AP Calculus and the Transition to College Mathematics

AP Calculus and the Transition to College Mathematics David Bressoud Macalester College, St. Paul, MN PowerPoint available at www.macalester.edu/~bressoud/talks T 3 Atlanta, GA March 6, 2010 MAA

The Chronicle of Higher Education January 17, 2010 The Rocky Transi/on From High School Calculus hep://chronicle.com/arhcle/high School Calculus The E/63533/

We need to: Get more information about what happens to students who study calculus in high school. Establish and enforce guidelines for high school programs offering calculus. Re-examine first-year college mathematics.

¾ Algebra II ⅓ Precalculus Advanced Mathema-cs and Science Coursetaking in the Spring High School Senior Classes of 1982, 1992, and 2004. NCES 2007 312

300 250 Fall Enrollments in Calculus I versus AP Calculus Exams (thousands) 2009: 305 550,000 600,000 students studied calculus in high school this year, roughly 1/3 of the 1.8 million who will go directly from HS to college. 200 150 100 1989: 74 1999: 158 Between 150,000 and 4 year 200,000 of these students colleges earn and take advantage of credit for Calculus I. 2 year colleges 50 AP exams (AB & BC) 0 1979: 25 CBMS and College Board data

120 Fall Enrollments, Calculus II (thousands) 100 80 60 40 4 year colleges 2 year colleges 20 0 1980 1985 1990 1995 2000 2005 It appears that a very significant number of students take their AP credit and use it to avoid taking any calculus in college. CBMS data

High school calculus should be pushing more students toward the study of higher mathematics in college. It appears to be having exactly the opposite effect.

Morgan & Klaric, 2007: study of 22 colleges and universihes in fall, 1994; grades weighted so that SAT scores are comparable Placed via average grade in Calculus II SAT Adjusted grade Passed Calculus I 2.43 3 on AB exam 2.69 2.64 4 on AB exam 2.90 2.78 5 on AB exam 3.34 3.15 Placed via average grade in Calculus II SAT Adjusted grade Passed Calculus I 2.50 3 on BC exam 3.00 2.92 4 on BC exam 3.45 3.35 5 on BC exam 3.46 3.27 Barnard College, Binghamton U., Brigham Young U., Carnegie Mellon U., College of William & Mary, Cornell U., Dartmouth, George Washington U., Georgia InsHtute of Technology, Miami U. (Ohio), North Carolina State U., Texas A&M, U. of California at Davis, U. of Illinois at Urbana/Champaign, U. of Iowa, U. of Maryland, U. of Miami, U. of Texas at AusHn, U. of Virginia, U. of Washington, Wesleyan College, Williams College

Keng & Dodd, 2008: study at UT AusHn, 1998 2001; students selected by strahfied random sample so that SAT distribuhon matched that of AP students. Prepara/on for Calculus II Average grade a) 3 or higher on BC exam 3.43 b) Passed Calculus I, SAT matched to BC students 3.16 c) 3 or higher on AB exam 3.13 d) Passed Calculus I, SAT matched to AB students 3.03 e) 3 or higher on AB exam and took Calculus I 2.96 f) Dual enrollment credit 2.93 g) BC course, no credit, took Calculus I 2.82 h) AB course, no credit, took Calculus I 2.45 StaHsHcally significant at 0.05 all four years: a) over b) two of four year: c) over d) one of four year: c) over e)

Two Current Studies: 1. Phil Sadler, Harvard, Factors Influencing College Success in Mathema-cs HS factors that influence success in Calculus I 2. MAA (Bressoud, Carlson, Pearson, Rasmussen), Characteris-cs of Successful Programs in College Calculus College factors that influence success in Calculus I and case study analysis of successful programs.

Of the high school students who graduated in 1992 and studied calculus while in high school, 31% took precalculus in college, and a further 32% took no calculus in college. The anecdotal evidence is strong that today s students who do not qualify for college credit (about 350,000) struggle to arhculate high school and college mathemahcs. From the transcript analysis of the National Education Longitudinal Study begun in 1988 and College Board and CBMS data.

W. L. Duren, Jr., University of Virginia, chair E. G. Begle, Stanford University A. A. Blank, New York University Ralph P. Boas, Northwestern University Leslie A. Dwight, Southeastern State College Marion K. Fort, University of Georgia Samuel Goldberg, Oberlin College Edwin E. Moise, Harvard University Henry O. Pollak, Bell Telephone Laboratories G. Baley Price, University of Kansas A. W. Tucker, Princeton University R. J. Walker, Cornell University Gail S. Young, Jr., Tulane University

AP Calculus AB AP Calculus BC MathemaHcs 1: Introductory Calculus 1. (7 lessons) IntroducHon: extreme value problems, slopes, velocity and rate of change, limits and approximahons 2. (14 lessons) Techniques and applicahons of differenhahon 3. (18 lessons) Techniques and applicahons of integrahon MathemaHcs 2 (ophon 1): 1. (9 lessons) Advanced understanding of differenhal and integral calculus 2. (15 lessons) Limits: sequences and series, Newton s method 3. (6 lessons) Parametric representahon of curves, arc length 4. (9 lessons) First order differenhal equahons MathemaHcs 2 (ophon 2): 1. (6 lessons) Vectors in 3 spce 2. (12 lessons) DifferenHal calculus of funchons of several variables 3. (9 lessons) Advanced understanding of integral calculus 4. (12 lessons) First order differenhal equahons

Trying to restrict the growth of AP Calculus is a non starter: Success in AP Calculus is the single most useful predictor of successful complehon of college. Students without access to a good program in AP Calculus are at a compehhve disadvantage. ExxonMobil and the Gates FoundaHon are strongly behind the spread of quality programs in AP Calculus.

Several states, including Hawaii, Mississippi, South Carolina, and Tennessee, have state high school standards that describe a watered down version of calculus. California s standards prescribe a course that is equivalent to a full year of college calculus. No state standards talk about the goals of the course or prescribe pre requisite knowledge.

Number 1 complaint from high school teachers: The focus of college placement tests is on what my students don t know, rather than on what they know. Colleges need to pay more attention to building on what students understand and can do.

2001 06 study at Arizona State University Of students who took pre-calculus and Their declared major required at least one semester of calculus, and They earned an A in pre-calculus, 43% chose not to enroll in calculus.

During the period fall 2001 through fall 2006, 43% of engineering majors, 54% of mathematics majors, 51% of physical science majors, and 50% of technology majors who enrolled in Calculus I at ASU and whose intended majors required Calculus II never earned credit for Calculus II.

A focus on teaching must avoid the temptahon to consider only the superficial aspects of teaching: the organizahon, tools, curriculum, content, and textbooks. The cultural achvity of teaching the ways in which the teacher and students interact about the subject can be more powerful than the curriculum materials that teachers use. Stigler and Hiebert (2004), Improving mathematics teaching, Educational Leadership

US Military Academy, West Point: MA 103: MathemaHcal Modeling and IntroducHon to Calculus, the first of our core courses for all cadets. This is the first of four courses in the USMA mathemahcs core curriculum. The focus of the course is to use effechve problem solving and modeling techniques to find soluhons to complex and oten illdefined problems. The course lays the foundahon for calculus and differenhal equahons through difference equahons. MA 104: Calculus I. This course builds upon the foundahon laid in MA103, as the cadet learns about differenhal calculus in single and mulh variable problems. This program has now been in place for twenty years.

Graphing Calculators 90% 80% 70% per cent of sections 60% 50% 40% 30% 20% 10% 0% 1990 1995 2000 2005 year Research Univ Comprehensive Univ Undergrad college 2-year CBMS data

Computer Assignments 45% 40% 35% per cent of sections 30% 25% 20% 15% 10% 5% 0% 1990 1995 2000 2005 year Research Univ Comprehensive Univ Undergrad college 2-year CBMS data

Writing Assignments 50% 45% 40% 35% per cent of sections 30% 25% 20% 15% 10% 5% 0% 1990 1995 2000 2005 year Research Univ Comprehensive Univ Undergrad college 2-year CBMS data

Group Projects 35% 30% 25% per cent of sections 20% 15% 10% 5% 0% 1990 1995 2000 2005 year Research Univ Comprehensive Univ Undergrad college 2-year CBMS data

12% Frac/on of incoming freshmen intending to major in... 10% 8% 6% 4% 2% 0% Engineering Bio Science Physical Science Computer Science MathemaHcs Source: The American Freshman 12% Frac/on of gradua/ng seniors who majored in... 10% 8% 6% 4% 2% 0% Engineering Bio Science Physical Science Computer Science MathemaHcs Source: NaHonal Center for EducaHon StaHsHcs

CRAFTY Curriculum Founda-ons Project: Voices of the Partner Disciplines Biology: Statistics, modeling and graphical representation should take priority over calculus.

American AssociaHon of Medical Colleges and the Howard Hughes Medical InsHtute, Scien-fic Founda-ons for Future Physicians September, 2009 Competency E1 Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world.

Learning ObjecHves: 1. Demonstrate quanhtahve numeracy and facility with the language of mathemahcs. 2. Interpret data sets and communicate those interpretahons using visual and other appropriate tools 3. Make stahshcal inferences from data sets. 4. Extract relevant informahon from large data sets. 5. Make inferences about natural phenomena using mathemahcal models. 6. Apply algorithmic approaches and principles of logic (including the dishnchon between cause/effect and associahon) to problem solving. 7. QuanHfy and interpret changes in dynamical systems.

The mathemahcs profession as a whole has seriously undereshmated the difficulty of teaching mathemahcs. Ramesh Gangolli MER Workshop May 31, 1991 PowerPoint available at www.macalester.edu/~bressoud/talks