Transcript Mathematical Modeling for Mathematics Education

Mathematical Modelling and
Mathematical Education – What,
why and how?
Dr Max Stephens
Graduate School of Education
THE UNIVERSITY OF MELBOURNE
[email protected]
The world of work in the 21st century
In her plenary at ICTMA 15, Lyn English identified competencies
that are now seen as important for productive and innovative
work practices (English, Jones, Bartolini, Bussi, Lesh, Tirosh, &
Sriraman, 2008). Her list included:
 Problem solving, including working collaboratively on complex
problems where planning, overseeing, moderating, and
communicating are essential elements for success;
 Applying numerical and algebraic reasoning in an efficient,
flexible, and creative manner;
 Generating, analysing, operating on, and transforming
complex data sets;
 Applying an understanding of core ideas from ratio and
proportion, probability, rate, change, accumulation, continuity,
and limit;
The world of work in the 21st century
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Constructing, describing, explaining, manipulating, and
predicting complex systems;
Thinking critically and being able to make sound judgments,
including being able to distinguish reliable from unreliable
information sources;
Synthesizing, where an extended argument is followed across
multiple modalities;
Engaging in research activity involving the investigation,
discovery, and dissemination of pertinent information in a
credible manner;
Flexibility in working across disciplines to generate innovative
and effective solutions;
Techno-mathematical literacy (“where the mathematics is
expressed through technological artefacts” Hoyles, Wolf,
Molyneux-Hodgson, & Kent, 2010, p. 14).
Implications for schools and schooling
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These changes to the world beyond school cause us
to reconsider what we ask children to learn in
school
Human resource development requires learning to
become more future oriented, interdisciplinary,
involving problem solving and modelling that mirror
similar experiences beyond school
More powerful links are needed between
classrooms and the real world where students can
apply their mathematics to solve authentic problems
Students for the
st
21
century
“I think the next century will be the century of
complexity.”----- Stephen Hawking (2000)
We need to develop students who are:
 Knowledge builders
 Complex, multifaceted and flexible thinkers
 Creative and innovative problem solvers
 Effective collaborators and communicators
 Optimistic and committed learners
Mathematical modelling
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Modelling is a powerful vehicle for not only promoting
students’ understanding of a wide range of key
mathematical and scientific concepts, but also for helping
them appreciate the potential of the mathematical sciences
as a critical tool for analysing important issues in their lives,
communities, and society in general (Greer, Verschaffel, &
Mukhopadhyay, 2007)
Importantly, modelling needs to be integrated within the
primary school curriculum and not reserved for the
secondary school years and beyond as it has been
traditionally. Research has shown that primary school
children are indeed capable of engaging in modelling
(English & Watters, 2005)
English, ICTMA 15, 2011
Mathematical modelling
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The terms, models and modelling, have been used
variously in the literature, including … solving word
problems, conducting mathematical simulations,
creating representations of problem situations
(including constructing explanations of natural
phenomena), and creating internal, psychological
representations while solving a particular problem
(English & Halford, 1995; Gravemeijer, 1999; Lesh &
Doerr, 2003)
English, ICTMA 15, 2011
Mathematical modelling
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One perspective on models … is that of conceptual
systems or tools comprising operations, rules and
relationships that can describe, explain, construct, or
modify … a complex series of experiences
Modelling involves the crossing of disciplinary
boundaries, with an emphasis on the structure of
ideas, connected forms of knowledge, and the
adaptation of complex ideas to new contexts
(Hamilton, Lesh, Lester, & Brilleslyper, 2008)
English, ICTMA 15, 2011
Mathematical modelling
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Modelling activities provide students with
opportunities to repeatedly express, test, and refine
or revise their current ways of thinking.
Modelling problems need to be designed so that
multiple solutions of varying mathematical
sophistication are possible and such that students with
a range of personal experiences and knowledge can
participate
In this way, the mathematical experiences of students
become more challenging, authentic and meaningful
English, ICTMA 15, 2011
Teacher’s role
How does a teacher cultivate students’
thinking of modelling?
Outside School
School mathematics
Modelling
Modelling
By scientists/experts
Contrast
By students
Problem is familiar to them and they
have clear reasons to solve a problem.
They can observe a situation/phenomena
for a long time. Abstraction is relative easy.
These three points
are quite different
for students.
They know the modelling process and have
good modelling skills.
Ikeda, ICTMA 15, 2011
Pedagogical aims of modelling
As an objective
Modelling for its own sake
Teacher’s role
How does the teacher cultivate students’
thinking about modelling?
As a means to an end
Mathematical knowledge construction
Relation between modelling and
mathematical knowledge construction
Where to locate modelling
in the teaching of mathematics?
Ikeda, ICTMA 15, 2011
Teacher’s role
How does teacher cultivate students’
thinking of modelling?
Scientists and other experts
Problem is familiar with them.
They have clear reasons to solve a problem.
In school,
How about for students?
Why do students solve a problem?
Selecting Material, Setting a situation
Ikeda, ICTMA 15, 2011
What is an appropriate modelling task?
Galbraith (2007)
Introducing real world modelling tasks
(a) the importance of using models based on experience
Further Questions
Does the problem situation concern the surroundings
of students at present, in the past or in the future?
Is it relevant to most students or to a few students?
Future
Is it concerned with situations they will confront as
citizens, as individuals or in their profession/vocation?
Ikeda, ICTMA 15, 2011
Compare with PISA context categories
What is an appropriate modelling task?
Galbraith (2007)
Introducing real world modelling tasks
(b) motivation
Two points
To clarify the reason why someone had to
solve the problem
To set the appropriate situation so that students
can accept the problem posed by someone else
as their own problem
Ikeda, ICTMA 15, 2011
(b) motivation
Clarify why someone had to solve the problem in the first place
Set an appropriate situation so that students can
accept a problem as their own problem
Observing or analyzing the phenomenon or action
How can we win in a relay in school sports?
(Osawa,2004)
It is important to consider the order
of runners, how to pass the baton, etc
One of the issues:
Focusing on the baton pass
When does the next runner begin to run to get the baton
from the previous runner, for the shortest baton pass time?
Ikeda, ICTMA 15, 2011
Distilling essential mathematical structure in
complex situations
Abstract content can be only understood
by connecting it with its concrete contents.
Concrete activity is essential!
A real world
Mathematical world
Students have limited experience to observe
a real world situation/phenomena.
Distilling essential structure
is difficult for students
Observation/Manipulation
by using Concrete Model
Ikeda, ICTMA 15, 2011
How can teacher make students realize
how to control many variables to solve
a real world problem?
Generating relating variables
Checking whether or
not generated variables
affect problem solving
Is it possible to solve
by using my acquired
mathematics knowledge?
Conflicting Situations
Meaningful conflicting situations so that students
can derive key ideas.
Setting up assumptions as simple as possible at
the beginning, after then modifying them into
more general situation gradually.
Ikeda, ICTMA 15, 2011
Communication on mirror problem
Formulating a real problem
What minimum size of mirror do you need
in order to see all your face?
(Shimada,1990; Matsumoto, 2000; Ikeda,2004)
Is the following sentence true or false?
Half size of mirror is needed at least in order to see my whole face
It might be true because it
seems to be half by drawing
a figure.
It might be false because if
the mirror is far from my face,
it is sufficient to use small mirror.
Let’s draw a figure to check their answer.
Ikeda, ICTMA 15, 2011
Communication on mirror problem
Please draw a figure on the blackboard.
How can we treat
these variables?
Setting Assumptions
How about the
width of the face?
Are three points, namely the point
of the eye, the point of head and
the point of chin, on a same line?
Are the two planes, namely
face and mirror, parallel or not?
Is the eye located at the midpoint between the
point of head and the point of chin?
Ikeda, ICTMA 15, 2011
Communication on mirror problem
Are the two planes, namely
face and mirror, parallel or not?
It seems easy to solve
the problem if the relation
of the two planes is parallel.
However, the relation of two planes
is not always parallel in a real
situation.
Conflicting Situations
If the relation of two planes is not parallel,
it is too difficult to solve the problem.
Let’s set up an assumption that the relation of two planes is parallel
at first. Regarding the case of not parallel, let’s consider that later.
Ikeda, ICTMA 15, 2011
side width
When we see one ear with two eyes
left ear
left eye
right eye
right ear
Mirror Size: Width between left eye and right ear
Ikeda, ICTMA 15, 2011
Side width
When we see one ear with one eye
Mirror Size: Width between left eye and left ear
Error elimination
Is it OK
in any situation?
left ear
left eye
right eye
right ear
Invisible
Assumption
Width between two eyes is shorter than
double of width between left eye and left ear
Ikeda, ICTMA 15, 2011
Pedagogical aims of modelling
As an objective
Modelling for its own sake
Teacher’s role
How does the teacher cultivate students’
thinking about modelling?
As a means to an end
Mathematical knowledge construction
Relation between modelling and
mathematical knowledge construction
Where to locate modelling in
the teaching of mathematics?
Ikeda, ICTMA 15, 2011
Thinking about the balance between
modelling and constructing math knowledge
Role 1
Build up the model to mathematize
in order to solve real world problems
Real world
Mathematical
world
Role 2
Build up the model to test the
validity of mathematical concepts
Real world
Mathematical
world
Clarifying
From which world is the problem derived ?
Ikeda, ICTMA 15, 2011
Spread Infectious Diseases – modelling
a natural disaster for senior high school
students
Dr Max Stephens
Graduate School of Education
THE UNIVERSITY OF MELBOURNE
[email protected]
Infectious disease
Movie Contagion (2011) –
“Don’t speak to anyone.
Don’t touch anyone!”
reflects the media frenzy
attaching to the perceived
threat.
Infectious disease
media images
Some have great potential to scare
Emerging Infectious Diseases (EIDs)
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A more careful study of the web gives a less
panicked view, and causes us to us some important
questions
Since 1940 more than 300 Emerging Infectious
Diseases have been identified. However, most do
not take off
So we have to ask why some do and some don’t
Emerging diseases go global
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Mark Woolhouse (2008) Centre for Infectious
Diseases at the University of Edinburgh:
Novel human infections continue to appear all over
the world, but the risk is higher in some regions than
others. Identification of emerging-disease 'hotspots'
will help target surveillance work
Nature 451, 898-899 (21 February 2008)
Global trends in emerging infectious
diseases
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Jones et al. Nature 451, 990-993 (21 February 2008)
Modelling Spread of disease
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One Sunday evening, five people with
infectious influenza arrive by plane in a large
city of about 2 million people
They then go to different parts of the city and
so the disease begins to spread
At first when a person becomes infected, the
disease is latent/incubating and he/she shows
no sign of the disease and cannot spread it
Modelling Spread of disease
About one week after first catching the disease
the person becomes infectious and can spread
the disease to other people
The infectious phase also lasts for about one
week. After this time the person is free from
influenza, although he/she may catch it again
at some later time
Modelling Spread of disease
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Scientists are trying to model the spread of
influenza. They make a simplifying assumption
that the infection progresses in one week units
That is, they assume that everyone who
becomes infected does so on a Sunday
evening, has a one week latent period, and
then becomes infectious one week later, and is
free of infection exactly one week after that
Modelling Spread of disease
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People who are free of the disease are called
“susceptibles” (= capable of catching it)
The scientists also assume that the city
population is large and so can be assumed to
be constant for the duration of the disease.
That is, they ignore births, deaths and any
movements into or out of the city
Modelling Spread of disease
It’s very hard to follow these descriptions. A
picture (Becker, 2009) shows the key stages:
Modelling Spread of disease
The scientists assume that each infectious person
infects a fixed fraction f of the number of
susceptibles, so that the number of infectious
people at week n + 1 is:
f × (number of susceptibles at week n) × (number of infectious at week n)
and the number of susceptibles at week n+1 is:
(number of susceptibles at week n) + (number of infectious at week n) – (the
number of infectious at week n + 1)
Modelling Spread of disease
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The modelling uses the variable ‘weeks’. This
simplification ensures that at any time there
are only susceptible people and infectious
people. The model excludes people who are in
a “latent” stage – i.e. infected but not
infectious
This allows the model to be investigated easily
Modelling Spread of disease
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The number of infectious people and the
number of susceptible people will be constant
from week to week.
Choosing values of f between 10-6 and
2 × 10-6 we can make a model showing how
the number of infectious people changes from
week to week
Modelling Spread of disease
Three equations connect In the number of
infectious people in each week n and Sn the
number of susceptible people at week n:
In+1 = f × Sn × In
Sn+1 = Sn + In f × Sn × In
In + Sn = 2 × 106, eliminating Sn to give
In+1 = f × [2 × 106 In ] × In
Modelling spread of disease
f = 10-6 means that each infectious person
spreads the disease to 2 people in a week.
It will be important to show how any limiting
values are connected to the size of f and to the
size of the population.
For what values of f will there be a situation
where the number of infectious people
eventually oscillates between two values?
Modelling Spread of disease
How can simple technology help us to investigate
In+1 = f × [2 × 106 In ] × In
One accessible way for senior students is to use
EXCEL to plot graphs for different values of f.
The recursion relation cannot be investigated
easily without technology.
Graphs for different values of f
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Remember that f = 10-6 means that each infectious
person spreads the disease to 2 people in a week
The following four graphs show what happens when
f = 0.1 × 10-6 , f = 0.5 × 10-6 , f = 0.8 × 10-6 , f
= 1 × 10-6
The first two show low rates of infection:
f = 0.5 × 10-6 means that only one person is
infected by each infectious person in a week, this
rate of infection is too low to spread the disease
Graphs for different values of f
y=0.1*10^-6*(2*10^6-x)*x
1
1
1
1
0
0
0
y=0.5*10^-6*(2*10^6-x)*x
5
5
y=0.1*10^-6*(2*10^6x)*x
5
y=0.5*10^-6*(2*10^6x)*x
5
5
0
5
10
15
5
20
0
10
20
30
f=
y=10^-6*(2*10^6-x)*x
y=0.8*10^-6*(2*10^6-x)*x
800000
600000
400000
y=0.8*10^-6*(2*10^6x)*x
200000
0
0
20
40
60
1200000
1000000
800000
600000
400000
200000
0
y=10^-6*(2*10^6-x)*x
0
10
20
30
Graphs for different values of f
y=1.1*10^-6*(2*10^6-x)*x
1200000
1000000
800000
600000
400000
200000
0
1500000
y=1.1*10^-6*(2*10^6x)*x
1000000
y=1.4*10^-6*(2*10^6x)*x
500000
0
0
10
20
30
40
0
y=1.5*10^-6*(2*10^6-x)*x
1500000
1000000
-y=1.5*10^-6*(2*10^6
x)*x
500000
0
0
y=1.4*10^-6*(2*10^6-x)*x
20
40
60
20
40
60
An interesting feature appears
for f = 1.5 × 10-6 , where the
graph begins to oscillate. This
occurs when the value for y at
any week is equal to the value
of y two weeks later.
Graphs for different values of f
y=1.9*10^-6*(2*10^6-x)*x
y=1.6*10^-6*(2*10^6-x)*x
2000000
2000000
1500000
1500000
1000000
y=1.6*10^-6*(2*10^6x)*x
500000
1000000
-y=1.9*10^-6*(2*10^6
x)*x
500000
0
0
10
20
30
40
0
0
20
40
The oscillating feature which appears for f = 1.5 × 10-6 ,
appears to continue for f = 1.6 × 10-6 , and possibly (?) for f =
1.9 × 10-6. But do we know if it starts at f = 1.5 × 10-6 ? We
need other technology to decide this. TI-Nspire CAS TE
worksheet can answer this question.
60
Utilising CAS to investigate further
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Only after looking at the different graphs and the
effect of different values of f does it make sense to
use CAS technology to explore the mathematical
relationships.
This cannot be done by hand. And should not be.
Yet a CAS solution to the equation provides a
powerful finding that students can anticipate from
their exploratory work using EXCEL
Where p = 2 × 106 , f > 3/p = 1.5 × 10-6
Implications for teaching
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Modelling the spread of disease requires much
more than traditional textbook resources
To explore the mathematical relationships students
need access to programs such as EXCEL
CAS capacity is highly useful
Web-based information is important for students to
understand the context
E-book formats integrate these different resources
in ways that students and teachers can easily use.
Concluding: What principles of curriculum design
are important when considering modelling
activities? How do they help us to think about the
balance between mathematical modelling and
mathematical education?
Principles of curriculum design
How will a modelling investigation help develop:
 Underpinning mathematical concepts and skills
from across the discipline (numerical, spatial,
graphic, statistical and algebraic)
 Mathematical thinking and strategies
 Appreciation of context
 Communicating to a wider audience
Principles of curriculum design
What tasks are suitable for modelling activities?
 Tasks that require information and resources that
are not easily available in textbooks or single
printed source
 Tasks that are extended in time
 Tasks that are interdisciplinary, crossing over and
integrating several curriculum areas
 Tasks that link mathematics to the real world
Principles of curriculum design
A modelling investigation changes teacher’s roles:
 Students moving in different directions
 Technological fluency is not the same as
mathematical fluency
 Ensuring that students understand and
communicate the key mathematical ideas
 Clear criteria on mathematical performance,
reasoning, and communication are needed
Principles of curriculum design
Teachers as designers:
 More than just using new technology
 Deciding what technology is mathematically
appropriate for students and whether students
are mathematically ready to use the technology
 Getting students to ask: what are we looking for;
framing and repeatedly testing conjectures;
justifying and communicating conclusions.
Principles of curriculum design
Teachers as designers:
 Writing tasks: investigative and problem-solving
tasks are very different from textbook tasks
 Developing new mathematical skills (especially in
graphing and data); representing and
interpreting graphs and data displays
 Exploring data sets; cleaning up large data sets;
sampling and Exploratory Data Analysis
Principles of curriculum design
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Why spend time on a modelling investigation?
Only if the opportunities and time invested in
designing and using a modelling investigation:
advance students’ mathematical knowledge
build their mathematical capacity in ways that
will inform their other school subjects, and
build habits of inquiry that they can carry
forward into their future study, life and work
Real world <=>Mathematical world
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In this activity students use mathematical
relationships that are partly new to them
These relationships can be manipulated using
technology using mathematical understanding
and understanding of the phenomena
Different mathematical behaviours can be
produced by careful variation of key terms
These variations are powerful because they can
help explain real world phenomena
Real world <=>Mathematical world
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These mathematical variations require careful
selection and analysis by students
This analysis depends on students being able to
connect mathematical behaviours to the
phenomena that they are trying to model
These mathematical variations help to explain
why some diseases take off while others
gradually die out
What is the balance between mathematical
modelling and mathematical education?
Role 1
Build up the model to mathematize
in order to solve real world problems
Real world
Mathematical
world
Role 2
Build up the model to test the
validity of mathematical concepts
Real world
Mathematical
world
Ikeda, ICTMA 15, 2011