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Elementary Mathematics, A Logical Approach--I thought I'd share about this text

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Here is a quote from the book, which was written in 1963,--



The aim of this book is to acquaint the student with the number systems of elementary algebra, to lead him to an appreciation of a deductive system through which he can gain a respect for logic and proof, and to show him how logic and proof are applicable in mathematics, particularly elementary algebra. A minimum of prior training in mathematics is assumed; indeed, the most primitive ideas of mathematics are taken as the starting point. This material has been used successfully over a period of three years in a required course for freshmen students.


Table of contents


chapter 1 discusses the roles of intuition, induction, and deduction in mathematics


chapter 2 discuss the rational number system and its subsystems


chapter 3 concerned with the meaning of proof


chapter 4 shows an example of the deductive system


chapter 5 and 6 review elementary algebra

However, in these chapters, the ideas and terminology developed to this point are used to show the reasons for the rules of algebraic manipulation.


chapter 7 deals with systems of linear equations


chapter 8 discusses proof using the principle of finite induction


chatper 9 treats probablity


From my brief perusal of this book, it seems like a very very basic course on logic and proof as applied to algebra. And of course, some of the answers are in the back.


When my oldest gets to pre-algebra next year, I'm thinking of using this for logic and a more traditional pre-algebra program at the same time. I think that this book may be able to bridge the gap between the New Math and today's math.


At the very least, it will introduce us to the terminology used in proofy math programs. If this sounds crazy, please let me know.:o And in case any one else is interested in it, I purchased this at alibris for about 5.00 plus shipping. The author is Paul Sanders.

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This looks interesting. It looks like this is for college freshmen who probably have already taken high school algebra but need to re-learn it with more depth and more theory (??). It's lacking the topics seen in abstract algebra such as group theory---would you consider this book some sort of hybrid between elementary algebra and abstract algebra?


I hope you can come back and give us some samples of interesting problems from the problem sets.


I tried to look up Paul Sanders online but could find no info about this person. Does the book tell you what institution he was affiliated with? High School? University?

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It looks like some kind of college level text maybe -- like pre-calculus of some sort or just an alternative to freshman calculus. No book that does mathematical induction and systems of linear equations is a pre-algebra text, I don't think. I think if you look at it more closely, you'll find you have to already know algebra for this. Maybe it's some sort of "pre-calc" or something.

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"...just an alternative to freshman calculus..."


We liberal arts types could use more alternatives to freshman calc. It would have been cool to have had the option of re-doing synthetic geometry in college, for example. It could be a terminal course, you'd really learn something, and the course wouldn't be constrained by the service needs of other departments.

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Paul Sanders was the Chairman of the Mathematics Department for Appalachian State Teachers College.


I guess I assumed pre-algebra because this book looks accessible by even me.


In the intro it says to use Chapters 1, 3, 4, 7,8, and 9 with a class of strong students.

It also says to cover Chapters 1,2, 3, 5, and 6 with weaker math students. Furthermore he writes, "Weaker students have been much more successful in this course than in a traditional college algebra course. Although this course is not intended as a preparation for calculus, those students who have taken it along with a strong modern trigonometry course have been quite successful in calculus. However, no attempt is made here to develop skill in algebraic manipulations."


So maybe this is a follow-up to algebra 1?


Anyway here a few examples, although they are hard to find because the book is mostly theorems postulates and proofs--


From Chapter 2,

1. What is the cardinal number of each of the following sets?

a) A={a, b, c};

b) B= {482, 689, 4,3, 2, 986};

c) A U B;

d) A^(couldn't do an upside down U)D;

e) A X B;

f) N = {x|x is a natural number}.


2. Decide whether each of the following sets is closed under the specified operation:

a) all natural numbers, under subtraction;

b) all natural numbers, under division;

c) all "squares" 1,4, 9, 16, 25, and so on, under addition'

d) all "squares," under multiplication.


Chapter 4


Theorem 14:1^(-1)=1



Statements Reasons

1. 1*1^(-1) = 1 1. Postulate VIII with a =1

2. 1*1^(-1) = 1^(-1) 2. Postulate VII with a = 1^(-1)

3. 1^(-1) = 1 3. Substitution.


(Postulates 1-12 for the exercise were given in the lesson labeled Postulates for Real Numbers


Chapter 5 covers the definitions of polynomials, monomials, degree of a term, and degree of a polynomial, complex numbers. Then it covers Principle 1 (sum of like terms), 2 (add two or more polynomials), 3 (multiply exprssions of the tyme ax^(m)y^(n), the fundamental principle of fractions, roots including i^2. Then it moves on to equations and equalities and systems of liner equations where matrices are introduced.



Chapter 8 Finite Induction and the Binomial Theorem


Theorem: If 1) p(sub1) is true and 2) p(subk)=>p(sub(k+1)) for any natural number k, then p(sub n) is true for all natural numbers.


To argue that this is true, let S be the truth set of p(subn).

(and it goes on to prove this with keys I don't have)


Much of this stuff, I've never seen before. But the sections for weaker students are really simplified and appear to teach the definitions needed to be able to understand the lingo in more proofy math. I have no problem at all doing the math, but I didn't know most of the terms and definitions, proofs and theorems that go along with them.


What do you think?

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Yes, it does seem to be a hybrid. Because the math problems are fairly easy, at least in portions of the book. But the lingo is very abstract and would leave me completely lost if I didn't know how to do the math. So for me, as a mom who wants this as an option for her kids, I can go through this book and really understand better, or maybe even use it as an intro for my kids to this type of math.

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It just dawned on me as I was responding to the earlier post about this being a hybrid, that was from Teachers College. Maybe it was from when they were educating teachers how to teach New Math?


Do it! lol.


No -- it isn't training on New Math, per se. It is just some good regular math is all. It isn't quite at the level of Abstract Algebra so much as more New-Math-esque and at the level of "college algebra" (ie before calculus but definitely probably a step up from most high school courses). Could you do it after Algebra I? Well, it looks kind of like you could theoretically do it as Algebra I or probably Algebra I and II combined or something. My guess, though, is that it probably goes too fast for a kid that age. Theoretically, you could do abstract algebra with 5 year olds -- I mean, it's not like there are any prerequisites. But, on a more realistic note, you probably have to wait for a lot more intellectual maturity to develop before you can really do that kind of stuff. This book looks like it is for college algebra. I would say that from what you have posted, it looks like it has some really good math in it -- proving that the multiplicative inverse of the multiplicative identity is the multiplicative identity, for instance (i.e. that 1*1^(-1) stuff).

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Well, I'm heading back to the K-8 board. My needs have been met. I found both a 1964 Doliciani student and teachers edition and a 1970 solutions manual which may or may not apply. Plus this Sanders dude and and Life of Fred which is modern math but in the context of one humongous word problem.


We'll probably do Sanders after pre-algebra with Fred and before Dolciani; I'm a little scared of teaching that.


My daughter asked, "What are we having for lunch?" I answered absentmindedly, "Polynomials."


Thanks for all the help!

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