The Changing Face of Calculus
42
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
description
The Changing Face of Calculus. 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. Where we are How we got here A closer examination of where we are Where we are going. - PowerPoint PPT Presentation
Transcript of The Changing Face of Calculus
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
0
50,000
100,000
150,000
200,000
250,000
300,000
1955195919631967197119751979198319871991199519992003
year
number
AB Calc BC Calc total Calc AP Stat
Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities)
0
50
100
150
200
250
1980–81 1985–86 1990–91 1995–96 2000–01
academic year
students (thousands)
4-yr colleges &universities
2-yr colleges
AP Calculus (AB & BC)
AP Calculus 2006: ~255,000
Currently growing at >15,000/year
Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities)
0
50
100
150
200
250
1980–81 1985–86 1990–91 1995–96 2000–01
academic year
students (thousands)
4-yr colleges &universities
2-yr colleges
AP Calculus (AB & BC)
Estimated # of students taking Calculus in high school: ~ 500,000
Estimated # of students taking Calculus I in college: ~ 500,000
(includes Business Calc)
AP Calculus 2006: ~255,000
Currently growing at >15,000/year
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
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 calculus
~330,000 retake calculus taken in HS
~170,000 will take calculus for first time
Mainstream Calculus II Enrollments(fall only for 2- & 4- year colleges and universities)
0
20
40
60
80
100
120
1980–81 1985–86 1990–91 1995–96 2000–01
academic year
students (thousands)
4-yr colleges &universities
2-yr colleges
AP Calculus BC only
BC exams for 2006: ~59,000
Still growing exponentially at over 9%/year (8 year doubling time)
Mainstream Calculus II Enrollments(fall only for 2- & 4- year colleges and universities)
0
20
40
60
80
100
120
1980–81 1985–86 1990–91 1995–96 2000–01
academic year
students (thousands)
4-yr colleges &universities
2-yr colleges
AP Calculus BC only
BC exams for 2006: ~58,000
Still growing exponentially at over 9%/year (8 year doubling time)
Last year, 12,500 students took the BC exam before their senior year. This year it was 13,800.
Calculus Before Grade 12
0
5000
10000
15000
20000
25000
30000
35000
40000
2002 2003 2004 2005 2006
year
number
AB Calc < grade 12
BC Calc, < grade 12
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
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
1955 1957 1959 1961 1963 1965 1967 1969 1971 1973
year
number
AB Calc BC Calc total Calc
AP Calculus
0
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
1973 1975 1977 1979 1981 1983 1985 1987 1989
year
number
AB Calc BC Calc total Calc
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
0
50,000
100,000
150,000
200,000
250,000
300,000
1990 1992 1994 1996 1998 2000 2002 2004
year
number
AB Calc BC Calc total Calc
1995, Graphing calculators1995, Graphing calculators
AP Calculus
0
50,000
100,000
150,000
200,000
250,000
300,000
1990 1992 1994 1996 1998 2000 2002 2004
year
number
AB Calc BC Calc total Calc
1997, New syllabus1997, New syllabus
AP Calculus
0
50,000
100,000
150,000
200,000
250,000
300,000
1990 1992 1994 1996 1998 2000 2002 2004
year
number
AB Calc BC Calc total Calc
2000, 3 non-calculator free response2000, 3 non-calculator free response
AP Calculus
0
50,000
100,000
150,000
200,000
250,000
300,000
1990 1992 1994 1996 1998 2000 2002 2004
year
number
AB Calc BC Calc total Calc
2001, AB subscore for BC exam2001, AB subscore for BC exam
Ratio of AB to BC exams
0
1
2
3
4
5
6
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
year
rati AB/BC
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,
Anti-derivative: The definite integral with variable upper limit is an anti-derivative,
F ' = f ⇒ f x( )dx=F b( )−F a( ).a
b
∫
d
dxf t( )dt
a
x
∫ = f x( ).
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
v t( )dt0
2
∫ =y 2( )−y 0( ); y t( ) =−1+ v x( )dx0
t
∫
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.
1997–98 exams based on new syllabus
•Understand both parts of FTC
y 2( ) =−1+ 1−tan−1 et( )( )dt0
2
∫ =−1.36069
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( )dt0
2
∫ =y 2( )−y 0( ); y t( ) =−1+ v x( )dx0
t
∫
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
