„In an increasingly complex world,sometimes old questionsrequire new answers!“

© Heugl

Sustainability of mathematics education

by using technologydemonstrated with the topic of

exponential growth

Sustainability

Sustainability of mathematics education

Source: Bärbel Barzel

General available thinking

technology

Sustainable learniung strategies

Sustainability of mathematics education?

Sustainable attitudes

and values

Sustainable learning results

The sustainability of an educational system can be recognized on longterm effects which

are caused by a learning or a developing process.

Perspective 1: The expectation of the society and the contri-bution of mathematics education to a higher education

Perspective 2: The potential of the tool for supporting the goals of mathematics education

Part 1: The expectation of the society and the contribution of mathematics education to a higher education

Roland Fischer

The main task of higher general education is to lead the human beings to the ability of a better communication with experts and the general public.

My amendment:

As important is to support human beings for becoming experts themselves.

The expectation of the society

Aspect 1:

While the focus of primary education is the living environment

of the human beings in the higher general education learners

should experience mathematics as a special way of worldly

wisdom, as spectacles for recognizing and modeling the world

around. That needs the acquisition of the thinking technology

which is significant for doing mathematics and which is the

base of a general problem solving competence.

The contribution of mathematics education

3 points of view

Aspect 2:

Mathematical Literacy is an individuals´ capacity to identify

and understand the role that mathematics play in the world, to

make well-founded mathematical judgements and to engage in

mathematics, in ways that meet the needs of individuals´

current and future life as a constructive, concerned and

reflective citizen.[PISA framework OECD 2006]

The main goal is to develop a relationship between

real life and mathematics

Real world Mathematical world

real problem

real modelmathemat.

model

mathemat.solution

realsolution

mathematizing

interpreting

str

ukt

uri

ng

op

eratin

g

val

ua

tin

g

Aspect 3: Mathematics Problem solving by reasoning

modelling

1

2

3

4

5The Spiral Principle[Bruner,J.S.,1967]

The same subject is treated at different dates with varying levels

Characteristics of the spiral method

The single steps must not be isolated from each other

The shift of the standpoint must be transparent, the profit must be

recognizable

Earlier steps must not impede further expansion

Didactical contributions to more sustainable results

Some mathematics becomes more important – because technology requires it

Some mathematics becomes less important – because technology replaces it

Some mathematics becomes possible – because technology allows it

Bert Waits

If the main task of mathematics is to train things which in one or two decades will be better done by the computer it will cause a disaster.

H. Freudenthal about 40 years ago

The contribution of technology

Contributions of a mathematical tool

© Heugl

Electronic

tool

A tool for modeling

A tool for visualizing

A tool for calculating

A tool for experimenting

Calculation competence is the ability of a human being to apply a given calculus in a

concrete situation purposefully[Hischer, 1995]

Part 2: Realizing the Spiral PrincipleExponential growth

as an example for a sustainable, technology supported learning process

11

Basic rule Iduplicationspercentage growth rate

Use of the basic rule I for problem solving

7th and 8th grade

2.1 Growth processes in secondary level I

Example 1:

The area needed by a waterplant doubles every day.

At the first day the plant needs 1 dm2.

• After how many days the half of a lake with 1 ha square is filled

• After how many days the lake is fully covered

Day Area in dm2

1.2.3.4.5.6.7.8.9.10.11.....

1248

163264

1282565121024????

+1

+1

+1

Growth of a plant

.2

.2

.2

Day Area in dm2

1.2.3.4.5.6.7.8.9.10.11.....

1248

163264128256512

1024????

+3

Growth of a plant

.8

+3

+3

.8

.8

Basic rules of exponential growth

Basic rule I:The same time period belongs to the same growth factor

Nnew = q.Nold

Percentage growth rate

Step 1: Translation rules

Step 2: Use of the percentage growth rate in tables produced by a numerical calculator

Step 3: Use of recursive models with graphic calculators, spreadsheets and CAS tools

English Mathematics

Vocabulary bookVocabulary book

3.4

.3

.100

p

.(1 )100

p

“equals“

“threefold of

“three fourth of“

“p % of“

“increase about p%“

A new growth rate: .(1 )100

p

Given is a capital K

Prove the „Word-formula“

„Increase K about p% (of K) multiply with

pK.(1 )

100p

K +K.100

=

.(1 )100

p

using the distributiv law

Example 2: Radioactive decay: Per hour 3% of the radioactive agent disaggregate. After what time the half is left if on Monday, at 10 a.m. the

quantity mo= 200 mg is available? („radioactive half life“)

Time Radioactive agent

Monday, 10h 200 mg

11h 194

12h 188.2

13h 182.5

14h 177.1

15h 171.7

16h 166.5

… …

… …

… …

Tuesday, 8h 102.3

9h 99.3

„decrease about 3% „„multiply with 0.97“

Example 2.1: After what time is less than 1 mg available?

Time Radioactive agent

Montag, 10h 200 mg

Dienstag, 9h 100

Mittwoch, 8h 50

Donnerstag, 7h 25

Freitag, 6h 12.5

Samstag, 5h 6.25

Sonntag, 4h 3.13

Montag, 3h 1.57

Dienstag, 2h 0.79

Half life„times 1/2“

solution

Example 3: Building saving

For bying a house one needs a loan of € 140.000 and wants

to pay off the loan in yearly installments in 30 years.

The bank offers an interest rate of 3.5% which could be

changed depending on the index Euribor. The maximum rate

is guaranted with 6%.

A loan payed in yearly instalments

Translation phase 1:„what happens every year?“

Interest is charged on the principal Kand the instalment is deducted

Knew = Kold.(1+p/100) - R

Translation phase 2:a recursive model

Modelling is a translation activity

A tool for modelling

B q r2

A tool for visualizing

A tool for operating „copy and drag “

A tool for experimenting„slider“

function f„evaluating “„storing“ xnew => xold

xnew

xnew = f(xold)Knew = Kold.q-R

xold

Recursive scheme a two phase process

xnew

Developing rule II by using tables

1

2

2Basic rule IIA first step to the term prototype of the exponential function

8th grade

Basic rule II:The n-fold time period belongs to the nth power of the growth factor

Basic rule I:The same time period belongs to the same growth factor

1

2

3

3Basic rules III and IV

From discrete to continuous description of growth processes

9th and 10th grade

2.2 Growth processes in secondary level II

Example 4: Earth population:Data material shows: The earth population growing exponentially has doubled during the last 40 years.The current population 2012 was estimated with 7.05 billion.

• How many people lived on earth at 1992?

Doubling time 40 yearsHow many people were living on the earth after the half of the doubling time? 

   

year population

   

   

1972 3.53 bn   

1992 ??

2012 7.06 bn

+40 .2

+20

+20

.x

.x

Basic rule III

Basic rule I:The same time period belongs to the same growth factor

Basic rule II:The n-fold time period belongs to the nth power of the growth factor

Basic rule III:The half time belongs to the square root of the growth factor

t1

f(t)

t2 t

.q. q

. q

. q

Conclusion:If there are given two points with different positiv function values, so exists exactly one growth function which is defined for all time points and assumes all positiv values.

Basic rules of exponential growth

Basic rule I:The same time period belongs to the same growth factor

Basic rule II:The n-fold time period belongs to the nth power of the growth factor

Basic rule III:The half time belongs to the square root of the growth factor

Basic rule IV:For any real number the n-fold time belongs to the nth power of the growth factor

Definition: Real functions with

f: R R+ xc.ax , a positivare called Exponential Functions

1

2

3

4Use of recursive models (difference equations) for problem solving

4

Grades 8 to 12

grow about r-foldincrease about 30%reduce about 15%direct proportional torelative rate ofabsolute change, relative changea.s.o.

Often used phrases whiche were developed in secondary level I

Interp

retin

gC

alc

ula

tin

g

ModelingGrowth process Difference equation

Explicitterm prototype

Recursive Models in traditional mathematics education

Mathematicalsolution

Calculating

Interpreting, Reflecting

Sim

ulat

ing

ModelingGrowth process Difference equation

Table,Graph

Recursive Models in technology classes

Several sorts of growth processes described by difference equations

Linear growth

Exponential growth

Limited growthLogistic growth

Interacting populations

Growth with intervention

Real world Mathematical world

real problem

real modelM athemat.

model

mathemat.solution

realsolution

mathematizing

interpreting

str

uct

uri

ng

op

eratin

g

val

ua

tin

g

Exponential growth

Real model

Characteristcs:- The rate of change is

proportional to the actual stock. The increase is not constant

- The same time period belongs to the same growth factor.

“Word-formula”“New population k = old population + increase” The increase is proportional to the actual stock

Mathematical model

Difference equationsy(n) - y(n-1) = r.y(n-1)growth rate r (per step),Starting value y(0)

y(n) = y(n-1) + r.y(n-1)y(n) = y(n-1).(1+r)growth factor q = (1+r); starting value y(0)

y(n) = q.y(n-1)

Real world Mathematical world

real problem

real modelmathemat.

model

mathemat.solution

realsolution

mathematizing

interpreting

str

uct

uri

ng

op

eratin

g

val

ua

tin

g

Logistic growth

Real model

Characteristcs:- Growth depending on the value

of the actual population and the free space.

- The relative change is decreasing with a growing number of individuuals

“Word-formula”“New population= old population + increase” The increase is proportional to the actual population and the relative change of the free space.

Mathematical modelDifference equations

growth rate r,growth limit (capacity limit) G, starting value y(0)

G y(n 1)y(n) y(n 1) r y(n 1)

GG y(n 1)

y(n) y(n 1) r y(n 1)G

Fis

h p

op

ula

tio

n

Real world Mathematical world

real problem

real modelmathemat.

model

mathemat.solution

realsolution

mathematizing

interpreting

str

uct

uri

ng

op

eratin

g

val

ua

tin

g

Limited growth

Real modelCharacteristcs:

-The rate of change is proportional to the available free space (e.g. living space for biological populations). The increase is not constant

“Word-formula”“New population = old population + increase” The increase is proportional to the available free space.

Mathematical modelDifference equations

y(n) - y(n-1) = r.(G - y(n-1)) growth rate r, growth limit G, starting value y(0)

y(n) = y(n-1) + r.(G - y(n-1))

A warming process

Real world Mathematical world

real problem

real modelmathemat.

model

mathemat.solution

realsolution

mathematizing

interpreting

str

uct

uri

ng

op

eratin

g

val

ua

tin

g

Growth with intervention

Real model

Characteristcs:The population is growing exponentially and is simultaniously increased or reduced by a certain amount

“Word-formula”“New population = old population + increase” The increase is proportional to the actual population an is increased or reduced by a certain value

Mathematical model

Difference equationsy(n) - y(n-1) = r.y(n-1) - egrowth rate r (per step),reduced amount estarting value y(0)

y(n) = y(n-1) + r.y(n-1) - e

y(n) = y(n-1).(1+r) – e

Fishing

Interacting populations2 populations Bk and Rk influence each other.

Predator-prey relationship The population Bk promotes the growth of Rk; on the other hand Rk impedes the growth of Bk

Predator-prey relationshipBk+1 = q1.Bk – d.Rk.Bk

Rk+1 = q2.Rk + c.Rk.Bk

Foxes and rabbits

SymbiosisEvery population Bk and Rk promotes the growth of the other

population.

Competition relationshipEvery population Bk and Rk impedes the growth of the other population.

SymbiosisBk+1 = q1.Bk + d.Rk.Bk

Rk+1 = q2.Rk + c.Rk.Bk

Competition relationshipBk+1 = q1.Bk - d.Rk.Bk

Rk+1 = q2.Rk - c.Rk.Bk

A link of several models of growth processesHIV and the immunesystem – a mathematical model[J. Lechner,1999]

The terrible fact is that HI-viruses are that `successful" because their

replication is susceptible to mistakes. For every mutated virus the immune

system must create new specific cytoxic T cell (cT-cells or former “killer cell”),

which can only fight this special kind. The resistant cells act as specialists.

On the contrary all mutating viruses can destroy all kinds of resistant cells

against HIV or at least impair their function. The HI virus work as

generalists.

AIDS Acquired Immune Deficiency Syndrome

HIV Humane Immundefizienz-Virus (English: human immunodeficiency virus),

Simulation 1: One Mutant is active.

Virus (type 1): vir1(n) = vir1(n-1) + r.vir1(n-1) – p.vir1(n-1).kill1(n-1)

Resistant cells (type 1): kill1(n) = kill1(n-1) + s.vir1(n-1) – q.vir1(n-1).kill1(n-1)

r: Increase rate of the virus (r=0.1)

p:„Efficiency“ of the cT-cells in their fight of resistance

(p=0.002)

s: The increase of the cT-cells which are

generated by the virus mutant 1 (s=0.02)

q: The agressiveness of the viruses (q=0.00004)

One step in time represents 0.005 years (i.e. 200 steps describe a year)Source of the parameter values: [LIPPA, 1997, NOWAK, 1992]

Simulation 2: Two mutants

Two mutants are active, the second of which shall appear after 60 steps of time (which means after about 3.6 months).

(The values of the parameters r,s,p,q are the same as in case 1)

Simulation 2: Two Mutants are active

Virus (type 1): vir1(n) = vir1(n-1) + r.vir1(n-1) – p.vir1(n-1).kill1(n-1)

Resistant cells (type 1): kill1(n) = kill1(n-1) + s.vir1(n-1) – q.vir1 (n-1).kill1(n-1)

Virus (type 2): vir2(n) = vir2(n-1) + r.vir2(n-1) – p.vir2(n-1).kill2(n-1)n60

Resistant cells (type 2): kill2(n) = kill2(n-1) + s.vir2(n-1) – q.virtot(n-1).kill2(n-1) n60

Total number of virus:virtot(n)=vir1(n) + vir2(n)

Total number of resistant cells:killtot(n)=kill1(n) + kill2(n)

Simulation 3: 11 Mutanten sind aktiv A program by J. Lechner implemented on the voyage 200

Virus

Resistant cells

1

2

3

4Additional mathematical aspects for modelling with difference equations

5

Grades 10 to 12

Geometric iteration: The use of the web plot

5

Limits or fixed points of a sequence defined by a difference equation

A sequence is defined by a difference equation

Geometric iteration: The use of the web plot

Two sorts of graphic representations

Time plot: xn = f(t)

Web plot: xn = g(xn-1)

Advanteges of the „web plot“:

Visualization of the two phases of a recursive scheme

Visual investigation of the convergence of the sequence

Investigation of the fixed points (invariant points)

xn-1

xn

1st Medianxn = xn-1

xn = g(xn-1)

x0

x1

x1

x2

x2 x3

evaluating

storing

„Geometric Iteration“

evaluating

storing

A fixed point x* (sometimes shortened to fixpoint, also known

as an invariant point) of a function f is a point that is mapped to

itself by the function f(x*) = x*

Is x* an atractive fixed point of a difference equation xn = f(xn-1)

than the sequence converges to x*: nn

limx x

The fixed point theorem

A fixed point x* of a difference equation xn = f(xn-1) (f is

continuous and differentiable) is an attractive fixed point, if

and is distractive, if f (x ) 1 f (x ) 1

Example: Sterile Insect Technique (SIT)

An insect population with u0 female and u0 male insects at the beginning may have a natural growth rate r.

To fight these insects per generation a certain number s of sterile insects is set free.

Investigate the effect of the method SIT by interpreting the growth function for several parameters u0, r, s.

• Model assumption: r=3; s=4• Initial values: u0=1.9; u0=2.2; u0=2.0

Modeling – a translation process

Unlimited growth

Relativ rate of fertile insects

1

2

3

4

5 12th

11th

10th

9th

8th

7th

Attributes and models

Base(growth factor)

Mathematical aspects of difference

equations

Mathematical needs: Algebra and Analysis

Difference Equations Real, especially e

TermSequence mode

Real, especially e

Basic rule 3 real

Basic rule II 2,

Basic rule I 2

Sustainability

Sustainability of mathematics education

Source: Bärbel Barzel

SustainablecompetenceUse of

technology