"A developmental course like Alain's is perfect for this approach."
Have you actually read Alain's book?
I cannot find any subset of students to which this treatment of arithmetic is applicable. And maybe I just don't know how developmentally challanged some students are. But if I had this much trouble following it, then I would suppose that it would be hopeless for a weak student to follow it. And I am using the term "follow" loosely. I was not actually able to follow most of it, or at least place it into mathematical reasoning.
The ontology is the most convoluted and complicated presentation of arithmetic I have ever witnessed. It reminds me of my tensor calculus book.
It is wrong on so many levels that it will probably take me a week to discuss its "wrongness".
I have to be honest here, this is an absolutely unintelligible treatment of arithmetic. I know it's title says algebra but it never even gets to algebra. This is probably due to its introductory statement being so far off the mark...
"To put it as briefly as possible, Arithmetic and Algebra are both about developing procedures to figure out on paper the result of real-world processes without having to go through the real-world processes themselves."
That is a definition of computation. Math is about REASONING not procedure.
Arithmetic is about understanding addition, subtraction, multiplication and division. It doesn't matter if we are talking about 6 dollars divided into 4 equal groups of 6 billion dollars divided into 4 equal groups. Once you recognize the DIVISION, the problem is solved. There is no reasoning left from the original problem of arithmetic, it is now a problem of computation.
As a rule of thumb, once you have reduced any problem to a procedure, the reasoning is OVER. However, you must understand the reasoning behind the procedures in order to be able to reduce a problem to a procedure. Thus, for students to use a procedure like long division, they must first develop it in the classroom.
This book isn't developing anything. It seems more interested in obfuscating simple arithmetic with a strange ontology and vocabulary. It is trying to equate mathematical reasoning to english statements, and is so over the top in this aspect that it even makes simple and intuitive concepts like "counting" incomprehensible.
I would really like to see someone that thinks that this is a "perfect" approach show me how a student of this book would get ANY questions right in the SAT math section. Show me where the algebraic reasoning was even developed in any section of the book to actually do algebra.
As I said before, I am totally clueless on what the purpose of this book is. I guess that leaves the possibility that there is another "math" out there that is incomprehensible by mathematicians but is still valid. Obfuscate the entire essence enough and you almost feel like you are discussing the existence of god.
Well, thank god we have authentic problems of algebra to test against.