Transcript A Long Step From Learning Organizations to Learning Society

Making a difference in
Mathematics Teaching and
Learning
Jozef Hvorecký
Vysoká škola manažmentu,
Bratislava, Slovakia
University of Liverpool
Liverpool, UK
Motto:
It is a miracle that curiosity
survives formal education.
Albert Einstein
To Remember or To Comprehend?
To remember:
To be able of reproducing the
learned fact
To comprehend:
To be able to apply the learned
fact in a case of need
Pupils are requested to
remember but not to
comprehend
Consequence
Public consensus:
Mathematics has no use for
common people
Good students’ opinion:
If I’d know everything, I could be
a Mathematics teacher.
Silently presuming: But no one
else.
Consequences of the
Consequence
1. Economists, physicians,
engineers do not expect that
mathematicians could
cooperate in solving their
problems.
2. Many job opportunities for
mathematicians are lost (by
their not-creating).
Can someone comprehend the
role of mathematics without
remembering everything?
• Mathematics as a goal
• Mathematics as a tool
Examples:
• Payments in shops
• Surfing the Internet
Hyperbolic functions
sinh x =
e x  e x
2
e x  e x
cosh x =
2
Why do we have them in our
curriculum?
Excursion to history
Hold the end of a chain in your
hands in such a way that it is
freely hanging.
Which curve does it form?
And what is the truth?
Galileo Galilei:
Parabola
Jungius (1669):
Catenary i.e. cosine hyperbolic
Hurray!
Mathematicians also make errors!
Curing by shock
http://mathworld.wolfram.com/Roulette.html
A model of logistics?
Mathematics education today
Typical formulations of problems to
solve:
• Find the maximum of the function:
F(x) = 693,8597 – 68,7672.cosh 0,0100333x
• Solve the equation:
693,8597 – 68,7672.cosh 0,0100333x = 0
Dreaming
about future
In 1965 in Saint Louis (USA) a huge arch
was built. It tracks the curvature of
(inverted) cosine hyperbolic
described by the formula
F(x) = 693,8597 – 68,7672.cosh 0,0100333x
How high is it?
How far are its pillars from each other?
Right moment for
applying information
technology
• Intelligent calculators
• CAS
• Specialized software
Max(693,8597 – 68,7672.cosh 0,0100333x)
Height of Arch: 192 m
Root(693,8597 – 68,7672.cosh 0,0100333x)
Distance of its pillars: 225 m
Not the end yet
Supporting curiosity:
• Why was the catenary
chosen?
– Set up hypotheses
– Search through information
sources (books, journals,
Internet)
– Set up discussion
• Local
• international
To be discussed among us
• How much do teachers/students have
to know about the background of the
problem before starting its solution?
• What relationships between Math and
real life do clarify its real-life role in the
most appropriate way?
• How to change:
–
–
–
–
–
the classroom atmosphere,
teachers,
pupils,
school administration,
parents?
Naïve vs. formal solution
A shoemaker has been asked to
make 100 special-purpose shoes.
During the first week he produces
nine of them, on the next week
eleven, on the third week thirteen.
He sees that due to his growing
experience and improved skills he
will be capable of producing two
shoes more during every next
week compared to the previous
one.
How long will it take for him to
produce all pairs?
“Naïve” Solution
Step 1
Step 2
Analytical Solution
n
S n  (b1  bn )
2
bn  b1  2(n  1)
n
S n  (2b1  2n  2)  b1n  n 2  1  n 2  b1n  1
2
2
100  n  9n  1
n 2  9n  101 0
Back to Our Calculator
Setting parameters
Getting 2 roots
Roots: 6.5113 and -15.51
Is naïve solution better than the formal one?
Real-life problem formulations
• Let us assume building a structure with many
floors. The basic floor is built for a certain
(basic) price. Building higher structures
becomes more expensive as the material
must be carried higher, the safety
precautions must be stronger, and the risk is
higher. For these reasons, insurance
companies usually request higher payments
to balance the risks.
• The ARITHMETIC insurance company gives
you its proposal: Our basic insurance per
floor is $15. For each floor above the ground
you pay $2 more than for the previous floor.
• How much is the insurance cost for a 10level building (i.e. for a building with a
ground floor and 9 floors above it)?
Continuing Series of Problems
• To have a choice of offers, we visited
another insurance company named
GEOMETRIC. Its manager gave as a
different proposal: Our basic
insurance per floor is $1. For each floor
above the ground you will pay 2-times
more than for the previous floor. Their
offer starts at much lower level than
the one from the ARITHMETIC. Will we
pay more or less for the whole
building?
• When is it cheaper to take an offer
from the ARITHMETIC and when from
GEOMETRIC?
Motivating formulation
A student body discusses a possibility
to organize a fund-raising dinner.
The presumed price of a ticket is
$24. One member of the organizing
committee has found an
appropriate space for the event
which can be rented for $350.
Another one knows a company
providing chairs for $1.50 per night
plus free tables.
The committee needs to know the
minimum number of people which
has to come for covering all these
expenses.
Capability of Reading Drawings
Costs:
Space rental $350
Chairs for $1.50
Break-even point
Revenue:
People for $24
Interpreting results
How many must come to the party to raise our
profit to $500?
P( x)  R( x)  C ( x)
P( x)  24x  (1,5 x  350)  22,5 x  350
22,5 x  350  500
x  37,77
The party should be visited by 37,77
persons.

Non-linear Models
Two girls want to make money to
buy Christmas gifts for their
relatives and friends. They see
their opportunity in making and
selling necklaces from glass
beans. They realized that first
they have to invest $50 to
various tools. For each
necklace they also need a set
of beans. The supplier offers
them for the basic price $2, but
the price declines by 1 cent
per set.
Costs
Fixed cost: $50
Variable cost per set: 2–0.1*x
(x is the number of sets)
C( x)  50  x(2  0.1x)  50  2x  0.01x 2
Revenue and Break-even point
Revenue: $3.99 per necklace
Intersection of costs and revenue
Further Analysis of the Solution
• Why are we interested in the x-coordinate?
• What is the meaning of y-coordinate?
•What if we would start selling with discounts,
too?
Why Should We Ever Mention the
Word “Quadratic”?
Fixed cost: $50
Variable cost per set: 2–0.1*x
(x is the number of sets)
C ( x)  50  x(2  0.1x)
Mathematics, Models and Reality
Colette Laborde (20 June)
– New types of tasks
– Joining mathematics and
technology
• We (22 June)
– New types of tasks
– Joining mathematics, models
and reality
Examples of a new category of tasks
About a model:
A horse runs with the speed of 20
km/hour. What is the speed of 20
horses?
About the interpretation of
results:
Our TV set has collapsed, we could not watch
any programs. So, daddy went to the shop
and bought 16 TV’s. Mommy went to the
shop and bought 27. How many TV sets do
we have?
Questionable interpretation
Peter went to the forest behind his
house and saw 5 giraffes, 4
gazelles and 7 lions.
How many animals did he saw?
In Mathematics, everything is absolute.
In real life, everything is relative.
No surprise that Mathematics does not
attract pupils.
Thank you for your attention
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