Martin Gardner, "The New New Math," New York
Review of Books, September 24, 1998, pp. 9-12.
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 Mathematics.
Focus on Algebra: An Integrated Approach by Randall I. Charles,
Alba Gonza1lez Thompson et al. 843 pages, $56.00 (hardcover)
published by Addison-Wesley.
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 strata.
These are troubling statistics because, in an advanced
technological society such as ours, a firm grasp of
basic mathematics is increasingly essential for
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) began 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 about 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 yearbook, Multicultural and Gender Equity
in the Mathematics Classroom: The Gift of Diversity.
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
important part of building a gender-equitable
multicultural classroom environment." Over 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 self-image."
"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
Ethnomathematics: 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 correctly.
Knowing how pre-industrial cultures, both ancient and
modern, handled mathematical concepts may be of
historical interest, but one must keep in mind 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 the symbols are identical. Two elephants
plus two elephants makes four elephants in every
African tribe, and the arithmetic of these cultures is a
1 Textbooks
emphasizing
multiculturalism are
proliferating rapidly.
Here are a few: Africa
Counts: Number and
Pattern in African
Culture, by Claudia
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 available
from teaching supply
houses. (back)
miniscule portion of the vast jungle of modern
mathematics. A Chinese mathematician is no more
concerned with ancient Chinese mathematics,
remarkable though it was, 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 identityan 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 division.
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 multiplication 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 calculator.
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 theory, 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 impossibility.
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
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).
(back)
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 areas of the two smaller squares.
According to fuzzy math, this is a terrible way to teach
the theorem. Students 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 shapes. 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 triangle 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 a2+b2=c2.
"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 Arizona State
University, whose daughter was getting an A in algebra
but had no notion 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 makes 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 fifty 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 was 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 something 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
professionals, 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 are 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 taxes 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 connection 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 other
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 money in the cook-off," and "How would
you set up a hotness scale for peppers?" This 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 first question is
"What role should zoos play in today's society?" The
book's index, 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 Rodas became vice-president of
a toy company, and how Linda Johnson Rice found a
creative way to market Ebon=E9 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 into the text.
The most outrageous page opens a section on linear
functions concerns the 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. However, it has long been known that the
Dogon made no such discoveries. They merely learned
these astronomical facts from missionaries and
other Western visitors. 3
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
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. 56-62; and
Terence Hines,
Pseudoscience and
the Paranormal
(Prometheus Books,
1988), pp. 216-219.
(back)
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 drop
of the mixture goes back to the wine. Is there now
more water in the wine than 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 blacks (water).
Shuffle the black half. From it remove from anywhere
the same number 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 reds 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 size; 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 value 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 glimpses 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 content.
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 the 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 "D=E9j=E0 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 Sandra and Valerie, are enthusiastic
users of new new math techniques. The other two
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!
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)
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 her 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 average.
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 deemed 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 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 variables 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 the 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 school. 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 system.
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 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 mathematicians 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 six.
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 dumbing
down. The National Science Foundation, which has
given more than $50 million to California districts
for research on new math teaching, is furious. It has
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 more than anything else
is having trained teachers who understand and love
mathematics, and are capable of communicating its
mystery and beauty to their pupils.
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
the 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 magic,
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.