Everybody's mountain; 13; The Final One-Hour Broadcast

Everybody's mountain a program in the recorded series written and produced by Robert Louis Shea on with the author as narrator. I was a citizen taxpayer on a mission behind the blackboard curtain of contemporary American education. I traveled throughout the United States for six months. I saw schools universities and educational experiments from Boston to Chicago from San Francisco to Miami. I began my journey in the valleys of generalization abstraction and controversy in education. I ended it on mountaintops of educational leadership and imagination. This broadcast is a final view of the whole range of mountain tops which have been reported individually in the series. It is a presentation in accepts from previous broadcasts of some of the most significant insights that I found into the needs and opportunities of 20th century American education. These insights are in the areas of mathematics the social studies special education and the career of teaching. No broadcast no

series of broadcasts could hope to encompass the breadth depth or immaterial subtleties of that boundless realm we seek to describe when we say the word education. To the continuing process of self education which is true education. These echoes from everybody's mountain are dedicated. The coldest room in education's house is commonly considered to be mathematics. One of the most active furnace is working to warm it is the University of Illinois Mathematics program in our Bana Illinois. The project breaks down the traditional patterns separating Algebra Geometry Trigonometry and calculus in the high schools and teaches the subject in a half dozen basic threads to preserve the essential unity of mathematics. Professor Max b been and is the director of the project. What about the public profess to be them and the great mass of adults for whom mathematics has always meant sudden intellectual death.

We need to have a new attitude on the part of the lay population that it is don't longer respectable to admit to illiteracy in mathematics. The time has passed when a person can say with a certain amount of pride that he can no longer add three and five. I don't think this is a good carry over for students to recognize that parents are proud of this fact. Generally speaking how shall laymen think about mathematics in the new age. They have to regard it as a subject which is constantly growing. They will find the children making discoveries in mathematics classes their children will be excited about mathematics and if they in some way can sympathize with this new wave of excitement and thereby lend support to the efforts of their children we should find a new kind of attitude among all a people taught mathematics in Palo Alto California at the Stanford elementary school. Two professors of Stanford University Newton Hallie mathematics and Patrick soup in his philosophy collaborated in an experiment teaching geometry to children in the first grade. Why geometry Dr. Holly.

There is in geometry an intrinsic beauty and intrinsic intellectual interest a deep satisfaction from having met unusual problems understanding what the significance of the problem is and finding a solution to this problem. It is certainly a creative effort in the deepest sense of the word. It seems to me that there is not sufficient awareness on the part of even the educated public that the creative aspects of science are genuinely in his deeply creative and those aspects of the art it is recognised by almost everyone and it's a great experience for an elementary school youngster to be given freedom to to paint to or to create in an artistic way. And I think that we should move toward a future in which the same thing can be said about his scientific education. The Holy See to peace premise in teaching Euclid to first graders was that children have an unsuspected maturity in many subjects. Mathematics was merely their own specialty. The children worked with thumbscrew compasses and straight edges. They learned to recognize

mathematical terms make elementary definitions and constructions. Brian Jones for example with Professor Hawley at the blackboard constructs an equal right triangle. First you had a straight line with a pencil and a straight it. And then you would take your compass. And measure that line. You mean I would put the point of the compass on one end of the line segment and the pencil of the compass on the other. Yes and then you would take take it up to the top there and swing a little arc drawn an arc on the board. Then you take your compass in put the point on the other side and set still the same length. Now bring it up in another. Are I now have the two arcs. And then you take

you straight in your pencil and you draw a line from one in the line to the two arcs where they meet. Top this point of intersection. Then you take us Steve over to the other side and place it on the other end line and draw a line to where the two arcs owners step to each other. Is this an equal lateral triangle. Yes it is because all. Three sides are the same professes wholly ensue peace talk that geometry to Mrs. Wilson's first graders for three months. Professor Sue PS What did you and Holly find what we found was that with a few emendation we could follow the classical program of the first book of Euclid's Elements. We did not ask the students to give formal proofs. We concentrated on what are known and you could as the constructions

we were after their capacity to reason and to think intuitively by geometrical problems. I would say there were three students of the twenty eight in the class who essentially got very little from the geometry. I think that Mrs. Wilson would confirm that these same three students got extremely little from the reading program. The average child will do average in geometry the way he does average so to speak in reading by the Hermes UPC experiment suggests that more rigorous mathematics could be taught in the elementary grades. It remains to be proved to child development experts that it should be taught. You know but I want to Sumption made by Professor Bieber man of the University of Illinois School Mathematics project. Is that a high school freshman will revel in mathematics. If he is permitted to play with it as an abstraction. Another assumption is that mathematics can be fun. This is a game called Guess the number I think about a certain number. And I tell you something about it and then you're supposed to tell me what number I think you already think of a number.

I am positive tend to work and I come out with zero very negative can negative to think of a number I multiply by itself and then I take that answer and multiply that answer by itself and I come out with he 1 What number was I think 3 body needed you need 3 are negative 3 Make him I would you believe me. Need if you can write negative 3. OK and the other number I multiply by itself. And that answer is negative 36. You know I wasn't thinking right. The maturity developed at our back in mathematics can be awesome. Let's turn now to a class of ninth graders high school freshman dealing with the notion of cartesian products a fairly contemporary development in mathematics which requires a good deal of sophistication in handling abstract notions.

OK let's say we start out with the set A and that's definitely the set B. And let's say there's an element in B which is not in a. OK now show me why the cartesian product of A on B is different from the cartesian product of B and hey Elizabeth the elements in B but it's not. That means that element will be the first component of an ordered pair. And if it's not in a why then a won't have it to be a first component therefore the two sets of ordered pairs will be different. OK. If you got change but are not quite House explain it. How would you explain if you got another element in set B. It won't be used. Component in the ordered pairs it won't be used as a first component so that you couldn't use it in set A and therefore you can't change it just because it wouldn't be the same ordered pair.

Let's see if suppose I have this problem and say let's give a name. What's a what's a nice name. 7 7 I have 7 which is in B but not in in a way that means in a cross B I'm going to have an ordered pair with a second component is 7 7 can be cross they are going to have an ordered pair whose first call on it is 7 7. But you wouldn't have A's first components if you wouldn't have wanted a crosspiece because 7 isn't an 8. OK anybody else like to talk about why Cartesian multiplication is not commuted to. Anyone. Run of Professor bee Bowman's associates on the school mathematics project is Professor haven't even won a magician. A philosopher of mathematics. He has taught math to engineering students at the University of Illinois for 20 years. The initial impetus for the project was a discovery that students coming to the university from the state's high schools while poorly prepared in mathematics. Professor

one do you expect that the graduates of this project will be better prepared when they reach you in college. I'm quite sure that this will be the case I expect them to be able to apply mathematics in situations in which the conventionally trained student merely throws up his hands in despair. This will help with the national problem of turning out say 50000 mathematicians per year for our highly complicated science and technological demands. Yes it certainly will do that but I think a much more important result will be that more people understand mathematics as a living subject. This I think is very important since our whole civilization is built on mathematics. I think it's also important because the more our kinds of aesthetic experience that a person is able to perceive the better the life of mathematics as a creative aesthetic experience. Will this also help to make children better citizens of a democracy. MS. nineteen clock head of the Department of Social Studies in Evanston Township High School

Illinois says no there is no automatic transfer in the mind of the average student and what he learns about the process of solving a theorem in geometry and the process through which he goes in a social studies classroom when he is asked to form a hypothesis and bounce an event that you may be discussing. These things are skills are techniques they need to be taught just as skills and techniques do in a laboratory are in a home economics room. The principles and skills of formal logic are taught in Evanston high. Evanston nutria Niles townships participated in the Illinois curriculum project for the improvement of critical thinking. The project directors were Bea Arthur Anil Smith and Kenneth B Henderson professor of education at the University of Illinois. PROFESSOR SMITH. What led you to seek a stronger emphasis on critical thinking in the public schools.

Well as we visited the public schools the high schools the elementary schools lesson to college classes broadcast over radio. Became obvious that there was loose thinking permitted on the part of the students far beyond what would be acceptable if the teaching was up to a higher standard of rigor as it ought to be in Lexington Massachusetts at the Franklin Elementary School. I witnessed a demonstration lesson in the social studies which stressed the problem solving rather than the formal logic approach to the teaching of critical thinking. They have purple of the staff of Harvard Graduate School of Education worked with a combined class of fifth and sixth graders. Who discovered America. He wrote on the blackboard. Columbus the children answered. Purple then asked them to suggest other words for who discovered an America. Many variants were offered and the Harvard experimenter wrote them on the blackboard. Now who knows all about Columbus who can tell me what Columbus did.

Peter do you know the OK to improve the world was round. He said he couldn't go around the world will end up in Spain but instead he bumped into an America into America. And when we landed we know Tanya. I don't think so Nancy. Not exactly from a truck I was right at Columbus I was at the pilgrims. OK how do you get mixed up about the Chino. And I want you to call this what I WANT WHEN I WANT. Richmond. San Salvador San Salvador. How many never hear the word San Salvador before we're here before us in a book. What the books say about it. You know who knows about San Salvador. Donald.

An island off the coast of North America. Right. The Columbus go to any other island around here they may know Steve. He might have gone to Florida but you know we do not know. Pat do you want on me. I mean maybe maybe you went to the moon but what do you know about the coast of Maine how many think you have the coast of me. Karen but I think you and I on the cusp of Boston clinical to Boston. Well Richmond I know that on those three trips to North America the landed somewhere around the Caribbean island all the time around the tip of Florida. OK well let's look at these things we have here. De Papel turn to the blackboard and pointed to the words which the fifth and sixth graders had suggested the synonyms for who discovered America. He was about to confront them with the vagueness ambiguity an overgeneralization and their thinking was Columbus the first explorer to bump

into us a path. No I don't think Nazi was he the first man who found us say Alan. No he wasn't. Was he the first person who saw North America. James. Oh. Was he the first human to become a poet. All region. DAVID No no. Well what Columbus Do you understand I thought you said these words were the same. What did he do. Nancy. He proved that the world was right. Oh let's forget about that. Let's just let's just talk about this business about discovering America. They discover America.

Well Eric the Red did but I think people forgot about him and then made that Columbus discovered it is something like that almost in this government. Now he didn't Charles he was the first civilized human that was on it because there were Indians at first and then you are not civilized. They weren't so the commies discover America. In a way did in what way. Well the Indians. Are. Well they were civilized. Well no one can. Columbus came. Well you know Maureen I think Columbus did discover was he the first person who saw United States is that true. We said before he went to the Caribbean islands but did he go to United States even if he had gone out of state who was here before him. And India so they must have seen it before Columbus. So I was glad was the first person of the United States you know. Now did Columbus discovered America didn't need. James would you say.

Yeah he didn't he just discovered islands off the coast just discovered islands of course the distance is wrong to say Columbus Covered America. We say Columbus didn't discover America. Right. Yes. Yes so while the books are wrong that's just what my father told me. Problem Robert I don't think to discover America. Eric discovered first he discovered it first and he was the first person to see our country. No but he was the first of the exposed spores to use the first explorer. How do you know the teachers were teachers said that Eric was the first explorer to bump into our country from the book from the book but we hear the books also say that Columbus discovered America. Which book do you believe.

That Peter discovered green. But I don't think that's considered the land comes discovered. So I think comes really did discovery really discover America and what I mean by America states about the islands off the coast off the coast because I consider North America I think but Columbus Oh you know a place except those islands. So there's I mean when you discover an island off the coast I mean you discover the whole continent with me. Well it means they generally pretty you pretty much discovered pretty much discovered so we also got America we say Columbus pretty much discovery. Yeah I guess. Maureen Columbus was the first one from Spain that discovered the islands. Columbus was the first person from Spain to discover the islands. That's not the same as Columbus discovered America though is it now. OK let

me tell you about this. Very quickly. When you say Columbus discovered America you're saying a lot of things that really aren't true. If you analyze each one of these words you take these words and break them down like we have. Columbus was the first Spaniard to discover San Salvador. Can anybody disagree with that. Not really. But the first name of Columbus discovered America. Lots of people here have disagreed with it. So what we have to do when we say statements like Columbus discovered America what kind of things should we think about Nancy chatted be specific. Right. Try to be more specific. So what happens when somebody asked you Peter who discovered America what would you say then I guess. What would you say Christine. I don't know you don't know what you say you care. Marco Paulo Marco Polo. What would you say. I'd say Columbus you say Columbus. What would you say Nancy. I'd say Columbus discovered the islands off America.

But he really didn't discover the home America. OK how about Steve up there. What would you say. The room you would Peter but you know a voter I think you just should just leave it at Columbus the Columbus discovered America. No but really feeling really good. Afterwards de Papel conferred with the children's teachers who had observed the lesson. What is needed is something to shake these kids up to blast them out of their complacency and their indifference to any concept of critical thinking you saw their reliance upon authority established so well he sounded good so that when even though the book may not be one I'd rather listen to the book as it's easier. Why are you implying then that we don't do the shaking up in any other area. Why that we haven't done it. Yes but I think it's our foremost objective in our teaching. Well how do you counter the kids being so resistant to these ideas this morning. I could have been you know as a new person.

But it seemed to me that they were relaxed enough to speak openly and yet the idea that the book may be wrong seemed not to be a new notion. One of the problems in traditional social studies teaching is that kids learn these things as a series of connected or unconnected events just rote and don't really know what to do when it when to face the situation where these things are made clear by getting them excited about Columbus discovering America without referring to any conceptual tool as a problem or something bothering them they have to find out what he's driving at. And there's some more motivation here to learn these new concepts not to solve this knotty problem. You know about the University of Illinois critical thinking project. Professor Smith and Henderson the directors make a distinction between the problem solving and the formal logic approaches to critical thinking. PROFESSOR SMITH What is that distinction in problem solving you get answers in critical thinking you assess you evaluate answers you decide whether you want to accept answers that other people give over you give yourself I think you also have to remember that the schools are probably the only institution in which the

students will get this kind of training. But the schools don't do this job. It probably won't be done at Evanston high. Nathan Clark head of the Department of Social Studies is aware of the fact that to some people the phrase critical thinking implies an attack on our fundamental democratic assumptions. What about that Miss Clark. The importance of these skills clear thinking let's call them is not to question and to doubt one's traditional beliefs but rather to strengthen these by giving them an underpinning an undergirding or a basis of reason rather than emotion. In so far as it's possible to do that in fact we don't think that those judgments we all stayed with people when they meet opposing viewpoints unless they do have these scales because they must be able to analyze the opposition. And this they have to be trained to do. Now listen to a demonstration of how the need for clear thinking can be met. And accept from a dean clock's honest class in American problems in Evanston high. The

juniors and seniors involved begin the course with an eight weeks unit in applied logic. Before that as freshmen and sophomores they have taken units on propaganda and learned about premises validity and fallacies. They have been taught how to apply what they have to the analysis of historical ideas for today you were supposed to have read the editorial that you have before you now I trust. I one of the leading political columnists in the United States from a popular magazine and see whether you can identify in any of the logical fallacies that we have been discussing in class. Now we should remember before we start the sacking in which this editorial was written it came out in the first issue of the magazine almost immediately after the launching of the first Soviet Sputnik and is intended to be a comment on this greatest hour one of the greatest events in the 20th century. Now looking at it in that

light and not forgetting its immediacy What do you think of the author's reasoning. His reasoning and maybe we'd better take it paragraph by paragraph or skip number one which is introductory Shepherd will write in paragraph 2 the Arthur starts off something like this. Just because the Russians manage to spin a ball in outer space a sort of hysterical terror sees too many persons in this country. Well the entire statement is an oversimplification and spin a ball is a misuse of emotional words in that it is using slanted language instead of reason and evidence to describe or discuss a proposition. It plays down their scientific achievement which certainly should be recognized as very great. Anything else there Mark. Also in this paragraph we have the statement it could circle the earth. Back to kill people. Now this is actually a sample of a diversion. The writer is straying from the subject at hand but the satellite can

do has nothing to do with the topic the importance of the satellite. Number 5 where I read there is no reason therefore to begin to yield increase in submission to the totalitarian enemy which is already demanding a change in our foreign policy. I think that's an excellent example of the misuse of emotional words the chap was talking about. And not only that it is also an exaggeration of what was going on in the United States at that time. It is an extension fallacy also emotional words in particular the craven submission to a totalitarian enemy. And the extension part then would be in implying that that's what you're doing if you are afraid of the Russian scientific achievement is it right. That's right. Any comment on number 7 Chevron a new fallacy which he has made here is begging the question. The definition of begging the question is that the author has wandered over a number of topics and now has returned yet in his

wanderings he offered up no evidence to support his proposition. And sometimes we call that circular reasoning don't we. You go around in circles but you never get anyplace because here you are back again having never arrived at any particular bit of evidence to support your position right. Yes let's look at number 12 Clare which also has a look. Forgive all the communist criminals have been doing and embrace them as trustworthy friends the peoples of the West in the neutral areas really selfish and so easily misled. These sentences as well sentences in the paragraph before I think are leading questions because the answer is already implied by the author. And not only that when he uses the phrase communist criminals he is trying to indirectly play down there even by implying that it is criminals are not capable of launching a satellite by poisoning the well of the achievement in other words out of this well from this source of communist criminality.

Even though it may be criminal and certainly as common as one cannot draw anything good is that what you're implying there. That's called poisoning the well. Anything else you want to say there. Sheppard a leading question is a question which produces the answer. It leads you right up to the door and then you have no choice but to open it. What is he saying behind the words he actually utters when he gives a leading question he's giving them a question that has to be answered one way or the other. Now when he says people from the west and of the neutral areas really so fickle and so easily misled what other answer is there but no. And therefore you must agree with the author with the author. Yes you must agree with me. What do you think. Love the author's reasoning here as a whole is there anything you can say about it as a whole reasoning What are you seeing. Oh you don't think there was any. Go ahead SHEPPARD Well this was said definitely not a mental reaction this is a glandular reaction to a crisis article reads like it's written.

Well what he would call back in the article hysterical Terry sounds like he's a victim of it himself. All right somebody else here but I believe the complete article in relationship to the whole thing is an example of special pleading. Or is what the propagandist calls card stacking. Write any other comment. Let's remember that this is not so far as we know a typical example of American editorializing. And let's be thankful that it isn't. But be alert for other examples that may resemble.