Dan Kennedy Baylor School Chattanooga, TN

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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
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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.
3
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.
4
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.
5
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.
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When Newton published his famous Principia in
1687
our ancestors were preparing to fight King
Williams War.
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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.
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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.
9
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.
10
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.
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Technology
Culture
Mental Discipline
Research(College Prep)
QuantitativeLiteracy
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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.
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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.
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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.
15
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.
16
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.
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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.
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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.
19
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.
20
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.
21
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.
22
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.
23
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.
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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.
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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.
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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
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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.
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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
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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.
30
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.
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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.
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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.
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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.
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• 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.

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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!
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Many here probably remember the New Math
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• 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

38
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.
39
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
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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.
41
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
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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.
43
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.
44
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
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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.
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Number of AP Exams Taken Per Student in May, 2004
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Cumulative AP Exams Per Student 2001-2004
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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.
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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!
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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
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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
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• 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.

53
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
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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.
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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.
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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.
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• 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

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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.
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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.
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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.
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Graphing calculators have brought the power of
visualization to young students of mathematics.
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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.
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Technology Intensive Calculus for Advanced
Placement (TICAP) was the launching pad.
John Kenelly
Clemson University
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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!
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Graphing calculators have liberated students,
teachers, and real-world textbook problems from
the tyranny of computation.
67
Graphing calculators have made more meaningful
data analysis accessible to young students of
mathematics
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Graphing calculators have made word problems more
accessible to students. The emphasis has shifted
much more toward modeling.
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An example of a problem that used to be hard for
students but that now is easy
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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.
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In 2000, the BC Calculus exam had two lengthy
modeling problems about an amusement park. They
appeared consecutively. Nobody complained much.
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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
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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
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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!
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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?
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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
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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.
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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?
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Example AL, OK, and CT want students to know how
to compute a 3-by-3 determinant.






0
2
1
(4)
(4)
0
11
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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!
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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.
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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
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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?
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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)
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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.
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AP Calculus Calculator Survey ResultsWhich
graphing calculator did you use?(percent of
students)
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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.
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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.
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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.
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dkennedy_at_baylorschool.org