David M. Bressoud
Macalester College, St. Paul, MN
Project NExT-WI, October 5–7, 2006
This PowerPoint is available at
www.macalester.edu/~bressoud/talks
1. Where we are
2. How we got here
3. A closer examination of
where we are
4. Where we are going
AP Calculus
300,000
250,000
150,000
100,000
50,000
0
19
55
19
59
19
63
19
67
19
71
19
75
19
79
19
83
19
87
19
91
19
95
19
99
20
03
number
200,000
year
AB Calc
BC Calc
total Calc
AP Stat
Mainstream Calculus I Enrollments
(fall only for 2- & 4-yr colleges & universities)
250
AP Calculus 2006: ~255,000
Currently growing at >15,000/year
students (thousands)
200
150
4-yr colleges &
universities
2-yr colleges
100
AP Calculus (AB & BC)
50
0
1
0–
98
81
1
5–
98
86
1
0–
99
91
1
5–
99
academic year
96
2
0–
00
01
Mainstream Calculus I Enrollments
(fall only for 2- & 4-yr colleges & universities)
250
AP Calculus 2006: ~255,000
Currently growing at >15,000/year
students (thousands)
200
150
4-yr colleges &
universities
2-yr colleges
100
AP Calculus (AB & BC)
Estimated # of students taking
Calculus in high school: ~ 500,000
50
0
1
0–
98
81
1
5–
98
86
1
0–
99
91
1
5–
99
academic year
96
2
0–
00
01
Estimated # of students taking
Calculus I in college: ~ 500,000
(includes Business Calc)
Bachelors degrees each year*
400,000 SMET + social & behavioral sciences
of which
210,000 Science, Math, Engineering
of which
100,000 physical, biological, &
ag sciences
60,000 engineering
50,000 math, stat, comp sci
of which
11,000 mathematics
*NSF: among 24-year olds in 2000
~170,000 arrive
with credit for
400,000 SMET + social & behavioral sciences calculus
~330,000 retake
of which
calculus taken in
210,000 Science, Math, Engineering
HS
Bachelors degrees each year*
of which
~170,000 will
100,000 physical, biological, & take calculus for
first time
ag sciences
60,000 engineering
50,000 math, stat, comp sci
of which
11,000 mathematics
*NSF: among 24-year olds in 2000
students (thousands)
Mainstream Calculus II Enrollments
(fall only for 2- & 4- year colleges and universities)
120
BC exams for 2006: ~59,000
100
Still growing exponentially at over
9%/year (8 year doubling time)
80
4- yr c olleges &
universities
2- yr c olleges
60
AP Calc ulus BC only
40
20
0
1980–81
1985–86
1990–91
1995–96
ac ademic year
2000–01
students (thousands)
Mainstream Calculus II Enrollments
(fall only for 2- & 4- year colleges and universities)
120
BC exams for 2006: ~58,000
100
Still growing exponentially at over
9%/year (8 year doubling time)
80
4- yr c olleges &
universities
2- yr c olleges
60
AP Calc ulus BC only
40
20
0
1980–81
1985–86
1990–91
1995–96
ac ademic year
2000–01
Last year, 12,500 students
took the BC exam before
their senior year. This year
it was 13,800.
Calculus Before Grade 12
40000
35000
30000
number
25000
AB Calc < grade 12
BC Calc, < grade 12
20000
15000
10000
5000
0
2002
2003
2004
year
2005
2006
Implications:
1. Students who 20 years ago would have arrived at college
ready to take calculus now take it in high school.
2. Students who take Calculus I in college either are
retaking a course taken in high school or have had to
overcome mathematical deficiencies. Calculus I is
increasingly taken as a terminal course.
3. Especially at elite institutions but increasingly
elsewhere, the traditional Calculus II which presupposes
Calculus I at that institution does not serve the needs of
the students who take it.
AP Calculus
16,000
14,000
12,000
8,000
6,000
4,000
2,000
year
AB Calc
BC Calc
total Calc
19
73
19
71
19
69
19
67
19
65
19
63
19
61
19
59
19
57
0
19
55
number
10,000
AP Calculus
90,000
80,000
70,000
50,000
40,000
30,000
20,000
10,000
AB Calc
year
BC Calc
total Calc
19
89
19
87
19
85
19
83
19
81
19
79
19
77
19
75
0
19
73
number
60,000
1983–84 scientific calculators allowed
1983–84 scientific calculators allowed
1986 Tulane Conference, birth of Calculus
Reform movement, “lean & lively calculus”
1983–84 scientific calculators allowed
1986 Tulane Conference, birth of Calculus
Reform movement, “lean & lively calculus”
1989 decision to revisit entire AP Calculus
curriculum and approach, bring in graphing
calculators; revisions led by Tom Tucker
(Colgate), John Kenelly (Clemson), Anita Solow
(Grinnell), Dan Kennedy (Baylor School)
1983–84 scientific calculators allowed
1986 Tulane Conference, birth of Calculus
Reform movement, “lean & lively calculus”
1989 decision to revisit entire AP Calculus
curriculum and approach, bring in graphing
calculators; revisions led by Tom Tucker
(Colgate), John Kenelly (Clemson), Anita Solow
(Grinnell), Dan Kennedy (Baylor School)
1993–94 scientific calculators required
1995 graphing calculators required, proposed
changes to AP syllabus agreed upon
AP Calculus
1995, Graphing calculators
300,000
250,000
150,000
100,000
50,000
year
AB Calc
BC Calc
total Calc
20
04
20
02
20
00
19
98
19
96
19
94
19
92
0
19
90
number
200,000
AP Calculus
1997, New syllabus
300,000
250,000
150,000
100,000
50,000
year
AB Calc
BC Calc
total Calc
20
04
20
02
20
00
19
98
19
96
19
94
19
92
0
19
90
number
200,000
AP Calculus
2000, 3 non-calculator free response
300,000
250,000
150,000
100,000
50,000
year
AB Calc
BC Calc
total Calc
20
04
20
02
20
00
19
98
19
96
19
94
19
92
0
19
90
number
200,000
AP Calculus
2001, AB subscore for BC exam
300,000
250,000
150,000
100,000
50,000
year
AB Calc
BC Calc
total Calc
20
04
20
02
20
00
19
98
19
96
19
94
19
92
0
19
90
number
200,000
Ratio of AB to BC exams
6
5
rati
4
3
AB/BC
2
1
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
year
AB subscore
New syllabus
1997–98 exams based on new syllabus
•Graphical, numerical, analytical, and verbal
descriptions of functions
1997–98 exams based on new syllabus
•Graphical, numerical, analytical, and verbal
descriptions of functions
1997–98 exams based on new syllabus
•Graphical, numerical, analytical, and verbal
descriptions of functions
1997–98 exams based on new syllabus
•Integral as limit of Riemann sums and as net
accumulation of rate of change
1997–98 exams based on new syllabus
•Understand both parts of FTC
Evaluation: If you know an anti-derivative for f,
you can use it evaluate the definite integral,
F'  f

 f x dx  F b  F a.
b
a
Anti-derivative: The definite integral with variable
upper limit is an anti-derivative,
d x
f t dt  f x .

a
dx
1997–98 exams based on new syllabus
•Understand both parts of FTC
1997–98 exams based on new syllabus
•Understand both parts of FTC
2004 AB3(d)
A particle moves along the y-axis so that its velocity v at time
t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle
is at y = –1. Find the position of the particle at time t = 2.
y '(t) = v(t) = 1 – tan–1(et)
y(t) = ?
1997–98 exams based on new syllabus
•Understand both parts of FTC
2004 AB3(d)
A particle moves along the y-axis so that its velocity v at time
t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle
is at y = –1. Find the position of the particle at time t = 2.
 v t dt  y 2  y 0; y t   1   v x dx
2
t
0
0
1997–98 exams based on new syllabus
•Understand both parts of FTC
2004 AB3(d)
A particle moves along the y-axis so that its velocity v at time
t ≥ 0 is given by v(t) = 1 – tan–1(et). At time t = 0, the particle
is at y = –1. Find the position of the particle at time t = 2.
 v t dt  y 2  y 0; y t   1   v x dx
2
t
0
0

 
y 2   1   1  tan1 et dt  1.36069
2
0
1997–98 exams based on new syllabus
•Be able to communicate mathematics: justify
local or absolute extremum, explain the meaning
of an answer
2005 AB5/BC5
Into the Future
•Use of CAS is coming - currently about 35% of BC
students, 20% of AB have TI-89 or comparable,
probably 5–10 years away.
•Exams administered via computer, probably 10–15
years away.
Into the Future
•Pressure to get college-bound students into an
AP Calculus class is going to intensify.
Into the Future
•Pressure to get college-bound students into an
AP Calculus class is going to intensify.
•The growth in AP Calculus is not about to end.
President’s American Competitiveness Initiative,
training 70,000 new AP math and science
teachers, Dept of Ed requesting $122,000,000 for
FY 2007 to support AP programs.
Into the Future
•Pressure to get college-bound students into an
AP Calculus class is going to intensify.
•The growth in AP Calculus is not about to end.
•% increase of BC Calculus will continue to
exceed that of AB
Into the Future
•Pressure to get college-bound students into an
AP Calculus class is going to intensify.
•The growth in AP Calculus is not about to end.
•% increase of BC Calculus will continue to
exceed that of AB
•% increase in # of students taking BC Calculus
before senior year will continue to exceed that of
BC generally
Into the Future
•Pressure to get college-bound students into an
AP Calculus class is going to intensify.
•The growth in AP Calculus is not about to end.
•% increase of BC Calculus will continue to
exceed that of AB
•% increase in # of students taking BC Calculus
before senior year will continue to exceed that of
BC generally
•More universities will see calculus as a high
school course.
Needed Response
•NCTM, MAA, AMS need to coordinate a strong
signal that calculus in HS is only appropriate when
students have a solid foundation in pre-calculus,
need to articulate what this foundation must be.
Needed Response
•NCTM, MAA, AMS need to coordinate a strong
signal that calculus in HS is only appropriate when
students have a solid foundation in pre-calculus,
need to articulate what this foundation must be.
•Need much greater collaboration between high
school and college teachers.
Needed Response
•NCTM, MAA, AMS need to coordinate a strong
signal that calculus in HS is only appropriate when
students have a solid foundation in pre-calculus,
need to articulate what this foundation must be.
•Need much greater collaboration between high
school and college teachers.
•Need to seriously address the question of what to
do with students who take (and pass) BC Calculus
before their senior year.
APCentral
at apcentral.collegeboard.com
SIGMAA TAHSM (Special Interest Group of the
MAA, Teaching Advanced High School
Mathematics)
at www.maa.org/SIGMAA/tahsm/
This PowerPoint presentation
at www.macalester.edu/~bressoud/talks