MAthematices and the SEA (and other things)

1
MAthematices and the SEA (and other things)

• Johan Deprez
• SEAMA-conference, Antwerp, 31/05/10
• www.ua.ac.be/johan.deprez gt Documenten

2
Overview

• Introduction
• Example 1 Journey of the drilling rig Yatzy
• Some comments
• Example 2 The future of the Belgian population
• Some comments

3
Introduction
second example comes from my classes

• about myself
• mathematics teacher
• for about 20 years
• higher education at university level (but not
  university)
• students in applied economics
• basic mathematics course

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Introduction

• about myself
• mathematics teacher
• mathematics educator
• related to secondary mathematics education
• informally Uitwiskeling magazine for secondary
  math teachers
• formally teacher education at university
  (KULeuven, Universiteit Antwerpen) for about 15
  years

first example comes from teachers with whom I work
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Introduction

• about myself
• mathematics teacher
• mathematics educator
• researcher in mathematics education
• for about 2 years
• applications in mathematics education
• concrete versus abstract in math education

6
Journey of the drilling rig Yatzy

• owned by Diamond Offshore
• operated in Brazil by Petrobras (NOC)
• built by shipyard Boelwerf Temse (1989)

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Journey of the drilling rig Yatzy
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Journey of the drilling rig Yatzy

• Yatzy was transported along the
• river Scheldt from Temse to
• Rotterdam on 11 January 1989
• problem passing power line
• hanging over river
• engineers problem is transformed into a problem
  for secondary school math classes (actually for
  teachers )
• Dirk De Bock and Michel Roelens, The journey of
  the Drilling Rig Yatzy Today on Television,
  Tomorrow as a Large-scale Modelling Exercise.
• in Jan de Lange et al. (1993) Innovations in
  Maths Education by Modelling and Applications,
  Ellis Horwood

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Introduction to the context
depth chart of the river Scheldt at the crossing
of the power line
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Introduction to the context
height of the power line
information about the tides
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First analysis

• conclusions
• problem is at the top
• pass near left bank (x180)
• pass at low water spring tide

• students add
• river profile under power line
• (lowest point of) power line
• water levels at high and low water spring tide
• and then experiment using scale model of Yatzy

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A mathematical model for the cable (engineers
version)

• power line follows catenary curve
• height is given by a function of the form
• using coordinates of lowest point
• c 422.92, d 79.80
• using left end (numerical methods)
• a 1491.09

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A mathematical model for the cable (students
version)

• power line approximately follows parabola
• height is given by a function of the form
• using coordinates of lowest point
• c 422.92, d 79.80
• using left end
• a 0.00033758

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A mathematical model for the cable
comparison of the two models
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Can Yatzy pass? How much margin?

• at x180 Yatzy has a margin of 2m
• on top of safety margin
• not realistic to pass at exactly x180!
• Solve quadratic inequality
• Mathematical solution
• In reality horizontal margin
• of about 15 meters

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How much time is there?

• not realistic to arrive at exactly 124 PM
  (forecasted low water)!
• water must not rise too much!
• mathematical model water level
• (a, b, c and d can be calculated from data
  below)
• solve inequality 1057 AM ? t ? 350 PM

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How much time is there?

• evaluation of the model (afterwards) using
  observed water levels
• in reality there was much more time than
  predicted by the model
• after all, sines are rather poor models for water
  level

18
Yatzy on television
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Comments

• In the 1970s and 1980s secondary math education
  in Flanders was strongly influenced by New Math
• very strong emphasis on deduction and proof,
  formalism, classical mathematical structures,
  pure mathematics,
• very little attention for problem solving,
  geometrical insight, applications of mathematics
  and mathematical models,
• Example 1 marks the start of an evolution towards
  inclusion of more applications in secondary
  mathematics education.

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Comments

• Example 1 was constructed before the integration
  of technology in secondary mathematics education.
• Nowadays, technology is used in nearly all
  Flemish secondary school math classes
• mainly graphing calculators
• graphs, numerical calculations, matrix
  calculations, statistics,
• nearly no symbolic calculations
• factoring, symbolic differentiation and
  integration,
• Example 1 would be different when used in class
  now, but not too much.

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Comments

• Two evolutions in secondary math education
• inclusion of more applications of mathematics
• integration of technology
• In general, there was no similar evolution in
  higher education
• sometimes technology is not admitted
• purely mathematical aspects are still more
  emphasized
• not much attention for relation between
  mathematics and main subject(s) of the students

Question How about your country?
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Comments

• Example 1 is an authentic application.
• Many applications in Flemish secondary school
  math are non-authentic.
• It is not easy to construct authentic
  applications
• and it is not evident to use authentic
  applications (time-consuming, not too easy for
  students, )
• but it is important that students meet an
  authentic application from time to time
• and it is possible to find/construct them
• in books, magazines, on the internet, in-service
  training,
• i.e. start from newspaper articles

23
Comments

• Example 1 contains important aspects of
  mathematical modelling
• translation of reality into mathematics
• i.e. from data to inequalities
• interpretation of mathematical results in reality
• i.e. solution of the quadratic inequality
• comparison of model and reality
• mathematical models do not match reality
  perfectly
• i.e. sine function is poor model for water level
• different models can be used for the same
  phenomenon
• comparison of different models
• i.e. parabolic versus cosh-model

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Comments

• Pure math and applications do not always match!
• Example 1
• sine functions in mathematics versus physics
• minor differences different letters used, no d
  in physics
• major differences different form for the
  argument of sine and, hence, different
  interpretation of c vs. ?
• both math teachers and physicists have good
  reasons to prefer their form (i.e. nice
  interpretation for c)
• My advice Do the translation twice both in
  math and physics classes!

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Comments

• Another example functions
• mathematics f is the function yx2
• other subjects st2
• first important difference standard names in
  math for variables (independent x, dependenty)
• implication
• inverse of a function in pure mathematics first,
  solve x in terms of y and next, interchange
  notations x and y
• inverse of a function in applications solve t in
  terms of s (and DO NOT interchange the notations
  for the variables)
• similarly, composition of functions is different
  in pure mathematics compared to applications

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Comments

• Another example functions
• mathematics f is the function yx2
• other subjects st2
• second important difference use of variables (x
  and y or other names) versus functions (f)
• implication different notations
• derivative using function (f) versus using
  variables (dy/dx)
• composition of functions has a notation using
  functions (g?f) but not using variables

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The future of the Belgian population
we do not take migration into account!
based on data of Belgian Statistical Bureau, cfr.
www.statbel.fgov.be
after 20 years 98 of the individuals in age
group I is still alive
during a period of 20 years an individual in age
group I is responsible for an average of 0.43
births
age 1 Jan. 2003 fertility rate survival rate
0-19 2 407 368 0.43 0.98
20-39 2 842 947 0.34 0.96
40-59 2 853 329 0.01 0.83
60-79 1 840 102 0 0.30
80-99 410 944 0 0
TOTAL 10 354 690
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The future of the Belgian population
age 1 Jan. 2003 feritility rate survival rate
0-19 2 407 368 0.43 0.98
20-39 2 842 947 0.34 0.96
40-59 2 853 329 0.01 0.83
60-79 1 840 102 0 0.30
80-99 410 944 0 0
TOTAL 10 354 690
number of age 0-19 in 2023
number of age 20-39 in 2023
number of age 40-59 in 2023
number of age 60-79 in 2023
number of age 80-99 in 2023
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A matrix model for the evolution of the Belgian
population
Leslie model
fertility rates
Leslie matrix
survival rates
population on 1 Januari 2003
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A matrix model for the evolution of the Belgian
population
number of age 0-19 jaar in 2023
number of age 20-39 in 2023
number of age 40-59 in 2023
number of age 60-79 in 2023
number of age 80-99 in 2023
2003
2023
from 2003 to 2023 L ? ...
from 2023 to 2043 L ? ...
from 2043 to 2063 L ? ...
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Leslie model for the internal growth of a
population
only death and birth, no migration!
population is subdivided in age groups of equal
width
fertility rates and survival rates
initial population column vector X(0)
Leslie matrix square matrix containing
transition perunages between age groups over a
period equal to the width of the age groups
population after n periodes X(n)
recursive equation
explicit equation
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The future of the Belgian population
long term graphs of all age groups show great
and common regularity
babyboomers
babyboomers
babyboomers
short term
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Long term first observation
After periods I ...
0 2003 2 407 368 ...
1 2023 2 030 304 ...
2 2043 1 702 458 ...
... ... ... ....
-15,7
growth percentages
-16,1
After periods I II III IV V
0
1 -15,7 -17,0 - 4,3 28,7 34,3
2 -16,1 -15,7 -17,0 - 4,3 28,7
3 -15,9 -16,1 -15,7 -17,0 -4,3
4 -16,0 -15,9 -16,1 -15,7 -17,0
5 -16,0 -16,0 -15,9 -16,1 -15,7
6 -16,0 -16,0 -16,0 -15,9 -16,1
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Long term first observation
0.84 is the long term growth factor

• in the long run the number of individuals in each
  age group
• decreases by 16 every 20 years
• is multiplied by 0.84 every 20 years

growth percentages
After periods I II III IV V
0
1 -15,7 -17,0 - 4,3 28,7 34,3
2 -16,1 -15,7 -17,0 - 4,3 28,7
3 -15,9 -16,1 -15,7 -17,0 -4,3
4 -16,0 -15,9 -16,1 -15,7 -17,0
5 -16,0 -16,0 -15,9 -16,1 -15,7
6 -16,0 -16,0 -16,0 -15,9 -16,1
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Long term first observation
if n is a very large number, then in each age
group
number of individuals at time n
number of individuals at time n-1
? 0.84 ?
in mathematical notation X(n) ? 0.84X(n-1)
equivalent forms X(n1) ? 0.84X(n) L?X(n) ?
0.84X(n)
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Long term second observation
in the long run the distribution over the age
groups stabilizes
long term age distribution
percentages give the distribution of the
population over the age classes
after ... periods 0-19 (I) 20-39 (II) 40-59 (III) 60-79 (IV) 80-99 (V)
0 23.25 27.46 27.56 17.77 3.97
1 20.22 23.50 27.19 23.59 5.50
2 19.06 22.27 25.35 25.36 7.95
3 18.91 22.04 25.24 24.84 8.98
4 18.91 22.07 25.20 24.95 8.87
5 18.91 22.06 25.22 24.90 8.91
6 18.91 22.06 25.21 24.92 8.89
7 18.91 22.06 25.22 24.91 8.90
8 18.91 22.06 25.21 24.92 8.90
9 18.91 22.06 25.21 24.91 8.90
10 18.91 22.06 25.21 24.91 8.90
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Long term second observation
n-th line in table on previous slide
distribution of population over the age classes
after n periods X(n)/t(n), where t(n) is total
population after n periods
stabilization of age distribution means if n is
a very large number, then X(n)/t(n)?X(n-1)/t(n-1)
a limit age distribution is defined by
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LT growth factor and LT age distribution

• LT growth factor and LT age distribution were
  observed in tables, found by massive calculations
• Can LT growth factor and LT age distribution be
  determined in a more elegant way?
• method to determine LT age distribution if LT
  growth factor is already known
• divide LX(n)?0.84X(n) by t(n) and take limit
  LX0.84X
• if you do not already know X, you can find X by
  solving the system LX0.84X
• (and adding the condition that sum of components
  is 1)

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LT growth factor and LT age distribution

• Can LT growth factor and LT age distribution be
  determined in a more elegant way?
• method to determine LT growth factor
• LX0.84X has non-trivial solutions
• this is exceptional!
• LT growth factor is the only strictly positive
  number ? for which LX ?X has non-trivial
  solutions
• i.e. for which det(L-?I)0

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Decontextualising

• long term growth factor is an eigenvalue of the
  matrix L
• long term age distribution is an eigenvector of
  the matrix L
• Definitions
• A a square matrix (n ? n)
• A number ? is an eigenvalue of A iff det
  (A-?In)0.
• A column matrix X (? 0) is an eigenvector of A
  corresponding to the eigenvalue ? iff AX ?X.

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Comments

• Experiences
• example 2 is not easy but feasible
• students report that it helps them to see that
  mathematical concepts are useful
• students master the mathematics at same level as
  with traditional approach
• decontextualising is necessary
• LT growth factors are strictly positive, but
  eigenvalues may also be negative or zero
• LT age distributions have sum of their components
  equal to 1, but eigenvectors need not satisfy
  this supplementary condition

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Comments

• Ex. 1 application AFTER mathematics has been
  covered
• Ex. 2 application INTRODUCES mathematics
• Rationale
• shows relevance of studied mathematics right from
  the start!
• you show abstraction process (instead of only
  result of abstraction process)
• I use this at several occasions
• speed of growth -gt derivative
• multiplier in economics -gt geometric series
• discrete/continuous dynamic market model -gt
  difference/differential equations

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Conclusion

• Relation between mathematics and the rest of the
  world is
• different from studying pure mathematics
• different from studying mathematics as a bag of
  tricks
• worth studying in higher education mathematics

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Thank you!