Discrete Math … It’s Not So Discreet!

1. Discrete Math … It’s Not So Discreet! Arizona Association of Mathematics Teachers October 19, 2009 Valerie A. DeBellis, Ed.D. debellis@discreteteaching.org

2. 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

3. 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

4. Women Industrial Workers August 1942 - Maryland June 1943 - CT 4 Made for Office of War Information

5. A “machine” 5

6. 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

7. 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

8. 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

10. The Internet Who uses it? What does the Internet look like? How do we fix it when it breaks? 10

11. Who uses it? As of 2009, an estimated quarter of Earth's population uses the services of the Internet. 11

12. http://en.wikipedia.org/wiki/Information_Age.html 12

13. 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

14. 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

15. Recommendations Systematic Listing and Counting (Combinatorics) Vertex-Edge Graphs Iteration and Recursion. … NCTM, 2009 15

16. 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

17. 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

18. 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

19. Do we have all possibilities? How do YOU know when you have them all? How do elementary school children know? 19

20. It is in the “process of organizing” that children begin to learn how to think systematically. 20

21. Building Graphs in Kindergarten 21

22. Exploring Properties of Graphs 22

23. Finding Paths in a Graph 23

24. Grades 6-12 Example: Recursive View of Functions 1 Describe patterns shown in the table. 24

25. Look down the column of B’s NEXT = NOW + 2, start at 5 Bn+1 = Bn + 2; B0 = 5 25

26. Look across from A to B B = 5 + 2A 26

27. 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

28. Recursive View of Functions 2 Describe patterns shown in the table. 28

29. Look down the column of B’s NEXT = NOW * 2 Bn+1 = Bn * 2 29

30. Look across from A to B B = 5 * 2A 30

31. 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

32. 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

33. A True Story North Carolina 2008-09 33

34. Times have changed … Have we? 34