For people who want to learn or teach mathematics on the basis of reason …
It seemed appropriate to me that there should be a page devoted to whatever I know of the reactions elicited by the texts on this site.
First and foremost, there is what the students say and here is a good place to check.
The reactions are, as should be expected, rather mixed. But I think that they are quite consonant with the three phases that I describe in .A Reasonable Sequence? For Real? II.
As for what little mail I have gotten so far about this site, it has been consistently favorable if to markedly varying degrees but I don't think that it would be proper for me to ask permission to reprint what were intented as personnal messages. If nothing else, there would be a "chilling effect".
On the other hand, here is, reprinted by permission, a decidedly unenthusiastic view of RBA that was posted on August 23, 2009 10:41:59 AM EDT by Robert Hansen in the thread "memorizing Math facts".
Yet on another hand, some time earlier, June 24, 2009 1:26:45 AM EDT to be precise, just about out of the clear blue sky, a post had appeared on mathedcc which its author just now kindly permitted me to reproduce here.
"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.
Review of RBA Before Revelation About True Mathematics Teaching.The author, though, just advised me that:
It is quite an interesting book and is clearly very different from commercial algebra textbooks. Here are some things I like about it:
Overall, I think this book does a great job of making algebra make sense to the student and encouraging students to think about algebra and to think about the differences between the real world and the paper world used to represent the real world. If students understand this book, they should have a good foundation for later algebra and precalculus and calculus courses. I think that perhaps portions of Parts II and III emphasize too much on procedure and not enough on understanding. However, other portions still do a very good job on focusing on understanding. I did find it interesting that Chapter 18 discusses what the book called the "Elementary School Procedure" and "Efficient Division Procedure" for long division. The Efficient Division Procedure is the procedure for long division I learned in elementary school, and I don't recall ever seeing the other method the book mentioned. I noticed some typos as well, but I don't remember all the ones I saw. But I posted this on mathedcc to try to encourage others to read the book and comment on it.
- The book does a great job of showing the difference between the real world and representing the real world on paper.
- The book does a great job of explaining that numbers by themselves (numerators without a denominator) cannot be represented accurately on paper because these have no concrete meaning. So we need number phrases, and these number phrases allow us to represent things that we can see and touch. Besides, number phrases give the distinction between the real world and the paper world we use to represent the real world. The book does a nice job of explaining that other books blur the distinction between the real world and the paper world.
- The book does a great job of making everything explicit and clearly stating which things "can go without saying" so that students aren't confused.
- Part I does a great job of explaining arithmetic so that it makes sense to the student. It does a good job of making comparisons and inequalities make sense to the student. In fact, these discussions do a good job of motivating the concepts of arithmetic to the students by showing them that arithmetic makes everything "work" in the way it's supposed to. I especially like the explanations of the arithmetic of signed number phrases since students often have trouble seeing why the rules for arithmetic of signed numbers work.
- Part II does a great job of explaining the meanings of equations and inequations and what solutions to these mean. I imagine many students in algebra can solve equations and inequations and not know what solutions to these really mean.
By the way, I had never heard of the term "inequation" until I saw it in this book. I've always heard the term "inequality" instead.
- Part II does a great job of explaining and using the Pasch Theorem for solving inequations.
I don't recall ever learning a name for this theorem. It's interesting to see the name in this book.
- Part II does a great job of finishing this part with reducing translation and dilation and affine problems to basic problems and solving double problems.
- Part III does a great job of introducing Laurent polynomials and Laurent monomials, rules of exponents when multiplying and dividing monomials, adding and subtracting and multiplying and dividing polynomials. The book does a great job of comparing the arithmetic of polynomials to that of arithmetic and shows why the arithmetic of polynomials is a generalization of addition, subtraction, multiplication, and division in arithmetric. Many algebra books today don't do this very well, if at all.
- The book does a great job discussing using the ideas of binomial powers to approximate (x_0+h)^n without having to multiply the long way and round off in the end. This idea is, of course, important in differential calculus, especially differential calculus of polynomials. Most algebra books today don't even discuss this at all.
- The book does a good job of explaining why we need polynomials written in descending powers of x for large x and ascending powers of x for small x. Most algebra books today don't even mention this.
- On a similar note, the book gives a good example of polynomial long division with the polynomials written in ascending powers of h where h stands for small x. Students don't usually see this in today's algebra books.
- The Epilogue does a good job of introducing functions to students and explaining some applications of where approximations arise in algebra and closes with a good example of a problem that needs to be solved by a differential equation. Such a problem gives motivation to the need for differential calculus and initial-value problems and that the algebra they learned in the book provides powerful tools for these areas of mathematics.
My views on teaching math have changed since I last looked carefully at RBA and wrote my review. And I didn't write the review as I would write it as a journal article since I had written it in just one draft. So I would like to find the time soon to reread RBA and reread my review and see if there are any changes I would like to make (I'm sure there are changes I would like to make)