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Challenges for the Undergraduate Curriculum

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. 80% precalculus and precollege. 53% introductory and precollege.

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Challenges for the Undergraduate Curriculum

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  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

  2. 80% precalculus and precollege

  3. 53% introductory and precollege

  4. 53% introductory and precollege, 72% if you count Calculus I as high school math

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

  6. 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

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

  8. Calculus Reform Reaches Maturity

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

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

  11. 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

  12. 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

  13. 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)

  14. 2003 AP Calculus exam AB4/BC4

  15. 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 ?

  16. 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.

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

  18. Challenges of the Movement of Calculus into High School

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

  20. 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)

  21. ~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

  22. ~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

  23. 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

  24. ~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

  25. ~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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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.

  31. ~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

  32. 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.

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

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

  35. 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”?

  36. 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”?

  37. 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.

  38. 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.

  39. 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?

  40. 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 ?

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