Sustainability of mathematics education
by using technology
demonstrated with the topic of exponential growth
„In an increasingly complex world,
sometimes old questions
require new answers!“
© Heugl
Sustainability of mathematics education
Sustainability
Source: Bärbel Barzel
Sustainability of mathematics education?
The sustainability of an educational system can be recognized on longterm
effects which are caused by a learning or a developing process.
Sustainable
attitudes and
values
General available
thinking technology
Sustainable
learning results
Sustainable
learniung strategies
Part 1: The expectation of the society and the
contribution of mathematics education to a
higher education
Perspective 1:
The expectation of the
society and the contribution of mathematics
education to a higher
education
Perspective 2:
The potential of the tool
for supporting the goals of
mathematics education
The expectation of the society
Roland Fischer
My amendment:
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.
As important is to support
human beings for becoming
experts themselves.
The contribution of mathematics education
3 points of view
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.
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
Aspect 3: Mathematics  Problem solving by reasoning
Real world
Mathematical world
mathematizing
mathemat.
model
interpreting
mathemat.
solution
operating
strukturing
real model
modelling
valuating
real
problem
real
solution
Didactical contributions to more sustainable results
The 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
5
4
3
2
1
The contribution of technology
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
Contributions of a mathematical tool
A tool for
modeling
A tool for
calculating
A tool for
visualizing
tool
for of a
Calculation competence is A
the
ability
experimenting
human being to apply a given
calculus in a
concrete situation purposefully
[Hischer, 1995]
© Heugl
Part 2: Realizing the Spiral Principle
Exponential growth
as an example for a sustainable, technology
supported learning process
2.1 Growth processes in secondary level I
7th and 8th grade
Basic rule I
1
duplications
percentage growth rate
Use of the basic rule I for problem
solving
1
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
Growth of a plant
+1
+1
+1
Day
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
....
Area in dm2
1 .2
2
4
8
.2
16
32
64
128
256 .2
512
1024
????
Growth of a plant
Day
+3
+3
+3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
....
Area in dm2
1
2
.8
4
8
.8
16
32
64
128
256
.8
512
1024
????
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
Vocabulary book
English
Mathematics
“equals“

“threefold of
.3
“three fourth of“
“p % of“
“increase about p%“
3
.
4
p
.
100
p
.(1 
)
100
p
)
A new growth rate: .(1 
100
Given is a capital K
Prove the „Word-formula“
p
)
„Increase K about p% (of K)  multiply with .(1 
100
p
K + K.
100
=
p
K.(1 
)
100
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
Monday, 10h
Radioactive agent
200 mg
11h
194
12h
188.2
13h
182.5
14h
177.1
15h
171.7
16h
166.5
…
…
…
…
…
…
Tuesday, 8h
9h
102.3
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
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,
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%.
Modelling is a translation activity
A loan payed in yearly instalments
Translation phase 1:
„what happens every year?“
Interest is charged on the principal K
and the instalment is deducted
Translation phase 2:
a recursive model
Knew = Kold.(1+p/100) - R
A tool for experimenting
„slider“
A tool for modelling
 B2  q  r
A tool for visualizing
A tool for operating
„copy and drag “
Recursive scheme  a two phase process
xnew
old
function f
„storing“
xnew => xold
„evaluating “
xnew = f(xold)
Knew = Kold.q-R
xnew
8th grade
Basic rule II
2
A first step to the term prototype of
the exponential function
2
1
Basic rule II:
The n-fold time period belongs to the nth
power of the growth factor
Developing rule II
by using tables
Basic rule I:
The same time period belongs to the
same growth factor
2.2 Growth processes in secondary level II
9th and 10th grade
3
2
3
Basic rules III and IV
From discrete to continuous
description of growth processes
1
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 years
How many people were living on the earth after the half of
the doubling time?
+20
+40
year
population
1972
3.53 bn
1992
.x
.2
??
.x
+20
2012
7.06 bn
Basic rule III
Basic rule III:
The half time belongs to the square root of
the growth factor
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
Conclusion:
If there are given two points with different positiv function values,
so exists exactly one
f(t) growth function which is defined for all time
points and assumes all positiv values.
. q
.
.q
q
. q
t1
t2
t
Basic rules of exponential growth
Definition:
Real functions with
Basic rule
IV:
f: R
R+ xc.a
, a positiv
For any real
number
the xn-fold
time belongs
are called
Exponential
to the
nth powerFunctions
of the growth factor
Basic rule III:
The half time belongs to the square root of
the growth factor
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
Grades 8 to 12
4
Use of recursive models
(difference equations) for
problem solving
4
3
2
Often used phrases whiche were developed
in secondary level I
grow about r-fold
increase about 30%
reduce about 15%
direct proportional to
relative rate of
absolute change, relative change
a.s.o.
1
Recursive Models in traditional mathematics education
Modeling
Interpreting
Mathematical
solution
Difference equation
Calculating
Growth process
Calculating
Explicit
term prototype
Recursive Models in technology classes
Growth process
Modeling
Table,
Graph
Difference equation
Several sorts of growth processes described by
difference equations
Growth with intervention
Limited growth
Logistic growth
Exponential growth
Interacting
populations
Linear growth
Exponential growth
Real world
Mathematical world
structuring
real model
mathematizing
M athemat.
model
Mathematical model
Real model
y(n) - y(n-1) = r.y(n-1)
growth rate r (per step),
Starting value y(0)
valuating
y(n) = y(n-1) + r.y(n-1)
y(n) = y(n-1).(1+r)
growth factor q = (1+r);
“Word-formula”
starting value y(0)
“New population k = old
population + increase”
y(n) = q.y(n-1)
The increase is proportional to
real
mathemat.
interpreting
the actual stock
solution
solution
operating
Characteristcs:
 The rate of change is
proportional to the actual stock.
real
The increase is not constant
problem
 The same time period belongs
to the same growth factor.
Difference equations
Logistic growth
Real world
Mathematical world
structuring
real model
mathematizing
mathemat.
model
Real model
Characteristcs:
Fish population
valuating
“Word-formula”
Mathematical model
Difference equations
y(n)  y(n  1)  r  y(n  1) 
 G  y(n  1) 
G
 G  y(n  1) 
y(n)  y(n  1)  r  y(n  1) 
G
growth rate r,
“New population= old population
growth limit (capacity limit) G,
+ increase”
starting value y(0)
The increase is proportional
to
real
mathemat.
interpreting
the actual population
and the
solution
solution
relative change of the free
space.
operating
 Growth depending on the value
of the actual population and the
free space.
real
 The relative change is
problem
decreasing with a growing
number of individuuals
Limited growth
Real world
Mathematical world
mathematizing
A warming
Real model
process
Characteristcs:
valuating
The rate of change is
real
proportional to the available free
problem
space (e.g. living space for
biological populations). The
increase is not constant
“Word-formula”
mathemat.
model
Mathematical model
Difference 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))
“New population = old
population + increase”
The increase is proportional
to
real
mathemat.
interpreting
the available free space.
solution
solution
operating
structuring
real model
Growth with intervention
Real world
Mathematical world
mathemat.
model
Real
model
Fishing
Characteristcs:
The population is growing
real
exponentially
and is
simultaniously increased or
problem
reduced by a certain amount
“Word-formula”
valuating
mathematizing
Mathematical model
Difference equations
y(n) - y(n-1) = r.y(n-1) - e
growth rate r (per step),
reduced amount e
starting value y(0)
“New population = old population
+ increase”
y(n) = y(n-1) + r.y(n-1) - e
The increase is proportional to
y(n) = y(n-1).(1+r) – e
the actual population
real an is
mathemat.
interpreting
increased or reduced
by a
solution
solution
certain value
operating
structuring
real model
Interacting populations
2 populations Bk and Rk influence each other.
Foxes
and rabbits
Predator-prey
relationship
Predator-prey relationship
The population Bk promotes the
Bk+1 = q1.Bk – d.Rk.Bk
growth of Rk; on the other hand Rk
R
k+1 = qthe
2.Rgrowth
k + c.R
impedes
ofk.B
Bk
k
Symbiosis
B
q1.Bk + d.R
Every
Bk and
k+1 =population
k.BkRk
promotes
the growth of the other
R
k+1 = q2.Rk + c.Rk.Bk
population.
Competition
Competition relationship
relationship
B
q1.Bk - d.R
Every
Bk and
k+1 =population
k.BkRk
Rk+1 = qthe
impedes
the
2.Rgrowth
k - c.Rof
k.B
k other
population.
A link of several models of growth processes
HIV and the immunesystem – a mathematical model
[J. Lechner,1999]
AIDS Acquired Immune Deficiency Syndrome
HIV Humane Immundefizienz-Virus
(English: human immunodeficiency virus),
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.
Simulation 1: One Mutant is active.
r: Increase rate of
the virus (r=0.1)
p:„Efficiency“ of the cT-cells in
their fight of resistance
(p=0.002)
Virus (type 1):
vir1(n) = vir1(n-1) + r.vir1(n-1) – p.vir1(n-1).kill1(n-1)
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)
Resistant cells (type 1):
kill1(n) = kill1(n-1) + s.vir1(n-1) – q.vir1(n-1).kill1(n-1)
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) n 60
Resistant cells (type 2):
kill2(n) = kill2(n-1) + s.vir2(n-1) – q.virtot(n-1).kill2(n-1) n 60
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
Grades 10 to 12
5
5
Additional mathematical
aspects for modelling with
difference equations
4
3
2
1
Geometric iteration: The use of the web plot
Limits or fixed points of a sequence defined by a
difference equation
Geometric iteration: The use of the web plot
A sequence is defined by a difference equation
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)
„Geometric Iteration“
1st Median
xn = xn-1
xn
storing
xn = g(xn-1)
storing
x2
x1
evaluating
evaluating
x0
x1
x2 x3
xn-1
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*: limx n  x

n
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

f (x )  1 and is distractive, if 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
5
12th
4
11th
3
2
10th
9th
8th
Attributes and
models
Base
(growth factor)
Mathematical aspects
of difference
equations
Mathematical needs:
Algebra and Analysis
Difference Equations
Real, especially e
Term
Sequence mode
Real, especially e
Basic rule 3
real
1
7th
Basic rule II
Basic rule I
2,
2
Sustainability of mathematics education
Use of
Sustainability
technology
Sustainable
competence
Source: Bärbel Barzel