**The Math Wars **

** **

**by Jerome Dancis**

** **

An
abridged version of this article appeared in the __Faculty Voice__ March
2000 (The faculty newsletter at the Univ. of Maryland).

"Now,
for at least the fourth time in a hundred years, school boards and university
professors and PTA leaders are engaged in a bitter debate over how to teach
arithmetic". ( Jay Mathews in his article on the Math Wars in the
Washington Post Magazine, Feb. 2000)
Secretary of Education Riley has called for an end to the Math Wars and
urged Math educators to work together in improving mathematics education; never
mind that the Department of Education continues to take the side of the
"Reformers". The NSF has awarded grants for Teacher Enhancement
Programs and Teacher Preparation programs for "Reform". When President Clinton called for a national
voluntary Grade 8 math test, the committee to write the specs, was stacked with
"Reformers".

In
December,1997, California's state Board
of Education was about to vote on math standards for public school students by
injecting more math into math -- actually expecting kids to memorize
multiplication tables in the third grade and master long division in the fourth^{[1]}. On , the day before the
final vote, Luther S. Williams, (then) assistant director of the National
Science Foundation, fired off a letter to board president Yvonne Larson. He
reminded Larsen that his group "cannot support individual school systems
that embark on a course that substitutes computational proficiencies for a
commitment to deep, balanced, mathematical learning.''

In September, 1999, the new California
Standards were published, standards which are very different from those of the
Reformers. In October, 1999, the
Department of Education announced a list of exemplary mathematics textbooks.
(Riley had repeated postpond making the announcement -- possibly because a
mathematician on his Expert Panel had written strongly against the panel's
recommendations.) In reaction, a
cryptic public letter^{[2]} to the Secretary of Education was published in the Nov. 18, 1999
Washington Post calling for the Secretary to "withdraw your premature
recommendations" and try again since several books on the list contained
"serious mathematical shortcomings". This letter was signed by about 200 professors, mostly of
mathematics, including Bill Adams, our Mathematics Department's Vice-chairman
for Undergraduate Educ and yours truly,
and four Nobel laureates.

**The failure of traditional
mathematics instruction and how it hinders college education.**

** **

Knowledge and understanding of arithmetic and
algebra is crucial for understanding the simple mathematics and other
quantitative concepts that arise in a wide variety of college freshmen courses
and lack of such student knowledge puts an unfair burden on both instructors
and students. (For an example, see
Problems 1 and 2 below.) It mucks up
many a student's education.

Some
high schools required that a student score only 30 points (on a 100 point test)
to pass and only 65 points for an A on
the 1999 Montgomery County Public Schools' Algebra 1. (All the middle schools required 60 points for passing). A passing score of 30 means students will
need remediation in college.

Elementary
Algebra (Math 001) was the largest single mathematics course on our campus in
the 1990s. Half the students therein,
had received a grade of B or A in high school Algebra II^{[3]}. Having to repeat two years of mathematics, already studied in
high school, is not limited to the weak students, it extends to many strong
students also. Dr. Frances Gulick^{[4]} , noted that about one third of the students in her __pre__calculus
class (Math 115) had already "completed" calculus in high school.

**Problem 1.** (from an elementary
nutrition course on our campus.) It is
a fact that fat has 9 calories per gram and protein has 4
calories per gram. If a piece of
meat consists of 90 grams of protein and 15 grams of fat, how many calories does it
have? (495)

Class
time, used for this instruction, reduces time available for teaching nutrition.

In
elementary sociology classes on our campus, students struggle with
"percents" in problems like:

**Problem 2**. A cohort
has 1000 males and 160 females. Suppose
that 25% of the males and 15% of the females have blue eyes. How many in this cohort have blue eyes?^{[5]} (Ans. 274)

The
lack of fluency in Arithmetic and Algebra I among students pressures colleges
and high schools to "dumb-down"
(course deflation) a variety of social science and mathematics courses.

In
the 1970s, Jack Goldhaber, then Chairman of the Mathematics Department,
assigned Professors David Schneider (now emeritus) and Larry Goldstein to write
a textbook for Math 110 (Finite Math).
Their text was used for many years on our campus. Later, when the students arriving in this
class were not sufficiently fluent in high school algebra to handle this text, it was replaced by a dumbed down
text.

"There
are other hidden, but measurable, costs. Laurence Steinberg, a psychology
professor at Temple University, noted last year that his institution's
requirement for two semesters of psychological statistics for majors is not a
cause to celebrate high standards. Rather, it is an admission that it now takes
two semesters to learn what used to be done in one".^{[6]}

Large numbers of college students change
majors under the duress of difficulty with the mathematics in a required
course. Often, a major reason for
their difficulties/failure is lack of fluency in high school mathematics.

"When
Grant Scott, a biology teacher, had to teach his chemistry students at Howard
High School [in Howard County] how to change centimeters to meters, he just
told them to move the decimal two places -- rather than illustrating the
concept. ... 'Forty-five minutes later, only three of them got it.' ", ^{[7]} (Not so hard since 100 __cent__imeters make a meter, just
like 100 cents make a dollar.) The new
California Standards require that students learn this in Grade 4.

At
the request of a local public school system (1980's), Prof. Jim Greenberg
(College of Education and Director of UMCP's Center for Teaching Excellence)
organized a series of discussions on the topic: What contributes to the failure of college freshmen? The participants in the initial discussions
were faculty members (who have much connection with freshmen) from 10
departments on our campus, together with high school teachers and UMCP
freshmen). The main conclusion was
that: "**The overemphasis on testing, skill development, and fact level content,
etc. [in high school] seems to have inhibited [student] interest in learning,
motivation, ability to work with and enjoy ideas, use creativity and attain
satisfaction from an educational experience**." In a later discussion among college faculty
members (mostly from departments of speech and communication including UMCP's
Andrew Wolvin, professor of Communication Arts and theatre and Barbara
Williams, then with our Institute for Urban studies), it was noted that: "**Entering
college freshmen appear severely limited in their ability to read critically,
synthesize information, interact effectively with both peers and instructors in
academic settings, and participate actively in discussions**."^{[8]} This is a natural consequence of these activities *not*
being included in the curriculum of most school systems.

**High school courses are
largely determined by the textbooks. **Now we discuss textbooks:

That
there is little value in reading school textbooks and that textbooks were going
from bad to worse was documented in the 1980's by Harriet Tyson-Bernstein's^{[9]} in her book: __The Textbook Fiasco; A Conspiracy of Good
Intentions__. Prof. Davis of
Worcester Polytechnic Institute wrote^{[10]} : "The [high school
precalculus textbook] ... is no more mathematics than the noise made by trained
seals is music. But the trained seal
approach abounds in textbooks and in classrooms. It never provides a foundation of fundamental ideas ... . **It
never offers intellectual challenges, or chances to build confidence and
problem-solving skills."**
Mathematics textbooks basically teach skills and calculation procedures
without teaching understanding, without teaching when and how to use the
skills, and without teaching how to think through problems. No wonder that students do not read their
calculus textbooks; only foreign
students read the textbooks, and they do so to improve their English! (Observed at U.C. Berkeley by Uri
Treisman.)

What
is the most basic rule of traditional American mathematics textbook
publishers? It is the "two-page
spread"; the entire day's lesson *must*
fit on two pages. In sharp contrast are
the Singapore mathematics textbooks which have a single day's lesson going on
for pages. The Singapore mathematics
textbooks are traditional mathematics taught properly. They will be on sale in the U.S. as soon as
the names of people and vegetables are changed to American ones.

Many
traditional texts/classes "educational beat-up" the students by
placing them in a non-viable pedagogical situation where chance of success is
small. This is why our otherwise great
country is largely populated by people who say (mostly without shame) "I
was never good at mathematics".

**End of
deductive reasoning.** Until the
1950's, students studied Euclidian geometry in high school. Starting with a small number of axioms, they
proved and watched the teacher prove many theorems. In this way they were
provided with *extensive* training in
deductive reasoning. *Students learned that a statement in plane
geometry was true because they had seen a proof of its validity.* Since then, 100 theorems have been renamed
axioms and their proofs (now being redundant) have disappeared from the
textbooks. *Now a statement is true in plane geometry because the book/teacher says
it is so* (It is an axiom.).
Deductive proofs have been exiled to the last quarter of the textbook;
not enough for students to learn this topic. The teaching of deductive proofs
in plane geometry is banned in the Montgomery County school system. Students now arrive in college with little
or no training in deductive reasoning, a serious educational handicap.

Prof.
Barry Simon, Chairman of the
Mathematics Department at California Institute of Technology in, "A Plea
in Defense of Euclidian Geometry "^{[11]} , "mourned this loss
of what was a core part of education for
centuries." as he noted
"what is really important is the exposure to clear and rigorous arguments.
... "They can more readily see through the faulty reasoning so often
presented in the media and by politicians". Also, they would have less difficulty adjusting to and understanding
college courses.

Math-Education Prof. Guershon Harel ^{[12]} wrote "It is imperative to reinstate Euclidian
geometry [*based on deductive
proofs*], in the high school curricula ... [it] is a concrete system where
students can learn the concept of a deductive system." I strongly agree, at least in those school
systems where the geometry teachers are fluent in deductive geometry. One could
obtain a State of MD high school mathematics teacher certification (before
1990) without having taken any course in geometry or in deductive
reasoning.

The Second International Mathematics
Study (SIMS) concluded that the** **major
reasons for the low achievement in mathematics in U.S. schools, are that* the mathematics curriculum is
underachieving, very repetitive,
ineffective and inefficient.** ^{[13]}* My children (in the fast academic track) were taught one
third less mathematics in high school than I was (in the standard academic
track) in the 1950s.

Summarized
as Traditional: pedantically rigorous drill and kill on skills, no thinking, or
conjecturing. Unreadable outline books
pretending to be textbooks. Excessive
memorizing, absence of reasoning things through.

All this cries out for dramatic solutions for
improving mathematics instruction. To
the rescue or semi-rescue or pseudo-rescue (depending on one's perspective)
came:

**The self-styled "Reform
Movement".**

** **

The
Reform movement was largely organized by professors of mathematics education.
It advocates much use of hand calculators, no drill, emphasizes on concepts,
group learning, students discovering mathematics for themselves which includes
much conjecturing, students inventing their own algorithms for arithmetic,
equity, mathematics education for all.

In
order to earn Department of Education approval, the Reform Movement textbooks
had to document that they were more effective somewhere than the traditional
textbooks/programs. Not hard to do considering that the traditional
textbooks/programs are so ineffective (as just noted).

The
Reform Movement is a package of many good ideas, many half-baked ones and many
counterproductive ones ; also many half-developed good ideas.

I
am a proponent of and have used discovery learning and group learning along
with lecturing in my teaching for decades. I accomplish this in college
mathematics courses , without reducing the quantity or quality or rigor of the
mathematics in the course.

The
Reform elementary school curriculum is far more interesting than the
traditional one. Having students work together in groups is far better than
each student working alone doing the traditional busywork. Having students
making conjectures and asking questions are valuable things that should be part
of any program. But conjectures should be educated guesses not distracting or
frivolous (time-wasting) ones. They should be checked and followed by a
rigorous justification. Checking that a conjecture works on a few examples is
good, but not enough. It is a pseudo-justification; it does not prove that it is
valid in all situations/problems. This type of mis-teaching will lead students
to believe all sorts of incorrect things and not just in mathematics.

Many
Reformers believe that a teacher should be the guide on the side instead of the
sage on the stage. But, just as with sports' coaches, it is useful for a
teacher to be _both_ a guide on the side and a sage on the stage.

Patricia
F. Campbell, Professor of Mathematics Education at UMCP, has developed a reform
style of pedagogy, which includes training in computation. She provided
training in her effective methods for teaching mathematics to the teachers at 3
of the 4 poor Montgomery County elementary schools cited in a May 16,1999 Post
article as doing exceptionally well. She organized Project IMPACT. Her reward
was to be assigned to do the same for the City of Baltimore public schools.

While
there is variety in reform, we can still talk about the Reform movement as the
movement exemplified by the NCTM standards and some of its spokesmen and
textbooks. This broad brush approach will misrepresent some of its
participants.

Steven
Leinwand is the co-chairman of the U. S. Dept. of Education's Expert Panel (on
textbooks) and the top mathematics adviser at Connecticut's Department of
Education. In his article "It's
Finally Time to Abandon Computational Algorithms"^{[15]}, he began:

"It's
time to confront those nagging doubts about continuing to teach our students
computational algorithms for addition, subtraction, multiplication, and
division [like 23 x 37]. It's time to acknowledge that teaching these skills to
our students is not only unnecessary, but counterproductive and downright
dangerous! And it's time to proclaim that, for many students, real mathematical
power, on the one hand, and facility with multi-digit pencil and paper
computational algorithms, on the other hand, may be mutually exclusive."
...

"Today,
real people in real situations regularly put finger to button and make critical
decisions about which buttons to press, not where and how to carry threes into
hundreds columns."

"No
longer simply perpetuators of the bell curve, where only some survive and even
fewer truly thrive, schools and their mathematics programs must instill
understanding and confidence in all". ... Most compelling to Leinwand is
the "sense of failure and the pain unnecessarily imposed on hundreds of
thousand of students in the name of mastering these obsolete procedures".

I
strongly agree with Leinwand's statements of parts of the problem in the last
paragraph. But I strongly disagree with his solution, namely, not to teach
children how to multiply 23x37. It is like throwing out the baby with the dirty
bath water.

Students
given this type of Reform math training will be able to Problems 1 and 2 above,
if they have their hand calculators handy. Without their hand calculators, they
will know which numbers to multiply, but will have difficulty with the
multiplication.

Similarly,
in their much quoted 1998 article, "The Harmful Effects of Algorithms in
Grades 1-4", Constance Kamii and Ann Dominick wrote: "Algorithms not
only are not helpful in learning arithmetic, but also hinder children's
development of numerical reasoning ..."

Invoking
Piaget's constructivism, Kamii and Dominick wrote: "Children in the
Primary grades should be able to invent their own arithmetic without the
instruction they are receiving from textbooks and workbooks."

As
Wu^{[16]} has noted: "Why not consider the alternate approach of
teaching these algorithms properly before advocating their banishment from
classrooms?"

Wu
also wrote: "What is left unsaid is that when a child makes up an
algorithm, the act raises two immediate concerns: One is whether the algorithm is correct, and the other is whether
it is applicable under all circumstances." To carefully check and correct many new student algorithms
periodically is a sizable task. Also as
Liping Ma has documented^{[17]}, many teachers do not have
sufficient knowledge of the mathematics they are teaching, to be qualified
algorithm checkers. Teachers currently
insist that students present fractional answers in simplest terms so that there
will be a unique right answer and they will not have to check whether each
student's unsimplified fraction is equivalent to the answer in the teachers
manual.

The
Reform movement also advocates an "integrated curriculum" which mixes
algebra and geometry. This makes as much sense as the teaching of European
history and American History in alternate months.

Dr.
Jerome Epstein gave the following (pre-algebra) problem to such a second-year
"integrated" algebra and geometry class. It was solved by none of the (mostly Grade 10) students.

**Problem 3**. Solve x/2 = (3/4)x +1.

The
reform movement professors of mathematics education largely organized and wrote
The National Council of Teachers of Mathematics (NCTM) Standards in the late
1980s. The NCTM is the professional society of school mathematics teachers.
Their standards and the AAAS criteria are for all students; there is no
separate higher standards for students going on to college. When adopted by a
school system, Reform methods and textbooks are used for all students even
though this is a dumbing down for college bound students.

The
reform movement professors of mathematics education largely organized and wrote
The National Council of Teachers of Mathematics (NCTM) Standards^{[18]} in the late 1980s. The NCTM is the professional society of
school mathematics teachers. Their
standards and the AAAS criteria are for *all*
students; there is *no* separate higher
standards for students going on to college.
When adopted by a school system, Reform methods and textbooks are
used for *all* students even though this is a dumbing down for college bound
students.

The
NCTM response, to the low level of students skill at using fractions, has been
to prescribe *decreased* attention to
fractions in algebra. The NCTM
standards state: "This is not to suggest that valuable time should be
devoted to exercises like (17/24) +
(5/18) or 5 3/4 x 4 1/4". ... "Division of fractions should be
approached conceptually".

This
is what the Grade 6 reform book "Mathland" does: "Rather than
relying on algorithms, where memorization of rules is the focus, the Mathland
approach relies heavily on active thinking.
To solve problems such as
1/4 — 1/2 , students
need to be able to visualize the question: How many halves go into one-fourth?
This kind of fluency enables students to use their own logical and visual
thinking skills to really know what the solution (1/2) means". Good, but I do not know how to
visualize 19/74 — 17/23?

In
Oct. 1998, the NCTM released its __Proposed Principles and Standards __(for
the next decade). This revision is less
revolutionary then its earlier Standards.
It has invited much feedback from a wide variety of organizations and
individuals. The final version will be
released at the NCTM convention this spring.

The
verbose, 700 page NCTM proposed
standards do not even consider the question of raising the content of the
mathematics curriculum back to the levels of the 1950s.

Several
reform textbooks have been written mostly by professors of mathematics educ,
some with support from the NSF or Dept. of Educ.

Many mathematics teachers are not fluent in
the sterile math curriculum they are currently teaching, never mind an enriched
curriculum. They will be overwhelmed by
a program, which has new topics or treats topics in a non-traditional manner,
especially if thinking is involved, even if the text is a big improvement. A short staff development session is not
likely to be sufficient.

The NCTM proposed Principles and Standards do
strongly argue for reasoning and proof in all grades. But the level being proposed is so low as to be
embarrassing. Details below.

The
NCTM Proposed Standards do not advocate the reinstatement of Euclidian
geometry [*based on deductive
proofs*], in the high school curricula.
In fact, the phrase "deductive proof" is not even mentioned in
the proposed Standard 7: Reasoning and
Proof for high school -- pages
316-8.

The
phrase "Formal proof" appears on line 31, Page 316 followed by a pictorial example of a useless
triviality: The sum of two consecutive
'triangular numbers' is a square. This
so called "formal proof" consists largely of looking at the diagram
on page 317. This is an example of an
elementary school, not high school level of proof. This demonstrates that the proposed standards are highly
underachieving. The concept of 'triangular numbers' is useless and distracting. It provides fodder to those who say school
mathematics will be useless to me after I escape high school.

Page
317 top paragraph. The only example of
a 'sophisticated' reasoning situation mentioned is to explain that rational
numbers may be converted to repeating decimals and vice versa, like 1/3 =
.33333.. . To me this *not so* 'sophisticated' reasoning
situation belongs in middle school not high school; at least in those school systems where the Grade 6-8 teachers are
fluent in fractions and decimals. The main use of these conversions is to
generate busywork exercises.

Page
246 Line 24 Problem solving for middle
school. Asks the useless
question: If we build semicircles, etc.
on the the sides of a right triangle, then the sum of the areas [of the
semicircles] on the legs equals the area [of the semicircle] on the hypotenuse. This encourages students to spend time on
useless, frivolous, trivial and time-wasting conjectures and to provide fodder
to those who say school mathematics will be useless to me after I escape high
school.

The
proposed standards Page 219 [Arithmetic]
Operations and their Properties __for Grades 6-8__, includes the arithmetic of decimals and
fractions like: 1.4 + .67 and
2/3 + 3/4. The new California
Standards require this teaching in Grades 4 and 5, resp. I agree, at least for those school systems
where the Grades 4-5 teachers are fluent in fractions and decimals -- *many are not.*.

The
proposed NCTM Principles are very verbose about emphasizing the importance of
equity. During the last 3 decades much
equity has been achieved in mathematics education as good consequences of the
civil rights and feminist movements. In
addition, much equity has been achieved
by the easy, cheap method of dumbing down the mathematics curriculum. If the proposed NCTM standards are
implemented, more equity will be achieved by simply dumbing down the
mathematics curriculum. Programs like Pat Campbells should be the rule, the
reality is that they are the exception.

Some
extremists believe that discovery learning means no lecturing and all group
work. They also believe lecturing and
group work are incompatible. A friend
(knowledgeable in the Math Wars) stated her amazement that discovery learning
and rigor could be part of the same teaching method. The polarization results in words carrying much (not always valid)
baggage. The terms
"Traditional" and "Reform" are too often interpreted
(or misinterpreted) to the bulk of the things I listed.

Faculty
members and other parents in Princeton NJ were so aghast at how the Reform
program was destroying their children's education that they organized a charter
school (Read "Why Charter Schools? -- The Princeton Story" by
research scientist and ex-school board member, Dr. Chiara Nappi, available on
the web at http://www.edexcellence.net/library/wcs/wcs.html).

Often
the choice is presented as restricted to either understanding or skills. Wu makes the point " ... as if thinking
of any sort -- high or low -- could exist outside of content knowledge. ...
in mathematics, skills and understanding are completely intertwined. ...
"Our children need courses which teach both skills and
understanding.

"They
really don't know anything about mathematics," said David Klein, a
mathematics professor at California State University, North Ridge, and
co-author of the letter to Riley, as he noted that the number of freshmen
needing remedial help in the California state university system has doubled
over the past 10 years. Two years ago, California stepped back from the
reforms, mandating more pencil-and-paper calculating and traditional drill and
practice mathematics.

To the rescue or semi-rescue or
pseudo-rescue from the Reformers, came:

**The new Calif. Standards (a
third way).**

In
opposition to the reform movement arose Mathematically Correct, a
politically-incorrect parents group and several professors of mathematics,
especially Dick Askey^{[19]} of the Univ. of Wisc., H. Wu of U.C. Berkeley, and Jim Milgram of
Stanford and Henry Alder.

The Fall 1999 issue of the American Educator,
(the quarterly professional journal of the American Federation of Teachers) has
four good articles. The one by Wu, "Basic skills versus conceptual
understanding: a bogus dichotomy in mathematics education", starts roughly
as follows:

"Education
seems to be plagued by false dichotomies. 'Facts vs.~higher order thinking' is
an example of a false choice that we often encounter these days, as if thinking
of any sort --- high or low --- could
exist outside of content knowledge. In mathematics education, this debate takes
the form of 'basic skills or conceptual understanding.' This bogus dichotomy
would seem to arise from a common misconception of mathematics held by a
segment of the public and the education community: that the demand for
precision and fluency in the execution of basic skills in school mathematics
runs counter to the acquisition of conceptual understanding. The truth is that
in mathematics, skills and understanding are completely intertwined. In most
cases, the precision and fluency in the execution of the skills are the
requisite vehicles to convey the conceptual understanding. There is not
'conceptual understanding' and 'problem solving skill' on the one hand and
'basic skills' on the other. Nor can one acquire the former without the
latter."

Mathematically
Correct won the latest battle in California. The new Mathematics Framework for
California Public Schools (K-12) reflects their views. Available on the Web at http://www.cde.ca.gov/cdepress/math.pdf.
None of the current textbooks, traditional or reform meet these new California
Standards.

The
new California Standards bring back two-column deductive proofs as the basis of
high school Geometry. The number of
axioms is reduced to a reasonable 16.
The new California Mathematics Framework emphasizes a balanced
mathematics curriculum. It stresses the
critical interrelationship among computational proficiency, problem-solving
ability and conceptual understanding of all aspects of mathematics.

Having
lessons end with a wide variety of problems on material that the students were
taught months and years ago is an important feature of the popular traditional
Saxon textbooks. These Saxon-reviews change the subject just when the students
are starting to internalize the day's new material; this disrupts the learning
process. The California Framework does not permit this type of busywork; it
states: "review must be developing automaticity or preparation for further
learning" (Page 231).

Testing.
The teacher allocated half of the time of my child's mathematics lessons in
Grades 6 and 7 just to testing. (just the teacher's tests, not counting
stansardized tests.) The California Framework does not permit this squandered
of class time; it states: "Minimize loss of instructional time [due to
testing]" (Page 197).

All
states would be wise to adopt the new Mathematics Framework for California
Public Schools (K-12).

These
Mathematics Framework for California Public Schools are the opposite of the
message delivered at a 1999 staff development session for Algebra I teachers in
Montgomery County, a rich suburb of Washington, D.C; it was summarized to me as
"Do not worry about the students understanding algebra -- Just be sure
they can put anything on their Hand calculators".

**<><><><><<><><>**

** **

** **

** **

**Partial Update
on "The Math Wars"**** **

** **

** **

** **

The
NCTM had solicited and received much feedback from a wide variety of
organizations and individuals on its 1998 __Proposed Principles and Standards__
(PPS). Many improvements were made as these standards were moved somewhat in my
direction. It is still on a considerably lower level than the new Calif.
Framework and Standards (1999). That the NCTM standards have been changed does
not mean that the Reformers have changed their views or that the number of
Reformers has been reduced. The final version, __Principles and Standards for
School Mathematics__ (PSSM)was released after my article, "The Math
Wars" appeared.

I
had noted that the only example of a formal proof for high school [in the PPS__]__
really belonged in elementary school.
This demonstrated that the NCTM writers held highly underachieving
standards for students. Fortunately,
the feedback persuaded the writers to move this example down to its appropriate
Grade 3-5 level (P. 159).

I
had criticized the only example of a 'sophisticated' reasoning situation for
high school students mentioned in the PPS; fortunately, this example appears to
have disappeared during the revision.

The
phrase "deductive proof" was not even mentioned in the PPS Standard
7: Reasoning and Proof for high school. This phrase has been added in the PSSM; but there is still no mention of deductive proofs for a geometric
theorem in the PSSM.

Much
of the low standards and foolishness remained.
The arithmetic of decimals and fractions like: 1.4 + .67 and 2/3 + 3/4
is still postponed to Grades 6-8.

In
the "Problem Solving for Middle School" section (Page 261 main
paragraph), the following useless and
frivolous question is still asked: What
if we build semicircles, etc. on the sides of a right triangle, will the sum of
the areas of the semicircles on the legs equals the area of the semicircle on
the hypotenuse. (adapted from Stephen
Brown and Marion Walters's book __The Art of Problem Solving__ [1983]).

The
following is an appropriate problem for Grade 3-5 students; but PSSM (Page 258
bottom) postpones it until Grades 6-8:

"[Teachers] can challenge students
with problems that have more than one answer, such as the following (adapted
from Gelfand and Shen's Algebra book [1993], p. 3): Make a sum of 1000, using some eights (8s) with some plus signs (+s)
inserted".

Answer: 1000=888+88+8+8+8.

That
a problem that has more than one answer does not a challenging problem
make. The thought involved in solving
this problem has nothing to do with the fact there are several answers.

^{[1]} Debra
Saunders San Francisco Chronicle essay of Dec. 19, 1997, called Man of Science
Has a Problem With Real Math.

^{[2]} The
letter may be found at http//www.mathmaticallycorrect.com/riley.htm, complete
with links to web sites containing (not
easily found) referenced documents.

^{[3]} Survey in the late 1980's by Math Lecturer Lynn
Cleary.

^{[4]} a lecturer in mathematics at UMCP and wife
of recent Campus Senate Chairman Prof. Denny Gulick.

^{[5]} I am collecting examples like these of
simple problems involving simple mathematics from non-hard-science
disciplines. Donations from readers
will be appreciated.

^{[6]} From Lois Cronholn's wonderful article
"Why One College Jettisoned All Its Remedial Courses" in Chronicles
of Higher Education (1999).

^{[7]} Linda
Perlstein's February 15, 1999,
Washington Post, front page article
"Right Teacher, Wrong Class".

^{[8]} John
L. Brown, Curriculum Dialogues, Prince Georges County Public School
System, (1987) Page 6

^{[9]} She was a member of the Board of Education
for the Montgomery County Public Schools.

^{[10]} Paul
Davis, Teaching mathematics and Training seals, SIAM News (the newsletter of
the [professional] Society for Industrial and Applied Math.), (March 1987) page
7.

^{[11]} in his February 6, 1998 Los Angeles
Times article.

^{[12]} of Purdue Univ. in his "Preprint "Greek vs Modern Mathematics Thought
...".

^{[13]} The SIMS
report was the booklet __The Underachieving Curriculum__ (1987) by Curtis C. McKnight et al. It was sponsored by the International
Association for the Evaluation of Educational Achievement. It was one of the most thorough analysis of
mathematics education in American schools in the 1980's. The conclusions of SIMS were not accepted by
school administrators, thereby justifying a
decade of inaction on mathematics education reform. A major purpose of the massive and
expensive Third International Mathematics Study (TIMS) was to double check the
SIMS results.

^{[14]} This can be documented by comparing
the New York State Regents high school mathematics exams from the 1950s and
1990s.

^{[15]} reprinted
in Wisconsin Teacher of Mathematics, Winter 1995

^{[16]} Dr. H.
Wu's great article, Basic Skills Versus Conceptual Understanding -- -- A Bogus Dichotomy in Math
Education" in __The American Educator__, American Federation of
Teachers, Fall 1999.

^{[17]}
Liping Ma Knowing and Teaching elementary Math, Lawrence Erlbaum Associates, Mahwah, N.J.
(1999), partially summarized by Dick Askey in
__The American Educator__, American Federation of Teachers, Fall
1999.

^{[18]}^{ } According
to the NCTM's __Curriculum and Evaluation Standards for School Mathematics__
(the Standards) (1989). On the web at http://www.enc.org/reform/journals/ENC2280/nf_280dtoc1.htm

^{[19]} member
of the National Academy of Sciences