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Topic: Martin Gardner on Fuzzy Math
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Wayne Bishop

Posts: 5,157
Registered: 12/6/04
Martin Gardner on Fuzzy Math
Posted: Sep 9, 1998 3:26 AM
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Martin Gardner, "The New New Math," _New York Review of Books_, Sept. 24,
pp. 9-12.

Gardner is the former mathematical puzzle editor of _Scientific American_
and a
prolific writer on science and math topics. The Cover headline is "The Fuzzy
New Math."

The article includes a detailed criticism of the textbook _Focus on
Algebra: An
Integrated Approach_, also known as "Rainforest Algebra."

An excerpt from Gardner's article:

"I seldom agree with the conservative political views of Lynne Cheney, but
she criticized extreme aspects of the new new math on the Op-Ed page of _The
New York Times_ on August 11, 1997, I found myself cheering."


The New New Math
September 24, 1998

Multicultural and Gender Equity in the Mathematics Classroom: The Gift of
Diversity (1997 Yearbook) edited by Janet Trentacosta and Margaret J. Kenney
248 pages, $22.00 (hardcover) published by National Council of Teachers of

Focus on Algebra: An Integrated Approach by Randall I. Charles and Alba
González Thompson, et al. 843 pages, $56.00 (hardcover) published by

Life by the Numbers: Math As You've Never Seen It Before narrated by Danny
Glover and Seven boxed videotapes produced by WQED, Pittsburgh $129.00

Surveys have shown for many decades that the mathematical skills of American
high school students lag far behind those of their counterparts in Japan,
Korea, Singapore, and many European countries. In the United States whites do
better than blacks, Hispanics, and Native Americans. Males outscore females.
Students from high socioeconomic backgrounds do better than those from lower

These are troubling statistics because, in an advanced technological society
such as ours, a firm grasp of basic mathematics is increasingly essential
better-paying jobs. Something clearly is wrong with how math is being taught
in pre-college grades, but what?

In the late 1960s the National Council of Teachers of Mathematics (NCTM)
to promote a reform movement called the New Math. In an effort to give
students insight into why arithmetic works, it placed a heavy emphasis on set
theory, congruence arithmetic, and the use of number bases other than ten.
Children were forbidden to call, say, 7 a "number." It was a "numeral" that
symbolized a number. The result was enormous confusion on the part of pupils,
teachers, and parents. The New Math fad faded after strong attacks by the
physicist Richard Feynman and others. The final blow was administered by the
mathematician Morris Kline's 1973 best seller Why Johnny Can't Add: The
Failure of the New Math.

Recently, the NCTM, having learned little from its New Math fiasco, has once
more been backing another reform movement that goes by such names as the new
new math, whole math, fuzzy math, standards math, and rain forest math. Like
the old New Math, it is creating a ferment among teachers and parents,
especially in California, where it first caught on. It is estimated that
half of all pre-college mathematics in the United States is now being taught
by teachers trained in fuzzy math. The new fad is heavily influenced by
multiculturalism, environmentalism, and feminism. These trends get much
attention in the twenty-eight papers contributed to the NCTM's 1997
Multicultural and Gender Equity in the Mathematics Classroom: The Gift of

It is hard to fault most of this book's advice, even though most of the
teachers who wrote its chapters express themselves in mind-numbing jargon.
"Multiculturalism" and "equity" are the book's most-used buzzwords. The word
"equity," which simply means treating all ethnic groups equally, and not
favoring one gender over another, must appear in the book a thousand times. A
typical sentence opens Chapter Eleven: "Feminist pedagogy can be an
part of building a gender-equitable multicultural classroom environment."
and over again teachers are reminded that if they suspect blacks and females
are less capable of understanding math than Caucasian males, their
behavior is
sure to subtly reinforce such beliefs among the students themselves, or what
one teacher calls, in the prescribed jargon, a student's "internalized

"Ethnomathematics" is another popular word. It refers to math as practiced by
cultures other than Western, especially among primitive African tribes. A
book much admired by fuzzy-math teachers is Marcia Ascher's
A Multicultural View of Mathematical Ideas (1991).1

"Critical-mathematical literacy" is an even longer jawbreaker. It appears in
the NCTM yearbook as a term for the ability to interpret statistics

Knowing how pre-industrial cultures, both ancient and modern, handled
mathematical concepts may be of historical interest, but one must keep in
that mathematics, like science, is a cumulative process that advances
steadily by uncovering truths that are everywhere the same. Native tribes may
symbolize numbers by using different base systems, but the numbers behind
symbols are identical. Two elephants plus two elephants makes four elephants
in every African tribe, and the arithmetic of these cultures is a minuscule
portion of the vast jungle of modern mathematics. A Chinese mathematician is
no more concerned with ancient Chinese mathematics, remarkable though it
than a Western physicist is concerned with the physics of Aristotle.

Fuzzy-math teachers are urged by contributors to the yearbook to cut down on
lecturing to passive listeners. No longer are they to play the role of "sage
on stage." They are the "guide on the side." Classes are divided into small
groups of students who cooperate in finding solutions to "open-ended"
problems by trial and error. This is called "interactive learning." The
use of
calculators is encouraged, along with such visual aids as counters,
geometrical models, geoboards, wax paper (for folding conic section curves),
tiles of different colors and shapes, and other devices. Getting a correct
answer is considered less important than shrewd guesses based on insights,
hence the term "fuzzy math." Formal proofs are downgraded.
No one can deny the usefulness of visual aids. Teachers have known for
centuries that the best way to teach arithmetic to small children is by
letting them "interact" with counters. Each counter models anything that
retains its identity an apple, cow, person, star. What's the sum of 5 and
2? A
girl who knows how to count moves into a pile five counters, then two more,
and counts the heap as seven. Suppose she first moves two, then five. Does it
make a difference? Similar procedures teach subtraction, multiplication, and

After a few days of counter playing it has been traditional for children to
memorize the addition table to at least 9. Later they learn the
table to at least 10. "Hands-on" learning first, then rote learning.
Unfortunately, some far-out enthusiasts of new new math reject anything
resembling what they call "drill and kill" memorizing. The results, of
course, are adults who can't multiply 12 by 12 without reaching for a

Aside from its jargon, another objectionable feature of the yearbook is that
its contributors seem wholly unaware that the best way to keep students awake
is to introduce recreational material which they perceive as fun. Such
material includes games, puzzles, magic tricks, fallacies, and paradoxes. For
example, determining whether the first or second player can always win at
tick-tack-toe, or whether the game is a draw if each player makes the best
moves, is an excellent way to introduce symmetry, combinatorics, graph
and game theory. Because all children know the game, it ties strongly into
their experience.

For what the yearbook likes to call a "cognitively challenging" task, give
each child a sheet with a checkerboard on it. Have each of them cut off two
opposite corner squares. Can the remaining sixty-two squares be covered by
thirty-one dominoes? After a group finds it impossible, see how long it takes
for someone to come up with the beautiful parity (odd-even) proof of

If new new math teachers are aware of such elegant puzzles, and there are
thousands, there is no hint of it in the yearbook. This is hard to understand
in view of such best-selling textbooks as Harold Jacobs's Mathematics: A
Human Endeavor (1970; third edition, 1994), which has a great deal of
recreational material; Mathematics: Problem Solving Through Recreational
Mathematics, a textbook by Bonnie Averbach and Orin Chein (1980); and scores
of recent books on entertaining math by eminent mathematicians.
I seldom agree with the conservative political views of Lynne Cheney, but
when she criticized extreme aspects of the new new math on the Op-Ed page of
The New York Times on August 11, 1997,2 I found myself cheering. As Cheney
points out, at the heart of fuzzy-math teaching is the practice of dividing
students into small groups, then letting them discover answers to problems
without being taught how to find them. For example, teachers traditionally
introduced the Pythagorean theorem by drawing a right triangle on the
blackboard, adding squares on its sides, and then explaining, perhaps even
proving, that the area of the largest square exactly equals the combined
of the two smaller squares.

According to fuzzy math, this is a terrible way to teach the theorem.
must be allowed to discover it for themselves. As Cheney describes it, they
cut from graph paper squares with sides ranging from two to fifteen units.
(Such pieces are known as "manipulatives.") Then they play the following
"game." Using the edges of the squares, they form triangles of various
The "winner" is the first to discover that if the area of one square exactly
equals the combined areas of the other two squares, the triangle must have a
right angle with the largest square on its hypotenuse. For example, a
of sides 3,4,5. Students who never discover the theorem are said to have
"lost" the game. In this manner, with no help from teacher, the children are
supposed to discover that with right triangles a^2+b^2=c^2.

"Constructivism" is the term for this kind of learning. It may take a group
several days to "construct" the Pythagorean theorem. Even worse, the paper
game may bore a group of students more than hearing a good teacher explain
the theorem on the blackboard.

One of the harshest critics of fuzzy math is the writer John Leo, whose
article on the subject, "That So-Called Pythagoras," was published last year
in US News and World Report (May 26, 1997). (His title springs from a
reference he found in a book on ethnomathematics to "the so-called
Pythagorean theorem.") Leo tells of Marianne Jennings, a professor at
State University, whose daughter was getting an A in algebra but had no
of how to solve an equation. After obtaining a copy of her daughter's
textbook, Jennings soon understood why.

Here is how Leo describes this book:

“It includes Maya Angelou's poetry, pictures of President Clinton and Mali
wood carvings, lectures on what environmental sinners we all are and photos
of students with names such as Taktuk and Esteban "who offer my daughter
thoughts on life." It also contains praise for the wife of Pythagoras, father
of the Pythagorean theorem, and asks students such mathematical brain
teasers as "What role should zoos play in our society?" However, equations
don't show up until Page 165, and the first solution of a linear equation,
which comes on Page 218, is reached by guessing and checking.”

Romesh Ratnesar's article "This is Math?" (Time, August 25, 1997) also
criticizes the new new math. It describes fifth-graders who were asked how
many handshakes would occur if everyone in the class shook hands with
everyone else. At the end of an hour, no group had the answer. Unfazed, the
teacher said they would be trying again after lunch. Professor Jennings
another appearance. She told Ratnesar that she became angry and worried when
she saw her daughter use her calculator to determine 10 percent of 470.

Curious about her daughter's textbook, which is now widely used, I finally
obtained a copy by paying a bookstore $59.12. Titled Focus on Algebra: An
Integrated Approach, this huge text contains 843 pages and weighs close to
four pounds. (In Japan, the average math textbook is two hundred pages.) It
is impossible to imagine a sharper contrast with an algebra textbook of
years ago.

"Integrated" in the subtitle has two meanings:

(1) Instead of being limited to algebra, the book ranges all over the math
scene with material on geometry, combinatorics, probability, statistics,
number theory, functions, matrices, and scatter graphs, and of course the
constant use of calculators and graphers. Fifty years ago high school math
given in two classes, one on algebra, one on geometry. Today's classes are
"integrated" mixtures.

(2) The book is carefully integrated with respect to gender and to ethnicity,
with photographs of girls and women equal in number to photographs of boys
and men. Faces of blacks and whites are similarly equal, though I noticed few
faces of Asians.

On the positive side is the book's lavish use of color. Only a few pages lack
full-color photos and drawings, all with eye-catching layouts. When it
comes to
actual mathematics the text is for the most part clear and accurate, with a
strong emphasis on understanding why procedures work, and on inducements to
think creatively. "After all," the text says on its first page, "what
good is
it to solve an equation if it is the wrong equation?" The trouble is that the
book's mathematical content is often hard to find in the midst of material
that has no clear connection to mathematics.

Not having taught mathematics myself, I have no opinion about the value of
students working in small groups as opposed to sitting and listening to a
teacher talk. Nor have I found research studies that make a decisive case in
favor of either method. Clearly a great deal depends on the qualities of
particular teachers, and these would be hard to appraise in any survey. The
authors justify the group approach by saying it anticipates the workplaces in
which students will find themselves as adults. John Donne's remark about how
no man is an island is quoted. The book's first "exercise" is a question: "In
general, do you prefer to work alone or in groups?"

An emphasis on ethnic and gender equity is, of course, admirable, though in
this textbook it seems overdone. For example, twelve faces of boys and girls
of mixed ethnicity reappear in pairs throughout the pages. Each has
to say. "Taktuk thinks..." is followed with "Esteban thinks...," "Kirti
thinks..." is followed by what "Keisha thinks...," and so on. These pairings
become mechanical and predictable.
The book jumps all over the place, with transitions as abrupt as the dream
episodes of Alice in Wonderland. I think most students would find this
confusing. Eight full pages are devoted to statements by adult
with their photographs. Each statement opens with a sentence about whether
they liked or disliked math in high school, followed by generally banal
remarks. For example, Diana Garcia-Prichard, a chemist, writes: "I liked math
in high school because all the problems had answers. Math is part of literacy
and the framework of science. For instance, film speed depends on chemical
reactions. I use math to model problems and design experiments. I like getting
results that I can publish and share." Presumably such statements are
intended to convince students that math will be useful later on in life.

Many of the book's exercises are trivial. For example, on page 20 students
asked to play forward and backward a VCR tape of a skier, then answer the
question: "How will this affect the way the skier appears to move?" On page
11: "A circle graph represents 180 kittens. What does 1/4 of the circle
represent?" (Answer, to be found in the back of the book: 45.) A chapter on
"the language of algebra" opens with a page on the origin of such phrases as
"the lion's share," "the boondocks," and "not worth his salt." It is not
clear what this has to do with algebra.

Many pictures have only a slim relation to the text. Magritte's painting of a
green apple floating in front of a man's face accompanies some problems about
apples. Van Gogh's self-portrait is alongside a problem about the heights and
widths of canvases. A picture of the Beatles accompanies a problem about
only because of the Beatles' song "Taxman." My favorite irrelevant picture
shows Maya Angelou talking to President Clinton. Beside it is the following
extract from one of her prose poems:
Lift up your eyes upon
This day breaking for you
Give birth again
Women, Children, Men,
Take it into the palm of your hands.
Mold it into the shape of your most
Private need. Sculpt it into
The image of your most public self.
Why is this quoted? Because the "parallel" phrases shown underlined are
similar to parallel lines in geometry! Is this intended to "integrate"
geometry and poetry?

The book is much concerned with how the environment is being polluted.
Protecting the environment is obviously a good cause, but here its
with learning math is often oblique, if not arbitrary. A chapter on functions
opens with a page headed "Unstable Domain." Its first question is "What
kinds of pollution besides air pollution might threaten our planet?" Page 350
has a picture of crude oil being poured over a model of the earth. It
accompanies a set of questions relating to the way improper disposals of oil
are contaminating ground water.

A page headlined "Hot Stuff" shows three kinds of peppers to illustrate how
they are used in cooking. Two of the "exercises" are: "The chili cook-off
raises money for charity. Describe some ways the organizers could raise
in the cook-off," and "How would you set up a hotness scale for peppers?"
page introduces a chapter on how to solve linear equations.
Another section on equations opens with pictures of zoo animals. It
discusses what can be done to prevent species from becoming extinct. The
question is "What role should zoos play in today's society?" The book's
under the entry "Animal study and care," lists thirty-two page references.

A section on mathematical inequalities is preceded by a page on how Mary
became vice-president of a toy company, and how Linda Johnson Rice found a
creative way to market Eboné cosmetics for black women. Under a photo of a
smiling Mary, the first questions are: "Would you like to own your own
business someday? Why or why not?"

On page 67, a picture of Toni Morrison is used to illustrate a problem about
how many ways four objects can be placed in a row. The text then introduces
four students who each read an excerpt from something Morrison has
written. In
how many different orders, the text asks, can the four excerpts be read? A
man from Mysore, India, who creates shadow pictures on the wall with his
fingers is featured on page 421. What this has to with the following section
on solving systems of inequalities is not evident. A photo of Alice Walker on
page 469 illustrates the question: "Is the time it takes to read an Alice
Walker novel always a function of the number of pages?" This and other such
references give the impression that well-known writers are being dragged
the text.

The most outrageous page opens a section on linear functions. It concerns
Dogon culture of West Africa. Students are told that this primitive tribe,
without the aid of telescopes, discovered that Jupiter has satellites, that
Saturn has rings, and that an invisible star of great density orbits Sirius
once every fifty years. Presumably the Dogon had supernormal powers.
it has long been known that the Dogon made no such discoveries. They merely
learned these astronomical facts from missionaries and other Western

Like the authors of the NCTM yearbook, those who fashioned this huge textbook
seem wholly uninterested in recreational material. The book's only magic
trick (page 246) is a stale, utterly trivial way to guess a number. Although
strongly favoring the use of calculators, the authors don't seem aware that
the hundreds of amazing number tricks that can be done with them provide
excellent exercises. A child can learn a lot of significant number theory in
discovering why they work. None is in the book.

An old brain teaser involves a glass of wine and a glass of water. A drop is
taken from the wine and added to the water. The water is stirred, then a
of the mixture goes back to the wine. Is there now more water in the wine
wine in the water, or vice versa? The surprising answer is that the two
amounts are precisely equal.

Students will be fascinated by the way this principle can be modeled with a
deck of cards. Divide the deck in half, one half consisting of all the red
cards, the other half consisting of all the black cards. Take as many cards
as you like from the red (wine) half and insert them anywhere among the
(water). Shuffle the black half. From it remove from anywhere the same
of cards you took from the reds and put them back among the reds. You'll find
the number of blacks among the reds is exactly the same as the number of
among the blacks. Students will enjoy proving that this is always the case.
But will it work if the two starting portions of the deck are unequal? (Yes.
It doesn't matter if the two glasses in the brain teaser are not the same
nor does it matter how many cards are in the black and red piles.)

This secondary math textbook has an index that is not very helpful. What
are more than 180 page references for the entry "Science"? What use is a
similar quantity of page numbers for the entry "Industry"?

WQED's boxed set of seven videotapes, Life by the Numbers, was funded mainly
by the National Science Foundation and the Alfred P. Sloan Foundation. The
photography is excellent. There are scenes of men and women mathematicians
seated at computer consoles, or driving a car, or walking down a street or
through the woods. There are many close-ups of their faces, dazzling
of mountains and skyscrapers, baseball games, martial arts contests,
blossoming flowers, wild animals, and everything imaginable that has little
to do with math. The tapes rate high on special effects, low on mathematical

The seventh tape covers a typical new new math class. To discover that the
longer a pendulum, the slower its swing, students tie weights to the ends of
string and swing the weights back and forth while other students keep charts
of string lengths and pendulum periods. After several days they learn that
the period of a pendulum is a function of its length. This discovery enables
them to calculate whether the victim in Poe's horror story "The Pit and the
Pendulum" has enough time to escape from the huge pendulum which threatens to
cut him in half as it swings lower and lower over his reclining body. It is
assumed that because students have fun swinging weights they will remember
function better than if a teacher takes a few minutes to demonstrate it by
swinging a weight and slowly lengthening the string. One of the most telling
attacks on new new math is Bernadette Kelly's article "Déjà Vu? The New 'New
Math,'" in the professional journal Effective School Practices (Spring 1994).
Kelly summarizes four case studies by four supporters of fuzzy math in which
they report on four fifth-grade teachers.4 Two of the teachers, called
and Valerie, are enthusiastic users of new new math techniques. The other
teachers use traditional methods.

Sandra was very good at getting students to cooperate in groups. However, in
one exercise she told students that one could obtain the perimeter of a
rectangular field by multiplying its length by its width! In another project
she calculated the volume of a sandbox by multiplying together its length and
width in yards, then multiplying the product by the box's height in feet!

In an interview Sandra said that while working on the sandbox problem her
pupils asked what a cubic foot was. "You know, the thing is that I couldn't
really answer that question. Then I thought and thought, then I remembered
how to measure a cube." Neither Sandra nor her students were ever aware of
two huge mistakes. In spite of these errors, the author of the article about
her said she was an "exemplary teacher." Sandra is praised for getting her
students to enjoy their cooperative efforts to solve problems "in the context
of real world situations." Finding a correct answer was less important than
having fun in working on the problem.

Valerie made an equally astonishing blunder. The task was to determine the
average number of times her thirty students had eaten ice cream over a period
of eight days. This was "solved," by dividing 30 by 8, to get 3.75, which
Valerie rounded up to 4!

As with Sandra, neither Valerie nor her students ever became aware that they
obtained a totally wrong answer. Nevertheless, the author of the paper about
her forgives her mistake on the grounds that she had succeeded so well in
getting her students to work on a problem in the context of their experience.
Moreover, the work had impressed on the students the "usefulness and
relevance of averages." No matter that they completely failed to find an

As for Jim and Karen, the two teachers who used more traditional methods, the
authors of the case studies are unimpressed by their students having scored
high on tests. Both are castigated for failing to appreciate the methods of
the new new math. What is deplorable, as Bernadette Kelly's article points
out, is not so much that the case studies revealed the incompetence of two
teachers, who come through as ignoramuses, as the authors' praise of Sandra
and Valerie for finding ways to get their pupils working joyfully on
problems. Little wonder that new new math is called fuzzy. Insights are
significant even when they are wrong.
The mathematician Sherman Stein, in his 1996 book Strength in Numbers,
devotes a chapter to a history of math reform movements. His hopes for the
new math are dim. "I am disturbed," he writes, that the authors of the [new
new math books] do not cite any pilot project or any school district as a
model to show that their goals can be achieved in the real world. That means
that they are proposing to change the way an entire generation learns
mathematics without checking the feasibility of their recommendations. A
manufacturer introduces a new soap with more care, first testing its
reception in a few stores or towns before committing to mass production.

But evaluating the efficacy of fuzzy math will not be easy. Too many
are involved, including the skill of teachers and the educational background
of parents, to mention only two. A glaring example of how research can be
biased is provided by a recent testing of pre-college math students around
world by the Third International Mathematics and Science Study. Results
announced last February revealed that American students did better than
students in just two other countries, Cyprus and South Africa. A cartoon in
The New York Times (March 8) showed a car's bumper sticker that said "My kid's
math scores beat kids in Cyprus and South Africa." Inside the car a
father is
giving a thumbs-up sign.

These statistics are worthless. In many cases the students in a foreign
country were much older than students here at the same grade level. More
significantly, in most foreign nations students in early grades who show no
aptitude for math are sent off to trade schools or to jobs, if they can find
them. In the US such students are required to continue attending high
Obviously our high school students will do less well on math tests than
students in countries where poor students are quickly moved out of the

Although we lack clear, systematic evidence that methods of fuzzy math are
inferior to older methods, education officials in California, the nation's
largest customer for math textbooks, have suddenly turned against the new
math. The change in state policy was mainly in response to the outrage of
parents who complained that their children were unable to do the simplest
arithmetical calculations. Their outrage was backed by many top
and scientists. Michael McKeown, for example, a distinguished molecular
biologist at the Salk Institute, heads a parental group called Mathematically
Correct. "We're not opposed to teaching concepts," he told Newsweek
("Subtracting the New Math," December 15, 1997). "I am opposed to failing to
give a kid tools to solve a problem."

In a vote of ten to zero (one person abstained) the eleven members of
California's Board of Education recommended this spring a broad return to
basics in math teaching. The decision is sure to have an effect in other
states. The board said students should learn the multiplication table by the
end of the third grade, and that fourth- graders should know how to do long
division without consulting a calculator. It banned the use of calculators on
state tests. Teachers were urged not to introduce calculators before grade

Defenders of fuzzy math are, of course, dismayed. They branded the board's
decisions a product of nostalgia, and a contribution to our country's
down. The National Science Foundation, which has given more than $50 million
to California districts for research on new math teaching, is furious. It
threatened to withdraw further funding to any California district that adopts
the board's recommendations.
The conflict is bitter and far from over. It may be many years before it
becomes clear how to sift out from the new new math what is valuable while
retaining worthy aspects of older teaching methods.5 My own opinion is that
the most important question concerning the teaching of math is not how big
and colorful textbooks are, how many visual aids are used, how the classroom
is physically arranged, or even what methods are used in it. The greatest
threat to good math teaching is surely the low pay that keeps so many
excellent teachers and potential teachers out of our schools. What matters
than anything else is having trained teachers who understand and love
mathematics, and are capable of communicating its mystery and beauty to their

1 Textbooks emphasizing multiculturalism are proliferating rapidly. Here
are a few: Africa Counts: Number and Pattern in African Culture, by
Zaslavsky (Lawrence Hill, 1997); Multiculturalism in Mathematics, Science
and Technology, by Miriam Barrios-Chacon and others (Addison-Wesley,
1993); Multicultural Mathematics: Teaching Mathematics from a Global
Perspective, by David Nelson, George Gheverghese Joseph, and Julian
Williams (Oxford University Press, 1993); Teaching with a Multicultural
Perspective: A Practical Guide, by Leonard Davidman and Patricia T.
Davidman (Perseus, 1996). Striking multicultural math posters are
from teaching supply houses. (back)

2 See also the letters in The New York Times of August 17, 1997, and an
earlier article by Cheney in the Weekly Standard (August 4, 1997).
3 On the myth of Dogon astronomy, see Carl Sagan, Broca's Brain (Random
House, 1979), pp. 63-64, and Chapter Six; Ian Ridpath, "Investigating the
Sirius 'Mystery,'" in The Skeptical Inquirer, Vol. 3 (Fall 1978), pp.
and Terence Hines, Pseudoscience and the Paranormal (Prometheus Books,
1988), pp. 216-219. (back)

4 R.T. Putnam, R.M. Heaton, R.S. Prewat, and J. Remillard, "Teaching
Mathematics For Understanding," in Elementary School Journal, Vol. 93
(1992), pp. 213-228. (back)

5 That the new new math has positive aspects goes without saying. It is
important that students understand the basic concepts of math and not just
memorize procedures that work; and to give students such conceptual
understanding teachers themselves must have such understanding. This is
theme of a recent monograph, Middle Grade Teachers' Mathematical Knowledge
and Its Relationship to Instruction, by Judith Sowder, Randolph Philipp,
Barbara Armstrong, and Bonnie Schappelle (State University of New York,
1998). The monograph reports on a two-year investigation of five teachers,
with a primary emphasis on how they taught fractions. Why, for example, in
dividing one fraction by another do you flip upside down the divisor
fraction, then multiply the numerators and denominators? Should
teachers be
content with letting students accept this as a trick that works like
or try to answer a student who asks, "Why is this division?" The monograph
defends the admirable aspects of math reform without going to fuzzy-math
extremes. (back)

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