AP Mathematics – An Idea Whose Time Has Come Again Dan Kennedy Baylor School NCTM Philadelphia April 2012.

AP Mathematics – An Idea AP Mathematics – An Idea Whose Time Has Come Whose Time Has Come

AgainAgain

Dan KennedyDan KennedyBaylor SchoolBaylor School

NCTM PhiladelphiaNCTM PhiladelphiaApril 2012April 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

Mathsucks.

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 ReportThis 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 PlansThis 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.

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.

Edward G. Begle of Yale directed the work of SMSG. He cited three goals:

Many of you probably remember the New Math…

Theorem: (b + c) + (–c) = b Statement Reason 1. b and c are real numbers Hypothesis 2. b + c is a real number Axiom of closure for addition 3. –c is a real number Axiom of additive inverses 4. (b + c) + (–c) = b + [c + (–c)] Associative axiom of addition 5. c + –c = 0 Axiom of additive inverses 6. b + [c + (–c)] = b + 0 Substitution principle 7. b + 0 = b Additive axiom of 0 8. b + [c + (–c)] = b Transitive property of equality 9. (b + c) + (–c) = b 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 toSMSG 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 top-level 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, problem-solving, and representation.

A major update and condensation was published in 2000: Principles andStandards 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 counter-productive 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

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!

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.

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 schoolmathematics, there is a newSheriff 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 pre-calculus 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 BressoudPast-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 understandingAn appreciation of mathMathematical literacyConfidence in their mathematical abilityNo 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 thatAP Calculus would be replaced by AP Mathematicswithin 10 years.

I thought that the impetuswould come from thecolleges 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?

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