"Honestly, I don't recall ever having seen a more offensive question here than how many *boys* in my district graduate with calculus behind them, to say nothing of its begging the question what it means to "have calculus behind you." You will need to ask a reasonable question if you expect me to give you an answer."
Would it be more in line with you high morals to ask the same question about our
12th grade girls?
"What percent of the girls in your district graduate with calculus behind them"?
For the record, this is my own answer to the question about the boys:
According to the College Board, a total of 151,578 or 2.8% of our 5,419,300
18yearolds "passed" [read: scored 3, 4, or 5) in Advanced Placement calculus
A/B last year.
Of those, only 83,391 or 3.2% of our 18yearold boys passed it, of whom 21,046
were Asian boys, who were overrepresented by (21,046 / 83,391 = 25.2%, vs. only
3.1% of the population) = 8 fold.
Without those Asian boys, most of whom learned calculus in their home countries
or in our private schools, only 60,306 or 2.3% of our 18yearold boys would
have passed a calculus placement exam which 70% of the 680,882 18 yearolds in
Germany, more than half of the 1,071,294 18yearold boys in Japan, and more
than half of the 415,311 18yearold boys in Korea, and more than half of the
189,250 18yearold boys in Taiwan: who all graduate from high school with
calculus behind them.
286,155 of our 2,601,408 18 yearold boys were in private high schools where
111,601 passed calculus exams which were equivalent to if not better than the AP
Calculus A/B exam.
535,647 Japanese 18 year oldboys
476,618 German 18 yearold boys
207,666 Korean 18 yearold boys
111,601 US 18 yearold boys who learned calculus in private schools
94,625 Taiwanese 18 yearold boys
60,306 US 18 yearold boys who learned calculus in public high schools
How do I and mine define "very good math teachers"?
Not too differently from teachers all the way from Korea, Japan, Taiwan, and
Singapore, to Germany and other European nations, all the way to our OWN private
schools.
But in our public schools, this is a taboo subject?
http://home.cc.gatech.edu/icegt/548
When you factor in that "passing" a calculus exam in all those other countries,
AND in our own private schools, is equivalent to scoring 5 on the AP Calculus
A/B exam, the number of graduates of our public schools who passed it is down to
less than 20,000:
535,647 Japanese 18 year oldboys
476,618 German 18 yearold boys
207,666 Korean 18 yearold boys
111,601 US 18 yearold boys who learned calculus in private schools
94,625 Taiwanese 18 yearold boys
20,000 US 18 yearold boys who learned calculus in public high schools
>Well, then, one person's "honesty" is
another person's style and taste.
>Why that's so bloody hard for you to grasp and accept is beyond me.
"Actually, it is more than bloody hard. It is impossible for me to grasp
how you compare honesty and taste."
Those without honor really can't tell the difference between honor and taste,
just like they can't understand that there are races, just like they can't
understand that claiming everyone who disagrees must by association be racists
(in the same breath "there are no races"), just like they take Wayne's clear
post and turn it inside out just so they can insult him, just like they fly off
the handle about how a mathrelated post is "off topic" in the same breath they
bend over backwards to change the subjct to every topic but math, just like they
claim "every public school student has the opportunity to take calculus" in the
same breath they actively undermine this opportunity for ALL students.
On Thu, 6/2/11,
Wayne Bishop <[email protected]> wrote: > From: Wayne Bishop
<[email protected]>
> Subject: Re: Korea: Test Focus Hurts Love of Learning
> To: [email protected]
> Cc: [email protected]
> Date: Thursday, June 2, 2011, 12:30 AM
> At 01:08 PM 6/1/2011, [email protected]
> wrote:
> > "Quote:
> > ... students who obtained a grade of 3 or above on
> either of the AP Calculus Exams obtained an average score of
> 596 (3.2) on the TIMSS Advanced Mathematics test,
> OUTPERFORMING ALL OTHER COUNTRIES [my emphasis].
>
> Here is another reason why that is misleading
> statistic... Most US students in "AP Calculus" would
> not score 3 or better; most do not even take the national
> exam. At least half to three fourths of all calculus students in the US take
the calculus exams. About two thirds pass with a score of 3 or better, and the
percent of the ENTIRE high school senior aged population of the US that passes
these calculus exams with a 3 or better is at least 5%. This is a very high
percentage, higher than what I stated before (I found that I overstated how many
high school senior aged people there are in the US). Here's the proof of all
this: http://en.wikipedia.org/wiki/Advanced_Placement_Calculus Grade
distributions for AP Calculus AB
In the 2010 administration, 245,867 students took the exam. The mean score was a
2.81.
The grade distribution for 2010 was:
ScorePercent
5 21.2%
4 16.4%
3 18.0%
2 11.2%
1 33.1% Grade distributions for AP Calculus BC
In the 2010 administration, 78,998 students took the exam. The mean score was a
3.86.
The grade distribution for the 2010 BC scores was:
ScorePercent
5 49.4%
4 15.4%
3 18.0%
2 5.8%
1 11.4% This means that 324,865 high school senior aged people in the US took
the exams in 2010. Passing is defined as scoring 3 or better. We have that 55.6%
of those who took the AB exam scored 3 or better. This means that about 136,702
people passed the AB exam with a 3 or better. We have that 82.8% of those who
took the BC exam scored 3 or better. This means that about 65,410 people passed
the BC exam with a 3 or better. This means that about 202,112 people passed the
calculus exams. "AP Calculus: What We Know"
http://www.macalester.edu/~bressoud/pub/launchings/launchings_06_09.html Quote:
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 7075% OF THE STUDENTS IN AP CALCULUS
TAKE THE EXAM [my emphasis], which suggests that this year 400430,000 students
took a course called AP Calculus." The below essentially says that there are
about 3.3 million high school seniors in the US: http://nces.ed.gov/fastfacts/display.asp?id=372
Quote: "The percentage of high school dropouts among 16 through 24yearolds
declined from 11.8 percent in 1998 to 8.1 percent in 2009. About 3,273,000
students are expected to graduate from high school in 201011, including
2,962,000 from public high schools and 311,000 from private high schools." With
a roughly 8% dropout rate, that means that there are roughly 3.6 million people
in the US of high school senior age. Given roughly 3.3 million high school
seniors and roughly 3.6 million people of high school senior age, the estimates
above are that roughly 400,000 to 600,000 students or roughly 1116% of the
ENTIRE high school senior aged population of the US (roughly 3.6 million) take
calculus as high school seniors. With 324,865 of the roughly 3.3 million high
school seniors and roughly 3.6 million people of high school senior age not only
taking calculus but actually taking the calculus exams and with 202,112 passing
the exams with a score of 3 or better, at least 8% of the ENTIRE high school
senior aged population of the US took not only calculus but also took the
calculus exams and at least 5% of the ENTIRE high school senior aged population
of the US passed the calculus exams with a score of 3 or better. Note that the
ENTIRE high school senior aged population of a country includes everyone  out
of school in school, where type of school does not matter whether it be a
mathbased subject area school or fine arts schools or music schools or general
vocational schools or whatever. At least 5% of the ENTIRE high school senior
aged population of the US passed the calculus exams with a score of 3 or better.
I doubt that there is a country on the planet outside of possibly those 4
exceptional East Asia countries of Japan, South Korea, Taiwan, and Singapore
that does better than this in terms of universal calculus education. Because
don't forget that the US is essentially the only country in the world that does
not ration its advanced math education opportunities after middle school  the
other countries via testing track only their higher performing students into
advanced math based high schools. The rest go into other high school based on
vocational training.
The last time we had a chance to compare the performance of our high school students to those in the rest of the industrialized world was with TIMSS in 1995, when out of 16 countries, we scored dead last in Numbers and Equations, Calculus, and Geometry. Our average calculus score of 450 was more than a standard deviation lower than Cyprus, France, Greece, Russia, Australia, Switzerland, Italy, and Denmark, and the calculus score for our girls of 399 was almost two standard deviations lower.
A score of 399 on this type of multiple choice test is too low to be able to measure anything. A person who knew nothing about the subject, signed their name, guessed on all the multiple choice questions, and answered a few giveaway questions, would score 440, or 41 points higher:
200 points for signing your name. 

200 points for guessing on all the multiple choice questions. 

40 points for answering the giveaway questions. 
http://home.cc.gatech.edu/icegt/548
Barb Ericson has been gathering the detailed data on pass rates, race, and gender for the Advanced Placement Computer Science A exam. The data is from the data for each state available at http://research.collegeboard.org/programs/ap/data/participation/2012.Texas had the most schools that passed the audit for AP CS A with 278 (vs 192 for California), but California had the most students who took the exam at 3,920 (vs 3,614 for Texas.  
California also had the most people who passed the exam with 2,862. Texas was second with 2,102.  
The state with the highest pass rate was Alaska with 92% with 12 out of 13 students passing.  
The state with the highest yield (number of students divided by the number of schools that passed the audit) of 20.4 was California. Wyoming had 2 schools pass the audit but only 1 student take the AP CS A exam in 2012.  
The average yield was 8.55 students per school. This is much lower than the average yield for Calculas AB (19 to 21)  
Texas had the most females take the exam with 838 vs 833 for California.  
Tennessee had the highest percentage of females at 31.25% with 50 females taking the exam. Some states had no females take the exam: Alaska, Mississippi, and Wyoming  
Maryland had the highest number of black (African American) students take the exam at 144. However, Texas had the most pass the exam at 52 (out of 142).  
Washington had the highest pass rate for black (African American) students at 60%, but that was only 6 of 10.  
The top 5 states with the highest number of black (African American) male students take the exam was: Maryland with 115, Texas 110, Georgia 98, Florida 78, and Virginia at 47.  
The top 5 states with the highest number of black (African American) male students who passed the exam was: Texas with 43, Maryland with 30, Georgia and California with 18 each, and New Jersey with 16.  
Only 9 states had 10 or more black (African American) females take the exam: Georgia 39, Texas 32, Maryland 29, Florida 27, New York 18, California 15, Illinois 14, South Carolina 12, and Virginia 11  
Texas had the most black (African American) females pass the exam with 9 (out of 32). Black females have a lower pass rate than black males.  
Texas had the most Hispanic students (Mexican American, Other Hispanic, and Puerto Rican) take the exam at 609 vs 314 for California.  
Texas had the most Hispanic female students take the exam at 153 versus California at 75.  
Texas had the most Hispanic female students pass the exam at 26 (out of 153) versus 22 for California (out of 75).  
The state that had the highest pass rate (35%) for Hispanic females was New Jersey with 7 out of 20 passing. 
The number of students that take the exam per teacher is much higher for AP Calculus AB at 21 students per teacher versus 11 for Computer Science A  
The number of schools that teach Calculus is 11,694 versus 2,103 for AP CS A  
AP CS A had a higher pass rate than Calculus  63% versus 59%  
AP CS A had a higher female pass rate than Calculus  56% versus 55%  
AP CS A had a higher Hispanic pass rate than Calculus  39.8% versus 38.4%  
AP Calculus had a higher black pass rate than CS  28.7% versus 27.3%  
Calculus had a much higher percentage of women take the exam than CS  48.3% versus 18.7%  
Calculus had a higher percentage of black students take the exam than CS  5.4% versus 4.0%  
Calculus had a higher percentage of Hispanic/Latino students take the exam than CS  11.5% versus 7.7% 
Of the 16% of American high school students who took calculus, 12% of them were in VERY EXPENSIVE private schools, leaving only 4% of public high school students graduating with calculus behind them. How else can it be explained that in the 12th grade TIMSS calculus subjects we scored MORE THAN ONE HUNDRED points lower than Cyprus, Greece, Russia, and France? Even lowly Italy scored 70 points higher!
Even this putative increase in the number of students taking calculus is FAR, FAR offset by an 8% DECREASE in students graduating with algebra behind them. While WE are teaching gender sensitivity and race baiting, China and Korea and Japan and Germany are teaching up to 95% of their high school students CALCULUS. If the Obama Administration had spent as much time promoting calculus education as it did race baiting Trayvon Martin, ONE HUNDRED PERCDENT of American high school kids (or at least the White ones) could have graduated with calculus behind them.
Hedrick Smith,"Rethinking America", July 15, 1995 points out that 6% of U.S. and 40% of German and 94% of Japanese students study calculus in high school. 

The US Department of Education reports that 10.1% of US high school graduates take calculus http://nces.ed.gov/pubsold/CoE95/2601.html 

SAT Scores are down 98 points since 1960. 
In calculus specifically, our advanced math students scored 450, lower than 13 other countries who participated in TIMSS that year, higher than only two countries (Czech Republic at 446 and Austria at 439, neither of which are statistically significant), and 111 points lower than Cyprus. Yet this is what we hear from ALL our educators, who are so sadly misinformed that no American child should be before them, much less under their authority:
"As for the performance of the calculus students in the US in 1995, it was in the middle of the pack:
'...the performance of U.S. twelfth graders with Advanced Placement calculus
instruction, who represent about 5 percent of the United States age cohort, was
significantly higher than the performance of advanced mathematics students in 5
other countries". This report shows the AP Calculus students with an average
achievement of 513 on the TIMSS Advanced Mathematics test.'"
"The middle of the pack"?
In advanced math in 1995, only Austria scored lower (436, vs. our 442, not even
statistically significant).
Our score of 442 (117 points lower than France, 100 points lower than Russia,
and SIGNIFICANTLY lower than TWELVE other countries, was not even VALID because,
in the words of TIMSS, the US was one of those "Countries Not Satisfying
Guidelines for Sample Participation rates (See appendix B for details)". iow, if
we HAD met the sampling requirements, our score would have been lower, MUCH
lower.
Our educators have been directed at NUMEROUS sources which PROVE that our scores
have DECREASED, not increased, since 1995, which is the ONLY reason the US
refused to take the risk of getting egg all over its face in 2008 just as we did
in the ABOVE sad performance.
The ONLY thing which improved since then is the ability of our educators to LIE
to you, and yoursand perhaps the fact that Beverly Hall et. al. (including,
unfortunately, Michelle Rhee) will go down in HISTORY as LIARS.
Most of the teachers we interviewed indicated that the goal of instruction is exposure, not mastery. Although most Japanese high school students are exposed to calculus, Japanese teachers indicated that they did not expect all their students to understand calculus. When asked, �What percentage of students, in your experience, are able to do calculus problems and understand calculus?�, one teacher said:
About 20 percent of students are able to both do and understand. Almost all students are able to do calculus, although they may not understand it. I am referring to science students here. As for the humanities students, all those that have taken math are able to do calculus, whether or not they understand it. But very few students understand the concept of change in calculus. That is why I think that there is an element of the horse learning to pull out the card when hearing two thumps. I do not think that this is a good way to teach math. I dislike this immensely. But I forget that and teach.
Retention (genkyutomeoki) is rarely practiced in Japanese schools, particularly during the compulsory years, or even during senior high school. In fact, one senior high school teacher could not recall a time in his entire teaching career when a student was retained. Other senior high school teachers reported �one case every 4 or 5 years.�
Instead of retaining students, Japanese teachers are encouraged by administrators to provide extra instruction in basic skills (fushinsya shido) to students having trouble meeting minimum school standards. This may involve the teachers spending time with students outside of class or providing remedial homework assignments during the summer or winter vacations. In spite of these efforts, some students still perform poorly on examinations. In these cases, Japanese teachers reported practicing what they call �letting the student put on a pair of geta (Japanese elevated wooden clogs)�. This expression refers to giving students extra points for good effort.
http://www.ed.gov/pubs/JapanCaseStudy/chapter2c.html
96% of Japan's students,
70% of Germany's students, and only 4% of US students take calculus in high school, another key factor
conveniently overlooked in this report
http://fathersmanifesto.net/depted.htm
Changes in mathematics education are currently taking place in Japan. The school week is changing from six days to five days per week to allow children to develop "competency for positive living." Changes in Japan seem to occur approximately every ten years, although the mathematics content over the last 30 years has been relatively stable. Reform is based on tradition and varies according to the times. Since 1994, Japan has had a "core and optional modules" model for upper secondary school mathematics. An overall objective of the entire curriculum is to foster students' abilities to think mathematically. "Open" methods of teaching  open process, using different ways to solve a problem; openended where problems have multiple correct answers; and open problem formulation where students pose new mathematical problems can help meet this objective. Traditionally, one important feature of learning mathematics was to develop the ability to calculate rapidly.
Miho Ueno
Tokyo Gakugei University Senior High School
The goal of mathematics education might be seen as learning the basic idea of calculus by the time students graduate from high school. Beginning calculus is taken by 82 percent of high school students in a traditional mathematics program. It is often difficult to implement a new course of study due to factors such as a lack of teachers in a small school system, which limits the courses offered to those needed for college entrance, or to a lack of technology. Thus, the intended curriculum may not get implemented.
The results of the Third International Mathematics and Science Study (TIMSS) indicated that Japanese students disliked mathematics. The new course of study addresses this by stressing mathematical activities aimed at helping students appreciate the importance of mathematical approaches and ways of thinking. What matters is finding principles in given phenomena in the world and finding materials that will allow students to use mathematics spontaneously. Reform should be realized through teachers' attempts to enrich the contents of the prescribed curriculum.
http://mathforum.org/pcmi/int2001report/page47.html
In 2006, the average U.S. score in mathematics literacy was 474 on a scale from 0 to 1,000, lower than the OECD average score of 498 (tables 3 and C7). Thirtyone jurisdictions (23 OECD jurisdictions and 8 nonOECD jurisdictions) had a higher average score than the United States in mathematics literacy in 2006. In contrast, 20 jurisdictions (4 OECD jurisdictions and 16 nonOECD jurisdictions) scored lower than the United States in mathematics literacy in 2006.
Between 1972 and 1995, the number of 18 year olds in the U.S. population decreased from 3,927,000 to 3,745,000 and the number of high school graduates decreased from 3,001,000 to 2,553,000, which means that the percent of 18 year olds who graduated from high school decreased from 76.4% to 68.2% in 23 years, while the percent of Japanese 18 year olds in high school remained consistent, at 94%.
Japanese teachers thus teach calculus to their citizens at a rate which is 12.8 times greater than the rate at which American teachers teach calculus to American citizens.
1972  1995  Difference  
Number of 18 year olds  3,927,000  3,745,000  182,000 
Number of high school graduates  3,001,000  2,553,000  448,000 
Percent of 18 year olds graduating from high school  76.4%  68.2%  8.2% 
Percent of high school graduates taking calculus  4.3%  10.1%  1.9% 
Number taking calculus  129,043  257,853  128,810 
Percent of 18 year olds taking calculus in the US  3.3%  6.9%  0.9% 
Percent of 18 year olds taking calculus in Japan    88.4%  
Ratio Japan:US  12.8X 
We agree with Charles A Fuller! 

Correlating standardized test scores with incomes and college grades. 

Download the spreadsheet with references from population18to24.xls 

and sat.xls 

SAT Graphs 

For 1960 SAT scores 

For 1995 SAT scores, http://nces.ed.gov/pubs/ce/c9622a01.html 

SAT Scores 19661999 
Comments? Telephone fax, or email
http://mathforum.org/kb/message.jspa?messageID=7758713
Engage
To Excel: Producing One Million Additional College Graduates with STEM
Degree Posted: Mar 31, 2012 9:08 PM 


(I prematurely hit the send button on my last reply) Here are the two main faults in your analysis that cause your arguments to fail (in other words that cause you to predict a result that everyone here knows first hand is false, that we have succeeded in teaching scores of students calculus). 1. You cite comparisons of very poor performance against very poor performance. The scores you cite are meaningless because those very same students you claim are successful cannot finish even one free response problem and they miss most of the MC problems. You claim that a student that scored a "1" is rocking because compared to other students in the world they score higher. If they were rocking then how come they failed to correctly answer over 90% of the AP exam? A contradiction like that Paul should be the end of that thread but instead you proclaim that 2 + 2 IS EQUAL TO 5! 2. You misunderstand how a "cumulative" exam works. A cumulative exam starts off with basic conceptual questions and works its way up through whole questions. The quality of the exam depends on how long it takes to get to the whole problems and where the cutoff scores lie. I did an extensive study of the 2003 and 2008 exams and found that the typical student scoring a "3" answered mostly just the basic MC questions and almost none of the FR problems. Remeber, they miss 60% of the test. They most certainly did not succeed at calculus. Cumulative exams are not a good fit for final exams and many courses do not use this format, especially if they are serious about whether a student succeeded or failed. This "cumulative" exam format became popular with mass education because it is easy and every student gets a prize. By the way, if you talked to AP teachers then you would understand that the students can't do algebra. The AP exam results are all the evidence I need. That should be all the evidence anyone needs. The questions are right there to examine and every 5 years they release the exams and some detail for the results. They are fraud just like Florida's test scores are fraud. Failing has become the new passing. And yes, this grade inflation started in earnest in colleges and now has infected high school. The end result of all this is that college is a bad investment for many students now. It is their version of the housing crisis. They will lose their entire investment and be in debt. And here is how it is going to play out from here on. Many of these students will default on that debt and the credit will dry up. We all saw this coming. Even when I went to college there was a trickle of students misusing loans to pay for an education (and other things not even having to do with education) that they were doing very poorly in. Well, like with the housing crisis, that trickle turned into a flood. The colleges had some reservations in the beginning but those reservations only delayed the inevitable. The flood is now here, the system is broke, and the defaults are on there way. This country has a very serious problem with socialism. The persons in charge of our socialistic programs simply refuse to make the tough calls that are required of them. We elect them to manage our government and our schools and they fail time and time again to do their job. When we really need them the most they balk. Medicare is unsustainable, their answer is to turn it over to private companies. Education is blowing up, their answer is to turn it over to private companies. These are not their answers because they are the right things to do. These are their answers because they do not want the responsibility they were elected to. Bob Hansen 
http://nces.ed.gov/fastfacts/display.asp?id=97
Question:
What are the recent trends in advanced mathematics and science coursetaking
among U.S. high school students?
Response:
The percentages of high school graduates who took mathematics courses in
geometry, algebra II/trigonometry, analysis/precalculus, statistics/probability,
and calculus while in high school were higher in 2009 than in 1990. Similarly,
the percentages of high school graduates who took science courses in biology,
chemistry, physics, both biology and chemistry, or in all three of these science
courses while in high school were higher in 2009 than in 1990. For example,
while in high school, 16 percent of 2009 graduates versus 7 percent of 1990
graduates took calculus, and 30 percent of 2009 graduates took biology,
chemistry, and physics in high school versus 19 percent of 1990 graduates. In
contrast, 69 percent of 2009 graduates took algebra I in high school versus
77 percent of 1990 graduates. Looking at more recent
changes, the percentages of graduates who took mathematics and science courses
were higher in 2009 than in 2005 for all courses except algebra I and the
combination of biology, chemistry, and physics, for which no measurable
differences were found.
Across subgroups, the percentages of high school graduates who had taken calculus and biology, chemistry, and physics were generally higher in 2009 than in 1990. For example, 9 percent of Hispanic 2009 high school graduates had taken calculus versus 4 percent of 1990 graduates. Also, 28 percent of female 2009 graduates had taken biology, chemistry, and physics versus 16 percent of 1990 graduates. Comparing 2009 with 2005, the percentages of graduates who had taken these courses were higher for some subgroups. For instance, 12 percent of 2009 graduates with disabilities had taken biology, chemistry, and physics versus 7 percent of 2005 graduates.
SOURCE: U.S. Department of Education, National Center for Education Statistics. (2012). The Condition of Education 2012 (NCES 2012045), Indicator 31.
Related Tables and Figures: (Listed by Release Date)
Other Resources: (Listed by Release Date)
http://www.nsf.gov/statistics/seind02/c1/c1s1.htm
For most of the 23 nations that participated in 8th grade in both TIMSS and
TIMSSR, including the United States, there was little change in the mathematics
and science average scores over the fouryear period. There was no change in
8thgrade mathematics achievement between 1995 and 1999 in the United States and
in 18 other nations. (See
Students� performance in the final year of secondary school can be considered a measure of what students have learned over the course of their years in school. Assessments were conducted in 21 countries in 1995 to examine performance on the general knowledge of mathematics and science expected of all students and on more specialized content taught only in advanced courses.
Achievement on General Knowledge Assessments. The TIMSS general knowledge assessments were taken by all students in their last year of upper secondary education (12th grade in the United States), including those not taking advanced mathematics and science courses. The science assessment covered earth sciences/life sciences and physical sciences, topics covered in grade 9 in many other countries but not until grade 11 in U.S. schools. On the general science knowledge assessment, U.S. students scored 20 points below the 21country international average, comparable to the performance of 7 other nations but below the performance of 11 nations participating in the assessment. Only 2 of the 21 countries, Cyprus and South Africa, performed at a significantly lower level than the United States. Countries performing similarly to the United States were Germany, the Russian Federation, France, the Czech Republic, Italy, and Hungary.
A curriculum analysis showed that the general mathematics assessment given to students in their last year of secondary education covered topics comparable to 7thgrade material internationally and 9thgrade material in the United States. Again, U.S. students scored below the international average, outperformed by 14 countries but scoring similarly to Italy, the Russian Federation, Lithuania, and the Czech Republic. As on the general science assessment, only Cyprus and South Africa performed at a lower level. These results suggest that students in the United States appear to be losing ground in mathematics and science to students in many other countries as they progress from elementary to middle to secondary school.
Achievement of Advanced Students. On advanced mathematics
and science assessments, U.S. 12th grade students who had taken advanced
coursework in these subjects performed poorly compared with their counterparts
in other countries, even though U.S. students are less likely to have taken
advanced courses than students at the end of secondary school in other
countries. The TIMSS physics assessment was administered to students in other
countries who were taking advanced science courses and to U.S. students who were
taking or had taken physics I and II, advanced physics, or advanced placement
(AP) physics (about 14 percent of the entire age cohort). The assessment covered
mechanics and electricity/magnetism as well as particle, quantum, and other
areas of modern physics. Compared with their counterparts in other countries,
U.S. students performed below the international average of 16 countries on the
physics assessment. (See
The advanced mathematics assessment was administered to students in other countries who were taking advanced mathematics courses and to U.S. students who were taking or had taken calculus, precalculus, or AP calculus (about 14 percent of the relevant cohort). Onequarter of the items tested calculus knowledge. Other topics included numbers, equations and functions, validation and structure, probability and statistics, and geometry.
The international average on the advanced mathematics assessment was 501.
U.S. students, scoring 442, were outperformed by students in 11 nations, whose
average scores ranged from 475 to 557. No nation performed significantly below
the United States; Italy, the Czech Republic, Germany, and Austria performed at
about the same level. (See
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.
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
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
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. 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.

Chinese Language  68.1%  13.8%  12.3%  3.1%  2.7%  June 25  80% of AP Chinese examinees are heritage speakers, the largest
percentage among any of the AP World Language exms. As a reminder, we
don't grade AP Exams on a curve; all students who meet the standard for
a 5 get a 5. AP Chinese scores show this clearly. Students performed much better on the multiplechoice questions re: interpretive reading than listening. Students generally did well on the writing; most difficult = cultural presentation (Q4): 8% got 0 pts ow.ly/yslMG 
Calculus AB  24.3%  16.7%  17.7%  10.8%  30.5%  June 24  I'm impressed by this year's AP Calculus AB results  a higher % of
5s as teachers expanded access to 13K more students. 1 student in the world earned all 108/108.. We'll notify her/him & the teacher/school this fall. Are too many students being rushed into AP Calc AB? On every freeresponse question, ~20% of students got 0/9 points ow.ly/yna8r The lowest avg score on the freeresponse questions was on #1 (modeling rate) & the highest was on #6 (separable DE w/slope field). 
Calculus BC  48.3%  16.8%  16.4%  5.2%  13.3%  June 24  There are large score increases in AP Calculus BC this year, w/ the
biggest gain being in % of 5s. Impressive work by teachers/students. 5 students worldwide earned perfect scores of 108/108 points on this year's AP Calculus BC Exam. Students tended to score higher on the multiplechoice questions about Differential Calc than Integral Calc. In the freeresponse questions, students performed best by far on Q3 (graphical analysis of F/FTC). The Taylor Series question (#6) was most difficult of the AP Calc BC questions; 22% of students earned 0/9 points: ow.ly/ynaXR. AP Calc BC: Q2 (polar) & Q4 were tied for 2nd most difficult question. On Q2, ~9% of students earned all 9 pts; 16% earned 0. 
English Literature and Composition  7.7%  17.8%  29.5%  33.0%  12.0%  June 23  15K more students, similar %5s, smalller %3s/4s; higher %1s/2s. Multiplechoice questions: as usual students scored higher analyzing 1) prose than poetry; 2) post1900 texts than pre. The most frequent score on each of the 3 essays was 4 out of 9 points possible. Average scores on each of the 3 essays: #1 (Gascoigne poem): 46%; #2 (Jones opening): 46%; #3 (sacrifice): 50%. Dr. Eileen Cahill of Salem Academy led the scoring of Q3 on sacrifice, collecting data re: the texts most frequently chosen by students. Any guesses on the novel/play most frequently chosen by students for their essays on values/sacrifice (question 3)? ow.ly/m8W2d The #1 & #2 most frequently selected texts for Q3 both center on female protagonists at odds with society. Besides Shakespeare, only one other author, Arthur Miller, showed up more than once among the top 10 texts chosen by students. The Scarlet Letter was the #1 most frequently selected text for AP Eng Lit Q3, followed by #2, The Awakening. The #3 most frequent text for AP Eng Lit Q3 was The Great Gatsby. It wasn�t on the list, but was a good, appropriate choice by students. The #4 and #5 most frequently selected texts for AP Eng Lit Q3 were both by AfricanAmerican female novelists. Most frequently selected texts for AP Eng Lit Q3: Beloved and Their Eyes Were Watching God were the #4 & #5 most frequently selected texts for AP Eng Lit Q3. #6: Othello; #7: Death of a Salesman; #8: Hamlet; #8: The Crucible; #10: King Lear. 