Author Topic: The Role of Psychology in the Quest for Mathematical Maturity  (Read 3214 times)

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Holden

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Sorry for the late reply but you do leave a lot of food for thought for me,a lot of chew on.
In 2014 ,for a whole year,I studied mathematics with a great deal of concentration. I was studying(not reading,mind) Lovecraft simultaneously. I wonder if that was merely a coincidence. I doubt it.

If I ever do get back to studying mathematics again seriously it must be as a kind of weird horror story. Why do I read horror stories? Because they leave me feeling astounded, flabbergasted. So, you are certainly right in saying that one may study mathematics as a tool of unearthing one’s own lack of understanding.

 I think I do see your point when you say,to paraphrase you,mathematics can be used as tool to unearth & bring to surface our unlimited ignorance or feebleness of the intellect/mind.

Yes,I agree ,you just might have done it again, by introducing me to Schopenhauer you turned my life up side down once & now you just might have introduced me to the greatest horror story of all time:MATHEMATICS.

Now, I would need to brood over it . But I think I might just have understood your point:
Calculus might just be weirder than Cthulu.

https://youtu.be/J1NIpXjuIPM
« Last Edit: April 14, 2017, 01:02:57 pm by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

The Indignant One

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DISCLAIMER:  I am sure our use of horror seems a little melodramatic, and the "horror story of mathematics" can't be compared with our more primal fears of suffering severe bodily harm (such as decapitation, having limbs chopped off, eyes gouged out, or eaten alive.)
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I think I do see your point when you say,to paraphrase you,mathematics can be used as tool to unearth & bring to surface our unlimited ignorance or feebleness of the intellect/mind.

Great paraphrase!  You articulate what I mean better than I can.

We are similar.  The way you escape the horror of everyday life by reading horror-literature/philosophical-horror, I attempt to escape my fear of ignorance by bringing it to the surface so that I face the feebleness of the intellect less fearfully, that is, with courage and an almost calm, resigned demeanor. 

I have done this by going back, and then further back still, until I reach a point where I can say, yes, even at this level, my ignorance is all too glaring (to myself, that is).

As I have said, such unlimited ignorance does not haunt those who never consider it.

Likewise, those who chant the mantra that "Life is good, God is good, or it gets better," well, they have solved the problem of horror by refusing to acknowledge it.

You take a more direct approach.

Now, as for mathematics, I escape the horror of my unlimited ignorance by taking myself back to school.  The significance of privacy in this pursuit cannot be overstated, for it is only in such solitude that one can muster the greatest degree of shameless honesty when facing the root of one's confusion.

I'm afraid that, when it comes to formal (institutionalized) education, there is just too much ego that comes into play, and a general lack of humility mixed with false pride, which sets many barriers preventing individuals from seeking out the source of their confusion.  Students in formal, structured settings, do not permit themselves to follow the trail of their doubts and confusion since there are time-restrictions, so there is a tendency not to go down into rabbit holes, but to instead, as I did at the university, study just enough to do well (relative to the herd) on the exams.   I - or we - thereby forfeit opportunities to build our confidence.  We forfeit confidence building in favor of acquiring credentials.

In solitude, there is no shame at confusion or ignorance, especially since one is actively trying to remedy the situation by "going back over things".  We are able to close one book 3/4 of the way through so as to go down the rabbit hole ... and to rebuild.


I am glad that we are on the same page with this, for although we often jest about certain things, I doubt we would ever joke around about someone's private attempts to study mathematics.   Who would suspect that so much psychology is involved?  We would be accused of complicating matters. 

The way I am dealing with this "horror of mathematics" is that I genuinely accept that I will never be a mathematician.  This helps me keep things in perspective.

Now, having said that, I very much long to improve my mind by developing certain skills even at an elementary level.  I would like to at least trust myself to begin jotting down some preliminary notes to constructing an elementary proof.  This means I can not allow myself to be ashamed at taking baby steps.   

People are wrong to call such obsessions a hobby, especially if the obsession keeps resurfacing throughout one's lifetime.  Is it mathematics madness?  Whatever it is, referring to it as a hobby seems to understate the facts.

I approach mathematics texts the way you approach horror ... Even, and perhaps especially, texts that I would have thought "beneath my level of interest" can be surprisingly challenging.  BOOM: IGNORANCE UNEARTHED!   - the horror and disillusionment in having to witness your own self-image destroyed.

I do not want to live a lie.   As I have said, I suspect many students might deceive themselves about what they think they know.   What goes on in other peoples' brains is none of my business.  The thing is, no matter how much I have studied, in the present moment, in the Now, when thinking is required, that's it, the intellect is feeble.  The brain is lazy.

And this is never a reason to stop, to choose a different path.  No, and again no, I will move as slowly as necessary. 

Such honesty requires social isolation.  Do you know why?

Because the world is generally full of shiit.

The kind of honesty I demand of myself is not to be found out there ... in polite society ... maybe, just maybe, such honesty is found on psychiatric wards.    :-\

While there is an element of horror in this, there is also the treasure of being intimate with the degree of one's own ignorance.   This treasure might be what is necessary for turning the horror of one's own ignorance upside down.  I mean, rather than plunging ahead stubbornly into Computational Physics, where my horror might become amplified, I switch gears and return to elementary analysis so that I can show myself where my knowledge is weak.

Hence, if we go about this process with care, although we risk being mocked by the herd as some kind of useless dreamer, we just might defy reason and common sense, finding great calm and tranquility when we at long last learn how to THINK.

We might learn to think more clearly, more slowly, and never resist the compulsion to scribble notes on paper or draw diagrams or whatever ...

In this way, even though we are forever haunted by the enormity of our ignorance, we will have become intimate with our mental capacity, so we will not overtax it.  We will have developed an intuitive grasp of the feebleness of the intellect, so our egos might not be so easily wounded when we make mistake after mistake.

We will learn to doubt ourselves and smash false confidence.

I still aim to develop humility, patience, and honesty in dealing with confusion.

So, when anyone questions the value of spending so many days and nights going over high school and undergraduate material while the world passes me by, I can say with dignity and self-respect that all this studying and thinking is helping to make me a more patient, humble, and honest man.

When I was a child, after the release of The Exorcist, some nights I would be very frightened that I was becoming possessed.

Do you know how I overcame the fear?

I became one with the Devil and the legion of demons.

They could not harm me if I merged with IT.

I am not demanding that I comprehend things far beyond my capacity.  I just want to know where I stand, and to learn how to work with what I've got between my ears, as feeble as it may be at times.

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related:  The epsilon-delta proof, the nemesis of many a calculus student, is an initiation rite into the mysteries of proof writing.

How To Construct a Delta-Epsilon Proof

* Note that my decision to devote myself to the text from high school that I had no recollection of doing any exercises in has to do with my anxiety in the face of writing proofs.  I seem to really enjoy "doing calculus" or "doing algebra".   I enjoy calculating and computing and mechanically deriving results via the "magic wizardry" of algebra, trigonometry, and calculus; and yet I would like to have more confidence in how to go about constructing proofs, even the simplest of proofs.  For whatever reason, when I see the word "Prove", I draw a blank.  I wonder if there was some kind of traumatic experience that is at the root of this mental block, or that I have encountered so many people who are getting through life without feeling compelled to learn how to read or write formal mathematical proofs.

I may never be officially initiated into this ancient craft, but it's something I just can't seem to shake.  There is no law against trying to understand things that seem so obscure or even pointless. 

There are those who might suggest I would be better off painting houses or digging ditches.   And yet, I don't want to paint houses or dig ditches.  I want to continue to unearth my unlimited ignorance. 

Maybe there is a certain degree of horror involved.  I suppose it has to do with the degree to which one is "haunted" by a sense of "lack" when it comes to their own understanding. 

Maybe I have come to associate studying mathematics with goofing off.  It's probably a reaction to how competitive our societies have become.  I just want to fart around.  It's not important to me to be doing anything useful.  I do not live a "purpose-driven" life.   I guess my only wish in all this studying is that I develop some latent math skills, to stretch my brain a little, to give me something to do with my life (other than writing a book - no, I do not want to write a book!  I just want to complain to Holden on this message board).   ;D



One problem seems to be that if I am working on a series of problems that are too straightforward, that is, too "easy", I find that there is a voice inside my head that mocks me, accusing me of avoiding more challenging material.   The thing is, though, in each section, the last several problems are real brain benders; but I don't want to just skip the easier problems either. 

WTF, Holden, it's ok to say it.  I'm pathetic.  I don't mind admitting it.  I'm down right ridiculous to the point of being comical.  Maybe, after all is said and done, I'm the Germanic Woody Allen ... a neurotic Raskolnikov who would not know what to do if he found himself homeless except to hide in the bushes drinking himself into unconsciousness ... sobbing himself to sleep like an ill-tempered child.
« Last Edit: April 22, 2017, 08:15:21 pm by Raskolnikov »
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 ~

The Indignant One

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OK, you know my story.  I graduated from the university at the ripe old age of 35 in 2002.

Fifteen years later, after collecting many books so that I could pick up where I left off, I stopped just shy of "where I left off" in 2002, and I have been engaged in this experiment with finally studying what they were trying to teach me in 1984/85 when I was 17/18.

Where I stand:  I will be satisfied if I can learn what I was expected to be retaining in high school.  The Solution: Lowering my expectations of how "Mathematically Mature" I will become might actually be a manifestation of maturing.  Schopenhauer counsels us not to try to be something we are not as it will only cause us grief.

I do not expect to become a mathematician, but I am prepared to spend my life "tinkering with mathematics".  I'm just a tinkerer ... my plate is full.

If I were ever condemned to be held prisoner for a long period of time, I would become severely depressed unless I could continue studying in and just beyond my comfort zone.

The way the zealots bring the New Testament, copies of the Quran, and Twelve Step literature (the State religion) into prisons, I day dream of bringing Algebra, Trigonometry, Calculus, and computer algebra systems into "Yard Out University".

When one is in such a place, you wish you had a double, for only a double would know what you need.  No one will go to the trouble of finding just the right book, and certainly not spend too much on a solution manual.   No, they will send cut-outs of prayers to "the Lord Jesus Christ" or some verse from the Bible.

My sister sent in a priest to talk to me, and I refused the visit.   He ended up talking to my cellmate, a hell raiser from Mexico.  Of course, the priest spoke Spanish.  No kidding.

Anyway, what was I rambling on about this time?  Oh, math.  That's right.   Math for the jailbirds!!!   Something to do in the cell ... to become a different kind of monk, a monk who does not believe in God, a monk who does not belong to any religion at all, but a monk just the same ...

I study math every day because I realize what a great joy it would be to have access to just a few of the mind-treasures I have on my shelf in the little room in my mother's domicile.  I will not call it a cell, as I can walk out the door when I choose to.

If the jail cell were to be transformed into the little room I dwell in today, I would have been filled with thankfulness.   I refuse to ruin the delight of learning by constantly biitching about how "I will never be a mathematician" or "I must have smoked too much crack and drank too much booze" or "maybe I am a little brain damaged". 

No!  Forget that bullshiit.   I still have to think very hard to figure out how to solve problems (especially involving writing a formal proof) that I am finding in these special books, "Modern Introductory Analysis" as well as "Introductory Analysis".

I wonder why this material is not covered in this manner in community college or university.   Maybe I am a rebel monk who has stumbled quite accidentally upon forbidden knowledge by stubbornly looking precisely where one would never look in a thousand years.  First of all, who is going to look in a "high school" text?   Secondly, who would think that the material would be presented in a superior manner way back in the 1970's and 1980's than it is now?

OK, so I am a little ridiculous and a bit comical, but I am going to "make believe" I am a toothless monk of an ancient cult, older than the Abrahamic triad, older than the Devil even.   To paraphrase Lovecraft, I daydream of being influenced by a qualitas occulta far older than even the aborigines of North America ... non-human ...

And I am guided by these specters ... guided to these old vintage math books which take on a glow ... saying, "When the student is ready, the Teacher appears."

This student is old.  This student is ready.  How I shun the elite academic mathematicians and just want to be left alone to tinker ... to allow myself to say, "I don't get it."

And to allow myself to be "retarded" ... and to begin to grasp something, and to make genuine breakthroughs that would seem pathetic to the snobs of academia.   

I become fixated on certain books and they become the Holy Grail to me ... in my own little world where I am this child growing to be an old man rather swiftly and almost happily.

.....extinction ....
« Last Edit: September 09, 2020, 06:58:14 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 ~

The Indignant One

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Obsession with Old Math Books as a Midlife Crisis
« Reply #48 on: April 23, 2017, 12:23:24 am »
Of course, it goes without saying that what I refer to as "an experiment in self-education" the gorts would categorically dismiss as "a midlife crisis".

Maybe it is my destiny to approach pure mathematics as a novice, and all my studying might prepare me to one day be qualified to help adults also approach this subject in a rigorous manner.   I can daydream about preparing my own specialized booklets made from scratch, where I make use of the old Dolciani series from the 1960's to 1980's, originally intended for "honors" high school students.   Perhaps this rigorous approach was abandoned in favor of the more user-friendly methods used in this twenty first century.

My humble mission would be to transform this curriculum to be geared not so much for "gifted" youth. but aimed at adults who only discover their interest in pure mathematics after they already went through the whole computational applied mathematics education only to find they would prefer to also approach the subject with more mathematical rigor and formality, something the educators were experimenting with in the 1960' into the 1980's with high school students.

Those days are gone, and quite frankly, as one who experienced that kind of exposure at an early age, it might have been too ambitious on the educators' part to expect more than a small handful of youth to be ready for such formality at that age.  i certainly was not one of the gifted ones.  Besides not being gifted, I was overwhelmed with the existential despair of being a teenager, the pressure of having to register with the selective service, the divorce of my parents, and my inclinations to escape with alcohol - as well as showing up to school after smoking herbs.  [HIGH school]

I lacked the discipline and was, to be honest, an emotional basket case.

Just maybe my obsession with retraining my mind with these old texts, allowing myself to study the solutions in the precious solution manuals, expanding the explanations, etc, might make me particularly qualified to one day assist adults who catch the bug for studying mathematics later in life, long after the pressures to "get into college" or "join the Army" have passed.

And even if these potential students do not exist, and it turns out that I am the sole recipient of the intellectual stimulation inspired by such a belated embrace of rigor and formality, then so be it.   It is quite possible that there is no "unified self" and that, as contradictory as this sounds, I am at once both Teacher and Student.

« Last Edit: April 23, 2017, 12:22:01 pm by Raskolnikov »
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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Herr Hentrich,thank you for all the links & the name of the books which you have been providing.
I am jotting down all of them,and I promise that I would make use of them as well,soon enough.
However, I just want to dig a little deeper into Schopenhauer's philosophy for some more time.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

The Indignant One

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Something odd is happening to me.  I know that life is pointless, and I know what a slow process it is to work diligently through math texts.  The odd thing is that I find myself hoping I live long enough to get through these books.  I just hope Life doesn't get in the way of studying math.  Life has a way of getting in the way, as though the gods took pleasure in frustrating our plans.

What is the point though?

I can't stop myself.  I think that really wanting to learn how to THINK again might be what is motivating me to abstain from imbibing alcohol.

I am obsessed with math ... even "manic" about it.  It is like a hard drug to me.  It's not a hobby; it's an obsession.

I feel like I am studying in defiance of the values of society.   The gort kids would be likely to think it is too late to start again, and most of society has the same attitude.  They love to write someone off as having "lost", as being a "loser".   And yet, like you, I am not trying to "win" the stupid games of society.

I just hope I can continue to spend my days this way ...

I keep repeating to myself, "To be conscious that you are ignorant is a great step to knowledge."

This way, when I catch myself feeling anxious over my brain not being too quick, rather than lash out at myself as "stupid," I can celebrate the process of becoming conscious of my ignorance.

I have to believe that I am not the first person this has happened to, to spend your life seeing yourself one way, but then coming to understand that a "self-image" is not going to THINK.

A representation of who we think we are is not able to think, since it is a thought itself.  A thought is not capable of thinking.

So what is it that thinks?   What is it that knows?  In order to continue studying in this manner, I just have to accept that I am going to spend a great part of my life feeling ignorant ... but imagine how much more ignorant I would be if I didn't spend my life trying to "fill in all these gaps"!
« Last Edit: April 27, 2017, 03:47:04 pm by Raskolnikov »
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 ~

The Indignant One

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On shipping books from US to India
« Reply #51 on: April 24, 2017, 11:38:56 am »
Quote from: Holden of India
thank you for all the links & the name of the books which you have been providing.  I am jotting down all of them,and I promise that I would make use of them as well,soon enough.  However, I just want to dig a little deeper into Schopenhauer's philosophy for some more time.

Take your time, Holden.  We are each on our own peculiar little paths in this life.  There is certainly no rush to get to where we are going, since we are, ultimately, going nowhere in particular.

When you are compelled to do so, do you think you would be interested in going through any of the old Docliani "experimental New Math"  high school books?

If so, I think it would be less expensive for you to order from amazon DOT com rather than amazon DOT in ( global shipping )

Example:  Modern Algebra: Structure and Method (1976) sells for about $7 plus the shipping.

or Teacher's Edition ... or another Teacher's Edition

As opposed to same book from India site over 7000 rupees, which is, as you must know, of course, over $120.

[UPDATE: 9 May 2017] Algebra 1 (Dolciani/Swanson/Graham c.1985 Teacher's Edition) covers proving theorems in more detail than any Algebra text I've ever seen.  That is reason enough to invest attention into this unique text.  [Note: disregard misprint of authors and page numbers.  Just verify with seller that it is ISBN-10: 0395343747 or ISBN-13: 978-0395343746]

And please do not be insulted if you think that such an Algebra book is beneath you.  I am ordering a copy myself just to see how the material is presented.  I had this book as a freshman in high school when I was 13 years old.    I suppose I was able to just waltz into a community college at age 27 and ace Calculus I and II after doing a little review was because I must have learned something in high school way back when.  I was very captivated with mathematics at that age until I lost interest in everything at age 17 and ... well ... Life got in my way and took me on a strange trip through the Amerikan Department of Corrections and into the State Park Service as a state slave ever so grateful to be employed by his masters.

So, when you are up to it, I'll be looking [working] through the Dolciani series again as well since I have decided to rebuild from the ground up.   In the past I had been obsessed with Calculus.  I even referred to it as "The Calculus".   Now I am more concerned with developing the ability to actually THINK in the way a pure mathematician would think.   I want to be more than a number crunching calculating machine, more than a "computer".  I want to learn how to become a man who can think as opposed to what I have become, just a man who can compute and calculate.

Don't get me wrong; I prefer calculating, computing, algebraic manipulations and arithmetic.  I love that shiit.  I really do.  I never tire of implementing the Euclidean algorithm (the Division Algorithm).  I enjoy writing computer code to do this, and making the program display the results at each step.  It's the simple things that make me smile.

The thing is, I am stubbornly facing the aspects of pure mathematics in which I feel a total lack of confidence, that aspect of mathematics that requires me to actually think, to construct a proof in a formal manner.   I would like to develop skills in this area if it is the last thing I do on earth.   Liberate mankind from the prison colony of existence?  I'm not the Buddha.  I've become a strange creature who is self-absorbed and preoccupied with my own peculiar agenda. 

When the guy who collects the "garbage" came through while I was smoking a cigarette and looking over a problem, he intrusively asked me, "What do you do?" 

I know he was asking me where I was employed, but I responded, "I do math."

"What do you mean, you do math?"

"I study math."

"Are you going to school?"

"Nope."

We left it at that.  I finished my cigarette with a grin.

It's not too late to start over.   Even though I'm 50 years old, there is nothing else I would rather be doing than retraining myself in pure mathematics, since now I have reached a level of maturity where I don't take offense when I find myself stumped.  Instead, I embrace the consciousness of my own ignorance as an invitation to learn to think in a different way than just diving into the arithmetic algorithm like a machine.

I know you prefer to think of math in terms of chaos and fractals, but there is something to be said for an axiomatic presentation of mathematics as a system of logic.

Anyway, as much as we might resent having to endure an existence we never asked for, it looks as though we both might be sustained by our intellectual development, and by "intellectual," I suppose I mean spiritual and psychological development.

In the meantime, it is great to have made contact with someone who is so captivated with the works of Schopenhauer.  I have been babbling about Schopenhauer for decades to people ... and if they have heard of him, they already write him off as a woman-hating atheist. 

"Life is evil."   

Wow.  And here we are, these living creatures who must kill to eat ...

Reading Schopenhauer is heavier than dropping LSD.  Awakening philosophic wonder, he is one of the true philosophers, like one of the ancients - not an academic philosopher.

I wish I could think in the manner of Euclid rather than just memorize his algorithms!

Would we create our own brute-force calculus if we were not taught the refined modern versions of Newton's and Liebniz's?

PS:

There is also an inexpensive teacher's edition of Algebra 2 and Trigonometry (revised edition 1983)
« Last Edit: May 09, 2017, 04:10:16 pm by Raskolnikov »
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 ~

The Indignant One

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Explanation of what "New Math" was
« Reply #52 on: April 26, 2017, 10:00:04 am »
The following was found in a review from Amazon about a 1965 edition of "Modern Algebra (2) and Trigonometry: Structure and Method" that is an ancestor to the edition linked to above (which is from 1983 with easier to find solution key).   It may shed some light into why I find these books so delightfuly peculiar, where mathematics that you think is so familiar becomes novel, where you begin to see some kind of unity instead of just random prroceduaral computations and formulas.

The following words are by Richard J. Pettion March 26, 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 1990, 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.


Now, I am leaving the following question I read on Mathematics Stack Exchange because I think I have an answer for this person, an answer that no one there suggested.  Why would they?  They might not have thought of this.  I do not want to sign up to answer this until I can organize a clear and coherent response, so I am leaving the question and a link to it here.   I believe the Dolciani series will bridge the gap this college gradulate is experiencing.  The way it presents the material is what bridges this gap.

Quote from:
Joshua Carmody
Is there a good “bridge” between high school math and the more advanced topics?

I love math and would like to know more of it. However, whenever I try to pick up a book on what I consider to be "advanced" mathematical topics, I often have a hard time understanding some of the terms and symbols right from the beginning.

I'm a computer programmer and I've used some complicated trigonometry, matrix math, linear algebra, etc in some of my programs. However I feel my understanding of some of these topics isn't very deep, and I don't always completely understand why certain processes work the way they do.

I've picked up some books and articles on calculus, linear algebra, and some other topics I feel I could use some brushing up on, but I often feel lost right out of the gate when I try to read them. They use terms and symbols that I'm not familiar with, and they don't provide any explanation. I feel (perhaps unjustifiably, but whatever) that I could do the math they're talking about if I just understood the explanation and notation, but I don't understand it.

However, when I pick up high-school level books on algebra or trigonometry, everything I read seems very simple and it bores me to death. I don't seem to find the math "vocabulary" I'm missing in them, and I don't find them challenging at all.

How can I get myself to the point where I can move on in my mathematical education when what I perceive to be "high school math" is too easy, but the more "advanced" subjects are incomprehensible? Do you know of a book I should read, a subject I could study, or some other resource I can use to prepare myself for the more advanced topics?

from Is there a good “bridge” between high school math and the more advanced topics?

I fully understand exactly what this person is saying when he writes, "However, when I pick up high-school level books on algebra or trigonometry, everything I read seems very simple and it bores me to death. I don't seem to find the math "vocabulary" I'm missing in them, and I don't find them challenging at all."

The vocabulary he is trying to understand and is missing from current high school curriculum is introduced and used throughout the Dolciani series.

I wonder if it is worth signing up on Mathematics Stack Exchange just to point this out.

Building this bridge will inevitably involve sitting quietly in a room. As Blaise Pascal said, “All the trouble in the world is due to the inability of a man to sit quietly in a room.”
« Last Edit: April 27, 2017, 05:29:17 pm by Raskolnikov »
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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The Indignant One

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I study mathematics for my mental health
« Reply #53 on: August 28, 2017, 08:32:24 pm »
Holden,

Just now while I was calmly working out a geometry problem, writing in essay form the details of the thought processes I would follow from what I needed to prove, backwards through the chain of reasoning, before I even attempted to write the formal two-column proof, I had a flash of understanding about why I could not encourage you to look for a way to find a profession which involved your spending a great deal of time studying math.

I realized that I do this for my mental health, and that I would resent anyone who had any kind of power over me (such as an employer) demanding I study the kind of math they felt was necessary for me to be a useful TOOL for them.

Because I know I would have a difficult time devoting my mental energies to a so-called superior (a manager, an employer, a supervisor), I could not encourage you to go down that path.

I guess I want you to be able to develop a relationship with mathematics outside of any professional or even academic capacity.

Even were I to imagine myself a "mental patient", were I permitted to devote several hours each day to a textbook, and to keep at it until I was through, and then be permitted to move on to the next textbook (all of which I choose myself), then I think this would have a calming effect on my mental state.

I want to enjoy the process of thinking.

It can get to the point that it is like solving some kind of puzzle, or learning the rules of a sophisticated game of sorts - one that takes place within the mind.

One reason I may not have encouraged you to seek a math-related career is because I do not think your being a student of mathematics should be dependent upon what you do to acquire lodgings and purchase the minimal amount of food, clothing, books, etc.   

Carefully choosing a text to commit yourself to, and then following through and pecking away at in honestly throughout the year during your free time when you are able to rest and rejuvenate, I really think this will help you develop a part of your life in which you can calmly space out and become intimate with the process of training your mind how to think about certain problems calmly and patiently.  You do not have to justify any time you spend studying mathematics. 

Even if you start out with material which seems too easy for you, you can move at a faster pace until you reach problems which force you to stop and think.  You will come to appreciate such problems.

I enjoy mathematics the most when I am in my own little world, where I actually pretend I am some kind of inmate in an asylum.

You may find that you do not have time to study at the slow pace you would like to.

Maybe one day you will have the time.

Until then, there are sure to be some days where you not be required to report to some manager, and then you can crack open the book you have chosen as though it were some sacred book speaking directly to you.

So many books sit on shelves with no one interested in reading them or studying them.

Surely there is a book that is your match.   

I thought that I might think less of myself were I to revisit some high school mathematics.  Not at all.  I acknowledge that I am a very honest man, and that the most crucial honesty is the honesty within myself about my own understanding or lack of understanding. 

The reason I may have trouble in the work force is because I suspect so many of our fellow human beings lack such psychological insight, and I have yet to develop the necessary patience required for interacting with them for extended periods of time.

One day, maybe I will develop such a capacity.

For now I am quite content to isolate.

If you are also destined to isolate from society for long periods of time, I think that planning to engage with mathematics over several decades will be beneficial to your mental health even if it seems to be having the opposite effect.

The more honestly I study mathematics, the more I feel like a "mental patient,"  the more intimate I may become with my own mental capacity.  I am not at all concerned about how I measure up with other human beings.  Thinking happens between our own ears. 

Maybe when we slow down enough to break a problem into baby steps, this is the point where we might feel awkward, but this may be precisely what it feels like TO THINK!

I speak to you as an "older brother", not to discourage you from trying to find a career which involves mathematics, but to reassure you that your involvement with mathematics is not in any way dependent on commerce or earning a living.
« Last Edit: August 28, 2017, 10:24:12 pm by { { } } »
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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Thank you for the posts.I will respond in the evening.I have been sleeping a lot over the last two days due to cough and cold-my immune system is shot I guess.I am getting down with flu every 20 days or so.
I am about to leave for the Salt Mines,would you find the following interesting:
What is the area of a triangle whose vertices are:(a,a), (a+1,a+1), (a+2,a).


Something you might like:

https://googleweblight.com/i?u=https://archive.org/stream/TheFogHorn/TheFogHorn.txt&grqid=fCQ3MjRT&hl=en-IN

La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

The Indignant One

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The Great Tiredness
« Reply #55 on: August 28, 2017, 11:37:58 pm »
I think that the problem is fairly straight forward.  Find the lengths of sides A, B, and C using the distance formula (Pythagorean Theorem):


A = sqrt((a+1-a)^2 + (a+1-a)^2) = sqrt(1 + 1) = sqrt(2)

B = sqrt((a+2 - (a+1)^2 + (a - (a+1))^) = sqrt(1 + 1) = sqrt(2)

C = sqrt((a+2 - a)^2 + (a - a)^2) = sqrt(2^2 + 0) = sqrt(4) = 2

So, the base of the triangle has length 2 and the two sides have length sqrt(2).

Drawing the altitude from vertex (a+1, a+1), the middle one, you see the altitude has length 1.

It's an isosceles triangle with sides sqrt(2), so the altitude separates it into two congruent right triangles with hypotenuse sqrt(2) and one leg length 1.

The other leg is the shared side, the altitude, H.   sqrt(2) = sqrt( 1^2 + H^2)

Squaring both sides, 2 = 1 + H^2

H^2 = 2 - 1 = 1, so H = 1 ---> altitude is 1.

AREA of triangle is 1/2 * (BASE * ALTITUDE) = 1/2 * (2 * 1) = 1

The area is 1.

Do you find such problems interesting?   Do you suspect the problem is intentionally confusing in that a beginner might doubt the area could be 1?

If you draw the diagram, you can intuitively see the two congruent parts of the triangle, the two right triangles, that is, could be visualized on top of each other forming a unit square, which is the way I think Schopenhauer would think of it.

Sorry you have been feeling ill.  You must be very tired.

I remember in "The H Files" I had read something into the recorder by Colin Wilson about that feeling of being so very tired you would prefer sleeping forever.

Do you take any kind of pain reliever?  Those "Contacts" can be helpful.  I know the medicine does not really treat the flu, just eases the discomfort of the symptoms.

Do you make chicken soup? 

I'm afraid this also only treats the symptoms.   Maybe only rest and sleep. 

Life sure isn't pleasant.  I imagine you are feeling miserable.  There is nothing to do but sleep as much as you can and keep hydrated.

The Great Tiredness is every bit as good as death.  There was no color here, no pain, no emotional weather at all, just an occasional oddness that was the outside world trying to puff itself up into significance when, of course, the secret of the Great Tiredness, the truth of this realm, was that everything was arbitrary and meaningless.  In the Great Tiredness, the transition from sleep to wakefulness was often blurred. – W.B. Spencer

Quote from: Colin Wilson
Our problems are fundamentally psychological. They spring from the fact that the complexity of our society tends to create a defensive attitude in many people, the sort of acknowledgement of defeat that a schoolboy might feel on looking into a volume of higher mathematics. The result is a sense of diffidence, a loss of the feeling of being self-determined. This diffidence gnaws into the nervous energies. It narrows the individual’s conception of his own abilities and values.

NOTE:
Quote from: I
I speak to you as an "older brother", not to discourage you from trying to find a career which involves mathematics, but to reassure you that your involvement with mathematics is not in any way dependent on commerce or earning a living.

While in living years I may represent a kind of "older brother" to you, culturally, as far as our roots go, India:Germany (via USA), you, Holden, also represent a kind of "older brother" to me.   :)

« Last Edit: August 29, 2017, 02:30:18 pm by { { } } »
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 ~

The Indignant One

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I found the following comment for Modern Algebra: A Logical Approach, Book One by Helen R. Pearson &Frank B. Allen (at Amazon) of interest:

I especially like the last paragraph about living 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 ...

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

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.

This may be why I feel like some kind of "mental patient".

Not only am I considered "disabled" (via a mood disorder), which means I would be seen as someone who may not work well with others, maybe not obey orders, report to work on time, etc.,

More than this though, the things I find to be so impressive, such as those old textbooks by Frank Allen or Mary Dolciani, this world ignores.   When I witness such things, I really feel in my heart, WHY BOTHER?

Why bother with anything?

I'll just continue smoking tobacco, drinking coffee, and studying, knowing full well that I am living the life of a second-class citizen, practically a ward of the State ...

It's so strange to consider oneself "crazy" or even having some kind of "mental disorder" when deep down inside I feel "intellectually superior" to many of those who serve society in some kind of administrative or managerial capacity.

I am simply resigned to being depressed and just have to get used to being generally disappointed with life.  Seeking a cure or a way out may only make us feel worse.

So, I am reading Frank Allen's book ... which was wrtten before I was born.

I was not the target audience, but happen to be one of the few who is interested.

It is what it is.
« Last Edit: December 18, 2017, 01:46:07 pm by Non Serviam »
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 ~

raul

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Hentrich,
Hentrich
I cannoy say anything about Frank Allen or Mary Dolciani. Beyond me, unfortunately. 

Good that you feel “intellectually superior to many of those who serve society in some kind of administrative or managerial capacity. I am not an intellectual. Not amateur even, too bad. In the years to come these who serve society in administrative or managerial capacity will find themselves jobless. They may not have the possibiblity to retire. What you reap, is what you sow.

As to the state, here, in Paraguay, this is a mafia state where even ruthless men like Al Capone, Frank Costello and Luciano would not feel at ease. These men had codes they followed to the letter. Unfortunately they gave birth,so to speak, to generation of hollow and cruel men and women.

I am a non citizen because I do not contribute in anything and do not serve society in any kind at all. Citizen is just another word for slave or prisoner. Were the Thracian Spartacus appear in this decade, he would find few eager to revolt. Vae victis

Stay well.


The Indignant One

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While I may not be an academic intellectual, since I find the culture here in the United States to be so extremely anti-intellectual, I go out of my way to refuse to suppress any intellectual or cerebral inclinations I might have.  In other words, I nurture my mental life in a spirit of rebellion against the state mafia and the vulgarity of the Mall Rat Consumerist Gort Culture.  I want to defy them all by finding some sanity in an old math textbook, the last place they would think to look for it.   Not in any God, not in any job.   

People are quick to give advise about "trusting God" and they love an opportunity to counsel someone to pray.

I am proud to be such a failure.  I am free to tinker about with math and computing accepting my role as a total reject.   I do not mean to sound arrogant when I point out my "intellectual superiority" to many of those who have some kind of position of authority or high social status or "job security".

No, in fact, by now I understand that showing any interest in such things such as mathematics, if it is not connected to advancing in some kind of career, is an invitation to be mocked.

The only reward I seek to compensate me for my efforts is to stumble upon many novel exercises along the way, exercises which may encourage me to enjoy my mental faculties and thereby transcend the anti-intellectual atmosphere of the culture at large.

It is good for us to prepare ourselves to be mocked.

Also, it is not a bad idea to view onself as ever so slightly mad, or leaning strongly in that direction.  When I allow myself to just focus in on my own little interests in mathematics, where I suddenly become enthusiastic about some math books from the 1960's and 1980's as if they were some kind of Holy Grail, it is during such moments, as long as the nerves in my decaying teeth are not throbbing in pain, that I am able to enjoy living in my own little world.

I may not believe in God or gods, but maybe my interaction with books is an interaction with ghosts.

Take care Raul.    I'm going to try to enjoy the depression coming over me.  Sometimes I fall into a state just beyond depression where I am actually quite content with what I have become.

I do not demand much from myself and I have no use for too much pride.

“Anti-intellectualism is virtually our civic religion. 'Critical thinking' may be a ubiquitous educational slogan -- a vaguely defined skill we hope our children pick up on the way to adulthood -- but the rewards for not using your intelligence are immediate and abundant.”   ~ A.O. Scott,
« Last Edit: December 19, 2017, 04:42:04 pm by Non Serviam »
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 ~

raul

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Hentrich,
Thank you for the response. In the 1980`s and the 1990´s I was always told that the US was the best place to study and get a university title.

Yes, it is much truth that "People are quick to give advise about "trusting God" and they love an opportunity to counsel someone to pray."

This reminds me of something I saw on TV. You see, a month ago a submarine from Argentina vanished in the Atlantic Ocean. Up to now there are many rumors that the submarine was sunk by the British. The thing is that one of the female relatives of the crew said on TV she was going to fast in order to ask God for the rescue of the 43 men and one woman.
Human beings are complex creatures.   

" I do not mean to sound arrogant when I point out my "intellectual superiority" to many of those who have some kind of position of authority or high social status or "job security"."

You are better human being that I thought.
But "many of those who have some kind of position of authority or high social status or "job security" are arrogant and they think they are the chosen ones. They forget that they are also cannon fodder. They are nemos. They are manipulated by the government or other forces.

I heard a politician from Argentina that 62 people make the money that four billion people get in the planet. 62 privileged trillionares or quatrillionaires,( no idea if this word is correct) almost own the resources of this planet. The pie is getting smaller.

"It is good for us to prepare ourselves to be mocked."
I have been mocked many times and I will still be. I would like to have been a Borneo head hunter. I would like to have my own gallery to see the reduced heads of all of those who mocked me. Ever heard of Enno Llolling? He was the SS medical officer in charge of the doctors in the concentration camps. This creature enjoyed the macabre. He ordered that all the human skins with tattooes be collected from the camps. Those prisoners with tattoes were injected with phenol in the heart, and their skins carefully removed and sent to the Kaiser Wilhelm Institute. Other samples were sent as handbags or cigarrette holders to his colleagues as gifts. He even gave instructions to the doctors in Buchenwald about how to reduce the heads.

"I may not believe in God or gods, but maybe my interaction with books is an interaction with ghosts."
Yes, God is the big guy, the fascist in heaven. You live in a very intriguing society where God is the main player. Charles Manson thought he was Jesus and Lucifer. I read that satanic cults use German shepherds for sacrifice in the US.
" I am being the the devil´s advocate" said Richard Nixon in the White House tapes. President Kennedy was allegedly killed by the Haitian Francois Papa Duvalier in November, the eleventh month. Duvalier was Baron Samedi in voudon, the Baron of the Cemetery, all dressed in black.

Take care and take the vitamins.