I was doing some research on Mary Dolciani and what was considered "New Math" in the 1960's and 1970's. Evidently, she died in 1985, the year I was a senior in high school where Modern Introductory Analysis was the text we used.
In 1987, the book called "Introductory Analysis" was published. She is listed as a co-author even though she had already deceased. Most of the material overlaps, but there is a some attention paid to Calculus.
I looked through some books on "Pure Mathematics" from Library Genesis, especially,
Understanding Pure Mathematics (from England?)
From what I can tell, it presents the material in a similar fashion but with far less formality (rigorous proofs).
So, I suspect that what was considered "New Math" was how the material is presented, using set theory notation, rigorous proofs, definitions, etc
Maybe when they "reformed" the curriculum, they included some applied mathematics.
All I know is that I am looking upon my current study as the manifestation of a desire to become more familiar with pure mathematics. For whatever reasons, I feel much more comfortable thinking of it as "pure mathematics" than as "pre-calculus". I don't need to bone up on pre-calculus as I have a firm grasp on the algebra, geometry, and trigonometry used in "the Calculus".
My entire reason for going through "Modern Introductory Analysis" [Mary P. Dolciani, Edwin F. Beckenbach, Alfred J. Donnelly, Ray C. Jurgensen, William Wooton] as well as "Introductory Analysis" [Mary P. Dolciani, David Myers, Robert Sorgenfrey, John Graham] is to, as I already stated, become more comfortable using formal notation, and to finally, over 30 years later, learn how to construct such proofs. This is where I am still a little baffled.
Why do you think it makes such a difference to me what I call it (referring to this as pure mathematics as opposed to "precalculus")? I think it is because the material is presented in such a different way than in texts that deal more with computational, mechanical drills. It has to do with the presentation being formal as opposed to just mechanical computational drills.
The exercises actually force me to think, and I realize this is the kind of mathematics that I find so challenging, as opposed to the calculation-obsessed drills of differentiation and integration, which I have certainly had my fill of.
One might wonder why I can't just study everything at once. Why do I feel I have to stop dead in my tracks, putting Multivariable Calculus, Vector Calculuc, and even Physics on the shelf?
This is just the way I am wired. As this exploration will take 10 to 15 years, I figure it is best to rebuild my foundations, retrain my brain to be more comfortable with the more "pure" approach, before continuing with the "applied mathematics" one finds in Physics, Differential Equations, Computational Physics, etc ...
I really sympathize with the youth who aspire to become "mathematicians". Do they yet realize how life can get in the way and totally mock their plans?
My main question is: why do you think it matters to me what I call it? I suspect that I would not be able to justify to myself studying "precalculus" as it might indicate something remedial - and it is not remedial, it is challenging to me to approach the material in such a formal manner.
Maybe clarifying that I am trying to better understand "pure mathematics" is a way to make it clear once and for all that I have no intentions of putting anything I learn to practical use. Besides that, since the Physics and other "applied mathematics" texts are so intimadatingly fat, I figure I have the rest of my life to get into them, and I don't want to drown in that swamp of books until I face my uncertainties concerning pure mathematics.
As long as I am here on this earth for no particular reason, I think I owe it to myself to spend as much time as possible concentrating on the aspects of mathematical training that are mysteriously absent in university education.
Even in courses such as Mathematical Reasoning, where the focus is on writing proofs, the foundations of the applied mathematics courses are never really explored.
In this world which rushes the youth from grade school to high school full steam ahead toward Calculus and Physics, there is not much "basic training" for those who need to be spoon fed "pure mathematics".
The reason I chose to major in Computer Science was because I was getting a grant from the goverment, and it was taken for granted that I would find some kind of job/career afterwards.
So I focused on applied mathematics.
Now I have the benefit of living without any hope of ever being employed as anything other than a janitor, so I am free of the delusions that I study so as to find a place in the so-called "modern day workforce". I simply don't have the necessary servile temperament to be some kind of "professional scientist".
My failure to find any kind of vocation has liberated me to do nothing with my life.
That is why I have this opportunity to return to the text that was presented to us (at age 17) as "AP Calculus", and to approach it with my own personal intitiation into "Pure Mathematics".
There seems to be a psychological aspect to the mind set of the student. If one approaches the text as an invitation to Pure Mathematics, then there is the potential for a quasi-religious encounter; but if one sees it as "precalculus" or a stepping stone to a career based in applied mathematics and science, well, it somehow gets contaminated by the student himself/herself with questions such as "what use is any of this?"
It doesn't have to have any use.
Student: "When am I going to USE any of this?"
Teacher: "When you find yourself on welfare living off foodstamps, if you choose to continue to explore these areas of knowledge, you can lock yourself in a fortress of solitude and resist the conspiracy against you, the conspiracy that wants to see you crawling on the floor looking for a speck of crack co-caine."
"Should you find yourself in a similar predicament as H.P. Lovecraft and countless others like him, and if you have no particular talent for writing stories, you can spend many years apparently doing nothing whatsoever while developing what we call mathematical maturity."
Student: "But, if I may be so bold to ask, oh wise teacher, what is the use of mathematical maturity? Will it put food on the table? Will it pay the landlord?"
Teacher: "Well, probably not; but you will develop an inner life of the mind in the process, and this might afford you enough detachment from practical concerns that you will be satisfied living on bean soup and sleeping on the floor."
"How's that for motivational encouragement?"