A BOOK REVIEW

When I volunteered to read the book, I never expected to take poison. The feeling I received from reading the student edition was irritation, frustration and pain. This is the feeling I receive from the world news these days, but as a mathematician, I usually enjoy reading math books with pleasure. At first I couldn't quite articulate the reason for my feeling. So I checked out the teacher edition (TE) from a local district teacher, and started to read. I then exploded.

It would take several pages to point out all the deficiencies of the book because I found a major deficiency in almost every page. Most of the problems of the book end with the word, "Explain." But the book or TE never offers any explanation. Students are forced to go through many explorations. They are led to discover some new formulas or patterns from the exploration. All these discoveries are left unexplained, without even a mathematical summary. Thus none of the students' efforts are rewarded. Let me quote from the page of TE where I exploded and couldn't continue for a while because of the pain I received. It is Lesson 5.9 (pages 190--191). The section covers what Gauss knew when he was about eight years old, namely, the summation formula for the arithmetic progression:

If you know just this formula, then the general enough formula the authors suggest can be derived immediately. The worst is that if a student found a formula, he/she would then be immediately discouraged by his/her teacher: "It is a silly formula, not general enough. Don't remember it!"

A purpose of algebra education is to let students derive a simple enough formula such as the above and memorize it, and then to guide them to generalization. Mathematicians and scientists don't remember complicated formulas. This particular formula is beautiful because it is simple. It deserves memorization, of course, because of its general applicability. The same sad situation repeats when the book deals with geometric sequences.

Now I understand the source of my feeling. The book is designed to destroy the mathematical sensibility of students.

Many explorations are leading to the discovery of completely unimportant things. For example, the book uses the dimensions of geometric shapes to define the degree of a polynomial. The dimensions are very important notion, but the degree of variables and constants are completely arbitrary in general, and dependent on the context. Mathematicians and scientists assign a degree to a variable that best describes the situation. There is no universal rule.

Some facts students are led to find are important, such as commutative, associative and distributive laws. But I felt they would simply waste time by the lengthy explanations and explorations. These laws are like "a red signal light means you have to stop.'' Nothing more.

The book contains many real-life problems. At least it appears to be so. A careful look at some of the sample problems shows, however, that it is not the case.

• A bank offers a 100% monthly interest rate that is compounded monthly with a monthly service charge of $100. The real mathematics behind the problem is the iteration of a linear function y = 2x-100. It is an interesting subject itself, but translating the iteration of this particular formula into the banking scenario is a mistake. It makes the problem look silly.

• Calculation of the difference between the distance of the Moon and Earth and the distance of Saturn and Earth. It reminded me a story in the book Surely You're Joking, Mr. Feynman! by Richard Feynman. Feynman was asked to examine math textbooks by the State Board of Education. He exploded when he found the following problem in a textbook: "What is the total temperature of the stars seen by John and his father?" The pink book problem is worse. Do the authors still believe that Earth is the center of the universe? The sin of the book is greater because it talks about effective accuracy of measured numbers in an earlier section. Not only the difference of these distances is absurd from a scientific point of view, the number (with a lot of digits) does not make any sense mathematically.

• Finding the page number of a book that requires 1992 digits to number all the pages.

• Finding a formula for the price of glass window panels from the sample price list of 11 windows of various rectangular shapes which I couldn't figure out. The TE solution is
  
  price = 2 * (perimeter of the window) + area.
  It is already bad enough, but the case is even worse: if the window is made by two panels put together, then the perimeter is calculated in the way that the intersection counts only once. The book admits the silliness of the formula. Several pages later, it says, "In Lesson 8 you figured out how window prices were determined in an imaginary store. Real prices are probably not determined this way."

• A company tries to sell widgets for $24 a piece, finding no customers. They found that every price reduction of a dollar would attract 10 more customers. But then they can attract only 240 customers altogether even they give away their product for free! Further worse, the problem suggests that the gross profit is equal to the price of the product times the number of customers. It is only a joke that this section is titled Business Applications.

• A problem on page 57. This is a problem I hate. It tries to explain the exponential function. The real-life material used here is a chain letter!

The book is totally anti-formula. The book comes with many plastic toys. One of them is the geoboard, which is a square integral lattice. Students have to find the area of figures created on the board. Since these figures are piecewise linear polygons whose vertices have integral coordinates, the area of any shape is calculated by the formula of a triangle:

Teaching mathematical laws such as the distributive law without a simple formula does not make sense at all. Even the pink book tries to avoid using

The Lab Gear is another set of toys. The authors seem to believe that it helps students understand such formulas as -(-x) = x. But they don't know the simple mathematical fact: any attempt to explain a basic formula necessarily leads to introduction of another set of rules. In the case of the college level algebra, the rules are for example the Peano axioms. In the pink book, the rules are: "if you have double-decker blocks, you can remove the matching upstairs and downstairs from the scene" and things like that. Why should students remember such stupid rules instead of -(-x) = x ?

The Lab Gear plays exactly the same role the fingers do to find 2 + 3 = 5. Throughout the year, students are forced to use fingers to do all these computations. The authors thus seem to believe that fingers are great tools to help creating algebra skills.

The introductory part of the book is plagued with Polyomino, that is described as the most important tool to learn algebra. Polyomino is a great invention of S.Golomb of the University of Southern California and is one of the best known puzzles along with Rubik's cube and Sam Loyd's Game of 15. These puzzles are interesting, but never easy. It is completely unrealistic and frustrating for students to find the perimeter as a function of the area of a polyomino. It doesn't teach any algebra.

The book asserts, "Mathematics is the science of patterns." What it means seems to be that pattern recognition is the most important thing in mathematics. The pink book actually has many interesting combinatorial problems. Unfortunately, the book stops where students come up with a guess of the right formula through the pattern recognition. The main deficiency is that the book or TE never offers any logical explanation or proofs. Students are left without mathematical solutions to these problems. Pattern recognition is for chimpanzees and seals in a circus, but definitely not for our children.

Because of the heavy use of combinatorics and toys, the book deals almost exclusively with positive integers. If one wants to enrich mathematical contents of a book that deals mostly with positive integers, then one is naturally led to combinatorics, number theory, and finite group theory. Indeed, the book explores into these three areas without any pedagogical reason. One of the number theory problems of the book is the following:

Let a and b be two relatively prime positive integers. Find the largest positive integer that cannot be expressed as

This is quite an interesting problem, and it teaches a lot about modular arithmetic, partition into equivalence classes, and the characteristic algebraic difference between the natural numbers and the whole integers nicely to graduate students of mathematics. But we don't have to teach any of these in an Algebra 1 class! Of course the only thing you get from the book is again pattern recognition. There is a simple formula for the number:

I'm not suggesting that the pink book is a good book for university level algebra courses. It is true that many problems can be used in the university level courses, but the lack of mathematical treatment of the subjects makes the book totally useless.

Even functions are treated combinatorially in this book. Students learn functions not through the graphs but with the function diagrams. Can one describe the derivative and the integral of a function with the function diagram? No, not at all. The graph is a great invention to visualize the effect of a function. It can help visualize the derivative and integral of a function. Unfortunately the use of the graphs of functions is very little and limited in the book. The function diagrams may be useful to explain compositions and iterations of functions. That leads to an interesting mathematics of fractals and complex dynamical systems. But that's not where the book leads us, so the function diagrams remain useless and nonsense. Meanwhile, the treatment of functions of this book leaves students planning to take further mathematics courses unprepared.

No matter what the book does, the subject is left unresolved. It creates, therefore, only frustration and irritation. The book seems to say, "Find the pattern. Make a guess. If that's the same as what TE gives, then you got the right answer. Bingo!" A student would then ask, "But why does the pattern occur? What is the mathematical explanation of all these patterns?" From the pattern we have observed, we can guess the correct answer a teacher would give.

Mathematicians and scientists remember subjects by simple formulas. The visual impression of key formulas should not be underestimated. When students read the book, they will feel lost. They won't know what they are doing and why they have to do all the silly tasks for learning mathematics. The teacher edition addresses this issue deceptively. Many introductory examples of the subjects of this book are too complicated, causing irritation. Analysis of a simple example is the most common way to teach mathematics, which is not used in this book. Combinatorics and number theory are not suitable for teaching algebra. They are way too hard. After finishing this book, students will have a completely distorted view of mathematics. Do they get any algebra education? I don't think so.

The book is full of destruction and discouragement to students. The teacher edition treats readers as fools, and encourages them to remain fools. A curriculum based on the book is a pure poison to the young mind. No matter how it is used, I do not believe that one can run a reasonable Algebra 1 course out of this book. The only message I obtained from the teacher edition is, Do not teach mathematics!

The mystery is: if the book is so bad mathematically, then why has it been in use in the Davis School District for the past three years? Following the subtlety of Hercule Poirot on board of the Orient Express, I would say, "My first solution to the mystery is that the teachers, parents, students and the district have been all fooled by the publisher of the book."