I'm often asked what algebra book I'd recommend.
The answer is Modern Algebra, Structure and Method by Mary Dolciani et. al. Copyright 1960 or 1962. But I can't recommend any edition of it written after 1970 without reservations, and the 1980s and later one should probably be pulped.
The same book title with same author has over 100 versions with over 100 different ISBNs. The old ones I recommend don't even have ISBNs.
What's the difference? Here is a tiny slice.
Modern Algebra, Structure and Method, Dolciani, Berman, and Freilich, Hougton Mifflin Co., Boston. c. 1965, 1962. No known ISBN.
This is a great book. Mathematically rigorous, 30+ oral exercises per lesson, 40+ written exercises per lesson, and ten problems as well per lesson and real content.
Chapter 3, Addition and Multiplication of Real Numbers(p. 69) begins with a discussion of axioms of equality (3.1). It states clearly:
"you learned to perform many operations with numbers because you abided by certain rules.These rules, which are statements accepted as true are called assumptions, axioms, or postulates...The first fundamental assumptions that you will meet are the axioms of equality which govern your work with equations..."
3.2 covers axioms of closure. The language is sophisticated--
"Any set S is said to be closed under an operation performed on its elements, provided that each result of the operation is an element of S. This is known as the closure property of a set under an operation. Calculations in arithmetic are based on the often unstated assumption that the set of numbers is closed under addition and multiplication."
The text goes on to list and explain the closure properties of addition and multiplication,
explains the set of numbers of arithmetic is not closed under subtraction.
A later but still similar edition is
Modern Algebra, Structure and Method, Dolciani and Wooton, Hougton Mifflin, Boston c. 1970,
(this one has the pendulum picture on the cover.)
Yet already content is removed and moved.
3.1 leaves out all discussion of axioms of equality, relegating them to one pink box, no explanation.
The bit I quoted above, "you learned.....fundamental assumptions..." is not here at all.
The content of 3.2 and the axioms of closure are in 3.1 instead, including what I quoted above, "Any set S..." so the language that is preset isn't always simplified, but some is missing.
A discussion of how the set of odd numbers is closed under multiplication but not addition stayed in, but the statement that the set of numbers of arithmetic is not closed subtraction is taken out.
3.3 and 3.4 in this 1970 edition are totally different, however. Their material is about
adding and subtracting real numbers on the number line, material in chapter 4 of the 1962 edition. still, it's similar, some of it the same.
This book is one I managed to procure ten of to teach with. I need more, but so does everyone else, whether they know it or not. Anyone know if HMH ever grants permission for a fee to someone to copy their out of print books?
Then we come to lousy. Algebra, Structure and Method, Dolciani, Wooton, Sorgenfrey, and Brown, ISBN 0-395-26637-8, c. 1979 is a horse of a different color.
Gone are the explanations. Gone are the depth of problems. Gone is the mathematically deep and proper language. Wholly new material is added and it is watered down.
3.1 is now "Rules for Multiplication". It states:
"To learn how to multiply negative numbers, notice that
2 x (-1) = -1 + (-1) = -2
3 x (-1) = -1 + (-1) + (-1) =-3
and so on."
The next sections go on to transform equations by multiplication and division, even though in the prior books, doing those sections came after introducing the axioms of equality, closure, opposites, etc.
In 3.8, the total explanation of axioms is given:
"Many rules or number properties have been stated earlier in this book. Some are axioms. Others are theorems. An axiom is a statement that is assumed to be true. A theorem is a statement shown to be true by using axioms, definitions, and other theorems in a logical development. The axioms that account for the rules or properties used in working with real numbers are listed below."
And then there is a list. No explanation of closure at all, no explanation about the use of axioms.
Exercises are far cry in content. Six oral exercises as opposed to 30, 20 written, except with nearly all of the work done for you (2 column proofs, steps provided, you fill in the property)
no problems at all about what sets are closed under what operations.
Nonetheless, I did find 30 of these, and we use them. It still has more problems and exercises than nearly any other book these days. But the teacher really uses the other 10 books as the base.
Even worse is Algebra, Structure and Method, Dolciani, Brown, Cole, ISBN 0-395-43053-4, c. 1988.
This book managed to drop an entire chapter by Chap 2. Chap 2 "matches" the old chap 3, Addition and Multiplication of Real Numbers. But it has no mathematical language. No use of "set", no explanation in general about closure, nothing of the kind.
Now, chap 2, Working with Real Numbers", begins with
"The rules used in adding and multiplying real numbers are based on several properties that you can take for granted. For example, the following statements are accepted as facts.
1. Every pair of real numbers has a unique (one and only one) sum that is also a real number.
2. Every pair of real numbers has a unique product that is also a real number.
3. When you add two real numbers, you get the same sum no matter what order you use in adding them.
4. When you multiply two you get the same product no matter what order you use in multiplying them."
It doesn't get much better. No precision, no formalism.
Worse, the whole thing above is misleading. Nowhere does it explain that various subsets of the reals *aren't closed* under addition or multiplication. Nowhere does it ask students to determine closure.
Nowhere does it even explain a set.
Now, I just compared 3.2 for ease. I didn't cherry pick for the most egregious. But every page, every sentence is rewritten in these later editions. The relationship to the 1970 work is not recognizable.
This phenomenon is known in college level math and physics books, certainly--my 2nd ed Thomas (no Finney yet) calculus is a thing of beauty. My 3rd edition Kittel Thermal Physics is too. The later editions are dreadful. But most people don't know this about high school texts.
Suffice to say, the 1970 books will be even rarer now that people know to look for them, but if I were homeschooling or running a small class, I would spend the premium for the oldest you can possibly find, and junk anything from the 80s on.