Taking calculus in high school gives a student one letter grade higher a score in Calculus II than taking it in college (3.43 vs. 2.45)

## AP® Calculus: What We Know

David M. Bressoud June, 2009

 Sam King of Loughborough University is conducting a study of the use of clickers. To participate in his survey, go to www.survey.lboro.ac.uk/clickers/

Close to one third of the 1.8 million students who this year will go directly from high school to either a 2- or 4-year college have taken calculus in high school. These constitute an overwhelming majority of the students we are likely to see in our calculus and advanced mathematics courses. Listed below are some basic questions and what I know about answers. I would greatly appreciate pointers to any other information that may be out there.

1.

2.    What happens to these students when they get to college?

3.    Does it make sense for students who have done well in AP Calculus to skip Calculus I in college?

How many students study calculus in high school and what kind of program do they take?

The most recent reliable data is from the high school graduating class of 2004. According to a large-scale transcript analysis by the US Department of Education [1], 14.1% of graduating seniors had taken a class that was called "calculus." That amounted to about 430,000 students. That spring, 225,000 or 52% of them took an AP Calculus exam [2]. By spring, 2009, the number of AP Calculus exams was just over 300,000. If the percentage has stayed constant, then about 575,000 of this year's graduating seniors have studied calculus while in high school. This does not count students who may have seen some calculus in a course that does not have calculus in its title.The College Board estimates that 70–75% of the students in AP Calculus take the exam, which suggests that this year 400–430,000 students took a course called AP Calculus.

AP Calculus comes in two varieties: AB Calculus, intended to cover one semester of college calculus, and BC Calculus, which covers a full year of college-level calculus. In 2008, 24% of the student who took an AP Calculus exam took the BC exam, just under 70,000 students. Participation rates in the exam tend to be much higher for BC Calculus students so that the number of students enrolled in BC Calculus is almost certainly less than 100,000, but close to that number.

In Spring, 2008, 192,000 students earned a score of 3 or higher on an AP Calculus exam, 61% of the AB Calculus students and 80% of the BC Calculus students. The College Board considers that these students have successfully completed work at the college level.

The number of students studying calculus in the US under the International Baccalaureate Program [3] is quite small. In 2008, only 8400 students throughout the world took the Higher Level Mathematics exam which includes a full year of college-level calculus. An additional 21,700 took the Standard Level Mathematics exam which has some but very little calculus. About 40% of IB schools are in the United States. Thus, the contribution to calculus in high school from IB programs is negligible.

What is very unclear is the effect of dual enrollment programs, programs where students simultanesouly earn both high school and college credit through an agreement between a specific college and a participating high school or school district. According to CBMS data for Fall, 2005 [4], this number was still fairly small, on the order of 30–35,000 a year. There is a great deal of anecdotal evidence that dual enrollment programs have spread widely since 2005. Thus, the number of students studying calculus in high school may be significantly higher than the estimated 575,000.

In summary, my best guess is that about 575,000 high school students took a calculus course offered in their high school this past year. About 40% believe that they have completed at least a semester's worth of college-level work, and for most of them this was through the AP program. It should be emphasized that these numbers are not static. In 1999, 158,000 students took the AP Calculus exam. In 1989, it was 74,000. In 1979, it was 25,000. The exponential growth rate has slowed, but it is still running at over 6.5% per year.

What happens to these students when they get to college?

Unfortunately, our ignorance of the answer to this question is vast. The most recent reliable data concerning all students who have studied calculus in high school come from the US Department of Education for the high school class of 1992 [5], back when AP Calculus enrollments were well under a third of what they are today. It showed that 31% of the students who had studied calculus in high school enrolled in precalculus when they got to college. This study said nothing about whether these students continued on to take calculus. A further 32% took no calculus in college. The remaining 37% took at least one course in college at the level of calculus or above. This study said nothing about success rates or the number of courses at the level of calculus and beyond. For the 350,000 or so graduating seniors this year who studied calculus but have not been certified as knowing calculus at the college level, we have no idea what effect their experience of calculus will have on their decisions whether or not to continue with mathematics, or, if they do continue, how this experience of calculus in high school will affect their performance in college mathematics.

There is a study from 2002 by Karen Christman Morgan [6] that investigated what happened to students who received a score of 3 or higher on an AP Calculus exam. The number of students in the study was fairly small: 435 for AB Calculus and 135 for BC Calculus, but the students were chosen from a randomized national sample. For AB Calculus, 74% received college credit. The distribution of credit broken down by exam score was as follows: 84% of those who received a grade of 5, 82% of those who received a grade of 4, and 60% of those who received a grade of 3 also received college credit. If credit was not received, about half of the students said this was because the college was not prepared to give credit (as is often the case for a 3 on the AB Calculus exam), the other half were entitled to credit but chose to enroll in Calculus I in college. For BC Calculus, 79% received credit for at least one semester's worth of college calculus. In this case, the numbers were too small to get meaningful estimates of the rate at which credit was awarded at each score level.

The Christman Morgan study also looked at the number of students who used or intended to use their credit in calculus to take advantage of advanced placement, that is, to go into the next mathematics class in the sequence. For a 5 on the AB exam, 92% took advantage of advanced placement; for a 4 it was 78%, and for a 3 it was 65%. On the BC exam, 92% of those who received college credit took at least one additional calculus class. Combining these numbers, weighted by the percentage of students at each of the scores, about 80% of the students who received credit for calculus also took the next course in the sequence.

These results are inconsistent with a study conducted by David Lutzer at William and Mary in the early 1990s [7]. There he found that among the students who received credit for AP Calculus (at least 4 on the AB exam or at least 3 on the BC exam), 60% took the next math course (Calculus II, Linear Algebra, or Calculus III). This is almost exactly the same as the 61% who completed Calculus I at William and Mary and continued on to the next course, but when Lutzer constructed a multiple regression model that controlled for SAT scores, he found that the difference was significant at the 95% level. Students who took Calculus I at William and Mary were more likely to take the next math class than those who arrived with AP credit for Calculus.

The two studies are very different, and it is not clear whether either of them is relevant to the situation today. The only other solid piece of evidence that we have is that—despite the dramatic increase in the number of students who receive credit for calculus studied in high school—the number of students taking Calculus II in the Fall term has remained essentially unchanged over the past two decades: 110,000 in 1990, 106,000 in 1995, 108,000 in 2000, and 104,000 in 2005 [8].

Does it make sense for students who have done well in AP Calculus to skip Calculus I in college?

This is the question for which we have the most evidence, yet even here the evidence is imperfect, most of it has been funded through the College Board or the Educational Testing Service, and most of it is at least ten years old. I will survey the four studies with which I am familiar. I also include a more recent but very small scale look at dual enrollment.

Morgan & Ramist, 1998 [9] This was a large-scale study conducted in Fall 1991 at 21 colleges and universities chosen from among those that receive the greatest number of AP Calculus scores [10]. It looked at students who received at least a 3 on an AP Calculus exam and chose to use this credit to skip at least one calculus class. It shows that even for students who scored a 3 on the AB Calculus exam, they did better in Calculus II then the average student who has passed Calculus I taken at that university. The study suffers from several flaws: All that is reported are averages taken across all of the universities; there is no attempt to compare students with a given AP score with students who received a particular grade in Calculus I; and there is no attempt to control for the possibility that the population of students who earn AP credit for and are sufficiently confident to skip Calculus I are not completely comparable to the population of those who take and pass Calculus I.

Nevertheless, this study does suggest that even at the most demanding universities, the student who chooses to take advantage of advanced placement is not putting him- or herself at a disadvantage.

 Placed via average grade in Calculus II Passed Calculus I 2.52 3 on AB exam 2.67 4 on AB exam 2.79 5 on AB exam 3.23

 Placed via average grade in Calculus II Passed Calculus I 2.51* 3 on BC exam 2.88 4 on BC exam 3.24 5 on BC exam 3.66

 Placed via average grade in Calculus III Passed Calculus II 2.74 3 on BC exam 2.93 4 on BC exam 2.88** 5 on BC exam 3.38

* The average grade in Calculus II for those who passed Calculus I is slightly different because not all universities could be used for the second table. ** The fact that a 4 on the BC exam predicts a lower score on Calculus III than a 3 is almost certainly a result of the fact that many universities do not allow a student with a 3 on the BC exam to place directly into Calculus III, and among those that do, many students with a 3 on the BC exam—especially those who are not confident of their ability—will choose not to place directly into Calculus III.

Morgan & Klaric, 2007 [11] This was a large-scale study conducted in Fall 1994 at 22 colleges and universities chosen from among those that receive the greatest number of AP Calculus scores [12]. The significant advantage over the previous study was that it adjusted the grades of those who took advantage of advanced placement, weighting their scores so that the distribution of SAT scores was comparable to that of students who had taken the previous course at that institution.

 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 III SAT Adjusted grade Passed Calculus II 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

Dodd et al, 2002 [13] This study was conducted at the University of Texas, Austin, over a five-year period: 1996–99. It looked at all of the students who used AP credit from the AB Calculus exam to place into Calculus II (M408D) and compared these to a stratified random sample of students in Calculus II who had passed Calculus I (M408C), stratifying the sample so that the SAT scores of the two groups were comparable. The average Calculus II grade of the AP students was 2.98. The average grade for students from the sample was 2.55.

Keng & Dodd, 2008[14] This study at the University of Texas, Austin, 1998–2001, compared students who had used AP credit to place into Calculus II (M408D) with four other groups: those who took an AP Calculus course but did not score a 3 or higher on the exam, those who did score a 3 or higher but chose to retake Calculus I, those who earned credit for Calculus I via dual enrollment, and those who had passed Calculus I (M408C) at UT-Austin. As with the previous study by Dodd, students in the last group were chosen via stratified random sample so that their SAT distribution matched that of the students who had used AP credit to place into Calculus II. Because this mainstream calculus sequence proceeds at a brisk pace, spending the second semester on sequences, series, and topics in multivariable calculus, students who brought credit from dual enrollment programs were only counted if they had passed courses covering both differential and integral single variable calculus.

 Preparation for Calculus II (M408D) average grade a) 3 or higher on BC exam 3.43 b) took Calculus I, SAT distribution matches 3+ on BC 3.16 c) 3 or higher on AB exam 3.13 d) took Calculus I, SAT distribution matches 3+ on AB 3.03 e) 3+ on AB exam and took Calculus I 2.96 f) dual enrollment credit 2.93 g) BC course but no credit for exam, took Calc I 2.82 h) AB course but no credit for exam, took Calc I 2.45

The following differences were significant at the 95% confidence level:

a) over b), all four years; c) over d), two of four years; c) over e), one of four years; a) over f), one of four years; a) over g), all four years; b) over h), all four years. Differences were always significant when comparing those who did with those who did not pass the AP exam.

The lack of significance comparing a) and f) is a result of very few students in category f). It would have more useful to compare c) and e) if the distribution of SAT scores were comparable, but the numbers were too small to allow for that.

Dual Enrollment Beyond the Keng & Dodd study, there is not much information on the preparation of students who arrive with credit for dual enrollment. However, Theresa Laurent at the St. Louis College of Pharmacy [15] did give a modified version of the Calculus Validation Exam developed at the US Military Academy [16] to the 143 incoming students who claimed to have had some experience with calculus while in high school. On a 16-point exam, students who had earned at least a 4 on the AB exam (22 students) averaged 12.14 . Those who arrived without any college credit for calculus (but who had taken some calculus, 93 students) averaged 4.17. Those who arrived with credit via dual enrollment (28 students) averaged 4.61. The performance of students with dual enrollment credit was not significantly different from that of students with no credit, even when controlling for ACT scores.

Conclusions

The most glaring observations from this survey are how little we know about the effects of our current calculus instruction in high school and how outdated what we do know is. Our most recent large-scale studies are from the Fall of 2001, back when the AP program was 60% of its current size. However, there are a few things that can be said:

1.    There is no evidence that taking calculus in high school is of any benefit unless a student learns it well enough to earn college credit for it, and there is some evidence—the high percentage of students who go from calculus in high school to precalculus in college—that an introduction to calculus that builds on an inadequate foundation can be detrimental.

2.    The AP Calculus program is doing what it was established to do: It identifies those students who have learned calculus well enough that they are ready to place into the next course. However, AP Calculus scores are not perfect predictors. In particular, there is some uncertainty about whether or not a 3 should be sufficient for advanced placement credit. While the evidence suggests that there is little or no benefit in retaking a calculus course for which the student is entitled to AP credit, there is some indication—the Morgan & Ramist study comparing performance of students with scores of 3 or 4 on the BC exam—that some students are better served by being allowed not to place as far ahead as they are entitled.

Bibliography and Endnotes

[1] US Department of Education. 2009. Education Longitudinal Study of 2002 (ELS:2002). nces.ed.gov/surveys/ELS2002/

[2] AP data can be found at professionals.collegeboard.com/data-reports-research/ap

[3] International Baccaluareate. 2008. The IB Diploma Program statistical bulletin. www.ibo.org/facts/statbulletin/dpstats/index.cfm

[4] Lutzer, David J., Stephen B. Rodi, Ellen E. Kirkman, and James W. Maxwell, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 2005 CBMS Survey, American Mathematical Society, www.ams.org/cbms/cbms2005.html

[5] US Department of Education. 2008. National Education Longitudinal Study of 1988 (NELS:88). nces.ed.gov/surveys/NELS88/

[6] Christman Morgan, K. 2002. The Use of AP Examination Grades by Students in College, preprint presented at AP National Conference, Chicago, 2002.

[7] Lutzer, D. 2007. private communication

[8] CBMS data combines 2- and 4-year college numbers, but the numbers are also essentially constant when considering just 2-year or just 4-year undergraduate programs. It is taken from

• Albers, Donald J., Don O. Loftsgaarden, Donald C. Rung, Ann E. Watkins, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences and Computer Science in the United States, 1990–91 CBMS Survey, MAA Notes Number 23, www.ams.org/cbms/cbms1990.html
• Loftsgaarden, Don O., Donald C. Rung, Ann E. Watkins, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 1995 CBMS Survey, MAA Reports Number 2, www.ams.org/cbms/cbms1995.html
• Lutzer, David J., James W. Maxwell, and Stephen B. Rodi, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 2000 CBMS Survey, American Mathematical Society, www.ams.org/cbms/cbms2000.html
• Lutzer, David J., Stephen B. Rodi, Ellen E. Kirkman, and James W. Maxwell, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 2005 CBMS Survey, American Mathematical Society, www.ams.org/cbms/cbms2005.html

[9] Morgan, R. and L. Ramist. 1998. Advanced Placement Students in College: An Investigation of Course Grades at 21 Colleges. Educational Testing Survey Report No. SR-98-13. Princeton, NJ. www.collegeboard.com/press/releases/50405.html

[10] The study was conducted at Boston College, Brigham Young University, Carnegie Mellon University, Clemson University, College of William and Mary, Cornell College (IA), Cornell University, Duke University, Michigan State University, Pennsylvania State University, Stanford University, Tulane University, UC-Davis, UC-Irvine, University of Georgia, University of Illinois, UNC-Chapel Hill, UT-Austin, University of Utah, University of Virginia, and Yale University.

[11] Morgan, R. and J. Klaric. 2007. AP® Students in College: An Analysis of Five-Year Acadmeci Careers. College Board Research Report No. 2007-4. New York. professionals.collegeboard.com/data-reports-research/cb/title

[12] The study was conducted at Barnard College, Binghamton U., Brigham Young U., Carnegie Mellon U., College of William & Mary, Cornell U., Dartmouth, George Washington U., Georgia Institute 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 Austin, U. of Virginia, U. of Washington, Wesleyan College, Williams College

[13] Dodd et al. 2002. An Investigation of the Validity of AP® Grades of 3 and a Comparison of AP and Non-AP Student Groups.College Board Research Report No. 2002-9. professionals.collegeboard.com/data-reports-research/cb/title

[14] Keng, L.and B. G. Dodd. 2008. An Investigation of College Performance of AP and Non-AP Student Groups professionals.collegeboard.com/data-reports-research/cb/title

[15] private communication

[16] see Retchless, T., R. Boucher, and D. Outing. 2008. Calculus Placement that Really Works!. MAA Focus. vol. 28, pp. 20–21. www.maa.org/pubs/jan08web.pdf.

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and President of the MAA. You can reach him at [email protected] This column does not reflect an official position of the MAA.

High School Calculus in the United States and in Japan

by Thomas W. Judson

In Japan, as in the United States, calculus is a gateway course that students must pass to study science or engineering. Japanese educators often voice complaints similar to those that we made about students' learning of calculus in the 1970s and 1980s. They believe that many students learn methods and templates for working entrance-examination problems without learning the concepts of calculus. University professors report that the mathematical preparation of students is declining and that even though Japanese middle school students excelled in mathematics in TIMSS-R, these same students expressed a strong dislike for the subject.

Japan has a national curriculum that is tightly controlled by the Ministry of Education and Science. In Japan, grades K–12 are divided into elementary school, middle school, and high school; students must pass rigorous entrance examinations to enter good high schools and universities. After entering high school, students choose either a mathematics and science track or a humanities and social science track. Students in the science track take suugaku 3 (calculus) during their last year of high school; most of them take a more rigorous calculus course at the university.

The course curricula for AP Calculus BC and suugaku 3 are very similar. The most noticeable differences are that Japanese students study only geometric series and do not study differential equations. The epsilon-delta definition of limit does not appear in either curriculum.

In the spring and summer of 2000, Professor Toshiyuki Nishimori of Hokkaido University and I studied United States and Japanese students' understanding of the concepts of calculus and their ability to solve traditional calculus problems. We selected two above-average high schools for our study, one in Portland, Oregon, and one in Sapporo, Japan. Our investigation involved 18 students in Portland and 26 students in Japan. Of the 16 Portland students who took the BC examination, six students scored a 5. We tested 75 calculus students in Sapporo; however, we concentrated our study on 26 students in the A class. The other two classes, the B and C groups, were composed of students of lower ability. Each student took two written examinations. The two groups of students that we studied were not random samples of high school calculus students from Japan and the United States, but we believe that they are representative of above-average students. We interviewed each student about his or her background, goals, and abilities and carefully discussed the examination problems with them.

Since we did not expect Japanese students to be familiar with calculators, we prohibited their use on the examinations. However, the students in Portland had made significant use of calculators in their course and might have been at a disadvantage if they did not have access to calculators. For that reason, we attempted to choose problems that were calculator independent. However, some problems on the second examinations required a certain amount of algebraic calculation.

We used problems from popular calculus-reform textbooks on the first examination. These problems required a sound understanding of calculus but little or no algebraic computation. For example, in one problem from the Harvard Calculus Project, a vase was to be filled with water at a constant rate. We asked students to graph the depth of the water against time and to indicate the points at which concavity changed. We also asked students where the depth grew most quickly and most slowly and to estimate the ratio between the two growth rates at these depths.

We found no significant difference between the two groups on the first examination. The Portland students performed as expected on calculus-reform-type problems; however, the Sapporo A students did equally well. Indeed, the Sapporo A group performed better than we had expected. We were somewhat surprised, since the Japanese students had no previous experience with such problems. The performance of Japanese students on the first examination may suggest that bright students can perform well on conceptual problems if they have sufficient training and experience in working such problems as those on the university entrance examinations.

The problems on the second examination were more traditional and required good algebra skills. For example, we told students that the function f(x) = x3+ ax2 + bx assumes the local minimum value—(2 )/9 at x = 1/—and asked them to determine a and b. We then asked them to find the local maximum value of f(x) and to compute the volume generated by revolving the region bounded by the x-axis and the curve y = f(x) about the x-axis. The Sapporo A students scored much higher than the Portland students did on the second examination. In fact, the Portland group performed at approximately the same level as the Sapporo C group and significantly below the Sapporo B group. Several Japanese students said in interviews that they found that certain problems on the second examination were routine, yet no American student was able to completely solve these problems. The Portland students had particular difficulty with algebraic expressions that contained radicals. Several students reported that they worked slowly to avoid making mistakes, possibly because they were accustomed to using calculators instead of doing hand computations.

Students from both countries were intelligent and highly motivated, and they excelled in mathematics; however, differences were evident in their performances, especially in algebraic calculation. One of the best Portland students correctly began to solve a problem on the second examination but gave up when he was confronted with algebraic calculations that involved radicals. On his examination paper he wrote, "Need calculator again."

Perhaps the largest difference between the two groups lies in the different high school cultures. Japanese students work hard to prepare for the university entrance examinations and are generally discouraged from holding part-time jobs. In contrast, students in the United States often hold part-time jobs in high school, and many are involved in such extracurricular activities as sports or clubs.

 Thomas Judson is a visiting assistant professor at the University of Puget Sound in Tacoma, Washington. He is interested in mathematics education and has spent parts of the last eight summers visiting Japan to learn Japanese and study mathematics education there.