Challenges for the Undergraduate Curriculum

1. Challenges for the Undergraduate Curriculum David Bressoud, Macalester College Visualizing the Future of Mathematics Education, USC, April 13, 2007 This PowerPoint will be available at www.macalester.edu/~bressoud/talks

3. 80% precalculus and precollege

5. 53% introductory and precollege

6. 53% introductory and precollege, 72% if you count Calculus I as high school math

7. Over the past quarter century, total enrollment has increased 42%.

9. College faculty cannot afford to ignore what is happening in K-12 education. • Initiatives to clarify expectations: • NCTM Focal Points • College Board Standards for College Success: Mathematics and Statistics • Achieve, Inc. (National Governor’s Association), Secondary Mathematics Expectations

10. Calculus Reform Reaches Maturity Challenges of the Movement of Calculus into High School Conclusion

11. Calculus Reform Reaches Maturity

12. January, 1986, Tulane — What has happened since?

13. January, 1986, Tulane — What has happened since? • Major NSF calculus initiative

14. January, 1986, Tulane — What has happened since? • Major NSF calculus initiative • Noticeable change in the texts: • Symbolic, graphical, numerical, written representations • Incorporation of calculator and computer technology • More varied problems, opportunities for exploration in depth

15. January, 1986, Tulane — What has happened since? • Major NSF calculus initiative • Noticeable change in the texts: • Symbolic, graphical, numerical, written representations • Incorporation of calculator and computer technology • More varied problems, opportunities for exploration in depth • No noticeable shift in the syllabus

16. January, 1986, Tulane — What has happened since? • Major NSF calculus initiative • Noticeable change in the texts: • Symbolic, graphical, numerical, written representations • Incorporation of calculator and computer technology • More varied problems, opportunities for exploration in depth • No noticeable shift in the syllabus • Advanced Placement Calculus embraced calculus reform in mid-1990s (Kenelly, Kennedy, Solow, Tucker)

17. 2003 AP Calculus exam AB4/BC4

18. 2003 AP Calculus exam AB4/BC4 What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

19. At 5-year intervals starting in 1990, CBMS has been tracking number of sections of mainstream Calculus I that use various markers of reform calculus: • Use of graphing calculators • Use of computer assignments • Use of writing assignments • Use of group projects • Results from 2005 are just in.

24. Use of online resources in mainstream Calculus I (2005): PhD: 9% MA: 2% BA: 2% 2-year: 5%

25. Challenges of the Movement of Calculus into High School

27. AP Calculus currently growing at >14,000/year (about 6%)

28. AP Calculus currently growing at >14,000/year (about 6%) Estimated # of students taking Calculus in high school (NAEP, 2005): ~ 500,000 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

29. ~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit) • ~300,000 retake calculus taken in HS • Some start by retaking the calculus they studied in high school • Some are required to take precalculus first • ~200,000 will take calculus for first time

30. ~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit) • ~300,000 retake calculus taken in HS • Some start by retaking the calculus they studied in high school • Some are required to take precalculus first • ~200,000 will take calculus for first time 1 2 3 4

31. 4 ~200,000 will take calculus for first time • Increasingly, these are students who neither need nor want more than a basic introduction to calculus (i.e. Biology majors). Challenge is to give them a one-semester course that • Acknowledges that they may not be our strongest students, but • Builds their mathematical skills, • Gives them an understanding of calculus, and • Does not cut them off from continuing the study of calculus

32. ~300,000 retake calculus taken in HS • Some are required to take precalculus first 3 We need a better solution for these students. Again, the challenge is to give them a course that enables them to overcome their deficiencies while challenging and engaging them. 4

33. ~300,000 retake calculus taken in HS • Some start by retaking the calculus they studied in high school 2 We need a better solution than having these students retread familiar territory, but at a much faster pace, in larger classes, and with an instructor who is unable to give them the individual attentionthat they experienced when they struggled with these ideas the previous year. 3 4

34. One approach (Macalester): • Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors: • Understanding of rates of change, meaning of derivative

35. One approach (Macalester): • Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors: • Understanding of rates of change, meaning of derivative • Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers

36. One approach (Macalester): • Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors: • Understanding of rates of change, meaning of derivative • Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers • Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns

37. One approach (Macalester): • Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors: • Understanding of rates of change, meaning of derivative • Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers • Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns • Integration as limit of sum of product and as anti-derivative

38. One approach (Macalester): • Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors: • Understanding of rates of change, meaning of derivative • Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers • Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns • Integration as limit of sum of product and as anti-derivative • Enough linear algebra to understand geometric interpretation of linear regression.

39. ~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit) • ~300,000 retake calculus taken in HS • Some start by retaking the calculus they studied in high school • Some are required to take precalculus first • ~200,000 will take calculus for first time 1 • These are our success stories but: • We need to worry about articulation with their high school experience. • We need to work at both challenging and enticing these students. 2 3 4

40. Dual enrollment In spring, fall 2005 (combined), 33,436 students studied Calculus I under dual enrollment programs: 14,030 in connection with 4-year colleges, 19,406 in connection with 2-year colleges.

43. BC exam, 8818 in 2002 13,809 in 2006 57% increase

44. Need for curricula that engage and entice E.g., Approximately Calculus, Shariar Shariari, AMS, 2006

45. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ? What do we mean by “point of inflection”?

46. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ? What do we mean by “point of inflection”? What do we mean by “concavity”?

47. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ? What do we mean by “point of inflection”? What do we mean by “concavity”? Graph of f is concave up on [a,b] if every secant line lies above (> or ≥ ?) the graph of f.

48. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ? What do we mean by “point of inflection”? What do we mean by “concavity”? Graph of f is concave up on [a,b] if every secant line lies above (> or ≥ ?) the graph of f. Graph of f ' is increasing over some interval with right-hand endpoint at 2, decreasing over interval with left-hand endpoint at 2.

49. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ? Is it possible to have g(x) < g(2) for all x < 2, but on every interval with right-hand endpoint at 2, there is a subinterval over which g is strictly decreasing?

50. What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?