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A 25th Anniversary Retrospective on American High

School Mathematics Education Change We Could

Sometimes Believe In

Dan KennedyBaylor SchoolChattanooga,

TN dkennedy_at_baylorschool.org

Mathematics education in America began humbly. In

the little red school house.

Early technology.

Early school.

Before the 1800s, not many American students

studied any mathematics at all.

What about the famous three Rs?

Reading

Riting

Religion

And, if you wanted to go to college, what you

really needed was Latin. Rithmetic didnt 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

Williams 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.

Technology

Culture

Mental Discipline

Research(College Prep)

QuantitativeLiteracy

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

desirableFortunately, 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.

CEEB

Originally, their only real objective was to

validate, through impartial testing, a students

ability to succeed in college.

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 curriculumexcept for the high school

mathematics teachers. There was an active

groupin Chicago, the ChicagoMens 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.

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 MAAs 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 truths 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 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.

1950 The National Science Foundation was

established. Now there would be money to fund

all this introspection.

1951 General Education in School and

College This 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 Mathematics This group

was formed by the CEEB to study the mathematics

needs of todays 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 didnt take a rocket scientist to figure out

what the governments 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 - Improve the school curriculum, preserving

important skills and techniques while providing

students with a deeper understanding of the

mathematics underlying these skills and

techniques - Provide materials for the preparation of

teachers, to enable them to teach the improved

curriculum - Make mathematics more interesting, to attract

more students to the subject and retain them.

Dozens of mathematicians worked with SMSG through

the 1960s 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

- 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

Dream on, math dudes!

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.

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.

u

Unofficial 2009 point

u

2008276,004 exams

2003212,794 exams

u

1993101,945 exams

u

198651,273 exams

u

196710,703 exams

1955285 exams

u

u

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

Cumulative AP Exams Per Student 2001-2004

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

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.

- CONTENT STANDARDS
- Number and Operations
- Algebra
- Geometry
- Measurement
- Data Analysis and Probability

- PROCESS STANDARDS
- Problem Solving
- Reasoning and Proof
- Communication
- Connections
- Representation

- PRINCIPLES
- Equity
- Curriculum
- Teaching

- 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.

(No Transcript)

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

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

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 students

mathematics education as a linear progression of

skills that must be mastered.

Arithmetic

Fractions

Factoring

Equations

Inequalities

Radicals

Geometry

Trigonometry

Proofs

Functions

Calculus

Statistics

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?

That does not count, Miss Nouveau. Put that thing

away.

Dr. Retro, Ive got it!

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, doesnt 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. Lets 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 Cramers Rule

in their state standards. The purpose of

Cramers Rule is to solve systems of linear

equations using determinants.

Recall

So, why would anyone still mandate the teaching

of Cramers Rule?

Example AL, OK, and CT want students to know how

to compute a 3-by-3 determinant.

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 Cramers Rule, go ahead and

teach Cramers 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!

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.

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! Im 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.

AP Calculus Calculator Survey ResultsWhich

graphing calculator did you use?(percent of

students)

Participation and Eligibility Both 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.

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.

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_at_baylorschool.org