The Changing Face of Calculus

1. The Changing Face of Calculus David M. Bressoud Macalester College, St. Paul, MN Bellingham, WA, May 19, 2006 This PowerPoint will be available at www.macalester.edu/~bressoud/talks

2. Where we are How we got here A closer examination of where we are Where we are going

4. AP Calculus 2006: >250,000 Currently growing at >14,000/year

5. AP Calculus 2006: >250,000 Currently growing at >14,000/year Estimated # of students taking Calculus in high school: ~ 500,000 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

6. Bachelors degrees each year* 400,000 SMET + social & behavioral sciences of which 210,000 Science, Math, Engineering of which 100,000 physical, biological, & ag sciences 60,000 engineering 50,000 math, stat, comp sci of which 11,000 mathematics *NSF: among 24-year olds in 2000

7. ~170,000 arrive with credit for calculus ~330,000 retake calculus taken in HS ~170,000 will take calculus for first time Bachelors degrees each year* 400,000 SMET + social & behavioral sciences of which 210,000 Science, Math, Engineering of which 100,000 physical, biological, & ag sciences 60,000 engineering 50,000 math, stat, comp sci of which 11,000 mathematics *NSF: among 24-year olds in 2000

8. BC exams for 2006: ~58,000 Still growing exponentially at over 9%/year (8 year doubling time)

9. BC exams for 2006: ~58,000 Still growing exponentially at over 9%/year (8 year doubling time) Last year, 12,500 students took the BC exam before their senior year

10. Implications: Students who 20 years ago would have arrived at college ready to take calculus now take it in high school. Students who take Calculus I in college either are retaking a course taken in high school or have had to overcome mathematical deficiencies. Calculus I is increasingly taken as a terminal course. Especially at elite institutions but increasingly elsewhere, the traditional Calculus II which presupposes Calculus I at that institution does not serve the needs of the students who take it.

14. 1983–84 scientific calculators allowed

15. 1983–84 scientific calculators allowed 1986 Tulane Conference, birth of Calculus Reform movement, “lean & lively calculus”

16. 1983–84 scientific calculators allowed 1986 Tulane Conference, birth of Calculus Reform movement, “lean & lively calculus” 1989 decision to revisit entire AP Calculus curriculum and approach, bring in graphing calculators; revisions led by Tom Tucker (Colgate), John Kenelly (Clemson), Anita Solow (Grinnell), Dan Kennedy (Baylor School)

17. 1983–84 scientific calculators allowed 1986 Tulane Conference, birth of Calculus Reform movement, “lean & lively calculus” 1989 decision to revisit entire AP Calculus curriculum and approach, bring in graphing calculators; revisions led by Tom Tucker (Colgate), John Kenelly (Clemson), Anita Solow (Grinnell), Dan Kennedy (Baylor School) 1993–94 scientific calculators required 1995 graphing calculators required, proposed changes to AP syllabus agreed upon

18. 1995, Graphing calculators

19. 1997, New syllabus

20. 2000, 3 non-calculator free response

21. 2001, AB subscore for BC exam

22. AB subscore New syllabus

23. 1997–98 exams based on new syllabus • Graphical, numerical, analytical, and verbal descriptions of functions

24. 1997–98 exams based on new syllabus • Graphical, numerical, analytical, and verbal descriptions of functions

25. 1997–98 exams based on new syllabus • Graphical, numerical, analytical, and verbal descriptions of functions

26. 1997–98 exams based on new syllabus • Integral as limit of Riemann sums and as net accumulation of rate of change

27. 1997–98 exams based on new syllabus • Understand both parts of FTC Evaluation: If you know an anti-derivative for f, you can use it evaluate the definite integral, Anti-derivative: The definite integral with variable upper limit is an anti-derivative,

28. 1997–98 exams based on new syllabus • Understand both parts of FTC

29. 1997–98 exams based on new syllabus • Understand both parts of FTC 2004 AB3(d) A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle is at y = –1. Find the position of the particle at time t = 2. y '(t) = v(t) = 1 – tan–1(et) y(t) = ?

30. 1997–98 exams based on new syllabus • Understand both parts of FTC 2004 AB3(d) A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle is at y = –1. Find the position of the particle at time t = 2.

31. 1997–98 exams based on new syllabus • Understand both parts of FTC 2004 AB3(d) A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle is at y = –1. Find the position of the particle at time t = 2.

32. 1997–98 exams based on new syllabus • Be able to communicate mathematics: justify local or absolute extremum, explain the meaning of an answer 2005 AB5/BC5

33. Into the Future • Use of CAS is coming - currently about 35% of BC students, 20% of AB have TI-89 or comparable, probably 5–10 years away. • Exams administered via computer, probably 10–15 years away.

34. Into the Future • Pressure to get college-bound students into an AP Calculus class is going to intensify.

35. Into the Future • Pressure to get college-bound students into an AP Calculus class is going to intensify. • The growth in AP Calculus is not about to end. President’s American Competitiveness Initiative, training 70,000 new AP math and science teachers, Dept of Ed requesting $122,000,000 for FY 2007 to support AP programs.

36. Into the Future • Pressure to get college-bound students into an AP Calculus class is going to intensify. • The growth in AP Calculus is not about to end. • % increase of BC Calculus will continue to exceed that of AB

37. Into the Future • Pressure to get college-bound students into an AP Calculus class is going to intensify. • The growth in AP Calculus is not about to end. • % increase of BC Calculus will continue to exceed that of AB • % increase in # of students taking BC Calculus before senior year will continue to exceed that of BC generally

38. Into the Future • Pressure to get college-bound students into an AP Calculus class is going to intensify. • The growth in AP Calculus is not about to end. • % increase of BC Calculus will continue to exceed that of AB • % increase in # of students taking BC Calculus before senior year will continue to exceed that of BC generally • More universities will see calculus as a high school course.

39. Needed Response • NCTM, MAA, AMS need to coordinate a strong signal that calculus in HS is only appropriate when students have a solid foundation in pre-calculus, need to articulate what this foundation must be.

40. Needed Response • NCTM, MAA, AMS need to coordinate a strong signal that calculus in HS is only appropriate when students have a solid foundation in pre-calculus, need to articulate what this foundation must be. • Need much greater collaboration between high school and college teachers.

41. Needed Response • NCTM, MAA, AMS need to coordinate a strong signal that calculus in HS is only appropriate when students have a solid foundation in pre-calculus, need to articulate what this foundation must be. • Need much greater collaboration between high school and college teachers. • Need to seriously address the question of what to do with students who take (and pass) BC Calculus before their senior year.

42. APCentral at apcentral.collegeboard.com SIGMAA TAHSM (Special Interest Group of the MAA, Teaching Advanced High School Mathematics) at www.maa.org/SIGMAA/tahsm/ This PowerPoint presentation at www.macalester.edu/~bressoud/talks