Dan KennedyBaylor School

Chattanooga, TN

dkennedy@baylorschool.org

A 25th Anniversary Retrospective on American High School Mathematics Education:

Change We Could Sometimes Believe In

Mathematics education in America began humbly.

In the little red school house.

Early technology.Early school.

Before the 1800’s, not many American students studied any mathematics at all.

What about the famous three R’s?

Reading ‘Riting Religion

And, if you wanted to go to college, what you really needed was Latin.

‘Rithmetic didn’t join the party until people perceived that it was needed. This would take some time.

After all, people did not come to the New World to study mathematics.

There were more important things to be done.

At about the time that Napier was discovering logarithms…

…colonists in Virginia were learning how to grow tobacco.

When Descartes published his famous Discours de la Méthode in 1637…

…the first school in America (in New Amsterdam) was all of four years old.

When Newton published his famous Principia in 1687…

…our ancestors were preparing to fight King William’s War.

While Leonhard Euler was changing the face of mathematics in the Old Country…

In the New World, a country was being born.

The Revolutionary War ended in 1783, the year that Euler died… while sipping tea and playing with his grandchildren.

So mathematics was alive and well, but America had basically been too busy to care.

Schools, however, were gradually spreading, and many of them believed that teaching arithmetic was a good way to develop young minds.

In 1745, Yale instituted an arithmetic requirement for admission.

Hey, it was a step.

Phillips Exeter Academy was founded in 1781 by merchant John Phillips, funded largely by the Gilman family.

The school has come a long way since then, but so have the United States of America.

The Gilmans were involved in both stories. Nicholas Gilman signed the United States Constitution in 1787.

In 1802, the United States Military Academy opened at West Point.

Harvard instituted algebra as an admission requirement in 1820. (Exeter, of course was on it.)

In 1821, the English High School was founded in Boston.

By this time, there was a serious debate brewing over why students needed to learn mathematics.

MATHEMATICS

Culture

QuantitativeLiteracy

Technology

Mental Discipline

Research(College Prep)

By 1857 there were enough teachers to form an organization: the National Teachers Association. This group spawned the National Education Association in 1870.

The college mathematicians, also feeling lonely, formed the American Mathematical Society in 1894.

Almost immediately, both organizations began to look into the American mathematics curriculum.

There were two main issues that both groups felt had to be confronted, particularly in light of the diverse student population in America:

1) High school – college articulation;

2) What mathematics should be taught to whom, how and when.

The first group to tackle the curriculum was the Committee on Secondary School Studies, appointed by the NEA in 1892. They came to be known as the Committee of Ten.

The chairman was Charles W. Eliot,the president of Harvard.

They published reports in 1893and in 1894, recommending a curriculum focused on mentaldiscipline and college preparation.

Much of it is still in place today, at least in mathematics.

In 1899 the NEA appointed the Committee on College Entrance Requirements, including members recommended by the AMS.

They recommended less drilland more emphasis on logicalstructure, making connections,and solving problems.

In 1915, college professors formed the Mathematical Association of America, which would concentrate more on teaching and less 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.

It also came to the following conclusion about the mathematical needs of college-bound students and students headed straight to the workplace:

“The separation of prospective college students from the others in the early years of the secondary school is neither feasible nor desirable…Fortunately, there appears to be no real conflict of interest between those students who ultimately go to college and those who do not, so far as mathematics is concerned.”

Since 1923, that philosophy has prevailed in the mainstream of American education.

Another group that would extend the influence of the colleges on the high school curriculum came along in 1901:

The College Entrance Examination Board.

Originally, their only real objective was to validate, through impartial testing, a student’s ability to succeed in college.

CEEB

The first CEEB tests were essay-type achievement tests in various subject areas, aligned with the 1923 NCMR report, like this 1928 exam in Elementary Algebra.

The first Scholastic Aptitude Test was given in 1926. The SAT-V and SAT-M structure began in 1930.

By this time there was an organization for just about everyone interested in the high school mathematics curriculum…except for the high school mathematics teachers.

There was an active groupin Chicago, the ChicagoMen’s Mathematics Club.

In 1920 they became the first charter members of a new corporation:

The National Council of Teachers of Mathematics.

Another group, the Association of Teachers of Mathematics in the Middle States and Maryland, had been publishing a journal called the Mathematics Teacher since 1908.

NCTM took it over in 1921, and today it is one of the most powerful voices in education at any level.

So, everyone was organized.

Everyone was also worried about mathematics education, and almost everyone had written or read a report about it. 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.

Mathematical historian E.T. Bell wrote the following sober assessment in a 1935 article in the MAA’s American Mathematical Monthly:

“It must now be obvious, even to a blind imbecile, that American mathematics and mathematicians are beginning to get their due share of those withering criticisms, motivated by a drastic revaluation of all our ideals and institutions, from the pursuit of truth for truth’s sake to democratic government, which are only the first, mild zephyrs of the storm that is about to overwhelm us all.”

Reform was badly needed, but the United States was, unfortunately, again 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.

From the 1923 NCMR report until the end of World War II, the main evolutionary force in American mathematics was in the direction of making it more socially useful.

Of course, there was still considerable confusion about how this was to be done.

A new day, however, was about to dawn…

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.

1950: The National Science Foundation was established. Now there would be money to fund all this

introspection.

1951: General Education in School and CollegeThis was an offspring of the Harvard Report that came from the faculties of Exeter,

Andover, Lawrenceville, Harvard, Yale, and Princeton.

It was notable for the following quote:

“No subject is more properly a major part of secondary education than mathematics. None has a more distinguished history or a finer tradition of teaching. Perhaps the very excellence of the topic has helped, in recent decades, to keep the content and order of its teaching largely unexamined. One of the most remarkable of our sessions was the one in which we consulted with a group of first-rate school and college teachers of mathematics and discovered, as the evening progressed, a very high degree of consensus on the view that school offerings in mathematics are ready for drastic alteration and improvement.”

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.

1955: The Commission on MathematicsThis group was formed by the CEEB to study “the mathematics needs of today’s American youth.” Its report did not come out until 1959, but its deliberations

greatly influenced other committees along the way.

This group specifically addressed the curriculum for college-bound secondary school students, deemed by the colleges to be the critical group most needy of educational reform.

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.

Although the reforms were well underway in mathematics education by October of 1957, they took on a new urgency in America 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.

E. 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.

Dozens of mathematicians worked with SMSG through the 1960’s to write material. In time, the SMSG pilot textbooks were replaced by books from mainstream publishers, often from the same authors.

There were other “reform projects” with similar goals and similar materials (not all of them in mathematics), but SMSG was certainly the biggest.

The “New Math” had arrived!

Many here 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.” He felt that the reformers has replaced the “fruitful and rich essence of mathematics with sterile, peripheral, pedantic details.”

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

It also did not help that a great many people had no understanding or appreciation of the “new” parts of the New Math.

Some authors tried to explain it to the masses, but their efforts were clearly doomed.

Even before blogs and talk radio, the New Math became a hot-button topic.

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.

Here is what they saw…

“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 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 this did not happen.

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

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.

Under the direction of Heinrich Brinkmann of Swarthmore College, the AP Mathematics Committee decided that the only mathematics course worth of the AP designation would be a full-year course in calculus.

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.

0

50000

100000

150000

200000

250000

300000

350000

1950 1960 1970 1980 1990 2000 2010 2020

1955285 exams

196710,703 exams

198651,273 exams

1993101,945 exams

2003212,794 exams

2008276,004 exams

Unofficial 2009 point

Once upon a time there were 11 AP courses.

One of them was in mathematics.

Today there are 37 AP exams in 20 subject areas.

Three of them are in mathematics.

Number of AP Exams Taken Per Student in May, 2004

Number of Exams Frequency Percent

1 630,198 57.2

2 273,282 24.8

3 121,411 11.0

4 49,852 4.5

5 18,475 1.7

6 6,061 0.6

7 1,849 0.2

8 478 0.0

9 133 0.0

10 44 0.0

More than 10 19 0.0

Cumulative AP Exams Per Student 2001-2004

Number of Exams Frequency Percent

1 530,411 48.1

2 245,248 22.3

3 134,252 12.2

4 77.946 7.1

5 47,039 4.3

6 28,106 2.6

7 17,340 1.6

8 9,903 0.9

9 5,509 0.5

10 3,015 0.3

11 1,471 0.1

12 742 0.1

13 412 0.0

14 165 0.0

More than 14 162 0.0

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.

What effect is this AP scramble having on the students?

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!

“Currently, the greatest growth in the high school curriculum is in courses that have traditionally been taught in colleges.

“The greatest growth in the college curriculum is in courses that have traditionally been taught in high schools.

“It is not clear that either institution is serving its clients very well.”

--Dr. Bernard Madison, Chair of the MAA Task Force on Articulation, 2002

But back to our history…

Buoyed by their success with the College admission exams and the AP program, the CEEB (which had now become simply the College Board) sought to clarify the secondary curriculum with another college study. It came out in 1983.

The basic competencies for mathematics were…

• The ability to perform, with reasonable accuracy, the computations of addition, subtraction, multiplication, and division using natural numbers, fractions, decimals, and integers.

• The ability to make and use measurements in both traditional and metric units.

• The ability to use effectively the mathematics of:− integers, fractions, and decimals;− ratios, proportions, and percentages;− roots and powers;− algebra;− geometry

• The ability to make estimates and approximations, and to judge the reasonableness of a result.

• The ability to formulate and solve a problem in mathematical terms.

• The ability to select and use appropriate approaches and tools in solving problems (mental computation, trial and error, paper-and-pencil techniques, calculator, and computer

• The ability to use elementary concepts of probability and statistics.

Ironically, it was that very same year, 1983, that another document was published, 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.

In 1985, Phillips Exeter began this conference, which has helped to write our history for 25 years:

25th Anja S. Greer Conference on Secondary School Mathematics Science and Technology

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.

PRINCIPLES:

•Equity

•Curriculum

•Teaching

CONTENT STANDARDS:

•Number and Operations

•Algebra

•Geometry

•Measurement

•Data Analysis and Probability

PROCESS STANDARDS:

•Problem Solving

•Reasoning and Proof

•Communication

•Connections

•Representation

•Learning

•Assessment

•Technology

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.

Meanwhile, the technology principlehad taken on a life of its own.

Indeed, technology in 1989 was about to change the entire landscape of mathematics education. The graphing calculator entered the market, and suddenly anybody could do what we once thought was higher mathematics.

The main catalyst for change in high school mathematics in recent years has been technology.

The passing of log tables and slide rules are obvious consequences.

Other changes have been more subtle.

Graphing calculators have brought the power of visualization to young students of mathematics.

1991: After much deliberation and careful study, the AP Calculus committee announced that graphing calculators would be required for the exam in 1995.

AP teachers would have four years to make the transition to Calculus for the New Century.

Incredibly, they actually did.

Technology Intensive Calculus for Advanced Placement (TICAP) was the launching pad.

John Kenelly Clemson University

TICAP training sessions were held after the AP Readings in 1992, 1993, and 1994. Every participant got free graphing calculators and textbooks.

TICAP graduates went on to conduct AP workshops across the country, exposing more and more teachers to the power of visualization for teaching AP Calculus.

And many of those teachers alsotaught other math courses!

Graphing calculators have liberated students, teachers, and real-world textbook problems from the tyranny of computation.

Graphing calculators have made more meaningful data analysis accessible to young students of mathematics

Data Shown in the table below is the population growth for the cities of Raleigh, NC and Mesa, AZ, using census numbers for 1980, 1990, and 2000, and estimates for 2004.

Year Raleigh Mesa 1980 150,255 152,404 1990 207,951 288,091 2000 282,956 397,776 2004 326,653 437,454

(a) Using a graphing calculator, find quadratic models for both populations as functions of time. (Use t = 0 for 1980, t = 10 for 1990, and so on.) Notice that quadratic models fit the data very well in both cases. (b) Graph the quadratic functions. The graphs suggest that the two cities will eventually have the same population. In approximately what year will this occur? (c) Discuss some reasons why this prediction is probably not very reliable.

Graphing calculators have made word problems more accessible to students. The emphasis has shifted much more toward modeling.

An example of a problem that used to be hard for students but that now is easy:

Three families order lunch at a fast food restaurant. The Jacksons pay $19.40 for 5 hamburgers, 3 small fries, and 5 soft drinks. The Garcias pay $11.05 for 3 hamburgers, 2 small fries, and 2 soft drinks. The Lorenzos pay $21.25 for 6 hamburgers, 4 small fries, and 3 soft drinks. How much would a person pay at this restaurant for one burger, one small order of fries, and one soft drink?

5 3 5 19.40

3 2 2 11.05

6 4 3 21.25

h f d

h f d

h f d

The former paradigm:

Learn the mathematics in a context-free setting, then apply it to a section of “word problems” at the end of the chapter.

In 2000, the BC Calculus exam had two lengthy modeling problems about an amusement park.

They appeared consecutively.

Nobody complained

…much.

For teachers, changes wrought by technology have not come easily.

We have made changes, hopefully for the better.

You might think we could pause, reflect, and enjoy what we have accomplished.

But history shows that we cannot.

As I look in my crystal ball, here are the changes that I see coming, many of them to be enabled by technology…

We need to stop thinking of a student’s mathematics education as a linear progression of skills that must be mastered.

Arithmetic Fractions Factoring

Equations Inequalities Radicals

Geometry Trigonometry

FunctionsCalculus Statistics

Proofs

If students who have not mastered our traditional mathematics skills can solve problems with technology, should it be our role as mathematics teachers to prevent them, or even discourage them, from doing so?

Dr. Retro, I’ve got it! That does not count, Miss

Nouveau. Put that thing

away.

We ALL must teach fundamental skills to our students, who probably will not have mastered them.

Patiently. Casually. As a matter of course.

Mr. Oiler, if there are twice as many dogs as

cats, doesn’t that mean that 2d = c?

Mr. Jones, if that is all you learned last year, you had better drop this course before it

drops you.

Good question, Mr. Jones. Let’s see what would happen if there

were 4 cats…

We must honestly confront the goals of our current mathematics curricula.

Just because it is good mathematics does not mean that we have to keep teaching it.

Nor is it necessary, advisable, or perhaps even possible to teach everything that is in your textbook.

Example:

AZ, OK and MA still have Cramer’s Rule in their state standards.

The purpose of Cramer’s Rule is to solve systems of linear equations using determinants.

Recall:

So, why would anyone still mandate the teaching of Cramer’s Rule?

Example:

AL, OK, and CT want students to know how to compute a 3-by-3 determinant.

2 1 1

1 4 2

1 1 0

2 1

1 4

1 1

+ + +– – –

0 + 2 + 1 – (–4) – (–4) – 0 = 11

Compare this to:

So how do we justify teaching a meaningless computational trick that is ONLY good for computing 3-by-3 determinants?

It does not generalize to higher orders.

It does not even suggest anything important about how determinants work!

We should treat every mathematics course as a history course – at least in part.

We will probably always teach some topics for their historical value.

In fact, if you love Cramer’s Rule, go ahead and teach Cramer’s Rule.

Just admit to your students that you are teaching it for its historical value.

Do not make them use it to solve simultaneous linear equations!

;

e b a e

f d c fx y

a b a b

c d c d

Cramer Himself

We must honestly assess every advance in technology for its appropriate uses in the classroom.

As noted before, we must also determine what is meant by important mathematics.

2 4

2

b b ac

a

Important?

Expendable?

The Skandu 2020:

It has the potential to scan any “standard” algebra textbook problem directly into its memory for an analysis of key instructional words, solve it with CAS, and display all possible solutions.

It will do the same for “standard” geometry textbook proofs.

The Skandu 2020

(Not its real name)

HA HA! I’m only kidding.

At least for now.

If there is no Skandu 2020 in our classrooms in five years, I doubt it will be because the design is impossible.

It will be because teachers do not feel that it would improve the teaching and learning of important mathematics.

  

Calculus ABCalculus AB Calculus BCCalculus BC

20022002 20032003 20042004 20052005 20062006 20022002 20032003 20042004 20052005 20062006

Casio 6300, Casio 6300, 7300, 7400, 7300, 7400, 7700; TI 73, 80, 7700; TI 73, 80, 8181

1.01.0 1.11.1 0.90.9 0.60.6 0.50.5 0.60.6 0.70.7 0.70.7 0.40.4 0.50.5

Casio 9700, Casio 9700, 9800; 9800; Sharp 9200, Sharp 9200, 9300; TI 9300; TI 82, 8582, 85

6.66.6 3.83.8 2.42.4 1.41.4 1.01.0 4.54.5 2.52.5 1.41.4 0.80.8 0.50.5

Casio 9750, Casio 9750, 9850, 9860, FX 9850, 9860, FX 1.0; Sharp 1.0; Sharp 9600, 9900; TI 9600, 9900; TI 83, 83 Plus, 83 83, 83 Plus, 83 Plus Silver, 84 Plus Silver, 84 Plus, 84 Plus Plus, 84 Plus Silver, 86Silver, 86

74.174.1 75.775.7 76.976.9 79.579.5 79.979.9 66.166.1 67.467.4 68.268.2 70.570.5 70.870.8

Casio 9970, Casio 9970, Algebra FX 2.0; Algebra FX 2.0; HP 38G, 39, HP 38G, 39, 40G, 48, 49; TI 40G, 48, 49; TI 89, 89 Titanium89, 89 Titanium

17.217.2 18.218.2 18.318.3 17.917.9 18.218.2 28.128.1 28.728.7 28.728.7 27.927.9 27.927.9

OtherOther 1.11.1 1.21.2 1.41.4 0.60.6 0.50.5 0.80.8 0.70.7 1.01.0 0.30.3 0.30.3

AP Calculus Calculator Survey ResultsWhich graphing calculator did you use?

(percent of students)

Participation and EligibilityBoth AMC 10 and AMC 12 are 25-question, 75-minute multiple-choice contests administered in your school by you or a designated teacher. The AMC 12 covers the high school mathematics curriculum, excluding calculus. The AMC 10 covers subject matter normally associated with grades 9 and 10. To challenge students at all grade levels, and with varying mathematical skills, the problems range from fairly easy to extremely difficult. Approximately 12 questions are common to both contests. Students may not use calculators on the contests.

AMC 12 / AMC 10: American Mathematics Competitions

Meanwhile, the CAS conversations continue.

They are not just about technology, nor should they be. They are about the teaching and learning of mathematics.

Stay tuned. Be informed. Join the conversation.

Is it another phase of our history? Time will tell.

THE NATIONAL COUNCIL OFTEACHERS OF MATHEMATICS

A History ofMathematics Education

in the United States and Canada

A major source for the early history in this talk was the 32nd yearbook of NCTM, published in 1970:

A History of Mathematics Education in the United States and Canada.

dkennedy@baylorschool.org