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from The Textbook Letter, November-December 1996

Rain-Forest Algebra
and MTV Geometry

Marianne M. Jennings

About a year and a half ago, when my daughter Sarah was in the 8th grade, she asked me to help her with her algebra homework. As I opened the textbook that she was using, I was prepared to read a little y = mx + b, but what I saw in this book was Dogon art, maps of South America, and tips about endangered species.

I tossed it back to my daughter and said, "Wrong book. This is geography." She rolled her eyes (as teenagers do when they want to signal "How can you be so dumb?"), and she pointed to the title on the book's cover: Addison-Wesley Secondary Math: An Integrated Approach: Focus on Algebra.

Since then I have spent a lot of time with Addison-Wesley's Focus on Algebra -- a book that easily could pass for a satire of contemporary American education and of the sociopolitical indoctrination that goes on in many of our classrooms. Even Saturday Night Live in the 1970s wasn't this good. Here are some of the problems that Addison-Wesley's writers wanted my daughter to attack in the name of algebra:

What other kinds of pollution besides air pollution might threaten our planet? [page 163]

Each year the Oilfield Chili Appreciation Society holds a chili cook-off. . . . 1. The chili cook-off raises money for charity. Describe some ways the organizers could raise money in the cook- off. 2. What is the hottest kind of pepper that you have eaten? People who have tasted them agree that cayenne peppers are hotter than pimento peppers. How would you set up a hotness scale for peppers? . . . . [page 217]

What role should zoos play in today's society? . . . . [page 233]

[A] zoo sponsors a creative writing contest for high school students. The topic for the essay this year is "Why should we save an endangered species?" . . . . What would you use as criteria for judging the essay? [page 253]

Welcome to rain-forest algebra! Under Addison-Wesley's wise tutelage, students master such important algebraic skills as measuring each other's armspans. They learn that fossil fuels are The Devil's handiwork. They discuss toxins in the environment. They read Maya Angelou's poetry. They write essays on why parallel sentence structure is similar to parallel lines. I would not be surprised to learn that they also sing Kum Ba Yah each day.

I wonder how the United States ever achieved its eminence in science and technology when students had to use algebra texts that were full of numbers and equations, instead of irrelevant fluff and lessons in social ideology. If Addison-Wesley's preaching is what passes for algebra these days, then I too could write a math book, with problems like this:

By cutting down beautiful trees in the forest, a logger person earns $20. What do you think of scumbags who make a living in this fashion? Break into groups and discuss how the forest birds, snakes and other creatures, who are our Earth brothers and Earth sisters in every sense, feel about this rapacious act. Write an essay expressing your outrage, and don't feel constrained to use correct spelling.

I'm sure that my book would make me rich beyond numbers -- or at least beyond any numbers that students of rain-forest algebra will ever recognize.

Addison-Wesley has sprinkled Focus on Algebra with photos of young people who tell us how to solve mathematical problems. These juvenile sages include a "Jason" and an "Elizabeth," but an inordinate number have odd names like "Esteban," "Taktuk," "Kirti" and "Keisha." This is a transparent and strained effort to exploit the educational fad that calls for plugging "diversity." Esteban, Taktuk, Kirti and Keisha obviously are diversity decoys, though I do not know who decided that "diversity" means having strange names.

For an example of how the decoys and the other juvenile sages are used, try page 377. A problem appears at the top of the page, then Kristin and Esteban show up to tell us how they will tackle it. First there is a photo of Kristin, with a report of her ruminations:

Kristin thinks . . .

I'll graph both equations on the same coordinate plane to decide.
I'll graph the first equation by sketching a line with slope -3 . . . .

Then comes a picture of Esteban, who is conducting ruminations of his own:

Esteban thinks . . .

First, I'll write the second equation in slope-intercept form. . . . The second equation written in slope-intercept form is the same as the first equation!

What the book is trying to teach is that a straight line is fully defined if we know two of its points, but Addison-Wesley's writers can't say this outright. Instead, they disguise it as The Wisdom of Kristin and then as The Wisdom of Esteban.

Aside from providing lots of opportunities to show the diversity decoys, these inane and condescending monologues serve only to break up the text and confuse the student. As my daughter said: "Who cares what Kristin thinks or Esteban thinks? Couldn't I just learn how to compute slope?"

From time to time, in a page-margin, one spots a little icon that looks like a Conehead. Or maybe it represents Jack, the hydrocephalic clown who serves as spokesman for a chain of hamburger stands. Or it could be a head wearing a dunce cap. Whatever it is, it signals an item that supposedly integrates algebra with other things. On page 316, for example, a Conehead marked "Science" points his cone at a problem that begins with: "Zoom! After flying at a constant altitude, a pilot decides to zoom upward. The graph shows the change in altitude each second." And what a graph it is! -- a biplane, reminiscent of the Red Baron, flying on a grid! With integration like this, there's no question that our youngsters' science scores will improve.

On page 348 a Conehead marked "Literature" aims his cone at a real brain-teaser. It seems that Patwin (Taktuk's twin brother, no doubt) has been reading Twenty Thousand Leagues Under the Sea, has wondered what a league is, and has done "research" which revealed that "1 league = 3.45 miles" and "1 mile = 5280 feet." Now the students must use Patwin's discoveries in answering questions: "What is a league in miles? in feet?" Could Addison- Wesley's writers make "Literature" any more challenging?

On page 361 an "Industry" Conehead turns up; and if students believe what he has to say, most American businesses will go into Chapter 11 bankruptcy within three years after these youngsters hit the workforce. The Conehead touts a problem in which students are supposed to graph the "profit" that a medical office makes from a blood-processing machine, given that the machine costs $12,000 and generates "a revenue of $135" each time it is used. But who is running the machine? Wouldn't it be lovely to have a business in which there were no labor costs (or maintenance costs or amortization charges) to affect profits?

My favorite, however, is the integration supplied on page 208, where a "Fine Arts" Conehead points his cone at this puzzler:

In the Beatles' song Taxman, the tax collector boasts of leaving the worker with only 5% of his income, and sings "that's one for you, nineteen for me." (What does nineteen refer to?) . . . .

That Conehead needs to learn that the Beatles, even in their "White Album," never reached the level of "Fine Arts." (As for the interchanging of fraction and percentages: This is something that I learned in grade 4.)

A "Very Poor" Book

It is difficult to focus on the text in Focus on Algebra, because the text is intermixed with problems, problems are mixed with discussions, and discussions are mingled with sermons and environmental alarms, among other things. There are "Self- Assessment" sections, "More Math Reasoning" sections, "Explore" sections, "Review" sections, "What Do You Think?" sections and "Performance Task" sections, not to mention the confusing way in which the book's chapters are laid out: Each chapter is subdivided into numbered segments, and each segment consists of Part A, Part B, etc. It's a two-Tylenol headache to find your homework assignment amid all the rubble -- which ultimately fills 812 pages.

Just why would anyone need 812 pages to present 8th-grade algebra? In Japan, a good math book for students of the same age has about 200 pages, as I have learned from Richard Askey, of the Department of Mathematics at the University of Wisconsin in Madison. Yet Japanese students regularly and decisively outperform their American counterparts. One reason for this, I'll bet, is that their slim math books concentrate on teaching math, instead of on vilifying fossil fuels, singing the praises of recycling, and trying to convince students that spilling crankcase oil onto the ground is worse than committing felony murder.

Richard Askey has seen Focus on Algebra and has described it to me as "very poor." He also has sent me his own review of Focus on Algebra, which he presented at an academic meeting held last summer. In that review, Askey emphasizes that Addison-Wesley's book purports to cover important concepts without telling where they come from or how they are rooted in mathematical reasoning. He observes, for example, that the treatment of irrational numbers is incomprehensible; the writers list some irrational numbers but fail to show why the numbers are irrational, and the student "can do no more than guess" in trying to answer a question that involves distinguishing the squares of irrationals from the squares of rationals. In the same way, Askey says, Addison-Wesley's writers introduce the Pythagorean Theorem (during a sort of parody of "discovery learning"), and then they present a lot of problems, but they never provide a proof of the Theorem. Even the quadratic formula, Askey says, is treated as a device of unknown origin:

[Addison-Wesley's book] gets around to quadratic equations on page 650. They are first solved using a graphing calculator. Then they are solved by looking at values . . . given in a table. Factoring is next. On page 677, the quadratic formula is stated and some "plug and chug" exercises are given. Then some word problems are given, so students have to translate to an equation before doing some calculations. That is it. There is no derivation of this important formula . . . .

After comparing Addison-Wesley's handling of these topics with the treatment given to the same topics in a Japanese book, Askey writes: "So, which book requires students to think? It is not the American one."

I don't share Askey's mathematical abilities or his knowledge of mathematical pedagogy, but I do share his low regard for Focus on Algebra. In my case, it has come from seeing my daughter frustrated by textbook-writers who emphasize dumb "application" problems instead of the mastering of basic principles and skills -- writers who ignore the orderly development of precepts but make much of social ideology, Conehead "integration," and fictitious tales of indigenous peoples. As my husband aptly asked one evening, after the task of helping Sarah with her algebra homework had fallen to him instead of to me: "What the hell kind of book is this anyway? And why is she studying African tribes in algebra class?"

In fact, she was not really reading about African tribes. She was reading about fictitious Africans who figure in some "Afrocentric" nonsense that is popular today among some of our particularly dull-witted educators. The nonsense in question appears on page 191 of Focus on Algebra, under the title "Dogon Astronomy":

The Dogon (Doh GAN) people live in a remote region of the West African nation of Mali. For thousands of years, these original inhabitants [sic] of the Niger Valley made their homes in isolated caves in the cliffs of the Hombori Mountains. Their knowledge of astronomy is extensive and, by the standards of modern science, baffling. Anthropologists studying the Dogon in the 1940s reported that without the aid of telescopes or other instruments, the Dogon had discovered that Jupiter has satellites, and that Saturn has rings. Neither fact is apparent to the naked eye.

Even more amazing, the Dogon claimed that an invisible star of enormous density orbits the star Sirius once every 50 years. Not until 1925 had astronomers discovered that a so-called "white dwarf" -- a dark, tiny, and incredibly dense star -- circled Sirius. . . .

This old, preposterous tale about the Dogon people has been repeatedly exposed and debunked, especially during the past few years. It is taken seriously only in the realm of "Afrocentric" political and racial ideology, where it is linked to other nonsense that is equally bizarre -- for example, the claim that black Africans, because their skins contain so much melanin, have an amazing ability to detect astral energy and cosmic vibrations! What really is amazing is that the writers of Focus on Algebra, who couldn't bother to explain the quadratic formula, have had the gall to devote a whole page to a story which has been thoroughly discredited. (Readers who are interested in this matter are in luck, because a new, definitive refutation of the Dogon story, with some informed suggestions about how the story may have originated, has just been published. It is Bernard Ortiz de Montellano's "The Dogon People Revisited," in the November-December issue of Skeptical Inquirer.)

Geometry Too!

Addison-Wesley hasn't stopped at algebra. Any students who survive their trip through Focus on Algebra can go on to Secondary Math: An Integrated Approach: Focus on Geometry.

Addison-Wesley's writers have managed to take all the analytical thought right out of geometry, and Focus on Geometry -- all 846 pages of it -- is another journey through the Land of Fluff. I see, however, that an "Atiba" has been added to the corps of diversity decoys who offer us their thoughts about matters mathematical.

Peripheral nonsense and "integration" do not seem to be as plentiful in the geometry book as in the algebra book, but they are plentiful enough. One of my favorite entries offers some lines from William Blake's renowned poem The Tyger, accompanied by this problem: "Describe or illustrate a line of symmetry for a tiger." Can you guess which Conehead appears beside the poem? Yes, of course -- the "Literature" Conehead! Another dandy item consists of a full-page picture of King Kong, Fay Wray and the Empire State Building, with this challenge to the student: "Name three other films in which small models might have been used to represent large creatures or objects."

The geometry book is worse than the algebra book, however, in its instructional approach. Focus on Geometry is like an MTV music video, cutting sharply from one thing to another. One minute the students are reading material about vectors, the next minute they are reading about congruence, and a minute later they are studying something else. When they encounter sets of problems to solve, no two problems are alike -- so students do not get an opportunity to master skills or principles.

Perhaps the most fascinating feature of this book is the order in which topics are presented. Students flit all the way through triangles, angles and intersecting lines before they learn that theorems and postulates have some importance. They go through half of the book (and presumably through half of the school year) before they encounter the concept of a mathematical proof. In fact, they already have been required to learn some proofs, but the book hasn't told them that this is what they are doing.

To help my daughter, I got a copy of the "Alternative Lessons" supplement that Addison-Wesley sells for use with Focus on Geometry. It has some 230 pages and it is remarkably good. This paperbound supplement, not Focus on Geometry, is the book that should be used in classrooms. In typical instances, it uses only two or three pages to cover material that Focus on Geometry fails to cover adequately in five or six (even with advice from Atiba). Key terms, definitions, illustrations, examples, and practice problems -- they're all here. This supplement must be a leftover from an earlier time, when Addison-Wesley had some writers who knew what they were doing and had some respect for students. Everything that my daughter knows about geometry has been learned with help from this paperback.

Where Are We Going?

Now that the results of the Third International Mathematics and Science Study (released in November) have shown that lots of American students are still floundering in their attempts to learn math, we should examine the direction in which we are going. We should also examine the heads of school officials who can keep a straight face while they adopt textbooks like Focus on Algebra and Focus on Geometry.

When I confronted some of our local school functionaries with the obvious defects of the Addison-Wesley Focus books, they responded with evasive claims about "studies" that allegedly had endorsed the books' "integrative" approach. Well, I've checked into those studies and I've found that all of them are strictly theoretical. The authors tout some notions, then say that it would be wonderful if were to use those notions in teaching math. They have not tested the notions in classrooms, and they do not have empirical evidence to suggest that the notions actually have pedagogic value.

Further, at least two of the studies that I've run across were done by people who now have affiliated themselves with schoolbook-publishers. Oh, there's a clever concept: Produce a "study" that promotes some odd idea about how to teach something, then hook up with a company that wants to exploit the "study" in contriving and promoting new schoolbooks. But where I come from, we call that a conflict of interest. Where I come from, we don't think that a person can do respectable work in the design or evaluation of pedagogic methods while he is thinking about endorsement contracts or royalty checks.

To me, it seems obvious that Focus on Algebra and Focus on Geometry are based on speculation, hype, and a wish to exploit a practice that has become alarmingly common in American education: Bureaucrats make faddish curriculum changes, and they refuse to turn back until it is clear to everyone that they have created a debacle. This is how California, during its fascination with the whole-language fad, produced a generation of illiterates. This is how California, more recently, has succeeded in producing a generation of innumerates -- people whose comprehension of numbers is nil, and whose principal mathematical skill consists of pressing the keys on a calculator. I suppose that it won't be long before the rest of the country catches up.

Is anyone listening? Does anyone care?


Schoolbooks cited in this article

Addison-Wesley Secondary Math: An Integrated Approach: Focus on Algebra
1996. ISBN of the student's edition: 0-201-86740-0.

Addison-Wesley Secondary Math: An Integrated Approach: Focus on Geometry
1996. ISBN of the student's edition: 0-201-86780-X.

The Addison-Wesley Publishing Company, Inc.,
2725 Sand Hill Road, Menlo Park, California 94025.


Marianne M. Jennings is a lawyer, a professor of legal and ethical studies at Arizona State University (in Tempe), and a columnist for The Arizona Republic.

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