#### Transcript The Positive Aspects of Modeling Process in Teaching Mathematics

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Natalija Budinski
Primary and grammar school ”Petro Kuzmjak”
Ruski Krstur
Serbia
[email protected]
Teaching Mathematics and Statistics in Sciences: Modeling and
Computer-aided Approach
May 2011
This
presentation
proposes
modeling based learning as a tool
for
learning
and
teaching
mathematics.

An example of modeling a real
world
problem
related
to
logarithims is described

In
this presentation we introduce
modeling process to students in a
belief that it could contribute toward a
better understanding of learning and
teaching mathematics.
We
place mathematics in real context
and focus on why mathematics exists
in the first place.
 Studying
a mathematical model of a real-world
situation can provide students with insights that are
hidden during a nonmathematical study of the same
situation.
 The modeling process in mathematical education
tends to follow the didactical cycle of activities and
reach a desirable level of accomplishments of
students’ activities and competencies.
 We use a diagrammatic representation (see next
Figure), which encompasses both the task
orientation, and the need to capture what is going on
in the minds of individuals as they work
collaboratively on modelling problems.
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Goals of modeling are
 prediction
 design
 testing of possibilities in order to
make a decision
 development of a deeper
understanding of a phenomenon
phenomenon

MARCH 11 WHIRLPOOL CAUSED BY
TSUNAMI RARE VIEW EVER_3.flv
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


Japan's most powerful earthquake since
records began has struck the north-east coast,
triggering a massive tsunami.
Cars, ships and buildings were swept away by
a wall of water after the 8.9-magnitude tremor,
which struck about 400km (250 miles) northeast of Tokyo.
A state of emergency has been declared at a
nuclear power plant, where pressure has
exceeded normal levels.
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Thousands of people living near the
Fukushima nuclear power plant have been
ordered to evacuate.
 Japanese nuclear officials said pressure inside a
boiling water reactor at the plant was running
much higher than normal after the cooling
system failed.
 Taken from:
www.bbc.co.uk/news/world-asia-pacific12709598

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


A tsunami is an ocean wave that is generated
by a sudden displacement of the sea floor. This
displacement can occur as a result of earthquakes.
Tsunami is a Japanese word for “harbor wave.”
The mathematics of logarithmic scales helps
us understand how earthquakes are measured.
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




An earthquake is the sudden release of energy in
the form of vibrations caused by rock suddenly
moving along fault lines.
This energy is extremely large and we repesent it
on a scale based on exponents
The idea to use a logarithmic earthquake
magnitude scale was first developed by Charles
Richter in the 1930s.
This scale is used to measure the magnitude of
earthquakes.
The Richter scale is an example of an “exponential
scale,” or a “logarithmic scale”.
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How many times more
intense was The Indian
Ocean earthquake (2004)
with a Richter magnitude
of 9.3 than The Great East
Japan earthquake (2011)
with magnitude 9.0?
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

An increase of 1 unit on the Richter scale roughly corresponds to a
multiplication of the energy released by a factor of 10.
In 1935 Charles Richter defined the magnitude of an earthquake to
be
I
M  log
S
where I is the intensity of the earthquake (measured by the amplitude of a seismograph
reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a
''standard earthquake'‘
(whose amplitude is 1 micron =10-4 cm).
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
Let I1 represents the intensity
of
The
Indian
Ocean
earthquake and I2 represents
the intensity of The Great
East Japan earthquake.
I1
9.3  log
S
I2
9.0  log
S

We are looking for the ratio
of the intensities:
I1
I2
Using the rules of logarithms we
isolate this ratio:
I1
I2
9.3  9.0  log  log
S
S
I1
0.3  log
I2
I1
 100.3  1.995262
I2
 The
Indian Ocean earthquake
was two times as intense as
The Great East Japan
earthquake
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At
this stage, the strengths and
weaknesses of the model should be
discussed. That involves reflecting
upon the mathematics that has been
used.
Students should verify the initial
conditions
Students should be encourage to use
 The
act of creating a model forces students to think
deeply about the problem. Translating an imprecise,
complex, multivariate real-world situation into a simpler,
more clearly defined mathematical structure such as a
function or a system of rules for a simulation, yields
several benefits.
 For the first step in this process, students identify a list of
variables. As they do so, they discover what they really
know about their problem and what information they
need to determine.
 Students must think about the connections between and
among variables, decide which relationships and
structures are the most important to capture
mathematically, and pick the mathematical realm that
offers the best possibilities for expressing all these
features.
 Once a mathematical model exists, the technical skills of
 This
teaching method tolerate different kinds of
activities such as research every day situation,
using computers and organizing the learning
process close fitting to contemporary students
and their field of interests
 It can be said that modeling based learning is out
of the box, but that is its great advantage over the
Mathematical Modeling: Teaching the Open-ended Application of Mathematics,
Joshua Paul Abrams, www.meaningfulmath.org
Geogebra in mathematics teacher education: the case of quadratic relations,
Lingguo Bu and Erhan Selcuk Haciomeroglu, MSOR Connections Vol 10 No 1.
February 2010 – April 2010
Boaler, J., Mathematical Modelling and New Theories of Learning (2001),
Teaching Mathematics and its Applications, Vol. 20, Issue 3,p. 121-128.
Doerr, H., English, L., A modelling perspective on students’ mathematical
reasoning about data, (2003) Journal for Research in Mathematics Education,
34(2), 110-136
Galbraith, P., Stillman, G., Brown, J., Edwards, I., Facilitating middle secondary
modelling competencies. (2007), In C. Haines, P. Galbraith, W. Blum, S. Khan
(Eds.), Mathematical modelling (ICTMA12): Edu., eng. economics, Chichester,
UK: Horwood, 130-140.
Henn, H.-W., Kaiser, G., Mathematical Modeling in school-Examples and Eperiences, (2005), In
Mathematikunterricht in spannungsfeld von Evolution und Evaluation, Fesband fur Werner Blum,
Hildesheim Franzbecker, 99-108 p.
Hohenwarter, M., Hohenwarter, J., Kreis, Y., Lavicza Zs., Teaching and calculus with free dynamic
software Geogebra, (2008), 11th International Congress on Mathematical Education, Mexico.
Kaiser, G., Schwarz, B. Mathematical modelling as bridge between school and university, (2006), ZDM, 38
(2), p. 196-208.
Lamon, S. J., Parker, W. A., Houston, S. K. (Eds.), Mathematical modelling: A way of life, Chichester, (2003),
UK: Horwood
Mason, J., Modelling modelling: Where is the centre of gravity of-for-when teaching modelling? (2001), In
J.Matos, W. Blum, K. Houston, S. Carreira (Eds.), Modelling and mathematics education, Chichester, UK:
Horwood.
Stillman, G., Brown, J., Challenges in formulating an extended modelling task at Year 9, (2007), In H.
Reeves, K. Milton, & T. Spencer (Eds.), Proc. 21. Conf. Austr. Assoc. Math. Teachers. Adelaide: AAMT .
Stillman, G., Galbraith, P., Towards constructing a measure of the complexity of applications tasks. (2003),
S.J. Lamon, W. A. Parker, & S. K. Houston (Eds.), Mathematical modelling: A way of life (pp. 317-327).
Chichester, UK: Horwood.
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