Project Application Microworlds
Mathematical microworlds are learning environments that contain objects and operations from a sub-area of mathematics. The constructive handling of the objects is intended to provide a better understanding and application of the underlying mathematics concepts. The term microworld was introduced by Seymour Papert in the early 1980s in connection with the LOGO environment.
In the mathematics application microworlds (MAM) created and tested in this project, the objects have an application meaning as well as a mathematical meaning. For example, in the Formula 1 microworld, straight lines and circles are interpreted as course sections and functions are interpreted as motion or speed functions. The application background provides interesting and meaningful scientific questions and data input for modeling. The micro-world forms a simplified design environment in which modeling and testing can be carried out, and feedback can be used to draw conclusions accordingly.
The application reference can also be extended by providing a real object in addition to the simulated world, from which data for the microworld can be obtained and on which constructions from the microworld can be tested. For the Formula 1 microworld, for example, a Carrera(R) track is such a real object.
A short description can be found on the flyer
Flyer Microworlds New (0.59 MB)605 KBPDF
How computer algebra systems can generally be used to create mathematics microworlds is described in the following article:
449 Alpers Microworld Ijcame2002 (0.32 MB)327 KBPDF
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The project was funded within the framework of the LARS program for Universities of Applied Sciences in Baden-Württemberg.
Two microworlds are being developed and tested in this project:
"Formula 1"
This is an environment in which racing courses (consisting of straight and circular sections) and motion functions can be modeled and tested.
Project Application Microworlds - Formula 1
The "Formula 1" microworld consists of a simulated environment in the Maple computer algebra system and a real Carrera(R) race track that can be controlled via a microprocessor. At the moment, the real course is used to obtain meaningful data for the construction of courses (geometry of the tracks) and the creation of driving functions (maximum speeds in course areas, maximum positive and negative acceleration). In a later expansion stage, the motion functions constructed in Maple will also be "downloaded" to the real course.
A more detailed description of the Maple microworld can be downloaded:
Description of the microworld (0.25 MB)254 KBPDF
A description of the basic learning possibilities in the microworld as well as a concrete use with students in grades 11 and 12 can be found in the following article:
The mathematics microworld "Formula 1" - Learning opportunities and use in the Student Engineering Academy (SIA) (0.33 MB)342 KBPDF
The microworld has also been used as part of mathematical application projects in the degree program "General Mechanical Engineering" (3rd semester"). This can be viewed in the MAPS project database.
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The project was funded as part of the LARS program for Universities of Applied Sciences in Baden-Württemberg.
"Billiards"
In this environment, carom situations can be constructed and tested.
Project Application Microworlds - Billiards
The billiard microworld consists of a simulation in the computer algebra system Maple and a real billiard machine with which the results of the simulation can be tested in reality. This machine is a converted three-axis machine with a self-built table underneath, as can be seen in the photo below. The impact cylinder can be used to perform centric impacts (without spin) with a predetermined direction and speed, and configurations can be "laid" (and thus reproduced), as a suction device can be used to lift balls from the impact cylinder and move them to a predetermined position. The rubber cushion, balls and cloth are original billiard accessories, the table is made of chipboard.
In the Simulation, you can set the system parameters (table size, ball size, number of shots, coefficient of friction) and then simulate shots. Impact tasks are to be processed and tested here (e.g. a certain collision is to be constructed). The feedback provided by the Simulation allows the learner to recognize incorrect ideas and correct them if possible. You can start with a simple "mathematics billiards" model without friction and energy loss during the shot and then use the real billiards machine to recognize the limits of the model. This could then provide a basis for refining the model. The following pictures show the surface of the microworld (the Maple code of the microworld is available from the author):
The underlying models and possible tasks are described in more detail in the following article:
Modeling and Problem Solving in Billiards Using DGS and Billiards Machine (0.25 MB)258 KBPDF
The use of dynamic geometry systems in billiards tasks is described in more detail in:
Using DGS for working on realistic Billiards tasks (0.44 MB)447 KBPDF
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The project was funded within the framework of the LARS program for Universities of Applied Sciences in Baden-Württemberg.
