to
do it and how to use it in theclassroom
Jean-Jacques DahanIREM of Toulouse
Universit Paul Sabatier ToulouseFrance
Lecture for the ACDCA Strand
Abstract
As Cabri 3D does not contain tools allowing the user to open cones or cylinders (itcan fold and unfold polyhedra), I have solved this problem and will first show brieflythe stages of my research before presenting the modelling I have realised. I have treatedthis problem to respond to the demand of a middle school teacher who wanted to use itto help her students to understand some misconceptions about the unfolding of a cone.What is surprising is the consequences of this work:
1. At a research level: I have used the files I have created as tools of investigationand have discovered some unknown theorems about cones (conjectured and proventhem).
2. At a teaching level: these files were used by the middle school teacher to treat themisconceptions pointed before but more than that, these files allowed me to createsome experiments for the students in paper and pencil environment to conjecturesome 3D properties of the intermediate stages of the unfolding of a cone.
These two points will be presented and discussed.
Keywords
Cones. Cylinders. Folding. Unfolding. Misconceptions.
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A037: CAS in the Swedish nationaltests in upper secondary level
G. Wastle and T. HellstromDepartment of Educational Applied Science
University of UmeaSweden
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
The introduction of calculators with CAS as an allowed tool in Swedish national testsin mathematics has resulted in several challenges in the construction and developmentof these tests. In our presentation we will share our experiences dealing with thesechallenges and present some solutions together with an analysis of their consequences.Furthermore we will present some conclusions from an ongoing developmental projectregarding a future introduction of computers as tools in students work with Swedishnational tests in mathematics.
CAS has been allowed in the national course tests in mathematics (NCT) for theSwedish upper secondary school since 2007. Its compulsory for the school to arrangeNCTs and according to existing regulations, teachers should use these course tests as anaid so that the bases for marking are as uniform as possible throughout Sweden.
One of the requirements for the NCT, which comprise a technology free part anda technology active part, is that the problems in the test must be done in a way thatthose who have a CAS-calculator are not allowed to have any advantage compared whitthose who have a graphic-calculator. Based on studies at our own department, and thework of others (see e.g. Flynn & MacRae, 2001), we know a great deal about whichtasks that are CAS-neutral, CAS-trivial, etc. This knowledge has caused us to movetasks that are directly solved by CAS (e.g. solve sin(2x) = 0.78 ) from the part of thetest where calculators are allowed to the ”calculator-free” part. However, this causesproblems because students often need to do some calculations in order to solve theproblem. Other challenges with respect to CAS-in the development of Swedish nationaltests in mathematics are how to signal to students that they are allowed to use thecalculator and how students can be expected to show their work when using advancedcalculators in solving problems. In the development of future national tests with CASwe must consider that lap-top computers will be common and students might not havea hand-held calculator. In addition, web-based CAS has developed and we will mostlikely see more and more of teaching based on these tools. In such a scenario it seemsunreasonable to expect students to have an advanced calculator for the sole purpose ofdemands on national tests. At the same time, web-based resources require web-accesswhich in turn opens for unwanted cooperation and cheating. Future tests do most likely
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need a part where no ICT is allowed. Another part could focus problem-solving allowingstudent to use any technology available to perform the necessary procedures for arrivingat an answer. These challenges have no immediate solution, but some suggestions willbe discussed.
Keywords
National tests, CAS, computers, fairness.
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A038: 3D-Dynamical Geometry inBuilding Construction
Raul Manuel FalconDepartment of Applied Mathematics I
University of SevilleSpain
Workshop for the ACDCA Strand
Abstract
In Architecture and Technical Architecture Degrees, students use CAD tools (Com-puter Aided Design) which are not capable, in general, of representing graphically curvesor surfaces starting from its corresponding equations. To get it, users have to definespecific macros or they have to create a table of points in order to convert a set of nodesinto polylines.
CAS tools used in Math classes allow this graphical representation of curves andsurfaces starting from their parametric equations. However, they lack the dynamicaldevelopment given by CAD tools, which plays a main role in the mentioned degrees.In this sense, the complementation of the algebraic and geometric tools included in thesoftware of dynamic geometry, Geogebra, is an attractive alternative to design and model,from a mathematical point of view, curves and rigid objects in the space. The use ofsliders related to the Euler’s angles and the possibility of generating tools which project3D into 2D, makes easier this kind of modeling.
In the current workshop, we will show how to construct 3D-models of several ar-chitectonical constructions which have been made in the context of the subject calledMathematics for Building Construction II , corresponding to the Building ConstructionEngineering of the University of Seville, which has been implemented this academic year2009-10.
Keywords
3D-modeling. Building Construction modeling. Perspectives and projections. Dynamicgeometry. Geogebra.
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A039: Those Magic Moments whenyou realise you could not do this
with chalk!
Douglas ButlerICT Training Centre
OundleUnited Kingdom
Workshop for the ACDCA Strand
Abstract
We have all be watching the evolution of teaching mathematics with ICT support fora number of years. Every now and again I think back to how it was in the chalk days,and whether we were effective or not.
My feeling is that we were probably very effective with the more able students, butthat ICT has enabled us to be considerably more effective and motivational with the lessable.
This workshop will give an opportunity to look at a number of lesson plans fromthis point of view. At the junior level: the straight line, parabolas, transformations anddata handling. At the advanced mathematics (calculus: differentiation and integration,differential equations, 2D and 3D vectors, probability).
We will try to concentrate on those moments where the ICT contribution can beconsidered to have materially improved the chances of the students visualising what isgoing on. There will be time to discuss each one.
The workshop will use the software Autograph, but most of the examples couldequally well be run in other software environments.
Keywords
Dynamic software, Coordinate Geometry, Statistics and probability, Autograph, Moti-vation.
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A040: Overcoming difficulties inunderstanding of the nonlinear
programming concepts
Alla StolyarevskaThe Eastern-Ukrainian Branch
of the International Solomon University,KharkovUkraine
Lecture for the ACDCA Strand
Abstract
The technology of education means something created to teach students by easiest,fastest, and best way. This paper presents several examples of using computer technologyto clarify the nonlinear programming concepts.
Nonlinear optimization problems occur in mathematical modeling of real processes.Analysis of students’ tries to solve these problems show the lack of understanding suchconcepts as level curve and level surface.
We had to step back and start with more simple problems. For example, to divide agiven number A into two summands so that their product would be maximal or to findthe radius and the height of a closed cylindrical surface of a given volume V having theminimal total surface, and so on.
At the beginning the students use methods, which are common in solving simpleproblems of finding extremum, afterwards they pass to method of Lagrange multipliers,and the gradient methods. To visualize the solution of nonlinear optimization problemsthe Maple functions contourplot, fieldplot, gradplot are proposed.
The active using of the graphic representation helps to overcome the difficulties inunderstanding of the basic concepts of nonlinear programming. CAS technologies givemore long knowledge retention compared to traditional methods of teaching.
Keywords
I Nonlinear programming, Maple, student’s motivation.
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A042: A Computational Measure ofHeterogeneity on Mathematical
Skills
E. M. Fedriani and R. MoyanoDepartment of Economics, Quantitative Methods and Economic History
University of Pablo de OlavideSevilleSpain
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
The educative fact is inherently multivariate, since there are lots of factors affectingeach student and their performances. Due to this, both measuring of skills and assessingstudents are always complex processes. This is a well-known problem, and a number ofsolutions have been proposed by different specialists. But, in most of cases, it is clearthat the different progress levels of students in the Mathematics classroom make alsodifficult the teaching work. We think that a measure of the heterogeneity of the differentstudent groups could be interesting in order to avoid such difficulties, or to prepare somestrategies to deal with this kind of problems.
The major aim of this work is to develop some new tools, complementary to thestatistical ones that commonly are used for these purposes, to study situations related toeducation (mainly to the detection of levels on mathematical education) in which severalvariables are involved. These tools are thought to simplify and better understand theseeducational problems and, through this comprehension, to improve our teaching work.
Several authors in our research group have carried out some mathematical theoretictools, to deal with multidimensional phenomena, and applied them in business models.These tools are based on multidigraphs. In this work, we implement all these toolsby using symbolic computational software and apply them to study a specific situationrelated to the mathematical education.
Keywords
Assessment, Graph Theory, Homogeneity, Multivariate.
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A043: Using Maxima in theMathematics Classroom
E. M. Fedriani and R. MoyanoDepartment of Economics, Quantitative Methods and Economic History
University of Pablo de OlavideSevilleSpain
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
Coming from the MACSYMA system and adapted to the Common Lisp standard,the symbolic calculus program called “Maxima” can be considered as a possible tool inthe Mathematics classroom. Several circumstances may lead us to this consideration.
Firstly, this software is a free program included in the Guadalinex distribution, sup-ported by the Junta de Andaluca, which makes it easily available at almost every schoolin Andalusia. Secondly, Maxima is able to manipulate and simplify algebraic expres-sions and it is also a programming language which allow us to create our own specificapplications. Besides, it has a lot of packages as a complement, making it suitable forworking in a lot of mathematical branches. And finally, this tool has been developedunder several operative systems. Particularly, it can be implemented under Windows,Linux and Macintosh, what can contribute to spread its use.
The main aim of this work is to show some of the possibilities of Maxima and itsgraphical interface as a tool for teaching Mathematics at University as well as at Sec-ondary school levels. Both Mathematics courses in Business degrees and A-levels can beprovided with a resource that will ease the learning of this subject to students.
As a conclusion, we also present a report of the main strengths and weaknesses ofthis software when used into the Mathematics classroom.
Keywords
Symbolic calculus, Mathematics teaching, algebraic expressions, management & businessdegrees, undergraduate Mathematics.
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A044: Applications of MultimediaTechnology to study the ordinal
thinking evolution of scholars from 3to 7 years old
P. Hernandez and J. L. GonzalezDepartamento de Didctica de la Matemtica
de las Ciencias Sociales y de las Ciencias ExperimentalesUniversity of Mlaga
Spain
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
One of the most interesting research field in Mathematics Education is the formationand evolution of numerical and arithmetical thought from comparison of quantities tothe concept of natural number and arithmetic operations with natural numbers. Nev-ertheless, many of the conducted studies have focused on the cardinal aspect of naturalnumbers and have devoted less attention to their ordinal and inductive side. Additionallythis sort of studies shows some methodological limitations produced by the age of infantswhich creates several difficulties to reach valid, objective and reliable information.
The main purpose of this communication is to present the master lines and first con-clusions of a research study on the analysis of ordinal thought evolution in prenumericaland preinductive levels among three to seven years old students. Our methodologicalscheme has been based on multimedia technology, automatic and objective record of theinformation and an outstanding of interaction between students and researcher.
Therefore a triple finality is fulfilled:
1. To find out the accuracy, validity and possibilities of multimedia technology as amean in Mathematics Educations research.
2. To check its usefulness for infants.
3. To seek the features and evolution of ordinal thought in a virtual, ludic and inter-active environment which have been adapted to the psycho-affective characteristicsof the infants.
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Our method is based on a set of multimedia tasks that is organised according toan evolutive model of ordinal competences and generated with the tool “MacromediaDirector”. These tasks have been conceived according to our theory on the production ofmultimedia items. In this theory the epistemic, ontological and onto-epistemic functionshave been redefined and some constructive principles about the multimedia learningfrom Mayer and other authors have been incorporated. Our first conclusions provethe usefulness of this methodology and show the existence of solid evolutive patternsin different stages of the proposed model. At the same time they are completing theprevious research results.
Keywords
Mathematics Education, Multimedia Technology, cognitive diagnosis and assessment, nu-merical thinking, development of mathematical concepts and skills, evaluation, prekinder-garten, infants, primary, order, natural numbers.
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A045: The Past and the Future ofComputer Algebra in Mathematics
Education – A personal (&nostalgic) perspective
Bernhard KutzlerACDCA (Austrian Center for Didactics of Computer Algebra)
Austria
Lecture for the ACDCA Strand
Abstract
In 1983 Klaus Aspetsberger and Gerhard Funk conducted the world’s first experi-ment with CAS in an Austrian high school. This was the beginning of a fascinatingdevelopment that changed so much for so many people. I review the past 27 years ofusing CAS as a teaching aid and sketch a vision for the future.
(Comment: This lecture is my personal farewell to the community after working inthis area for very many years. I will give a personal perspective of this time and sharemy ideas of what the future could look like.)
Keywords
Computer algebra in education, history.
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A046: Technology and theYin&Yang of Mathematics
Education
Bernhard KutzlerACDCA (Austrian Center for Didactics of Computer Algebra)
Austria
Lecture for the ACDCA Strand
Abstract
We develop a model comprising six teaching and learning archetypes and use thismodel to look at the various roles that technology, in particular computer algebra systems(CAS), can play for each.
(Comment: This is a repetition of my keynote lecture from the TIME conferencein South Africa. At the same time this is a summary of my personal research workin this area of the previous 25 years. So, in a way, this complements my farewell tothe community as is done with my other lecture entitled “The Past and the Future ofComputer Algebra in Mathematics Education”.)
Keywords
CAS, teaching and learning models.
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A047: An example of learning basedon competences: Use of Maxima in
Linear Algebra for Engineers
A. DıazDepartamento de Matemtica Aplicada.
Universidad Nacional de Educacin a DistanciaSpain
A. GarcıaDepartamento de Matemtica Aplicada. E.U. Informtica
Universidad Politcnica de MadridSpain
A. de la VillaDepartamento de Matemtica Aplicada y Computacin. ETSI (ICAI).
Universidad Pontificia Comillas.and
Departamento de Matemtica Aplicada. E.U.I.T.IUniversidad Politcnica de Madrid
Spain
Lecture for the ACDCA Strand
Abstract
The adaptation to the European Area of Higher Education (EAHE) implies a newteaching and learning model, with active methodologies and learning based on compe-tences. Therefore, it will be necessary to adjust all methodological resources (and inparticular the use of a CAS in the new process) to this new scenario.
We will emphasize the need to provide, in a cognitive and instrumental way, genericskills such as:
• Self Learning
• Planning and organization
• Decision-making and problem solving
• Critical Thinking
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• Teamwork
We will restrict our paper to the topic Linear Algebra for Engineers. Several exampleswill be included with trials, proposals and activities, trying to evaluate the main LinearAlgebra competences. The experience has been carried out in Open University withengineering students. The CAS used has been Maxima. We will report its contributionto the acquisition of skills.
We will also justify the choice of Maxima as support, considering the characteristicsof the software, the easiness of use and the University where it’s going to be implemented.
Keywords
Learning based on competences, Maxima, Linear Algebra for Engineering, MathematicalSoftware.
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A048: Learning Math in the contextof European Space for Higher
Education
Salvador MerinoDepartment of Applied Mathematics
University of MlagaSpain
Lecture for the ACDCA Strand
Abstract
In this study we specify the main aspects of the implementation of the EuropeanHigher Education Area to the teaching of Mathematics.
In a new technological and scientific framework, the interest of teachers in the knowl-edge and application of current tools encourages us to reflect on which are the mainchallenges we face. Teachers training in the field of “Distributed Education” and theestablishment of strategic plans for teaching math, form the foundation for the successfulimplementation of new educational paradigms.
We will analyze the methodology for the implementation of subjects and how toachieve the proposed objectives, starting from basic descriptors established by law forall engineering degrees.
Finally, we will describe our analysis about the educational tools needed to makeMathematics more and better accessible for students, when and where these tools shouldbe applied, and which are the departmental resources and teacher training required toachieve the most appropriate use of educational tools.
Keywords
Distributed Learning, Mathematical Tools, Bologna Process, European Space, StrategicPlanning.
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A049: Using intelligent adaptiveassessment models for teaching
mathematics
J. Galvez, E. Guzman and R. ConejoDepartment of Languages and Computer Science,
University of MlagaSpain
[email protected], [email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
The utilization of CAS nowadays has been applied in a wide range of domains, includ-ing maths, to assess theoretical student knowledge or, also called, declarative knowledge.Despite this successfully utilization of CAS, assessing students using reliable and well-founded methods has still room for improvement when the knowledge being assessed isprocedural. This kind of knowledge occurs when the student must solve a problem witha serial of steps involved on it to achieve a solution.
Problem solving environments are normally used for learning by practising knowl-edge but not for assessment. This gets even worse in complex domains where the over-specificity problem in the knowledge representation makes difficult not only to assessingstudents, but even the creation of a suitable model capable to be used in a learningsystem.
In this work we will present an adaptive and intelligent model that pursues two maingoals: first, to assess students by means of well-founded assessment methodologies on do-mains where an evaluation is difficult to be performed using traditional CAS; and second,to allow students learning in such domains by applying a constructivist methodology (i.e.using the assessment for learning approach). The theoretical foundations that make ourapproach to be a valid for both, assessing and learning purposes, lie on combining ItemResponse Theory, a well-founded assessment theory, with Constraint-Based Modelling, amethodology developed to overcome the over-specificity problem when building learningenvironments and modelling student knowledge. We will discuss how to build concretelearning environments in mathematical domains that use this generic methodology andhow the process of assessing students is done.
Between the currently existing systems that incorporate this methodology, it will bepresented an example in the linear optimization topic, which has been applied with realstudents from the University of Mlaga.
Keywords
Assessment, Web Intelligent Learning environments, CAS, learning by doing.
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A050: How to assess the quality ofinteractive dynamic geometry
resources? The Intergeo experience
C. Laborde S. Soury-LavergneDIAM-LIG EducTice
University Joseph Fourier INRPGrenoble LyonFrance France
[email protected] [email protected]
Workshop for the ACDCA Strand
Abstract
There is a great heterogeneity of available digital resources for teaching mathematics.As a result, it may be difficult for maths teachers to resort to explicit and objective crite-ria for analyzing a digital resource and estimating the value of using it in the classroom.
One of the aims of the Intergeo project (http://i2geo.net) is to gather all digitalresources based on dynamic geometry in European languages and to make them availableto teachers (see Fioravanti and Recio workshop). As more than 2000 resources areavailable, it was important to also make available comments and quality evaluation inorder to help teachers in finding resources appropriate for their needs and aims. Anevaluation tool was thus developed within the Intergeo project under the form of aquestionnaire dealing with several aspects of a resource.
The workshop will discuss the theoretical foundations and the choice of the topicsaddressed by the questionnaire: metadata, technical aspect, mathematical dimension ofthe content, instrumental dimension of the content, potentialities of Dynamic Geometry,didactical implementation, pedagogical implementation, integration of the resource intoa teaching sequence, ergonomics.
After an introduction to the questionnaire, the participants will experience themselvescompleting the questionnaire for one resource. Then in a collective phase, the relevanceof the criteria will be discussed as well as the questions faced by the participants whencompleting the questionnaire. The contribution of the questionnaire to teacher educationwill also be addressed.
Keywords
Dynamic geometry, interactive geometry, digital resources, quality.
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A051: Free Dynamic Software forExploring Multi-Dimensional
Relations
Sharon WhittonDepartment of Curriculum and Teaching/Mathematics Education
Hofstra UniversityHempstead, NY 11549-1190
U.S.A.
Workshop for the ACDCA Strand
Abstract
This workshop introduces participants to Winplot, a free computer-graphing util-ity, available to everyone on the web. Winplot is simple to use and far surpasses thefunctionality of many other graphing utilities. This workshop will use Winplot for ex-ploring mathematical concepts spanning pre-algebra to calculus III. Illustrations willcome from algebra, pre-calculus, trigonometry, systems of equations, calculus, surfacesof revolution, and 2- and 3-dimensional graphs. Winplot can draw, shade, and animatecurves and surfaces. The slider feature of Winplot will be applied for guiding studentsto discover the impacts of parameters on a graph’s shape, extreme values, and roots.Participants will engage in a series of activities associated with each of the above topicsand gain competence in using Winplot in mathematics education. It is a powerful anddynamic piece of instructional software that also produces gorgeous hard copies of 2- and3-dimensional graphs. Winplot is a stand alone computer graphing utility for Windows(95/98/ME/2K/XP/Vista). In addition to English, it comes in the following languages:Croatian, Dutch, French, German, Hungarian, Italian, Korean, Lithuanian, Portuguese,Russian, Slovak, and Spanish.
Keywords
Free, computer-graphing utility, 2- and 3-dimensional graphs, algebra to calculus III.
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A052: Taking advantage ofSherman’s march
P. Guerrero M. TorilDepartment of Applied Mathematics Department of Communications Engineering
University of Malaga University of MalagaSpain Spain
[email protected] [email protected]
Lecture for the ACDCA Strand
Abstract
During the simulation of a mobile telecommunications system, a sequence of systemsof linear equations must be solved. In this sequence, the coefficient matrix of the (k+1)thsystem is of order one greater than that of the kth, and the former is constructed byenlarging the latter with a new column and a new row. All matrices involved are strictlydiagonally dominant, but the condition number suffers a heavy worsening as k increases.In this lecture we show that taking advantage of this diagonal dominance property iscrucial to be able to obtain as much as a 30% improvement on average in the CPU timeto complete the whole process in Matlab v7.5.
Keywords
LU decomposition, updating, leading principal submatrix, Sherman’s march, strictlydiagonally dominant.
Observations
This work has been supported by the Spanish Ministry of Science and Innovation(grant TEC2009-13413). It is available at
http://www.matap.uma.es/investigacion/tr.html
as Tech. Report MA-10/01, Dept. Applied Mathematics, Univ. Mlaga, 29th January2010.
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A053: GeoGebra Workshop for theInitial Teacher Training in Primary
Education
Natalia RuizFacultad de Formacin de Profesorado y Educacin
Universidad Autnoma de Madrid (UAM)Spain
Lecture for the ACDCA Strand
Abstract
This paper is part of a wider research we are conducting at the Faculty of Educa-tion (UAM) to discuss how to involve the dynamic geometry software GeoGebra in thedevelopment of geometric and pedagogic competences in the degree Teacher in PrimaryEducation.
We have designed a GeoGebra Workshop for the course “Mathematics and its Peda-gogy II” (2nd year in the degree) to work the main geometric competences that primarypre-service teachers need to develop.
We will show examples of constructions made by the students in which we can analyzethe development of skills such as reproduction of geometric figures, verification andgeneralization of properties, geometric problem solving, etc.
In addition, we try to know the impact of using dynamic geometry software in thestudents’ attitudes and motivation.
Keywords
GeoGebra, Geometrical Competences, Pre-services Teachers, Dynamic Geometry.
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A055: Problem solving usingGeogebra
Kaja MaricicMechanical Engineering High School
Novi SadSerbia
Lecture for the ACDCA Strand
Abstract
Problem solving is an important component of teaching mathematics. It is a way topresent mathematics and it is also a skill which enhances logical reasoning. In this paperthe two different approaches toward the problem solving will be presented. The first oneis classical and the other one is by using Geogebra. The steps of both parallel solutions ofseveral math problems of different levels will be analyzed in order to illuminate specificbenefits from each. The group of students from the third grade who are familiar withGeogebra took part in number of exercises and their experience in this specific problemsolving will also be presented.
Problems of different levels and several mathematics topics (such as geometry, ana-lytic geometry or algebra) have been considered in this work. The students (seventeenyears old) chosen are those who are a bit more interested in mathematics then the oth-ers. They are studying in the third grade (out of four) of mechanical engineering high(secondary) school in Novi Sad.
Keywords
Math teaching, logical reasoning, problem solving, Geogebra.
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A056: Dynamic Applets forDifferential Equations and
Dynamical Systems
Robert DeckerDepartment of Mathematics
University of HartfordUSA
Workshop for the ACDCA Strand
Abstract
Dynamic/interactive graphing applets can be used to supplement standard computeralgebra systems such as Maple, Mathematica, Derive, or TI calculators, in courses suchas Calculus, Differential Equations, and Dynamical Systems. The addition of this typeof software can lead to discovery learning, with students developing their own con-jectures, and possibly proofs (as has happened when students explore geometry withinteractive software). Also, topics which have traditionally been considered too ad-vanced for standard undergraduate courses, such as the Poincare map in differentialequations/dynamical systems, can be introduced at an intuitive level.
The presenter has both developed and used such applets in the teaching of vari-ous mathematics courses. Furthermore, in conjunction with several students, tools foreasily creating new applets have been developed, so that mathematics educators cancustomize applets to their courses without having to become java programming ex-perts (see the website created by these students at www.dynamicgrapher.com). In thisworkshop, participants will be lead through some guided investigations in differentialequations/dynamical systems (described below) which can be used with students. Also,participants will be shown how to develop their own applets.
Possible investigations (depending on participant interest):
1. Interesting relationships between various numerical methods for solving the logisticdifferential equation y′ = ay(1− y) are discovered. Euler’s method, second-orderRunge-Kutta (modified Euler) and fourth-order Runge-Kutta are compared as aparameter is smoothly changed in the differential equation. Computer algebrais used to confirm conjectures. This investigation was done by a student of thepresenter and given as a talk by the student at professional meetings.
2. The equation for a damped pendulum y′′ + cy′ + sin(y) is investigated in the caseof a large damping constant c. Varying the initial conditions interactively revealsa relationship between the solution curves in the phase plane and the sin(y) termin the differential equation. This investigation can then be extended to generalequations of the form y′′ + cy′ + f(y).
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3. A form of the logistic differential equation with constant harvesting, given by y′ =ay(1 − y) − d, is investigated both interactively with an applet (the parameterd and the initial conditions are varied smoothly) and using computer algebra, inorder to understand the bifurcation structure of the system as d is varied. In anextension, periodic harvesting is introduced via the model y′ = ay(1− y)− d(1 +cos(2πt)). In order to locate periodic solutions, rather than constant ones as in theprevious model, the Poincare map is introduced via an interactive applet, which isthen used to compare the bifurcation structure of the two models.
Keywords
Interactive applets, computer algebra, differential equations, dynamical systems.
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A057: Lessons learned usingPublishers’ Web-based software to
assess student workJune Decker
Department of MathematicsThree Rivers Community College
Norwich, CT.USA
Lecture for the ACDCA Strand
Abstract
The presenter will discuss the use of publishers’ web-based text-specific softwareMyMathLab and WebAssign in remedial algebra, statistics, precalculus and calculus.Course software delivers and grades student homework and quizzes and may containan electronic copy of the text, video lectures, applets and animations. Some studentsembrace the software, others insist on using a traditional textbook. Among the benefitsto the student, the software provides instant feedback and just-in-time homework assis-tance. Student accountability and accuracy improves. Student engagement increases.The remedial math success rate using MyMathLab increased from 41% to 57%. Somestudents embrace the computer interface while others resist it, claiming it takes timeaway from doing mathematics, and grades answers inaccurately.
The professor can see from the software what problems students did and did not docorrectly, and can then tailor the day’s lesson to the students’ needs. The software facil-itates communication between the professor and the student about the math problems,thus increasing student learning and motivation. The software is fairly easy to use, so ittakes only a few hours to compile the teacher-chosen homework for the semester. Thesoftware provides options to the professor as to how to use classroom time. Quizzes maybe taken outside of class. Several versions of tests and homework are available usingrandomly generated numbers in the problems which enables a professor to institute amastery learning approach, requiring students to repeat the lesson until he achieves aminimum grade. Diagnostic tests can be used to determing which content the studentmust learn, so that a student may progress more quickly through remedial courses bynot having to repeat what he already knows.
Use of web-based, text-specific software for homework and other assessments providesboth the instructor and the student with immediate information about what the studentknows and does not know. The professor can adjust his instruction to student needs,and the student knows what he needs to study.
Keywords
Course management software, online homework systems, computer assisted instruction(CAI).
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A058: Interactive TeachingMaterials on the Web – Their
Diversities and Variety
Vlasta KokolUniversity of Ljubljana
Slovenia
Lecture for the ACDCA Strand
Abstract
Today a lot of materials and tools are offered on the web for supporting the teachingand learning of mathematics. Most of them come with an attractive name “interactiveteaching materials”.
What makes a teaching material an interactive teaching material? Is it enough that“it is available from the internet”?
What are the characteristic and basic features of an “interactive teaching material”?The TIME community is an appropriate group to discuss design and quality as well as
give suggestions for the quality standards of and/or criteria for CAS and other interactiveteaching materials.
This lecture–workshop is meant to be a discussion group for exchanging ideas aboutthe currently available interactive teaching materials.
Keywords
Web tools for the teaching and learning of mathematics. Interactive teaching materials.
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A059: On the Visualization of theFunction
R. Vukobratovic and D. TakaciDepartment of Mathematics and InformaticsFaculty of Sciences, University of Novi Sad
Novi Sad,Serbia
[email protected] and [email protected]
Lecture for the ACDCA Strand
Abstract
We present the introduction of the notions of function, at the beginning of the fourthgrade of Grammar school by using program package GeoGebra.
The emphasis of the work is on testing three different classes in fourth grade ofsecondary school. The questionnaire, considered in paper of David Tall, is given to ourstudents and the obtain results are analyzed.
In order to check if students had a previous similar knowledge, the first step was toinvestigate the student’s knowledge of definition and notions of function at the end ofNovi Sad Grammar school.
Two of the three groups were considered as experimental groups. In these two exper-imental groups, the topic was taught by using program package GeoGebra. The thirdgroup was considered as a control group and this concept was introduced without usingthe computer.
After analyzing the obtained results, we will describe the different outcomes thathave been achieved concerning students level of knowledge. We will show that stu-dents from experimental groups, who used program package GeoGebra, afforded betterachievements in visualization of Function than the students from the control group.
Keywords
CAS, visualization, teaching-learning process, calculus, computers, limit process, func-tions.
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A060: Using Mathematics Journalsto Enrich the Methods Course
Experiences of ProspectiveMathematics Teachers
Reda AbuMathematics Education
Sultan Qaboos UniversityOman
Lecture for the ACDCA Strand
Abstract
This paper illustrates ways to employ Mathematics Teacher Journals to improvethe quality of methods course experiences for prospective mathematics teachers. Basedupon research conducted in an undergraduate teacher preparation program, this studydescribes how the author used Mathematics Teacher Journals to mentor prospectiveteachers in new ways. The study describes the author’s experiences through this educa-tional activity, but did so in ways that highlighted strategies for change to the methodscourse. Through the Mathematics Teacher Journals reading and analyzing activities theauthor identified specific ways that the prospective teacher were impacting their teach-ing practices, a result that enabled them to better performances in the methods coursereflect on their teaching.
The main topics of Methods of teaching mathematics course were: Problem solv-ing and posing, Teaching Fractal Geometry, Using dynamic geometry software (Cabri3D, Geometric Sketchpad) in teaching, Using Graphing Calculator (Casio cfx9850+) inteaching Functions, and teaching mathematical proof. Selected articles were chosen fromMathematics Teacher Journal focused on these topics.
The activities of reading and analyzing of Journal articles were dividing in threedimensions: Articles related to course content, articles in new ideas in teaching math-ematics; and articles in mathematics problem solving. Student-teachers were asked todevelop new ideas based on the articles context for their teaching practices.
By providing a detailed account of the feedback process that led to this result, this pa-per illustrates how mathematics student-teacher can use Mathematics Teacher Journalsactivities to enrich the quality of their methods courses.
Keywords
Mathematics education, mathematics teachers, Technology content in math teachingmethods course.
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A061: Teaching and LearningGeometric Transformations in the
Context of a Dynamic Environment
H. Bahadir YanikDepartment of Elementary Education
Program of Mathematics Teacher EducationAnadolu University
EskisehirTurkey
Lecture for the ACDCA Strand
Abstract
Dynamic Geometry Environments provide learners opportunities to construct geo-metrical objects and explore relationships between them. Within these environments,one can create and explore conjectures and develop explanations.
This presentation focuses on the implications of using Dynamic Geometry Technologyfor teaching and learning of geometric transformations which includes translations, re-flections and rotations. Specifically, the presentation reports on some data from a studywhich focused on the effects of a dynamic geometry software on prospective elementaryteachers’ understanding of geometric transformations. Some examples of teacher candi-dates’ initial understandings of geometric transformations will be provided and changesin their preconceptions in the context of technological environment will be discussed.
Keywords
Dynamic geometry environment, teaching and learning geometric transformations, teachereducation.
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TI-Nspire & Derive Strand
D001: C++ as a programminglanguage for CAS
Lluıs ParcerisaProgrammer for Casio
Flamagas Didactical DepartmentSpain
Lecture for the TI-Nspire & Derive Strand
Abstract
Calculator add-in applications, although effectively fulfill their purpose, usually presentsome drawbacks that could be overcome by a flexible and commonly known program-ming language like C++. With an appropriate software development kit to program andtransfer the application to a calculator-embedded C++ interpreter, the customizationand its usability will be boosted. Hence, both the user and the programmer experienceare improved.
From the programmer side, the re-usage of third party code or modification ofcommunity-provided source code is a clear advantage, in consonance with the 2.0 philos-ophy. Moreover, due to the code organization in classes and objects, the comprehensionin a lower level is easier, as it is the learning curve at initial programming stages. Al-though the application performance can be slightly decreased, the overall process is worthenough and the drawback is imperceptible in the adequate hardware.
From the user point of view, the advantages are also notorious. First of all, thegraphical user interface (GUI) provides a much intuitive and easy-to-use application ex-tensions. The inclusion of web inspired GUI elements (e.g., buttons, text fields, bitmaps,check boxes, lists) naturally drives the user to the correct application usage without prac-tically any extra help or tutorial instructions. Furthermore, the automation of frequentlyused equations is envisaged to assist the user in several everyday tasks.
In this lecture, the aforementioned features will be highlighted and a typical ap-plication structure will be shown. After some developed examples explanation, a list ofinternet addresses containing both resources to develop C++ applications and directionsto already implemented software will be provided.
Keywords
C++, programming, GUI, user experience, embedded functions, CAS calling.
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D002: Coding Theory for theClassroom
Josef BohmACDCA &
International DERIVE & CAS-TI User GroupAustria
Workshop for the TI-Nspire & Derive Strand
Abstract
The curriculum for the Austrian “Handelsakademie” (= College for Business Ad-ministration) does not contain only “Cryptography” but also “Coding Theory”. I tookthese chapters as one of the co-authors of a textbook series without knowing exactly theintentions of the authors of the curriculum.
It was a challenge to find issues in Coding Theory which are beyond of only discussingthe ASCII-Code and presenting ISBN- and EAN-Code and which are suitable for thestudents (age 17).
We will explain and work with data compressing like the Huffman-Code and withself correcting codes like the Hamming-Code. We will also show many questions whichlead the students to a better understanding of the problems.
CAS- and spreadsheet tools will support the activities.
Keywords
Coding Theory, proofs, CAS, spreadsheet.
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D003: 20 Years InternationalDERIVE & CAS-TI User Group
Josef BohmACDCA &
International DERIVE & CAS-TI User GroupAustria
Lecture for the TI-Nspire & Derive Strand
Abstract
It was in 1988 when an Austrian school teacher found an announcement of “DERIVE– A Mathematical Assistant” on his desk. His headmaster ordered the software and theteacher was immediately fascinated and had visions how his teaching could be changed.
Short time later he contacted the distributor of DERIVE, Bernhard Kutzler, whopromised supporting establishing a “DERIVE User Group” (DUG). In 1990 the DERIVENewsletter #1 (DNL) was sent to a list of DERIVE users together with the invitationto subscribe the membership to the DUG. The number of subscribers was satisfying, sothe DUG was born.
When the symbolic calculators appeared on the market many of the TI-CAS usersjoined the group which was then renamed in DERIVE and TI-CAS User Group. Now theDUG is a world wide spread association of CAS enthusiasts and the DNL is its bulletin.Nearly 80 issues of the DNL have been published so far.
Within the DUG a special “spirit” was created which is responsible that even a coupleof years after taking DERIVE off the market the DUG is still growing and the membersare active as ever before.
We will present some impressive data, highlights and pitfalls from the 20 years. Wewould also like to discuss aims and visions for the future – the next 20 years? – of thisremarkable group of people connected by the “Spirit of DERIVE”.
Keywords
DUG, DNL, DERIVE, TI-CAS.
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D008: Teaching Linear Regressionin the PC Lab
Karsten SchmidtFaculty of Business and Economics
Schmalkalden University of Applied SciencesGermany
Workshop for the TI-Nspire & Derive Strand
Abstract
The Bachelor program of the Faculty of Business and Economics at SchmalkaldenUniversity of Applied Sciences includes a good deal of statistics education as each studentmust take two courses in statistics. The first course (Introductory Statistics) is given inthe traditional way, i.e. it takes place in a classroom equipped with blackboard andoverhead projector; the only technology used are pocket calculators. The second course(Intermediate Statistics) was redesigned in recent years. The course is now held in the PClab throughout the semester and its title was changed to Computer-assisted Statistics .In addition to statistical software (SPSS), a Computer Algebra System (Derive) is usedthroughout the course. Students are then already familiar with Derive as this softwarewas used one semester earlier in a (required) course in matrix algebra. The facultyacquired a special Derive license that permits its use on the students’ own PCs, suchthat all students have permanent access to Derive, be it in class, at home, or during theexam.
This presentation will be about new possibilities emerging from the fact that thecourse is taught in the PC lab, concentrating on topics from linear regression (which infact is the main topic of the course). SPSS and Derive are used both for numerical andgraphical analyses. Since the creation of graphs is more challenging in Derive than inSPSS, a utility file was developed to facilitate such tasks. This file includes functions forthe preparation of a 2D scatterplot with the associated least squares regression line forany two-variable, or simple, linear regression model, as well as functions for a 3D scat-terplot with the associated least squares regression plane for any three-variable model.In addition, the so-called system of normal equations, which plays a major role in thederivation of the Ordinary Least Squares (OLS) estimator, is a system of two straightlines in the case of simple regression, which again can be viewed in a 2D plot: the OLSestimator is the point (a vector in R2) where the two lines intersect. In the case of athree-variable model, the OLS estimator is the point (a vector in R3) where the threeplanes, defined by the system of normal equations, intersect. It is also interesting tolook at the graph of the sum of squared residuals (SSR) which is the objective functionthat is minimized by the OLS estimator. The SSR function is a convex function of thetwo (three) elements of the parameter vector in simple (three-variable) linear regression.
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In the case of simple regression, a 3D plot reveals that the OLS estimator is the pointwhere the three-dimensional SSR function has its minimum.
We will also take a quick look at some files and applications from the World WideWeb which are interesting and helpful in teaching linear regression.
Keywords
Coding Theory, proofs, CAS, spreadsheet.
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D009: Solving Second Order ODEs,Two Non-analytical Methods
Revisited
M. Beaudin and G. PicardService des enseignements generaux
Ecole de Technologie SuperieureCanada
[email protected] and [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
Mathematics can still be taught without using a CAS and this is probably the casein most schools and universities. Although CAS and technology are often used by in-structors to demonstrate or illustrate mathematical concepts, they are rarely used bystudents. When we consider our mathematics curriculum, “Differential Equations” isone course that we firmly believe can and should benefit from the use of CAS. In thistalk, we will report how our ODE course has evolved, as our engineering students haveaccess to technology (Voyage 200 symbolic calculator) in the classroom at all times. Thistalk will show examples of what students still do by hand and what CAS allows us todo now to enrich the learning experience.
We will consider the series solutions of a second order equation with variable coeffi-cients and numerical solution of the first order equivalent system. Many textbooks donot show the relation between these two subjects. With a CAS on every desk, we canask students to compare results obtained with both methods. Of course, technology isa must to support this. As teachers, we still want our students to be able to do somespecific computations manually. For example, they have to find the recurrence formulaby hand for the coefficients of the series solution. However, we also want students to beable to compute, with some accuracy, the value of the solution at a certain point usingpartial sums or even graph this approximate solution. Then, converting the same equa-tion into a first order system, they can plot the numerical generated curve obtained bythe built-in RK method in the Voyage 200 or create a table of values for the approximatesolution.
Keywords
CAS, Second order differential equations, power series solutions, RK method.
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D010: Using Rational Arithmetic toDevelop a Proof. “What Josef and
Carl Saw”
J. Bohm C. LeinbachACDCA & Gettysburg College
International DERIVE & CAS-TI User Group Gettysburg, PennsylvaniaAustria USA
[email protected] [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
It all began with an article in DNL #22 entitled Finding a Limit via GeometricReasoning authored by Marvin Brubaker and Carl Leinbach. In that paper the limitof a recursively defined sequence of points was found by connecting successive pointswith straight lines, thus creating a nested sequence of triangles that seem to convergeto the point (0.4, 0.4). While editing an archival edition of DNL #22, Josef correctlypointed out that the paper did not really have a proof of the limit, only a collection ofheuristic evidence gained by zooming in on the suspected limit. He wrote to Carl askingif he had a mathematical proof that sequence did, in fact, converge to its claimed limit.Both Josef and Carl began independent work on the problem. Their initial step was thesame. They each wrote a small DERIVE program to print out the first n terms of thesequence using the CAS’s rational arithmetic display of the points. After this their twoapproaches differed.
In this presentation both Josef and Carl will discuss their approaches to constructinga proof that the sequence converges to its’ claimed limit, thus supporting the visualevidence. They will also discuss the value of using the Rational Arithmetic to supportthe discovery of a strategy to accomplish their mathematical goal. If time permits, thepresenters will investigate applying their approaches to other sequences of points.
Keywords
geometric intuition, rational Arithmetic, formal proof, limit, sequences
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D011: Beyond Newton’s Law ofCooling Updated Methods forEstimating Time Since Death
Carl LeinbachDepartment of Computer Science, Retired
Gettysburg CollegeUSA
Lecture for the TI-Nspire & Derive Strand
Abstract
The relationship between body temperature and time since death has been noted forcenturies. The earliest attempt to describe this relationship used the usually inaccurateformula of 1oC per hour. The first scientific breakthrough occurred in 1868 when H.Rainey (Glasgow Medical Journal) applied Newton’s Law of cooling to the relationship ofdeep inner body temperatures (rectal or liver temperatures) of deceased bodies. However,Rainey noted that during the first hours after death the rate of cooling is slower thanthe rate at later times. He described this phenomenon as a “plateau”. It took almostan entire century until this problem was seriously approached by Marshall and Hoare in1962 (Journal of Forensic Science) who proposed a double exponential model for makingthe estimate. This method has proved to be unwieldy and difficult to implement in thefield because of the need for an extended period of accurate temperature observations.In 1988 Henssge (Forensic Science International) developed a nomogram based on thework of Marshall and Hoare for use in the field. This tool, though making the jobeasier, has had little acceptance because of the need to estimate body weight, weight ofclothing, body position, and other factors. In 1985, Green and Wright (Forensic ScienceInternational) proposed another attack on the problem, which looks at the percentageof possible cooling done by the deceased’s body. Their method requires only a twotemperature measurements on the scene where the body is discovered. Despite all ofthese attempts at accuracy, many coroners still use the old 1oC per hour rule.
In this presentation, an overview of using DERIVEr to explore the methods de-scribed above and compare their results using temperature data from actual coronerscenes. The presenter will also present a method that may be applied to situationswhere the body is located outdoors. This method will use daily temperature data and aminimum of body temperature measurements.
Keywords
Time since death, body temperature, ambient temperature, exponential models.
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D012: Using DERIVEr’S GraphicsTools For Geographic Profiling of
Serial Offenders
Carl LeinbachDepartment of Computer Science
Gettysburg CollegeUSA
Lecture for the TI-Nspire & Derive Strand
Abstract
While crimes committed by serial offenders do not constitute the majority of most po-lice investigations, their occurrence attracts much media attention and has also been thesubject of research by a committed group of social scientists. This presentation is basedon the work began in 1981 by Brantingham and Brantingham (Environmental Criminology)who developed the idea of a “buffer zone” that offenders seem to establish about theirhome base or base of operations and the subsequent extension in 1987 to the idea ofworking from data on crime scene locations and nature of the crime back to the homebase by D. Kim Rossmo (PhD Thesis, Simon Frazer University). Rossmo uses an expo-nentially distributed probability model based on the size of the offender’s buffer zone, thesize of the hunting area, scaled exponential functions, and the number of sites includedin the model. In 2000, David Cantor and colleagues (Journal of Quantitative Crimi-nology) explored various models for quantitative profiling, including Rossmo’s model.Cantor examined 70 examples of serial offences and found that the exponential modelusing a specifically designed normalization parameter produced a dramatic reduction inthe search costs of location the operational base of the serial offender. In both Rossmo’sand Cantor’s approaches maps were placed on a grid, and the locations of the offenseswere marked. Probabilities were assigned to several points on the grid based on theexponential probabilities that the point was the offender’s base.
In this presentation maps from several known serial criminals’ exploits will be placedon the DERIVE graphics screen and grid points uniformly chosen and assigned proba-bilities using scaled exponential decay functions. Those with a suitably high probabilitywill be shaded and help to define the search area for locating the home base of the serialoffender.
Keywords
Serial crime, negative exponential function, regression axes, graphical display.
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D013: Using Mathematical Tools InForensic Investigations
C. Leinbach P. LeinbachDepartment of Computer Science Office of the Coroner, Retired
Gettysburg College Adams County, PennsylvaniaUSA USA
[email protected] [email protected]
Workshop for the TI-Nspire & Derive Strand
Abstract
During this hands on workshop participants will choose one of four hypothetical crimescenes and work as teams to construct a classroom learning environment for studentsto gather evidence and analyse it using mathematical techniques and technical toolsthat are available to them in their classroom. Each team will discuss their pedagogicalobjectives and strategies for presenting the scenarios and how to guide their studentswhile performing their mathematical investigations. After discussing their plans withthe presenters, the teams will conduct their crime scene investigation. If time permits,the teams will review the scenarios and strategies of other teams.
Topics from forensics that will be part of individual scenarios will include: bloodspatter analysis, estimating time since death, estimating vehicle speed from the lengthof tire skid marks, fingerprint analysis, force of impact, locating exact positions, relatinga person’s height to their stride length, and other topics. The technological tools required
to solve the mathematical portions of the analysis will be solved using DERIVEr orTI- Nspire.
Keywords
Forensics, pedagogy, applications, student motivation
Observations
The presenters will supply access to materials that describe the mathematical peda-gogy necessary to analyze the forensic situation and bring tools such as thermometers,materials to simulate blood spatters, GPS’s, measuring tapes, etc. A room large enoughfor teams to lay out their “crime scene” is required for this workshop. It is also impor-tant that the room also contain computers with DERIVE or that the participants haveaccess to TI-Nspire handhelds.
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D014: Laplace Transforms, ODEsand CAS
G. Picard and C. TrottierService des enseignements generaux
Ecole de Technologie SuperieureCanada
[email protected] and [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
Mathematics can still be taught without using a CAS and this is probably the casein most schools and universities. Although CAS and technology are often used by in-structors to demonstrate or illustrate mathematical concepts, they are rarely used bystudents. When we consider our mathematics curriculum, “Differential Equations” isone course that we firmly believe can and should benefit from the use of CAS. In thistalk, we will report how our ODE course has evolved, as our engineering students haveaccess to technology (Voyage 200 symbolic calculator) in the classroom at all times. Thistalk will show examples of what students still do by hand and what CAS allows us todo now to enrich the learning experience.
Laplace transform in differential equations is a great subject where CAS can be usedin an efficient manner. Starting with the classic definition of the Laplace transform, wealready have a nice area to use our CAS, reviewing with students when an improperintegral converges. In engineering, we often have to work with piecewise continuousfunctions, so we introduce the unit step (Heaviside) function and show them how tocreate it on their Voyage 200 so they can easily plot these type of functions. We stilldemand that our students learn and work with a basic Laplace transforms table. We havethem work, by hand, many of the classic properties of Laplace transform. When comestime to calculate inverse Laplace transforms, we let them use an “Expand” command ontheir Voyage 200 so they can get directly the result of the partial fraction expansion ofrational functions. This being done, they still have to finish manually the inverse process,going back to their table and doing, when needed, the completion of the square forcomplex roots or using the appropriate property of the table (time-shifting for example).As for solving differential equations with the aid of Laplace transforms, we will show anexample where, even with a discontinuous forcing function, the solution can still easilybe found with the aid of the CAS and we insist on having them plot this solution. Wewill complete this presentation by solving a system of differential equations using theLaplace transform method, which is rarely done due to the amount of manual calculationsinvolved.
Keywords
CAS, Laplace transform, Differential equations, Unit step function.
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D015: Could it be possible toreplace DERIVE with MAXIMA?
A. Garcıa and F. GarcıaDepartamento de Matemtica Aplicada. E.U. Informtica
Universidad Politcnica de MadridSpain
[email protected] and [email protected]
G. RodrıguezDepartamento de Matemtica Aplicada.
E.P.S. de Zamora.Universidad de Salamanca
Spain
A. de la VillaDepartamento de Matemtica Aplicada y Computacin. ETSI (ICAI).
Universidad Pontificia Comillas.and
Departamento de Matemtica Aplicada. E.U.I.T.IUniversidad Politcnica de Madrid
Spain
Lecture for the TI-Nspire & Derive Strand
Abstract
For about twenty years, a large number of Spanish teachers are using DERIVE inthe apprenticeship of mathematical topics in the High Schools and in the Universities.In 1993 the group of “DERIVE users in Spain”, was established. This group has beenactive until 2005.
Many Spanish Universities have included DERIVE system as a support tool in Cal-culus, Mathematical Analysis and Linear Algebra Courses mainly for Engineering. Cur-rently, taking into account the commercial situation of DERIVE, we are balancing thepossibility to use public domain software like Maxima, and we have done a compara-tive study of DERIVE and Maxima as support tools in a Calculus course for first yearstudents of Engineering. The work has been divided in different steps:
1. In the first step a comparative general study (documentation, accessibility, suit-ability, portability, ease of use, interface, feedback, etc), has been realized.
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2. In the second step the practices of Mathematical Analysis of Computer Engineeringat the Polytechnic University of Madrid have been carried out during 2009-10with Maxima. The obtained results (specifically: student’s knowledge, satisfaction,marks, etc.) are compared with those of similar practices carried out in previousyears with DERIVE.
3. In the third step we take a detailed view over Calculus skills, by solving withMaxima the proposed problems in our text book (Clculo I. Teora y problemas deAnlisis Matemtico en una variable) and analyzing the similarities and differenceswith DERIVE.
The paper will present the results of this study.
Keywords
Derive, Maxima, Calculus for Engineering, Mathematical Software.
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D016: WIRIS – Derive, enablingcompatibility
E. Badia, S. Egido, R. Eixarch and D. MarquesMaths for More
BarcelonaSpain
Lecture for the TI-Nspire & Derive Strand
Abstract
WIRIS is a suite of mathematical tools. WIRIS CAS and WIRIS Desktop are, re-spectively, the online and offline versions of a CAS with Dynamic Geometry capabilities,2D and 3D graphics, advanced formulae typesetting, and other mathematical features.
Since Derive was discontinued, some of its users have shown their interest in con-verting their work into WIRIS. This is particularly the case in those European countrieswhere teachers and students have free access to WIRIS on a national license. In orderfor migration to be as simple as possible, WIRIS team has been working to offer thepossibility of importing Derive files (both .mth and .dfw) into WIRIS format. Sinceboth CAS do not have exactly the same functionalities, WIRIS is being enlarged untilincluding as many of Derive’s options as possible. The goal of this project is, in fact, toease the use of WIRIS by former Derive users, who want to keep having available theirfavourite features. In time, they will get to appreciate the new capabilities provided byWIRIS.
The strategy used to read Derive files is similar to that of a compiler; after running apre-processor, a tokenizer and a context-free grammar are used to interpret the instruc-tions, which can then be rewritten in terms of WIRIS functions. Since both CAS arefairly different systems, complete compatibility is not possible. The output of resultsis done in WIRIS fashion; in particular, formulae typesetting and plot styles are quitedifferent to those of Derive. The results of some algebraic manipulations, such as thesimplify() function, may vary. Other aspects have a slightly different behaviour; for in-stance, numerical functions do not provide exactly the same results. And, because of itsdesign philosophy, WIRIS libraries are very different to Derive’s.
In spite of the differences that will remain, it is expected that almost all small piecesof software written in Derive, such as educative resources, calculation examples, andgraphical illustrations, will be able to run under WIRIS. Teachers and students will findthat migration to WIRIS is reasonably comfortable, amenable to improvements, andachieves a high success rate of teachers previous work reutilization. Furthermore, sinceWIRIS CAS is an online tool integrated into several Virtual Learning Environments thatwill enhance the use of their existing work in new contexts.
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Keywords
Derive, WIRIS, software reutilization.
Observations
This is a work in progress. Initial conversions of Derive files into WIRIS have beensuccessful.
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D017: Construction of mathematicalknowledge using graphic calculators
(TI-84 plus & CAS) in themathematics classroom
Fernando HittDepartement de Mathematiques et Faculte de Psychologie et des Sciences de
l’EducationUniversite du Quebec a Montreal & Universite de Geneve
Canada & Switzerland
Lecture for the TI-Nspire & Derive Strand
Abstract
Mathematics education researchers are asking themselves about why technology hasimpacted heavily on the social environment and not in the mathematics classroom. Theuse of technology in the mathematics classroom has not had the expected impact, as ithas been its use in everyday life (f.e. cell phone). Mathematics teachers can be dividedinto three categories: Those with a boundless overflow (enthusiasm) that want to usethe technology without worrying much about the construction of mathematical concepts,those who reject outright the use of technology because they think that their use inhibitsthe development of mathematical skills, and others that reflect on the balance that mustexist between paper-pencil activities and use of technology. The mathematics teachernot having clear examples that support this last option about the balance of paper-pencilactivities and technology, opt for one of the extreme positions outlined above. In thispaper, we show the results of research on the implementation of activities in an environ-ment of paper-pencil and calculator (TI-84 Plus & CAS) in the mathematics classroom,regarding mathematical modeling in secondary school and the training of teachers ofmathematics in that level. We note also that with the development of technology on theuse of electronic tablets and interactive whiteboards, these activities will take on greatermomentum in the near future.
References
Hitt F. (2004). Reflexions sur les potentialites des logiciels et des calculatrices symbol-iques pour l’enseignement des mathematiques. Une approche didactique. Proceed-ings/Tagungsband TIME-2004 : International Symposium on Technology and itsIntegration into Mathematics Education. Montreal, Canada, pp. 1-10.
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Hitt F. (2007). Utilisation de calculatrices symboliques dans le cadre d’une methoded’apprentissage collaboratif, de debat scientifique et d’auto-reflexion. In M. Baron,D. Guin et L. Trouche (Editeurs), Environnements informatises et ressources nu-meriques pour l’apprentissage. conception et usages, regards croises (pp. 65-88).Editorial Hermes.
Hitt F. & Cortes C. (2009). Planificacion de actividades en un curso sobre la adquisicionde competencias en la modelizacin matematica y uso de calculadora con posibili-dades graficas. Artıculo por invitacion, Revista Digital Matematica, Educacin etInternet. www.cidse.itcr.ac.cr/revistamate/. Vol. 10u, No 1, pp. 1-30.
Hitt F. & Kieran C. 2009. Constructing knowledge via a peer interaction in a CASenvironment with tasks designed from a Task-Technique-Theory perspective. In-ternational Journal of Computers for Mathematical Learning.DOI number: 10.1007/s10758-009-9151-0.http://www.springerlink.com/content/657wt76n04x43rk8/.
Keywords
Learning Mathematics, calculators, environment, mathematical activity.
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D018: 3D Analytic geometry inusing the link between CAS andGeometry applications of the TI
N’Spire
Jean-Jacques DahanIREM of Toulouse
Universit Paul Sabatier ToulouseFrance
Workshop for the TI-Nspire & Derive Strand
Abstract
We will start this workshop in showing how to represent in the Graphs and Geometryapplication of the TI N’Spire, in parallel perspectives, points given by their coordinates,lines and circles also generated by points given by their coordinates. We will continuein showing how to construct cones and cylinders (in using coordinates) having circlesas basis but also having other curves given by their parametric equations as basis. Wewill have the opportunity to use the possible links between the Calculator page and theGeometry page. We will point the power of the tools “Slider”, “Store”, “Link”, “Lock”as well as the difficulty for both teachers and students to deal with the different sorts ofapproach of the notion of “variable” and “unknown” in this environment. We will usefor the workshop, the last version of the handheld (with the touchpad) as well as thenew version of the Teacher Edition version of the TI N’Spire software.
Keywords
Analytic geometry. Cones. Cylinders. Variables. Unknowns.
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D019: WIRIS, a complete CAS withDGS
Ramon EixarchMaths for More
BarcelonaSpain
Workshop for the TI-Nspire & Derive Strand
Abstract
WIRIS is a suite of mathematical tools. WIRIS CAS and WIRIS Desktop are, re-spectively, the online and offline versions of a CAS with Dynamic Geometry capabilities,2D and 3D graphics, advanced formulae typesetting, and other mathematical features.
WIRIS tools focus on usability and are tailored for educational purposes. The work-shop will take a tour though the basics of WIRIS but also on the newest features, suchas region plotting or ODE solving, for those assistants already aware of WIRIS.
WIRIS CAS is integrated into several Virtual Learning Environments. In particular,it is part of WIRIS quizzes, a mathematical assessment tool developed over Moodle.The last part of the workshop will be devoted to a brief presentation of the assessmentsystem.
Some links to test:
• WIRIS CAS www.wiris.com/demo/en/
• WIRIS quizzes www.wiris.com/demo-moodle/
Keywords
WIRIS, CAS, DGS, VLE, LMS, assessment.
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D020: Experiments with geometricloci
Wolfgang MoldenhauerThuringian Institut of Inservice Teacher Training, Curriculum Development and Media
(ThILLM)Bad BerkaGermany
Lecture for the TI-Nspire & Derive Strand
Abstract
In elementary geometry, geometric loci define a set of points that exhibit a certaingiven property. In planar geometry they are usually curves. Occasionally, geometric lociare discussed in school mathematics within the scope of curve sketchings. Due to theavailability of dynamic geometry software (DGS) and computer algebra systems (CAS)the question arises if this topic should be viewed from a different perspective. This couldpotentially stimulate the mathematical education in the following aspects:
• linking geometry, analysis and algebra,
• experimental work,
• learning by discovery.
The study of geometric loci can, at least, be an enrichment in the course of talentedstudent programs and thereby improve distinction. An assessment of the difficultiesto solve the respective question is not always, a priori, possible since already for easylooking problems it occasionally turns out that e.g., finding an algebraic descriptionof the geometric locus is difficult. On the contrary, there are also examples that aresolvable with elementary school methods. The task for the lecturer is to be aware ofthe difficulty rating of the respective problem to really provide his students with thedesired competency gain. The presentation tries to enlighten this problematic topic andto simultaneously present some examples of geometric loci from triangular geometry.
References
[1 ] Wolfgang Moldenhauer and Wilfried Zappe. Wanderungen. TI-Nachrichten,2:27-31, 2008.
2 ] Wilfried Zappe, Wolfgang Moldenhauer, and Sonnhard Graubner. Eine Ortskurvedes Schnittpunkts der Winkelhalbierenden eines Dreiecks. TI-Nachrichten, 1:26-28,2007.
Keywords
Mathematical Education, Mathematical Problem Solving, Secondary School.
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D021: Mathematical Models inBiology
Mazen ShahinDepartment of Mathematical Sciences
Delaware State UniversityDover, Delaware 19901
USA
Workshop for the TI-Nspire & Derive Strand
Abstract
In this workshop we will share the pedagogy and methodology of a course based onMathematical Models in Biology. The course utilizes difference equations, matrix algebraand Markov chains as the main mathematical tools and integrates a computer algebrasystem and cooperative learning. The main theme of the course is modeling of biologicaland ecological systems using linear and nonlinear difference equations as well as systemsof difference equations. The target audiences of this course are freshmen life scienceand mathematics majors. Students work in small groups on carefully designed activitiesthat guide them in discovery of mathematical concepts on their own and exploration ofconnection between biology and mathematics.
In this workshop the participants will be introduced to and will work on various mod-els in biology. They will utilize the computer algebra system Derive or/and a spreadsheetto investigate and analyze the models that are represented by a single difference equation,a system of difference equations, or a matrix equation. We will investigate situationsthat are modeled by first-order linear difference equations such as population dynamicsof a single species and drug administration. Models represented by nonlinear first-orderdifference equations such as logistic growth model, maximum sustainable yield, and har-vest strategies will be explored. We introduce Markov chains and investigate a modelof population movement between a city and its surrounding suburbs, as well as a modelthat utilizes Markov chains in genetics. We will discuss an innovative way of introduc-ing eigenvalues and eigenvectors through age-structured population models. We willconclude with some population models of interacting species.
Keywords
Mathematical models, difference equations, matrix algebra, CAS and cooperative learn-ing.
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D022: The ElGamal Public-KeyCryptosystem – A Full
Implementation Using DERIVE
Johann WiesenbauerDepartment of Discrete Mathematics
Vienna University of TechnologyAustria
Lecture for the TI-Nspire & Derive Strand
Abstract
The ElGamal cryptographic scheme is along with RSA one of the most widely usedpublic-key cryptosystems, in particular when it comes to the use on smartcards and smalldevices due to its relatively small need of resources in some variants. From a mathemati-cal point of view, the underlying “hard” mathematical problem is the discrete logarithmproblem (DLP). While it can be formulated for any group, usually the multiplicativegroup of a finite field or groups emerging from elliptic curves are used in this context.
In my talk using DERIVE a full implementation of the ElGamal cryptosystem isprovided using elliptic curves over finite residue class rings mod p, where p is a primewith at least 160 bits, including the signature scheme as proposed by NIST. As forthe problem of determining the order of the involved elliptic curves we make use of so-called CM-curves for this purpose. Furthermore the most common attacks on DLP likeBaby-step-giant-step algorithm, Pollard’s rho-method and the Pohlig-Hellman methodare discussed, along with implementations and a lot of examples.
Keywords
ElGamal cryptosystem, Public-key cryptosystems, discrete logarithm problem, ellipticcurves.
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D023: A virtual laboratory forblended-learning: Numerical
Methods using WIRISA. Mora and E. Merida
Department of Applied MathematicsUniversity of Mlaga
Spain
[email protected] and [email protected]
R. EixarchMaths for More
BarcelonaSpain
Lecture for the TI-Nspire & Derive Strand
Abstract
Numerical Methods is a hard subject for the students of the Computer Science Engi-neering in the Mlaga University. Since 2003-2004 academic course, we have been movingto a blended-learning framework and introducing the new technologies as a method toimprove the understanding of the subject. In a first state, we designed a virtual course inthe previous platform of the university and we observed a crucial change in the relationwith students (motivation, number of students attended to exams, results, etc.). Thenext step was to adapt the lecture material for the new Moodle site of the virtual coursein our university.
Nowadays, Numerical Methods is a blended-learning subject supported with a widematerial developed in Moodle: forums, questionnaires, lessons, tasks, wikis, glossaries,books, chats, etc. Therefore, the Moodle platform is the meeting point for working onour subject.
This subject needs special materials and activities: firstly, the material must includemathematical formulas and secondly, the activities included in the virtual course must beable to do math computation. We emphasize, for these proposals and for the developmentof new and more interesting learning units, the help of WIRIS (www.wiris.com) as analternative to Matlab and Derive used in other academic courses. WIRIS is a powerfultool with the ability of edition, calculation, easy design of graphics, etc. WIRIS hastwo versions: WIRIS CAS (web version) and WIRIS Desktop (local version) especiallyadequate for educational environments. It has libraries for calculi, algebra, geometry,etc, but it has not a library for the learning of numerical methods.
In this work, we present the design of a library for WIRIS for the teaching-learningprocess of the numerical method subject. Moreover, as a complement of this library, wehave developed a web portal that connects with WIRIS. In this virtual laboratory, weprovide the students with different materials for each unit:
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• theoretical aspects
• exercises to be solved with WIRIS
• auto-evaluation activities in order to evaluate the knowledge and understandingacquired in each unit.
Keywords
Blended-Learning, LMS, CAS, WIRIS, Virtual Laboratories.
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D024: Integrating a new Derive 6Video User Guide into Virtual
Teaching
J. Bustos, J. L. Galan, Y. Padilla and P. RodrıguezDepartment of Applied Mathematics
University of MalagaSpain
[email protected], jl [email protected], [email protected] [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
The main goal of this Derive 6 online tutorial is to set up a video user guide forits content, functionality and applications. Our initial objective was to provide theTelecommunication Engineering College students at the University of Malaga with a toolto help them to cope with the incremental homework overload after the implementationof the new Bologna academic syllabus for 2010-11. However, this online tutorial hasproven to be an excellent tutorial for any person willing to use Derive 6 in a daily basis.
The Derive 6 online video user guide has been firstly drafted in Spanish because itis a program widely used in courses such as “Ampliacion de Matematicas”, “Fundamen-tos de Calculo”, “Analisis Vectorial y Ecuaciones Diferenciales” and the online courseof “Fundamentos basicos de variable compleja con un programa de calculo simbolico”.But afterwards, it was translated into English to make accessible to any Derive 6 userinternationally.
This Derive 6 online video user guide environment is within a web-page format forthe user to browse through its content and/or to look for specific applications in an easyway. The Derive 6 online tutorial consists of an index dividing its content in differenttabs. Furthermore, the user can find some videos that explain how to use Derive 6 tomake different calculations. These videos have the same structure of previous Derive 5videos already available in internet such as the ones developed by Terence Etchells andTheresa Shelby allocated at:http://www.cms.livjm.ac.uk/etchells/Derive%20Avi’s/DeriveTutorials2.htm,
but in Spanish. The next stage of this Derive 6 video user guide will be to translatethese videos into English. The Spanish version is allocated at:
http://www.matap.uma.es/profesor/jl galan/Derive/VideoTutorial/.
In this lecture we will show how this Derive 6 video user guide has been integratedinto the completely online subject “Fundamentos bsicos de variable compleja con unprograma de clculo simblico”. We will describe how this subject has been developed inthe last three academic years and the resources, together with this video user guide, that
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have been used. The tasks developed, the evaluation method and the different resourcesused within this online subject, could be considered as a model for teaching in the newBologna syllabus which will be start next academic course.
We will finish with the conclusions obtained from the point of view of both: studentsand teachers.
Keywords
Derive user guide, Video tutorials, Online and Virtual teaching Resources, Bologna syl-labus.
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D025: A CAS routine for obtainingeigenfunctions for Bryan’s effect
S. V. Joubert1,3, C. E. Coetzee1, and M. Y. Shatalov1,2
1Department of Mathematics and StatisticsTshwane University of Technology
2Material Science and Manufacturing,Sensor Science and Technology, CSIR
South Africa
Lecture for the TI-Nspire & Derive Strand
Abstract
Vibratory gyroscopes are used in the navigation of submarines and space-shuttles.The operation of these gyroscopes is based on Bryan’s effect, namely: “when a vibratingring is subjected to rotation in inertial space, the vibrating pattern rotates in the ring ata rate proportional to the inertial rotation rate”. Consequently, an easy mathematicaltreatment of the principle on which they operate is a nice real-life application of math-ematics suitable for senior undergraduates. We expand on the lecture given at DresdenTIME 2006. In that lecture, unknown eigenfunctions were used to derive an expressionfor Bryan’s factor (the constant of proportionality referred to above). At that stage wefound that the derivation of these eigenfunctions was too advanced for junior postgrad-uates to understand. However, in the interim we have discovered an easy routine foriteratively determining these functions using the NDSolve routine of the computer alge-bra system (CAS) Mathematica. All mathematical expressions are simplified using theCAS from basic formulae that can be found in standard textbooks. With this approach(using the CAS at every step) it is possible for senior undergraduates to effortlesslycalculate the eigenfunctions that are necessary to determine Bryan’s factor.
Keywords
Bryan’s factor, Iterative process, Senior undergraduates, CAS, Mathematica.
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D026: Interactive Explorations ofMathematics with TI-Nspire
Technology
K. Stulens and G. BrothersTexas Instruments
[email protected] and [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
TI-NspireTM technology provides the ability for users to view different representationsof mathematical concepts at the same time: graphical, geometric, numeric as well assymbolic. These representations are dynamically linked to help students see connectionsand to get a better understanding.
TI-Nspire technology can be used to improve algebraic reasoning. Its algebraic –CAS – functionality is interactive and dynamic and linked to graphical and numeric(table) representations. It makes TI-NspireTM technology a real mathematical teachingand learning tool and can be used as a calculation & graphing assistant.
TI-Nspire Technology is document based. The document structure with independentproblems and linked pages within a problem gives students and educators the possibilityto save their work in one document, even aligned to the content of textbooks they areusing.
To meet different classroom needs and technology use, TI-Nspire Technology offersthe same functionality both on handhelds and as computer software:
• Teachers can use the computer software to create materials for students usinghandhelds and share the materials easily with the students in the classroom,
• Students can easily transition their work between the handheld and computer andshare the work with teachers.
We will show, via classroom examples from mathematics to statistics, the educationalvalue of exploring multiple representations of mathematical concepts using TI-Nspiretechnology.
Keywords
TI-Nspire, dynamic, interactive, CAS, statistics, multiple representations.
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D027: 11 Years of Master Theses inEngineering using DERIVE in the
University of Malaga
G. Aguilera, J. L. Galan, and P. RodrıguezDepartment of Applied Mathematics
University of MalagaSpain
[email protected], jl [email protected] and [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
The development of a master thesis (MT) is a must for students of Engineering inSpain (as in many other countries along the world) in order to obtain their Degree. Themain purpose of a MT is the development of a personal work by students where theycan apply and integrate their knowledge (theoretical and technical) acquired along theCurriculum and also to foment their capacity for creativity and originality. In a MTone or two teachers must participate acting as advisors. The main topics chosen bystudents for developing their MT are directly related with applications of their studiesto a specific purpose.
We will present the MTs developed using Derive in different Engineering Degreesin the University of Mlaga in which at least one of the authors of this lecture has beenthe advisor. We will present chronologically these MT which will also lead to show theevolution in the use of the software Derive: from our old friend, the Derive 3.13 Ms-Dos version, to Derive 6.1 and its integration in a Java environment or its use as aPeCas (Pedagogical Computer Algebra System).
Specifically, we will present the MTs developed using Derive for topics as MultipleIntegration, Complex Analysis, Discrete Mathematics, Graph Theory, Automatic The-orem Proving for Classical Logic, Probabilistic Logic, Statistics and even ContrapuntalMusical Compositions. We will also present the MTs which are now in developmentsuch as a Numerical Analysis course, a Math Bachelor course and different extensions ofprevious works by means of using Derive 6.1 as a PeCas. These MTs have been thefoundations for many lectures developed in previous TIME and other Conferences andfor different papers published in specialized Journals.
Finally, it is our hope to develop a very important project which would consist inthe developing of a friendly environment which can translate Derive to other actualsoftware such as Maxima. This will be done by integrating different future MTs andwith the main goal of being able to use the work developed in Derive in other software.
Keywords
Derive, Master Thesis, Computer Algebra System (Cas), Pedagogical Computer Al-gebra System (PeCas), Engineering Syllabus.
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D028: Teaching DifferentialEquations and its Applications
Using Derive 6 as a PeCasG. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla and P. Rodrıguez
Department of Applied MathematicsUniversity of Malaga
Spain
[email protected], jl [email protected], [email protected], [email protected] [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
In this lecture we will describe the file diferential equations.mth, created in De-rive 6 in order to be used in mathematical subjects which deal with differential equa-tions, aimed at Engineering students. Such file contains a series of programs whichpermit to solve differential equations problems.
The programs contained in the file can be grouped within the following blocks:
• First order differential equations: separable equations and equations reducibleto them, homogeneous equations and equations reducible to them, exact differentialequations and equations reducible to them (integrating factor technique), linearequations, the Bernoulli equation, the Riccati equation.
• First order differential equations and nth degree in y’.
• Generic programs to solve first order differential equations.
• Cauchy problems for first order differential equations.
• Higher orders differential equations.
• Cauchy problems for higher orders differential equations.
• Applications of differential equations.
We will also describe in this lecture some examples of applications that have beencarried out with our students of Telecommunication Engineering.
The programs have been developed using the Display function in order to be usedas didactical tools with explications of what the programs do step by step. In this way,Derive 6 is used as a Pedagogical Cas (PeCas) or as a white-box Cas.
Finally, we include the conclusions obtained after using this file with our studentsand also some future work on this and other related subjects.
Keywords
Differential Equations, Derive, Pedagogical Computer Algebra System (PeCas), Math-ematics Teaching Techniques, Engineering.
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D030: Teaching Calculus andNumerical Analysis using CASaccording to Bologna Process
J. M. Gonzalez1, T. Morales2, M. L. Munoz1, M. A. Atencia1 and S. Merino1
1Department of Applied MathematicsUniversity of Malaga
Spain2Department of Mathematics
University of CordobaSpain
[email protected], [email protected], [email protected], [email protected] [email protected]
Lecture for the TI-Nspire & Derive Strand
Abstract
The objective of this work is designing, experimenting and evaluating a particularproposal for Calculus and Numerical Analysis teaching in university education. We aremainly interested in the teaching of these subjects in engineering schools, following theEuropean Higher Education Area agreements. Our purpose is to show students the closerelation between both subjects. This is achieved by facing real problems in engineeringwhich are described by means of a mathematical model and need the use of numericalmethods for their practical resolution. We introduce mathematical models that rely onconcepts previously studied in Calculus. These models will be by themselves a practicalmotivation for the definition of those concepts. Usually, the exact resolution of thosemodels by formal calculations is not possible. Thus, the use of numerical methods toobtain approximate solutions is required. This enlightens the need of the NumericalAnalysis subject. In fact, both subjects have been merged into one in the new structureof some engineering degrees.
In this lecture, we propose new course materials, computer lab activities and didac-tical strategies, all of them based on the use of mathematical software such as Matlab,Derive and/or free software.
On the other hand, Moodle e-learning platform provided by the University of Malagaand some other universities in Spain will be a useful tool for the coordination and devel-opment of the course. We will encourage the use of this web platform and the integrationof the different course materials, activities and other resources in it.
Keywords
Calculus, Numerical Analysis, Bologna process, CAS, web based education.
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D031: E-Learning and Joomla
S. Merino, J. Martınez, J. L. Galan, P. Rodrıguez, M. L. Munoz, J. M.Gonzalez, P. Cordero, Y. Padilla, G. Gutierrez, A. Mora, E. Merida and F.
RodrıguezDepartment of Applied Mathematics
University of MalagaSpain
{smerino, jmartinezd, jl galan, prodriguez, mlmunoz, jgv, pcordero}@uma.es,{ypadilla, gloriagb, amora, merida, fjrodri}@ctima.uma.es
Lecture for the TI-Nspire & Derive Strand
Abstract
We present a work developed by a group of teachers in the Department of AppliedMathematics at the University of Malaga. Since 2002, FERMAT1 Project has becomean important meeting point between teachers and students for the subjects of Numeri-cal Analysis, Algebra, Calculus, Vectorial Analysis, Differential Equations and DiscreteMathematics in the Degree of Telecommunication Engineering.
Our main goal was the development of an educational environment complementaryto the classical model of teaching, in order to get a gradual adaptation to the EuropeanSpace for Higher Education (ESHE) according to Bologna Declaration. Our methodconsists on theoretical lessons using multimedia technologies and also computer lab lec-tures using different CAS (among others: Derive, TI-Nspire, Matlab, Mathematica andScilab). FERMAT also provides different resources to students such as lecture notes forclassroom and for lab, proposed and solved problems and exams, computer tasks, ques-tionnaires, videos, and also on-line resources such as: mail, forum, chat, videoconference,. . .
In a first step we developed FERMAT as a website where we included some of theseresources useful for students. The suggestions made from students and teachers led usto the development of a new framework: FNOVA2. Thanks to these suggestions we haveintroduced important improvements in this version of our tool.
In this lecture, we will describe this framework as a model for the new Bolognasyllabus.
Keywords
Mathematics, E-learning, Moodle, Joomla, ESHE, HECACEJ, CAS, Bologna Syllabus.
1FERMAT: Forum about Experiences and Resources of Mathematic Applied to Technologies.FERMAT web (http://www.fermat.uma.es) has been recognized as the second winner of theaward for the III International University Competition in Research and Teaching in the web:http://www.campusred.net/certamen.
2FNOVA: FERMAT NOVA, arises of integration between our website and Joomla.
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D032: Using Notes with InteractiveMath Boxes in TI-Nspire Software
Version 2
Wade Ellis, Jr.Department of Mathematics
West Valley CollegeSaratoga, California
USA
Workshop for the TI-Nspire & Derive Strand
Abstract
The recently announced TI-Nspire Version 2 software contains a new feature in theNotes application called Interactive Math Boxes with which interesting and powerful CASdocuments can be created. In this workshop, participants will create several documentsthat exploit this feature to develop student understanding of calculus and linear algebratopics. Then the workshop facilitator will demonstrate and participants will work with avariety of documents that use this and other features of Version 2 in topics from calculus,matrix theory, and differential equations.
In addition, a discussion of the types of questions that such interactive softwaredocuments afford for increasing student engagement and mathematical understanding.
Although no previous experience with TI-Nspire is expected, participants should befamiliar with the Microsoft Windows and Word interfaces to feel comfortable in thisworkshop.
Mathematics topics covered will include:
• L’Hospital’s Rule
• Riemann Sums
• The epsilon-delta definition of the limit of a function
• Types of Solutions to Systems of Linear Equations
• Eigen values and Eigen vectors
• The Fundamental Theorem of Calculus
• The solutions to differential equations
Keywords
Interactive mathematics, TI-Nspire, inquiry-base learning, calculus, linear algebra.
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D033: Exploring Rose Curves withthe TI-Nspire Calculator
James PettyDepartment of Educational StudiesUniversity of Tennessee at Martin
USA
Workshop for the TI-Nspire & Derive Strand
Abstract
This workshop will utilize the TI-Nspire calculator to gain an understanding for therole of the values of a and n in the equation r = a sin(n). It will demonstrate how toprovide students an opportunity to explore changing the values and their impact uponthe curve. By examining the changes that occur students will be able to predict thenumber of petals and their length by examining the polar equation. This will enablestudents to explore and understand the relationship between the equation of a rosecurve and the equation of a sinusoidal function. By using sliders to observe the effectof changing the values of a and n students will learn to generalize the roles of aand n in the polar equation. This is accomplished by grabbing a point and dragging italong a sinusoidal function. As the point is dragged, the corresponding polar equationwill be formed. This allows students to compare the equations of the function and therose curve and make generalizations about the relationship between the two equations.We will also discuss the impact of having students write equations of rose curves whengiven information about the petals of the curve.
Keywords
Exploring with TI-Nspire, Rose Curves, Trigonometric translations.
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TICEMUS 2010 (Spanish Strand)
T001: La calculadora ClassPadcomo recurso didactico para laensenanza de las Matematicas.
Interpretacion matematica a travesde la calculadora
Agustın CarrilloInstituto de Ensenanza Secundario Sierra Morena
Andujar (Jaen)Espana
Taller para TICEMUS
Resumen
La calculadora grafica y simbolica, en especial la ClassPad, constituye un excelenterecurso didactico para el aula de matematicas que facilita el trabajo y permite al alum-nado adquirir los conocimientos del currıculum de ESO y sobre todo de Bachillerato ytambien para su uso en Universidad, de una manera mas acorde con la epoca actual enlas que las TIC estan presentes en cualquier tarea o proceso.
A partir de ejemplos y actividades el profesorado conocera las posibilidades que ofreceesta herramienta como recurso didactico. En cada una de las propuestas de trabajode este taller se mostraran las ventajas de la calculadora grafica para la resolucion deproblemas sobre todo de problemas no graficos pero en los que las ventajas de tenerlas graficas de manera facil y directa, pueden ser de una gran ayuda para trabajar lainterpretacion matematica y favorecer el desarrollo de los contenidos en el aula.
Por ejemplo, para descomponer en factores un polinomio lo normal es utilizar Ruffini,dejando a un lado el significado de la representacion grafica de la funcion polinomica quese quiere descomponer. Al no relacionar la grafica con la funcion polinomica hace queel alumnado no relaciones conceptos como factor, raız, ceros de una funcion o cortescon los ejes. Por lo que en el taller que proponemos intentaremos relacionar conceptosaprovechando las posibilidades graficas que nos ofrecen las calculadoras.
Otro ejemplo podrıa ser el calculo de lımites. A ningun alumno se le ocurre repre-sentar la funcion para determinar el valor del lımite por lo que lo mas rapido es aplicarel metodo correspondiente para hallar su valor. Por lo que disponer de una calculadoragrafica permite trabajar el concepto de lımite a partir de la representacion de la funcion.
Ejemplos similares a los anteriores se propondran en esta presentacion sobre la calcu-ladora Classpad para la que dispondremos de las calculadoras necesarias facilitadas porla Division Didactica de CASIO.
Palabras clave
Calculadora, currıculum, ESO, Bachillerato.
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T002: Esquemas de Argumentacionen la Matematica Escolar
Homero FloresColegio de Ciencias y Humanidades
UNAMMexico
Comunicacion para TICEMUS
Resumen
La capacidad de razonamiento matematico en la que se incluye el razonamiento de-ductivo y el reconocimiento de demostraciones matematicas esta en la base de la va-lidacion del conocimiento matematico. Ademas, este tipo de razonamiento, fuera delambito escolar, sirve como una ayuda en la toma fundamentada de decisiones. Por elloes de suma importancia que se desarrolle esta capacidad en nuestros estudiantes desdelos niveles basicos. Un esquema de argumentacion es la forma de razonamiento que unindividuo pone en juego durante una practica argumentativa, entendida esta como elconjunto de acciones y razonamientos que se ponen en juego cuando se trata de validaro de explicar un resultado. En la presente comunicacion se caracterizaran los esquemasde argumentacion que se utilizan cuando estudiantes y profesores se enfrentan a situa-ciones de construccion geometrica con software dinamico, en las que se deben explicaro justificar sus resultados. Ası mismo, se vera como se relaciona el uso de la funcion dearrastre del software con los esquemas de argumentacion.
Palabras clave
Esquemas de argumentacion, Demostracion matematica, Software dinamico.
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T003: Matematicas en EntornosVirtuales de Aprendizaje. Caso
WIRIS en Moodle
Ramon EixarchMaths for More
BarcelonaEspana
Comunicacion para TICEMUS
Resumen
La familia de herramientas WIRIS es ampliamente conocida en Espana especialmentepor WIRIS CAS, sistema de calculo matematico y representacion grafica, disponibleen diferentes portales educativos publicos. Esta familia de soluciones para educacionmatematica incluye ademas un editor de formulas y un sistema de creacion de cues-tionarios matematicos.
Estan en marcha en Espana y resto de Europa planes para fomentar el uso de con-tenidos y herramientas digitales en la mejora de la educacion. Los Entornos Virtualesde Ensenanza-Aprendizaje (EVEA) deben jugar un rol importante en este proceso decambio. En especial, las herramientas de evaluacion del alumno se presentan como unode los campos donde mas actividad y progreso se producira.
En nuestra presentacion daremos una vision general de las herramientas WIRIS. Paramostrar las posibilidades de las nuevas herramientas digitales en un EVEA se presentarael uso de estas herramientas en Moodle, el EVEA de mayor implantacion en Espana.
Palabras clave
WIRIS, EVEA, Moodle, evaluacion automatica.
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T004: Manuales Electronicos para laEnsenanza de la Geometrıa en 2o de
Bachillerato
G. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla y P. RodrıguezDepartamento de Matematica Aplicada
Universidad de MalagaEspana
[email protected], jl [email protected], [email protected], [email protected] [email protected]
Comunicacion para TICEMUS
Resumen
En los ultimos anos, con la implantacion de Centros TIC en Andalucıa, la utilizacionde manuales electronicos en las asignaturas de Matematicas de se ha impuesto a la de lostradicionales libros de texto. Sin embargo, si se analizan los materiales disponibles en elArea de Matematicas (vease Descartes, libros de texto electronicos. . . ), todos presentanla misma problematica: la edicion de las formulas matematicas. El hecho de estarelaborados en formato de paginas web hace que el aspecto no sea el mas deseable paraunos materiales de calidad: las formulas hay que extraerlas del texto en renglones aparteen formato dibujo, lo que crea incluso deformaciones en los textos para “cuadrar” lasformulas, ademas del tamano considerable de los ficheros (lo que retarda la carga de laspaginas). Los componentes de este equipo de profesores siempre hemos tenido claro lanecesidad de crear “autenticos” manuales electronicos de calidad, que tengan un disenoque permita su utilizacion en pantalla y que resulten atractivos para su uso, tanto enclase para las explicaciones como para el estudio personal del alumno.
Este grupo de profesores lleva varios cursos academicos experimentando, tanto enniveles de Secundaria como en niveles universitarios, con diversos materiales que palıenestas deficiencias. La solucion que nosotros hemos adoptado es utilizar el procesador detextos cientıficos LATEXcombinado con el formato PDF. Con ello, se consiguen presenta-ciones de alta calidad, “agradables” a la vista, que “pesan” poco y preparadas para serutilizadas en la docencia de las asignaturas. Nuestras experiencias iniciales con alumnosuniversitarios nos ayudaron a pulir y definir el modelo de manuales que ahora presenta-mos. Como ejemplo, hemos elegido para esta comunicacion el tema de Geometrıa parael alumnado de 2o de Bachillerato.
Palabras clave
Manuales electronicos, Bachillerato, LATEX, Geometrıa.
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T005: Utilizacion de vıdeos en lasasignaturas de Matematicas de 2o de
Bachillerato
G. Aguilera, J. L. Galan, M. A. Galan, Y. Padilla y P. RodrıguezDepartamento de Matematica Aplicada
Universidad de MalagaEspana
[email protected], jl [email protected], [email protected], [email protected] [email protected]
Comunicacion para TICEMUS
Resumen
La modalidad de ensenanza b-learning esta generando una nueva forma de apren-dizaje, cobrando una gran importancia la planificacion, la pedagogıa, la didactica, eldiseno y el desarrollo de los contenidos a ensenar. El aprendizaje con b-learning situa alalumno en el centro de una formacion independiente y flexible. Se esta evolucionandoo migrando de un aprendizaje instruccionista a un aprendizaje constructivista, donde elalumno construye su propio aprendizaje de una manera independiente con el apoyo deun tutor o de una manera grupal, construyendo ası el conocimiento compartido.
Cuando para transmitir a un alumno un contenido concreto se desarrolla una clasepresencial, luego se puede complementar esa clase con una serie de materiales adicionales,o con un foro en el cual se debata o se resuelvan dudas sobre el contenido en cuestion.Sin embargo, cuando se quiere que el alumno realice un estudio autonomo de un tema,se hace necesario buscar alternativas que sustituyan a esa clase. La eleccion que se hagadependera, en gran medida, de la materia en cuestion que se este abordando.
Por las caracterısticas especiales del caso concreto de las Matematicas, la simpleentrega de un material escrito para que el alumno lo trabaje (solucion que se adopta enmuchas ocasiones en asignaturas virtuales) no es suficiente para que el alumno alcanceel nivel suficiente de conocimiento de un concepto determinado. Este hecho se agravacuando el alumno no tiene ningun conocimiento previo sobre la materia en cuestion.
La solucion pasa, en nuestra opinion, por la utilizacion de vıdeos a modo de “pizarravirtual”, con el fin de que los alumnos reciban las explicaciones como si de una clase setratase. La utilizacion de vıdeos tiene muchas de las ventajas de una clase presencial, conel aliciente para el alumno de que es el quien marca el ritmo de la explicacion, pudiendorepetir, parar o avanzar segun su propia velocidad de aprendizaje.
En esta comunicacion presentaremos los vıdeos que estamos elaborando para la asig-natura Matematicas II de 2o de Bachillerato.
Palabras clave
Vıdeos educativos, Bachillerato, Ensenanza virtual, pizarra virtual.
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T006: Criterios topologicos en laevaluacion y promocion delalumnado en Secundaria
M. T. DavilaI.E.S. Antonio Machado
SevillaEspana
E. M. FedrianiDepartamento de Economıa, Metodos Cuantitativos e Historia Economica
Universidad Pablo de OlavideSevillaEspana
R. MoyanoI.E.S. Miguel de Cervantes
SevillaEspana
Comunicacion para TICEMUS
Resumen
La evaluacion es un aspecto muy concreto y relevante en el proceso de ensenanzay aprendizaje. En el caso concreto de la ensenanza secundaria, la legislacion actualestablece que la evaluacion del alumnado lo es en competencias. Este novedoso sistemaconvierte en multivariante un hecho educativo que, hasta ahora, habıa sido univariante.Por otra parte, la legislacion tambien establece que la representacion final del procesoevaluativo en competencias debe ser un numero entero entre uno y diez, algo que noparece concordar con la multidimensionalidad del fenomeno que se quiere medir.
La traduccion mas usual de la evaluacion en competencias (multivariante) a un en-tero entre uno y diez (univariante) se lleva a cabo ponderando cada variable. En estetrabajo proponemos trabajar con todas las competencias simultaneamente, sin ponderar,pero aportando la posibilidad de reducir el fenomeno a una unica variable. Se utilizan,para ello, tecnicas topologicas que han sido recientemente aplicadas en el terreno de laEconomıa y que nosotros hemos adaptado para el estudio de fenomenos educativos, trasimplementarlas en el ordenador.
Asimismo, podemos contemplar la decision de promocion o titulacion del alumnocomo un caso particular de evaluacion en funcion de varias variables, por lo que tambienes un fenomeno tratado a la luz del metodo que proponemos.
Palabras clave
Evaluacion, promocion, multivariante, algoritmo, topologico.
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T007: Explorando la Geometrıa enEducacion Secundaria con los
Graficos de la Tortuga
Rosario VeraI.E.S. Jarifa
Cartama, MalagaEspana
Comunicacion para TICEMUS
Resumen
La programacion funcional es un estilo de programacion alternativo al paradigmade programacion mas habitual, conocido como estilo imperativo. Desde el punto devista funcional, un proceso de computo se describe como una funcion matematica quea partir de unos argumentos (correspondientes a los datos del problema) produce unresultado. La evaluacion de un programa funcional consiste simplemente en la evalua-cion de una expresion. En este trabajo usaremos Haskell, un lenguaje de programacionfuncional estandar utilizado habitualmente en el mundo academico. Comparados conlos programas imperativos, los funcionales resultan mas concisos y faciles de entender yse expresan con una notacion muy cercana a la notacion matematica habitual. Es porello que consideramos estos lenguajes mas adecuados para presentar conceptos propiosde la programacion de ordenadores a alumnos y alumnas de secundaria, contribuyendoademas a que entiendan la importancia del uso de la notacion formal matematica.
Los graficos de la tortuga fueron introducidos originalmente por Seymour Paperten el lenguaje de programacion LOGO. Basicamente, permiten describir graficos di-rigiendo los movimientos de una tortuga virtual la cual deja un rastro correspondientea su movimiento sobre un plano. El modelo geometrico en el que se basan estos graficoses euclıdeo, ya que se usan traslaciones y rotaciones relativas a la posicion actual de latortuga, lo que permite al estudiante describir un grafico como si estuviese inmerso enel, facilitando de este modo la descripcion del mismo.
En este trabajo presentaremos una librerıa funcional para los graficos de la tortugaque hemos desarrollado para el lenguaje de programacion funcional Haskell. En primerlugar, describiremos las distintas funciones que constituyen el interfaz de nuestra librerıa.Posteriormente, mostraremos a traves de ejemplos como se puede usar esta para ilustrargraficamente distintos conceptos matematicos, fundamentalmente del area de geometrıa,que forman parte del currıculo de ESO y bachillerato.
El objetivo de nuestra propuesta es doble: por un lado, contribuir al desarrollo de lacompetencia digital, en tanto y cuanto estamos trabajando con el ordenador, utilizandoloen este caso, como una herramienta grafica que ayuda a que nuestro alumnado visualice
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mucho mejor el significado de los movimientos en el plano y en el espacio. Por otro ladopretendemos con esta actividad intentar desterrar de la cabeza de nuestros alumnos yalumnas la vision de que las matematicas son algo meramente academico. Les mostramosque conceptos matematicos, en este caso de geometrıa, son la base para la sıntesis deimagenes por ordenador, y por tanto resultan esenciales para el desarrollo de materialesaudiovisuales cotidianos, como pueden ser los vıdeo-juegos, pelıculas de animacion, etc.
Palabras clave
Competencia digital, Geometrıa, graficos por ordenador, programacion funcional.
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T008: TutorMates: Un nuevoparadigma en la ensenanza de las
Matematicas
A. Pascual y A. LimonEquipo de desarrollo TutorMates
Addlink Research, Universidad de Cantabria, Universidad de La RiojaEspana
Taller para TICEMUS
Resumen
¿Que es TutorMates? Es una herramienta unica para la docencia de las Matema-ticas dirigida a profesores de la Educacion Secundaria y el Bachillerato. El desarrollo delmaterial docente y laboratorios se adecua estrictamente a los planes de estudio, ası como alas orientaciones pedagogicas vigentes. Se presenta como la aplicacion completa para serusada en el aula ya que integra todos los contenidos curriculares en una plataforma queincorpora, a la vez, herramientas de calculo cientıfico, estadıstico y geometrico ası comola gestion educativa de dichas herramientas en lecciones del currıculo de Matematicas.Fomenta la interactividad y el aprendizaje con el profesor ası como el aprendizaje atraves de la experimentacion en el aula ya que ademas, contempla la posibilidad de usode perfiles diferenciados para profesor y alumno: el alumno en su ordenador personalusa TutorMates para hacer el seguimiento de una leccion correspondiente a su nivel,y el profesor, para disenar o adaptar dicha leccion. Es una aplicacion multiplataformay multiidioma que integra teorıa, ejercicios, problemas y tests de autoevaluacion detodo el currıculo de las Matematicas de la E.S.O. y Bachillerato. Aporta infinidad deejemplos, siendo un programa intuitivo, facil de usar que permite alcanzar las com-petencias de la educacion matematica e informatica a traves de comandos y escenasinteractivas. TutorMates ha sido desarrollado por la empresa Addlink junto con laUniversidad de Cantabria y la Universidad de La Rioja y ademas tiene un acuerdo es-trategico con el Instituto GeoGebra Internacional en el cual se establece el derecho autilizar la aplicacion GeoGebra dentro del software de TutorMates. Cabe destacar queTutorMates recibio el reconocimiento del Ministerio de Ciencia e Innovacion median-te una subvencion a traves del Proyecto CONSOLIDER y otra, a traves del ProyectoTRACE. Ambos proyectos premian la innovacion e integracion de la actividad empresa-rial en colaboracion con la Universidad publica, para la incorporacion del conocimientoen el mercado.
El taller que se propone tiene como objetivo presentar la aplicacion a profesores dematematicas y demostrar como se puede utilizar TutorMates como entorno de apren-dizaje y de trabajo en el aula. TutorMates adapta el entorno de trabajo a los diferentes
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temas y niveles: editor de ecuaciones, representacion de graficos, geometrıa, tratamientosestadısticos, etc. Cada tema esta compuesto por modulos interactivos que integran teorıa,ejercicios, problemas y autoevaluacion. El taller se iniciara con una breve presentaciongeneral del producto y a continuacion se familiarizara con la aplicacion TutorMatesmostrando siente modulos de ensenanza, adaptados a cada uno de los temas previstos.La estructura de estos modulos es la siguiente:
1. Primeros pasos con TutorMates.
2. Numeros.
3. Iniciacion al Algebra.
4. Ecuaciones.
5. Funciones y graficas.
6. Estadıstica y probabilidad.
7. Geometrıa.
Se trata de trabajar, mediante la utilizacion de TutorMates, los diferentes bloquesde contenidos que componen el curriculum en la ensenanza de las Matematicas en laE.S.O.
Palabras clave
Matematicas, educacion secundaria, currıculo, libro digital, software, ensenanza.
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T009: Matematicas 2.0 conDescartes
F. J. Rodrıguez, J. R. Galo y J. A. SalgueiroRed de Buenas PracTICas 2.0 y Proyecto Descartes
Instituto de Tecnologıas Educativas del Ministerio de EducacionEspana
[email protected] y [email protected]
Taller para TICEMUS
Resumen
En este taller, cada asistente disenara y elaborara su propio espacio virtual deMatematicas 2.0, aprendiendo a embeber en un blog escenas interactivas de Descarteso a integrarlas como recurso en Moodle, lo que le aportara autonomıa para realizar suplanificacion digital “didacTICa” a partir de objetos de aprendizaje existentes en la Webbajo licencia Creative-Commons.
Las actividades a realizar durante el Taller estaran secuenciadas en un curso Moodlecon acceso para invitados, organizado en los siguientes temas o secciones:
1. Descargas: Los participantes podran descargarse a priori, durante el desarrollodel taller o a posteriori, el tutorial y los materiales necesarios para la integracionde Descartes en Blogger y Moodle.
2. Tutoriales: Acceso al tutorial en lınea.
3. Ejemplos de materiales de Descartes en Blogger: Desde aquı puede apre-ciarse que se pretende y que se espera conseguir entre Descartes y Blogger, conejemplos disenados para los distintos niveles educativos a los que se dirige esteTaller, a saber, Tercer Ciclo de Primaria, Secundaria Obligatoria, Bachillerato yUniversidad.
4. Practicas en Blogger: Actividades concretas para los participantes, orientadaspara la diversidad a la que se dirige, pues deberan llevarla a cabo en su propioblog de Blogger o en el creado al efecto. Ademas, se proponen enlaces a cursosespecıficos para crear un blog en Blogger una vez concluido el taller, motivando yasesorando al profesor o profesora que no lo posea.
5. Ejemplos de materiales de Descartes en Moodle: Se muestra que puede ha-cerse con Descartes en Moodle: escena incluida mediante etiqueta, unidad didacticacompleta en ventana emergente, uso del editor de contenido, Descartes en un cues-tionario, actividad de aprendizaje colaborativo con un foro agregado para el debatey tareas individuales.
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6. Practicas en Moodle: Actividades concretas para practicar la inclusion de ma-teriales de Descartes en un curso moodle, el del participante o el ofrecido por laorganizacion de este Taller. Finalmente, se enlaza a un curso Moodle para auto-formacion del profesorado que desconozca la administracion con perfil docente enMoodle.
Palabras clave
Escuela 2.0, TIC, Proyecto Descartes, Interactividad, Blog, Moodle.
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T010: Descartes en Wikispaces
F. J. Rodrıguez, J. R. Galo y J. A. SalgueiroRed de Buenas PracTICas 2.0 y Proyecto Descartes
Instituto de Tecnologıas Educativas del Ministerio de EducacionEspana
[email protected] y [email protected]
Comunicacion para TICEMUS
Resumen
Gracias a los materiales del Proyecto Descartes, podemos ensenar matematicas conla ayuda de una serie de propuestas interactivas de aprendizaje, llamadas escenas, quehacen posible un aprendizaje activo de las matematicas, en el que tiene cabida la expe-rimentacion, la relacion de conceptos y la investigacion.
Descartes, por su capacidad de interaccion y su flexibilidad, se adapta a los distintosestilos y ritmos de aprendizaje de los alumnos y alumnas, por lo que permite dar respuestaa la diversidad que encontramos en las aulas.
Podemos incrementar todavıa mas el potencial educativo de Descartes, haciendo usode metodologıas basadas en la cooperacion. Con estas metodologıas la investigacion y eldescubrimiento, propio del trabajo con Descartes, no se realiza de forma individual sinocolectiva, y al realizarse en grupo debe ir acompanada de reflexion y dialogo, ası comode la sıntesis de lo aprendido en un lenguaje adecuado. De este modo, los alumnos yalumnas son capaces de llegar mas lejos de lo que lo harıan individualmente.
En esta comunicacion se expone el modelo teorico que permite usar materiales delProyecto Descartes con metodologıas basadas en la colaboracion, y que fomentan el de-sarrollo de competencias, en especial las relacionadas con el tratamiento de la informaciony la autonomıa e iniciativa personal. Se muestra como integrar Descartes en un entornocolaborativo, Wikispaces. Por ultimo, se describe una experimentacion con Descartesen wikispaces, en la que los alumnos y alumnas crean materiales de forma cooperativa yproponen actividades.
Palabras clave
Escuela 2.0, TIC, Proyecto Descartes, Interactividad, Blog, Moodle.
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T011: Programacion Recreativaversus Matematica Recreativa
M. Ruiz y B. C. RuizIES Fuente Lucena
Alhaurın el Grande, MalagaEspana
Comunicacion para TICEMUS
Resumen
La Programacion Recreativa (PR) es la disciplina que motiva el estudio de la pro-gramacion de computadores a traves de problemas ludicos. Los problemas tıpicos queestudia esta disciplina son similares a los de la Matematica Recreativa (MR), lo que llevaa veces a identificar ambas disciplinas. Sin embargo, los metodos de una y otra puedenllegar a ser muy diferentes. El objetivo de la PR es escribir programas, mientras que enMR podemos ayudarnos de estos para enunciar conjeturas sobre la solucion.
Desde una vision educativa, son muy interesantes los puzles logicos y los problemasde teorıa de numeros elemental (aquellos que se formulan con un conocimiento elementalo basico del algebra). El estudio de ciertos puzles logicos conduce al estudio de pro-blemas de teorıa de numeros elemental, resolviendose a traves de resultados conocidosde los segundos, o incluso siendo equivalentes a problemas NO-elementales de teorıade numeros elemental (la conjetura de Goldbach o la existencia de numeros perfectosimpares). Exponemos en este artıculo un par de problemas que ilustran esta conexion:El problema PS de Freudenthal, y la forma general de los numeros de Zumkeller (aquellosnaturales tales que sus divisores positivos pueden repartirse en dos grupos disjuntos conidentica suma).
La escritura de programas de ordenador utilizando un lenguaje de programacionproximo a la notacion matematica (como por ejemplo Haskell) constituye el primerpaso a la solucion del problema, ya que es posible escribir con poco esfuerzo sencillosy elegantes programas tan proximos a la descripcion del problema que su correccion esinmediata. Por ello, la escritura de estos programas permite realizar un analisis rapido delproblema. Sin embargo, para los problemas que analizamos, tales programas pueden sercasi inutiles si son extraordinariamente ineficientes. A veces, un estudio previo siguiendopautas de la MR conduce a propiedades de facil comprobacion computacional, y que“casi” caracterizan a las soluciones; tales propiedades permiten escribir programas muyeficientes.
Palabras clave
Programacion recreativa, matematica recreativa, lenguajes funcionales, Haskell.
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T012: Los materiales de Descartescomo catalizadores de la reflexion
metodologica
Jose R. GaloCoordinador del Proyecto Descartes
Instituto de Tecnologıas Educativas del Ministerio de Educacion de Espana
Taller para TICEMUS
Resumen
Descartes es un proyecto educativo, promovido por el Ministerio de Educacion delGobierno de Espana. El fin basico del proyecto Descartes es aprovechar las ventajas delordenador y de Internet para ofrecer al profesorado y al alumnado una nueva forma deensenar y aprender Matematicas, promover la innovacion en Educacion Matematica me-diante las Tecnologıas de la Informacion y de la Comunicacion (TIC). Como medio parala consecucion de este objetivo se ha desarrollado una herramienta: el nucleo interactivopara programas educativos (nippe), denominado tambien “Descartes”, con el que se con-sigue disenar escenas en las que mediante controles numericos y graficos se interactua conel sistema informatico obteniendose una respuesta acorde a la accion realizada, permi-tiendo la simulacion o modelacion de conceptos y procesos en un contexto de aprendizajeactivo y un entorno de representacion grafica bi y tridimensional. Desde que en julio de1998 se inicio el proyecto Descartes, el nippe ha sido objeto de un desarrollo continuo yprogresivamente ha ido incrementado su potencial creativo y formativo, y el interes enel profesorado y alumnado. Se contabilizan mas de un millon de visitas mensuales en elservidor web del proyecto: http://recursostic.educacion.es/descartes.
Con Descartes se han desarrollado materiales educativos interactivos que contribuyenal aprendizaje significativo y a la formacion en competencias, materiales que puedenser modificados y adaptados por el profesorado –tambien de manera interactiva y sinnecesidad de aprender un lenguaje de programacion– segun su interpretacion docenteo necesidad didactica. En la actualidad se cuenta con un recubrimiento curricular dela Educacion Secundaria Obligatoria y del Bachillerato, avanzandose ahora tambien enPrimaria y Universidad.
En la labor de implementacion didactica de las TIC este recubrimiento curricular seplantea desde dos perspectivas o lıneas de actuacion y accion docente no excluyentes:
• Como banco de recursos auxiliar a partir del cual el profesorado selecciona, orga-niza, secuencia y programa las actividades adecuadas a las necesidades formativasde su alumnado, es decir, contextualizando los materiales. Esto requiere ciertainmersion y tarea informatica ajena al fin educativo basico y consecuentemente latecnologıa se hace visible en la labor docente, limitando su uso generalizado.
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• Como libros digitales interactivos, es decir, con secuencias didacticas y planes deformacion pre-elaborados, que incorporan y hacen suyos atributos propiamente TICcomo la interactividad, integracion de recursos audiovisuales y de codigos, registrode la informacion, capacidad de simulacion, etc. Esto aporta cierta “invisibilidadtecnologica”, contribuye al uso iniciatico de las TIC en el aula e introduce demanera natural un cambio metodologico.
Los materiales de Descartes se adaptan a las diferentes metodologıas y modelos edu-cativos. En esta comunicacion se muestra la diversidad de estos recursos incidiendoprincipal y esencialmente en las propuestas metodologicas que de manera natural surgenal usarlos en el aula. Partiendo de ejemplos donde el recurso se usa solo como elemento derepresentacion grafica en modelos tradicionales transmisores –como mera pizarra digital–,progresivamente se ira detallando y profundizando en el potencial didactico que la inter-actividad alumnado-maquina aporta en la construccion del conocimiento significativo,potenciando simultaneamente tanto la “autonomıa personal”, el “aprender a aprender”y la formacion competencial propia, como la necesaria colaboracion y cooperacion enla consecucion de objetivos colectivos o sociales. Igualmente se expone como la uti-lizacion de los recursos de Descartes adentra al profesorado en una reflexion sobre sulabor docente, cuestiona acerca de las rutinas profesionales, promueve de manera natu-ral un soporte para el analisis y desarrollo metodologico y establece nuevas e intensasrelaciones docentes-discentes.
Palabras clave
Proyecto Descartes, TIC, materiales educativos, metodologıa.
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T013: Nuevas tecnologıas yensenanza: Introduccion
constructiva, geometrica y dinamicadel concepto de derivada
C. Caballero y J. BernalI.E.S. Portada Alta
MalagaEspana
Comunicacion para TICEMUS
Resumen
Mucho se habla de las TIC y de las nuevas posibilidades que abre en educacion,en particular es especialmente destacable el papel que puede jugar en las clases comouna herramienta util para mejorar la comprension de conceptos matematicos, poderrealizar conjeturas, economizar el tiempo del docente en la busqueda de recursos y servircomo una nueva vıa de comunicacion que retroalimente mas rapidamente este procesode mejora continua del proceso.
En esta comunicacion vamos a presentar una unidad interactiva que se centrara enla introduccion del concepto de derivada en bachillerato, su asimilacion y adquisicioncontraponiendolo con la vıa clasica de la derivada motivada por el estudio de la tangente ala grafica de una funcion; que es un tanto artificial a nuestro juicio abordar ası la cuestion,y que normalmente termina chocando para el alumnado con el clasico teorema que nosdice que “derivabilidad implica continuidad” y esas ideas intuitivas en forma de recetamagica como “suavidad”, “forma un pico”, etc. Este nuevo enfoque sera posible graciasal uso de las TIC, en particular sera muy destacado el uso del software gratuito Geogebrapues permite profundizar si uno lo prefiere en el propio enfoque como lımite, pero ademaspermite mostrar al alumnado conceptos como cercanıa, entorno o pendiente infinito sinesfuerzo imaginativo. Nuestra unidad con un afan construccionista, donde el alumnadodebera aprender mediante la experimentacion, consistira en una pseudoinvestigacionguiada donde se repase desde la elemental recta (dos puntos en el plano, punto y vectordirector, expresiones . . . ) en el que la propia formula cobrara vida. Seguidamente semezclara con una experiencia fısica (girar una pelota atada a una cuerda) lo que llevaraa profundizar en la relacion vector-recta y que finalmente podremos ver modelada ymanipulable tambien. La ventaja de este soporte es que los propios alumnos podrancolgar sus dudas y a su vez permitirıa mejorar el propio formato, algo que es impensableen el formato tradicional de libro.
Finalmente se introduce el concepto de funcion derivable en un punto como ser lo-calmente como una recta que se basa en la idea que si pudieramos ampliar un entorno
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de la grafica que fuera cada vez mas pequeno, claramente vemos una recta (por lo queser continua y tener pendiente sera entonces algo natural). Este modelo, cuyo interesen su construccion ya es de por sı interesante para el alumnado de este nivel lo hemosbautizado como “la lupa matematica”. Este enfoque en cursos superiores universitariosde hecho este es el que se suele usar en Calculo diferencial. Despues de observar ejemplosbasicos de discontinuidad y como consecuencia de haber ido trabajando sobre entornoscon la lupa, el hecho de plantear la busqueda de dicha recta tangente como un lımite esentonces natural y hasta la infinitud tiene una interpretacion nada forzada y obvia. Sedestierran coletillas como “suavidad” y “picos”, para tener un significado construido y ala vez hemos repasado varias ideas de geometrıa, con toques de algebra y desarrollandoel sentido crıtico que han de tener los alumnos en esta rama del conocimiento comouna oportunidad para aprender. Seguidamente se presentan modelos de los resultadosclasicos de derivabilidad.
Palabras clave
TIC, interactividad, derivada, conceptos, Geogebra.
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T014: Matematicas Manipulativascon DESCARTES
J. L. Alcon, J. R. Galo y J. G. RiveraProyecto Descartes
Instituto de Tecnologıas EducativasMinisterio de Educacion de Espana
Comunicacion para TICEMUS
Resumen
En etapas formativas tempranas el calculo, el fomento del razonamiento y en generalel desarrollo cognitivo se fundamenta y formaliza mediante juegos, retos y actividadesmanipulativas que promueven el ingenio y la elaboracion de estrategias. La manipulacionnos mantiene en nuestro entorno y contexto inmediato, llevando al alumnado a la inte-riorizacion y al aprendizaje significativo. Se aprenden los conceptos mediante canciones,dibujando, recortando, moldeando plastilina. . . se llega a las Matematicas a traves deotra matematica. En etapas formativas posteriores esos procedimientos suelen decaeraconteciendo un predominio de la formulacion abstracta que se centra en el aprendizajede tecnicas y planteamientos generales, abordando una resolucion global y generica. Peroen este modelo la abstraccion aleja de entorno y contexto proximo, lo que dificulta lainteriorizacion y el aprendizaje de los conceptos.
En la situacion descrita se ubica la experiencia que presentamos y en la que seplantea como la utilizacion de mediadores virtuales favorece globalmente tanto la for-macion matematica abstracta como la concreta. Los mediadores virtuales facilitan ypermiten un aprendizaje significativo basado en la propia experimentacion, como basepara adquirir experiencia, todo canalizado y/o catalizado por ellos. Uno de esos mode-los se construye desde la aplicacion de los mediadores virtuales desarrollados dentro delproyecto Descartes del Ministerio de Educacion de Espana, un proyecto cuyo objetivoprincipal es promover nuevas formas de ensenanza y aprendizaje de las Matematicasintegrando las TIC en el aula como herramienta didactica, pero entendiendo un aula nosolamente como el grupo de alumnos y alumnas ubicados en un espacio fısico cerrado,propio de una ensenanza presencial y reglada, sino tambien como un espacio virtualque abarca la “aldea global” nombrada por McLuhan y Powers. Utilizando diversasunidades didacticas interactivas de Descartes mostramos como se produce el aprendizajeen el modelo manipulativo planteado.
Palabras clave
Descartes, mediador virtual, razonamiento, manipulacion, aprendizaje, TIC.
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T015: Un experimento de ensenanzapara la construccion del concepto de
integral definida usando unprograma de geometrıa dinamica
C. Aranda y M. L. CallejoI.E.S. Pere M Orts i Bosch
Benidorm, AlicanteEspana
Facultad de EducacionUniversidad de Alicante
Espana
[email protected] y [email protected]
Comunicacion para TICEMUS
Resumen
En esta comunicacion se presenta el diseno de un experimento de ensenanza sobrela introduccion del concepto de integral definida con estudiantes de Bachillerato (16-18 anos), utilizando el programa de geometrıa dinamica Geogebra. El objetivo de esteexperimento es que los estudiantes construyan el concepto de integral definida desde unenfoque centrado en la resolucion de problemas. Se han disenado tareas interactivas enlas que se usan simultaneamente representaciones analıticas y geometricas, que permitenmodificar los parametros manipulando controles graficos. Se utiliza una hoja de calculopara que los estudiantes puedan aproximar las areas bajo una curva, con el objetivo deque sean conscientes del proceso que lleva a obtener el area buscada con el error fijado yaproximarse de manera intuitiva al concepto de lımite, subyacente al de integral definida.Se propone tambien tareas de lapiz y papel, en las que juega un papel fundamental laresolucion de problemas y los procesos de generalizacion.
Siguiendo el proceso historico, se parte del calculo del area de figuras geometricas en eldominio de la geometrıa sintetica, para despues abordar el area bajo una curva en el de lageometrıa analıtica. Se aproximan las areas por metodos geometricos y series numericas,introduciendo el lımite de manera formal solo al final, definiendo la integral. Una vezestablecida la diferencia entre area e integral, se aborda el estudio de sus propiedades yal final se llega al Teorema fundamental del Calculo integral y a la regla de Barrow.
Palabras clave
Experimento de ensenanza, Integral definida, Geometrıa dinamica.
178
T016: Las Matematicas ante elproblema de incomprension de un
idioma
Ana LopezI.E.S. Los BolichesFuengirola, Malaga
Espana
Comunicacion para TICEMUS
Resumen
Los grupos de Compensatoria estan formados por alumnado extranjero que por desco-nocimiento del idioma no pueden afrontar nuestro sistema educativo. Estan integradospor un maximo de 15 alumnos. La idea que se persigue es que para facilitarles laincorporacion al sistema, se les realiza una adaptacion. Esta adaptacion consiste enque se les reduce el numero de asignaturas (tambien el numero de profesores): las dosasignaturas de mas peso pasan a ser ambito cientıfico y ambito lingıstico. En ellas, elobjetivo es aprender el idioma lo mas rapido posible, de modo que en cuanto puedanseguir el desarrollo de una clase, se incorporen a su grupo original. El valor de laevaluacion en compensatoria solo es algo informativo para los padres; no tiene valoracademico. Las asignaturas del curso estan evaluadas negativamente hasta que el alumnono se incorpora al grupo de referencia.
Durante el curso 06/07 impartı el ambito cientıfico de un grupo de 1o ESO de Com-pensatoria. Para el desarrollo de esta labor, en el marco de un grupo de trabajo llevado acabo en colaboracion con otros Centros Educativos, elaboramos material especıfico paraestas clases.
El desarrollo de mi labor docente en ese curso academico me hizo reconocer un graninconveniente: este tipo de alumnado requiere de la atencion continua por parte delprofesor. No “saben” trabajar sin las indicaciones y ayuda constante del profesorado.
Esta situacion se agravaba cuando el alumno volvıa a su clase de referencia. Solo losalumnos realmente motivados, o aquellos que conseguıan un cierto dominio del idioma,lograban superar las materias del curso. El resto no adquirıa la suficiente autonomıapara seguir el desarrollo normal de una clase.
Con el fin de paliar estas deficiencias, decidı utilizar las Tecnologıas de la Informaciony la Comunicacion como fuente y recurso de autonomıa para el alumnado. Esta expe-riencia se puede extender, evidentemente, a todo tipo de alumnado que requiera de unaadaptacion curricular especial.
Las actividades que se elaboraron no fueron muy distintas, en cuanto a contenidos,a las que habıamos utilizado en el curso anterior. Sin embargo, diferıan sustancialmenteen cuanto a formato y tratamiento.
179
El alumnado trabaja las actividades en un formato mas atractivo (por ejemplo ela-boramos cuestionarios con Hot Potatoes). Con el fin de que la realizacion de este tipo deactividades fuera lo mas util posible, despues les exigıamos que lo reflejasen por escrito.En estas actividades introducimos tambien el uso de Computer Algebra Systems comoDERIVE, ası como traductores y diferentes recursos de Internet. Ası los alumnos podıantrabajar tanto en el aula como desde casa.
La disponibilidad actual en nuestros centros de aulas TIC ası como de acceso aInternet en las casas, hace que nuestro planteamiento pueda llevarse a cabo de manerafactible.
En esta comunicacion nos centraremos en mostrar el tipo de material con el que estu-vimos trabajando, para finalizar analizando las conclusiones obtenidas tras su utilizaciony comentando las expectativas futuras.
Palabras clave
Compensatoria, Secundaria, TIC, DERIVE, recursos en Internet.
180
T017: Un Acercamiento al Calculodesde la Realidad Virtual con
DESCARTES
J. L. Alcon, J. R. Galo y J. G. RiveraProyecto Descartes
Instituto de Tecnologıas EducativasMinisterio de Educacion de Espana
Comunicacion para TICEMUS
Resumen
Representar las aplicaciones del calculo matematico es un reto permanente de nosotroslos docentes. A veces recurrimos a las metaforas o a las analogıas, otras veces copiamoslas representaciones de un texto y las comunicamos a traves de dispositivas o fotocopias,pero, en la mayorıa de las veces, nos atrevemos a esbozarlas en una pizarra. En esteultimo caso la comunicacion pierde efectividad por su alejamiento notorio de la realidadque se pretende representar.
En esta comunicacion demostraremos como es posible romper con esta practica tradi-cional que, increıblemente, poco se diferencia de las realizadas en siglos pasados. Uti-lizando una unidad didactica interactiva de Descartes mostraremos como se representanvolumenes de revolucion (con o sin seccion hueca) y su vinculacion a la integral definida.La representacion de la realidad desde la virtualidad muestra su poder representacionala traves de una escena interactiva del calculo estructural, que permite identificar losatributos imposibles de lograr en una imagen plana. Es el retorno al pensamiento 3Dcon Descartes que pretendemos recuperar, en tanto que el pensamiento espacial es unelemento basico en el pensamiento cientıfico. Ingenieros, escultores y arquitectos sonpensadores espaciales que requieren representaciones mas cercanas a esa realidad queintervienen.
Palabras clave
Descartes, calculo, representacion, pensamiento espacial, TIC.
181
T018: Influencia del empleo de unametodologıa didactica basada en eluso de las TIC en el tratamiento
efectivo de la diversidad en el aulaordinaria de matematicas
J. J. Larrubia (coordinador),Y. Padilla, M. Alvarez, J. A. Argote, L. Troughton, N. Puertas, V. Perez, M.D. Domene, M. A. Medina, E. de la Plata, R. Ros, R. Loring y E. Garcıa,
I.E.S. Universidad LaboralMalagaEspana
Comunicacion para TICEMUS
Resumen
El trabajo que presentamos tiene por objeto explorar y evaluar la eficacia de medidasdidacticas especıficas para la atencion educativa a la diversidad en el aula de matematicasde ESO y Bachillerato, en contextos educativos ordinarios, desde una perspectiva inclu-siva, a traves de la eliminacion/reduccion de posibles barreras de aprendizaje y del au-mento de la participacion curricular de todo el alumnado. Estas medidas estan basadasen el uso de herramientas tecnologicas para la informacion y la comunicacion.
El estudio pertenece a la lınea investigacion para la innovacion curricular en laaccion en el aula de matematicas, que esta orientada a la mejora de las competen-cias matematicas de todos los alumnos y alumnas y a las necesidades generadas por ladiversidad del alumnado dentro del marco curricular ordinario, a traves de un procesode investigacion basado en una metodologıa mixta que se desarrolla en varios pasos.Entre ellos destaca la realizacion de un estudio empırico sobre la eficacia del tratamientodidactico inclusivo a traves de un proceso de innovacion curricular en la accion en el aulade matematicas (Gonzalez, 2003; Galan, 2003; Padilla, 2003; Gonzalez, 2004; Rodrıguez,2004; Larrubia, 2006).
Se han obtenido resultados que prueban suficientemente la bondad de las hipotesisestablecidas y han permitido alcanzar los objetivos propuestos: a) Identificar y delimitarlas posibles barreras de aprendizaje y de participacion curricular del alumnado del grupoobjeto de estudio. b) Construir un modelo didactico de caracter inclusivo y curricular,basado en el empleo complementario de un diseno curricular virtual de la asignatura, yuna metodologıa docente centrada en la utilizacion de recursos y materiales interactivos.c) Comprobar que el modelo didactico es viable para el desarrollo curricular ordinarioen el aula de matematicas.
182
Palabras Clave
Atencion a la diversidad, inclusion, equidad, Mathematical literacy, competencias basicas.
Observaciones
Proyecto de Investigacion Educativa PIV-075/06. Subvencionado por la Consejerıade Educacion de la Junta de Andalucıa. Realizado por el grupo de investigacion en elque colaboran los profesores investigadores perteneciente a la UMA: J.L. Gonzalez, J.L.Luque, J. L. Galan y P. Rodrıguez.
183
T019: Introduccion geometrica delos numeros reales con Geogebra
A. Caro y A. M. MartınI.E.S. La Campina, Arahal (Sevilla),
Departamento de Economıa, Metodos Cuantitativos e Historia EconomicaUniversidad Pablo de Olavide de Sevilla
Espana
[email protected] y [email protected]
Comunicacion para TICEMUS
Resumen
En esta comunicacion se presenta una propuesta didactica basada en una web dina-mica que vertebra el proceso de ensenanza-aprendizaje de los numeros reales en el marcode Bachillerato. La aplicacion Web interactiva esta desarrollada con el software libreGeogebra. La propuesta repasa los conceptos de numero natural, entero y racional desdeel punto de vista geometrico. Se introduce un primer numero irracional, raız de 2, yse demuestra geometricamente que no es racional. En el mismo sentido, se presentandemostraciones geometricas de la irracionalidad de las raıces de 3, de 5 y de 6.
Por otro lado, del hecho de que las raıces de numeros naturales son soluciones deecuaciones algebraicas y aplicando el metodo de los intervalos encajados de Cantor juntocon la interactividad de Geogebra con HTML a traves de Javascript, se aplica una herra-mienta que posibilita que el alumnado descubra como pueden aproximarse los numerosirracionales algebraicos. Este metodo permite introducir los conceptos de errores deaproximacion relativos y absolutos.
Finalmente se utilizan metodos de Webquest para que el alumnado investigue si losnumeros irracionales π o e pueden ser aproximados por el metodo utilizado anteriormente,llegando al descubrimiento del concepto de numero irracional trascendente. Completandoası el estudio de los numeros reales.
El uso de las TICs, particularmente de las aplicaciones Web y de la herramienta degeometrıa dinamica Geogebra, busca despertar el interes del alumnado por el estudio dela materia. Ademas, el hecho de que el alumnado sea el actor principal del proceso deaprendizaje induce este sea significativo ayudando a la interiorizacion de los conceptospresentados. Tambien se favorece la atencion a la diversidad del alumnado al adaptarsea diversos ritmos de aprendizaje y niveles iniciales de competencia.
Palabras clave
Geogebra, Web dinamica, numeros reales, demostraciones geometricas, Bachillerato.
184
T020: Elaboracion de paginas webdinamicas con Geogebra
A. Caro y A. M. MartınI.E.S. La Campina, Arahal (Sevilla),
Departamento de Economıa, Metodos Cuantitativos e Historia EconomicaUniversidad Pablo de Olavide de Sevilla
Espana
[email protected] y [email protected]
Comunicacion para TICEMUS
Resumen
Geogebra es un software interactivo de matematica, libre, para aprender y ensenar.Reune dinamicamente geometrıa, algebra y calculo. Es elaborado por Markus Hohen-warter junto a un equipo internacional de desarrolladores. Con el se generan graficosinteractivos y son relacionados con el algebra obteniendo planillas dinamicas. Cubretodos los niveles educativos, desde el escolar mas basico al universitario, y permite laelaboracion de materiales de aprendizaje libres y gratuitos.
Se muestra lo esencial para crear paginas Web que permitan la interaccion de losusuarios con el Applet de Geogebra a traves de formularios HTML. Se analizan algunosejemplos de paginas interactivas con GeoGebra y se muestran como construir planillasdinamicas y exportarlas a HTML usando los comandos existentes en el software. Seanaliza el HTML necesario para la manipulacion del applet de GeoGebra y se abordanalgunos metodos de JavaScript que permiten comunicar las applets de aplicaciones deGeoGebra con las paginas web.
Palabras clave
Geogebra, Web dinamica, HTML, Javascript.
185
List of Participants
List of Participants
1. Abu Elwan, Reda. Oman. [email protected]
2. Ada, Tuba. Turkey. [email protected]
3. Aguilera Venegas, Gabriel. Spain. [email protected]
4. Alcon Camas, Jose Luis. Spain. [email protected]
5. Antrobus, Peter. Australia. [email protected]
6. Aranda Lopez, M Carmen. Spain. [email protected]
7. Atencio, Dariana Analeidy. Spain. [email protected]
8. Barzel, Brbel. Germany. [email protected]
9. Beaudin, Michel. Canada. [email protected]
10. Bogun, Vitaly. Russia. [email protected]
11. Bohm, Josef. Austria. [email protected]
12. Brothers, Gosia. United States. [email protected]
13. Budinski, Natalija. Serbia. [email protected]
14. Burkhead, Melissa. United States. [email protected]
15. Bustos Heredia, Jairo. Spain. [email protected]
16. Butler, Douglas. United Kingdom. [email protected]
17. Butler Lucena, Ana Isabel. Spain. [email protected]
18. Caballero Gonzalez, Carlos. Spain. [email protected]
19. Canet Espinosa, Carlos. Spain. [email protected]
20. Caro Chaparro, Andres. Spain. [email protected]
21. Carrillo de Albornoz Torres, Agustın. Spain. [email protected]
22. Casas Garcıa, Luis Manuel. Spain. [email protected]
23. Dahan, Jean-Jacques. France. [email protected]
24. Davis, Jon D.. United States. [email protected]
25. De la Vila, Agustın. Spain. [email protected]
26. Decker, June. United States. [email protected]
27. Decker, Robert. United States. [email protected]
189
28. Diaz, Olimpia. Spain. [email protected]
29. Dogan, Mustafa. Turkey. [email protected]
30. Duhaini, Hamzeh. Qatar. [email protected]
31. Eixarch, Ramon. Spain. [email protected]
32. Ellis, Wade. United States. [email protected]
33. Falcon Ganfornina, Raul Manuel. Spain. [email protected]
34. Fernandez de Guardia, Benigna Elena. Panama. [email protected]
35. Figueroa Ardila, Silvia Carolina. Spain. [email protected]
36. Fioravanti, Mario. Spain. [email protected]
37. Flores, Homero. Mexico. [email protected]
38. Flores Martos, Lydia. Spain. [email protected]
39. Galan Garcıa, Jose Luis. Spain. jl [email protected]
40. Galan Garcıa, Ma Angeles. Spain. [email protected]
41. Galo Sanchez, Jose R.. Spain. [email protected]
42. Galvez Cordero, Jaime. Spain. [email protected]
43. Garcıa Lopez, Alfonsa. Spain. [email protected]
44. Garcıa Mazarıo, Francisco. Spain. [email protected]
45. Garrido, Pedro. Spain. [email protected]
46. Gomez Sanchez, Isidro. Spain. [email protected]
47. Gomez Santiago, Francisco. Spain. thunder91 [email protected]
48. Gonzalez Marı, Jose Luis. Spain. [email protected]
49. Gonzalez Vida, Jose Manuel. Spain. [email protected]
50. Guerrero Garcıa, Pablo. Spain. [email protected]
51. Gulliksson, Jan. Sweden. [email protected]
52. Gutierrez Barranco, Gloria. Spain. [email protected]
53. Guzner, Claudia. Argentina. [email protected]
54. Harrison, Martin. United Kingdom. [email protected]
55. Hasani Nejad, Abdul-Sahib. Iran. [email protected]
56. Heinrich, Rainer. Germany. [email protected]
190
57. Hellstrom, Timo. Sweden. [email protected]
58. Hernandez Hernandez, Pedro. Spain. [email protected]
59. Herweyers, Guido. Belgium. [email protected]
60. Heugl, Helmut. Austria. [email protected]
61. Hitt, Fernando. Switzerland. [email protected]
62. Hofbauer, Peter. Austria. [email protected]
63. Hoffmann, Ronit. Israel. [email protected]
64. Hvorecky, Jozef. Slovakia. [email protected]
65. Jimenez Mazure, Daniel. Spain. [email protected]
66. Joubert, Stephan. South-Africa. [email protected]
67. Karnasih, Ida. Indonesia. [email protected]
68. Keleher, Robert. Australia. [email protected]
69. Klein, Ronith. Israel. Ronit [email protected]
70. Kleive, Per-Even. Norway. [email protected]
71. Kokol Voljc, Vlasta. Slovenia. [email protected]
72. Kurtulus, Aytac. Turkey. [email protected]
73. Kutzler, Bernhard. Austria. [email protected]
74. Laborde, Colette. France. [email protected]
75. Larribia Martınez, Juan Jesus. Spain. [email protected]
76. Leinbach, Carl. United States. [email protected]
77. Limon Moron, Antonio. Spain. [email protected]
78. Lokar, Matija. Slovenia. [email protected]
79. Lopez Banos, Ana Marıa. Spain. [email protected]
80. Lopez Rubio, Ezequiel. Spain. [email protected]
81. Lorente Moreno, Carlos. Spain. [email protected]
82. Luksic, Primo. Slovenia. [email protected]
83. Mammana, Maria Flavia. Italy. [email protected]
84. Maricic, Kaja. Serbia. [email protected]
85. Martın Cetina, Pilar. Spain. [email protected]
191
86. Martınez del Castillo, Javier. Spain. [email protected]
87. Merida Casermeiro, Enrique. Spain. [email protected]
88. Merino Cordoba, Salvador. Spain. [email protected]
89. Metzger Schuhker, Heidi. Austria. [email protected]
90. Moldenhauer, Wolfgang. Germany. [email protected]
91. Mora Bonilla, Angel. Spain. [email protected]
92. Morales de Luna, Tomas. Spain. [email protected]
93. Moyano Franco, Rafael. Spain. [email protected]
94. Munoz Ruiz, Marıa Luz. Spain. [email protected]
95. Mustafa, Hassan Mohamed. Saudi Arabia. Mustafa [email protected]
96. Neuper, Walther. Austria. [email protected]
97. Nodelman, Vladimir. Israel. [email protected]
98. Oates, Greg. New Zealand. [email protected]
99. Ockerman, Regis. Belgium. [email protected]
100. Padilla Domınguez, Yolanda. Spain. [email protected]
101. Parcerisa Gine, Lluıs. Spain. [email protected]
102. Pennisi, Mario. Italy. [email protected]
103. Perez Gallardo, Miguel Luis. Spain. mlpg [email protected]
104. Petty, James. United States. [email protected]
105. Picard, Gilles. Canada. [email protected]
106. Pihlap, Sirje. Estonia. [email protected]
107. Roanes Lozano, Eugenio. Spain. [email protected]
108. Rodrıguez, Gerardo. Spain. [email protected]
109. Rodrıguez Cielos, Pedro. Spain. [email protected]
110. Ruiz, Natalia. Spain. [email protected]
111. Ruiz Munoz, Manuel. Spain. [email protected]
112. Salgueiro Gonzalez, Jose Antonio. Spain. [email protected]
113. Sanchez Jimenez, Nuria. Spain. [email protected]
114. Schmidt, Karsten. Germany. [email protected]
192
115. Shahin, Mazen. United States. [email protected]
116. Siklosi, Lena. Sweden. [email protected]
117. Silfverberg, Harry. Finland. [email protected]
118. Soucie, Tanja. Croatia. [email protected]
119. Stolyarevska, Alla. Ukraine. [email protected]
120. Stulens, Koens. Belgium. [email protected]
121. Svedrec, Renata. Croatia. [email protected]
122. Takaci, Djurdjica. Serbia. [email protected]
123. Tarrant, Barry. Australia. [email protected]
124. Ternander, Michael. Sweden. [email protected]
125. Trottier, Chantal. Canada. [email protected]
126. Van Wonterghem, Sandy. Belgium. [email protected]
127. Vera Ruiz, Rosario. Spain. [email protected]
128. Vukobratovic, Ruzica. Serbia. [email protected]
129. Warthmann, Dirk. Germany. [email protected]
130. Wastle, Gunnar. Sweden. [email protected]
131. Whitton, Sharon. United States. [email protected]
132. Wiesenbauer, Johann. Austria. [email protected]
133. Yanik, H. Bahadir. Turkey. [email protected]
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Authors Index
A,Abu, R., 119Ada, T., 69Aguilera, G., 151, 152, 162, 163Al-Hamadi, A., 88Alcon, J. L., 177, 181Alvarenga, K., 54Alvarez, M., 182Aranda, C., 178Argote, J. A., 182Atencia, M. A., 153
B,Badia, E., 136Barzel, B., 47Beaudin, M., 48, 128Bernal, J., 175Bogun, V., 64Bohm, J., 60, 124, 125, 129Brothers, G., 150Budinski, N., 59Bustos, J., 147Butler, D., 97
C,Caballero, C., 175Callejo, M. L., 178Caro, A., 184, 185Carrilo, A., 159Carvalho, J. L., 67Casas, L., 67Coetzee, C. E., 149Conejo, R., 108Cordero, P., 154
D,Dahan, J. J., 93, 140Davila, M. T., 164Davis, J., 85de la Plata, E., 182de la Villa, A., 105, 134Decker, J., 116
Decker, R., 114Dıaz, A., 105Dogan, M., 81Domene, M. D., 182Domınguez, E., 63
E,Egido, S., 136Eixarch, R., 136, 141, 145, 161Ellis, W., 155
F,Falcon, R. M., 96Fedriani, E. M., 99, 100, 164Fioravanti, M., 73Flores, H., 54, 160
G,Galan, J. L., 147, 151, 152, 154, 162, 163Galan, M. A., 152, 162, 163Galo, J. R., 169, 171, 173, 177, 181Galvez, J., 108Garcıa, A., 105, 134Garcıa, E., 182Garcıa, F., 134Godinho, V., 67Gonzalez, J. L., 101Gonzalez, J. M., 153, 154Guerrero, P., 111Gutierrez, G., 154Guzman, E., 108Guzner, C., 71
H,Harrison, M., 76Hasani, A. S., 61Heinrich, R., 66Hellstrom, T., 94Hernandez, P., 101Heugl, H., 57Hitt, F., 138Hofbauer, P., 80
Hoffmann, R., 84Horvat, B., 77, 78Hvorecky, J., 75
I,Icel, R., 81
J,Joubert, S. V., 149
K,Karnasih, I., 55Klein, R., 84Kokic, I., 53Kokol, V., 117Kortam, M. H., 88Kurtulus, A., 69Kutzler, B., 103, 104
L,Laborde, C., 49, 109Larrubia, J. J., 182Leinbach, C., 129–132Leinbach, P., 132Leon, O., 71Limon, A., 167Lokar, M., 77–79Lopez, A., 179Lopez, E., 63Loring, R., 182Luengo, R., 67Luksic, P., 77, 78
M,Maksic, J., 91Mammana, M. F., 74Maricic, K., 113Marques, D., 136Martın, A. M., 184, 185Martınez, J., 154Medina, M. A., 182Merida, E., 145, 154Merino, S., 107, 153, 154Micale, B., 74Moldenhauer, W., 142Mora, A, 154Mora, A., 145Morales, T., 153Moyano, R., 99, 100, 164Munoz, M. L., 153, 154
Mustafa, H. M., 88
N,Neuper, W., 58Nodelman, V., 83
O,Oates, G., 87
P,Padilla, Y., 147, 152, 154, 162, 163, 182Parcerisa, Ll., 123Pascual, A., 167Pennisi, M., 74Perez, V., 182Petty, J., 156Picard, G., 128, 133Pihlap, S., 92Puertas, N., 182
Q,
R,Radovic, N., 53Ramos, G., 63Recio, T., 73Rivera, J. G., 177, 181Ro, R., 182Roanes-Lozano, E., 50Rodrıguez, F., 154Rodrıguez, F. J., 169, 171Rodrıguez, G., 134Rodrıguez, P., 147, 151, 152, 154, 162,
163Ruiz, B. C., 172Ruiz, M., 172Ruiz, N., 112
S,Salgueiro, J. A., 169, 171Schmidt, K., 126Shahin, M., 143Shatalov, M. Y., 149Silfverberg, H., 70Sinaga, M., 55Smirnov, E., 64Soucie, T., 53Soury-Lavergne, S., 109Stolyarevska, A., 98Stulens, K., 150
196
Svedrec, R., 53
T,Takaci, D., 91, 118Toril, T., 111Trottier, C., 133Troughton, L., 182
U,
V,Vera, R., 165Vukobratovic, R., 118
W,Wastle, G., 94Whitton, S., 110Wiesenbauer, J., 144
X,
Y,Yanik, H. B., 120
Z,
197