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Discrete Math … It’s Not So Discreet!. Arizona Association of Mathematics Teachers October 19, 2009. Valerie A. DeBellis, Ed.D. [email protected] The Roots of American Education …. Until 1840s, education was highly localized, for wealthy

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Discrete math it s not so discreet

Discrete Math … It’s Not So Discreet!

Arizona Association of Mathematics Teachers

October 19, 2009

Valerie A. DeBellis, Ed.D.

[email protected]


The roots of american education
The Roots of American Education …

Until 1840s, education was highly localized, for wealthy

“Reformers” wanted all children to gain education benefits arguing that “common schooling” could create good citizens, unite society, and prevent crime and poverty

As a result, free public education was available by the end of the 1800s, at the elementary school level, for all American children

2


A View of America … during the birth of public education

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

  • 1760 – 1920s (1800s)

  • The major technological, socioeconomic and cultural change resulting from the replacement of an economy based on manual labor to one dominated by industry and machine manufacture.

  • 1920 - ?

  • Period of industrial implementation

Industrial Age

Industrial Age

3


Women industrial workers
Women Industrial Workers

August 1942 - Maryland

June 1943 - CT

4

Made for Office of War Information



General life skills for an industrial age
General life skills for an Industrial Age

MIND SET: Build to last, local perspective (America)

WORK ENVIRONMENT: A fixed system – work at a company, punch a clock, managers & workers – white collar, blue collar

WORKERS: needed the ability to listen and learn (from management – i.e., receive instructions) and carry out precise repeated procedures (on the assembly line)

6


A shift in the world
A Shift in the World …

A Shift in the World …

A Shift in the World …

A Shift in the World …

A Shift in the World …

A Shift in the World …

A Shift in the World …

  • 1760 – 1920s (1800s)

  • The major technological, socioeconomic and cultural change resulting from the replacement of an economy based on manual labor to one dominated by industry and machine manufacture.

  • 1920 - ?

  • Period of industrial implementation

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Revolution

Industrial Age

Industrial Age

Industrial Age

Industrial Age

Industrial Age

Industrial Age

Industrial Age

1936 – Alan Turing’s math paper on “computable numbers” outlined rudimentary ideas of the programmable computer

Information Age

Information Age

  • 1975 - present

  • A period when information became easily accessible and manipulated through computers and computer networks

7


If one purpose of public education is to create functional citizens, what life skills do educators need to provide children in an information age?

8


The internet
The Internet citizens,

Who uses it?

What does the Internet look like?

How do we fix it when it breaks?

10


Who uses it
Who uses it? citizens,

As of 2009, an estimated quarter of Earth's population uses the services of the Internet.

11



General life skills for an industrial age1
General life skills for an Industrial Age citizens,

MIND SET: Build an open architecture so development can continue, global perspective (The World)

WORK ENVIRONMENT: A dynamic system – flexible working hours & locations; productivity measure rather than “supervised”

WORKERS: co-construct knowledge; need personal accountability; ability to imagine, create, design, question, explore, cooperate, & collaborate. Problem solving now a skill!

13


What is discrete mathematics
What is Discrete Mathematics ? citizens,

“Three important areas of discrete mathematics are integrated within the Standards: combinatorics, iteration and recursion, and vertex-edge graphs. … Combinatorics is the mathematics of systematic counting. Iteration and recursion are used to model sequential, step-by-step change. Vertex-edge graphs are used to model and solve problems involving paths, networks, and relationships among a finite number of objects.” NCTM, 2000

14


Recommendations
Recommendations citizens,

Systematic Listing and Counting (Combinatorics)

Vertex-Edge Graphs

Iteration and Recursion. …

NCTM, 2009

15


Discrete mathematics
Discrete Mathematics … citizens,

is the branch of mathematics that deals with arrangements of distinct objects

is the mathematics used by decision-makers in our society; from workers in government to those in health care, transportation and telecommunications

is the mathematics behind computing

16


Five major themes of discrete math
Five major themes of discrete math … citizens,

systematic listing and counting

using discrete mathematical models

applying iterative patterns and processes

organizing and processing information

finding the best solution using algorithms

17


A grades 3 5 example
A Grades 3-5 citizens, Example:

How many outfits

can you create using

four types of shirts

(red, green, blue, & yellow);

two types of pants (dotted & striped) and

three types of shoes (boots, loafers, & sneakers)?

18


Do we have all possibilities
Do we have all possibilities? citizens,

How do YOU know when you have them all?

How do elementary school children know?

19


It is in the “ citizens, process of organizing” that children begin to learn how to think systematically.

20





Grades 6 12 example recursive view of functions 1
Grades 6-12 Example: citizens, Recursive View of Functions 1

Describe patterns shown in the table.

24


Look down the column of b s
Look down the column of B’s citizens,

NEXT = NOW + 2, start at 5

Bn+1 = Bn + 2; B0 = 5

25


Look across from a to b
Look across from A to B citizens,

B = 5 + 2A

26


Recursive view of linear functions
Recursive View of Linear Functions citizens,

Explicit form: B = 5 + 2A

Recursive form:

NEXT = NOW + 2, start at 5

Bn+1 = Bn + 2, B0 = 5

Slope seen concretely in recursive form

Rate of Change seen concretely

Note also: arithmetic sequence

27


Recursive view of functions 2
Recursive View of Functions 2 citizens,

Describe patterns shown in the table.

28


Look down the column of b s1
Look down the column of B’s citizens,

NEXT = NOW * 2

Bn+1 = Bn * 2

29


Look across from a to b1
Look across from A to B citizens,

B = 5 * 2A

30


Recursive view of exponential functions
Recursive View of Exponential Functions citizens,

Explicit form: B = 5 * 2A

Recursive form:

NEXT = NOW * 2, start at 5

Bn+1 = Bn * 2, B0 = 5

Comparison to linear – add constant vs. multiply by constant at each step

Note also: geometric sequence

31


How where dm in grades 6 12
How/Where DM in grades 6-12? citizens,

Integrate DM into other strands and courses

Vertex-Edge Graphs  geometry

Recursion  algebra and geometry

Math and voting  Social Studies

DM topics  richer Consumer or General Mathematics courses

Separate DM course

4th year course alternative to precalculus

DM and Stat course

32


A True Story citizens,

North Carolina

2008-09

33


Times have changed

Times have changed … citizens,

Have we?

34


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