Teaching with mathematical modeling: providing multiple entry points
and connections Mary Beth Searcy
Appalachian State UniversityBoone, North Carolina USA
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013
10 January 2013Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Chile • Currículum Nacional• Current Implementation: Educación Básica 1° a 6° Básico Matemática
United States• Almost a national mathematics curriculum for K-12 • Common Core State Standards for Mathematics (CCSSM) http://www.corestandards.org• 2012 Implementation: North Carolina one of 13 states in Phase 1
Mathematics Curriculum Changes
10 January 2013Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Comparing Grades 1-6 Curricula: Mathematical Thinking in the Classroom
North Carolina (CCSSM) Mathematical Practices
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
Chile (Currículum Nacional) Habilidades and Actitudes
Problem Solving Argumentation and Communication Modeling Representation Demonstrate an organized and methodical working
style Be flexible and creative when solving problems Be curious about and interested in learning
mathematics Have a positive attitude about yourself and your
abilities. Be hard-working and persevere Express yourself and listen attentively to others
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Teaching with Mathematical Modeling -- My journey started with questionsWhy do I want to use modeling in my classroom?
How can I include modeling in the curriculum?
How can I make sure that my students make the mathematical connections within modeling?
What is teaching with modeling anyway?
10 January 2013
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Why Teach with Mathematical Modeling?Reason #1: The Experience of Modeling
Students discover something new
Generates excitement about answering a question when “THE answer” is not known
Students see modeling as a complex process
Promotes creativity and communication
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Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling Example #1: How many barrels of water did Columbus bring on his 1492 journey to
the “New World”?
10 January 2013
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling: Columbus’ Journey and Water Voyage manifests were lost – we do not know the answer
Can be modeled by different levels of students
Students Ask questions – what impacts the need for water? Learn more about Columbus’ journey and the world in the late 1400’s Justify their choice of mathematical tools Communicate their solution processes
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Why Teach with Mathematical Modeling?Reason #2: Reinforce Mathematical Concepts
Allows more opportunities to use mathematic concepts students have learned.
Helps draw connections to other mathematics concepts
Allows student to see how mathematical concepts are interpreted in terms of real world situations.
Promotes curiosity
Helps build foundation for more complex mathematical ideas.
10 January 2013Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling Example #2: How much medicine do I have to take to make my sore throat feel better?
10 January 2013
(tonsillitis)
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling: How much medicine do I need to take?
10 January 2013
I will start by thinking about how I will take my medicine each day.
What must I know to answer my question?
Each day •I take a dose when I get up in the morning and•I take a dose when I go to bed.
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling in Grade 1: Tonsillitis and Medicine
So what does this tell us about the amount of medicine I need to take for my sore throat?
10 January 2013
2
4
6
Number of Doses
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
What happens if …?• you start your first dose when
you go to bed on the first day?
• you take a dose when you get up in the morning, a dose at lunch, and a dose when you go to bed?
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Number of Doses
3
6
9
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Foundational Thinking for Later Grades
10 January 2013
Counting by Twosleads us to Rates“2 doses per day”
2
4
6
Days taken
Total Doses
0 0
1 2
2 4
3 6
4 8
5 10
6 12
7 14
2
2
2
2
2
2
2
1
1
1
1
1
1
1 0 1 2 3 4 5 6 7 80
2
4
6
8
10
12
14
16
Number of Days TakenN
umbe
r of D
oses
Making a Table and Graphleads to a Linear Function Model: y = 2x where x = Number of Days &y = Total Number of Doses
Next day’s Total = Yesterday’s Total + 2Also …Total Number of Doses= sum of groups of 2 doses
Total Number of Doses = (Number of Days Taken) X 2
Why Teach with Mathematical Modeling?Reason #3: Introduce New Concepts
Explore a familiar situation with mathematics
Analyze situation and uncover the need for a new mathematical concept
Promotes further research on the situation
10 January 2013Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling Example #3: What happens if I am in a classroom with 20 students and one of those
students has the flu?
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Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling: What will happen to the class when one person has the flu?
10 January 2013
Let us start with a simple idea …
Each hour, an infected person will come in contact with two people and thus spread the flu germ to two people.
What do I know about the spread of germs?
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Let us explore this situation with an activity.
Give each of the twenty students a natural number, starting with 1 up to 20 .
Now you have Student No. 1, Student No. 2, Student No. 3, Student No. 4, … , Student No. 20.
Using a random number generator from natural numbers 1 to 20, we will select which student comes into class with the flu.
Continue to use random number generator to see who comes in contact with “sick” students.
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Grade 8 Modeling: Catching the flu!
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Grade 8 Modeling: Catching the flu!
10 January 2013
Number of Hours in Class Student Number -- Those who are Infected
0 19
1 19 11 18
219 11 18
5 10 5 9 20 33 19 11 18 5 10 9 20 3
8 7 18 3 9 10 12 17 13 18 18 12 11 1 3 10
Continue until all 20 students are “sick” and sitting down at their desks
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Grade 8 Modeling: Catching the flu!Introducing Logistic Function
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0 1 2 3 4 5 60
5
10
15
20
25
Number of Hours
Num
ber I
nfec
tedOur modeling activity
leads us to a new idea ofbounded growth.
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Modeling: Catching the FluMore questions What happens if we change the number of contacts that people have?
What happens if we only infect a fraction of the people contacted?
Are there infectious diseases that we can model where the entire population does not become infected?
What happens if we allow for people to “recover” from their infectious state while others continue to “infect” the population?
10 January 2013
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
And so my journey with teaching with modeling continues.
And it always leaves me with more questions to ask.
10 January 2013
Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile
Muchas Gracias Dr. Roberto Araya de Universidad de Chile
Ministerio de Educación de Chile
Dr. Eric Marland
10 January 2013