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Improving the General University Physics Course with Computational Modelling 1
Rui Gomes Neves Jorge Carvalho Silva 1 Vítor Duarte Teodoro 2
1
Unidade de Investigação Educação e Desenvolvimento (UIED), FCT-UNL 2 Centro de Física e Investigação Tecnológica (CEFITEC), FCT-UNL
Comunicação apresentada por Rui Neves em Junho de 2008, na Gordon Research Conference (GRC) on Physics Research and Education, Bryant University, Rhode Island, USA.
Abstract General physics is an extremely difficult subject for first year university students. The level of failure is high across the world and, unfortunately, students in Portugal are no exception. Most students admitted to university are not well prepared nor motivated to learn physics and the standard physics course format based on theoretical lectures and laboratories is clearly unable to change this gloomy picture. In this work, we describe and discuss a new step forward in improving general physics as an educational experience: the implementation of a set of innovative workshop activities based on computational modelling in the general physics course taken by first year students of the Biomedical Engineering course at the Faculty of Sciences and Technology of the New Lisbon University. The workshop activities were organized as interactive and exploratory learning experiences where students worked as teams of scientists on a small number of problems about challenging but easily observed physical phenomena. All activities were created as mathematical modelling experiments with Modellus, a computer software tool specially designed to construct and explore mathematical models based on functions and differential equations. Each activity consisted of a set of tasks, presented in PDF documents, with embedded video guidance to support students both in class and/or at home in a collaborative online context based on the Moodle learning environment. Special emphasis was given to cognitive conflicts in the understanding of physical concepts, to the manipulation of multiple representations of mathematical models and to the interplay between analytical and numerical approaches to the solution of physical problems.
1. Introduction Although a central theme in the curricula of science and engineering university courses, physics continues to be an extremely difficult subject for students which need it to complete their first and second year plan of studies. Due to a lack of understanding of fundamental concepts in physics and mathematics, the number of students unable to pass the course examinations is usually very high. For example, in the Faculty of Sciences and Technology of the New Lisbon University, on average only less than thirty per cent of the students are able to take the course on the first time. What is worse is that many of those students that do actually succeed also reveal several weaknesses in their understanding of elementary physics (Hestenes, 1987; McDermott, 1991). Clearly, a solution for this problem requires changes in the way physics is taught and learnt. In recent years, many results of physics education research have shown that the process of learning is effectively enhanced when students are involved in the learning activities as scientists are involved in research (Mazur, 1997; McDermott, 1997; Beichner et al, 1999; Handelsman et al., 2005). In this kind of class environments students learn physics working through interactive and exploratory activities which approximate the collaborative action frame of scientific research. Consequently, knowledge performance is better promoted and common sense beliefs as well as incorrect scientific notions can be more effectively fought. Another important aspect of these research inspired learning environments is the possibility to give a central role to computer software tools and computational modelling. Modelling physics in computer learning environments started with an emphasis on programming languages. Using, for example, languages such as Fortran (Bork, 1967), Pascal (Redish and Wilson, 1993) and Python (Chabay and Sherwood, 2008), this approach requires students to develop a working knowledge of programming. The same is true when professional scientific computation software like Mathematica or Matlab is used. To avoid overloading students with programming notions and syntax, computer modelling systems such as Dynamic Modelling System (Ogborn, 1985), Stella (High Performance Systems, 1997), Easy Java Simulations (Christian and Esquembre, 2007) and Modellus (Teodoro, 2002) were developed to focus the learning activities on the understanding of the concepts of physics and mathematics. Of these, Modellus (see the software webpage at the site http://modellus.fct.unl.pt) is especially interesting for allowing a model to be conceived almost as it is on a piece of paper, for having the possibility of creating animations with interactive objects that have mathematical properties expressed in the model, for permitting the analysis of experimental data in the form of images and for allowing simultaneous exploration of multiple model representations. In spite of these important advances, interactive and exploratory learning environments which integrate computational modelling remain largely absent from the majority of the general physics curricula. In this work, we report on an action we have organized to introduce this approach in a general university
physics course in Portugal. Following recent results of science education research (Handelsman et al., 2005; Slooten et al., 2006) we have created and implemented a new course component based on computational modelling with Modellus, in the general physics course taken by first year biomedical engineering students at the Faculty of Sciences and Technology of the New Lisbon University. In the following sections we describe how this computational modelling course was organized, what was the pedagogical methodology implemented and the educational aims of the computational modelling activities offered to the students in the first part of the course. We also discuss the student’s reaction to this new approach and their knowledge performance on key physical and mathematical concepts of the general physics course.
2. Course organization and methodology In the 2008 academic year, the general physics course for biomedical engineering involved a total of one hundred and fourteen students among which fifty were taking the course for the first time. It was a one semester course and it was divided into theory lectures where the general physics subjects were first introduced, standard physics laboratories and computational modelling classes based on interactive computer workshop activities. The students taking the course for the first time had to attend all three course components while the others, already repeating the subject, took only the theory lectures and the physics laboratories. In the computational modelling classes, the students were organized in groups of two, one group for each computer in the classroom. During each class, the student teams worked on a workshop activity with a selected set of problems about challenging but easily observed physical phenomena. The teams were instructed to analyse and discuss the problems on their own using the physical, mathematical and computational modelling guidelines provided by the workshop protocols. To ensure adequate working rhythm with appropriate conceptual, analytical and computational understanding, the students were continuously accompanied and helped during the exploration of the activities. Whenever it was felt necessary, global class discussions were conducted to keep the pace and to clarify any doubts on concepts, reasoning or calculations. All workshop activities were created as modelling experiments with Modellus. Modellus is a computer software tool designed to introduce students and teachers to scientific computation through the construction and exploration of mathematical models based on functions, iterations and differential equations. This software was chosen for the development of the computational modelling activities of the course because of its emphasis on modelling focusing on the meaning of models. Indeed, with Modellus, modelling experiments involving multiple representations (tables, graphics, animations and images) can be done almost without the need to introduce new symbols or syntax, following the same notation and reasoning already familiar when attempting to solve physical and mathematical problems on paper.
Each workshop activity was presented in a PDF document, with embedded video support to help students in class and at home in a collaborative online context centred on the Moodle learning environment. The course was globally conceived to give special emphasis to cognitive conflicts in the understanding of the relevant physical concepts, to the manipulation of multiple representations of mathematical models and to the interplay between analytical and numerical approaches to the solution of physical problems. At the end of the semester, the students answered a questionnaire to access their receptivity degree to this new computational modelling component of the general physics course.
3. Computational modelling activities with Modellus The program of the general physics course given to the biomedical engineering students was based on the book by Young and Freedman (2004). Following its structure, the computational modelling activities covered eight basic themes in mechanics (Teodoro, 2006): 1) Vectors. 2) Motion and parametric equations. 3) Motion seen in moving frames. 4) Newton’s equations: analytic and numerical solutions. 5) Circular motion and oscillations. 6) From free fall, to parachute fall and bungee-jumping. 7) Systems of particles, linear momentum and collisions. 8) Rigid bodies and rotations. During all classes, the students groups would read the PDF protocols and follow the instructions of the embedded videos as well as our personal guidance to carry out all the modelling activities. In what follows we discuss two example themes from the first part of the course: a) motion seen in moving frames and b) Newton’s equations: analytic and numerical solutions.
3.1 Motion seen in moving frames The description of the motion of an object depends on the reference frame considered to describe it. For example, two cars moving in straight lines side by side with the same constant velocity, relative to a reference frame fixed on the road, are at rest in reference frames which are attached to the cars. The choice of frames is very important to solve many problems in physics. However, care must be taken in the physical interpretation of the corresponding motion, a point students should be made clearly aware of. At stage in their instruction, students were only introduced to the concept of inertial reference frame, a coordinate frame moving with constant velocity, and to the motion kinematics associated with inertial frames moving with different velocities. Building upon the previously acquired knowledge about vectors, uniform motion and the corresponding parametric equations, the interactive exploration of the Modellus computational modelling activities about motion seen
in different inertial frames lead students to experience several cognitive conflicts and realise that observers in different moving frames can indeed have different views of the motion. For example, students modelled and constructed the animation of the motion of a swimmer in a river with a downstream current with magnitude equal to five meters per minute (see figure 1).
Mathematical Model and Animation
Figure 1. Modelling relative motion with linear functions of time. When the swimmer tried to move up stream with the same speed as the downstream current it would not move at all relative to an observer on the river margin. However, for an observer on a boat dragged by the current, the swimmer would move up stream with a speed of five meters per minute. When observing this animation for the first time, students were generally startled for a moment. This cognitive conflict is rooted in the inability to make a clear distinction between the descriptions of the motion in the two different inertial frames. After careful analysis of the mathematical model and the associated animation, all students were able to understand that there are two different points of view and that they are related by the Galilean vector transformation. After completing the analysis of the motion of the swimmer in a river with current, the students groups were able to successfully explore with Modellus the similar problems of a boat crossing a river with current and of the motion of a plane when there is wind. An example was the following: a plane flies from west to east. The pilot chooses a velocity of 400 km/h pointing in the direction 080. If the wind blows at 100 km/h, determine the wind direction and the plane ground speed (see figure 2). In this context, the modelling process with Modellus was of special relevance in helping students to realise that many different, everyday life physical situations can be explained using the same simple mathematical model. These computational modelling activities with Modellus also allowed students to consolidate their knowledge about vectors and resolve persistent difficulties in the
clear identification of the position and the velocity as vector quantities characterized by a magnitude and a direction. For example, many students at first consider only the magnitude of a velocity. After realising that to specify a velocity the direction must also be given, students showed difficulties in defining it as angle whose value depends on the angle unit (degrees or radians) and on the chosen measuring convention. Modellus uses the navigation convention where the angle varies between zero and three hundred and sixty degrees clockwise starting from the North.
Mathematical Model and Animation
Figure 2. Modelling the motion of a plane in a windy atmosphere. This convention is not equal to the convention used for the argument of the trigonometric functions, which varies between zero and three hundred and sixty degrees counter clockwise starting from the positive horizontal axis. As a consequence, students were frequently unable to make the vectors point in the right direction. The ability to correct the mathematical models and simultaneously visualise the effect of the change in vectors of the animation, was for the students an essential advantage of the modelling process with Modellus in helping them to reify vectors as concrete mathematical objects defined by a magnitude and a direction and to solve these learning difficulties.
3.2 Newton’s equations: analytic and numerical solutions At this stage our students were introduced to dynamics. The starting question was: what must happen for the velocity to change during motion? If the velocity is changing during motion there must be an acceleration vector and at least one applied force. The acceleration is the vector that measures the instantaneous rate of change of the velocity with time. According to Newton’s second law of motion, this vector is obtained dividing the sum of all the forces that act on the particle by the mass of the particle. If there are no net forces then there is no acceleration and the velocity is constant. This is the statement of Newton’s first
law of motion or law of inertia. Our students began the exploration of these laws with an activity where the objective was to change the velocity of a particle in the perpendicular to make an Aristotle’s corner (see figure 3) (diSessa, 1982). Mathematical Model and Animation
Figure 3. Interactive Euler-Cromer model to make an Aristotle’s corner. In the Aristotle’s corner model, Newton’s equations of motion were written in the form of Euler-Cromer iterations. The students were thus introduced to a simple numerical method to solve the equations of motion and determine the velocity and position of the particle knowing its mass and the net applied force. The model animation was constructed with three objects: the particle, a vector representing the velocity attached to the particle and a vector representing the net force. Because the coordinates of the net force vector are independent variables and the model is iterative, students were able to manipulate this vector at will and in real time control realistically the motion of the particle. Soon, it became clear that to do an “Aristotle’s corner” it was first necessary to break and stop, and only then accelerate in the perpendicular. During this modelling activity with Modellus, students were helped to resolve still another cognitive conflict: to break is not that different from accelerating, it is just to accelerate in the direction opposite to the direction of the velocity. They also realised that the choice of a small time step was important for a numerical method to work and generate a good simulation of the motion and that this is the same as determining a good numerical solution of the equations of motion. The same numerical model was then used to throw a ball up in the air with Modellus (see figure 4). In this modelling activity, students were taught to interact with the net force vector applied to the ball and to simulate the throw as well as the following motion under the earth’s gravity. The choice of the appropriate time step was once more an important source of difficulties. These were solved using the possibility of changing the mathematical model and immediately observe the effect of the correction on the animation. With the position and velocity time graphs, students were able to determine how long it
takes for the ball to reach the highest point of the trajectory, what is the height of that point and when is the ball three metres up in the air. Comparing with the animation of the throw, students were able to clearly distinguish between the physical trajectory of the ball and the position time graphs. Mathematical Model, Animation and Graph
Figure 4. Throwing a ball up in the air with iterative Newton’s equations. The students were also able to create vectors and draw on the screen the vector diagrams representing the forces acting on the ball, the velocity and the acceleration during the whole motion. Another important learning difficulty students were helped to address during this modelling activity was the need to choose appropriate scales for the animation objects and graphs. Finally, by observing the net force graph as a function of time, they were able to estimate the duration of the throw. The next step was to compare the analytic solution for the motion just after the throw with the corresponding numerical solution obtained using, for example, the Euler-Cromer method. With the same sum of applied forces and the same initial conditions students were able to find out that the analytic solution is not equal to the numerical solution. The difference is the numerical error and is only present for the position and not for the velocity, which changes linearly with time.
4. Conclusions In this work we have presented the computational component of an introductory physics course based on a set of innovative workshop activities with Modellus
given to first year Biomedical Engineering students of the Faculty of Sciences and Technology of the New Lisbon University. After a description of the course organization and methodology, we have analysed two thematic examples of the computational modelling activities to illustrate how they were designed to generate cognitive conflicts relative to important physical concepts, to promote the manipulation and correct interpretation of multiple model representations and to analyse the interplay between analytic and numerical solutions of physical problems. 1. Introducing the computational component was useful for the learning process of Physics 1B. 2. The percentage of computational classes was adequate. 3. The computational classes in Physics 1B are important in the context of the Biomedical Engineering course. 4. Doing the activities in groups of 2 or 3 has more advantages than doing the activities individually. 5. During classes, the teacher’s guidance and support to the several groups was sufficient and adequate. 6. The problems analysed in the computational activities with Modellus were interesting and motivating. 7. The activities with Modellus in PDF format with video support are well conceived and interesting. 8. The activities with Modellus contributed to a better understanding of the mathematical and physical concepts introduced in Physics 1B. 9. The activities with Modellus allow the concretization of abstract mathematical and physical concepts. 10. The activities with Modellus allow the manipulation of the multiple representations of the mathematical and physical models. 11. The activities with Modellus were helpful in clarifying the interplay between the analytical and numerical processes used to solve concrete physical problems. 12. Modellus is a useful program to help in the learning of mathematical and physical models. 13. Modellus is easy to learn and is user-friendly. 14. With adequate activities, Modellus should be used in other disciplines of the Biomedical Engineering course.
Table 1: Physics 1B questionnaire. We have concluded that during class the computational modelling activities with Modellus were successful in identifying and resolving several student difficulties in key physical and mathematical concepts of the course. Of crucial importance to achieve this was the possibility to have a real time visible correspondence between the animations with interactive objects and the object’s mathematical properties defined in the model, and also the possibility of manipulating
simultaneously the several different model representations (mathematical model, animation, graphs and tables). The implementation of this set of computational modelling activities with Modellus was thus successful. This was indeed reflected in the student answers to the questionnaire given at the end of the course (see table 1 and figure 6). In this questionnaire, students gave their opinion to a set of assertions about the characteristics and relevance of the new Modellus computational modelling component created for the general physics course, using a scale starting at –3 and ending at +3, where –3 stated complete disagreement and +3 complete agreement. The remaining negative values stated partial disagreement and the remaining positive values partial agreement. The number 0 corresponded to a no opinion position.
Figure 6: Physics 1B questionnaire results. Globally, students reacted positively to the activities based on modeling with Modellus, considering them to be important in the context of the biomedical engineering course. A natural sense of caution in relation to novelty and to evaluation procedures was also clearly detected. Students showed a clear preference to work in teams in an interactive and exploratory learning environment, with our proper guidance and support. The computational activities with Modellus presented in PDF format with embedded video guidance were also considered to be interesting and well designed. In this work, Modellus was also successfully tested as a software tool that allows students to work as authors of mathematical physics models and simulations, not as simple browsers of computer simulations. Models can be presented as differential equations solved by simple numerical methods and students can appreciate the differences between numerical solutions and analytical solutions. For the students, Modellus was
indeed seen as helpful in the learning process of mathematical and physical models. It was also found to be easy enough to learn and user-friendly.
Acknowledgements Work supported by Unidade de Investigação Educação e Desenvolvimento (UIED) and Fundação para a Ciência e a Tecnologia (FCT), Programa Compromisso com a Ciência, Ciência 2007.
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