The following three quotes are from ''The nature of Proof'', published in 1938 -Fawcett. I show these to allow people to see how the arguments were
couched at the time. That and to provide evidence that critical thinking used to be understood as almost inseprable from geometry. Instead of
belaboring the point, I'll let these quotes speak for themselves.
"The purpose of geometry is to make clear to students the meaning of demonstration, the meaning of mathematical precision and the pleasure of
discovering absolute truth. If demonstrative geometry is not taught to enable a pupil to have the satisfaction of proving something, ..., then it is
not worth teaching at all." W.D. Reeve (1930)
"Geometry achieves it highest possibilities if, in addition to direct and practical usefulness, it can establish a pattern of reasoning; if it
can develop the power to think clearly in geometric situations, and to use the same discrimination in non-geometric situations." H. C. Christofferson
"I firmly believe that the reason we teach demonstrative geometry in our high schools today is to give pupils certain ideas about the nature of
proof. The great majority of teachers of geometry hold this same point of view. 2 Our great aim in the tenth year is to teach the nature of deductive
proof and to furnish pupils with a model of all their life thinking." C. B. Upton, (1930)
Once trained in the fundamentals of geometry, and how it can be used to show absolute proof (the square fits into the squre hole, or it does not),
only then can ...
Critical thinkers can gather such information from:
Critical thinking has its basis in intellectual criteria that go beyond subject-matter divisions and which inclued:
Notice that logic is merely one category within a sub division, of applied critical thinking. It is not the foundation, nor is it the capstone of
critical thinking. Merely another brick. The first sentence in this paragraph cannot, I think, be repeated often enough. Logic is merely one
category within a sub division, of applied critical thinking.
Let's take a look at Jr High, and High school back in the old days. Here is some information on Indiana that is relevant. Let's take it as an
example of what the whole country used to be like. First of all there is a strict moral code.
The Teachers will endeavor to combine mildness with firmness, kindness with justice. Strict regard will also be had to the language, the
manners, the whole deportment of the pupils, in the school room and out of it, so as to secure, if possible, not only that they may become good
scholars, but also good men and women. Students who will not refrain from drunkenness and other vicious indulgences, will be promptly dismissed, as
dangerous to the morals of the school, and unworthy of a place in it or any other.
At the Indiana Seminary and Normal School a classical education was taught. The important thing to note here is that all of this just to get a
student ready for college. Back in those days these were considered mere prepratory courses. To get one ready for college, where the advanced work
In addition a student had to take a test just to get into this prep school. The things tested are...
Orthography (that is, spelling)
Mental Arithmetic, Written Arithmetic,
Descriptive Geography (including Map Drawing),
History of the United States,
English Grammar (including Syntactical Analysis),
also, for those who intend to pursue a classical course, Latin Grammar and Latin Reader, or French Grammar. Remember, this was prerequisite to
entering the Graduating Course.
So it was assumed that if a student paid attention in Jr High they would pass the above tests, and now they are ready to enter High School. I don't
know about you, but the above reads like advanced college course work these days.
In the following list, remember that the non math books would have been studied in their original languages. Greek, latin, or what-have-you. German
in the case of Henraid and Corinne.
Here is the High School course work.
Junior Class, First Term:
Caesar's Commentaries or Fasquelle's French Reader; Greek Grammar and Reader or German Grammar; Ray's Algebra, Part
1st; English Literature.
Junior Class, Second Term:
Cicero's Orations or Telemaque; Greek Reader finished and Anabasis commenced (Xenophon’s account of the Younger
Cyrus’ expedition into central Asia) or Woodbury's German Reader; Ray's Algebra, 2d Part, commenced; [color=gold]Geometry, Five Books.
I just gotta point out my personal favorite up there. Geometry, Five Books. Yeah... That's the missing key stone. You know. The central stone in
the top of an arch that keeps it from falling in on itself, and gives the whole thing strength.
Middle Class, First Term:
Virgil's Aeneid or Henriade and Corinne ; Anabasis, finished or Schiller's Wilhelm Tell; Ray's Algebra, Part 2d,
finished; Geometry, finished.
Middle Class, Second Term:
Odes of Horace or Racine; Homer's Iliad or Marie Stewart and Klopstock; Chemistry; Trigonometry and
[color=gold]Surveying or Botany and Music or Drawing.
Senior Class, First Term:
Mental Science (roughly speaking, psychology); Conic Sections and Analytical Geometry or Music or Painting; Rhetoric;
Senior Class, Second Term:
Moral Science (generally this means philosophy); Natural Philosophy (this means science, just like Newtons
"Philosophiae Naturalis Principia Mathematica"); Astronomy; Geology.
Exercises in English Composition will be continued throughout both Courses. Classical students will somewhere in the Course study Latin Prose
Composition and the rules of Hexameter verse; and all will be required to write frequently exercises in Latin or Greek, or in French and German.
Students may study [color=gold]Differential and Integral Calculus, and thus complete their mathematical studies, if they have time and
inclination to do so.
Daily Lessons are given in the elements of Vocal Music, with appropriate practice. For these lessons no extra charge will at present be made. Besides
this there is a daily exercise in Singing, engaged in by the whole school.
The School is provided with a fine rosewood Steinway PIANO and an excellent MELODEON. Those wishing to become correct and tasteful performers on these
instruments will here find excellent opportunities. The Music Classes are open to all, whether they attend the other classes or not.
This is the course of study for all students. The mathematics requirement consists of three semesters of Algebra, two semesters of Geometry, one
semester of Trigonometry and Surveying, and one semester of Conic Sections and Analytical Geometry. Calculus was not required, but could be taken by
those students with the "time and inclination to do so."
All of the above gets one a diploma. And now they are qualified to go to college to get an actual Degree. This is light years away from what a
degree means these days.
So how was the process of teaching critical thinking through the study of Geometry removed from planet earth? Well... it wasn't that hard actually.
There was a long tradition of critizing geometry that went all the way back to the greeks themselves.
The followers of the Greek pilosopher Epicurus, who esteemed feeling over reasoning, had no patience for the arguments of Euclid. His science
is ridiculous, they said, pointing to a proposition half way through the first book of the Elements, in which Euclid labours to show that no side of a
triangle can be longer than the sum of the other two sides.
'It is evident even to an jackass.' For a hungry jackass, standing at A (Fig. 1.1.2) will go directly to a bale of hay at B. without passing through
any point C outside the straight line AB. The beast's geometrical intuition tells him that AB must be shorter than AC + CB.
The charge that Euclid stops to prove propositions evident even to an ass has ecoed thorugh the ages.
One of the echoers was the seventeenth-century philosopher and mathematician Blaise Pascal, who accused his fellow geometers of six perennial faults,
among them 'proving things that have no need of proof'. He took as his example Euclid's compusion to demonstrate that two sides of a triangle taken
together exceed the third. To this objection the Greek philosophical geometer Proclus, who wrote a lengthy commentary on the Elements early in the
fifth century, had replied that a proposition evident to the senses 'is still not clear for scientific thought'.
So, um, yeah. Standards have fallen.
I don't even want to show you what the entrance requirements
used to be for college back in the old days. The info I found in a Carnegie Mellon
reasearch book, blew my hat off and made me so sad I wanted to weap. Is America a lost civilization?
Links of interest:
www.noindoctrination.org (This site has been removed from the net)
edit on 13-11-2010 by davidgrouchy because: (no reason given)