Cheryl Ooten, Ph.D. Mathematics Professor Emerita Santa Ana College

1. Through the Eyes of the MathematicianThrough the Eyes of the MathematicianThrough the Eyes of the MathematicianThrough the Eyes of the Mathematician Cheryl Ooten, Ph.D. Mathematics Professor Emerita Santa Ana College Bernie Russo, Ph.D. Mathematics Professor Emeritus University of California at Irvine

2. Preview I. Set the Stage—Some Models & Myths II. What’s It All About? III. Sampling the Branches IV. Stories about The Big Ones— Solved & Unsolved

3. I.Set the Stage—Some Models & Myths

4. The math world is like a restaurant…

5. The math world is like a restaurant… with “a kitchen & dining room”

6. Math is like gossip. Devlin, K.

7. Math is like gossip. It’s about relationships. Devlin, K. ∆

8. Models for Managing the Mean Math Blues

9. a) Fight negative math rumors. 1. Some people have a math mind and some don’t. 2. I can’t do math. 3. Only smart people can do math. 4. Only men can do math. 5. Math is always hard. 6. Mathematicians always do math problems quickly in their heads. 7. If I don’t understand a problem immediately, I never will. 8. There is only one right way to work a math problem. 9. It is bad to count on fingers. 10. Negative math-experience memories never go away.

10. b) Use reframes. • What can a reframe do? • Affect attitude & change feelings. • Neutralize negativity. • Change a helpless victim to an in-charge owner. • Remind us of the “YET.” Ref: Ooten & Moore, Managing the Mean Math Blues

11. Cognitive Psychotherapy Model of how people “work” THOUGHTS EMOTIONS BEHAVIORS BODY SENSATIONS Ref: Ooten & Moore. Managing the Mean Math Blues

12. EMOTIONS I am frightened by math THOUGHTS I can’t do math. BEHAVIORS I avoid numbers I don’t practice math BODY SENSATIONS My stomach tenses when I see numbers Ref: Ooten & Moore, Managing the Mean Math Blues, p 154

13. EMOTIONS Relief Curiosity about what else I can learn Joy with skills I have THOUGHTS I can do some math. I can learn more. I don’t need to get it all right now. BEHAVIORS Take a deep breath Write a possible solution. Try something new. Ask questions BODY SENSATIONS Relax Become calmer Heart rate slows Ref: Ooten & Moore, Managing the Mean Math Blues, p 154

14. c) Check your Mindset. (Carol Dweck, Stanford Psychology Prof) vs • Growth Mindset • Your smartness increases with hard work. • Fixed Mindset • You can learn but can’t change your basic level of intelligence Ref: Boaler, Mathematical Mindsets; Dweck, Mindset: The New Psychology of Success

15. Mindset Model vs • Growth Mindset • Focus on effort • Skip judging • Ask: • What can I learn? • How can I improve? • What can I do • differently? • Fixed Mindset • Focus on ability • Evaluate & label Good-bad • Strong-weak Choose this! Ref: Boaler, Mathematical Mindsets; Dweck, Mindset: The New Psychology of Success

16. “Just because some people can do something with little or no training, it doesn’t mean that others can’t do it (and sometimes do it even better) with training.” Reframe anxiety by focusing on effort, not ability. Math skills are learnable! Ref: Dweck

17. d) Anxiety comes from being required to stay in an uncomfortable situation over which we believe we have no control.

18. References for Managing the Mean Math Blues Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass: San Francisco, CA. Boaler, J. (2015). What’s math got to do with it? Penguin Books: New York, N.Y. Dweck, C. S. (2006). Mindset: The new psychology of success. Ballantine Books, N.Y. Ooten, C. & K. Moore. (2010). Managing the mean math blues: Math study skills for student success. Pearson: Upper Saddle River, N.J.

19. Watch for math myths everywhere! Math Writer John Derbyshire wrote: “I don’t believe [this topic] can be explained using math more elementary than I have used here, so if you don’t understand [it] after finishing my book, you can be pretty sure you will never understand it.” Ref: Derbyshire, Prime Obsession, p. viii ∆

20. UC Berkeley Mathematics Professor Edward Frenkel says: “One of my teachers…used to say: ‘People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.’ My goal is to explain this stuff to you in terms that you will understand.” Ref: Frenkel, Love and Math, p 6

21. Our goal is to talk about a few math things we think are cool that were not taught in school and hopefully to talk about them in ways that make sense! II.What’s It All About? Math’s beautiful, Math’s everywhere, and Math’s huge! ∆

22. Our goal is to talk about a few math things we think are cool that were not taught in school and hopefully to talk about them in ways that make sense! II.What’s It All About? Math’s beautiful, Math’s everywhere, and Math’s huge!

23. “[The world of mathematics is] a hidden parallel universe of beauty and elegance, intricately intertwined with ours.” Ref: Frenkel, Love and Math, p 1 These are mathematical models of fractals.

24. Example: Neuroscientist/musician Daniel Levitin says: Music is organized sound. We say: Mathis the organizing tool. Music is “intricately twined” with math. Example: Dave Brubeck with Paul Desmond’s “Take Five” Ref: Levitin, This Is Your Brain on Music ∆

25. How much math does one person know? “By the [late 1800s], math had passed out of the era when really great strides could be made by a single mind working alone. [It became] a collegial enterprise in which the work of even the most brilliant scholars was built upon, and nourished by, that of living colleagues.” Ref: Derbyshire, Prime Obsession, p 165

26. Four Branches of Mathematics: Arithmetic Algebra Geometry Analysis

27. Four Branches of Mathematics: 1. Arithmetic (Counting) 2. Algebra (Symbolic Manipulation) 3. Geometry (Figures,Drawing) 4. Analysis (Calculus, Limits)

28. 60+ Mathematics Research Specialties (A.M.S.) Combinatorics Number Theory Algebraic Geometry K-Theory Topological Groups Real Functions Potential Theory Fourier Analysis Abstract Harmonic Analysis Integral Equations Functional Analysis (Bernie) Operator Theory Calculus of Variations Optimization Convex/Discrete Geometry Differential Geometry General Topology Algebraic Topology Manifolds & Cell Complexes Global Analysis Probability Theory Statistics Numerical Analysis Computer Science Fluid Mechanics Quantum Theory Game Theory; Economics Operations Research Systems Theory Mathematical Education (Cheryl) AND MORE…

29. Each specialty is related to one or more branches. 1.Arithmetic: Combinatorics Number Theory Statistics ··· 2.Algebra: Algebraic Geometry K-Theory Group Theory ··· 4.Analysis: Fourier Analysis Differential Equations Functional Analysis ··· 3.Geometry: Convex/Discrete Geometry Differential Geometry General Topology ··· Deeper results are possible when different research specialties are connected.

30. Mathematical research is either: • Pure (theory for its own sake) • Applied (e.g. credit cards security)

31. III.Sampling the Branches

32. School math is: • centuries old • tiny part of the whole field of math. Let’s see some cool things that aren’t always shown in school. ∆

33. 1.Arithmetic (Counting or Number Theory)

34. Number Theory & prime numbers have given us cryptography &, thus, the ability to securely use credit cards online.

35. Prime numbers are a rich source of ideas.

36. The FUNdamental Theorem of Arithmetic: Every positive integer greater than 1 is either a prime or it can be factored as the unique product of prime numbers.

37. Review: What is a prime? The numbers 2, 3, 4, 5, … are either “Composite” or “Prime” Composites: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30… Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … Primes cannot be factored in an interesting way.

38. Review: What is a prime? The numbers 2, 3, 4, 5, … are either “Composite” or “Prime” Composites: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30… Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … Primes cannot be factored in an interesting way. Even numbers after 2 are composite.

39. The FUNdamental Theorem of Arithmetic: Every positive integer greater than 1 is either a prime or it can be factored as the unique product of prime numbers. e.g. 2 and 3 are prime 4 = 2·2 5 is prime 6 is 2·3 7 is prime … ∆

40. The FUNdamental Theorem of Arithmetic: Every positive integer greater than 1 is either a prime or can be factored as the unique product of prime numbers. Let’s prove it: Good Numbers = primes or can be factored as the unique product of prime numbers Bad Numbers = all the others

41. The FUNdamental Theorem of Arithmetic: • Consider the smallest bad number. • It can’t be prime (it’s “bad,” not “good.”) • That means it’s composite and can be factored into the product of smaller numbers. BAD Smaller number times Smaller number

42. The FUNdamental Theorem of Arithmetic: • These “smaller numbers” must either be prime or able to be factored as primes. • Whoops! The BAD number isn’t “bad” after all. BAD Smaller number times Smaller number

43. Since the smallest bad number couldn’t be bad, we continue “up” to the next smallest bad number but the same thing happens. And on and on. That means that there are no “bad” numbers and our theorem is true. IT’S TRUE: Every positive integer greater than 1 is a prime or can be factored as the unique product of prime numbers.

44. The FUNdamental Theorem of Arithmetic: IT’S TRUE: Every positive integer greater than 1 is either a prime or can be factored as the unique product of prime numbers.

45. The FUNdamental Theorem of Arithmetic: This proof is done by induction which is like setting up dominoes so each domino could push the next over & then starting them to fall. ∆

46. “Arithmetic has the peculiar characteristic that it is rather easy to state problems in it that are ferociously difficult to prove.” • Example: • In 1742, Christian Goldbach (age 52) made a conjecture. • It is not yet proved or disproved • But it is the subject of a novel called Uncle Petros and Goldbach’s Conjecture. Ref: Derbyshire, Prime Obsession, p 90

47. “Arithmetic has the peculiar characteristic that it is rather easy to state problems in it that are ferociously difficult to prove.” • Example: • In 1742, Goldbach made a conjecture. • It is not proved or disproved yet. • But it’s the subject of a novel called Uncle Petros and Goldbach’s Conjecture. Ref: Derbyshire, Prime Obsession, p 90

48. Goldbach’s Conjecture: Every even number greater than 2 is the sum of two primes. 4=2+2 6=3+3 8=3+5 10=3+7 12=5+7 14=3+11 16=5+11 18=7+11 20=? 22=?