AP Mathematics – An Idea
Whose Time Has Come Again
Dan Kennedy
Baylor School
NCTM Philadelphia
April 2012
Everyone is talking about the Common
Core State Standards these days.
Some are excited.
Some are skeptical.
Many are terrified.
But is it really a new thing?
This is not the first attempt of the (mostly
collegiate) mathematics community to define
the high school curriculum.
In 1899 the NEA appointed the Committee on
College Entrance Requirements, including
mathematicians recommended by the AMS.
They recommended less drill and
more emphasis on logical structure,
making connections, and solving
problems. (Sound familiar?)
In 1915, college professors formed the
Mathematical Association of America, which would
concentrate more on teaching than on research.
They promptly formed a committee to study the
American high school curriculum.
The MAA formed the
National Committee on
Mathematics
Requirements in 1916.
They published their
report in 1923.
This was to stand as the
definitive study for more
than three decades!
Among other things, it
gave us the unifying idea
of functions.
So the college professors were constantly trying to
tell the American teachers how to prepare students
for their mathematical futures.
Nonetheless, mathematics education was not going
very well in the actual schools.
This led everyone to complain about it.
In other words, it was a lot like today.
Math,
yuck
Math
sucks.
The percentage of high school students taking
algebra declined steadily from 56.9% in 1910 to
24.8% in 1955.
In that same period, the percentage taking geometry
declined from 30.9% to 11.4%.
Many schools could not have taught more
mathematics if they had wanted to. As late as 1954,
only 26% of schools with a twelfth grade even offered
trigonometry.
College preparatory mathematics was hanging on in
enough schools to keep the colleges fed, but it was
available to a dwindling proportion of students.
Reform was badly needed, but the United States
was, unfortunately, too busy to deal with it.
World War I
Depression
World War II
While these events did delay education reform, they
also served to convince many people that American
mathematics education mattered to their welfare.
So things began to happen fast after the war.
1945: The Harvard Report
This report emphasized college
preparatory mathematics, although it was
also big on its cultural value. Not much
attention was paid to the non-college-bound.
1944-47: The Commission on Post-War Plans
This NCTM report gave the mathematics
education reaction to other reports. It was
more specific about content and pedagogy,
and it paid more attention to psychology and
student development.
1951: The University of Illinois Committee on
School Mathematics (UICSM)
“The progenitor of all current curriculum
projects in mathematics” was funded by the
Carnegie Foundation, the NSF, and the
USOE. It created curricula and materials,
field-tested them, and refined them. It had
great credibility among all the professional
organizations, and it showed how change
could actually be effected.
1958: The School Mathematics Study Group (SMSG)
This group, the culmination of ten years of
simmering reform, was formed by
mathematicians. Every set of professional
initials was in on it: AMS, MAA, NSF,
NCTM, etc. They had the minds, and they
had the money.
Quite unexpectedly, they also had the full
attention of the American people.
In October of 1957, mathematics education reform
took on a new urgency when the Soviet Union
launched Sputnik I into orbit.
It didn’t take a rocket scientist to figure out what the
government’s new priority would be:
rocket scientists!
And rocket scientists needed to know mathematics.
Edward G. Begle of Yale directed the work of
SMSG. He cited three goals:
1. Improve the school curriculum,
preserving important skills and
techniques while providing
students with “a deeper
understanding of the mathematics underlying
these skills and techniques”;
2. Provide materials for the preparation of
teachers, to enable them to teach the improved
curriculum;
3. Make mathematics more interesting, to attract
more students to the subject and retain them.
Many of you probably remember the New Math…
Theorem: (b + c) + (–c) = b
Statement
1.
2.
3.
4.
5.
6.
7.
8.
9.
b and c are real numbers
b + c is a real number
–c is a real number
(b + c) + (–c) = b + [c + (–c)]
c + –c = 0
b + [c + (–c)] = b + 0
b+0=b
b + [c + (–c)] = b
 (b + c) + (–c) = b
Reason
Hypothesis
Axiom of closure for addition
Axiom of additive inverses
Associative axiom of addition
Axiom of additive inverses
Substitution principle
Additive axiom of 0
Transitive property of equality
Transitive property of equality
There were critics from the start.
Morris Kline, a mathematician
and author himself, called it
“wholly misguided” and
“sheer nonsense.”
Other, less polemical critics concentrated on three
shortcomings:
•Disregard of the purposes of secondary education
•Neglect of important concomitant outcomes (e.g., the
ability to solve real-world problems)
•Neglect of differential needs of various pupil groups
Undaunted, the mathematicians continued to meet,
and the NSF continued to pick up the tab.
The Cambridge Conference in 1962 convened 25
mathematicians to discuss where the reforms would
eventually lead. W. T. Martin (MIT) and Andrew
Gleason (Harvard) chaired the committee.
Their 1963 report, Goals for School Mathematics,
tried to look ahead thirty years.
Based on the early commitment to
SMSG reforms in high school, here
is what they saw for 1993…
“A student who has worked
through the full thirteen years of
mathematics in grades K to 12
should have a level of training
comparable to three years of toplevel college training today; that
is, we shall expect him (sic) to
have the equivalent of two years
of calculus, and one semester
each of modern algebra and
probability theory.”
Dream on, math
dudes!
There are many reasons why the predictions of the
Cambridge Conference did not come to pass.
One of them began in 1954 with the report of the
School and College Study of Admission with
Advanced Standing.
This was a task force, funded
by the Ford Foundation,
charged with coming up with
an equitable way to award
credit and/or advanced
standing to students who had
done college-level work in
high school.
Kenyon College
While other subject areas designed courses
that were advanced versions of senior high
school courses, Swarthmore professor
Heinrich Brinkmann, chairman of the
Mathematics committee, rejected
precalculus as a course worthy of advanced
standing.
The committee decided that
the course would be a
full year of calculus.
Product
Placement
In 1955 this program was taken over by the
Committee on Advanced Placement of the
College Entrance Examination Board.
It became, of course, the Advanced
Placement program.
Originally there were only
11 AP courses, one of which
was called AP Mathematics.
The exams included more
pre-calculus mathematics
than they do today.
For example, this was Problem #2 (of 7) on
the 1965 test in “AP Mathematics”:
An ellipse has its center at the origin and
its foci F and G on the x-axis. The point M
(4, 3) lies on the ellipse and angle FMG is a
right angle. Find an equation of the ellipse.
Would AP students do well on this today?
In 1969, AP Calculus became two courses: AP
Calculus AB and AP Calculus BC.
The phenomenal growth of
AP Calculus may have done
more to affect the secondary
mathematics curriculum
than any of the previous
reforms.
Of course, there were other
AP subjects as well, and their
impact was also felt.
Nobody at the Cambridge Conference in 1963
would have seen this coming.
Our best students could not possibly accumulate
as much mathematics as they were predicting.
Instead, they would become AP scholars, taking
AP courses in as many subjects as possible.
It is how they would get into their colleges.
So how did AP Calculus (and later AP Statistics)
become so successful while the quality of
mathematics education by every other measure was
declining?
One possible answer: Teachers everywhere were
teaching the same course, with the same goals, and
with the same externally-administered assessment.
Teacher development, course materials, textbooks,
and resources were coherent and focused.
This could not possibly happen with any other
course in the high school mathematics curriculum!
So, how bad were things
elsewhere in the math
curriculum? Well, in
1983 a document was
published that was
destined to change the
rules for high school
academic preparation
for years to come…
A Nation at Risk: The
Imperative for
Educational reform
From A Nation at Risk:
“If an unfriendly foreign
power had attempted to
impose on America the
mediocre educational
performance that exists
today, we might well have
viewed it as an act of war.”
Response to A Nation At Risk was immediate,
reminiscent of the post-war angst that led to the
New Math.
NCTM had published An
Agenda for Action in 1980. It set
into motion the movement that
would result in the Standards in
1989.
Another 1989 document,
Everybody Counts from the
National Research Council,
sought to mobilize the public.
And, of course, in 1989
NCTM published
Curriculum and
Evaluation Standards for
School Mathematics,
continuing the long
tradition of the
American mathematics
community trying to
boost its own
educational standards.
NCTM worked long and hard on the Standards,
hoping to produce national standards for a
country averse to national standards.
Perhaps their greatest successes were raising
teacher awareness of equity, assessment, problemsolving, and representation.
A major update and condensation
was published in 2000: Principles and
Standards of School Mathematics.
In 2001, Congress and President George W. Bush
rolled out the No Child Left Behind Act.
Although this did not seek to redefine high school
mathematics, the emphasis on standardized testing
and accountability did have some unintended
consequences on how mathematics was taught. This
has gradually eroded its credibility.
But, despite its flaws, NCLB has focused
everyone’s attention on one paradox of local
accountability: If you try to cure a national
problem with local punishments, who is going to
admit to having a problem?
Thus it was that the National Governors
Association decided to face their educational
shortcomings together, beginning with the
adoption of Common Core Standards in English
language and mathematics.
As of today, 48 of the 50 states
have joined the Common Core
initiative, and all but 4 of those
have formally adopted the
standards.
By 2015, the adopting states expect
to base 85% of their curricula on
the common core.
So what are they? And how do
they differ from all the movements
we have seen that have preceded
them?
What’s not so different:
1. Many of the ideas come from college professors.
2. College readiness is still the primary goal.
3. Drill is bad; conceptual understanding is good.
4. Much is added; little is subtracted.
5. Teachers may not be ready to handle it. Yet.
6. Same goes for the students. Yet.
7. As with all the previous ideas, it will take about
ten years to put it into perspective.
What IS different this time:
1. Everyone will be teaching the same stuff.
2. The textbooks must have the same goals.
3. There will be an external assessment, so it’s
serious.
4. Teachers can collaborate. The AP model!
5. Quantitative literacy is a genuine goal.
6. This could actually end the hugely counterproductive math wars. (Sooner or later.)
Whatever happens, the CCSS initiative will be a
success if everyone buys into the eight Standards
for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics
5. Use appropriate tools strategically.
6. Attend to precision
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
The Mathematical Practice standards are based
on the NCTM Process Standards from 1989 and
2000:
•Problem solving
•Reasoning and proof
•Communication
•Connections
•Representation
The CCSS also seek to give focus and coherence to
the K--12 mathematics curriculum by identifying
six Conceptual Categories:
•Number and Quantity
•Algebra
•Functions
•Modeling
•Geometry
•Statistics and Probability
The Conceptual Categories are similar to the
NCTM Content Standards:
•Number and Operations
•Algebra
•Geometry
•Measurement
•Data Analysis and Probability
If the CCSS authors hoped to address the apt
criticism of the mathematics curriculum as a
mile wide and an inch deep, they apparently
did not do this by eliminating topics. Most
algebraic and geometric topics are still there,
along with additional clusters in statistics and
probability throughout.
The Mathematical Practices also
dictate a less efficient, more reflective
method of teaching. It takes TIME to
explore, think, understand, critique,
prove, and make connections!
So what did they hope to cut? Redundancy!
For example, they assume that a student raised
on CCSS mathematics will be well-acquainted
with linear functions before 9th grade: graphs,
slope, proportionality, modeling, and (yes) linear
association of bivariate data.
Schools have tried to cut redundancy through
curriculum mapping, but that’s a local remedy
for a national problem.
With CCSS, it just might work!
It is past time to honestly confront the goals of
K—12 mathematics education.
If we want understanding, we might have to let go
of mastery. We must all teach the Big Ideas!
The same Big Ideas can be
taught every year, but not by
teaching the same details. The
Common Core assumes that
students will progress.
With that assumption, we can
eventually make our textbooks
smaller!
Now…what are the implications for the AP
program?
I think they will be profound!
Most obviously,
when it comes to measuring
success in high school
mathematics, there is a new
Sheriff in town.
CCS
Here’s a provocative thought…
Compared to a successful
AP score, a successful
CCSS score is likely to be:
1) More important to the student;
2) More important to the school;
3) More predictive of colleges success;
4) Harder to achieve (at least at first).
The truth is that the narrow goal
of “calculus preparation” has had
a stifling effect on the high school
curriculum for many years. A goal
of “Quantitative Literacy” for all
students would require a different
set of priorities.
So now we have some.
Lynn Steen, a strong
proponent of QL,
worried 15 years ago
about what he called
“the teleological
influence of calculus”
on secondary
mathematics education.
The AP Teleological
Effect is doubled:
AP Calculus drives a
more focused precalculus preparation.
Taking more AP courses
drives algebra into the
lower grades.
What happens as
students scramble to
take more AP courses?
On the one hand, they
are condensing or
skipping foundational
courses, so they are
less prepared for
advanced courses.
On the other hand, they are taking more
advanced courses, assuring that their lack of
preparation will be exposed.
“…a high school calculus course should not be
the be-all and end-all of mathematics, nor
should it be the only transition path from high
school to college mathematics. High school
mathematics should prepare students not just
for further, specialized study in mathematics
but also for the variety of STEM careers and
other professions that will be open to them in
the future.”
-- J. Michael Shaughnessy,
President, NCTM, April 2012
“… too many students short-change their
preparation in algebra, geometry,
trigonometry, and other mathematical topics
in order to stay on a fast track to calculus.
Too many otherwise talented students arrive
at university without the mathematical
foundation that is needed to succeed in the
college-level mathematics required for their
intended major.”
-- David Bressoud
Past-President, MAA, April 2012
“Although calculus can play an important
role in secondary school, the ultimate goal of
the K–12 mathematics curriculum should not
be to get students into and through a course
in calculus by twelfth grade but to have
established the mathematical foundation that
will enable students to pursue whatever
course of study interests them when they get
to college.”
-- Joint MAA-NCTM Statement on High School Calculus
April 2012
College Mathematicians want all their incoming
students to have:
Deeper understanding
An appreciation of math
Mathematical literacy
Confidence in their mathematical ability
No rush through foundational courses
College Mathematicians want our best high
school students to have:
Deeper understanding
More proofs
Harder problems
Challenges like the AMC Competitions (which
deliberately avoid calculus)
No rush through foundational courses
Potential employers want job-seekers to have:
Mathematical literacy
Statistical savvy
Self-confidence
Creativity
The ability to solve real-world problems
No rush through foundational courses
By all outward appearances, the CCSS
are designed to please all these
constituencies.
If they can succeed.
Can the AP program help?
I believe they can help a lot!
By bringing back AP Mathematics.
As AP and pre-AP have grown, the
College Board has promulgated them
more and more as an ideal preparation
for college and beyond. Good point.
But is “advanced placement” really
part of that goal?
Or do the Common Core Standards in
mathematics have a better vision? Recall:
Deeper understanding
An appreciation of math
Mathematical literacy
Confidence in their mathematical ability
No rush through foundational courses
So why not have an
AP Mathematics program that
embraces (and assesses, and
professionally develops) all
of the Common Core topics and practices?
It would emphasize the top-end material,
and it could still hold AP students to a higher
standard. But it would encourage them to
have a broader mathematical background!
Think of the advantages:
•Consistent national goals
•Better-prepared students
•Less tracking for acceleration
•Mathematics more like other APs
•Rewards a good mathematical foundation
•College Board professional development
•College Board assessment expertise
More advantages:
•Employers should be happy
•College professors should be happy
•Secondary teachers should be happy
•Students should be happier
•Common Core will have a chance
•College Board will benefit in every way
Potential downsides:
•A lot of work for ETS and College Board
•Fewer students in HS calculus courses
•Will AP Mathematics lack gravitas?
•How will college departments react?
Questions to ponder:
•Should AP Mathematics
include some AB Calculus
and/or AP Statistics?
•What other AP courses in the
Mathematical Sciences should
survive?
•Does AP need a new name?
The College Board has a history of
preserving acronyms while changing
names to fit the times. Take the SAT:
Scholastic Aptitude Test
Scholastic Assessment Test
SAT Reasoning Test
If AP courses are really just used to
enhance a student’s chance for
college admission, how about:
The College Board’s
Admission Possible ©
Testing Program?
If AP courses are really the ideal
preparation for college and beyond,
how about:
The College Board’s
Academic Preparation ©
Testing Program?
If AP is really a predictor of
college achievement, how about:
The College Board’s
Academic Potential ©
Testing Program?
If we want to emphasize AP as the
uniquely American answer to the
International Baccalaureate, how
about:
The College Board’s
American Preparatory ©
Program?
Note that my record as a prophet is not all
that good.
I predicted in 1992 that
AP Calculus would be
replaced by AP Mathematics
within 10 years.
I thought that the impetus
would come from the
colleges and universities!
I had predicted that colleges
would be embarrassed to let
their students graduate
without knowing how to use
half the mathematical power
of their graphing calculators.
Apparently, they were not.
In fact, they still aren’t.
They did reform college Calculus for the new
century … and AP responded accordingly.
Then, AP led the way with an introductory
Statistics course for the new century … and the
colleges responded accordingly.
Now the Common Core State Standards have
redefined pre-college mathematics for the new
century.
How will AP respond?
dkennedy@baylorschool.org