Helping Students
Navigate the
Modeling Process
Maria Hernandez and John Sheridan
NCSSM
Goals for the presentation
 Introduce Modeling
 Present the Suez Canal problem: “How can
we optimize the shipping traffic through the
Suez Canal?” and its parameters.
 Consider the Classroom Approach – How
can we introduce the problem to students
and make the solution accessible to them?
Goals for the presentation
 Anticipate student struggles - How can
we help students navigate the modeling
process?
 Summarize main ideas and provide time
for questions
What is Mathematical Modeling?
“Modeling links classroom mathematics and
statistics to everyday life, work, and decisionmaking. Modeling is the process of choosing and
using appropriate mathematics and statistics to
analyze empirical situations, to understand them
better, and to improve decisions.
Quantities and their relationships in physical,
economic, public policy, social, and everyday
situations can be modeled using mathematical and
statistical methods. When making mathematical
models, technology is valuable for varying
assumptions, exploring consequences, and
comparing predictions with data.”
Taken from CCSS High School Modeling
Mathematical Model Building
Mathematics is studied because it is a
rich and interesting discipline, because
it provides a set of ideas and tools that
are effective in solving problems which
arise in other fields, and because it
provides concepts useful in theoretical
studies in other fields.
From Indiana University
http://www.indiana.edu/~hmathmod/modelmodel.html
Modeling Cycle
From “A First Course in Mathematical Modeling”
Giordano, Weir, and Fox
From CCSS Math
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Introduction to the
Suez Canal problem
 The Suez Canal is an
artificial waterway
that connects the
Mediterranean Sea
and the Red Sea
 It provides an
alternate shipping
route to traveling
around the southern
tip of Africa
 It accommodates
approximately 7% of
worldwide sea
transport
A Brief Suez Canal History
 The Canal opened in 1869.
 In 1967, the Canal closed for 8 years due
to Arab-Israeli conflicts. It did not open
again until 1975.
 At that time, the Egyptian Canal
Authority hired a team of
mathematicians from Cardiff University in
Wales to optimize daily shipping traffic –
the first time this had been done.
Why Optimize Canal Traffic?
Suez Canal’s Impact on Egypt
 In 2011, Egypt’s Treasury received
approximately $5.2 billion from canal
shipping fees
 The Egyptian Canal Authority oversees
the Canal and charges, on average,
$250,000 per ship
 Adding 4 extra ships per day means an
additional $1,000,000 per day
Parameters of the Problem
 The Suez Canal is
about 163 km long
 At a width of less
than 300 meters,
only one-way
travel is permitted
 There are two
docking areas:
Ballah Bypass and
Bitter Lake
Suez Canal Time Lapse
 http://www.youtube.com/watch?v=L0JVIvKLsc
 The video shows the use of Convoys as a
way to organize shipping traffic
Traffic Flow on the Canal
 South-North traffic has the
right-of-way and does not
stop.
 North-South Convoy must
pull over to let South-North
traffic pass.
 Two docks for N-S ships are
Ballah Bypass and Bitter
Lake.
The Canal Authority sets
requirements for the solution
The solution must be a 24-hour
transportation schedule.
The solution must have an equal
number of ships pass from North to
South as South to North in a 24hour period.
A First Step:
Deciding How to Use the Pull-off Areas
 Because there are 2 pull off areas, there are
3 possible convoys that can occupy the
canal at one time
 Two convoys (A and B) leave from the North
A stops at Bitter Lake
B stops at Ballah Bypass
 Convoy C will leave the South after an initial
delay and travel non-stop
A First Attempt at a
Visual Model
 GSP Suez Canal
The Visual Model’s Limitations
 There is no accurate count on the
number of ships in each convoy
 There is no way to tell when Convoy B
and C depart
 There is no way to tell when ships arrive
at pull-off areas or the end of the canal
What other information
do I need to get started?
 Distances?
 Velocities?
 Pull-off Areas?
 Other quantities?
Distance markers along the Canal
Try some calculations
 Approximately how long does it take for
1 ship to pass through the Canal nonstop?
 Approximately how long does it take for
10 ships to pass through the Canal nonstop?
 If Mondays are “One-Way Traffic Day,”
how many ships can travel South-toNorth in 24 hours?
Recapping Our Goals
 Determine the total number of ships that
can pass through the canal in a 24-hour
time period.
 Determine a schedule for two NorthSouth convoys and one South-North
convoy.
A Model for consideration
10-ship Convoy A
Convoy C leaves too early!
(Do you see why?)
Convoys A and C complete
a successful Canal trip
Reflect on What Students
Know so far
 Students can put the problem in their
own words, describe parameters, and
perform simple calculations
 Students understand the graph paper
model
 Students understand how ships “move”
and what a failed solution looks like
Attempt a First Try
 Choose convoy sizes
 Convoy A has 36 ships
 Convoy B has 17 ships
 Convoy C has 53 ships
 More failure lurks around the corner,
but an “A-ha” moment awaits
A Dynamic Geogebra
Solution
Classroom Approach
 How can we support students as they
navigate through the modeling process?
 What is the role of questioning?
 When and how do we “help” kids with
the solution?
What does lesson planning look like?
 How do you engage the students with the problem?
 What kinds of questions do you pose?
1. Questions that help students understand the context
and what they are being asked to do
2. Questions that help students get started
3. Questions that help teachers understand students’
reasoning.
 How do you anticipate road blocks and plan next
steps?
 How will students share their models and solutions?
 How will you assess your students?
Supporting Students
 How can we support each other as we
introduce new modeling problems in our
classrooms?
 How do we get students to try new
approaches?
 How can we encourage students to
persevere through the process?
 How can we promote student discourse
and collaboration?
Student Reflection
“I enjoyed the convoy problem because it required
me to think and test solutions instead of quickly
finding the answer. Although I may not have found
the best answer, but finding a viable solution took a
while and was rewarding. It was also good to get
practice clearly explaining my thought process. I’m
used to not showing each step and clearly defining
my variables. It was new and interesting to start with
a visual representation to create equations and test
its accuracy.”
References and Further Reading
 The Consortium for Mathematics and Its Applications
(COMAP)
http://www.comap.com/index.html
 “How Not to Talk to Your Kids: The Inverse Power of
Praise”, Po Bronson, New York Magazine, 8/3/2007:
http://nymag.com/news/features/27840/
 PISA 21012 Mathematics Framework
http://www.oecd.org/pisa/pisaproducts/46961598.pdf
 Phillips Exeter Academy Math, Science and
Technology Conference – Exeter, NH
June 22- 27, 2014
http://www.exeter.edu/summer_programs/7325.aspx
Productive Struggle
 NPR Report
http://www.npr.org/blogs/health/2012/11/12/164793058/struggle-forsmarts-how-eastern-and-western-cultures-tackle-learning
Nicaragua partners with China
to build a canal
to rival the Panama Canal
(NY Times; June 13, 2013)
Contact Information
John Sheridan
[email protected]
Maria Hernandez
[email protected]