teaching with mathematical modeling: providing multiple entry points and connections

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

http://www.corestandards.org/


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


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

10 January 2013


Modeling Example #1: How many barrels of water did Columbus bring on his 1492 journey to

the “New World”?

10 January 2013


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

10 January 2013

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.



Modeling Example #2: How much medicine do I have to take to make my sore throat feel better?

10 January 2013

(tonsillitis)


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.


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


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?

10 January 2013

Number of Doses

3

6

9


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



Modeling Example #3: What happens if I am in a classroom with 20 students and one of those

students has the flu?

10 January 2013


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?


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.

10 January 2013

Grade 8 Modeling: Catching the flu!


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


Grade 8 Modeling: Catching the flu!Introducing Logistic Function

10 January 2013

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.


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


And so my journey with teaching with modeling continues.

And it always leaves me with more questions to ask.

10 January 2013


Muchas Gracias Dr. Roberto Araya de Universidad de Chile

Ministerio de Educación de Chile

Dr. Eric Marland

10 January 2013

teaching with mathematical modeling: providing multiple entry points and connections

Documents

mathematical modeling

chile modeling

mathematical connections

mathematical conceptsallows

mathematical thinking

choice of mathematical

columbus journey

connections teaching