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Topic: Math Wars
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Jerome Dancis

Posts: 8
Registered: 12/6/04
Math Wars
Posted: Apr 29, 2001 11:32 PM
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Sorry, the footnotes are not included, just their numbers.


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 fourth1. 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 letter2 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 II3. 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 Gulick4 ,
noted that about one third of the students in her precalculus 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 centimeters 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's9 in her book: The Textbook Fiasco; A Conspiracy of Good
Intentions. Prof. Davis of Worcester Polytechnic Institute wrote10 : "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.14 Serious training in
deductive proofs and word problems have disappeared. Prof. Barry Simon
noted: "The dumbing down of high school education in the United States,
especially in mathematics and science, is a crime that must be laid at the
doorstep of the educational establishment".
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
ones 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 Wu16 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
documented17, 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:
Problem 3. Solve x/2 = (3/4)x +1.
It was solved by none of the (mostly Grade 10) students. 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) Standards18 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
1/4ÊÖÊ1/2 , 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) meI do not know how to visualize 19/74ÊÖÊ17/23?
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 www.edexcellence.net/library/wcs/wcs).
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 Askey19 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".





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