Transcript CAS as Environment for Mathematical Microworlds Example

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
– Formula 1 as a CAS microworld
• Conclusions / Questions
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)
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“
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)
„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
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
Positioning microworlds within
the CAS usage „space“
CAS
„as is“
Real world
scenario
Simplified
model
Higher-level
Objects and ops.
Creation of
Production
Toolbox by Math.
Student
Project
(math.)
Application
reduction
Student
Project
(eng.)
Production
Toolbox for
Engineer
Student
microworld
CAS model
reduction
(Industrial
usage)
(Educational
usage)
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, ...
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
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
Formula 1 – Project with CAS (c‘d)
• Project result:
– Provision of structured worksheet
– Graphics and animation
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 )
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
Formula 1 - Microworld with CAS
(III)
• Example for possible activities:
– Produce a v(s) plot with restrictions
v
–
–
–
–
s
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
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 ...“