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)
http://www.maa.org/columns/launchings/launchings_06_09.html
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 4year 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.
How many students study calculus
in high school and what kind of program do they take?
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 largescale 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
collegelevel 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 collegelevel
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 collegelevel 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
largescale 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
largescale 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 fiveyear 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 UTAustin. 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 16point 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.
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 largescale 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.
[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/datareportsresearch/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 4year college numbers, but the numbers are also
essentially constant when considering just 2year or just 4year undergraduate
programs. It is taken from
[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. SR9813.
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,
UCDavis, UCIrvine, University of Georgia, University of Illinois, UNCChapel
Hill, UTAustin, University of Utah, University of Virginia, and Yale
University.
[11]
Morgan, R. and J. Klaric. 2007. AP® Students in College: An Analysis
of FiveYear Acadmeci Careers. College
Board Research Report No. 20074. New York. professionals.collegeboard.com/datareportsresearch/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 NonAP Student Groups.College
Board Research Report No. 20029. professionals.collegeboard.com/datareportsresearch/cb/title
[14]
Keng, L.and B. G. Dodd.
2008. An Investigation of
College Performance of AP and NonAP Student Groups professionals.collegeboard.com/datareportsresearch/cb/title
[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.
Access pdf files
of the CUPM Curriculum Guide 2004
and the Curriculum Foundations Project:
Voices of the Partner Disciplines.
Purchase
a hard copy of the CUPM Curriculum Guide 2004 or the Curriculum Foundations Project: Voices of the Partner
Disciplines.
Find
links to coursespecific software resources in the CUPM Illustrative Resources.
Find
other Launchings
columns.
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.
http://www.nctm.org/resources/content.aspx?id=1580
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 entranceexamination
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 TIMSSR,
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 epsilondelta 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 aboveaverage 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
aboveaverage 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
calculusreform 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 calculusreformtype 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) = x^{3}+ ax^{2}
+ 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 xaxis
and the curve y = f(x) about the xaxis. 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 parttime jobs. In contrast, students in the
United States often hold parttime jobs in high school, and many are involved
in such extracurricular activities as sports or clubs.
