Transcript STEM Modeling in the Secondary Setting to Deepen Understanding

STEM Modeling in the
Secondary Setting to
Deepen Understanding
AMTNYS Summer Workshop
August 3, 2011
Brandon Milonovich
The College of Saint Rose
Syracuse University
Overview
O What is modeling?
O Why use modeling?
O How does modeling relate to the Common
Core?
O How can I use modeling in my classroom?
What is modeling?
O “The real situation usually has so many
‘angles’ to it that you can’t take everything
into account, so you decide which aspects
are most important and you keep
those…Now you have a mathematical model
of the idealized question” (Pollak, 2011, 7).
What is modeling?
O Modeling differs from solving word problems
because it is a process.
O After modeling we have to ask, “are the
results practical, the answers reasonable,
the consequences acceptable” (Pollak,
2011, 7)?
O Word problems focus more on solving a
given problem with known variables, using a
specifically aimed at algorithm and carrying
it out successfully.
What is modeling?
O “[I]mportant aspect of mathematical
modeling: real-world situation comes first,
the mathematics follows naturally” (Tam,
2011, 67).
O “The heart of mathematical modeling, is
problem finding before problem solving”
(Pollak, 2011, 7).
Word Problem vs. Model
O Word Problem: How long does it take to drive
20 miles at 40 miles per hour?
O Answer: 30 minutes
O Modeling Problem: When you live 20 miles
from the airport, the speed limit is 40 miles
per hour, and your cousin is due to arrive at
6 PM. Does this mean you leave at 5:30?
Word Problem vs. Model
O To tackle the modeling problem, in reality, you
probably won’t leave at 5:30.
O For example,
“This is rush hour. There are those intersections at
which you don’t have the right of way. How long will
it take to find a place to park? If you take the back
way, the average drive may take longer, but there is
much less variability in the total drive time…But
don’t forget the arrival time they give you is the
time the plane is expected to touch down on the
runway, not when it will start discharging
passengers at the gate” (Pollak, 2011, 6).
Why use modeling?
O Modeling allows students to make deeper
connections to the mathematics they are
learning through relationships.
O Since our brain learns best through
connecting new material to old material,
modeling aids in the process of learning.
O Modeling promotes critical thinking skills
necessary both in higher-order mathematics
and in other disciplines.
Why use modeling?
O “In sum, the main reasons for teaching
modeling are that every child can benefit
from its power of application, and that
mathematics cannot only be learned in an
isolated way but also be seen in the realworld” (Tam, 2011, 29).
O Life is complicated, modeling turns a
complicated situation into a simpler
situation to analyze.
The Process
1.
2.
3.
4.
5.
6.
Understand and identify the issue in the real world
Formulate the structure of real-world situation
Translate to a mathematical model
Derive some mathematical facts from the model
Translate the resulting facts back to the real-world
Validate the results
Two Viewpoints
O “Mathematical
modeling provides rich
examples through
which students can
retain the
mathematics that they
have learned, and can
extract important
mathematical content”
(Tam, 2011, 30)
O “…the modeling
process [is] a key part
of mathematical
content that needs to
be taught and grasped,
with recognition that
the process is
comprised of a
different set of skills
from what is needed
for ‘pure’
mathematics” (Tam,
2011, 30)
How does modeling relate to
the Common Core?
O “Standard 4. Model with Mathematics
Mathematically proficient students can
apply the mathematics they know to solve
problems arising in everyday life, society,
and the workplace”(CCSS, 2011).
O Although not explicitly a standard in middle
school, almost all of the standards reflect
steps in the modeling cycle.
A Deeper Look…
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in
solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the
reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.
How can I use modeling in
my classroom?
Predicting the weather
O Algebra—Linear
Relationships
O Can I approximate
the unknown
temperature
somewhere based
on three places I
know? (Gould,
2011, 66)
The Process
1.
2.
3.
4.
5.
6.
Understand and identify the issue in the real world
Formulate the structure of real-world situation
Translate to a mathematical model
Derive some mathematical facts from the model
Translate the resulting facts back to the real-world
Validate the results
Predicting the Weather
Activity
O Take a few minutes to work through the
modeling process, either individually or as a
group, and find an approximate temperature
of our unknown location.
My own solution…
O Use GSP to create a triangle, and construct
a ray from St Rose through my unknown
point intersecting the line between Siena
and Albany International.
O We see the ratio from the Siena to the new
point is REALLY close to 1/3 of the distance
from Siena to ALB, therefore, accounts for
REALLY close to 1/3 of the change, so must
be right around 82.
My own solution…
O Now, to do the same work on our final point.
The recent point calculated, call that point C,
takes the place of Siena. The ratio of C with
our final point to C with St Rose is again
close to 1/3.
O The difference in temperature is 7,
multiplied by 1/3 is about 2.31, we’ll just
say 2. This suggests our final point is roughly
80 degrees.
An alternative…
O Find the midpoints to each of the lines →
78, 80, and 83
O The average of these rounds to 80 degrees.
O You notice that this average is the same if
you just average the three temperatures, but
finding the midpoints helps us determine
the best way to round. If we are skeptical in
how to round, continue finding more
midpoints as smaller triangles develop.
Predicting the Weather
O NOT a word problem!
O Multiple solutions can arise, one of the keys
to this lesson would be in the discussion
that would follow, it’s NOT about the answer!
Examining the Heart
O Can be used throughout high school in
conjunction with science classes from
grades 9-11
O The heart’s AV node decides to either beat
or skip a beat based on the strength of the
signal arriving from the SA node.
O Students can then work to answer the
question “will the heart beat?”
Examining the Heart
The heart acts as a pump which converts electrical
energy to mechanical energy. The heart is divided
into four chambers, two atria and two ventricles.
The atria receive blood from throughout the body.
Blood is then pumped through the ventricles and
out of the heart to the rest of the body. The
Sinoatrial (SA) node is the pacemaker of the heart
and sends signals to the Atrioventricular (AV) node.
The AV node receives the signal from the SA node
and sends a signal to the ventricles if the heart
should beat.
Examining the Heart
Examining the Heart
Examining the Heart
O Situation:
O The threshold voltage of the AV nodes is 20
mV.
O The potential at the AV node is 30 mV now.
O The potential at the AV node reduces to half
its original between the two signals arriving
from the SA node.
O The SA node sends signals of strength 10
mV.
Translating into the Model
Substituting
Symbolizing
O
O
Examining the Heart
O We can continue this examination through
modeling further activity of the heart using a
basic Microsoft Excel Spreadsheet
O Working with various situations leads
students to finding equilibrium values
allowing the heart to beat steadily.
Equilibrium of the Heart
O
Does Equilibrium Exist?
O
O
Does Equilibrium Exist?
Yes, the heart does become stable (with an
equilibrium value of 20 mV.
The heart will steadily beat in our previous
situation.
Final Remarks
O Modeling essentially takes word problems to
a new level—thereby developing deeper
understanding of not only the solution, but
also the problem and process itself.
O When teaching lessons with modeling, our
primary goal is the process of modeling,
learning math follows naturally.
Final Remarks
O Although some models may be complicated
to have students to develop on their own, we
can adjust models to meet all levels of
students in our classroom, i.e., the heart
model simplified to the high school level
O Modeling provides a medium through which
we can facilitate deeper thinking skills,
make interesting uses of technology, and
arrive at interesting mathematics.
References
Blum, W., & Leiss, D. (2005). “Filling up”—The problem of independence-preserving
teacher interventions in lessons with demanding modeling tasks. In
Proceedings of the Fourth Congress of the European Society for Research in
Mathematics Education (1623-1633). Presented at the European Research in
Mathematics Education IV, Sant Feliu de Guixols, Spain.
Champanerkar, J., & Eladdadi, A. (2011). Mathematics and the Heart Workshop for
High School Teachers. 2011 NHLBI-VCU World Conference on Mathematical
Modeling and Computer Simulation in Cardiovascular and Cardiopulmonary
Dynamics. Williamsburg, PA.
Common Core State Standards Initiative in Mathematics. (2010). Common Core State
Standards Initiative. Retrieved July 5, 2011, from
http://www.corestandards.org/
Gould , H. (2011). Meteorology: Describing and Predicting the Weather - An Activity in
Mathematical Modeling. Journal of Mathematics Education at Teachers
College, 66-67.
Pollak, H. O. (2011). What is Mathematical Modeling? In H. Gould, D. R. Murray, A.
Sanfratello, & B. R. Vogeli (Eds.), The Mathematical Modeling Handbook.
Bedford, MA: The Consortium for Mathematics and Its Applications.
Tam, K.C. (2011). Modeling in the Common Core State Standards. Journal of
Mathematics Education at Teachers College, 28-33.
Tam, K. C. (2011). Packing Oranges. Journal of Mathematics Education at Teachers
College, 67.
Questions and Comments?
Brandon Milonovich
[email protected]
Entire presentation available at http://www.bmilo.com.
Feel free to visit http://matheducate.wordpress.com.