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Mathematical modelling in science and mathematics education

Scientific research involves mathematical modelling in the context of an interactive balance between theory, experiment and computation. However, computational methods and tools are still far from being appropriately integrated in the high school and university curricula in science and mathematics. In this paper, it is discussed the relevance of mathematical modelling and illustrated how a computer modelling tool (Modellus, a free tool available on the Internet and developed at FCTUNL) can be used to embed modelling in high school and undergraduate courses. Modellus allows students to create and explore mathematical models using functions, differential and iterative equations, and visualize the behaviour of mathematical objects.

Learning with Computer-Based Exploratory Environments in Science and Mathematics

Computer-based exploratory environments are becoming more and more popular in schools. Exploratory software has its roots in a constructivist view of learning and in a revolution in the design of computer interfaces. It is argued that exploratory software should be used very carefully because a student can only explore what s/he knows but s/he is not familiar with. Some issues (didactic, psychological and technical), on the design of exploratory software, illustrated with two examples, are discussed.

Direct Manipulation of Physical Concepts in a Computerized Exploratory Laboratory

Computer microworlds are becoming more and more powerful for learning and teaching science. However, even a powerful computer microworld is not enough, by itself, to enable students to explore and learn about a formal domain. It is argued that it should be integrated with other media, especially books. In a computer microworld, direct manipulation techniques allow the implementation of direct manipulation of physical concepts, as it is shown with an example from physics — NEWTON, a computerized exploratory laboratory. In this computerized exploratory laboratory — a conceptual laboratory — the user can explore and experiment with concrete-abstract objects, confront multiple representations and pose and devise strategies to learn about the most fundamental phenomenon of Nature: motion.

Playing Newtonian games with Modellus

This article is a short introduction on how to use Modellus (a computer package that is freely available on the Internet and used in the IOP Advancing Physics course) to build physics games using Newtons laws, expressed as differential equations. Solving systems of differential equations is beyond most secondary-school or first-year college students. However, with Modellus, the solution is simply the output of the usual physical reasoning: define the force law, compute its magnitude and components, use it to obtain the acceleration components, then the velocity components and, finally, use the velocity components to find the coordinates.

Improving Learning in Science and Mathematics with Exploratory and Interactive Computational Modelling

Scientific research involves mathematical modelling in the context of an interactive balance between theory, experiment and computation. However, computational methods and tools are still far from being appropriately integrated in the high school and university curricula in science and mathematics. In this chapter, we present a new way to develop computational modelling learning activities in science and mathematics which may be fruitfully adopted by high school and university curricula. These activities may also be a valuable instrument for the professional development of teachers. Focusing on mathematical modelling in the context of physics, we describe a selection of exploratory and interactive computational modelling activities in introductory mechanics and discuss their impact on student learning of key physical and mathematical concepts in mechanics.

Modellus: Interactive computational modelling to improve teaching of physics in the geosciences

Many aspects of modern research and other professional activities in the geosciences require advanced knowledge about mathematical physics models and scientific computation methods and tools. In-depth meaningful learning of such knowledge skills is a difficult cognitive process which involves developing strong background knowledge of physics, mathematics and scientific computation appropriately contextualised in the geosciences themes. In this paper we describe an interactive engagement teaching approach that is based on Modellus, a freely available computer software system allowing (1) mathematical modelling ranging from explorative to expressive modelling, (2) the introduction of scientific computation without requiring the development of a working knowledge of programming and (3) the simultaneous manipulation and analysis of several different model representations, namely, tables, graphs and animations with interactive objects having properties defined in a visible and modifiable mathematical model. As examples of application, with insights for the development of other activities in a wide range of geosciences courses, we discuss a set of interactive computational modelling activities for introductory meteorology we have implemented in undergraduate university courses.

Improving science and mathematics education with computational modelling in interactive engagement environments

A teaching approach aiming at an epistemologically balanced integration of computational modelling in science and mathematics education is presented. The approach is based on interactive engagement learning activities built around computational modelling experiments that span the range of different kinds of modelling from explorative to expressive modelling. The activities are designed to make a progressive introduction to scientific computation without requiring prior development of a working knowledge of programming, generate and foster the resolution of cognitive conflicts in the understanding of scientific and mathematical concepts and promote performative competency in the manipulation of different and complementary representations of mathematical models. The activities are supported by interactive PDF documents which explain the fundamental concepts, methods and reasoning processes using text, images and embedded movies, and include free space for multimedia enriched student modelling reports and teacher feedback. To illustrate, an example from physics implemented in the Modellus environment and tested in undergraduate university general physics and biophysics courses is discussed.A teaching approach aiming at an epistemologically balanced integration of computational modelling in science and mathematics education is presented. The approach is based on interactive engagement learning activities built around computational modelling experiments that span the range of different kinds of modelling from explorative to expressive modelling. The activities are designed to make a progressive introduction to scientific computation without requiring prior development of a working knowledge of programming, generate and foster the resolution of cognitive conflicts in the understanding of scientific and mathematical concepts and promote performative competency in the manipulation of different and complementary representations of mathematical models. The activities are supported by interactive PDF documents which explain the fundamental concepts, methods and reasoning processes using text, images and embedded movies, and include free space for multimedia enriched student modelling reports and te...

The Computer as a Conceptual Lab: Learning Dynamics with an Exploratory Environment

Computer simulations will change the way science is learned In this paper we will discuss the rationale and some problems facing the development of a computer tool for exploring newtonian mechanics—a particle movement simulation called NEWTON 4. The development was based on common misconceptions found in students before and after they have been exposed to newtonian dynamics. The goal of the program is to “extend the range of manipulable objects” allowing the user to confront “reality” against “simulation and description of reality”. It can be considered as a “computer-based conceptual laboratory” where physical constructs (like velocity or force) can be controlled and manipulated. The user interface is based on the technique of “direct manipulation”.

Integrating Computational Modelling in Science, Technology, Engineering and Mathematics Education

Modelling is a central aspect of the research process in science, technology, engineering, and mathematics (STEM), which occurs in the cognitive context of an interactive balance between theory, experiment, and computation. The STEM learning processes should then also involve modelling in environments where there is a balanced interplay between theory, experiment, and computation. However, an adequate integration of computational themes in STEM high school and undergraduate university curricula remains to be achieved. In this chapter, we present an approach to embed computational modelling activities in the STEM learning processes which may be fruitfully adopted by curricula at secondary and introductory university levels, as well as be a valuable instrument for the professional development of teachers. To illustrate, we consider the example of physics.