Mathematical Modelling and Mathematical Education – What, why and how? Dr Max Stephens Graduate School of Education THE UNIVERSITY OF MELBOURNE [email protected]

Mathematical Modelling and Mathematical Education – What, why and how? Dr Max StephensGraduate School of Education THE UNIVERSITY OF [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 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 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 21st 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

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


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)



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)



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


Outside School

Modelling

School mathematics

Problem is familiar to them and they

have clear reasons to solve a problem.

By scientists/experts By students Contrast

They can observe a situation/phenomenafor a long time. Abstraction is relative easy.

They know the modelling process and have

good modelling skills.

These three points are quite differentfor students.

How does a teacher cultivate students’thinking of modelling?

Teacher’s role

Modelling

Ikeda, ICTMA 15, 2011

Pedagogical aims of modelling

Modelling for its own sakeAs an objective

Mathematical knowledge construction As a means to an end

How does the teacher cultivate students’thinking about modelling?

Teacher’s role

Where to locate modellingin the teaching of mathematics?

Relation between modelling and mathematical knowledge construction


How does teacher cultivate students’thinking of modelling?

Teacher’s role

Scientists and other experts

In school, How about for students?

Why do students solve a problem? Selecting Material, Setting a situation

Problem is familiar with them. They have clear reasons to solve a problem.


Further Questions

Introducing real world modelling tasks

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?

What is an appropriate modelling task?

(a) the importance of using models based on experience

Galbraith (2007)

Is it concerned with situations they will confront as citizens, as individuals or in their profession/vocation?

Future

Compare with PISA context categoriesIkeda, ICTMA 15, 2011

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

(b) motivation

Introducing real world modelling tasks

What is an appropriate modelling task?Galbraith (2007)


Observing or analyzing the phenomenon or action

It is important to consider the order of runners, how to pass the baton, etc

When does the next runner begin to run to get the baton from the previous runner, for the shortest baton pass time?

Clarify why someone had to solve the problem in the first placeSet an appropriate situation so that students can accept a problem as their own problem

(b) motivation

(Osawa,2004)How can we win in a relay in school sports?

Focusing on the baton passOne of the issues:


Distilling essential mathematical structure in complex situations

Abstract content can be only understood by connecting it with its concrete contents.

A real world Mathematical world

Distilling essential structure is difficult for students

Observation/Manipulation by using Concrete Model

Students have limited experience to observe a real world situation/phenomena.

Concrete activity is essential!


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.

Conflicting Situations

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 acquiredmathematics knowledge?


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.

Communication on mirror problemFormulating a real problem What minimum size of mirror do you need in order to see all your face? (Shimada,1990; Matsumoto, 2000; Ikeda,2004)


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?

Please draw a figure on the blackboard.

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?

How can we treat these variables?

Setting Assumptions

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.

If the relation of two planes is not parallel, it is too difficult to solve the problem.

Conflicting Situations

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.

However, the relation of two planes is not always parallel in a real situation.

Communication on mirror problem


side width

When we see one ear with two eyes

left eye

right eyeleft ear

right ear

Mirror Size: Width between left eye and right ear


Mirror Size: Width between left eye and left ear

When we see one ear with one eye

Is it OKin any

situation?

Error elimination

Width between two eyes is shorter than double of width between left eye and

left ear

Assumption

left eye

right eyeleft ear

right ear

Invisible

Side width


Pedagogical aims of modelling

Modelling for its own sakeAs an objective

Mathematical knowledge construction As a means to an end

How does the teacher cultivate students’thinking about modelling?

Teacher’s role

Where to locate modelling in the teaching of mathematics?

Relation between modelling and mathematical knowledge construction


Build up the model to mathematize in order to solve real world problems

Role 1

Build up the model to test the validity of mathematical concepts

Role 2

Real world Mathematical world


Thinking about the balance between modelling and constructing math knowledge

From which world is the problem derived ?Clarifying


Spread Infectious Diseases – modelling a natural disaster for senior high school students Dr Max StephensGraduate School of Education THE UNIVERSITY OF [email protected]

Infectious disease

Movie Contagion (2011) –

“Don’t speak to anyone. Don’t touch anyone!”

reflects the media frenzyattaching to the perceived threat.

Infectious disease media images

Some have great potential to scare

Emerging Infectious Diseases (EIDs)

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

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

Jones et al. Nature 451, 990-993 (21 February 2008)

Modelling Spread of disease

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


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


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


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


It’s very hard to follow these descriptions. A picture (Becker, 2009) shows the key stages:


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)


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


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


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 × InSn+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?


How can simple technology help us to investigate In+1 = f × [2 × 106

In ] × InOne 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

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


y=0.1*10^-6*(2*10^6-x)*x

0001111

0 5 10 15 20

y=0.1*10^-6*(2*10^6-x)*x

y=0.5*10^-6*(2*10^6-x)*x

5

5

5

5

5

5

0 10 20 30

y=0.5*10^-6*(2*10^6-x)*x

y=0.8*10^-6*(2*10^6-x)*x

0

200000

400000

600000

800000

0 20 40 60

y=0.8*10^-6*(2*10^6-x)*x

y=10^-6*(2*10^6-x)*x

0200000400000600000800000

10000001200000

0 10 20 30

y=10^-6*(2*10^6-x)*x

f =


y=1.1*10^-6*(2*10^6-x)*x

0200000400000600000800000

10000001200000

0 10 20 30 40

y=1.1*10^-6*(2*10^6-x)*x

y=1.4*10^-6*(2*10^6-x)*x

0

500000

1000000

1500000

0 20 40 60

y=1.4*10^-6*(2*10^6-x)*x

y=1.5*10^-6*(2*10^6-x)*x

0

500000

1000000

1500000

0 20 40 60

-y=1.5*10^-6)*2*10^6x*(x

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.


y=1.9*10^-6*(2*10^6-x)*x

0

500000

1000000

1500000

2000000

0 20 40 60

-y=1.9*10^-6)*2*10^6x*(x

y=1.6*10^-6*(2*10^6-x)*x

0

500000

1000000

1500000

2000000

0 10 20 30 40

y=1.6*10^-6*(2*10^6-x)*x

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.

Utilising CAS to investigate further

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

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


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


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


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.


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


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

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

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

Build up the model to mathematize in order to solve real world problems

Role 1

Build up the model to test the validity of mathematical concepts

Role 2



What is the balance between mathematical modelling and mathematical education?


Documents

Mathematical Modelling and Mathematical Education – What, why and how? Dr Max Stephens Graduate School of Education THE UNIVERSITY OF MELBOURNE [email protected]