Discrete math it s not so discreet
This presentation is the property of its rightful owner.
Sponsored Links
1 / 34

Discrete Math … It’s Not So Discreet! PowerPoint PPT Presentation


  • 91 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Discrete Math … It’s Not So Discreet!

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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


Discrete math it s not so discreet

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


A machine

A “machine”

5


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


Discrete math it s not so discreet

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

Who uses it?

What does the Internet look like?

How do we fix it when it breaks?

10


Who uses it

Who uses it?

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

11


Http en wikipedia org wiki information age html

http://en.wikipedia.org/wiki/Information_Age.html

12


General life skills for an industrial age1

General life skills for an Industrial Age

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 ?

“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

Systematic Listing and Counting (Combinatorics)

Vertex-Edge Graphs

Iteration and Recursion. …

NCTM, 2009

15


Discrete mathematics

Discrete Mathematics …

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 …

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 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?

How do YOU know when you have them all?

How do elementary school children know?

19


Discrete math it s not so discreet

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

20


Building graphs in kindergarten

Building Graphs in Kindergarten

21


Exploring properties of graphs

Exploring Properties of Graphs

22


Finding paths in a graph

Finding Paths in a Graph

23


Grades 6 12 example recursive view of functions 1

Grades 6-12 Example: 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

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

B = 5 + 2A

26


Recursive view of linear functions

Recursive View of Linear Functions

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

Describe patterns shown in the table.

28


Look down the column of b s1

Look down the column of B’s

NEXT = NOW * 2

Bn+1 = Bn * 2

29


Look across from a to b1

Look across from A to B

B = 5 * 2A

30


Recursive view of exponential functions

Recursive View of Exponential Functions

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?

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


Discrete math it s not so discreet

A True Story

North Carolina

2008-09

33


Times have changed

Times have changed …

Have we?

34


  • Login