Critical thinking in Geometry

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 (1930)

"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:

reflection,
observation,
experience,
reasoning,
and/or reading,
writing,
speaking,
and listening.

Critical thinking has its basis in intellectual criteria that go beyond subject-matter divisions and which inclued:

Clarity,
accuracy,
precision,
relevance,
depth,
breadth,
[color=gold]logic,
significance,
fairness.

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.








[atsimg]http://files.abovetopsecret.com/images/member/2a41e0873fd2.gif[/atsimg]

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 was done.

In addition a student had to take a test just to get into this prep school. The things tested are...

Orthography (that is, spelling)
Reading, Writing,
Mental Arithmetic, Written Arithmetic,
Descriptive Geography (including Map Drawing),
Physical Geography,
History of the United States,
English Grammar (including Syntactical Analysis),
Physiology,
[color=gold]Book-Keeping,
Vocal Music,
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; General History.

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'.

-Geometry Civilized

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?


David Grouchy


Links of interest:

www.noindoctrination.org (This site has been removed from the net)
www.criticalthinking.org
www.badscience.net
www.sciencedaily.com

There is a lot in that post OP.

I agree education in this country has lost its way. I doubt what we spend billions of dollars on as a country prepares our kids for what they will face.

Perhaps it is my poor education but I am not really sure where you are going with the whole thing. Maybe it is pure comparative analysis and I am looking for a solution where none was presented.

This illustrates the geometry between the rich and the poor possibly far better than a statistical approach ever will! Instead of bickering over inadequacies, or making fine-sounding arguments for a blind equality, those successful have had the opportunity and dedication to take the hard steps required to learn their way to a practical evolution of thought. That doesn't simply equate to money, although it could. A classical education wrought a well-rounded character who could succeed at any field they attempted thereafter, and bred men and women who were not prone to fault and exploitation. I've always considered the difference between colleges and universities to be that of feet-running compared to a strong theoretical background.

University is a means to an end, but it takes such self-discipline to undergo the same training at a self-imposed basis, and it usually becomes a much more leisurely process. Part of the expected learning trial is to stamp everything with time limits, and measured burdens of stress. Today's society largely breeds specialists instead, trumps people in their field, and lets them fill in the missing blanks later.

Great topic, and very inspiring!

are you talking about deductive reasoning (base for geometry) or education? or is this thread about morals?

It's amazing that with such great schooling in critical thinking and reading Latin that in the 1890's over 13% of the population were illiterate, and now with the terrible modern education system the rate is way below 1%

i take it mr David is just training to be a writer. geometry is not removed from school curricula.
Euclidean geometry is still taught in schools for purpose of developing mathematical reasoning skills, but is not significant for science and engineering anymore. there`s much more powerful and precise tool called calculus.

besides the OP lacks some logical glue between paragraphs. have you taken geometry course before?

Originally posted by ABNARTY
I am not really sure where you are going with the whole thing. Maybe it is pure comparative analysis and I am looking for a solution where none was presented.

I only have one point to make.
It is the first sentence in the first quote.
The rest is just supporting evidence that this used to be true.

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.

That is to say that geometry was not taught just so people could bisect a figure, or calculate the length of one side of a triangle, but geometry was taught as the foundation of critical thinking. That quite literally thought is best understood and analyzed when it has a definet shape. This may sound quite alien in this day and age. But consider the legal profession as shown on TV.

Even in the media we still hear the phrases "to the point," "line of reasoning," and "Don't be obtuse." all of which come from geometry.

Originally posted by Northwarden
Today's society largely breeds specialists instead

Boy isn't that the truth.
The renaissance ideal, was an individual who was accomplished in many fields of science and art.
Many of the signers of The Unanimous Delclaration of Independance were of this type.

The Media would have us believe that there is just too much to know.

But that's not the whole picture is it. It is easier to get information now. Just at the point in our history where everyone could be of the renaissance ideal, all of a sudden everyone has an emotional commitment to being a specialist. Myself included. I want to be good at something. Like everyone else I worry about only being medeocre at several things, and gifted at none. We get this message a lot, subtly and with repetition. I have to keep reminding myself that not only is it false, but it was never true.

Originally posted by delicatessen
are you talking about deductive reasoning (base for geometry) or education? or is this thread about morals?

I'm speaking to the subject of education in this way. What permission do we give ourselves, in learning. Do we give ourselves permission to learn everything? Why not. If no one is putting this expectation on us from the outside, are we being fair with ourselves to say "I'm entitled to know everything." I hazard a yes.

Originally posted by davespanners
It's amazing that with such great schooling in critical thinking and reading Latin that in the 1890's over 13% of the population were illiterate, and now with the terrible modern education system the rate is way below 1%

DaveSpanners,

I don't know what to add to that statment. I wish I had written it myself.

After a hard freeze one winter an old Polymer Physicist friend of mine was back in town for Christmas to visit his family. I was underneath a raised Louisian house fixing a pipe that had ruptured. I learned how to fix pipse in damage control classes onboard ship in the Navy. Having effected the patch I crawled out, covered in mud and carrying tools. There he was, large as life, looking at me.

Feeling emberassed I said "I was fixing a pipe that ruptured from the freeze."

He looked wistfull and said "I wish I knew how to fix pipes."

And we went inside to talk physics.

Originally posted by davidgrouchy

That is to say that geometry was not taught just so people could bisect a figure, or calculate the length of one side of a triangle, but geometry was taught as the foundation of critical thinking. That quite literally thought is best understood and analyzed when it has a definet shape. This may sound quite alien in this day and age. But consider the legal profession as shown on TV.

Even in the media we still hear the phrases "to the point," "line of reasoning," and "Don't be obtuse." all of which come from geometry.

excuse me, but how does it "sound alien in this day and age"? last i checked Euclidean geometry or the purpose it`s taught didnt change. you cant just get by without deductive reasoning in calculus. besides calculus is much trickier than straightforward simple logic of Euclidean geometry. let alone calculus is much younger than classical geometry if this thread`s goal is just to be a cheap shot at today`s society and it`s alleged low standards.

reply to post by davidgrouchy


Ah true, it's not simply the feeling of mediocrity, but the potential envy of the specialization, and the pride of intellectual achievement that come into play. As with my last line though, specialists usually are "trumped" by the system, no matter how far they aspire

I'm thinking about a passage as one that provided good direction for me years ago, the essay in whole, and this passage more specifically. I could put it into personal context for conformity to a hope in originality, but I usually prefer to pull up the original and give due credit.

Self-Reliance
The Complete Works of RWE

"Ne te quaesiveris extra."

If our young men miscarry in their first enterprises, they lose all heart. If the young merchant fails, men say he is _ruined_. If the finest genius studies at one of our colleges, and is not installed in an office within one year afterwards in the cities or suburbs of Boston or New York, it seems to his friends and to himself that he is right in being disheartened, and in complaining the rest of his life. A sturdy lad from New Hampshire or Vermont, who in turn tries all the professions, who teams it, farms it, peddles, keeps a school, preaches, edits a newspaper, goes to Congress, buys a township, and so forth, in successive years, and always, like a cat, falls on his feet, is worth a hundred of these city dolls. He walks abreast with his days, and feels no shame in not `studying a profession,' for he does not postpone his life, but lives already. He has not one chance, but a hundred chances. Let a Stoic open the resources of man, and tell men they are not leaning willows, but can and must detach themselves; that with the exercise of self-trust, new powers shall appear; that a man is the word made flesh, born to shed healing to the nations, that he should be ashamed of our compassion, and that the moment he acts from himself, tossing the laws, the books, idolatries, and customs out of the window, we pity him no more, but thank and revere him, -- and that teacher shall restore the life of man to splendor, and make his name dear to all history.

www.rwe.org...

Originally posted by delicatessen
i take it mr David is just training to be a writer. geometry is not removed from school curricula.
Euclidean geometry is still taught in schools for purpose of developing mathematical reasoning skills, but is not significant for science and engineering anymore. there`s much more powerful and precise tool called calculus.

besides the OP lacks some logical glue between paragraphs. have you taken geometry course before?

I accept everything said as a valid criticism.
Yes, I could have, and should have glued the post together better.
Calculus is not only more powerful and precise, it is easier than other maths.
All this I learned in High School with my physics nerd friends, and I should have related some personal stories to this end.


Found guilty but still trying to plead my cause I can only say that; while geometry is still taught in school. It is not taught as the foundation of critical thinking. The kind used in a court of law. The kind used to define what is, and is not, evidence.

What I have so spectacularly failed to convey is that geometry, once apon a time, was a tool to teach critical thinking. The kind used in Law, buisness, contracts, government, and to make people immune to being manipulated by hearsay or excessive emotionality.


I think the Media has done a lot help make sure that geometry is seen as obsolete, and doing what feels good as ultimate justice.


In way of supporting evidence of what Geometry used to be I present the five common notions. Not for you Delicatessen, as I'm confident you have more than a passing knowledge of them, but for the benefit of anyone reading who may be thinking, "what the H are they talking about."

Common notion 1.
Things which equal the same thing also equal one another.

Common notion 2.
If equals are added to equals, then the wholes are equal.

Common notion 3.
If equals are subtracted from equals, then the remainders are equal.

Common notion 4.
Things which coincide with one another equal one another.

Common notion 5.
The whole is greater than the part.

Originally posted by davidgrouchy

Originally posted by delicatessen
are you talking about deductive reasoning (base for geometry) or education? or is this thread about morals?

I'm speaking to the subject of education in this way. What permission do we give ourselves, in learning. To we give ourselves permission to learn everything? Why not. If no one is putting this expectation on us from the outside, are we being fair with ourselves to say "I'm entitled to know everything." I hazard a yes.

man, are you smoking something? so much for the thread with the word geometry in the title. this rhetoric probably may impress others, but it would be helpful if you stayed logical. what does all of this have to do with geometry?

Originally posted by delicatessen

Originally posted by davidgrouchy

That is to say that geometry was not taught just so people could bisect a figure, or calculate the length of one side of a triangle, but geometry was taught as the foundation of critical thinking. That quite literally thought is best understood and analyzed when it has a definet shape. This may sound quite alien in this day and age. But consider the legal profession as shown on TV.

Even in the media we still hear the phrases "to the point," "line of reasoning," and "Don't be obtuse." all of which come from geometry.

excuse me, but how does it "sound alien in this day and age"? last i checked Euclidean geometry or the purpose it`s taught didnt change. you cant just get by without deductive reasoning in calculus. besides calculus is much trickier than straightforward simple logic of Euclidean geometry. let alone calculus is much younger than classical geometry if this thread`s goal is just to be a cheap shot at today`s society and it`s alleged low standards.

Owch.
I may be guilty of preaching to the choir in your case.
I did not, and do not, mean to be be taking cheap shots at today's society.

I only said that calculus is easier,
to encourage anyone who is close enough,
but thinking about stopping their education prior to calculus.

Originally posted by delicatessen
man, are you smoking something? so much for the thread with the word geometry in the title. this rhetoric probably may impress others, but it would be helpful if you stayed logical. what does all of this have to do with geometry?

Ok,
then don't be impressed.

As to the last sentence quoted above I respond with the first sentence quoted in the OP.

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.

All I can do is emphasise the word pleasure.

Originally posted by davidgrouchy

I accept everything said as a valid criticism.
Yes, I could have, and should have glued the post together better.
Calculus is not only more powerful and precise, it is easier than other maths.
All this I learned in High School with my physics nerd friends, and I should have related some personal stories to this end.


Found guilty but still trying to plead my cause I can only say that; while geometry is still taught in school. It is not taught as the foundation of critical thinking. The kind used in a court of law. The kind used to define what is, and is not, evidence.

What I have so spectacularly failed to convey is that geometry, once apon a time, was a tool to teach critical thinking. The kind used in Law, buisness, contracts, government, and to make people immune to being manipulated by hearsay or excessive emotionality.


I think the Media has done a lot help make sure that geometry is seen as obsolete, and doing what feels good as ultimate justice.


In way of supporting evidence of what Geometry used to be I present the five common notions. Not for you Delicatessen, as I'm confident you have more than a passing knowledge of them, but for the benefit of anyone reading who may be thinking, "what the H are they talking about."

Common notion 1.
Things which equal the same thing also equal one another.

Common notion 2.
If equals are added to equals, then the wholes are equal.

Common notion 3.
If equals are subtracted from equals, then the remainders are equal.

Common notion 4.
Things which coincide with one another equal one another.

Common notion 5.
The whole is greater than the part.

sorry for coming off like an ahole, but seriously man.

classical geometry is still well and alive just because it teaches deductive reasoning. and it`s not a today`s conspiracy that students dont perceive the subject as such. for instance, for many students physics is not a study of nature and the ways it works, but a mere set of rules meant to be memorized and calculations required by the course. the same goes for geometry. and it`s not exclusive to students of today.

I guess I'm basing it on my personal experience.

One day in high school three math classes were hearded into a sigle classroom.
The teacher had an overhead projector, and proceeded to show us the proofs and theorems.
We copied down the answers, and the test was the next day.

I asked around asking what had happened.
This is what I was able to peice together.

Apparently the other two geometry math teachers had refused to teach. The order had come down from on high that geometry was not to be taught as the foundation of either deductive reasoning, or logic, any more. Theorems and proofs only. The teachers were outraged, but were advised that they would be fired if they dissobeyed. So that year they obeyed by omission.

Logic is now considered a seperate subject.
No longer do public school students learn how to construct the entire mountain of logic from the 5 common notions listed above.



Later I began to notice an increase in the use of fuzzy logic in advertising.
I do believe there was a conspiracy.

reply to post by davidgrouchy


you cant teach or learn Euclidean geometry without logical manipulations of "givens". i dont think US department of Education approved/approves 'learning" geometry by just memorizing bunch of postulates. that would be/is outrageous and sounds fantastical. i am feeling lazy to go dig up info on methods of geometry teaching in US or elsewhere, but feel positive it isnt the case. 10 years ago in a post Soviet high school that i graduated geometry was taught the traditional way. everything seems to be the same since, on that side of the world. since there is such a thing as "World Championship Mathematics Competition for High School students" there has to be a standard. so it`s logical to assume math and all its forms are taught the same everywhere.

Originally posted by delicatessen
10 years ago in a post Soviet high school that i graduated geometry was taught the traditional way.

Interesting.

My experience was in Louisiana in the early 80's.

Something else shocking I have learned since then about public education of math in Louisiana. In long division students are no longer taught to carry-the-remainder. They use something called clusters. For instance.

80 divided by 10
the correct answers are 7, 8, or 9.
As long as the student understands the general idea.
For precision they use a calculator.

I discovered this while helping a friend of mines' daughter do her math homework. When I tried to show her how to carry the remainder, she said "I'll get in trouble if I do it that way." I did some checking around and found out that she was for real about this. Clustering, not carrying the remainder.

A year later I was having lunch with her grandfather and I told him about it. He was angry at me. And I recieved quite a lecture.

A month later He took me to lunch and appologized. He had checked into it. It was true.

They don't even know how to divide around here any more. I guess that's not fair to say really. They know how to load a division problem into a calculator. It would be more accurate to say, the students these days don't know how to do math accurately enough to determine if the calculator is giving the correct answers or not.


David Grouchy