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Topic: Alain Schremmer's "Reasonable Basic Algebra"
Replies: 36   Last Post: Nov 30, 2009 8:24 PM

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Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05
Alain Schremmer's "Reasonable Basic Algebra"
Posted: Jun 24, 2009 1:26 AM
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Alain,

Your book "Reasonable Basic Algebra" is quite an interesting
book and is clearly very different from other commercial
algebra textbooks. Here are some things I like about it:

1. The book does a great job of emphasizing modeling by
showing the difference between the real world and representing the
real world on paper.

2. 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.

3. 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.

4. 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.

5. 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 your book. I've always heard the term "inequality" instead.

6. 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.

7. 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.

8. 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.

9. 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.

10. The book does a good job of explaining why we prefer 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.

11. On a similar note, the book gives a good example of polynomial
long division with the polynomials written in ascending powers of x.
Students don't usually see this in today's algebra books.

12. 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.

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 you 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 you mentioned.

I had noticed some typos as well, but I don't remember all the ones I
saw. I might have to e-mail them to you sometime.

But I had posted this on mathedcc to try to encourage others to read
your book and comment on it.

Jonathan Groves



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