Why not use Singapore's NEM?

This was the topic of a recent thread on the WTM message forums. The reasons given by many parents had nothing to do with the content of the book, but addressed student and parental factors. One person finally addressed content:

NEM does not provide lists of theorems and postulates for algebra and do "proofs or show tha this equals this" which I think is really improtant in algebra.

For that matter, the only American algebra text still in print that I have seen that does so in Foerster's Algebra. [I have no idea about Lial's, Chalkdust, Teaching Textbooks, or any other book you may be using. I do have an idea about Jacob's: It's not there. If you go flipping through your book to look for it also make a mental note, or even better a written list of all theorems proved in the book and make a note if the student is then required to prove theorems in the exercises] It lists out the field axioms in a list at the back of the book--something like getting assigned a basement office--but that at least is better than being completely absent.

Now that I've had time to think about it, I think what was missing from the ensuing conversation (monologue actually on my part) that these properties of the numbers, when considered together are very, very, very important in math. They aren't important because they help a 14 year old become a better rune manipulator and solver for x , they are important because they create a new universe of math called group theory. Asked one person puzzled about the need of such a list of properties at such a late date in the student's math career,

But aren't those properties all taught it most elementary mathematics programs? Or am I being naive? Certainly they've been featured and reviewed constantly in the program we've used. ... I'm not saying they're immaterial -- just that they're basic enough principles that I would expect students to know them well before they reach algebra (if not expressed in exactly the way that they are in your link). Am I wrong?

And so this is how I responded:

They aren't "basic" in the sense of being simple. They are more like foundational. Mathematicians have made careers talking about the differences between things (groups) that follow some of these rules and things that follow all of the rules. Essentially all of these various properties and what happens when they are or aren't true is the sum total of what one learns in Abstract Algebra so this isn't something that can be covered in its entirety in a K-8 arithmetic program. Usually what you might see in arithmetic is the appearance of the distributive, commutive, and associative properties and the book states it and then has the kid apply it in the simplest way possible or identify the use of it in a very simple way.

However, what you don't see (but if your arithmetic book does this I'd be interested) is go in depth into distinguishing between properties of numbers, axioms, definitions, theorems, and notation. It's all thrown in there and all mixed up. It's all ad hoc. [I should have said here is that I'm talking about algebra books, not arithmetic] And the students aren't required to use them as axioms, but instead they are taught "facts." [Axioms play the role of the premise in a logical syllogism. Think of algebra as having 11 premises, you will get the bigger picture in the long run if in algebra you start thinking of them as assumptions rather than as "facts." What follows from accepting these assumptions, if you do your derivations and proofs correctly, are all things that you use to do algebra arithmetic such as the algorithm you use to find equivalent fractions, multiplying double digit numbers together, etc). The sceptic then asks, "What happens if I reject one of these premises and instead only use these five? What sort of a math world would that create? You ask the question that open ups a worm hole and you leave the 9th grade algebra universe and go straight into group theory. Another sceptic might say, how can those 11 properties be proven? And that takes you to a different area of math as well, Back to the original post...]

That 2 X 3 = 3 X 2 is true is not "learning the commutative law". In fact, even learning that such a property is called "the commutative law" is not "learning the commutative law.""Learning the field axioms" entails being able to write down the general statement of the principle involved, not merely as a fact about the integers, but as a property that any set of elements together with any kind of operater [+ - x ÷ √] on that set may or may not satisfy.

"Learning the field axioms" also entails being able to use the list of axioms/properties that some given set and its operators might satisfy and actually derive significant mathematical results from that list through an unbroken chain of completely valid logical deductions.

What seems to happen in most algebra books (Saxon, Singapore not excluded) is that the student never is expected to make a distinction between the field axioms and other "facts" in algebra. They don't understand that multiplication distibutes over addition is axiomatic, but that it distributes over subtraction is not. Or, that the natural numbers are closed under addition is axiomatic but that they are closed under subtraction must be proved. And, they would "know" and use the rules of order of operations but they wouldn't know how those fit in with the other rules they've memorized. The rules about order of operations are not on "the list" of field axioms, for example. So, unless there is a master list ,the student is likely to go through his algebra book lumping it all together because he's never been given an indication that there is some sort of structure or hierarchy to all these facts that he's memorizing. [But then again, unless you have theorems for the student to prove or something else to say about the field axioms, nothing magically will come of simply memorizing it. ]

Once this list is memorized then there can follow discussion about what happens if one of the properties is missing. So even beginning in ninth grade algebra, with a text like Allen or Dolciani the student is getting an introductory taste of groups, rings, and field theory along with the skills it takes to prove theorems. This maturity of proving theorems, to gauge the appropriate amount of rigor for a given problem, constitute "true" math skills that cannot be acquired over night, skills that are absolutely necessary in doing in higher math (not to be understood as engineering calculus), For instance, after a diet of Allen, Gelfand, Oakley and Allendoerfer even an average student has practiced the skills needed for a Calculus text such as Spivak. Even a really good program like Singapore would leave a student unprepared for Spivak.

That's our story and we're sticking to it.