Post on 27-Dec-2015
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„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
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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
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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
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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
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op
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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
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ng
op
eratin
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tin
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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
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op
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tin
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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
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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