Author Topic: The Math Modules  (Read 1859 times)

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Kaspar Hauser

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The Math Modules
« on: October 18, 2020, 01:40:51 am »
I have scanned and uploaded the first of the "preliminary" notebooks,
Fundamental Mathematical Preliminaries, Book 1 



I will begin scanning handwritten solution keys for exercises as well, but there are so many of them, so it will be very slow going.

Anyone who wishes to get into these exercises can download (get it while the iron is hot even if you just leave it with these solutions for a rainy day) the original Modern Introductory Analysis, circa 1964/1970, at Library Genesis. 

(click on book!)



Note that the text is not truly necessary as you can just use my notebooks, but it sure would help you keep track of my notes.  It would also allow you to go through the exercises on your own without the solutions present.

It is a shame that the first handful of the corresponding (long) series (of solutions to all exercises) are lined notebooks of the cheapest quality, so it is with a little shame that I must scan these first, while by the fourth or fifth I had made the switch to unruled notebooks for the sake of posterity.  The unlined kind have since allowed me to organize "the work" (solutions, steps, notes) in a more aesthetically pleasing manner.

The link above goes to the Wayback Machine, where I also stored my psychobabble scribblings (The H-Diaires).  The "fundamental preliminaries" notes were placed in better notebooks for posterity.  I have a feeling they will take on some kind of glow postmortem, although I will not be around to experience that glow.   Maybe this will help the hardcopies glow for me should I live long enough.   Although, if there is no bread, water, or decent shelter for "the future," I suppose only some bored eccentric math student might stumble upon them.

the full URL address is not difficult to remember: 

https://archive.org/details/@gorticide?tab=uploads

The background explanation and motivation for this project is sort of outlined in the thread, Old School "New Math"

Note:
Quote from: Richard J. Petti
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.
« Last Edit: June 25, 2021, 10:20:06 pm by "No No Bad Dog!" »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Kaspar Hauser

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Re: The Math Modules
« Reply #1 on: October 26, 2020, 09:28:40 pm »
Analysis Scratchpad 1

Solutions to exercises chapters 1 and 2 of Modern Introductory Analysis
(1: Statements and Sets, 2: Ordered Fields)
(see first post in this thread for text and FMP1 (extra notes on material))


It may take awhile to load, but then it can be downloaded.  I would not suggest reading these with a smartphone or e-reader.  To do yourself justice, it is best read from notebook or desktop computer.   I think the human interface is important.  Of course, with a phone, if you have access to a printer, you might print out specific sets of exercises and work in a manner suited to your personal preferences and circumstances.

I have only been able to scan the first scratchpad for the introductory analysis series (solutions to exercises).  I am beginning with the original Modern Introductory Analysis (1964) text, but further along, I begin to alternate between chapters, eventually covering all of both the original as well as the 1988 edition, called simply "Introductory Analysis."

Also note that I have about 50 other notebooks filled with solutions from texts covering Geometry, Algebra, and Trigonometry, but I thought it best to start with these, the final texts in the curriculum.    Then, if someone works a little through the "Modern" Introductory Analysis and feels they need more detail into preliminary material, they can go through those texts/exercises/notes first.   They will just have to be patient and wait for the earliest, as I will most likely upload in reverse order, with most challenging material first; that is (1) Analysis, (2) Algebra 2 & Trigonometry ["precalculus"] & Limits, (3) Geometry, (4) "Algebra 1".  These other notebooks began as what I called "Disciple of the Weird".

It is intended to present these "high school" topics in such a way as to bridge the gap into advanced mathematics.  I fully understand that this material is not "popular."  If this entire project is pointless, this shall not discourage me, since it can be no more pointless than life itself, or the schemes projecting what kinds of "jobs" will be available for future sapiens of the industrial or post-industrial world (after some cataclysmic collapse).

This is something I would like to leave for some set-theoretical weirdos of the future.

I am not sure how much will get uploaded, but it is something I do hope to be able to keep pecking away at on a daily basis for future students not yet born.  Consider it my own little Vastarien tome, a kind of over-reaching, a forbidden book (encyclopedic reference/series) - in the sense of being something ALIEN to the 21st century education entrepreneurs' whole "Common Core gold rush".

There may be absolutely no interest in it whatsoever, and this is just so wonderfully depressing to me; but the depressing feeling is balanced out by absurdity.  I want to leave something behind that may inspire some kind of very small cult following.  That is, a formal presentation of the structure and method of modern mathematics - in a series of hand-written modules - may possess the potential to not only elucidate the structure and methods therein, but to also serve as an example of how one might go about getting through a life not worth living.   ;)

I do not want my commitment or devotion to getting these notebooks scanned and uploaded to The Wayback Machine (archive DOT org) to alienate any of our registered members even though I am not in the least bit anxious over the possibility of further alienating myself from the diaper-sniffing masses.    Maybe none of those babies being hatched will ever have any use for these modules.  No matter.  I am the servant of my Muse, and I toil where She commands.
« Last Edit: November 01, 2020, 01:41:40 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

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Re: The Math Modules
« Reply #2 on: November 01, 2020, 01:37:19 am »
Analysis Scratchpad 2



Solutions to all exercises
Modern Introductory Analysis
ch3:  Mathematical Induction, Sequences & Series
ch4:  The Algebra of Vectors
« Last Edit: November 01, 2020, 02:14:40 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

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Re: The Math Modules
« Reply #3 on: January 24, 2021, 04:06:35 am »
There must be nearly 100 notebooks, with far greater detail and care, which I may never be able to scan.  I would be able to employ a handful of drunken monks to do the task were I of a Clan of Learned Scholars or Oracles, but I am a man of no means whatsoever, so must play all roles in this absurd comedy.

If I could end it all with a desire, the pain associated with organic existence would have been enough to destroy my will to live.  The ultimate solution is to end it all somehow by whatever means necessary - and yet!

Therein lies the rub, the ridiculous nature of existence, that the brute terrestrial man [who walks upon the earth] that I am witnesses his core organism-as-a-whole-in-environments manage to consume food over the pain; that is, a Presence observes itself as Lebenswelt des Schmerzes (Lifeworld of Pain).

This Presence most certainly would not choose to exist as endless intense pain and agony.  Such distress is beyond the dignity of any organism-as-a-whole in any-environment-whatsoever --- and yet! (again)   ::)

The bottom line is that The Razor's Edge of Conscious Existence, the one we are aware of in the ever-elusive present here-and-now is a mysterious place we take for granted as our ATTENTION.  The brain gets this Presence's attention with PAIN, intense and horrifically redundant.  The patterns of pain-inducing inner/subjective neurological communications involved in organic existence can be classified as Eldritch.  The patterns themselves are older than mankind, older than the most ancient Civilizations or Empires of Earth, existing long before the most prehistoric ancestors of the Natives, Aborigines, and Natural World Peoples of the Earth, built into the very fabric of material existence itself.

Pain is essential.

Pain is all there is.

Pain is never missed.   Only if I am able to subdue the pain of the here and now will I be able to follow through with any "projects."

There is a level of ridiculous in this state of affairs which screams ridiculous.

In such moments of intense subjective experience, when all attention is focused inward on the personal Hell of organic existence, I get a glimpse of Schopenhauer's skill as a Teacher of Mankind, a true "holy man."   

We are trapped in processes which manipulate us with severe pain.  Our so-called very own neurological organism torments itself into action.   This is by design. 

Is it not apparent that the concept of a Creator would incriminate the said Creator in that this design shows no sympathy for the poor Creature itself who must suffer the Creation in all its "subjective" detail!

It would not seem ethical to create organisms without their concent.

Is the Christ, Son of God, Story supposed to be some kind of consolation, that the Wizard takes human form and suffers organic existence?   If so, then, I can appreciate the analogy, but then we would all be Sons and Daughters of this God of Abraham, who a Gnostic would call Lucifer, the Creator-God of the Material World, the World-as-Representation, Maya.

At some point in raw experience, part of the mind must surely discard all the garb inherited from our Elder Brothers of this Family of Man and face the world as representation, our Lebenswelt [LIFEWORLD] head on.  This may happen most directly during intense attention pain to pain.

What could possibly over-ride severe pain?  Desire for food and satisfaction of eating, however powerful, is the heading off of the extreme agony of starvation.

The world of mathematics textbooks and technical manuals never mention this raw, Elephant in the Room, thing-in-itself, we call life, or "the world," existence itself.

It's a given, I suppose.
« Last Edit: January 24, 2021, 04:32:43 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

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Re: The Math Modules
« Reply #4 on: February 09, 2021, 01:50:12 pm »


Solutions to Exercises to Modern Introductory Analysis
 ch 4: The Algebra of Vectors (4.6-4.8 ... )
 ch 5: Plane Analytic Geometry of Points and Lines (5.1-5.9)



Exercises 38,39, 40 of 5-9 will continue in Scratchpad #6 of this series.
I will be skipping modules 4 and 5 as they contain notes and exercises from the 1987 edition which included computer programming exercises I wished to explore and rejuvenate with Modern C++ and Python, rather than with its Pascal and BASIC.  So far, for now, my plan is to hold off on scanning Math Modules for the Introductory Analysis circa 1987 edition; that is 4 [1, 2], 5 [2.2-2.8], 7 [up to and including 3-7 Infinite Series], the "deep extension to module 7 DIRECTLY LINKED TO Limits Revisted  (Complete Solutions to Exercises, Buchanan 1966 Limits: A Transition to Calculus), Limits Revisted 2 with APPENDIX in back (A Discussion of the Real Number System), 8 [Functions and Limits = 3.8 The Axiom of Completeness, 3.9 Integral and Rational Exponents], 9 [ch 5 Theory of Polynomial Equations] ... etc !

I will scan modules 6 and 10 (which continue Solutions to Exercises for oriignal 1964 edition of Modern Introductory Analysis, God willin' and the crick don't rise.  Eventually I would wish to scan the work from the other edition as well, since those contain documentation of computer programming symbiotic "events" --- what Schopenhauer might politely refer to as "being possessed by the Holy Ghost" - ? - ! -

I will forge ahead into module 16 as far as "scratching the paper" goes.   >:( ;D



Since this has been such a long-standing project, I have been forced to reinvent similar modus operandi as Robert M. Pirsig.  During these periods when my Mom seems to drain my energies (see Einstein's demands of wife), I find that scanning the original trail of notebooks belonging to this series would be a great way to expend thwarted energies, not to mention coping with lack of gumption due to reawakened interests in Listening to Music While Walking and Sitting, Smoking and Shitting [just kidding].

I am just now catching up with some December and January notes which might help explain to myself just why I stopped where I did in the exercises and began scanning the pages themselves.  This is how I am dealing with a particular way of presenting particular types of exercises, where students are walked through the construction of a formal proof.   I wish to be most focused for this, without an elderly woman pestering me about her next meal or her latest hyperphobic meltdown.  One morning I will awaken and know the time is right for that kind of work.  For now, I am in some kind of transition or simple regrouping.  No code-mode, no math-mode.   As I scan the pages, I too often get lost in the content.  The notation would require some effort to become familiar with, so there is this timeless "Lovecraftian" glow of eldrich intelligence at work - in this case, a group of dedicated "mathematicians" presenting modern mathematics to high school students wishing to understand the structure underneath the symbols and applications.

This culture of well-being has given full reign to the mediocre pygmies.  Excellence of any kind will surely suffocate.

There is only so much time for Being to Be in.  When I give my attention to something, I wish to have that Full Power of Concentration and Focus.  Maybe I have paused to scan simply to allow myself to process the significance of these exercises.  I may have to refresh my memory by looking over the previous module (15) before proceeding with the "scratching of the paper" - but I do intend to scan pages from old notebooks just as a way of validating the efforts (to myself).  If not me, who?   Future teachers/students might benefit.  Most importantly, it might give me some dignity and the Fortress of Mental Solitude required.

I don't understand why I feel compelled to send musical vibrations into the atmosphere.   Part of me has always been like this. 
« Last Edit: February 09, 2021, 05:30:00 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Kaspar Hauser

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Re: The Math Modules
« Reply #5 on: June 14, 2021, 01:04:47 am »
I was able to spontaneously (almost organically) dive back into some computer programming for the "Introductory Analysis" Solution Keys PROJECT, and I sense that I may be getting back into the mathematics.  I have never stopped thinking about it, but I have been reluctant to get into formal exercises - and I have wanted nothing to do with editing and compiling code for many moons.

Well, I finished the exercise set, and I left detailed documentation about the process of writing the code from the mathematics.   The diagrams for 6 possible outcomes (2 when the triangle is right or obtuse, 4 if the triangle is acute) transform into code organically and symbiotically.

I had been discouraged, thinking I had lost all interest in this long project I had once been so devoted to.   And yet, the quality of the notes in the current volume show that my thinking is clear.   There is little room for confusion in the explanations of the code.   The C programs are elegant and to the point in comparison with the old BASIC code with its "GOTO 80, GOT 110, etc ... " ::)

Generally, solutions attempted in BASIC in the 1980s have been written using the C programming language, whereas those solutions attempted in Pascal (in the 1980s) have been written using the C++ programming language.  I do not use a sledge-hammer to swat at flies.

Actually, C and C++ are close siblings.  One has not evolved from the other.

I thought Holden might be glad to know that I have been crawling around dragging some books around.   I carry a small pad that fits in my pocket, I use scrap paper on a clipboard ... I transfer notes.   I have even been writing in my main "diary," trying to figure out when and why I stopped "keeping track of my life" ...

It is mostly my struggles to organize my interests in literature (Madness Theory), mathematics and programming which have usually forced me to continue some kind of "Main Diary" - even if that Diary is not used daily.   I like to keep track of it "over the years" ...

There is something of the ambience of a low-grade existentialist science fiction dystopian world to my moment-by-moment existence.  To find spiritual nourishment from an old forgotten mathematics textbook with obscure computer programming exercises in it (from the 1980s) is peculiar in a world such as ours, you know, with Hollywood, Bollywood, the Olympics, American Idol, and Professional Organized Sports.
« Last Edit: June 14, 2021, 01:09:54 am by Deep Truth »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: The Math Modules
« Reply #6 on: June 14, 2021, 04:02:03 pm »
So, Aslan is on the Move:



Very happy that you are back with your maths books. I am also doing the best I can.I am sorry you are suffering due to your injury and hope that you get better very soon.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.