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Mathematical Modeling:
What it is;
What it looks like in the classroom;
Why it is so important
Facilitators:
Eric Robinson
Teri Calabrese-Gray
Presentation Overview:
What is Mathematical Modeling?
Examples of mathematical modeling
problems
Summary: What mathematical modeling
isn’t. Why mathematical modeling is so
important in school mathematics.
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In the NYSCCLS Mathematical Modeling is:
• One of the eight Standards of Practice (that
span the grades)
• One of the Conceptual Categories that span
the high school content areas
Why is it both??
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“Mathematicians are in the habit of
dividing the universe into two parts:
mathematics, and everything else, that is,
the rest of the world, sometimes called
“the real world”.
People often tend to see the two as
independent from one another
– nothing could be further from the truth…”
--- Henry Pollak
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“When you use mathematics to understand a
situation in the real world, and then perhaps
use it to take action or even to predict the
future, both the real-world situation and the
ensuing mathematics are taken seriously.”
-- Henry Pollak
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The “practice”
“Mathematical modeling begins in the unedited
real world, requires problem formulation before
problem solving and once the problem is solved,
moves back into the real world where the results
are considered in their original context. Are the
results practical, the answers reasonable, the
consequences acceptable? If so, great! If not,
take another look at the choices made at the
beginning, and try again.
This entire process is what’s called mathematical
modeling.”
-- Henry Pollak
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Warm Up
Your grandmother will be arriving at
the airport at 6:00 pm. You live 20
miles from the airport. The speed limit
is 40 miles per hour. When should you
leave to get her?
-- Henry Pollak
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Mathematical Model
Real World
Formulate
Clearly identify
situation
Include assumptions and
constraints
Pose (well-formed)
question
Revise
List key features of
situation
Simplify the situation
Modeling
Paradigm
Build math model :
(strategy, concepts, data,,
variables, constants, etc.)
Compute
process
deduce
Apply:
Do results:
make sense?
Interpret
(Valid) Mathematical
results
satisfy criteria?
Are results sufficient?
Real World Conclusions
Mathematical Conclusions
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Note that the
Formulate and
Revise
components are
about “Problem
Posing.”
Real World
connection
“Inside Mathematics”
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Interpret: Contextualize mathematical results and see if
the model results make sense or works (e.g. if the
results satisfies certain criteria).
Revise:
• Because the results do not seem to fit what
does/would actually happen.
• You want to generalize your results thus far and
this might affect your modeling approach.
•You want to remove some of the simplifying
features and/or add other features.
OR Validate: You decide this is good, accept the
model results and write a report.
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4. Model with Mathematics Mathematically proficient students
can apply the mathematics they know to solve problems arising
in everyday life, society, and the workplace…. Mathematically
proficient students who can apply what they know are
comfortable making assumptions and approximations to simplify
a complicated situation, realizing that these may need revision
later. They are able to identify important quantities in a practical
situations…. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret
their mathematical results in the context of the situation and
reflect on whether the results make sense, possibly improving
the model if it has not served its purpose.
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The (High School) Conceptual
Category
Modeling Standards: Modeling is best
interpreted not as a collection of isolated
topics but rather in relation to other
standards. Making mathematical models is a
Standard for Mathematical Practice, and
specific modeling standards appear
throughout the high school standards
indicated by a star symbol (★)
--NYSCCLS pg. 62
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Examples of Content standards and modeling:
Algebra: Seeing Structure in Expressions A-SSE
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its
context.★
Functions Interpreting Functions F-IF
Interpret functions that arise in applications in terms of
the context.
4. For a function that models a relationship between two
quantities, interpret key features of graphs and tables in terms
of the quantities. Key features include: intercepts; intervals
where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end
behavior; and periodicity. ★
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Standards for Mathematical Practice
1.
Make sense of problems and persevere in
solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the
reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated
reasoning.
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Storm Models
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Thanksgiving Table Example: 6th grade
https://vimeo.com/46127286
Questions to consider as you watch the video:
Is the attention to the real world realistic?
Is the modeling question clearly stated?
Is the modeling question phrased in a way that there value to answer in the minds of students?
How does the material in the video fit with the modeling cycle? (Refer to the modeling cycle
graphic on the handout.) Specifically,
• What work/information/evidence, if any, which is in the video would you put under
“formulation?” step in the cycle? Who is providing the information, the teacher or the
student? (Refer to the top right box in the graphic.)
• What work/information/evidence, if any, would put under the “compute/process/deduce”
step in the cycle? (Refer to the lower right box in the graphic)
• What work/information/evidence, if any, would you put in the “interpret” step of the cycle?
(Refer to the lower left box in the graphic.)
• What work/information/evidence would you put in the “revise” step in the cycle?
What content standards are evident in the student activity? Give evidence.
What mathematical practices are evident as students work? Give evidence.
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PISA –Like Assessment Item
Rock Concert
For a rock concert a rectangular field of size 100 m by 50 m
was reserved for the audience. The concert was completely
sold out and the field was packed with all the fans standing.
Which one of the following is likely to be the best estimate of
the total number of people attending the concert?
A 2,000
B 5,000
C 20,000
D 50,000
E 100,000
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Activities: Fun, Fun, Fun
Questions common to all Activities:
Is the attention to the real world realistic?
Is the modeling question clearly stated?
Is the modeling question phrased in a way that there value to
answer in the minds of students?
How does the work on the activity fit with the modeling cycle? Be
specific.
What content standards are evident in the student activity? Give
evidence.
What mathematical practices are evident as students work? Give
evidence.
Could you use or modify this problem for the grade level at which
you teach?
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“Math Class Needs a Makeover”
Speaker: Dan Meyer
http://www.youtube.com/watch?v=NWUFjb8w9Ps
Question to consider while watching:
What is the role of mathematical modeling in the
suggested “makeover?”
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Summary regarding what mathematical modeling is.
(a) Problems in which both the real world and mathematics are taken
seriously. With modeling problems, student need to think about both
the real world and the mathematics.
(b) Modeling is about problem posing as well as problem solving.
(c) Modeling is often open-ended requiring decisions about what
assumptions, information and simplifications are to be included.
(d) Different models of some problems are viable.
(e) Solutions to modeling problems usually suggest actions or
predictions.
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And maybe most importantly:
(f) The practice of modeling includes a multistep process: Formulating the problem,
building the mathematical model, processing
the mathematics, interpreting the
conclusions, and often revising the model
before writing a report.
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What Mathematical Modeling is not.
(a) Just a fancy name for traditional textbook applications.
(b) An incidental context for the teaching of the
decontextualized “mathematics.”
(c) Accomplished by simply “covering” the NYSCCLS
content standards that are marked with a .
(d) A learning goal that can be accomplished without
student understanding of the modeling cycle.
(e) Only possible if you know a lot of complicated math.
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Why mathematical modeling is important.
(a) Modeling serves many everyday situations.
(b) Some entire careers revolve around a single modeling problem.
(c) Eliminates questions regarding “what good is this stuff?”
(d) Standards from multiple mathematical domains (and multiple
grade levels) can occur together in modeling problems. This serves
to make connections between mathematical content.
(e) It fosters flexible (mathematical) thinking and use of concepts.
(f) Full scale modeling often engages many of the Standards for
Mathematical Practice.
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Additional important reasons:
(g) Modeling serves as an environment that
promotes deeper understanding of concepts.
(h) Modeling problems provide context for the
application of mathematics students know. In
addition, such problems sometimes serve as a
context to introduce new concepts in a
meaningful way.
(i) It is consonant with what we know about
student learning.
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Resources mentioned today:
1. Teachers College Mathematical Modeling
Handbook, COMAP Inc. 2011
(www.comap.com)
2. Mathematics Modeling Our World
(MMOW); COMAP Inc. 2010 - 2012
(www.comap.com)
3. NCTM Reasoning and Sense Making Task
Library http://www.nctm.org/rsmtasks/
4. NCTM Focus on Reasoning and Sense
Making series
5. Illustrative Mathematics Project
http://www.illustrativemathematics.org/
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Sample modeling questions:
• Where do you put the fire station? (2)
• Why do trucks get stuck going under bridges?(4)
• Is the testing of pooled blood samples an effective technique
for detecting which athletes are using drugs? (2)
• When will the Moose population in the Adirondacks reach the
right size? (2)
• You just won the “Gasoline for Life” prize. Should you take
the option of a lump sum of $50,000 instead? (3)
• When should you fill the bird feeder? (1)
• Can you move that huge sofa around the hallway corner? (1)
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Presentation Take-aways:
Answers to: what is Mathematical
Modeling?
Examples of mathematical modeling
problems intended for students that
illustrate all or part of the modeling cycle
What mathematical modeling isn’t. Why
mathematical modeling is so important in
school mathematics.
EngageNY.org
Mathematical Model
Real World
Formulate
Clearly identify
situation
Include assumptions and
constraints
Pose (well-formed)
question
Revise
List key features of
situation
Simplify the situation
Modeling
Paradigm
Build math model :
(strategy, concepts, data,,
variables, constants, etc.)
Compute
process
deduce
Apply:
Do results:
make sense?
Interpret
(Valid) Mathematical
results
satisfy criteria?
Are results sufficient?
Real World Conclusions
Mathematical Conclusions
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Framing you own Modeling Cycle
Problem(s): Ways to start
Create a new modeling cycle problem from an
interesting real world question.
Make a list of potential contexts for modeling.
Make a list of real world questions.
Create a modeling cycle problem/activity from a
favorite “application.”
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Thank You!
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