1 / 16

CAS as Environment for Mathematical Microworlds Example: Formula 1

CAS as Environment for Mathematical Microworlds Example: Formula 1. Burkhard Alpers CAME Meeting, Utrecht, July 2001. Contents. What constitutes a mathematical microworld? How can CAS be used to create projects and microworlds? Example: Formula 1 as a CAS project

mabyn
Download Presentation

CAS as Environment for Mathematical Microworlds Example: Formula 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CAS as Environment forMathematical MicroworldsExample: Formula 1 Burkhard Alpers CAME Meeting, Utrecht, July 2001

  2. Contents • What constitutes a mathematical microworld? • How can CAS be used to create projects and microworlds? • Example: • Formula 1 as a CAS project • Formula 1 as a CAS microworld • Conclusions / Questions

  3. Aspects of Microworlds (I)(Kent) Computational environments which • represent a particular knowledge domain • are constructed for the purpose of learning • contain comput. objects embodying math. ideas • offer activities for these objects to „evoke“ encounter, recognition and exploration of ideas (P. Kent: Expressiveness and Abstraction with Computer Algebra Software, Journées d'étude: Environnements informatiques´de calcul symbolique et apprentissage des mathématiques, Rennes, France. June 2000)

  4. Aspects of Microworlds (II)(Edwards) „Functional view“: How do learners interact with microworlds? • Manipulate objects and execute operations „with the purpose of inducing or discovering their properties ... Experimentation, hypothesis generation and testing, and open-ended exploration are encouraged“ • „Interpret feedback ... in order to self-correct or debug ... understanding of the domain“ • „Use ... objects and operations to create new entities or to solve specific problems or challenges“

  5. Aspects of Microworlds (II)(Edwards, continued) • „This experimentation-feedback cycle is a hallmark of computer microworlds when viewed from the functional perspective“. • „In fact, microworlds are created precisely in order to surface and challenge students‘ current ... understanding ...- if the learner fully understood the mathematics ... the program would have little value“ (L.D. Edwards: Embodying Mathematics and Science: Microworlds as Representations, Journal of Mathematical Behavior, 17 (1), 53-78)

  6. „Types“ of Microworlds • Intra-mathematical microworlds: objects and operations do not carry any application (real world) meaning • Ratio and proportion (Noss, Hoyles) • Transformation geometry (Edwards) • ODEWorld (?) (Kent) • Application microworlds: math. objects and operations carry application meaning which provides a motivating („situated“) context for problem solving • „Rainbow bridge“ (Kent) • Formula 1 microworld for functions (see below) • Simulation environment with mathematical content • Contain not only different intra-mathematical representations but also extra-mathematical ones

  7. Usage of CAS for microworlds • Potential • CAS provide a variety of mathematical objects and operations (functions, expressions, equations, ...) • CAS provide different representations (numeric, symbolic, graphical, animation) • CAS provide language for programming higher-level abstractions • Limitations (for direct usage) • CAS require – sometimes – usage of complicated syntax • CAS require programming for getting larger tasks done • CAS provide only limited feedback • Mathematical objects are for general purpose and not „tuned“ for usage in a specific application context

  8. Positioning microworlds within the CAS usage „space“ CAS „as is“ Higher-level Objects and ops. CAS model reduction Real world scenario Production Toolbox for Engineer Creation of Production Toolbox by Math. (Industrial usage) Student Project (eng.) Student Project (math.) Student microworld (Educational usage) Simplified model Application reduction

  9. Example: Formula 1 • Application context: • Main application objects: course and the cars driving on a course. • Goal: achieve „good“ (optimal) lap times. • Engineers and mathematicians industry (e.g. Porsche) use sophisticated simulations of courses and motions for tuning cars „optimally“. • This context stands prototypically for many motion design problems in mechanical engineering (cf. VDI) • Application reduction (simplification) • Model course as curve consisting of line and arc segments (cf. real world: „curve problem“) • Consider only one-dimensional motion • Simple acceleration scheme, no gears, no friction, ...

  10. Example: Formula 1 (continued) • Relevant mathematical objects and operations: • Curves in parameter representation (particularly line segments and arcs) • Function classes to construct motion functions (part. piecewise defined functions, polynomials, square roots) • Differentiation, integration, diff. Equations (e.g. getting from s(t) to v(t) or from v(s) to s(t)) • Prerequisites: Students have „knowledge“ of ... • the above mathematical concepts and operations • relations between distance, velocity, acceleration

  11. Formula 1- Project with CAS • Project task: • Model the Hockenheim motodrom with Maple and construct a „realistic“ motion function • Student activities • Retrieve course data from internet (simplified) • Construct course piecewise (composition on plot level, then on functional level) • Make „realistic“ assumptions on acceleration, max. velocity • Construct piecewise defined function v(s) • Compute s(t), lap time • Create animation

  12. Formula 1 – Project with CAS (c‘d) • Project result: • Provision of structured worksheet • Graphics and animation

  13. Formula 1 -Microworld with CAS (I) • Technical simplification (easier usage of CAS) • Mathematical simplification (hide unknown math.) • Types: • Offer higher-level objects (e.g. „course“ with parameters) • Offer higher-level operations (e.g. „create“ s(t), v(t), a(t) from v(s)) • Offer higher-level feedback (e.g. restriction violation as functions or in animation, comparison with „good“ function) • Offer higher-level representations (motion animation, real motion on toy course, sound )

  14. Formula 1 - Microworld with CAS (II) • Assumed activities and learning scenarios promoting conjecturing and understanding • think about functions and their properties • try out functions and learn more • think about continuity, differentiability • Assumed difficulties • Restrictions are not fulfilled • Lap time is bad

  15. v s Formula 1 - Microworld with CAS (III) • Example for possible activities: • Produce a v(s) plot with restrictions • Try linear connection, think about continuity • Check a(t) (set max. acceleration so that the restriction is violated) • Try other functions • Compare with „good“ function plot or animation

  16. Conclusions / Questions • CAS as such are production tools and need a pedagogical context when using them for didactical purposes • CAS provide a rich environment to construct such a pedagogical context • The context reaches from providing simply meaning to providing high-level, easy to use abstractions • Relationship between concept development and concept application: Is application a deepening or a constitutive part of concept understanding?Edwards: „... use and thereby learn ...“

More Related