1. MAA Intermountain Section Meeting March 24, 2007 PETER E. TRAPA Department of Mathematics University of Utah Organizing a High School Math Circle

2. University of Utah March 24 rd, 2007 What is a Math Circle? Provisional definition:

3. University of Utah March 24 rd, 2007 What is a Math Circle? Provisional definition: a program for K-12 students devoted to mathematical training and exploration. Particular emphasis is placed on direct interaction with mathematicians.

4. University of Utah March 24 rd, 2007 What is a Math Circle? Provisional definition: a program for K-12 students devoted to mathematical training and exploration. Particular emphasis is placed on direct interaction with mathematicians. Typically the methods and materials used are non-standard, at least compared with usual K- 12 curriculae.

5. University of Utah March 24 rd, 2007 Examples Three that I’ll discuss today: –The Boston Math Circle –The Berkeley Math Circle –The Utah Math Circle

6. University of Utah March 24 rd, 2007 Examples But many others exist, including: –Davis (UC Davis) –New York (Lehman College, CUNY) –Miami (Florida International U) –Mobile (U of Southern Alabama) –Palo Alto (Castilleja School) –San Diego (UCSD, Art of Problem Solving) –San Franscisco (USF) –San Jose (collaboration with AIM) –St. Louis (Washington U)

7. University of Utah March 24 rd, 2007 Toward a National Network? MSRI is interested in building and maintaining a national-level “Math Circle Infrastructure”.

8. University of Utah March 24 rd, 2007 Toward a National Network? MSRI is interested in buliding and maintaing a national-level “Math Circle Infrastructure”. Small “seed grants” from MSRI are now available for starting new Math Circles.

9. University of Utah March 24 rd, 2007 Why all the interest in Math Circles?

10. University of Utah March 24 rd, 2007 Why all the interest in Math Circles? Typical high school math curriculae are almost exclusively template based.

11. University of Utah March 24 rd, 2007 Why all the interest in Math Circles? Typical high school math curriculae are almost exclusively template based. This is extremely effective for some purposes.

12. University of Utah March 24 rd, 2007 Why all the interest in Math Circles? Typical high school math curriculae are almost exclusively template based. This is extremely effective for some purposes. But it’s terrible for others.

13. University of Utah March 24 rd, 2007 Downsides of template-based learning

14. University of Utah March 24 rd, 2007 Downsides of template-based learning Encourages very short attention spans.

15. University of Utah March 24 rd, 2007 Downsides of template-based learning Encourages very short attention spans. Implemented algorithms take precedence over their conceptual formulation.

16. University of Utah March 24 rd, 2007 Downsides of template-based learning Encourages very short attention spans. Implemented algorithms take precedence over their conceptual formulation. Creativity is often limited to the confines of the template.

17. University of Utah March 24 rd, 2007 But this is not what mathematicians do!

18. University of Utah March 24 rd, 2007 But this is not what mathematicians do! Instead we are trained to think deeply about simple things.

19. University of Utah March 24 rd, 2007 But this is not what mathematicians do! Instead we are trained to think deeply about simple things. This idea is completely absent in template-based learning.

20. University of Utah March 24 rd, 2007 A Different Model: The Russian Experience In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists.

21. University of Utah March 24 rd, 2007 A Different Model: The Russian Experience In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists. Early Identification Special Schools Math Circles

22. University of Utah March 24 rd, 2007 A Different Model: The Russian Experience In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists. Early Identification Special Schools Math Circles The objective is not limited to producing only research mathematicians.

23. University of Utah March 24 rd, 2007 Example (courtesy of Josh Zucker)

24. University of Utah March 24 rd, 2007 Example (courtesy of Josh Zucker) St. Petersburg’s School 239: 2500 sixth-graders take the district Olympiad. 25 or so are tracked into their own math course and math circle.

25. University of Utah March 24 rd, 2007 Example: 8 th grade algebra class

26. University of Utah March 24 rd, 2007 Example Problems If it is true that (a 2 - 1) x ≥ 3a - 1 for all x, what value(s) of a are possible. What is an inequality whose solution set will be (-infinity, -2) union [3, infinity). Solve |2x - 1| > 3x.

27. University of Utah March 24 rd, 2007 Example of Math Circle Problems Vlad draws diagonals of unit squares in an 8x8 grid of squares. Nikita sees to it that they have no common points (including ends). What is the maximum number of diagonals that Vlad can draw?

28. University of Utah March 24 rd, 2007 Example of Math Circle Problems Asterisks are arranged in some squares of an n by n grid. In each vertical, horizontal, and diagonal (even diagonals of only one square count), the number of asterisks in it is known. For which n is it possible to determine where the asterisks are?

29. University of Utah March 24 rd, 2007 Can any of this be emulated here?.

30. University of Utah March 24 rd, 2007 The Boston Math Circle

31. University of Utah March 24 rd, 2007 The Boston Math Circle Started in 1994 by math educators Bob and Ellen Kaplan.

32. University of Utah March 24 rd, 2007 The Boston Math Circle Started in 1994 by math educators Bob and Ellen Kaplan.

33. University of Utah March 24 rd, 2007 The Boston Math Circle Started in 1994 by math educators Bob and Ellen Kaplan. No official university affiliation (although resources are provided by Harvard, MIT, Northeastern, Tufts,….)

34. University of Utah March 24 rd, 2007 The Boston Math Circle Started in 1994 by math educators Bob and Ellen Kaplan. No official university affiliation (although resources are provided by Harvard, MIT, Northeastern, Tufts,….) Grown into a rather large enterprise

35. University of Utah March 24 rd, 2007 Typical Spring Schedule Four pre-algebra courses –Divided, roughly, into grades 3-4, 5-6 –10 sessions, one hour each

36. University of Utah March 24 rd, 2007 Typical Spring Schedule Four pre-algebra courses –Divided, roughly, into grades 3-4, 5-6 –10 sessions, one hour each One advanced course –for high school students –10 sessions, three hours each

37. University of Utah March 24 rd, 2007 Approach for younger students A “seed” topic is a springboard for open-ended explorations.

38. University of Utah March 24 rd, 2007 Approach for younger students A “seed” topic is a springboard for open-ended explorations. Sample topics for kids aged 5-10: –Are there numbers between numbers? –Primes –Infinity

39. University of Utah March 24 rd, 2007 Approach for younger students A “seed” topic is a springboard for open-ended explorations. Sample topics for kids between 5 and 10: –Are there numbers between numbers? –Primes –Infinity Ages 12-14 (some algebra): –complex numbers –Fibonacci sequence –continued fractions

40. University of Utah March 24 rd, 2007 Approach for High School Students Almost exclusively devoted to a single topic.

41. University of Utah March 24 rd, 2007 Approach for High School Students Almost exclusively devoted to a single topic. Sample courses: –Algebraic Geometry –Complex Analysis –Knot Theory –Hyperbolic Geometry –Group Theory –Quantum Mechanics

42. University of Utah March 24 rd, 2007 The Boston Math Circle charges tuition Each pre-algebra courses –$225 tuition Each advanced course –$450 tuition About $20/hour per student

43. University of Utah March 24 rd, 2007 To start your own Math Circle… From the Boston Math Circle webpage: “Want to start your own Math Circle? Founders Robert and Ellen Kaplan can be hired as consultants to help get you going. Feel free to contact Bob and Ellen Kaplan directly to learn more about what they can do for you. They are willing to travel to your locale to help out. Their per-diem fee is $1000 plus expenses.”

44. University of Utah March 24 rd, 2007 The Berkeley Math Circle

45. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

46. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others). Closely followed the Russian-type model.

47. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others). Closely followed the Russian-type model. In particular contests were absolutely central.

48. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others). Closely followed the Russian-type model. In particular contests were absolutely central. This led to the creation of the Bay Area Math Olympiad (BAMO).

49. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others). Closely followed the Russian-type model. In particular contests were absolutely central. This led to the creation of the Bay Area Math Olympiad (BAMO). Draw heavily on faculty from Berkeley, Stanford, UCSF, etc.

50. University of Utah March 24 rd, 2007 The Berkeley Math Circle Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others). Closely followed the Russian-type model. In particular contests were absolutely central. This led to the creation of the Bay Area Math Olympiad (BAMO). Draw heavily on faculty from Berkeley, Stanford, UCSF, etc. Free.

51. University of Utah March 24 rd, 2007 BMC: Schedule Excerpt Ravi Vakil, “The Mathematics of Doodling” Sam Vandervelde, “Geometric Mad-Libs” Tom Rike, “Infinite Series” Zvezda Stankova, “AMC 10/12 Preparation” Tom Davis, “BAMO Preparation” Alexander Givental, “Solving Problems with Vectors” Vera Serganova, “Combinatorics and Geometry”

52. University of Utah March 24 rd, 2007 Very Successful! Although the BAMO program only started in 1999, the 6-member team at the International Math Olympiad (held in Washington, D.C., July 2001) included 3 members from this program: –Gabriel Carrol –Tiankai Liu –Oaz Nir

53. University of Utah March 24 rd, 2007 Footnote: San Francisco Math Circle Specifically targets students (and teachers) from urban schools who may not have access to after-school enrichment programs

54. University of Utah March 24 rd, 2007 The Utah Math Circle

55. University of Utah March 24 rd, 2007 The Utah Math Circle Lauched in 2001 as part of Utah’s first VIGRE grant. Hugo Rossi had returned from MSRI and his experience with the Berkeley Math Circle was instrumental.

56. University of Utah March 24 rd, 2007 So We Started

57. University of Utah March 24 rd, 2007 So We Started Sent out application materials to a hundred or so high schools statewide.

58. University of Utah March 24 rd, 2007 So We Started Sent out application materials to a hundred or so high schools statewide. Received a healthy response.

59. University of Utah March 24 rd, 2007 Feeling in the Dark Immediately met some challenges:

60. University of Utah March 24 rd, 2007 Feeling in the Dark Immediately met some challenges: –Had to get acquainted with high school students.

61. University of Utah March 24 rd, 2007 Feeling in the Dark Immediately met some challenges: –Had to get acquainted with high school students. –Our expertise and resources differed from those in Berkeley.

62. University of Utah March 24 rd, 2007 Feeling in the Dark Immediately met some challenges: –Had to get acquainted with high school students. –Our expertise and resources differed from those in Berkeley. –Gradually we polished our own model.

63. University of Utah March 24 rd, 2007 The Utah Math Circle Model

64. University of Utah March 24 rd, 2007 The Utah Math Circle Model Weekly for two hours

65. University of Utah March 24 rd, 2007 The Utah Math Circle Model Weekly for two hours Two consecutive weeks devoted to the same topic

66. University of Utah March 24 rd, 2007 The Utah Math Circle Model Weekly for two hours Two consecutive weeks devoted to the same topic Typically led by faculty or graduate students

67. University of Utah March 24 rd, 2007 The Utah Math Circle Model Weekly for two hours Two consecutive weeks devoted to the same topic Typically led by faculty or graduate students Monthly contests

68. University of Utah March 24 rd, 2007 The Utah Math Circle Model Weekly for two hours Two consecutive weeks devoted to the same topic Typically led by faculty or graduate students Monthly contests

69. University of Utah March 24 rd, 2007 A typical session….

70. University of Utah March 24 rd, 2007 A typical session…. Some lecturing…

71. University of Utah March 24 rd, 2007 A typical session…. Some lecturing… But mostly problem solving

72. University of Utah March 24 rd, 2007 Some Examples The 15 puzzle:

73. University of Utah March 24 rd, 2007 Some Examples Higher dimensional projections:

74. University of Utah March 24 rd, 2007 Some Examples Higher dimensional projections:

75. University of Utah March 24 rd, 2007 Getting Started

76. University of Utah March 24 rd, 2007 Getting Started Targeted Mailing (identification)

77. University of Utah March 24 rd, 2007 Getting Started Targeted Mailing (identification) Settle on a model, collect materials

78. University of Utah March 24 rd, 2007 Getting Started Targeted Mailing (identification) Settle on a model, collect materials Open enrollment

79. University of Utah March 24 rd, 2007 Getting Started Targeted Mailing (identification) Settle on a model, collect materials Open enrollment Seed Money

80. University of Utah March 24 rd, 2007 Getting Started Targeted Mailing (identification) Settle on a model, collect materials Open enrollment Seed Money Release Time

81. University of Utah March 24 rd, 2007 Have Fun!