{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

General Category => Why Mathematics? => Topic started by: Nation of One on December 20, 2017, 08:47:01 pm

Title: Old School "New Math"
Post by: Nation of One on December 20, 2017, 08:47:01 pm
Holden,

I think I made some kind of cognitive breakthrough which may be significant.  It is merely an observation about two different ways of solving inequalities.

It has to do with how using the formal and rigorous notation of the old school "new math" (with set builder notation as well as logic symbols) makes complicated problems less complicated and very organized.  I want to run this by you to see which method you find is more clear and straightforward.

Let's consider an example.  On cymath, the online algebra calculator which shows every step of the calculation, you will see that they use the idea of intervals.  Notice how they go about solving (1-x)/(x+4) < 0 (https://www.cymath.com/answer?q=%281-x%29%2F%28x%2B4%29%3C0)

They change the inequality to an equality and find the 3 intervals to test.  All this will appear overly complicated once I show you the old school method that Frank Allen uses in the c.1966 Modern Algebra: A Logical Approach, Book Two.

I think you may find this very significant once you see how much simpler the process becomes when we use the formal notation.  There is no need to find intervals or methodically test values in those intervals or declare when x is undefined (such as at x = -4).

I will use words and, or, union, and intersection rather than the symbols (as we don't have those symbols here).

Also, <-> will mean "bidirectional implication" or "equivalence", and, when using set-builder notation, {x | } is read "x SUCH THAT ..."   The | (such that) is sometimes written as : (colon).

EXAMPLE (with formal notation rather than finding intervals and critical values):

Solve (1-x)/(x+4) < 0

(a/b < 0) <-> (a and b have opposite signs)

Therefore, (1-x)/(x+4) < 0 <-> (1-x and x+4 have opposite signs)

<-> [(1 - x > 0) and (x + 4 < 0)] or [(1 - x < 0) and (x + 4 > 0)]

Therefore {x | (1-x)/(x+4) < 0} = S union T, where
S = {x | (1 - x > 0) and (x + 4 < 0) } and T = {x | (1 - x < 0) and (x + 4 > 0) }

Now S = {x | x < 1} intersection {x | x < -4} = {x | x < -4}
and T = {x | x > 1} intersection {x | x > -4} = {x | x > 1}

Therefore, {x | (1-x)/(x+4) < 0} = S union T = {x | x < -4} union {x | x > 1}

What do you think?

Both methods are cool, but the more formal method (which I doubt is taught anymore) seems more elegant to me.   i suppose it is a good idea to get a feel for both ways.


Another example:  (3*x + 2)*(2*x - 5) < 0

This time, let's look at the formal (old school "new math") way first.

{x | (3*x + 2)*(2*x-5) < 0} = {x | (3*x + 2 > 0 and 2*x - 5 < 0) or (3*x + 2 > 0 and 2*x - 5 < 0)}
= {x | (3*x > -2 and 2*x < 5) or (3*x < -2 and 2*x > 5)} = {x | x > -2/3 and x < 5/2) or (x < -2/3 and x > 5/2)} = {x | -2/3 < x < 5/2}

That was very orderly and straightforward, right? 

Now look at the current and more mainstream approach for solving the same inequality:  (3*x + 2)*(2*x-5) < 0 (https://www.cymath.com/answer?q=(3*x%20%2B%202)*(2*x-5)%20%3C%200)
Title: AHA! Score 1 point for formal rigor
Post by: Nation of One on December 21, 2017, 11:12:49 am
I found a scenario where the computer automated method fails, which means any students following that method would find no solution for this particular problem were they to follow that step by step procedure for solving inequalities.

example:  5/(2*m + 8 ) + 4/(3*m +12) > 0 (https://www.cymath.com/answer?q=5%2F%282%2Am%2B8%29%2B4%2F%283%2Am%2B12%29%3E0)

Their results:  23 = 0 therefore, NO SOLUTION.

This is incorrect.

Observe the more formal method:   We still simplify to 23/6(m+4) > 0

{m | 5/(2*m + 8 ) + 4/(3*m +12) > 0} = {m | 23/6(m+4) > 0} = {m | m+4 > 0}
= {m | m > -4}

That is the correct result, that is, m > -4.  What do you think of those apples?

Verified with Sage (http://sagecell.sagemath.org/):   solve(5/(2*x + 8 ) + 4/(3*x + 12) > 0, x)   ----> [[x > -4]]

Then you have situations such as 5/(x+7) + 3 >= 8/(x+7) (https://www.cymath.com/answer?q=5%2F(x%2B7)%20%2B%203%20%3E%3D%208%2F(x%2B7)) where cymath says, "Oops! No solution was found.  Sorry for the inconvenience."

Whereas with Sage:  solve(5/(x+7) + 3 >= 8/(x+7),x)
[ [ x < -7 ], [ x >= -6 ] ]

And by hand with formal notation?

Let's try it.

(5/(x+7) + 3 >= 8/(x+7)) <-> (5/(x+7) - 8/(x+7) + 3 >= 0) <-> (-3/(x+7) + 3(x+7)/(x+7) >= 0)
<-> ((3*x+21-3)/(x+7) >= 0) <-> ((3*x+18)/(x+7) >= 0) <-> ( 3*(x+6)/(x+7) >= 0)

Therefore we can write {x | 3*(x+6)/(x+7) >= 0} = {x | (x = -6) or [(x+6 > 0 and x+7>0) or (x+6 < 0 and x+7 <0)]} = {x | x = -6 or [(x > -6 and x > -7) or (x < -6 and x < -7)]}
= {x | (x >= -6 or x < -7)} = {x | x < -7} union {x | x >= -6}




Title: Review of SMSG inspired textbooks (part 1: Frank Allen's Logical Approach)
Post by: Nation of One on January 13, 2018, 04:31:52 pm
CLASSIC TEXTS TO BRIDGE THE GAP BETWEEN HIGH SCHOOL AND ADVANCED MATHEMATICS

Suprisingly, some of the most honest and heartfelt opinions concerning the old "Modern Mathematics" movement that came and went all too abrubtly can be found in the reviews posted as comments at Amazon.

At the risk of repeating quotes, I am compelled to gather these reviews here in a somewhat organized manner in an attempt to shed some light upon my current fascination/obession.  It is also worth mentioning that the classic Modern Introductory Analysis (Dolciani, Beckenbach, Donnelly, Jurgensen, Wooton), circa 1964, was the text used in my senior year of high school in 1984.

My mind was not tuned into it as I was struggling through nervous breakdowns and emotional difficulties during that period of my life.  I had not looked at that text again until last year, and even though I have been exposed to mathematics at a higher level, there is something about the approach taken that I wish to revisit and honor.

I will most likely be coming back to this post/thread to reorganize and edit.  The reason I am preserving these reviews here is that I am afraid that once there are absolutely no copies available, maybe the reviews might vanish.  Who knows?

As for a Geometry text, I used an older edition of Jurgensen text (https://www.amazon.com/gp/offer-listing/0395977274/ref=tmm_hrd_used_olp_sr?ie=UTF8&condition=used&qid=&sr=).

So, here we go, for posterity:

Modern Algebra A Logical Approach, Book One
Frank B Allen, Helen R Pearson
c.1964

(Impossible to track down solution key [Teacher's Key], but worth studying, for what it's worth; that is, it is novel, unique, with fanatical notation).   8)

Quote from: A. Chong
Why aren't math texts like this anymore?

(2015)

One of the few preliminary math texts that are (or rather were) designed to train college bound engineers and scientists. As stated in the preface, this book is intended to show algebra as a logical system, and not a bunch of disjointed operations. And it succeeds, showing a coherency lacking in all the modern texts.

I've got bookcases of math textbooks, and of the intro algebra books, this one and the Dolciani/Brown ones are the best (both the old 1965 and new 1990+ Dolciani, aka Structure and Method). I really like the Art of Problem Solving, but to some degree, it's too much a math team/problem solving series.

On a presentation level, I love that this book treats the student as a mature thinking individual.

1) lack of pictures and cutesiness. The few pictures are a major historical contrast with the still relevant text -- tape reel computers

2) lack of distractions. the text is almost devoid of drawings and diagrams.

3) there is no dishonest attempt to justify algebra with contrived problems trying to show how the average person will use algebra in real life.  (note to students: you won't; your teachers are lying to you). 

There is a dearth of word problems, which is good and bad (many students lack the reading skills to translate these; however, the typical word problems found in most texts don't teach problem solving well)

4) real unadulterated mathematical notation is used copiously.

Content-wise, everything in the book relates to something the student will be doing later in a mathematics, engineering, or science career. There is none of the cruft or pretend mathematical
terms and concepts you get in books by a committee of people all with doctorates in education.

1) good introduction to set theory

2) excellent introduction to logic and proof, something that is often covered only in geometry in current texts. and a good 70 pages of it rather than the usual short chapter.

3) the properties section is written from a theoretical mathematical perspective.  It's rare to see an intro algebra book discussing for instance, that "the numbers are closed under addition". (though the engineer in me never found it very important ;) )

4) more than most texts, it relates operations on real numbers to operations on variables.

5) a few fairly advanced topics are introduced, for example, proof of irrationality of sqrt(2).

Incidentally, the section on radicals is the most thorough and mathematically rigorous of any text I have.

6) intro number theory, aka divisibility rules are covered, where most texts leave them out.

Now for a few quibbles (and they might be strengths depending on your perspective).

1) section of analytic geometry (lines, etc) and inequalities is short but mostly complete

2) section on systems of equations is short

3) section on functions is short, and curiously covers quadratics and completing the square rather than in the polynomial/factoring section.  The quadratic formula is absent from the book (and pushed to the 2nd one)

4) 2 advanced topics, polynomial long division and introduction to trig are both covered too early (and trig doesn't even NEED to be covered in geometry)

However, the pluses far outweigh the minuses. I think this text is exemplar of the New Math, which for better or worse came and went in the 60s and 70s. For anyone entering a STEM career, you need New Math.

The next I have quoted from before.

Quote from: Adrian S Durham
A still born child of New Math the loss of which has left us with little hope
(2007)

I have an MS in Math from Ohio State, and my wife and I home school our three children. We've been home schooling now for several years and it is approaching time for us to figure out our algebra/geometry/trig (or the equivalent) program. As a former graduate student in math, I know about this thing out there lurking under the surface of college math. It is the proof, of course. Somehow what a "proof" is, what "math" is, and what it is all good for has all gotten extraordinarly lost in a way that goes far beyond even the scope of secondary school education.

This basic problem can be heard reverberating in ancient videos of Feynman lecturing to the public on the role of mathematics in physics (and how rigor is not particularly useful). It can be seen in the mathematics curricula of undergraduate programs all over the nation that pander to other departments' needs. cutting out most of the actual math content and reducing the math major to a generalist in the mathematical sciences rather than a specialist in mathematics. At any rate, it is much, much bigger than even math ed or math ed reform and will stop any meaningful progress in math ed reform, for that matter, since it is a basic disagreement on the necessity and/or intellectual value of rigor (and, in many cases, what "rigor" even is for that matter).

At any rate, it's too bad these books are out of print -- victims of a war far greater in magnitude than even the math wars. The New Math of the 60s was as close as it gets to mathematics being handed down to society by its mathematicians, and we threw it all away. Frank Allen's books are not just books written to pay lip service to the movement, but truly written in the spirit of the times by a real advocate of the New Math. In any case, these books are probably the very best algebra books I have ever seen as of this writing. If you put them together with a good geometry program that at the very least proves the Pythagorean Theorem, you will have youself one first class high school education.

Unfortunately, Frank Allen will never receive the vindication he deserved. But, perhaps he imagined that there might be people like me that would happen upon his work and find it immeasurably valuable in an anti-intellectual world so dominated by politics that only the most vulgar displays of superficial mechanical proficiency are ever even noticed while everyone frantically attempts to "Beat the Joneses" with whatever latest gimmick they can get their hands on.


Modern Algebra: A Logical Approach Including Trigonometry - Book Two
Frank B Allen, Helen R Pearson
  c.1966

(Note that the Teacher's Keys are nowhere to be found as these were not really mass produced after initially not selling well.)

There were no reviews for this text, but I am leaving it in the list in case I come across anything related directly.  I may also add some of my own comments as I am currently engaged with this volume documenting many of the exercises with SageMath in Jupyter Notebooks.
___________________________________________________________________________
Title: Review of SMSG inspired textbooks (part 2: Modern Algebra)
Post by: Nation of One on January 13, 2018, 04:35:13 pm

Modern Algebra: Structure and Method, Book 1
 Mary P. Dolciani,‎ Simon L. Berman,‎ Julius Freilich,‎
Jr. Albert E. Meder (Editor)
  c.1962

(Note that there is some overlap with the second volume of this series, and that you can probably safely jump right into Book 2) 

Quote from: Lorenzo
Great Preparation for University Mathematics

(2018)

This is an older SMSG text, School Mathematics Study Group, produced as an answer to the Soviet Union's orbiting of Sputnik. The text is concise and to the point, incorporates set theory, symbolic logic, and develops the structure of the subject very well. It prepares students for university study of mathematics. This program was sadly abandoned by so-called educators who came up with emphasizing programmable calculators or using a cookie cutter approach which lost sight of the mathematical structure of the subject. Other approaches such as Saxon resorted to rout review of concepts and lost again the mathematical structure. These older text might well be harder to find, but they are perfect for home schooling students who are not subject to the politics of education. The earliest editions of this series are the best. Some publishers offer newer versions but are more expensive and lack the concise presentation found in the older editions.


Quote from: beejaybee
5.0 out of 5 starsSOMEONE PLEASE REPRINT THIS SERIES

(2015)

I wish this book was still in print. It is a sound algebra text free of twaddle and graphic distractions and peppered with BRIEF notes of value to young scholars: the etymology of math words such as "algebra," "fraction," etc.; biographies of mathematicians; thumbnail sketches of math-related careers; etc. There are more practice problems than in many texts and a sufficient number of challenge questions per lesson. The lesson length and number of lessons are just enough for a year of solid algebra instruction. Many modern day texts include far more material than can be taught in a year which leaves the effort of deciding what to teach to the already overburdened instructor. I might hope that a wise textbook publishing company would obtain rights to reprint this series, beginning with the text that precedes this one as it is an excellent pre-algebra book.

Quote from: Barry Garelick
I use this book to teach my 8th grade algebra class

(2018)

Although the explanations are many times too formal (it was written at the time of the 60's New Math which was steeped in set theory), the sequencing of topics in the book is excellent as are the problems. The problems are graduated in difficulty; they start out easy and ramp up to moderate difficulty to extremely difficult. There are many types of word problems, and such problems intersperse each chapter, so that a particular algebraic technique learned in that chapter is used to solve the word problems. (In the chapter on algebraic fractions, for example, the word problems involve fractional equations, such as mixture, work, and distance/time problems).

I use this book to teach algebra in 8th grade. The official book we are supposed to use is "Big Ideas" which should be avoided at all costs. The first year I taught the course I started with Big Ideas and then switched to Dolciani after purchasing many of them via the internet. The class remarked how much more they liked Dolciani than Big Ideas. And I heard from parents too.

See also: Confessions of a 21st Century Math Teacher Paperback – September 8, 2015 (https://www.amazon.com/Confessions-21st-Century-Math-Teacher/dp/1517274451/ref=asap_bc?ie=UTF8)

Quote from: Gerald L Calabrese
A Classic!
(2014)

I used this book as a student in 1971. I did not realize it at the time, but the book was a classic, and was written by a foremost mathematician of this generation. This is reflected in the book. It has a logical presentation of algebra, but along the way there are historical side points, and nifty mathematical formulas (for example, it has one formula to determine the day of the week for a given date of history, and it works!) I did struggle early on in the course, but then, with this publication, things clicked, it all made sense and I did much better. I now have fond memories of the course, and am very glad this book was used.
____________________________________________________________________________

Modern Algebra and Trigonometry: Structure and Method, Book 2
Dolciani, Berman, Wooton
  c.1964/1965

Quote from: Richard J. Petti
Best version of best Algebra II textbook of all time
(2013)

After the Soviet Union launched Sputnik into orbit in 1957, America feared a potential missile base on the moon in the hands of a hostile superpower that America lacked the technology to reach. The federal government did something unusual in America: it asked top universities what should be taught in high schools to optimize the education of future scientists and engineers; and it used it influence to gain adoption for the new curriculum. In mathematics, this book and its cousins with first Author Mary Dolciani were the results.

The basic approach is to blend a set-theory approach to the foundations of mathematics with procedural math for doing computations. (That is why the subtitle is "Structure and Methods.") For example:

(a) A function is a mapping from one set (the domain) to another (the range), and the set of all points that get mapped onto is called the "image." (Current high school terminology unfortunately uses the term “range” for what mathematicians call the image, which is a bad attempt to "simplify" the ideas.)

(b) Addition and multiplication of real or complex numbers are associative commutative binary operations on pairs of real numbers to the real numbers.

The power of this approach is that students’ intuitions about integers and real numbers serve as foundations for higher mathematics, starting with matrices and linear algebra, calculus, function spaces, probability and literally everything.

A bit more history about how this occurred: Late in the nineteenth century, mathematics had outgrown use of equations and variables as the fundamental language of most of mathematics. There were two competing approaches to providing a stronger foundation: “lambda calculus” which formalized symbolic computation using symbolic logic (and which became the foundation of LISP), and set theory. Set theory proved far more flexible and powerful and it became the universal language for all mathematics in the twentieth century.

From 1960 to 1999, this approach to secondary mathematics was called “the new math.”


The Math Wars

In the 1970s and 1980s, this approach to high school mathematics was virtually eliminated from American high schools. Evidence clearly shows that most students do worse with the “new math” approach based on sets and mappings, and I believe this is accurate. You can see a review of the math wars at [...]

The unmentionable elephant in the room is that the mythical top 10% or so do much better with the new math approach. That is why the universities recommended this approach to the federal government after Sputnik, and why the federal government encouraged its adoption in schools. That is why all my friends at MIT found it a great help. That is why as a math tutor today, I find clients who like it and benefit from it. This approach works well because (a) it explains the complexities of ordinary math in such elementary terms that some students say it feels like they already know it and just have to rediscover that they know it (Plato’s concept of innate knowledge); and (b) when you move on to more advanced math, it is based on the same abstract concepts of sets and mappings, as are integers and real numbers.

I say “mythical top 10%” because the students who can benefit from the sets and mappings approach are not necessarily the ones with the best math grades. The key determinant of who benefits is ability to think abstractly and to relate the abstractions to concrete procedures. My experience is that some students, who have these abilities but do not do very well in math, like this approach benefit greatly from it.

The sets-and-mappings approach impairs performance of the majority of students because they do not make the connection between the abstractions and procedural math. As a result, these students have more material to learn, the new material does not help them understand and perform math procedures, and math procedures get a smaller share of student time than with the traditional approach.

It is a losing proposition to introduce curricula that meet the needs of the top 10% but impairs the learning of perhaps 70% of the student population, unless you restrict that curriculum to people who can benefit. That restriction is an important reason why AP courses survive in the current environment. Similarly, if you tried to introduce an MIT undergraduate curriculum in most colleges, it would be resoundingly rejected.

If you know a student or a whole class who can benefit from this, this book is the way to go.

Quote from: Jimmy Jones
Answers to Exercises are Available in Some Versions
(2013)

In its day, this was a widely used book across the US for teaching high school Trig and Algebra II. Mary P. Dolciani was a Math professor at Hunter College, NY, but more famously, an influential educator of Math teachers. Today, both an education center and an endowment bear her name in recognition her legacy. This is a review of the 1963 and 1965 editions of the book. The only substantive differences I noticed were in the last chapter on p. 599 and pp. 603-604 (practice tests). "Modern Algebra and Trigonometry Structure and Method Book 2" was published in four different versions. These are,

Teacher's Manual ("Teacher's Edition" is printed on the spine) - This has a separate teacher's manual (usually on green paper) at the beginning of the book along with teacher notes throughout the rest of the student text. Answers to exercises are in the back of the book.

Student Version ("With Answers" is printed on the spine) - Answers to odd and even-numbered exercises are in the back of the book.

Student Version ("With Odd-Numbered Answers" is printed on the spine) - Answers to odd-numbered exercises are in the back of the book.

Student Version (no additional annotation on the spine) - No answers to exercises.

However, those answers in the back exclude four types of exercises: 1) graphs, 2) proofs and "show thats," 3) exercises in the twelve "Extra for Experts" sections, and 4) exercises in two of the three transparency insert sections. If you desire the answers for 1, 2 and 3, they are included in a separate publication, viz., "Solution Key for Modern Algebra and Trigonometry Structure and Method Book 2" (no ISBN). I found it devilishly difficult getting hold of a copy of the Solution Key at a reasonable price.

There is significant overlap in the topics in Ch. 1-8 and Book 1 of this series. Ch. 9-16 are essentially new material with respect to Book 1. The plentiful exercises are all quite doable and are ordered in very gradually increasing levels of complexity. There are almost no compromises to mathematical rigor. Axioms and definitions are clearly presented, and proofs given for all theorems - except for one in Ch. 14, "Every polynomial equation with complex coefficients and nonzero degree n has exactly n complex roots." Sections dealing with interpolation for increasing accuracy when using four digit log and trig tables are now passé.

Ch 1 Sets of Numbers; Axioms
Ch 2 Open Sentences in One Variable
Ch 3 Systems of Linear Open Sentences
Ch 4 Polynomials and Factoring
Ch 5 Rational Numbers and Expressions
Ch 6 Relations and Functions
Ch 7 Irrational Numbers and Quadratic Equations
Ch 8 Quadratic Relations and Systems
Ch 9 Exponential Functions and Logarithms
Ch 10 Trigonometric Functions and Complex Numbers
Ch 11 Trigonometric Identities and Formulas
Ch 12 The Circular Functions and Their Inverses
Ch 13 Progressions and Binomial Expansions
Ch 14 Polynomial Functions
Ch 15 Matrices and Determinants
Ch 16 Permutations, Combinations and Probability

______________________________________________________________________
Modern School Mathematics Algebra 1
 Mary P. Dolciani,‎ William Wooton,‎ Edwin F. Beckenbach,‎ Ray C. Jurgensen,‎ Alfred J. Donnelly
  c.1967

Quote from: Where're my Glasses?
relief from sensory overload
(2015)

Because of the age of this book (book and I are similar age) it is written in an older style with more detailed, "old school" explanations. There are chapter summaries which include vocabulary definitions and page references. If you are easily distracted by the flash. color and pizazz of new books, this might be up your ally. I tend to be a visual learner and generally like colors and graphics, especially when I'm trying to understand math concepts, but have to say this is a welcome relief from sensory overload.



Modern school mathematics: Algebra and trigonometry 2
Dolciani, Wooton, Beckenbach, Sharron
c. 1968
_____________________________________________________________________________
Title: Review of SMSG inspired textbooks (part 3: Algebra 2 and Trigonometry)
Post by: Nation of One on January 13, 2018, 04:40:19 pm
Algebra and Trigonometry: Structure and Method, Book 2
Mary P. Dolciani,‎ Robert H. Sorgenfrey,‎ Richard G. Brown,‎ Robert B. Kane
c.1986

While there were 118 reviews for this particular edition, the real enthusiasts seem to gravitate towards the original 1960's editions.

I found the comment to the following review right on point.  Just substitute the word "gort" for the word "twit."

Quote from: Jane Doe
Great book-no need for a teacher!
(2010)

I used this textbook last year for my Algebra II class. I personally loved this book. It had very good, worked-examples in each section that were exactly the same as the practice problems given. Most textbooks start out with some information explaining what is going on, and then the student is just supposed to put the pieces together on how to do the problems, whereas the examples in this book show the student exactly how to do the problems. My algebra teacher was not very good, and while she did work out examples on the board for the class, they just never made sense. By the end of the year, I stopped even listening to her lessons and I just did the problems in the book on my own. I made an A in the class, and even though math is one of my weaker subjects, I think I learned enuogh to do fine on the SAT and ACT.

However, my teacher disliked this book. She said that it really needed to include more graphing, and that it was outdated.

To each his own, I guess. I definitely think that its many clearly layed out examples helped me.

COMMENT (by B.McCall): I'm not sure how math can be outdated, especially since its foundations are thousands of years old. Your teacher sounds like a twit who thinks math must he "relevant" to pop culture.

Quote from: Richard J. Petti
Later edition of best Algeba II textbook of all time
(2013)

This is an updated version of the original "Algebra and Trigonometry Structure and Method Book 2" from the mid 1960s. These are the best high school algebra books for students with an aptitude for math.

I prefer the older version. I think the newer version was made as a compromise with the opponents to the new math, and it is not quite as effective.

See the longer review for "Algebra and Trigonometry Structure and Method Book 2" by Mary Dociani et al from the 1960s (https://www.amazon.com/Modern-Algebra-Trigonometry-Structure-Method/dp/B000J4LEJS/ref=sr_1_5?s=books&ie=UTF8&qid=1515870735&sr=1-5&keywords=modern+algebra+and+trigonometry+structure+and+method+book+two) for more information.

Quote from: M. Davis
Great program!
(2013)

This is still one of the best math books out there for Algebra 2/trig. It has three tracks for your students: basic, average, and advanced, and all three types of learners can take this course and get a lot from it. If you are a home educator, you will probably want to get the solutions manual as well as the student book for this program, but the teacher's manual does have the answers to all the questions written in red, as well as extra problems for practice and hints for how to approach certain topics. It is one of the few decent teacher's editions out there - most seem to only have a bunch of silly extension projects without providing clear hints about the material in the actual book. This book has practical hints.

I have taught from this book and its Algebra 1 companion for years, and will do so again this coming year. I am so sad that this program is now so difficult to get at a reasonable price and that the solutions manuals are harder to find, but the search is worth it. Most of the students I teach using this program excel and score well on college exams, while not feeling overwhelmed. I do recommend this program to those who have a decent understanding of math. I have heard home teachers who are not very math-oriented sometimes have a hard time teaching from this book without some extra preparation, although I have also heard that many self-learning math-oriented homeschoolers have loved using this book without a teacher's help.
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COMMENT(M. Davis 2017):  I see that several other customers have gotten different editions of this book - one even got a student edition. The review I wrote was based on the 1990 version of the teacher's edition, although I also referred to the whole program, which included the solution's manual and the student edition. I agree that it is essential to make sure you are getting the correct edition before you buy when you are ordering older books. I don't remember for sure, but I probably contacted the seller first and verified the copyright date before buying, because that is what I usually do if at all possible.

Very good advice.  I will be exploring 4 incarnations (1965, 1968, and two different 1986 editions, one HONORS [see below]).  Hence, in a year or so, God willin' and the crick don't rise, I will be able to add some commentary to these reviews. (Gorticide)


Quote from: Little Al
Another Classic Math Textbook
(2017)

Features many, many exercises and problems. The materials in the first third of the book overlaps with Volume 1, but at a higher level of difficulty and in greater depth. Although some of the materials (for example, computer programming exercises and "explorations") are a bit dated, the explanations are very clear and easily understood.

COMMENT (by your's truly) : I find it fun to implement these old [BASIC] computer programming exercises in Python, and some of the more robust exercises into C/C++ to create execuatables. While working through the text, if you store all these in a folder in your PATH, you will gather a handy toolbox of functions you might use without even firing up a computer algebra system like Sage or SymPy.

It is a satisfying exercise in itself to transform old BASIC code into a current language since it drives the point home that the mathematical ideas underneath the code are ancient.



I couldn't resist commenting and inserting a link to Stepanov's From Mathematics to Generic Programming (https://www.amazon.com/Mathematics-Generic-Programming-Alexander-Stepanov/dp/0321942043/ref=sr_1_1?ie=UTF8&qid=1515869959&sr=8-1&keywords=Stepanov)

Quote from:
The Best Algebra II/Trigonometry Text Available
(2000)

I used this book (when it was written by Mary P. Dolciani) in 1973-74 as a junior in high school and loved it. The recent revisions continue the excellence of Dolciani and the series. The theory is sound and the examples are easy to follow. Especially important is the numerous exercises graded in difficulty that help the student master the material. This book actually makes students THINK and does not spoon feed the material.

I can't say enough great things about this text. My school district uses this text for the honors tract (9th/10th grades) but uses another text for the "regular" 11th grade Algebra II sequence. I feel this is unfortunate because all students can benefit from the rigors of this book.

Quote from: Martha Jo Fleischmann
A powerful, comprehensive, and lucid textbook
(2000)

Faithful to the highest standards of traditional math, this rigorous and thorough tour of algebra and trigonometry is presented with meticulous clarity. It is an invaluable tool in its own right, but is especially worthwhile in preparation for advancing to upper level math concepts. The sections dedicated to word problems have particularly relevant real world applications, addressing the "When are we ever gonna have to use this?" issue with convincing examples. The organization is intelligent, uncluttered, and methodical. As a resource, this volume represents an investment of substantial educational merit, and uncompromising quality.

Quote from: Mark Gottlieb
A superb text

As a teacher of mathematics I have used a number of different algebra texts and evaluated many others. After an extensive search for a text that would meet the needs of students preparing for college-level mathematics, I decided that this book had no equal. The exposition is both clear and rigorous, and the problems are of real mathematical depth without being too difficult for most students. One of the best aspects of the book is the extensive drill on algebra skills included in every section.
Title: Review of SMSG inspired textbooks (part 4: Algebra 1 & 2 Honors Editions)
Post by: Nation of One on January 13, 2018, 04:41:24 pm
_____________________________________________________________________________
The Honors editions

ALGEBRA 1 (1986 Dolciani, Swanson, Graham)
  c.1985/1986

Teacher's Edition: ISBN 0-395-34374--7
Solution Key: ISBN 978-0395535912  (039553913) (https://www.amazon.com/Algebra-Solution-Key-Mary-Dolciani/dp/0395535913/ref=sr_1_1?s=books&ie=UTF8&qid=1515873441&sr=1-1&keywords=9780395535912) Might as well be unavailable at that price!  ($896???)

ISBN : 9780395343739 (https://www.ebay.com/itm/ALGEBRA-1-By-Mary-P-Dolciani-Hardcover-Excellent-Condition/182999901745?epid=120290600&hash=item2a9ba4e631:g:h04AAOSwSPBaTuq2)


Algebra 2 and Trigonometry
Dolciani, Graham, Swanson, Sharron
ISBN: 0-396-34378-X


THIS EDITION HAS PROGRAMMING EXERCISES in PASCAL, which is easily translated into C/C++.  You can also code directly with Lazarus [with simple console programming].

Quote from: Fred Hicks
Best text book ever written for this subject
(2016)

Best text book ever written for this subject. I taught Math for 36 years and this is my favorite Algebra 2 textbook.

Quote from: Dave F[/quote
One of the classic, challenging math texts
(2013)

It'd s pity this text is no longer being published. Guess it fell victim to dumbing down of math education.
Title: Review of SMSG inspired textbooks (part 5: ANALYSIS ++)
Post by: Nation of One on January 13, 2018, 04:43:43 pm

Modern Introductory Analysis
Dolciani, Beckenbach, Jurgensen, Donnelly, Wooton
c.1964, 1980

Quote from: Jimmy Jones
Review of the 1964 and 1967 editions
(2014)

"Modern Introductory Analysis" was published in four different versions:

1. Teacher's Manual ("Teacher's Edition" is printed on the spine) - This has a separate teacher's manual (usually on green paper) at the beginning of the book along with teacher notes throughout the rest of the student text. In particular, short answers are provided as dialog to the pedagogical (Why?), which is encountered continuously in the student text. Answers to exercises are in the back of the book.

2. Student Version ("With Odd-Numbered Answers" is printed on the spine) - Answers to odd-numbered exercises are in the back of the book.

3. Student Version ("With Answers" is printed on the spine) - Answers to odd and even-numbered exercises are in the back of the book.

4. Student Version (no additional annotation on the spine) - No answers to exercises.

However, those answers in the back exclude both graphs and proofs. If you desire answers for graphs and proofs, they are included in a separate publication, viz., "Solution Key for Modern Introductory Analysis." Nota bene: the "1980 Impression" (reprint) of the Solution Key, ISBN: 0395255554, is compatible with the 1964 and 1967 editions of the text.

Though its title may suggest otherwise, "Modern Introductory Analysis" does not concern itself with epsilon-delta calculus proofs. Rather, it is a kind of pre-calculus text originally intended for high school seniors who had completed a course in Algebra II and Trigonometry in the previous year. It was the follow-on book to Dolciani's "Modern Algebra and Trigonometry Structure and Method Book 2."

There is substantial overlap in the topics of these two books, although "Modern Introductory Analysis" covers them at a higher level. Other topics in "Modern Introductory Analysis" are new material, for example, mathematical induction, vectors, limits, derivatives, transformations and polar coordinates. There are > 4400 exercises. Most of the exercise sets contain at least a handful of proofs to whet your appetite. Considerable added insight is gained by doing these proofs, which serve to extend the basic material. The presentation is axiomatic. Yet, you will find a fair amount of intuitive development preceding the theorems. Some of the theorems are stated without proof, either left as exercises, or with a citation (in the Teacher's Manual) to another book containing a proof.

The 1964 and 1967 editions of the book are nearly identical. The 1964 ed. has several easy-to-spot errors that are fixed in the 1967 ed. However, be advised of certain less obvious errata in the 1964 ed. You may wish to revise your 1964 ed. as follows:

1) Exercise 36, P. 182, "T - S and P - S" should read "T - P and S - P"
2) Exercise 11, P. 312, "the length" should read "a function whose minimum is the length"
3) Exercise 12, P. 312, "what is" should read "let L be" and "(-2, 5)?" should read "(-2, 5). Find a function whose minimum is L."
4) Exercise 13, P. 312, "which for a fixed volume requires the least material (surface area)" should read "of greatest volume having a total surface area of 96π sq. cm."
5) Exercise 18, P. 313, should read "Solve Exercise 17 if triangle POQ has minimum area."
6) Exercise 34, P. 327, "b = 1" should read "b ≠ 1"
7) Exercise 14, P. 362, "log 2 (base e)" should read "log 4 (base e)"
8) Exercise 23, P. 362, "b = √a " should read "b = a² "
9) Exercise 24, P. 362, "b = ³√a " should read "b = a³ "
10) Exercise 42, P. 424, "csc θ" should read "-csc θ"
11) Exercise 47, P. 424, "sin X1 - sin X2" should read "sin X1 + sin X2"
12) Exercise 41, P. 441, "0 < x < π/2" should read "π < x < 3π/2"
13) P. 451 (also 1967 ed.), "y' = acos x" should read "y' = acos ax " and "y' = -asin x" should read "y' = -asin ax"
14) Exercise 20, P. 453, "a sin(ax + b)" should read "-a sin(ax + b)"
15) P. 461, incorrectly drawn graph. Point A should be in the fourth quadrant.
16) Exercise 18, P. 472, "8 m.p.h." should read "2 m.p.h."
17) Exercise 21, P. 472, "2r(180/n)º" should read "2r sin(180/n)º"
18) Exercise 13, P. 481, "bearing 250º" should read "bearing 265º" and "B and 265º" should read "B and 250º"
19) Exercise 30, P. 493, change ending to "... such that r* = f(θ*) and for some integer n, either r* = g(θ* + 2nπ) or -r* = g(θ* + π + 2nπ).
20) Exercise 26, P. 502, "θ1 = θ2." should read "θ1 = θ2 + 2nπ for some integer n."
21) Exercise 29, P. 524, " 5x² - 4y² " should read " 3x² - y² "
22) Exercise 11, P. 536, " 9x² + 24xy + 6y² = 0; θ = Sin‾¹ (-4/5)" should read " 2x² + 3xy + 2y² = 7; φ = π/4"
23) Exercise 4, P. 541, " x² - 3xy + 3y² + 6y = 7" should read " 7x² + 6xy + 15y² = 144"
24) Exercise 5, P. 541, " y² - 4xy - 4y + 8x = 0" should read " 8y² + 6xy - 26y -12x + 11 = 0"
25) Exercise 6, P. 541, " x² - xy - x + y - 2 = 0" should read " x² + xy + y² = 8"
26) Exercise 16, P. 546 (also 1967 ed.), change to " 20x² - 24xy + 27y² + 20x - 55y - (4615 / 396) = 0"
27) Exercise 33, P. 568, "rv = vr" should read "If rv = vs and v ≠ 0, then r = s."
28) Exercise 7, P. 616, "n - 1" should read "r - 1"
29) Exercise 14, P. 616, "12 or more." should read "12 or more balls."
---
Contents:
Ch 1 Statements and Sets in Mathematics
Ch 2 Ordered Fields
Ch 3 Mathematical Induction - Sequences and Series
Ch 4 The Algebra of Vectors
Ch 5 Plane Analytic Geometry of Points and Lines
Ch 6 Functions
Ch 7 The Field of Complex Numbers
Ch 8 Graphs of Polynomial Functions
Ch 9 Exponential and Logarithmic Functions
Ch 10 The Circular Functions and Trigonometry
Ch 11 Properties of Circular Trigonometric Functions
Ch 12 Vectors, Trigonometry and Complex Numbers
Ch 13 Analytic Geometry and Matrices
Ch 14 Space Geometry
Ch 15 Probability
Appendix (1967 edition) Area Under a Curve

Quote
Grade-12 Text...
(2000)

Miracle! I finally got a copy of Dolciani's "MODERN INTRODUCTORY ANALYSIS", and it is everything I remembered it to be thirty three years ago in my 12th grade College Prep (CP-12) Math course. I still endorse this text as 5-stars!

Quote from:
The finest pre-calculus text ever.(go with the older version and not the revised 1980s version)
(2003)

This text, along with all the Dolciani books by Houghton Mifflin were the corner stones of any high school math program in the 60's-70's and early 80's. This book particularly was the book that I learned pre-calculus from, and made me become a math major. The books were literate in their context, never watered down, but not so abstract that a high school student couldn't read and follow. What makes this book so unique is the fact that mathematical induction is introduced in Chapter 3 and is carried throughout the book. Normally mathematical induction is included at the end of a precalculus text and is never covered. No other precalculus book prior or since has used this approach. Mathematical induction is a proof driven treatment for the topics which follow. It makes the students think in a logical manner and enables them, by proof, to understand the full argument of why certain "things happen" in math as they do. The Teacher's Edition's were the best of any series (until Houghton Mifflin changed the format, and made them less teacher friendly). I still have 2 TE's of this book and the solutions key, and still refer to it when I need to...it beats any precalculus book out today. Once Dr. Dociani passed away, the entire Houghton Mifflin series went down the tubes, their current texts DO NOT hold muster to this old classic!!!


Quote from: Ebasco Engineer
1967 edition, of course
(2014)

The gold standard of high school textbooks. Has never been approached in organization, content, and depth. Every edition after the 1967 series got worse, until ... what we have in every high school math book today. No organization, no content, no depth, and ten pounds of glossy paper loaded with full color pictures unrelated to the math. Do your high school student [a favor] and bite the bullet, pay any price for a used copy. Your student is worth the very best. The price says it all.

COMMENT: You are correct. Dr. Dolciani did set the gold standard, and what we are left with today in terms of secondary math texts in nothing but application with NO theory or proof to understand why what happens...happens. I actually reviewed a Geometry text that did not introduce proof until the final chapter??? WTF...the whole purpose of Geometry IS proof.

_______________________________________________________________________
revised 1991:

Introductory Analysis
Mary P. Dolciani, Robert H. Sorgenfrey, John A. Graham, David L. Myers
  c.1988/1991


Quote from: V. Rao
Good precalculus book for strong math students
(2013)

This is a serious math book with no fluff. Problems are divided based on difficulty into levels of difficulty A, B, and C (the hardest, often asking the student to prove something). There are computer exercises in BASIC. Here is the table of contents:

p001 Ch 01: Foundations of Real Analysis
p041 Ch 02: Analytic Geometry
p089 Ch 03: Sequences, Series, and Limits
p141 Ch 04: Functions and Limits
p185 Ch 05: Theory of Polynomial Equations
p221 Ch 06: Introduction to Differential Calculus
p271 Ch 07: Trigonometric Functions and Triangle Solving
p313 Ch 08: Trigonometric Identities and Graphs
p361 Ch 09: Applications of Trigonometry
p399 Ch 10: Exponential and Logarithmic Functions
p441 Ch 11: Vectors
p479 Ch 12: Matrices, Determinants, and Systems of Linear Equations
p525 Ch 13: Further Vector Topics
p557 Ch 14: Introduction to Integral Calculus
p589 Ch 15: Probability and Statistics

Quote from:
Best Math Book I've ever used!!!
(2014)

This book is the best pre-calc book on the market! I had a terrible teacher that year in school, but was still able to learn everything very well from the book's explanations. Definitely buy this book!

Quote
The finest secondary pre-calc book
(2017)

Whether it is this edition or the "original" edition, there has never been a pre calculus text for secondary school as rigorous or mathematically sound as this text. The main reason is that the concept of mathematical induction is introduced early and is carried through the entire text..this means it is an algebraic proof driven book. All other precalculus books have mathematical induction at the end of the text and is almost never covered...well, it is never covered. By covering this concept of abstract algebra early in the book and then applying it in all further chapters, it enables the student not just to think but understand why things are occurring.

This is the most rigorous precalculus text for secondary school ever on the market. It is clearly geared for those students who are in advanced courses (precalculus is taught in 11th grade and calculus taught in 12th) and prepares them extremely well for college math. For non-advanced students this book will be a challenge, but if approached correctly a worthwhile challenge.

The only other thing I would add is that the earlier edition (the "black/blue" cover) and can be viewed and purchased on Amazon, is still the better of the two editions. I have never felt there is a need to introduce calculus in a precalculus course and by doing so you take away from other precalculus topics...as this book does. If you want to purchase the text, just search and get the "blue/black" cover first introduced in 1963 and then additional printings in 1967, 1972, 1977.

COMMENT (2017):  Agreed. I was a math major in the early 70s and taught only a short time. I learned on Dolciani and I taught on Dolciani. MIA, be it this edition or the original "blue/black" edition were the finest pre calculus texts ever written for secondary school students.
___________________________________________________________________________

Modern Analytic Geometry
Beckenbach, Wooton, Fleming  c.1983


ISBN-10: 0395340594
ISBN-13: 978-0395340592

____________________________________________________________________________

Limits: A Transition to Calculus
Lexton O Buchanan
c.1966,1985, 1990

ISBN-10: 0395340454
ISBN-13: 978-0395340455
_____________________________________________________________________________

Analysis of Functions
Beckenbach, Sorgenfrey
c.1970, 1974, 1984


Quote from: V. rao
advanced but concise precalculus text
(2012)

Sorgenfrey co-authored some math books with Mary Dolciani, and this book has the same feel as the Dolciani texts, with few distractions from the mathematics. It covers the differential (but not integral) calculus of elementary functions. Here is the table of contents:

Chapter 1 p1 The Algebra of Functions
Chapter 2 p55 Polynomial Functions
Chapter 3 p109 Introduction to Differential Calculus
Chapter 4 p151 Applications of Differential Calculus
Chapter 5 p197 The Natural Logarithm and Natural Exponential Functions
Chapter 6 p229 General Logarithmic and Exponential Functions: Applications
Chapter 7 p265 The Circular Functions Sine and Cosine
Chapter 8 p305 Circular Functions: Derivatives; Other Functions

Each section has oral exercises and written exercises, the latter with three levels of difficulty, labelled A, B, or C.

There is a 3-page section of computer exercises to be programmed in BASIC.

There are answers for the odd-numbered exercises.
Title: Re: Old School "New Math"
Post by: Nation of One on March 22, 2018, 10:29:22 pm
I found myself daydreaming about the possibility that these texts might one day in the dark future become highly coveted collectors' items among a cult of amateur mathematicians in the tradition of Abdul Alhazred and the fabled Necronomicon.

Title: A Cult of One
Post by: Nation of One on March 23, 2018, 12:58:15 pm
For now, I myself am such a cult.   8)
Title: Re: Old School "New Math"
Post by: Nation of One on July 30, 2018, 02:03:21 pm
Holden,

I have to say, over this past week I have found myself searching our humble message board for specific posts of yours for one reason or another, and I have come to the conclusion that our idea in 2014 to switch from email correspondence to an Old School style message board for the purpose of organizing the Chaos of the Mind was a wise decision.

Yesterday I received a subtle insult from an acquaintance of my moTHEr in the form of a very boring and somewhat remedial book he gave her to give to me called "Mathematics Made Simple: An Introduction to Algebra, Geometry, and Trigonometry".

Needles to say, since you are well aware of how I tracked down very specific texts which contain novel exercises as well as a rigorous treatment of the subject matter including actual PROOFS (written by mathematicians rather than by math educators), you can surmise that I was more than a little "put off".

Today I was still slightly ticked off about it wondering why this arrogant blowhard would think I would be in need of such a book, and my mind zoomed into something you had once brought to my attention.   I remember reading it out loud into a tape recorder while drunk nearly four years ago!

And now I can't seem to find it.  It would have been around the time of this post, Omega Man Cometh-Eats Alpha Gorts for Breakfast (http://whybother.freeboards.org/what-now/omega-man-cometh/msg695/#msg695), in the earliest months of our communications. 

It had something to do with the perception of another [of us] being limited by their capacity for even recognizing certain traits.   Someone who is not interested in proofs will presume they know all there is to know about a  subject area simply because they can apply some formulas summarized in a "Mathematics Made Simple" Introductory summary; and when such a person hears you are studying such-and-such, they will mistakenly assume that you are studying material presented in such a manner, where they just use formulas or apply these formulas in drill exercise calculations.   They cannot even imagine what it is you are even studying.  Proofs and rigor are lost on them.  And they will accuse you of "complicating the subject matter" by involving such things as "set builder notation".   

The thing is, what really irks me, is that such ignoramuses think they are hot shiit for knowing what a "Cloud" is and reading about Quantum Physics in a Popular Science magazine, or for having set up an account on FakeBook.   A Confederacy of Dunces, Holden, that's what it is. 

"When a true genius appears, you can know him by this sign: that all the dunces are in a confederacy against him."  ~ Jonathan Swift

It is not my intention to insult those without any mathematical training.  That's not what gets my goat.  What ticks me off is someone who puffs themselves up or looks down at others, making others uncomfortable, but fails to recognize someone who actually has far more understanding than he can even imagine.  You see, this man delights in showing off his smartphones and tablets, and he supposedly teaches some kind of class for seniors on "how to use computers".  He continually belittles my mother's intellect, as she is not all that swift, although I have taught her how to navigate the Internet so that she is constantly using it fro medical advice and the rest.  I suppose my gripe is against those who think they know a great deal, but don't know enough to realize how little they know.   It's one of those situations where I sense I may have greater knowledge than someone who thinks they know a great deal, even as I am painfully aware of how little I know.


Anyway, I gave up searching post by post for what i was searching for.

I did find this, though: 

Henry Fool: I can't work for a living, Simon, it's impossible. I've tried once. My genius will be wasted trying to make ends meet. This is how great men topple, Simon.

The greats all say the same thing: little. And what little there is to be said is immense. Or, in other words, follow your own genius to where it leads without regard for the apparent needs of the world at large.

Simon Grim: I worked, while you sat back and comfortably dismissed the outside world as too shallow, stupid and mean to appreciate your ideas.

Henry Fool: Is that such a priority? Is that some sort of measure of a man's worth? To drag what's best in him out into the street so every average slob with some pretense to taste can poke it with a stick?

This is how I feel about the kind of mathematics I am so very focused on.  I would not want someone with a pretense to mathematical knowledge poking what i am studying with a stick.

Two books can be of the same subject matter and yet be worlds apart as far as the nature of their content. 

One of the above reviews of Frank B. Allen's Modern Algebra: A Logical Approach, Book 2 Including Trigonometry c.1966 speaks volumes about why I was insulted by another's misconceptions of what it is I actually study these days:

Quote from: Adrian S. Durham
A still born child of New Math the loss of which has left us with little hope
(2007)

I have an MS in Math from Ohio State, and my wife and I home school our three children. We've been home schooling now for several years and it is approaching time for us to figure out our algebra/geometry/trig (or the equivalent) program. As a former graduate student in math, I know about this thing out there lurking under the surface of college math. It is the proof, of course. Somehow what a "proof" is, what "math" is, and what it is all good for has all gotten extraordinarily lost in a way that goes far beyond even the scope of secondary school education.

This basic problem can be heard reverberating in ancient videos of Feynman lecturing to the public on the role of mathematics in physics (and how rigor is not particularly useful). It can be seen in the mathematics curricula of undergraduate programs all over the nation that pander to other departments' needs. cutting out most of the actual math content and reducing the math major to a generalist in the mathematical sciences rather than a specialist in mathematics. At any rate, it is much, much bigger than even math ed or math ed reform and will stop any meaningful progress in math ed reform, for that matter, since it is a basic disagreement on the necessity and/or intellectual value of rigor (and, in many cases, what "rigor" even is for that matter).

At any rate, it's too bad these books are out of print -- victims of a war far greater in magnitude than even the math wars. The New Math of the 60s was as close as it gets to mathematics being handed down to society by its mathematicians, and we threw it all away. Frank Allen's books are not just books written to pay lip service to the movement, but truly written in the spirit of the times by a real advocate of the New Math. In any case, these books are probably the very best algebra books I have ever seen as of this writing. If you put them together with a good geometry program that at the very least proves the Pythagorean Theorem, you will have yourself one first class high school education.

Unfortunately, Frank Allen will never receive the vindication he deserved. But, perhaps he imagined that there might be people like me that would happen upon his work and find it immeasurably valuable in an anti-intellectual world so dominated by politics that only the most vulgar displays of superficial mechanical proficiency are ever even noticed while everyone frantically attempts to "Beat the Joneses" with whatever latest gimmick they can get their hands on.

 >:(

By the way, I appreciate the respect you show to me, as in when you once determined that I put more time into my "studies" than a graduate student (and am not hindered by the mostly unnecessary stress of exams and jumping through professors' hoops).    I appreciate that you have more respect for the way I spend my days than were I driving a new VW Passat paid for by money earned from building "apps" for mobile devices and smart TVs.   ::)

Forgive me if I sound petty, trite, and overly snarky.  You have to read much of what I type with a good sense of humor.

I may not be an actual "graduate student," but the significant aspect of my particular path is that I have hunted down a set of texts which, while may have been too advanced an approach for high school students when published, do in fact present the material with mathematical rigor, something extremely novel at that level.   This means that I am going over areas in a manner that is no longer common at that level, and hasn't been for several decades.   If you think they present this material with this kind of rigor in the universities or community colleges, you are sadly mistaken.

So, basically, I am fine tuning or revamping my entire approach to mathematics, from the calculation-oriented approach which has come to be what we think of as mathematics, to the set-builder notation, rigorous proof-based approach which one is supposed to magically become acquainted with if one wishes to study "pure mathematics".

I don't know why I even bother to document the details of such qualms other than to help drive the point home to myself about the possible significance of what it is I am daily obsessed with and committed to.

I'm fairly certain that I know what I'm doing, and that I am not totally delusional.

If I live into old age, by Schopenhauer's example, if I continue to document what I am going over, then, with the assistance of a basic inexpensive scanner, I may be able to upload a massive amount of notes which might be appreciated by peculiar high school and university students, as well as older, informal independent learners who want an encyclopedic yet relatively concise guide to what mathematicians like Dolciani and educators like Frank B. Allen had attempted to teach in a radically different way during the 1960's.    My targeted audience may be an even smaller segment than Schopenhauer's target.  After all, I am not tackling the riddle of existence itself, but just interested in a novel presentation material often taken for granted without any such rigorous treatment.

Schopenhauer is known as the artist's philosopher, whereas what I am up to is anything but artistic, although it does have to potential to spawn creativity in that it may unleash in the "student" a certain degree of freedom in their own lifelong interaction with the abstract realm of mathematics and mathematical notation.  This is nothing in comparison to the content of Schopenhauer's life's work.  When I say I am following Schopenhauer's example, I only mean to imply that he showed me that a man tinkering away day after day, without wife, children, or employer, is bound to leave some kind of notes behind which a handful of like-minded scholars are bound to find interesting, to say the least.

It is precisely because I am not a mathematical genius that I think my notes and work will be helpful to students who are sincerely drawn to mathematics but who may feel shut out by the classist politics and the ranking systems of formal educational institutions which have driven many students to suicide.

Incidentally, I would like the math notes I one day upload to the Interwebs to become some kind of "cult phenomenon" which might attract a small following of metaphysical mutants.

May it infect them with a desire to quit their jobs, drop out of their respective schools, and live in a yurt with solar panels, catching the rain in large barrels, so that they might devote themselves to their actual and true education.
Title: Re: Old School "New Math"
Post by: Nation of One on July 30, 2018, 09:52:21 pm
After all that, I have to confess that I may have over-reacted.  You see, I actually already have a 1962 edition of the original 1943 text, Mathematics Made Simple by Abraham Sterling and Monroe Stewart.  I had gotten this back in February when I was working on the Square Root Algorithm code because I was interested in how students used to be taught how to do this by hand before calculators were readily available.

Maybe the guy wasn't insulting me, and simply thought I might want a 1991 revision.  I don't know.

I guess I shouldn't be so quick to jump to conclusions.  What a sensitive bum I am!

Man, Holden, I guess that woman neighbor I had back in 2006 might have been on to something when she accused me of being a little arrogant at times.     :-[
Title: Re: Old School "New Math"
Post by: Holden on July 31, 2018, 08:13:46 am
Quote
Man, Holden, I guess that woman neighbor I had back in 2006 might have been on to something when she accused me of being a little arrogant at times.     :-[

You are not arrogant by any stretch of the imagination. I do think math is a sort of reflection of the eternity.I also study a bit of math after working hours.But I don't like to write much about it as it is quite basic and may not interest you much.
In the office my colleagues turn my life quite difficult and then I study mathematics and feel sort of better. I try to understand their motives. I do not believe in literal reincarnation & yet I am not sure if death could erase all the pain. Its better to try to understand it then to wash it away.

I do not makes friends easily. Come to think of it- I hardly talk with anyone except you and Señor Raul. You see,what you are doing is extremely hard for a man caught up in the net of Maya to understand. It think if I were you I don't even greet the man who you mentioned when he visits your home & would just hide away in my room.

It is important to recognise that much as we may admire Schopenhauer  we are not him. You are Herr Hauser & I am Holden of India & we must each sort things out in our respective circumstances.I have learnt a bit of math-I have been trying to do so for the last 10 years despite working full time. Sort of like Kafka writing his stories at night. If it is destined,then certainly one day I might be in a position to share codes and proofs with you. But also if it is not destined then,well, I could only say that I am sorry for sort of letting you down,but one cannot really fight with destiny. My math notes are really child like & if I were to get killed  today in a motor vehicle accident then they would just be chucked away. But you see, Herr Hauer,this would have made me very sad indeed sometime  back-it no longer does ,for I am learning to see the cause & effect relationship.






Title: Re: Old School "New Math"
Post by: Nation of One on July 31, 2018, 12:18:17 pm
In reference to your statement from the Creatureliness (http://whybother.freeboards.org/what-now/creatureliness/msg5788/#msg5788) thread:

Quote from: Holden
What Herr Hauser has done is to tear a hole in the collective hypnosis of the early 21st century ,creating a wormhole of the imagination through which I have followed him.

... and related to this statement from this thread:

Quote from: Holden
I have learnt a bit of math-I have been trying to do so for the last 10 years despite working full time. Sort of like Kafka writing his stories at night. If it is destined,then certainly one day I might be in a position to share codes and proofs with you. But also if it is not destined then,well, I could only say that I am sorry for sort of letting you down,but one cannot really fight with destiny.

Seeing as you have been the sole recipient of my "Salvaged [digitized] Scribblings" from 1987 to 2015, God willing and the crick don't rise, I think it is quite probable that, before too many years pass, you will also be the main target audience for motivating me to one day finally get around to putting my nose to the grind stone and scanning each and every page of the multi-volumed series of math notes I am gathering during this strange phase of my life.

I don't think we need to be concerned about being on the same page simultaneously, and the fact that you are presently kept jumping through hoops at the office, which is why I refer to you as the Kafka of Northern India, does not mean that this will be your Fate indefinitely.

Considering that you may one day find yourself with many consecutive days of leisure at some point in your [distant?] future helps to motivate me to keep my notes well organized and as legible as possible.  Even though all the notes are in pencil (for I constantly have need to erase), I tested some pages and they are legible in the pdf digitized format.

Mind you, I have no intention of scanning these any time soon; and hence, it is just as well that you are studying at the only pace you can steal into the nights.

While this is not the main motivation, for the joy is in the intellectual journey itself, it would be something special if these notes end up some kind of spiritual treasure trove in a later phase of your own life.  It would be quite novelesque, even if you end up being the only one, besides possibly my own nephew, who gives any attention to, and thereby benefits from, these years I have to "fill in the gaps" in my topsy-turvy "mathematical training".

I will not speak much about this mathematical element of our spiritual connection, but want you to be cognizant of the fact that your commitments to your employer and your daily struggle to deal with the burden of your own existence should never be construed in any way as "letting me down."

Should you ever be on the brink of ending it all, I urge you to let me know so that I might drop everything and get down to the thankless task of starting the process of digitization of the "math notes", which will be very dissimilar to the batch of "real life diaries" I shipped out to you on the flash drive.

As I alluded to before, it may be just as well that I am free to do what I am doing now without any kind of commitment to explain any particular page of the notes.

Should something happen where my life unexpectedly ends, I am not sure the notes will ever be scanned, and I am not sure anyone in my family would even know what to do with my books but to donate to a church of some kind.

I would give you my nephew's email, but as he is on the other end of the continent, he will not have much power to salvage anything.

He sometimes does not respond to emails as his box is flooded with spam.

Still, now that I think of it, it would not be a bad idea to send an email to you with his email.  I would send him an email to request permission to do so, and then, if you don't mind, I could also send him your email so that he can contact you should he hear that I had been involved in what Ligotti calls "a vehicular misadventure".

Let me know if this sounds like a good idea (the email to nephew idea).
Title: Re: Old School "New Math"
Post by: Nation of One on August 01, 2018, 07:37:09 am
Of course, at the end of the day, I have to realize that my notes are mainly the main medium for helping my mind focus on this particular discipline, and it is in no way, shape, or form dependent upon anyone else sharing an interest, whether in the future or not ever.

An audience is not required.

It is difficult to remain focused and to resist becoming overwhelmed with the Fear inherent in the human condition.  We are not very efficient animals, perhaps even quite flawed by our very design.

People like to think "God" or "Nature" knows what it is doing.

I like to reflect on Kurt Vonnegut's assessment that it would appear that evolution doesn't know what it's doing.

Let us live in the moment, then, and face the possibility that the universe itself is a failure.

Let us doubt.

I will not assume anyone would be the least bit interested in my insignificant documentation of my study of mathematics, since, well, there are more pressing issues swirling around in peoples' heads.

I must learn to become more selfish, learn to not care too much at all about what happens to my notes should I die or become homeless.

After all, nothing really matters.

My nephew has become more difficult to reach and hardly responds to email.  Hell, he could be dead for all I know.

So, I will stop kidding myself about "leaving a trail of notes" that someone might be interested in.  If I get around to scanning them before I disappear from the grid via death or homelessness, then I suppose someone may eventually appreciate them, but whether that happens or not makes no difference, really.   It makes no difference to me.

The main purpose of the notebooks is to help me remain focused, helping me to overcome daily anxieties enough to possibly be in some kind of control of what goes on in my own head.

I do not want to put any pressure on anyone to take an interest in these things.  I am content to admit that what I do with my days is totally insignificant and of no consequence or importance.

Maybe the reason I am not fond of the "Made Simple" approach to mathematics is that, if you ever wish to engage with mathematics in a rigorous manner, then it is going to get complicated, and to many people, such complexity appears ugly.  It all may seem so unnecessary.

It is difficult to discuss such things meaningfully since, unlike Holden, few are willing to admit ignorance on a subject, and they feel that if they have been exposed to something on even the most shallow level, that that "know about that."

I am used to working alone, and I understand that the only requirement for me to continue studying is that I maintain my personal interest in what it is I study. 

Discussing things with others is not a requirement.  Please note that I say this without any kind of resentment or anger or disappointment.  I am not disappointed.   

I am like Holden in that I am very accustomed to being alone, and very much accustomed to thinking about things few people are interested in.   

I am not deluded into thinking I am somehow preserving the efforts of folks like Mary Dolciani and Frank B. Allen.  My obsession with their books is simply a consequence of my estimation of the value these books have for me personally, that they do not sugar-coat things that are complicated.   I would not expect others to share my excitement, enthusiasm, or passion; and I appreciate the encouragement Holden and Raul have given me.

It is difficult to speak to people about what I study as I may become angry if they were to misconstrue what it is I am actually up to.

I read too many comments on the Internet by those who call themselves "web developers" who find most mathematics quite unnecessary. 

Sometimes I feel like a volcano about to erupt, but I am just a human animal destined to be eaten by the worms.  This seems to be the only activity that helps me stay calm.  I mean, it's something I never would have been able to do were I registered in some formal university "Physics" class.  I suppose that, for whatever reason, proving trigonometric identities is more meaningful to me at this time than applying calculus to solve physics problems.   My formal education seemed always to be geared to higher and higher levels, when I know from personal experience that I could benefit from a more rigorous treatment of trigonometry and analytic geometry before proceeding any further into an informal study of physics.

I am haunted by the fact that one can pull A's in physics and multivariable calculus, and then, years later, find proving trigonometric identities in a formal manner quite challenging.

The reason I take the liberty of describing the details of the kind of discipline and radical devotion it takes to face such challenges is because I sense that this will also be a lifelong issue in the background of Holden's life.   The yearning to understand does not fade over time.  If anything, you begin to realize that, if it is deeper, more intuitive understanding you are after, it really is your own responsibility to do what it takes to understand.  Sometimes this amounts to forsaking any notions of formal education and tracking down the textbooks suitable to how you want to approach your personalized education.

I'm sure this is not a merely a Hentrich thing, but a human thing.

Quote from: Holden
My math notes are really child like & if I were to get killed  today in a motor vehicle accident then they would just be chucked away.

You may not think I understand what that feels like, but I do.  In order for me to overcome that feeling of writing "child-like" math notes, I had to have some notebooks with a little heavier paper which would be used when neat diagrams were required.  Now I mostly use the least expensive composition books I can find (50 cents for 100 sheets), but I would not hesitate to use a 10 dollar sketch pad were I to find an area I was going through required constant use of rulers and compasses, etc.

Yes, I understand how critical we can be of ourselves.  That is one of the great hurdles, and why I encourage you to continue to pursue your interests in math privately.

Henry Fool was right about this.  Do not give people an opportunity to poke you with a stick.

Try not to be too critical of your notes, however child-like they appear.

One thing I love about set-builder notation is writing the curly braces, and I like to use <----> symbol between corresponding statements which imply each other.

As pointed out in that video I linked to in another thread (http://whybother.freeboards.org/math-diary/fishing-for-pointers-on-self-teaching/msg5796/#msg5796), Each individual learner has to discover mathematics for themselves.

It is a private matter.   Even more crucially, I see it as a sacred matter.  After all, we are born alone, we die alone, and - for the most part - we learn alone.

Learning, like dying, is a solitary activity.

We don't need to be studying the same material.  I would only like to keep encouraging you to continue discovering mathematics for yourself.   

That feeling that your notes are child-like, well, this is the sensitive ego chiming in.   Ego was one of the things which prevented me from revisiting material from old high school textbooks, for you see, I saw myself as a "university graduate who pulled A's in Physics-II and Calculus-III" 

I never allowed myself to experience the "humiliation?" or "ego-deflation" of realizing that there are very many gaps in my "mathematical training."

There is a reason I keep repeating myself over and over again about how important it is to nurture a Secret Private Inner Life of the Mind.  I know, from being a maintenance worker wearing a monkey suit for the park for 10 years, or even when jobless, cleaning toilets at gas stations for beer and tobacco money, that a scholar is not really identifiable by garment or position in society.

I am fairly certain there are many scholars wearing State issued prison garb, or hospital gowns on psychiatric wards in hospitals.

I was thinking about this hole in the collective hypnosis of the early 21st century you credit me with tearing, thereby creating a wormhole of the imagination.   

I am quite fond of this metaphor.  While Hollywood and Sports Culture lay claim to the imaginations of the youth, my imagination is fertile ground for a New Math Revival.  I don't need society's permission to honor these ignored or despised texts.  Maybe they were too rigorous for high school students, but they are perfect for me at this stage of my life, when I am calm enough to take "mature" notes and really document my engagement with the exercises.

You may want to use scrap paper to work out what you call child-like notes, and then transcribe a more organized and detail version into a more permanent notebook.

I must encourage the use of notebooks and to learn always with pencil in hand, even if the marks you make on the paper appear child-like.  You will witness maturity, but you have to be very patient and understand that it is a lifelong learning process.

I have found that it is also helpful to keep a few color pencils handy as well, for boxing, circling, pointing things out with arrows, topic headings, etc. [red, blue, green, black, or whatever you choose].  It helps to separate your work for each exercise with a colored line if you feel like it all looks like it is running together.  You want to be able to refer to your work to see where you made any errors if your results are different from answer key or computer algebra system.  Sometimes, I change an answer in a key if I am very confident, else I error on the side of caution and hunt down where I might be mistaken.

I would caution against the use of ink pens for "doing" math.  There are those elite snobs who write their proofs in pen.  I am certainly not one of them.
Title: How do you know what you're going to do till you do it?
Post by: Holden on August 02, 2018, 08:23:11 am
Well,I would love to get your math notes someday.By all means,feel free to provide my contact details to your nephew ,provided he does not have any problems with regard to it.You mentioned in one of the other threads that circa 2014 I was into maths a bit-well, I ,indeed was.
You know what I did then was quite something. I really was into maths.
There have been many false starts as regards mathematics for me. I mean, I thought many a times that now I'd be able to study it on a more or less continuous basis but it never really materialised.

When I was at school, at one point of time, I wanted to study mathematics. But I ended up majoring in a subject which really makes folks laugh when they hear about it so I won't mention it. But I think it is not so much as lack of aptitude as the psychological issues that have held me back.Your nephew is only a few years older than me I think and yet unlike him I know almost nothing about programming and very little about mathematics.
But you see, I have been doing a lot of research into the roots of mathematics & why it makes some people tick & others don't like it.
Will I be able to study mathematics in the future? Honestly I do not know. I'd quote "Holden":

" A lot of people, especially this one psychoanalyst guy they have here, keeps asking me if I’m going apply myself when I go back to school next September. It’s such a stupid question, in my opinion. I mean how do you know what you’re going to do till you do it? The answer is, you don’t. I think I am, but how do I know? I swear it’s a stupid question."
 

I often say this to myself.
Title: Re: Old School "New Math"
Post by: Nation of One on August 13, 2018, 11:51:42 am
I realize that I am considered a "poor man" in the USA, and such men, with no financial resources to speak of, do not generally write up any kind of "will;" but I have been considering where the safest place to donate my humble little library along with "the notebooks" would be.

It would be impractical and all-too-expensive to have them transported either to Northern India or the north-western United States (Seattle area), and I sometimes think, in the event of an untimely death, the best place to donate these books and notebooks and computer programs to might actually be the high school I attended in my youth; that is, good ole' Christian Brothers Academy on Newmansprings Road, across the street from Brookdale Community College in Lincroft, New Jersey, an institution uncomfortably reminiscent of the school in Salinger's Catcher in the Rye.

It is there that, perhaps, some "teaching brother" (not a priest) in a black uniform might take an interest, specifically in the Dolciani collection, but also the notebooks, as an aid to math instructors.

To my nephew, even if they could be shipped, the books and notebooks might represent a burden, and to Holden of India, where physical shipment is even more infeasible, they would become an anchor chaining him in a room/cell for a couple decades.   No, I'm afraid, as depressing as it may sound, the recipients of this secret treasure most likely will be a religious order of pseudo monks called "Christian Brothers."

I just wish someone besides myself could benefit, or that I might be one small link in a fragile chain preserving this kind of treatment of this level of the discipline.

I am sure I must sound like a real nerd, if that word is even taken seriously anymore.  It's such a disgusting word, I think.  Back in my high school days I ran track, smoked herb, and wished I understood the math better.  I did not fit into any of the "jock, druggie, nerd" roles, although I was leaning strongly in the direction of druggie and totally sympathized with the "nerds".

Truth be told, I have a special kind of hatred for those who use the word "nerd" in a derogatory fashion, even if I personally do not identify myself with such shallow labels.
Title: Core Books in Advanced Mathematics [U.K]
Post by: Nation of One on September 11, 2018, 01:31:38 pm
Holden,

You may want to save and store these somewhere for future reference.  This series from around 1983-1985 seems very focused.

Core Books in Advanced Mathematics (http://libgen.io/search.php?req=Core+books+in+advanced+mathematics&lg_topic=libgen&open=0&view=simple&res=25&phrase=1&column=def)  (Plumpton)

Title: Re: Old School "New Math"
Post by: Nation of One on November 15, 2018, 12:57:53 am
I realized that even if the future finds me a drunken Herr Hyde, a mere Igor to my current Dr. Funkenstein mental state, that I am leaving plenty of grunt work to be handled by a less clear-minded version of myself.

My little life goal is quite humble but incredibly time-consuming.   As Henry Fool said, it takes a lifetime to achieve, really.  My life's work will not be some deep philosophical magnum opus, but simply a huge collection of hand-written notebooks with evidence of how I would work through the full spectrum of high school and undergraduate "Preliminary Mathematics."

If I do leave a small philosophical treatise which gets the attention of a small cult following of the future's youth, a handful of those youth might be genuinely inspired to spend several years going through the "math notes" I intend to one day scan into digital format to send like some kind of arrow into cyberspace before I die.

That's why, as Dr Jeckyl of today, I work so diligently while my mind is clear, sharp, focused, leaving instructions to a possible future-self I will call Herr Hyde.  I trust that he will be a diligent, even if somewhat drunk and bad-tempered, loyal counterpart to my present self.  In other words, Dr. Jeckyl trusts that Herr Hyde will follow through and scan the math notes even if he is too drunk to care about or understand the contents therein.

Dr Frankenstein will be an Igor to his own notes should the future throw me a few curve balls that send me into a downward spiral.

I had this thought today, and it warmed my heart, really.   I took this as evidence that I am "whole," that even the most flawed and broken aspects of my psyche have a certain appreciation and respect for this humble desire to help some curious seeker of the future in an underground, nontraditional manner from beyond the grave in a manner that may or may not have the blessings of the established educators of the future.

The math notes are void of philosophical reflections, and this is what may make them all the more novel.

The youth will not be able to deny that this man, whatever society may say about him after he is dead, really was into "his old school classical and so-called MODERN mathematics" ... even if he was not a very good role model in a society which scorns the thoughtful and philosophical man who would prefer to contemplate equations to shooting guns or praising the Lord.
Title: Re: Old School "New Math"
Post by: Silenus on November 15, 2018, 02:24:35 pm
I am not sure where else to post this, but since you bring up the potential for "downward spiral" in your life, and seeing as I too will be changing my way of getting through this life soon, I wanted to let all of you know that in a recent video Gary Inmendham has stated that he is likely losing his humble hermit's abode due to the landlord selling the land he rents on.  :-\

https://www.youtube.com/watch?v=K8GgTMPtru4 (https://www.youtube.com/watch?v=K8GgTMPtru4)

Sorry to take this thread off of the rails a bit.  We simply do not know what is around the corner for us.
Title: Re: Old School "New Math"
Post by: Nation of One on November 15, 2018, 03:28:31 pm
This would present quite a dilemma.  Landlord selling the land - sounds like a conspiracy with some Big Money behind it.  "They" are trying to "fuuck with G's life ...

Big Money (https://www.youtube.com/watch?v=-KqKMJmvk4s)

I could not sit through an hour of watching anyone, not just Inmendham, read through "comments" while smoking on vaporized nicotine.   

It makes me thankful that I can amuse myself with old math books. 

I sympathize with the stress that this selling of the land to an ass-hole is going to introduce into Gary's life.   One thing I appreciate about Inmendham is the insight he has into his own low frustration tolerance.  He knows that such a "curve ball" will have him seriously contemplating suicide.   I have somewhat of a game plan but very basic:  place books, notebooks, cot, blankets, and computers in very small ($50 per month) storage and roll with the punches, hoping that I might land somewhere where I might transplant the "study cell".

Oh, and I can't refrain from stating that I am more than a little creeped out by the background images in that V-Blog.   It leads me to question where the hell Inmendham's head is at.   Does he really think people want to see images of his body in tight shorts like that?

Maybe I should keep such comments to myself.   Maybe I am just uptight.  Actually, I don't think I would want to be walking in the woods without long pants on as I would often run into patches of stickers and find myself crawling in areas where you would prefer some clothes to protect the skin.

Eh ... whatever, I suppose.   
Title: Re: Old School "New Math"
Post by: Nation of One on December 15, 2018, 01:12:54 pm
I have locked this topic as the essential information pertaining to the "New Math" movement, i.e., "Modern Mathematics" ===>  "Abstract Mathematics", is covered on the first page of the thread (for posterity).

Last point: What is the difference between "Discrete Mathematics" and "Finite Mathematics" (https://eric.ed.gov/?id=ED313082)?

I wanted to add to this thread but will redirect it to a new thread called "Opposition to Pure Mathematics (http://whybother.freeboards.org/math-diary/opposition-to-pure-mathematics/msg6384/#msg6384)".
Title: Some Classic Dolciani Texts on Library Genesis
Post by: Nation of One on October 26, 2020, 09:19:33 pm
(https://covers.zlibcdn2.com/covers299/books/58/f7/a1/58f7a13397a4cd23c03af395a34aa2c5.jpg) (https://b-ok.cc/book/5236165/cbe672)
Modern Introductory Analysis
Mary P. Dolciani, Edwin F. Beckenbach, Alfred J. Donnelly, Ray C. Jurgensen, William Wooton



(http://gen.lib.rus.ec/covers/2477000/b8c827096a49d6f05f6211a7c099884b-g.jpg) (http://gen.lib.rus.ec/book/index.php?md5=B8C827096A49D6F05F6211A7C099884B)
Modern Algebra - Structure and Method Book One
Author(s):   Mary P. Dolciani, Simon Berman, Julius Freilich



(http://gen.lib.rus.ec/covers/2175000/0d5195dd602e3de478df9470bb674620-d.jpg) (http://gen.lib.rus.ec/book/index.php?md5=0D5195DD602E3DE478DF9470BB674620)
Modern geometry   Volume:
Author(s):   R.C. Jurgensen, A.J. Donnelly, Mary P. Dolciani, A.E. Meder



(https://covers.zlibcdn2.com/covers200/books/c8/62/6a/c8626aaeec5bb488e81c7707d9c16628.jpg) (https://b-ok.cc/book/2577298/d17306)
 Geometry
Houghton Mifflin McDougal Littell
Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

That's it for now.