MODEL WITH MATHEMATICS
 Is this mathematical modeling?
An engineer may make a mechanical model,” with
weights and springs, of an electrical circuit with
capacitances and resistances.
 One real-world object is represented by another.
 Is this mathematical modeling?
A switching circuit may be a “model” for a
Boolean function.
 A mathematical object is represented by a real-world
object
 Is this mathematical modeling?
A real number may be a “model” for a point
on a line, and vice versa.
 A mathematical object by a different mathematical
object
 Is this mathematical modeling?
A logistic function may be a “model” for the
growth of a bacterial population.
 Represents a real-world situation by a mathematical
one.
 The first three all represent meanings of a
model in a scientific or mathematical
setting. However, we are concerned with the
forth and last meaning, is a mathematical
model, which represents a real-world
situation by a mathematical one.
 “Mathematical modeling is the link between
mathematics and the rest of the world.” (Meerschaert,
M., Mathematical Modeling, Elsevier Science, 2010)

The process of beginning with a situation and
gaining understanding about that situation is
generally referred to as “modeling”. If the
understanding comes about through the use of
mathematics, the process is known and
mathematical modeling.
Problems in
everyday life…
…reasoned using
mathematical methods
Mathematically proficient students
• make assumptions and approximations to simplify a
situation, realizing these may need revision later
• interpret mathematical results in the context of the
situation and reflect on whether they make sense
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© Institute for Mathematics & Education 2011
 Identify important quantities in practical situations
and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and
formulas.
 Analyze those relationships mathematically to draw
conclusions.
 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.
 The CaCCSS expects mathematically
proficient students to be able to
apply the mathematics they know
to solve problems arising in
everyday life, society, and the
workplace.
 To prepare students for the modeling they will do in
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high school, students must have practice in:
Translating from real world context to mathematics
(“mathematizing”)
Selecting and using multiple representations and
manipulatives
Using academic language for the purpose of
communication
Checking the validity of solutions and adjusting or
changing representations when necessary
 Talk given by Dan Meyer on March 6, 2010 where he
blends real-life modeling with tips on how to take a
textbook problem and create a problem more suitable
for modeling.
 He teaches high school math outside of Santa Cruz,
CA. He received his Masters of Arts from the UC Davis
in 2005 and Cable in Classroom’s Leader in Learning
award in 2008.
Dan Meyer
1.
2.
3.
4.
5.
6.
Lack of initiative.
Lack of perseverance.
Lack of retention.
Aversion to word problems.
Eagerness for a formula.
Jeff Linder picks his nose.
 Reorder the layers of the problem
1. Begin with the visual only.
2. Immediately ask the question.
3. Throw down the mathematical structure.
4. Students develop sub –steps.
1.
2.
3.
4.
5.
Use multimedia.
Encourage student intuition. “How long will it take
to fill the water tank?”
Ask the shortest question you can.
Let students build the problem.
Be less helpful.