MAthematices and the SEA (and other things)

Johan DeprezSEAMA-conference, Antwerp, 31/05/10

www.ua.ac.be/johan.deprez > Documenten

2

Overview

1. Introduction2. Example 1: Journey of the drilling rig Yatzy3. Some comments4. Example 2: The future of the Belgian population5. Some comments

3

Introduction

about myself:• mathematics teacher

for about 20 years higher education at university level

(but not university) students in applied economics basic mathematics course

• …

second example comes from my

classes

4

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

5

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)

7

Journey of the drilling rig Yatzy

8

Journey of the drilling rig Yatzy

• Yatzy was transported along theriver Scheldt from Temse toRotterdam on 11 January 1989

• problem: passing power linehanging 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

9

Introduction to the context

depth chart of the river Scheldt at the crossing of the power line

10

Introduction to the context

information about the tides

height of the power line

11

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

conclusions:•problem is at the top•pass near left bank (x=180)•pass at low water spring tide

12

A mathematical model for the cable (engineer’s 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

da

cxay

1cosh

13

A mathematical model for the cable (student’s 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 enda = 0.00033758

dcxay 2)(

14

A mathematical model for the cable

comparison of the two

models

15

Can Yatzy pass? How much margin?

• at x=180 Yatzy has a margin of 2mon top of safety margin

• not realistic to pass at exactly x=180! • Solve quadratic inequality

• Mathematical solution:

• In reality: horizontal marginof about 15 meters

72.9280.79)92.422(00033758.0 2 x

33.653or 59.192 xx

16

How much time is there?

• not realistic to arrive at exactly 1:24 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: 10:57 AM t 3:50 PM

dctbah )(sin

17

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

Yatzy on television

19

Comments

• In the 1970’s and 1980’s 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.

20

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 calculationsfactoring, symbolic differentiation and integration, …

• Example 1 would be different when used in class now, but not too much.

21

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?

22

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

24

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!

dcxbay )(sin tAy sin

25

Comments

• Another example: functions mathematics: f is the function y=x2

other subjects: s=t2

first important difference: standard names in math for variables (independent: x, dependent:y)

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

26

Comments

• Another example: functions mathematics: f is the function y=x2

other subjects: s=t2

… 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

27

The future of the Belgian population

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

during a period of 20 years an individual in age group I is responsible for an average of 0.43 births

after 20 years 98% of the individuals in age group I is still alive

based on data of Belgian Statistical Bureau, cfr. www.statbel.fgov.be

we do not take migration into account!

28

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:

329853201.0947842234.0368407243.0

368407298.0

947842296.0

329853283.0

102840130.0

29

A matrix model for the evolution of the Belgian population

V

IV

III

II

I

030.0000

0083.000

00096.00

000098.0

0001.034.043.0VIVIIIIII

toL

from

944410

1028401

3298532

9478422

3684072

)0(X

population on 1 Januari 2003

Leslie matrix

survival rates

fertility rates

Leslie model

30

A matrix model for the evolution of the Belgian population

944410

1028401

3298532

9478422

3684072

030.0000

0083.000

00096.00

000098.0

0001.034.043.0

102840130.0

329853283.0

947842296.0

368407298.0

329853201.0947842234.0368407243.0

20032023

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:

329853201.0947842234.0368407243.0 368407298.0

947842296.0 329853283.0

102840130.0

from 2003 to 2023: L ...

from 2023 to 2043: L ...

from 2043 to 2063: L ...

31

Leslie model for the internal growth of a population

population is subdivided in age groups of equal width

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:

only death and birth, no migration!

explicit equation:

)1()( nXLnX

)0()( XLnX n

fertility rates and survival rates

32

The future of the Belgian population

Evolution per age class

0

500000

1000000

1500000

2000000

2500000

3000000

0 1 2 3 4 5 6 7 8 9 10 11 12

after ... periods

I

II

III

IV

V

long term: graphs of all age groups show great and common regularity

babyboomers

babyboomers

babyboomers

‘short’ term

33

Long term: first observation

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%

growth percentages:

After … periods I ...

0 2003 2 407 368 ...

1 2023 2 030 304 ...

2 2043 1 702 458 ...

... ... ... ....

-15,7%

-16,1%

34

Long term: first observation

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%

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:

0.84 is the long term growth factor

35

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) 0.84·X(n-1)

equivalent forms:

X(n+1) 0.84·X(n)

LX(n) 0.84·X(n)

36

Long term: second observation

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%

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

37

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

...089.0

...249.0

...252.0

...220.0

...189.0

)(

)(lim

nt

nXX

n

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

38

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?

1.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: LX=0.84X if you do not already know X, you can find X by solving

the system LX=0.84X(and adding the condition that sum of components is 1)

39

LT growth factor and LT age distribution

• Can LT growth factor and LT age distribution be determined in a more elegant way?

1.…2.method to determine LT growth factor

LX=0.84X has non-trivial solutions this is exceptional! LT growth factor is the only strictly positive number λ for

which LX= λX has non-trivial solutions i.e. … for which det(L-λI)=0

40

Decontextualising

• long term growth factor is an eigenvalue of the matrix L

• long term age distribution is an eigenvector of the matrix L

DefinitionsA 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.

41

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

42

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 -> derivative multiplier in economics -> geometric series discrete/continuous dynamic market model ->

difference/differential equations …

43

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

Thank you!