Glencoe Pre-Algebra
The experience of recording Glencoe Pre-Algebra left me
wondering how anyone could learn any math from this book. It looks
like a one-stop-shopping depot of blunders, and it displays nearly
all of the undesirable didactic practices that I can think of.
The fundamental reason for this is quite obvious -- Glencoe
Pre-Algebra is an 800-page tribute to the fad called "fuzzy
math." Fuzzy math (which is known by several other names as well,
including "new new math," "maybe math," "rain-forest math" and "math
appreciation") is the replacement for the disastrous "new math" fad
of the 1960s and 1970s. New math was a collection of corrupted
factoids drawn from real mathematical disciplines, such as set
theory and number theory. Fuzzy math is a corruption of notions
drawn from pop culture and especially from pop psychology.
I must point out that fuzzy math is not the same thing as fuzzy
logic. Fuzzy logic is a formal branch of mathematical inquiry,
usually taught at the graduate-school level. It is based upon a
rigorous body of theory, and it has important applications in
systems-analysis work. Fuzzy math is not formal, is not rigorous,
has little to do with mathematics, and is barely applicable to
anything. The similarity of the names "fuzzy logic" and "fuzzy
math" is unfortunate.
Marianne Jennings described some fuzzy math to readers of The
Textbook Letter in her 1996 article "Rain-Forest Algebra and MTV
Geometry." That article was Jennings's analysis of fuzzy-math
books that her daughter had been forced to use in public schools in
Arizona [see note 1, below]. For a broader account of fuzzy math
-- with information about its history, its notions, and some people
who have promoted it -- I recommend Martin Gardner's essay "The New
New Math," which appeared in The New York Review of Books
[note 2]
As Gardner observes, fuzzy math is strongly influenced by
multi-culti, by environmentalism and by feminism, and the devotees of
fuzzy math are preoccupied with "equity" and "internalized
self-image" and other pop notions. To the extent that they concern
themselves with math lessons, they emphasize such things as
"interactive learning" and making students work on "open-ended"
problems. Those are code-phrases. "Interactive learning" means
that students work in groups and try to perform tasks by
trial-and-error, instead of listening to a teacher instruct them in
established principles of mathematical analysis and computation.
Working on "open-ended" problems means trying to reinvent math that
has already been invented. Let me quote Gardner:
According to fuzzy math, this is a terrible way to teach the
theorem. Students must be allowed to discover it for themselves. .
. . Using the edges of [squares cut from paper], 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 two other 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 the teacher, the
children are supposed to discover that with right triangles a2 +
b2 = c2.
Fuzzy-mathers purport to think that this sort of thing is better
than having a real teacher of mathematics demonstrate the art of
constructing mathematical proofs. They purport to believe that
making students fumble around for hours or days, trying to discover
what we already know, is preferable to having a real teacher of
mathematics explicate the mathematical knowledge that humanity has
accumulated over the centuries. Memorization -- even the
memorization of multiplication tables and other indispensable
information -- is anathema to fuzzy-mathers, who refer to
memorization tasks as "drill and kill."
It is easy to recognize fuzzy-math textbooks. They fairly glow with
racial awareness and political correctness, as manifested in
frequent, self-conscious, irrelevant references to blacks, women and
other members of Victim groups. The books also display a vivid,
though sometimes confused, awareness of health issues and
environmental issues, and they have swarms of bright-colored
pictures that bear little relation, or no relation at all, to the
text. Gardner, describing a fuzzy-math book produced by
Addison-Wesley, remarks that "the book's mathematical content is often hard
to find in the midst of material that has no clear connection to
mathematics."
One could say the same thing about Glencoe Pre-Algebra, if
one wanted to make an extreme understatement.
The parade of irrelevancies, discontinuities and non sequiturs (both
visual and verbal) continues into the body of the book. Chapter 1
("Tools for Algebra and Geometry") starts out with a lengthy
advertisement for blue jeans manufactured by Levi Strauss & Co. --
and now the phrase product placement springs forcefully to my
mind. Product-placement advertising is well established in the film
and television industries: Corporations pay fees to secure the
"placement" (i.e., the displaying or mentioning) of their products
in films, television dramas, or game shows. Now, it seems,
corporations can also buy placements in Glencoe schoolbooks.
The Levi Strauss ad, thinly disguised as a news report, tells
students that
Apparently, the young student of mathematics is supposed to infer
that there is a branch of math called "fancy figuring."
The ad is augmented by a time-line which tells about great moments
in the history of jeans. The student learns, for example, that 1926
saw the introduction of "First jeans with a zipper instead of
buttons." (This item is illustrated not with a picture of a zipper
but with a picture of a button, and the button displays the name
"Levi Strauss & Co.") Similarly, the student learns that 1939
brought the introduction of "Lady Levis, the first name-brand jeans
made specifically for women." However, one illustrated item on the
time-line is plainly absurd: "1974 Beverly Johnson, the first black
model on the cover of a major fashion magazine."
What does Beverly Johnson or her blackness have to do with Levi
Strauss & Co.? What is she doing here? She certainly isn't wearing
jeans.
It is interesting to compare the Levi Strauss advertising with
certain other items that appear in Glencoe Pre-Algebra. Page
155 has a photograph of a package of dental floss, but an artfully
placed toothbrush covers the brand-name. On page 405, a photograph
shows packages of such familiar products as Kraft grated cheese and
Dinty Moore Beef Stew -- but here a Glencoe illustrator has gone to
the trouble of obliterating the brand-names. These observations
lead to an inference: Kraft Foods, Inc., and Hormel Foods
Corporation (the manufacturer of Dinty Moore stew) were unwilling to
pay product-placement fees to Glencoe, but Levi Strauss & Co. sent
Glencoe a check.
Remember that we have merely looked at the first few pages of the
first chapter. We still have far to go and many chapters to
traverse.
On our way to attaining a proper comprehension of pre-algebra, we
have to absorb more doses of irrelevant advice about careers.
(After all, there's more to life than retailing!) We must digest an
advertisement for Reebok shoes, read about the tallest recorded
house of cards, and apprehend that Chinese players of bridge are
turning into strong competitors. We must try to find the
arithmetic in an arithmetic problem that begins: "Bertha Thangelane
of Soweto, South Africa used the political changes in South Africa
to start her business. She is the co-owner and managing director of
RNB Creations (Pty) Ltd., which makes school uniforms." (The
arithmetic finally emerges when we determine how many uniforms this
woman can cut from a 50-yard bolt of cloth. "Pty" and "Ltd." are
never explained.) We must learn that in 1974 Alex Haley finished
writing his book Roots, which "traced his family's history"
back to an African named Kunta Kinte, and that "A TV miniseries of
Roots made in 1977 won a record-setting 9 Emmys." (However,
we don't have to learn that Haley's book was a fake, or that Haley
promoted both Roots and himself by making dubious claims
[note 3]. Glencoe's writers don't mention these points.) We must
learn that potato chips were invented in 1853 by an Amerindian named
George Crum, that "the first African-American to have a speaking
role in a nationally televised commercial" was a woman named Gail
Fisher, and that a black man named Lewis Howard Latimer made the
drawings that accompanied Alexander Graham Bell's application for a
patent on the telephone.
And so on, and so on, and so on. I could list many more of the
diversions, decoys and irrelevant bagatelles that I have seen in
Glencoe Pre-Algebra, but I'm sure that you have caught the
overall nature of the book. Glencoe Pre-Algebra seems more
likely to induce attention-deficit disorder than to impart any
mathematical knowledge. An article that appeared four years ago in
the American Educational Research Journal pointed to research
which showed that the addition of vivid, distracting material to a
book can often diminish students' recall of information that is
important [note 4]. That research was published in 1992, but
Glencoe's writers apparently don't know about it or don't care.
Perhaps that is the reason why I also am troubled by the
end-of-chapter "Alternative Assessment" sections that occur throughout
Glencoe Pre-Algebra. Read, for example, the material at the
end of chapter 10. In the section headlined "Study Guide and
Assessment," the exercises demand that students plot data, recognize
permutations and combinations, do calculations involving factorials,
and "Find the range, median, upper and lower quartiles, and
interquartile range" of a given list of numbers. But in the section
headlined "Alternative Assessment," students are sent away to write
an article titled "Are you average?" or to ponder whether they are
using their time wisely, and each student is directed to "Select an
item from your work in this chapter that shows how your
understanding of the concepts improved and place it in your
portfolio." Here Glencoe is clearly declaring two expectations.
First, some students will be completely defeated by the material in
chapter 10 and will be unable to perform the exercises in the
"Study Guide and Assessment" section. Second, teachers will want
some pretext for awarding good grades to such students and for
pushing the students ahead to be defeated by chapter 11. The
"Alternative Assessment" section provides the desired pretext.
Glencoe's two expectations reflect a basic tenet of fuzzy math: The
"reformers" who have promoted fuzzy math have explicitly renounced
the idea that it is necessary for each student to acquire certain
basic skills, at certain points in the math curriculum, before
moving forward [note 5]. This renunciation coincides with a broader
fad that extends beyond math education and has affected the teaching
of many subjects in many school districts. Impelled by
pseudoegalitarian social ideology, many districts deliberately mix
together, in the same classes, students who vary greatly in their
capacities and their degrees of preparation [note 6]. The backward
students in such "heterogeneous" classes cannot hope to handle the
work or the assessment exercises that the better students can
handle, so the backward students are assessed by a different set of
standards. The term alternative assessment is an established
code-phrase for the process of creating the impression that backward
students have succeeded when they actually (and predictably) have
failed. The other code-phrase that applies here is social
promotion. I pity the alternative-assessment students who are
to be dragged along, bewildered, through curricula that clearly do
not match their skills or their needs.
In Glencoe Pre-Algebra, the "Alternative Assessment" sections
seem to become weaker and less mathematical as the chapters roll on.
In a pathetic way, this makes sense. As the chapters roll on, the
alternative-assessment students will fall farther and farther
behind, and will grasp less and less.
In keeping with a dictate of fuzzy math, students are expected to
reinvent the laws of arithmetic on their own, and to do this while
"Modeling with Manipulatives." The term manipulatives is
jargon. The manipulatives are small toys (such as counters or tiles
or sticks) that usually represent quantities, and the students are
supposed to move these objects around in ways that represent
mathematical operations or that will reveal mathematical
relationships. This soon becomes ridiculous. There is no doubt
that colored counters can help young children learn to add and
subtract, but can you imagine using counters for showing how to find
the product of two negative numbers? Believe it or not, Glencoe
puts forth (on page 98) an ornate procedure for doing this:
Students are supposed to use positive and negative counters,
including "as many zero pairs as you need" -- but the procedure
seems to be circular, because knowing the proper number of "zero
pairs" that "you need" evidently requires foreknowledge of the
product that is being sought.
As another example of Glencoe's contrived confrontations with
manipulatives, let me cite "Representing Polynomials with Algebra
Tiles" (page 710). The tile representing x is a square whose
side is about three times as great as the side of the tile
representing 1. Does this mean that x equals 3? Does
it mean that x is equal to the square of 3? Is it supposed
to imply that the sum of x + x + x is equal to x
squared? I shudder to think of the tiles that would be needed for
"representing" polynomials that contain third-power or fourth-power
terms.
As they toil to reinvent arithmetic and elementary algebra, the
students "verify" their discoveries by using something that already
has been invented: a calculator. On pages 94 and 95, for example,
the students' task is to multiply 74 by 76. You or I would do this
by using long multiplication, which we learned in the fifth grade --
but long multiplication definitely isn't fuzzy, and fuzzy-math
teachers might find it baffling. Accordingly, Glencoe has devised a
procedure in which students begin by examining a table of factors
and products (4 x 6 = 24, 14 x 16 = 224, 24 x 26 = 624, and so on).
Then the students are supposed to find numerological patterns in the
groups of digits forming each product, and then they are supposed to
divine the groups of digits in the product of 74 x 76. Finally,
they "Use a calculator to verify the result." This is but one of
many exercises in which students do fuzzy things and then seek
verification from the oracular calculator.
Though it is reasonable for Glencoe to teach something about the use
of calculators, I have qualms about Glencoe's approach:
The emphasis on calculators in Glencoe Pre-Algebra leads me
to make some comments about costs. One reason why this book has
more than 800 pages is that it is swollen with pointless
illustrations, tangential tales, racial frolics and other gimmicks
-- gimmicks that may hypnotize and comfort fuzzy-math teachers but
can only distract, confuse and impede students. Glencoe has
incurred costs by contriving all those gimmicks, and by printing all
the extra pages that are required for displaying them, and now such
costs are reflected in the selling price of the book. Moreover, a
school district that buys Glencoe Pre-Algebra must also buy
Glencoe's toy-boxes of algebra tiles, "geoboards," equation mats and
other fuzzy-math props. And then come the calculators -- perhaps to
be bought by the district, perhaps to be bought by the students. At
retail, the calculators will cost about $90 apiece [note 7]
School-district officials should carefully weigh the expenses
associated with using Glencoe Pre-Algebra, including the
expenses imposed on students and the expenses imposed on teachers
who use their own money to buy some of the instructional materials
they use in their classrooms. Is Glencoe's swollen product
worth all that money?
Problem 41 carries the label "Social Studies." This signals that
Glencoe's writers not only can produce ethnokitsch but also can
comply with the fad that calls for devising "connections" between
different subjects:
b. List five years that are Years of the Rat.
As nearly as I can tell, the writers derived their material from a
place-mat in a Chinese restaurant. They write as if the people who
live in China today were still reckoning time by using the
traditional Chinese lunar calendar. Not so. In their daily lives,
today's Chinese use the same Gregorian calendar that we use.
After stating that a Chinese year starts "in late January or early
February" of a Gregorian year, the writers go on to say that "1996
was the Year of the Rat." Let me paraphrase that: A Chinese year
doesn't start when a Gregorian year starts, but Gregorian 1996 was
identical with the most recent Year of the Rat [note 8]. Surely
even an alternative-assessment student can see that 1996 couldn't
have been a Chinese Year of the Rat if 1996 wasn't a Chinese year to
begin with.
Now the writers ask a question: "How old is the youngest person
born in a Year of the Tiger?" Let me explain why that question is
absurd. Students who confront the Year-of-the-Tiger question today
can't answer it because Glencoe's writers have failed to give
necessary information: The most recent Year of the Tiger began on 28
January 1998 and ended on 15 February 1999. How about students who
confronted the Year-of-the-Tiger question before 28 January 1998
arrived? [note 9] Those students could not have answered the
question either, because they needed to know that the previous Year
of the Tiger had ended on 28 January 1987. This information, too,
is absent from Glencoe's book.
All of that silliness and absurdity could have been avoided if the
Glencoe writers had possessed some real respect for Chinese culture
and had written their problem wholly in terms of the Chinese lunar
calendar. They could have given legitimate information about the
Chinese calendar, and could have asked some legitimate questions,
within that calendar's own framework. The pretense that Chinese
lunar years correspond to Gregorian solar years is fortune-cookie
stuff, and it just doesn't work -- as math, as history, or as
anything else.
Readers of The Textbook Letter are familiar with the rule
which states that the lame "history" invented by textbook-writers is
never as interesting as the real history that the writers don't
know [note 10] In Glencoe Pre-Algebra, this rule is
confirmed by the Chinese-calendar problem and by many other items as
well. On page 94, for instance, Glencoe's writers turn a famous
story about Carl Gauss into colorless, pointless pap. Let me first
show you how the story is told in a real book -- Paul Hoffman's
The Man Who Loved Only Numbers [note 11]. Where Hoffman uses quotation marks, he is quoting the 20th-century mathematician Paul Erdos:
And he certainly could calculate. At the age of ten, he was a
show-off in arithmetic class at St. Catherine elementary school, "a
squalid relic of the Middle Ages run by a virile brute, one
Büttner, whose idea of teaching the hundred or so boys in his charge
was to thrash them into such a state of terrified stupidity that
they forgot their own names." One day, as Büttner paced the room,
rattan cane in hand, he asked the boys to find the sum of all the
whole numbers from 1 to 100. The student who solved the problem
first was supposed to go and lay his slate on Büttner's desk; the
next to solve it would lay his slate on top of the first slate, and
so on. Büttner thought the problem would preoccupy the class, but
after a few seconds Gauss rushed up, tossed his slate on the desk,
and returned to his seat. Büttner eyed him scornfully, as Gauss sat
there quietly for the next hour while his classmates completed their
calculations. As Büttner turned over the slates, he saw one wrong
answer after another, and his cane grew warm from constant use.
Finally, he came to Gauss' slate, on which was written a single
number, 5050, with no supporting arithmetic. Astonished, Büttner
asked Gauss how he did it, "And when Gauss explained it to him,"
said Erdos, "the teacher realized that this was the most important
event in his life and from then on worked with Gauss always," plying
him with textbooks, for which "Gauss was grateful all his life."
What was Gauss' trick? In his mind, he apparently pictured
writing the summation sequence twice, forward and backward, one
sequence above the other . . . . Gauss realized that you could add
the numbers vertically instead of horizontally. There are 100
vertical pairs, each summing to 101. So the answer is 100 times 101
divided by two, since each number is counted twice. Gauss easily
did the arithmetic in his head.
Now here is Glencoe's version. Purged of all human interest, it is
the most forgettable version that I ever have seen:
That's all. The Glencoe writers then hasten to bury Gauss's
wonderful insight under some questions and calculator exercises.
Sometimes, the "history" in Glencoe's book is not just vague and
dull -- it's wrong. On page 618, for example, a geometric figure
is accompanied by this caption:
The Swedish mathematician Niels Fabian Helge von Koch (1870-1924)
developed the "snowflake" in 1905 or so. He evidently wanted it to
exemplify figures that are continuous everywhere but differentiable
nowhere. A figure is continuous if it holds water -- i.e., if there
are no gaps in it. A figure is differentiable (loosely speaking)
wherever it is smooth. A circle, for example, is smooth everywhere,
while a square is smooth everywhere but at its sharp corners. In
the Koch snowflake, however, no part is smooth. The entire figure
consists of sharp corners, without any smooth lines connecting them.
The claim that Koch "worked on fractal geometry" exemplifies a kind
of anachronistic distortion that occurs often in revisionist
pseudohistory. (Here is another example: The "abstract
expressionist" artists of the 1960s claimed that Pierre Auguste
Renoir (1841-1919) was one of their band.) In Koch's day, the term
fractal did not exist. That term was coined by Benoit
Mandelbrot in the 1970s, when fractal geometry emerged, with
considerable help from Mandelbrot, as an important field of study.
By the way: It is not always true that "if you look at any part of a
fractal, you will see a miniature replica of the larger design."
Nor is a broccoli plant a fractal.
All of this notwithstanding, I believe that Glencoe
Pre-Algebra could have been worse. Unlike the fuzzy-math texts
that Marianne Jennings encountered, Glencoe Pre-Algebra does
have some math in every chapter. Exercises and problems seem to be
plentiful enough to enable students to practice and master the
techniques that Glencoe's writers recommend. By regularly relating
mathematical constructs to the physical world, the writers help to
answer students' "Who cares?" questions. And the writers generally
avoid the mistake of presenting problems which involve units,
definitions or abbreviations that haven't yet been introduced -- a
mistake that is common in some other books. These desirable
features of Glencoe Pre-Algebra are offset, however, by the
book's lack of rigor. Students who are starting to learn algebra
should acquire some idea of what proofs are and why proofs are good
things.
I'm glad that Glencoe pays attention to "mental math" and
estimation. Learning to estimate correct answers is important, and
it grows even more important as students tackle increasingly
complex problems. I wish, however, that Glencoe's approach to
developing mathematical intuition were more consistent. On page 417
a calculator "activity" shows a 45-degree line whose slope seems to
be 2 instead of 1. I had to look carefully to discern that the
scale used on the vertical axis differs from the scale used on the
horizontal axis. This surely cannot help students to develop an
intuitive appreciation of how the m in y = mx + b
affects a line's appearance.
I recently heard about a math teacher who was scouring used-book
sales and looking for out-of-print math textbooks. He felt that
the newer books were nearly worthless, so he was assembling his own
private stock of textbooks that really taught math. After reading
Glencoe Pre-Algebra, I can understand why.
Notes
Tom VanCourt is a software engineer. His review in this issue
originated from his work with a charitable organization that makes
audiotapes of textbooks, for use by blind or dyslexic students. He
lives in Charlestown, Massachusetts.
Reviewing a mathematics textbook
An Integrated Transition to Algebra & Geometry
1997. 843 pages. ISBN of the student's edition: 0-02-825031-1.
Glencoe/McGraw-Hill, 936 Eastwind Drive, Westerville, Ohio 43081.
(Glencoe/McGraw-Hill is a division of the McGraw-Hill Companies.)
Glencoe's Manual of Fuzz
Tom VanCourt
I read textbooks aloud. I am affiliated, as a volunteer reader,
with an organization that produces audiotapes of textbooks, for use
by blind or dyslexic students. During my work as a reader, I
recite what each book says, and I describe aloud any equations,
important illustrations, or other substantive appurtenances that
accompany the main narrative. I specialize in recording books that
deal with the natural sciences, with computer science, or with
mathematics -- and this is how I encountered Glencoe
Pre-Algebra.
[T]eachers 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 areas of the two
smaller squares.
Selling "Dream Jeans"
While jeans may be the most popular fashion of all time,
sometimes finding a pair that fits is like searching for buried
treasure. But Levi Strauss, the original maker of jeans, is coming
to the rescue. You can now go to a store and get a pair of jeans
made to measure. . . . After some fancy figuring, the computer
chooses a trial pair of jeans for you to try on. . . . In three
weeks, your dream jeans arrive. Being in fashion has never been so
easy!
Bertha Thangelane, Alex Haley, Potato Chips . .
.
"Alternative Assessment"
Reinventing Arithmetic
Ethnokitsch and Forgettable "History"
Social Studies Chinese New Year, the most important
holiday of the Chinese year, comes in late January or early
February. The Chinese calendar follows an annual cycle, which
repeats every twelve years. For example, 1996 was the Year of the
Rat, so 1996+12 or 2008 and 1996-12 or 1984 are also the Year of the
Rat.
a. 1998 is the Year of the Tiger. How old is the youngest person
born in a Year of the Tiger?
Insights and connections -- that's what mathematicians look for.
Carl Friedrich Gauss, who was born in 1777 in Braunschweig, Germany,
the son of a masonry foreman, was a master at exposing unsuspected
connections. . . . Gauss was a mathematical prodigy, and in his old
age he liked to tell stories of his childhood triumphs. Like the
time, at the age of three, he spotted an error in his father's
ledger and stopped him just as he was about to overpay his laborers.
Like the fact that he could calculate before he could read.
Carl F. Gauss (1777-1855), a famous German mathematician, taught
himself to read and to make mathematical calculations. When he was
10 years old, his teacher asked the students in his class to find
the sum of the numbers from 1 to 100. The teacher thought that
this problem would keep the students busy for a long time, but young
Gauss wrote the answer almost immediately.
The design shown at the right is called the Koch snowflake. It
is made up of three rotated copies of the Koch curve, introduced by
Helge von Koch, a mathematician who worked on fractal geometry.
Fractals are self-similar. That is, if you look at any part of a
fractal, you will see a miniature replica of the larger design. A
tree of broccoli is an example of a fractal in nature. Each branch
of the tree is a miniature replica of the entire tree.
Saving the Fishes
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