As you are probably aware, there are those mathematicians and mathematics
educators who are opposed to what has been proposed by NCTM.  One group,
Mathematically Correct, has some pretty distinquished members -- including a
former president of the Mathematical Association of America and  Frank Allen,
a past president of NCTM!
 
I just returned from the NCTM Annual Meeting in San Diego where I saw evidence
of the opposition.  I thought you would be interested in this statement from
Mathematically Correct which I got over the WWW.  Note that MC has its own Web
site.  (Sorry about the lousey formatting.)
 
Bob
=================
 
 
 
April 20, 1996
 
 
 
FOR IMMEDIATE RELEASE
 
 
 
In 1989 the National Council of Teachers of Mathematics (NCTM) promulgated
 
a set of Standards for mathematics education in the United States.
 
Although these standards have been highly controversial since their
 
introduction, advocate unproven methods of teaching mathematics, and lack
 
true bench marks of student learning, they have had substantial influence
 
on mathematics curricula at all levels.  Many concerned parents and
 
mathematicians view these changes as highly negative.
 
 
 
In conjunction with the national meeting of the NCTM in San Diego,
 
Mathematically Correct of San Diego, in cooperation with other groups,
 
today releases an alternative proposal for the improvement of secondary
 
math education in the United States.  Frank Allen, former President of
 
NCTM wrote this statement and a number of eminent mathematicians, as well
 
as concerned citizens, have endorsed it.
 
 
 
For more information about the changes in mathematics education, as well
 
as links to other sites, please see the 2+2=4 -  - The Mathematically
 
Correct Homepage at
 
 
 
http://ourworld.compuserve.com:80/homepages/mathman/
 
 
 
Michael McKeown
 
for Mathematically Correct
 
 
 
____________________________
 
Mathematically Correct
 
 
 
   A PROGRAM for RAISING the LEVEL of STUDENT
 
   ACHIEVEMENT in SECONDARY SCHOOL MATHEMATICS
 
 
 
           Submitted by Frank Allen
 
 
 
   All students should be given the opportunity
 
   to master academic subject matter
 
   calibrated against world norms.
 
                  National Alliance for Business
 
 
 
Many of the serious problems we face in secondary
 
school mathematics today are due to deteriorating social conditions in the
 
home and community over which our teachers and school administrators have no
 
control, and for which they should not be faulted. In some school districts,
 
"students" attend school sporadically, and are often in no condition to
 
learn when they do attend. Problems arising from this situation should not
 
be attributed to deficiencies in curriculum, teaching or assessment.  We
 
believe that much can be done to raise the level of student
 
achievement in secondary school mathematics without
 
making radical changes in any of these areas. We present our recommendations
 
under four headings.
 
 
 
I.    We state our objectives in terms of the mathematics that we expect the
 
student to learn.
 
 
 
II. We describe a learning situation in which these objectives can be
achieved.
 
 
 
III. We state the essential properties of an effective grading system.
 
 
 
IV. We reestablish the teacher's role as an expositor and the student's role
 
as a learner.
 
 
 
I. Objectives.
 
 
 
A clear statement of objectives in terms of subject matter requires that
 
there be provided, for each course, an accurate description of the
 
mathematics we expect students to learn. These descriptions could take the
 
form of course syllabi of the kind utilized in most of the other
 
industrialized countries in the world. The successful completion of each
 
course should contribute to the student's understanding of the symbolic
 
language of mathematics, and to his ability to employ this language in the
 
construction of valid mathematical arguments (proofs) and in the solution of
 
problems.
 
 
 
This is not the place for a detailed description of course syllabi. We
 
believe that they should be consistent with our traditional curriculum in
 
secondary school mathematics, with which we are reasonably content. We
 
insist that these course syllabi be comparable with those employed in other
 
countries where students have reached high levels of achievement. To
 
insure this we specify that these syllabi shall be prepared by committees of
 
mature, well-established mathematicians appointed on a regional or
 
preferably national basis, with some input from experienced teachers of
 
secondary level mathematics.
 
 
 
II. Description of a Learning Situation in Which Our Objectives Can Be
Achieved.
 
 
 
1. Differences in learning rates must be recognized and provided for. In
 
most secondary schools the student population presents wide variations in
 
student interest, aptitude and training with respect to mathematics and
 
there is no point in pretending otherwise. Schools are to be commended for
 
making curricular provisions for these variations. It is not unusual for our
 
larger high schools to provide as many as six levels of instruction for
 
grade nine. These are not established by faculties bent on making invidious
 
comparisons. It is simply impossible to meet the needs of the entire school
 
population without them.
 
 
 
2. We reassert the principle that the secondary school mathematics
 
curriculum must be organized around its own internal structure, and not
 
around problem solving as the NCTM s "Agenda for Action" requires. There
 
are, of course, many such structures, but all of them are logically
 
consistent in the sense that the prerequisites for each new concept are
 
available when it is introduced. This logical structure is the very
 
essence of mathematics. The idea that mathematics is a hierarchy of
 
propositions forged by logic on a postulational base should begin to form
 
in the student's mind about grade nine and be thoroughly established by
 
grade twelve.
 
 
 
3. The teaching of mathematics should be regarded as an extension of the
 
teaching of language, in which facility in two-way translation between
 
language and mathematical symbolism is emphasized. Efforts to develop an
 
awareness of the intimate relationship which exists between grammar,
 
mathematics and logic should begin with games of "How do we know" in the
 
early grades, continue with the introduction of formal proof not later than
 
grade nine and culminate in the ability to read and write valid essay proofs
 
not later than grade twelve.
 
 
 
This linguistic approach to secondary mathematics has several advantages. It
 
helps the student to develop the gradually formalized natural language so
 
essential for mathematical discourse, and indeed for the expression of
 
rational thought in any field.
 
 
 
It tends to build bridges between mathematics and other subjects. It fosters
 
the idea that reasoning in mathematics has much in common with reasoning
 
elsewhere. This will help to counteract the idea that there is something
 
esoteric and arcane about
 
mathematics. It will help to dispel the notion, already too widely held,
 
that one must have a mathematical mind in order to deal with the peculiar
 
thinking required in mathematics. This idea has served to isolate our
 
subject from the main stream of public consciousness to an extent that we
 
cannot possibly counteract by our current over-emphasis on real world
 
applications.
 
 
 
The two-way translation between language and mathematical symbolism is
 
important for problem solving. First, we express the problem situation in
 
mathematical language (L to M) thus bringing it under the sway of the
 
powerful laws that govern mathematical calculations. Second, we apply these
 
laws to perform these calculations. Here we can use a computer or
 
calculator, but we should understand the nature of the operations being
 
performed. Finally, we must interpret our mathematically expressed result in
 
terms of the original problem (M to L).
 
 
 
4. We must take a balanced view of the role of problem solving in school
 
mathematics, lest our preoccupation with it causes us to fragment and
 
distort the very mathematics that makes problem solving possible.
 
 
 
Problems are the life blood of mathematics. But we must not fail to convey
 
to our students that the body of mathematics is given structure and
 
coherence by the bones and sinews supplied by definitions, postulates and
 
proof. Make no mistake, a person's problem solving ability depends on how
 
much mathematics he understands. Moreover, one of the principal objectives
 
of problem solving in high school is to inculcate a better understanding of
 
the basic mathematical theory. It is this understanding that will enable the
 
student to deal with problems that are today unforeseen and unforseeable. A
 
student who solves a problem has devised a key that will open a specific
 
lock. A student who understands the mathematical theory underlying his
 
solution has a master key that will open many locks. Each problem should be
 
placed in its proper mathematical context by citing the principles used in
 
its solution.
 
 
 
5. We must also try to develop a little more confidence in the idea that
 
mathematics is interesting for its own sake, and that a problem can be
 
interesting, challenging and instructive without being obviously attached to
 
some real world application. These ideas open the door to an appreciation of
 
the recently much neglected cultural and aesthetic values of mathematics.
 
As Davis and Hersch put it, Blindness to the aesthetic values is widespread
 
and can account for the feeling that mathematics is dry as dust, as exciting
 
as a telephone book. Contrariwise, appreciation of this element makes the
 
subject live in a wonderful manner and burn as no other creation of the
 
human mind seems to do. Teachers who have this appreciation should try to
 
transmit it to their students.
 
 
 
6. We must promote the idea that mathematics, properly taught, makes unique
 
and indispensable contributions to the development of the student's ability
 
to think and communicate in a logical manner. For many students the value of
 
this ability, which is by no means confined to the field of mathematics, far
 
transcends the value of the mathematical facts learned. The eminent
 
mathematician Jean A. E. Dieudonne has eloquently expressed the concept that
 
enhancement of the student's ability to think effectively as a major goal of
 
mathematical study. For what good do we seek? Certainly it is not to
 
introduce them (students) to collections of more or less ingenious theorems
 
about the bisectors of the angles of a triangle or the sequence of prime
 
numbers, but rather to teach them to order and link their thoughts according
 
to the methods mathematicians use, because we recognize in this exercise a
 
way to develop a clear mind and excellent judgment. It
 
is the mathematical method that ought to be the object of our teaching, the
 
subject matter being only well-chosen illustrations of it.
 
 
 
7. We need to take a more balanced attitude regarding the building of basic
 
mathematical skills. Jeremy Kilpatrick describes our problem very well in
 
the July 1988 issue of the NCTM's research journal, One of the most
 
venerable and vexing issues in mathematics education today concerns the
 
trade-off between proficiency and comprehension, between promoting the
 
smooth performance of a mathematical procedure and understanding how and why
 
the procedure works. He continues Researchers -- should not dismiss too
 
lightly the question of how and where skill development fits into the school
 
mathematics curriculum. Recent research in cognitive science suggests that a
 
strong knowledge base is needed for problem solving, and surely some of this
 
base should be composed of procedural knowledge. Furthermore, conceptual
 
knowledge both supports and is supported by what Brownell termed 'meaningful
 
habituation,' the almost automatic performance of a routine that is based on
 
understanding.
 
 
 
We want our teachers to feel free to seek this balance on their own,
 
uninhibited by strictures against A tight focus on manipulative facility
 
with which the NCTM Standards are replete, or the notion that technology has
 
supplanted the skills students need.
 
 
 
8. We urge caution in the use of calculators and dynamic geometries.
 
 
 
* Taking time to develop skill in the use of these devices is justified only
 
to the extent that it improves the students' understanding of mathematics.
 
(Thousands of people have learned to use these devices without formal
 
instruction; very few have learned mathematics that way.)
 
 
 
* There is grave danger that calculators, used too early, can seriously
 
impair the student's ability to perform and understand simple calculations.
 
The need for this understanding is not obviated by the availability of
 
calculators and computers.
 
 
 
* The use of dynamic geometries tends to blur the distinction between
 
illustration and proof. For example the dissection proofs of the Pythagorean
 
theorem in which the two smaller squares are partitioned in such a way that
 
their parts can be reassembled to apparently exactly cover the largest
 
square, and where no previous theorems are invoked, are not proofs at all.
 
They are only experiments which make the Pythagorean Theorem seem plausible.
 
 
 
* Electronic dynamics make little contribution to the analysis of
 
construction problems in Plane Geometry and none at all to the proof
 
necessary to show that the construction is correct.
 
 
 
III. The grading system.
 
 
 
First we consider the internal grading system which has long been used by
 
teachers to determine course grades and to provide feed-back for improving
 
instruction.
 
 
 
At mid-century the following principles were widely accepted as valid by
 
high school teachers. They are still valid today.
 
 
 
1. Standardized tests can be designed that provide valid indicators of the
 
student's degree of mastery of specified course content. (Consider New York
 
Regents and the Advanced Placement tests.)
 
 
 
2. The grading system (GS) must be as simple as possible so that it can be
 
understood by the students.
 
 
 
3. The GS must be as OBJECTIVE as possible. The students course grade should
 
NOT be influenced by the
 
teacher's perception of the student's attitude, mathematical disposition or
 
social status.  Considerations based on gender, race, or linguistic
 
handicaps must not be allowed to affect the course grade which should
 
measure mastery of subject matter only. There are other, more appropriate
 
ways to deal with attitude, diligence, etc.
 
 
 
4. The GS must be economical of teacher time, so that secondary school
 
mathematics teachers working under
 
normal teaching conditions (five classes and at least one extra-curricular
 
assignment) can cope with the heavy demands made upon them.
 
 
 
5. In every unit of instruction there are some basic facts that the student
 
should be expected to remember and some basic skills to be habitualized. The
 
GS should require the student to demonstrate that he knows these facts and has
 
     mastered these skills. Other than that, the GS in mathematics should
 
make minimal demands on the student's memory.
 
 
 
6. The GS must include measures of Symbol/Language translation and writing
 
skills.
 
 
 
7. The GS must have a strong diagnostic component.
 
 
 
Now we turn to the external grading system.
 
 
 
Every student must have the opportunity to take the externally set and
 
externally graded examinations covering the standard syllabus that has been
 
adopted for each course on a regional/national basis. There will, of course,
 
be variations in the amount of time that students need to prepare for these
 
examinations. For example, some students may need two years of
 
study to prepare for the regionally administered examination in first year
 
algebra (roughly algebra through quadratics).
 
 
 
These externally set examinations provide a basis for comparing the
 
student's performance with regional, national or, perhaps even world norms.
 
They invoke what John Bishop calls The Power of External Standards. In his
 
article by that title in the Fall 1995 issue of American Educator, the AFT
 
magazine, Bishop says: Such standards are established when the student's
 
mastery of a common curriculum taught in high school is assessed by
 
examinations that are set and graded on the national or regional level. He
 
continues: The standards reflected in these exams are visible and public
 
as well as demanding. In France and the Netherlands, for example, questions
 
and answers are published in the
 
newspapers and available in video texts. And the grades a student gets are
 
extremely important because they signal the student's achievement to
 
colleges and employers. They influence the jobs that graduates get and the
 
universities and programs to which they are admitted. How well graduating
 
seniors do on these exams also affects the reputation of a school and, in
 
some cases, the number of students applying for admission to the school.
 
 
 
This describes the situation where high school seniors take batteries of
 
examinations covering the basic courses, for which they have prepared over a
 
four or five year period.
 
 
 
We would begin the establishment of externally set and graded examinations
 
on a course-by-course basis. There would be one such examination covering
 
the prescribed syllabus in algebra, another for geometry, etc. Exam results
 
would be publicized on a regional/national basis. Such exams do much more
 
than compare classmates with each other. They set performance levels for
 
each course, measure the extent to which each student has mastered the
 
prescribed syllabus and compare his performance with regional norms. These
 
examinations will be prepared by committees of mature, well-established
 
mathematicians appointed on a regional/national basis.
 
 
 
IV. We Reestablish the Teacher's Role as an Expositor and the Student's Role
 
as a Learner.
 
 
 
The dictionary defines a teacher as ... a director, one who imparts
 
knowledge, an instructor. The teacher in secondary school has always been
 
regarded as one who explains. We believe that the teacher of secondary level
 
mathematics should be confirmed in his centuries-old role as an expositor
 
and director of learning.
 
 
 
There are, of course, many ways to direct learning and each teacher should
 
use those that work best for him. Cooperative learning can be used
 
occasionally as it has been in the past. But teachers should not be under
 
pressure to use it because, used excessively, it tends to relegate the
 
teacher to the role of facilitator. To facilitate means to make easy. This
 
sends the wrong message. For most people, the learning of mathematics is not
 
easy. It requires hard work, sustained effort,
 
intense concentration and diligent attention to homework. The student needs
 
the kind of experience that the individual study of mathematics provides in
 
order to learn how to learn. Without it, he may graduate, steeped in self
 
esteem, but totally unprepared to meet the ever-changing demands of an
 
intensely competitive world, where there are no facilitators to make
 
things easy.
 
 
 
We want our teachers of secondary mathematics to have at least an
 
undergraduate major in mathematics. They should be encouraged to continue
 
their study of mathematics at the graduate level and in graduate math
 
education courses that emphasize mathematical content, if such can be found.
 
 
 
We would give them the wide discretion that well-trained professionals
 
deserve. We would set performance goals for students and would rely on the
 
ingenuity of our mathematically competent classroom teacher to find many
 
ways to achieve them.
 
 
 
As Bishop observes, with the establishment of external standards, the
 
teacher's situation improves dramatically. The teacher with high standards
 
is no longer regarded as a taskmaster whose demands are to be evaded, but
 
rather as ... a coach or mentor whose advice and expertise help students to
 
achieve a goal they care about. Moreover, they are released from the
 
pressure now exerted by students, administrators and parents to grade on the
 
curve, lower standards and inflate grades. This pressure is fast becoming
 
intolerable and it is destroying education in America.
 
 
 
The learning situation for students is also vastly improved. No longer rated
 
against each other, they would be striving to reach a standard that has been
 
externally set. Diligent students would no longer be resented and derided as
 
"nerds" who are lifting the curve and thus making it more difficult for
 
their classmates. As Bishop notes: ... students will work very hard and
 
achieve at a high level if you make clear what you want them to learn and if
 
there are serious consequences attached to their achievement.
 
 
 
We believe that it is time for the gradual introduction of external
 
standards in secondary school mathematics, on a voluntary course-by-course
 
basis. We believe that the public will support this. We cite the first
 
recommendation in the report, The Challenge of Change: Standards to Make
 
Education Work for All Children, recently produced by the National Alliance
 
for Business. All students should be given the opportunity to master
 
academic subject matter calibrated against world-class norms.
 
 
 
It is the duty of teachers and parents to call upon the best advice
 
obtainable (in this case, from mathematicians) to set standards of student
 
performance and then demand that students meet these standards. When
 
standards are held firm and the student is required to adjust to them, we
 
have a process that can be accurately described as education. In recent years
 
we have seen a distressing reversal of this process. Students don't listen
 
very well? Adjust by downgrading oral exposition by the teacher, and,
 
perhaps resort to cooperative learning. Students don't like the curriculum?
 
Change the curriculum, perhaps by emphasis on practical applications in an
 
effort to recapture student interest. Students don't do well on
 
standardized tests? Try to discredit these tests by proclaiming that they
 
are not and cannot be valid measures of student achievement. This
 
stultifying process where changes take place in the system rather than in
 
the student is education turned on its head. It is destroying education in
 
America, and it must be stopped.
 
 
 
We have described a way to stop it, and to put math education in America on
 
a par with that in other industrialized countries. We realize that the
 
establishment of external standards will encounter powerful opposition. But
 
it is long overdue.
 
With the support of concerned parents throughout the nation, we believe that
 
it can be achieved.
 
 
 
 
 
Submitted by
 
Professor Frank B. Allen
 
Advising Member, Mathematically Correct
 
Former President, National Council of Teachers of Mathematics
 
Former Chair, Division of Natural Sciences and Mathematics, Elmhurst College
 
 
 
April 1996
 
 
 
 
 
The following individuals have indicated their support of this position
 
statement.
 
 
 
Henry L. Alder
 
Department of Mathematics
 
University of California, Davis
 
Former President of the Mathematical Association of America
 
Former Member of the State Department of Education
 
Former Chair of the University of California Board of Admissions and
 
Relations with Schools
 
 
 
George E. Andrews
 
Evan Pugh Professor of Mathematics
 
Department of Mathematics
 
Penn State University
 
 
 
Terry Antl
 
Instructor, General Math Studies Department
 
San Diego State University
 
 
 
Richard Askey
 
Dept. of Mathematics
 
Univ. of Wisconsin-Madison
 
 
 
Alicia Booth
 
 
 
Rebecca Bryson-Kissinger
 
Parent
 
Associate Dean of Science
 
San Diego State University
 
 
 
Gunnar Carlsson
 
Professor and Chair
 
Department of Mathematics
 
Stanford University
 
 
 
Lisa Churchill-Roth
 
Parent, San Diego
 
Member, Mathematically Correct
 
 
 
Jamie Clopton
 
Parent, San Diego
 
Member, Mathematically Correct
 
Dept. of Behavioral Science, Palomar College
 
 
 
Paul Clopton
 
Parent, San Diego
 
Member, Mathematically Correct
 
Research Department, VA Medical Center, San Diego
 
 
 
Pam Davis
 
Computer Resource Specialist
 
University of California, Davis
 
 
 
Bill Evers
 
Parent, Palo Alto, Calif.
 
Research Fellow, Hoover Institution, Stanford University
 
Steering Committee, Honest Open Logical Debate (HOLD) on math reform
 
 
 
Peter M. Fischer
 
Parent
 
Member Brea Open Logical Debate (BOLD)
 
Software Engineer, Hughes Aircraft
 
 
 
Carol Gambill
 
Mathematics Teacher
 
 
 
Leonard Gillman
 
Professor Emeritus of Mathematics, University of Texas at Austin
 
Former President, Mathematical Association of America
 
 
 
Larry Gipson
 
Engineer
 
Member, Parents for Math Choice
 
 
 
Detlef Gromoll
 
Department of Mathematics
 
State University of New York
 
 
 
Bonnie Grossen
 
Researcher
 
College of Education
 
University of Oregon
 
 
 
Jeffrey Haydu
 
Parent, San Diego
 
University of California, San Diego
 
 
 
Douglas L. Inman
 
Research Professor of Oceanography
 
Scripps Institution of Oceanography
 
University of California, San Diego
 
 
 
Fred Jaeger
 
Parent, San Diego
 
Member, Mathematically Correct
 
 
 
Heide Jaeger
 
Parent, San Diego
 
Member, Mathematically Correct
 
 
 
John Kissinger
 
Parent
 
Mechanical Engineer
 
General Atomics
 
 
 
William J. Kurdziel
 
Math Educator
 
 
 
Joseph Lipsick, M.D., Ph.D.
 
Concerned Parent and H.O.L.D. Member
 
Associate Professor of Pathology
 
Stanford University
 
 
 
Patricia M. Masters
 
Parent, Member of Mathematically Correct
 
La Jolla, California
 
 
 
Mike McKeown
 
Molecular Biology and Virology Laboratory, The Salk Institute
 
Member, Mathematically Correct, San Diego
 
 
 
Robert McPherson, Jr.
 
Hemet USD Board President
 
Retired Probation Officer
 
 
 
Motohico Mulase
 
Professor and Vice Chair
 
Department of Mathematics
 
University of California, Davis
 
 
 
R.A. Raimi
 
Professor Emeritus of Mathematics
 
The University of Rochester
 
 
 
Chuck Rathbone
 
Hemet USD Trustee
 
Director of Finance, Eastern Muncipal Water District
 
 
 
Debra Rathbone
 
Parent
 
 
 
F. A. Jack Roach
 
 
 
Patricia Rorabaugh
 
 
 
Jerry Rosen
 
Professor of Mathematics
 
California State University, Northridge
 
 
 
Marina Rumiansev
 
Founding Member, Parents Advocating for Children's Education
 
 
 
Dennis L. Stanton
 
Mathematics Dept. Chairman
 
Soquel High School
 
 
 
Andre Toom
 
Department of Mathematics
 
University of the Incarnate Word
 
 
 
Hung-Hsi Wu
 
Professor of Mathematics
 
University of California, Berkeley
 
 
 
Ze'ev Wurman
 
Parent, Palo Alto, Calif.
 
Director of Software, Dyna Logic Corporation
 
Steering Committee, Honest Open Logical Debate (HOLD) on math reform
 
 
 
--
 
 
 
The opinions expressed are my own and not those of the Salk Institute or its
lawyers.