This book has a 1996 publication date, and like almost all school texts, lists many authors. The Lead author is Randall I. Charles, and Alba Gonzalez Thompson is Associate lead author. John A. Dossey is listed as a Conceptualizer, and is a Lead author of another book in this series. Dossey was President of the National Council of Teachers of Mathematics when the idea of doing Standards was acted on, and he appointed the committee which wrote the Curriculum and Evaluation Standards.
In Ms Jennings's article, she quotes a teacher saying We don't plug and chug anymore. We're teaching them to think. The message NCTM likes to say about their standards is the same. Let us look at some things in this book to see if the claim of teaching thinking is born out. First, the book is 812 pages long, which is much longer than the texts in other countries, although not unheard of here. A very good Japanese text for students the same age is about 200 pages long [2]. Some supplementary problems are added in Japan, so maybe our text is only three times as long. If students are taught more about how to think, there might be justification for some extra length. So, let us look at three topics which are included in both books and contrast the treatments.
In elementary school, the numbers treated are integers like 0, 1, 2, ... and -1, -2, ..., and rational numbers like 1/2, 13/17 and -2/3. However, there are other numbers, and they must be introduced eventually. Numbers like the square root of 2 arise when looking at the length of the long side of a right triangle whose other sides are each 1. Square root of 2 is not a rational number, i.e., it can not be written in the form p/q with p and q integers. The Japanese ninth grade book starts with the problem of finding the side of a square whose area is 2, and then gives an approximate value for it. They introduce a method of computing square roots by finding numbers whose squares are larger than and then smaller than the number sought. On page 9 of the English translation, they raise the question of whether square root of 2 can be expressed as p/q with p and q integers, and show that this is not the case. The next thing they do is to show that rational numbers have either a terminating decimal expansion or a repeating one. They also indicate why repeating decimals always represent rational numbers. I do not like the method they use for this last result, since there is another way which will generalize to other very important settings, but at least something was done. The details are not given completely, so the student is expected to do some further thinking to see why the specific examples given are generic. The responsibility to see that this is done rests with the teacher. All of this takes 13 pages, and the students have learned some important mathematics.
Focus on Algebra gets to square roots on page 476. Rational numbers had been treated in an earlier book, so the definition as the quotient of two integers is all that is said about them here. [The Japanese program has also done rational numbers earlier. Repeating decimals are currently treated in fifth grade there.] Next there is the claim: We cannot find two integers whose quotient is square root of 2, so square root of 2 is not a rational number. We call it an irrational number. Then there is an illustration with two columns. The one on the left is titled Rational numbers and under that is can be written as a ratio of two integers.
Three numbers are given, an example of an integer which is given as square root of 4 and as 2, an example of a terminating decimal which is given as square root of 5.76, as 2.4 and as 12/5, and a repeating decimal, square root of 1/9, also given as 0.3333333333333 and as 1/3. The column on the right is labeled Irrational numbers and subtitled cannot be written as a ratio of two integers. Two examples are given, pi or 3.1415926535897 and square root of 2 or 1.4142135623730. Nothing is said about why these are irrational.
The next page has some problems, including the following.
Which are squares of rational numbers?
d. 5 e. 29 f. 16/9 g. 48 h. 2.56
The student can do no more than guess that d, e, and g are not.
If the same type of argument given in the Japanese book had been given
here, then the problem could have been answered after the student thought
a bit. There really is a difference between a guess and a well thought
out answer. Our students are not given enough information to give
a well thought out answer. All they can do is guess, and even then they
are relying on authority as expressed by the claims that 2 and 5
have square roots which are irrational.
In the next section of the same chapter, the Pythagorean Theorem is considered. There is a short section where the students are supposed to discover what the Pythagorean Theorem is. There is so much guiding that it is not what one would call discovery, but being led to something. That is not much use since it is unlikely someone will be led in such an obvious way to something of interest, but discovery learning is a fashionable topic now so authors try to force everything into this mold. The Pythagorean Theorem is stated, as is its converse. In case your geometry is rusty, the Pythagorean Theorem says that if you have a triangle with a right angle, the square of the side opposite the right angle is equal to the sum of the squares of the other two sides. The converse says that this happens only for right triangles. Many problems about the Pythagorean Theorem and its converse are given. Some are what the teacher above called plug and chug, others are given in words so that a translation to a formula is needed before doing calculations. What is missing is a proof of either the Pythagorean Theorem or its converse.
The Japanese text also treats the Pythagorean Theorem. They start with two problems for the student to do. Pictures are given of a right triangle and squares drawn on each side. Extra triangles are drawn about the square on the longest side, and the students are asked to find the areas of the squares drawn on each side. This is a significantly harder problem than the one posed in the Addison-Wesley book about how to discover the Pythagorean Theorem, yet the extra triangles drawn make it possible for many students to do it. The Pythagorean Theorem is stated, and a proof is given. Then a proof is given of the converse. Many problems using the Pythagorean Theorem are given, and at the end of the chapter a different proof is outlined, with the students expected to fill in the missing steps. At the end of the book there are two pages which contain six more pictures of proofs of the Pythagorean Theorem. Four are without words, and two contain a one line equation each. There are other problems students are asked to do where they need to prove something. They use what I would call a local proof. No formal axioms have been stated, but students use facts which they consider obvious. I have yet to meet anyone who claims that the Pythagorean Theorem is obvious, and I have asked many mathematicians.
There is another place where square roots appear, in solving quadratic equations. The Japanese book builds up to the general quadratic equation in steps, and on the sixth page of the text of this chapter gives a derivation of what is called the quadratic formula. This is a formula for the two roots of the general quadratic equation. The derivation of this general formula is given on the right hand side of a page, and on the left hand side of that page a specific equation is solved. The method used is exactly the same for the specific equation and the general one.
The American 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 the function takes on as 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, and the tool that is usually introduced to derive it, and can be used directly on any quadratic equation and in other settings, is not introduced.
So, which book requires students to think? It is not the American one. The last chapter is titled Functions and the Structure of Algebra.
At last one can hope that something will be done to show students
what can be developed in algebra, and use it for something interesting.
The first section starts with The Golden Ratio. This comes from a
rectangle which is similar to another rectangle inscribed in the first
one with the longer side of the small rectangle the same as the shorter
side of the longer triangle. Similarity has been treated in this book
as well as the previous one in this series. However, it is not used
to define the golden rectangle. The ratio of the two sides is pulled
out of the air, and given as
g = [1 + 5^{1/2} ] / 2.
The Fibonacci numbers are then introduced. After having the
students compute the ratio of successive ones, expressions for the
even and odd Fibonacci numbers are given in terms of the
number g.
These have to be given in words, since adequate notation for even such simple functions as (-1)^n has not been introduced. This is a bit surprising, since the whole book deals with algebra as the study of functions. However, in this book, functions are essentially those things you can draw pictures of on a graphing calculator. The quadratic formula could have been used here to derive formulas for the Fibonacci numbers, but was not. It also could, and should, have been used along with symmetry to derive the value for g given above. The formulas for the even and odd Fibonacci numbers were given from on high, without any explanation where they came from.
Later in the chapter there is a section where similarity is mentioned, and students are led through a treatment of the golden rectangle. Since this comes most of the way through the last chapter of a book of more than 800 pages, it is unlikely to be used in most classes.
In the review problems for this chapter, there is the following one.
Page 736, problem 7.
Explain why 4^{1/2} is rational while 5^{1/2} is irrational.
The answer, given in the teacher's edition, is:
4^{1/2} = 2 which is rational.
5^{1/2}, in its decimal form, does not terminate
or repeat and therefore cannot be written as an
integer over an integer.
I am offering $100 for a proof of this last claim, i.e.
for a proof that the decimal expansion if 5^{1/2} has a decimal
expansion which does not repeat without first showing that it is
irrational.
I told this problem and answer to two number theorists at a meeting at Illinois State, and they both laughed. I would too except this is too serious to laugh.
Richard Askey
Dept. of Mathematics
Univ. of Wisconsin-Madison
[1] Randall I. Charles, Alba Gonzalez Thompson and others, Focus on Algebra, Addison-Wesley Secondary Math, An Integrated Approach, Addison-Wesley, Menlo Park, CA, 1996.
[2] K. Kodiara, ed., Japanese Grade 9 Mathematics, 1984, translation published by Univ. of Chicago School Mathematics Project, Chicago 1992.