Mathematical Modeling with Dynamic Geometry Software

1
Mathematical Modeling with Dynamic Geometry
Software

• Tai-Yih Tso
• Department of Mathematics
• National Taiwan Normal University

2
Research Interests

• My major research field is mathematics education,
  especially in how to integrate technology into
  mathematics learning and teachers education.
• Earlier works
• Dynamics windows for ellipse instruction.
• Cognitive characteristic of senior high school
  students in constructing multiple representations
  of ellipse.
• The design of junior high school geometry based
  on van Hiele model.
• The study of interrelationship between spatial
  abilities and van Hiele levels of thinking in
  geometry of eigth-grade students.
• The study of junior high school students
  symmetry concept image.
• The critical phase curve of Van der pol equation.
  
• The derivation of two parallel zero-finding
  algorithm of polynomials.

3
Current research topics of mine

• One of my current research topics is studying
  mathematical modeling in mathematics education.
• I am interested in investigating the theory and
  practice of teaching and learning mathematical
  modeling, particularly by using dynamic geometry
  software in the process of mathematical modeling.

4
Contents

• I. Introduction
• II. Theoretical Framework
• III. A framework of mathematical modeling
• IV. Case studies
• V. Concluding remarks

5
Introduction

• In this presentation, I will talk about my
  experience of integrating computer into a class
  for potential teachers. The goal of this course
  is to provide an opportunity for potential
  teachers to explore geometry by using the dynamic
  geometry software, such as Carbri, The Geometers
  Sketchpad. Moreover, they are expected to use
  this technical tool and their geometry knowledge
  to teach geometry in middle schools in the
  future. Usually, I use mathematics modeling as my
  teaching strategy in class.

6
Why students need to learn mathematical modeling

• Why students ought to learn mathematical modeling
  is that it provides a means for understanding the
  world around us, for coping with everyday
  problems, for developing mathematical knowledge,
  and for preparing for future professions.
• Today mathematical model and modeling have
  invaded a great variety of disciplines.
• Mathematical modeling has been substantially
  supported and accelerated by the availability of
  powerful electronic tools, such as computers with
  their enormous communication capabilities.
  (Werner Blum ET AL. 2002)

7
Structural Aspect of Mathematical Modeling

• A model is a structure R(S,P,M,T). In this
  relation structure, M is a model made by a
  subject S from a real world prototype T for the
  particular purpose of P (Leo Apostel, 1960) .
• Thus the outcome model M of a certain T varies by
  two factors which are the subject S and purpose
  P. In other words, the same prototype T would
  lead to different models M due to different
  purposes and subjects.

8
Structural Aspect of Mathematical Modeling

• A model as subjects taking the object as model of
  the object during the time and in view of the aim
  (H. Stachowiak,1973) .
• He mentioned that in addition to subject and
  purpose, there is still another factor affecting
  the outcome model, which is time.
• He pointed out three important properties of a
  model the mapping property, the shortening
  property, and the pragmatic property.

9
Conceptual Aspect of Mathematical Modeling

• Models are conceptual systems (consisting of
  elements, relations, operations, and rules
  governing interactions) that are expressed using
  external notation system, and that are used to
  construct, describe, or explain the behaviors of
  other systems perhaps so that the other system
  can be manipulated or predicted intelligently.
• A mathematical model focuses on structural
  characteristics (rather than, for example,
  physical or musical characteristics) of the
  relevant system. (Richard Lesh Helen M. Doerr,
  2000)

10
Mathematical Modeling

• Mathematical models are mathematical structure
  describing a real world phenomenon that would
  differ according to different subjects, purpose
  and time being.
• Mathematical modeling is a process that attempts
  to match observation with symbolic statement by
  creating a mathematical representation of some
  phenomenon in the real world in order to gain a
  better understanding of that phenomenon.

11
Theoretical Framework

• ????????

12
(No Transcript)
13
(No Transcript)
14
Theoretical framework
15
environments supporting
16
Reflection-on-action

• A action of metacognition involves stepping back
  and representing a difficulty, searching for
  solution, and conjecturing action in the
  construction or reconstruction knowledge
• (Noddings,1984).

• Reflection includes a state of doubt and an act
  of resolution.
• (Deway, 1933)

• The complex schema of knowledge is active in the
  long-term memory during performance of
  reflection-on-action

• Reflective abstraction is a process in which
  individual try to construct abstract structures
  in long-term memory, operations by reflecting on
  their activities, and the arguments used in
  social interaction.
  (piaget, 1976)

17
A framework of mathematical modeling
Mathematical modeling as learning activities for
reflection-on-action

• Mathematical modeling is a process of using
  mathematical language to describe, communicate,
  express, and think about the real world

• Learning mathematics is like learning language.
  It is not only to learn vocabulary (symbol,
  definition), grammar(property, rule, procedure)
  but more important to use it (mathematical
  modeling).

• Mathematical modeling has usually been applied in
  the research field, students have seldom
  experience in the process of building a
  mathematical model.

• A major benefit in learning how to do
  mathematical modeling for students is to
  encourage them developing a particular way of
  reflecting and acting on mathematics through
  making connections between mathematics and real
  world.

• Mathematical modeling is incorporated into the
  learning activities for two reasons for
  examplification and application (Nesher, 1989).

18
Mathematical Modeling in multiple representation
window
19
An example for learning mathematical modeling
A Real World Problem There are three fire
stations and a fire center(119) in a city. A fire
is reported to the fire center at a certain
address, which of the three fire stations should
send firefighters to the scene? How do we know
which is closest?
20
Two Approaches for the Fire Problem

• Suppose we had a city map on which the sites of
  the fire stations are indicated with dots.
• We could use a ruler measuring the distance from
  the fire to each station on the map.
• We could divide the map to three regions, each
  region containing one fire station and consisting
  of all the points closer to that station than to
  any other.

21
Mathematical model

• (Voronoi Diagrams)
• A,B,C are three distinct points on a plane.
• How do we divide the plane into three regions,
  each region containing a point and consisting of
  all the points closer to that point than to any
  other points?
• 2. (Inverse problem)

22
A case study on modeling a heart-shaped figure at
the bottom of a cup

• Due to the fact that applying computer to
  mathematical modeling is important for students
  and they seldom have the chance to experience it.
• It is necessary to provide them with a
  computational environment. In order to design
  this environment, we need to understand the
  mathematical modeling process of students, and
  thus a case study is done on potential teachers,
  for teachers need to have the experience before
  they can teach students.
• The major concerns of this case study are the
  process of potential teachers doing mathematical
  modeling, and the role of the dynamic geometric
  software in the process.

23
Experimental Design

• The subjects of this experiment are 24 potential
  teachers divided into five groups.
• The study of this paper would be focusing on the
  mathematical modeling process of one group came
  up with a more complete result.
• Only a more complete mathematical modeling
  process, it can help the researcher draw out the
  features of this process .

24
Experimental Design

• The setting of this experiment was in the
  computer room where each student had his own
  computer to work on. The students gathered into
  groups and were encouraged to discuss and solve
  the task. The classroom was videotaped and the
  discussion of each group was recorded separately.
  

25
A case study on modeling a heart-shaped figure at
the bottom of a cup

• As an example, I shall illustrate students
  modeling process with the Geometers Sketchpad
  (GSP).
• It starts generally from specifying the real
  world phenomena or problems in the process of
  mathematical modeling.

26
Specifying real-world phenomena

• For the experiment that day I brought a
  cylindrical mug and a flash light to class and
  let students look at the bottom of mug with the
  flash light as a point source of light overhead.
• By experimenting and observing this phenomenon,
  they wanted to understand two modeling problems
  (1)Why did the bottom of the cup display a heart
  shape? (2)What is the equation of the curve of
  the heart shape in mathematics?

27
Making Assumptions and Simplifications

• The students analyzed the bright pattern within
  group discussion and found that the bottom of the
  mug is illuminated by rays directly from the
  flash light and by rays reflected one or more
  times.
• They proposed a hypothesis that the hart-shaped
  pattern is mainly contributed by the reflections
  of rays at the bottom.
• They simplified the cylindrical mug, a three
  dimension model, to the circular rim, a plane
  model.

28
Examining assumptions with GSP

• After making the assumption, a student said that
  we could use GSP to examine how the reflections
  of rays work.

They found that the computer graph is similar to
the heart-shaped figure at the bottom of the mug
and could explain the real world phenomena.
29
Exploring the Mathematical Model

• Students used computer simulation to develop a
  useful representation system for mathematical
  model (creating a coordinate system and choosing
  suitable parameters to describe the curve).

30
Modifying the Mathematical Model

• Students constructed a conceptual system, which
  is expressed by mathematical symbols, formulas,
  and rules in order to interpret the behaviors of
  the real world.

31
Experiment Observations and Findings

• From the above observations, I would like to
  conclude them into the following three findings.
• Potential teachers explore in between three
  worlds, which are the real world, the
  mathematical world, and the computer world.
• The transition between the three worlds is based
  on reflective acting.
• The dynamic geometric software not only
  simulates the mathematical model, but also
  connects two aspects of mathematics, which are
  the experimental mathematics and the reasoning
  mathematics.

32
Experiment Findings

• Real world
• Observing physical phenomenon
• and forming modeling task
• 2.Making assumptions and
• simplification to form real-world model.
• 11.Examining the mathematical model.

Mathematical world 6. Creating
coordinate system and setting representation. 7.
Choosing parameters and studying the relations
among them. 8. Forming the equation
(mathematical model) 10.Modifying the
mathematical model. 13. investigating the
mathematics history abort the mathematical
model.
Reflecting and Acting
Computer world 3.Simulating the real-world
model and forming the computing
model. 4.Examining the assumptions and
simplification 5.Developing conceptual tool to
think mathematically. 9.Simulating the
mathematical model. 12.Simulating the modified
mathematical model.
33
Concluding remarks
making meaningful situations into mathematical
descriptions
Mathematical World
Real World
Using symbolically described situation to
interpret the real world
Experiment Geometry
Bridge
Reasoning Geometry
observing, manipulating
conjecturing, justifying
GSP Simulation
Computing World
34
BELIEVES Other Knowledge Fundamental assumptions
METACOGNITION Monitoring, evaluation, planning
and controlling.
PROBLEM DOMAIN Domain-specific reasoning
MATHEMATICAL DOMAIN Domain-specific reasoning
COMPUTATION DOMAIN Simulation reasoning
From the cognitive psychology point of view, we
can use this figure to explain.
35
Research issues of mathematical modeling with DGS
in mathematics education

• What authentic modeling materials are available?
• How does the authentic materials affect students
  mathematical learning and modeling?
• What are the process components of modeling?
• What is the meaning and role of abstraction,
  proof, proving, formalization, and generalization
  in the process of modeling?
• What are the additional roles and functions of
  computer in the process of modeling?
• What training strategies can be used to assess a
  future teachers ability to teach mathematical
  modeling?

Mathematical World
Real World
Computing World
36

Thank you Comments are welcomed
37
Two Questions about GSP

• What is the difference between the circle drawing
  tool in GSP and the mechanical compass?
• Do we have any tools for drawing a straight line?
  In other words, how can I guarantee that the line
  segment I get from a ruler is a definite straight
  line?

38
Human intellectual activities

• According to Vygostkian socio-cultural
  perspective, the formation of human intellectual
  activities requires the use of technical and
  symbolic tools as mediators of acting and
  thinking.
• In order for the students to solve problems
  systematically, I believe to introduce the
  culture and history of human development in
  mathematics is helpful for increasing their
  interest in both geometry and history in
  mathematics.

39
What is the difference between the circle drawing
tool in GSP and the mechanical compass? Study
history of mathematics before Build up a
mathematical model

• The students studied Euclids Elements in order
  to investigate the Euclidean construction tools.
• They noted that the first three postulates of
  Euclids Elements is to state the primitive
  construction tools. Let the following be
  postulated
• to draw a straight line from any point to any
  point.
• to produce a finite straight line continuously in
  a straight line.
• to describe a circle with any center and
  distance.
• The first two postulates state what we can do
  with a ruler(Euclidean straightedge) and
• the third postulate tell us the circle drawing
  instrument(Euclidean compass).
• The Euclidean straightedge is a tool to draw the
  straight line of passing through given two
  distinct points and the Euclidean compass is a
  tool to draw a circle of given a center and
  passing through a given point. Both of these two
  primitive construction tools can not be marked.

40
The Euclidean compass and the modern mechanical
compass

• The students noted that the construction tools of
  the Geometers Sketchpad are just the Euclidean
  construction tools after studying the postulates
  of Elements.
• They also found that the Euclidean compass
  differs from the modern mechanical compass which
  can draw a circle of given center and radius.
• The Euclidean compass (computer compass) is often
  referred as a collapsing compass that if either
  leg is lifted from the paper the tool immediately
  collapses.
• The students asked naturally a problem if the
  Euclidean compass can serve as the modern
  mechanical compass.

41
Build up a mathematical model

• This is not easy for our students to obtain the
  idea of the construction procedure(mathematical
  model ).
• They continuously read the Elements and found
  that the proposition 2 of Book I.
• To place at a given point a straight line equal
  to a given straight line is useful for the
  model.
• The construction of the proposition 2 shows that
  the Euclidean compass the Euclidean
  straightedge is equivalent to the modern compass.

42
students report

• In our original knowledge, we believed that the
  history of mathematics and the drawing tool of
  mathematical software are unrelated.
• Now we find that the history of mathematics can
  give us hints to overcome the restriction of
  computer technology.
• We understand much more about the Euclids
  Elements.

43
Do we have any tools for drawing a straight line?

• The second question is particularly practical.
• James Watt (1736-1819) had a problem when he
  worked on improving the efficiency and power of
  steam engines.
• His problem was how to turn the linear motion
  into the circular motion of a wheel and vice
  versa.

44
Study history of mathematics before making
computer simulation

• The students read the book A Survey of Geometry
  by Howard Eves, and the paper How to Draw a
  Straight Line by Daina Taimina.
• They noted that An outstanding geometrical
  problem of the last half of the nineteen century
  was to discover a linkage mechanism for drawing a
  straight line.

45

• A solution was finally found in 1864 by a French
  army officer, A. Peaucellier (1832-1913), and an
  announcement of the invention was made by A.
  Mannheim (1831-1906), a brother officer of
  engineers and inventor of the so-called Mannheim
  slide rule, at a meeting of the Paris Philomathic
  Society in 1867.
• But the announcement was little heeded until
  Lipkin, a young student of the celebrated Russian
  mathematician Chebyshev (1821-1894),
  independently reinvented the mechanism in 1871.
• Lipkin received a substantial reward from the
  Russian Government.
• Peaucelliers merit was finally recognized and he
  was awarded the great mechanical prize of the
  Institut de France.

46
students report

• Form the History, we noted that a pure
  geometrical problem can be connected with a
  mechanical problem.
• We also learnt what is the Peaucellier-Lipkin
  Linkage.

47
Conclusion Remarks

• 1.Through the history of mathematics they can
  understand where the mathematic questions came
  from and in which direction they will develop.
• 2.It gives students a complete concept and a
  picture as a whole of geometry.
• 3.While tracing the paths of mathematics history,
  they can get ideas for solving their problems.
• 4.It provides an environment for students to
  experience reflective thinking.

48
Research issues of mathematical modeling with DGS
in mathematics education

• What authentic modeling materials are available?
• How does the authentic materials affect students
  mathematical learning and modeling?
• What are the process components of modeling?
• What is the meaning and role of abstraction,
  proof, proving, formalization, and generalization
  in the process of modeling?
• What are the additional roles and functions of
  computer in the process of modeling?
• What training strategies can be used to assess a
  future teachers ability to teach mathematical
  modeling?

Mathematical World
Real World
Computing World
49

Thank you Comments are welcomed
50

• The floor is open for suggestions and comments.
• Thank you for your participation.

51
(No Transcript)
52
(No Transcript)
53
(No Transcript)
54
(No Transcript)
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)
58
Research issues of mathematical modeling with DGS
in mathematics education

• What authentic modeling materials are available?
• How does the authentic materials affect students
  mathematical learning and modeling?
• What are the process components of modeling?
• What is the meaning and role of abstraction,
  proof, proving, formalization, and generalization
  in the process of modeling?
• What are the additional roles and functions of
  computer in the process of modeling?
• What training strategies can be used to assess a
  future teachers ability to teach mathematical
  modeling?

Mathematical World
Real World
Computing World