Learning Physics with Mathematical Modelling
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Modellus: Learning Physics with Mathematical Modelling Vitor Duarte Teodoro Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2002 1 2 Modellus: Learning Physics with Mathematical Modelling Vitor Duarte Teodoro Dissertação apresentada para obtenção do grau de Doutor em Ciências da Educação – especialidade de Teoria Curricular e Ensino das Ciências, sob orientação conjunta da Professora Doutora Maria Odete Valente e do Professor Doutor Cândido Marciano da Silva Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2002 1 2 Table of Contents Table of Contents [3] List of Figures [7] List of Tables [11] Abstract [13] Resumo [15] Résumé [17] Chapter 0 Summary, in Portuguese (Modellus: aprendizagem da Física e modelação matemática) [19] 0.1 Aprendizagem, software educativo e a natureza da Ciência [19] 0.2 Modellus: uma ferramenta cognitiva [23] 0.3 Currículo: uma visão interdisciplinar [27] 0.4 Modellus: algumas opiniões de alunos e professores [29] 0.5 Coda [31] Chapter 1 Introduction [33] 1.1 Why this Thesis? [33] 1.2 Cognitive Artefacts and Physics Education [36] 1.3 Technological Change and Science Education [42] 1.4 The Importance of this Thesis: Research and a Guide to the Thesis [48] 1.4.1 Can we do it? [48] 1.4.2 Research questions [50] 1.4.3 The nature of research in education and research in computers in education [51] 1.4.4 A guide to the thesis [53] 1.4.5 Who can benefit from this thesis? [54] Chapter 2 Computers in Physics Education (with a Note on Mathematics Education) [55] 2.1 Introduction [55] 2.2 From Programming Languages to Educational Applications [57] 2.2.1 Changes in computer technology and in views of education [57] 2.2.2 Papert’ s Logo and its successors [58] 2.2.3 Pascal and programming utilities [61] 2.2.4 Spreadsheets and mathematical tools [62] 2.2.5 Data logging (computer-based laboratory systems) [66] 2.2.6 Simulation tools [68] 2.2.7 Modelling tools [71] 2.2.8 Modelling vs. simulation: classifying the use, not the tool [75] 2.2.9 The rise of educational applications [75] 2.3 Research on Learning Physics with Computers [76] 2.3.1 Which methodology? [76] 2.3.2 Research on physics learning using data logging [77] 2.3.3 Research on physics learning using computer simulations [78] 2.3.4 Research on physics learning using modelling [80] 2.3.5 Other studies of physics learning with computers [81] 3 2.4 Computers in Mathematics Education [82] 2.4.1 Mathematics education under change [82] 2.4.2 The calculus reform movement [82] 2.4.3 NCTM principles in mathematics education and its importance to science education [84] 2.5 Coda [85] Chapter 3 Modellus: A Tool for Doing Experiments with Mathematical Objects [87] 3.1 Introduction [87] 3.2 Issues in Software Design [87] 3.3 Some Contributions to the Creation of New Computer Tools for Learning Physics (and Mathematics) [91] 3.3.1 The motivation: students’ learning difficulties and misconceptions [91] 3.3.2 Early titles: Kinematics (“Cinemática”), Projectiles (“Projécteis”), Space (“Espaço”) and Functions (“Funções”) [92] 3.3.3 Dinamix and Change (“Variações”) [94] 3.3.4 Galileo (Macintosh), Galileo (Windows), Mouse in Motion (“Ratos em movimento”) [95] 3.3.5 Descriptive geometry (“Geometria descritiva”) and Functions and derivatives (“Funções e derivadas”) [98] 3.3.6 Function games I & II (“Jogos de funções I & II”), Field force games (“Jogos de campos de forças”) and Vrum... vrum: motion games (“Vrum... vrum... Jogos de movimento”) [98] 3.3.7 Thales and Statistics: exploring basic concepts (“Estatística: explorar conceitos básicos”) [100] 3.3.8 Faraday, Another world (“Outro mundo”) and Signals and oscilloscope (“Sinais e osciloscópio”) [102] 3.3.9 Mendeleieff [102] 3.3.10 A note and a synthesis [103] 3.4 Modellus: the Concept and a Technical and Pedagogical Description [105] 3.4.1 The concept [105] 3.4.2 Modellus: a direct manipulation interface [107] 3.4.3 The structure of the program [108] 3.4.4 Creating and running a file [109] 3.4.5 Modellus: a pedagogical description [117] 3.5 A Model to Develop Exploratory Software for Learning Science (and Mathematics) [120] 3.6 Coda [124] Chapter 4 Modelling and Modellus in the Physics Curriculum: Some Examples and a Framework [127] 4.1 Introduction [127] 4.2 How Can Modellus Help Change the Physics Curriculum? A Few Examples [128] 4.2.1 Graphs as “story tellers” [128] 4.2.2 Models from data: a different approach with Modellus [133] 4.2.3 Exploring phasors: using Modellus to “think with” [138] 4.2.4 From simple multiplication to calculus and differential equations [140] 4.3 Modelling in the Physics Curriculum: A Framework [145] 4.3.1 The physics curriculum: old and new challenges [145] 4.3.2 Experimentation and meaningful learning [147] 4.3.3 Experimenting with computers [149] 4.3.4 Which computer tools for the physics curriculum? [150] 4.3.5 Physics curriculum and modelling (with some notes on the mathematics curriculum) [151] 4.4 Coda [155] 4 Chapter 5 Teachers, Modelling and Modellus [157] 5.1 Introduction [157] 5.2 Questionnaire about Modellus and Modelling [158] 5.3 A Reading of the Answers to the Questionnaire [161] 5.3.1 Modellus as a tool for learning [161] 5.3.2 Integrating physics and mathematics [162] 5.3.3 Working with “formal objects” in a concrete way [164] 5.3.4 How good is the user interface of Modellus? [166] 5.3.5 Students using Modellus [168] 5.3.6 Making conceptual experiments [170] 5.3.7 New features suggested by users [172] 5.4 Coda [173] Chapter 6 Students Using Modellus [175] 6.1 Introduction [175] 6.2 Some Methodological Reflections [175] 6.3 Pilot Work [177] 6.4 First Study: Modellus Used by Secondary School Students [178] 6.4.1 Subjects, research questions and description of the intervention [178] 6.4.2 Questionnaire and results [180] 6.4.3 Discussion [181] 6.5 Second Study: Modellus Used by Undergraduate University Students [182] 6.5.1 Subjects, research questions, description of the intervention, and questionnaire [182] 6.5.2 Results [184] 6.5.3 Discussion [185] 6.6 Comments on the two Studies [186] Chapter 7 From Theory to Practice: Computers, Modelling and Modellus in Education [189] 7.1 Introduction [189] 7.2 Modelling in Physics Education: the Way Ahead [190] 7.2.1 The place of models in science and mathematics education [190] 7.2.2 Modellus: making “conceptual experiments” with a computer [192] 7.2.3 Modelling in the curriculum: a horizontal and a vertical perspective [193] 7.2.4 Modellus: a preview of the next version [194] 7.3 Students and Teachers using Modellus [196] 7.3.1 What teachers say... and do [196] 7.3.2 What students say... and can do [197] 7.4 Curriculum Change and Computers [198] 7.4.1 Forty years of curriculum change [198] 7.4.2 Curriculum development and support [199] 7.4.3 Some policy and organizational considerations [200] 7.5 Coda: old and new processes, old and new technologies [203] Appendix Results from Students Questionnaires [207] A.1 Results from the First Study [208] A.2 Results from the Second Study [211] A.2.1 The “Braking car problem” [211] A.2.2 Questionnaire about the work done with Modellus [214] Author Index [221] Index [225] References [229] 5 6 List of Figures Figura 0.1 Explorando múltiplas representações (equações, tabelas, gráficos, trajectórias, estroboscopia, vectores) de um movimento acelerado no Modellus. [24] Figura 0.2 Construção de um modelo a partir da análise de dados experimentais. O gráfico correspondente aos dados experimentais foi utilizado para obter informação para a construção do modelo. Uma vez construído o modelo, este foi comparado com os dados experimentais. [25] Figura 0.3 Construção de um modelo a partir de um vídeo. O modelo foi, em seguida, comparado com o movimento do pêndulo. O vídeo foi colocado no fundo da janela de “Animação”. A imagem da direita é uma cópia da imagem da esquerda. Apenas se pode fazer medidas e colocar objectos sobrepostos na imagem da direita. [26] Figure 1.1 How to estimate sin 35º? [40] Figure 2.1 Spreadsheet used in an investigation in the Active Physics curriculum. In the cell E4 and E5 the formulas are (B5*B3)-B7 and (B5*B4)+(B5^2)/(2*B6), respectively. [63] Figure 2.2 An Excel worksheet from Workshop Physics to compute the maximum height of a projectile as a function of the launch angle. Cells in column C have the formula h = v02sin(q) /(2g). [63] Figure 2.3 A Mathcad file showing the computation of the range of a projectile, the representation of the trajectory and the vertical co-ordinate as a function of time. [64] Figure 2.4 A Mathematica file showing the numerical solution of the height of the fall of a parachutist, before and after opening the parachute. [65] Figure 2.5 Using a motion sensor to understand position-time graphs. In this experiment, the student tries to match the graph on the left moving in front of the sensor (from Pasco Science Workshop). [67] Figure 2.6 Video analysis of a projectile motion using VideoPoint (Luetzelschwab, Laws, Gile, & Cooney, 1997). Clicking on the projectile, at the different images of the video sequence, generates a table of data. This table was used to create two graphs, x(t) and y(t). [68] Figure 2.7 A simulation of a collision of two disks made with Interactive Physics. Both the disks and the centre of mass of the system are shown in stroboscopic representation.