Everybody's mountain; 7; Geometry in the First Grade at Stanford Elementary School in Palo Alto, California

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 cutting 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 report from one of those mountaintops geometry in the first grade at Stanford Elementary School in Palo Alto California. It's an isosceles time anybody tell me. Brian it's a triangle with I. It's equal to their link. It's very good. What is an equal lateral. Triangle

charity. I need color is all three sides. All right. How many points determine a line. So two points determine I learned Tricia or Patricia Sue PS is the daughter of the man asking the questions. Professor Patrick SU Pais a member of the philosophy department of Stanford University in Palo Alto. Ordinarily he teaches mathematical logic to graduates and undergraduates. Here he was instructing first graders who hadn't lost their first teeth in a branch of mathematics more than 2000 years old. If I just give you one point say the point how many lines can we draw through it charity. You know you it's very good if I give you two points though. Exactly how many lines can we draw through it. Mike why

now suppose I gave you three points. Could I draw one line through those three points. Not in general could I. So we want to repeat how many points determine a line. Let's all say it. Stanford elementary school is on the university campus quiet tree shaded streets curve among some that houses. The children many from faculty homes had worked with Dr. Sue Pease during the spring for periods varying from 15 to 30 minutes a day. Now at the beginning of the new front he was testing their recall of elementary Euclidean definitions. I could we define a circle now we all know what a circle looks like. But how would you say what a circle is. Brian it's really a guard when one arc four arcs and that's a sort of an arc but is that a circle.

That is not a circle that is unknowable. I suppose we think about as the center of the oval and here is the center of the circle. What's the difference between the center of an oval and the center of a circle. Well in the all not all this sides are the same length from the middle. Very good some are closer then there has a spare at Stamford elementary like most campus schools in the United States has a special relationship to its university. It is closed to new developments in teaching and to scholars who are strong on content. The fact that many university professors like Dr Sue PS are parents of the elementary school children motivates them as the educators say to a keen interest in what their children are being taught. And open mindedness on the part of the school staff and willingness to challenge traditional practices and a determination to meet the needs of its capable group of primary great is provides a fertile soil for enrichment within the framework of the basic curriculum.

Members of Stanford faculty are often intrigued during the spring semester to come in and work with the students in their specialties. Dr. Sue Peters you taught the elements of mathematical logic for six weeks to fourth fifth and sixth graders here at Stanford. What did you find. Well I found that if the material were brought down to their level in terms of advance examples taken from television programs and the like that they essentially had no difficulty with the material. But here you are experimenting with the possibility of teaching elementary geometry to first grade students. What are you trying to find out. We are trying to determine if it is possible to add to the curriculum of first grade students a considerable body of geometrical material which as you probably know is not now taught in the elementary schools. Well how did Noah come about. How did you get involved in this experiment. A more than a year ago I went to a meeting in which several social scientists came to the elementary school to discuss the role of the social sciences in the elementary school curriculum. I felt that the relative role of mathematics and the

physical sciences was not sufficiently emphasized in this discussion. And I made some remark to this effect in the general meeting the teacher of the first grade Mrs. Wilson was present at this meeting and she asked me if I would care to put in practice some of the range ideas I set for this meeting. I agreed to do this before attempting anything. In terms of actual work with the students I asked of Newton HAWLEY Because he has a daughter in the class as well. If he would like to participate with me in this project what is the Hollies background. Ali is a mathematician who took his Ph.D. at Princeton University who is now on the faculty at Stanford University. Newton Holly's daughter in Mrs. Wilson's class is Melissa Hawley. Professor Holly's early studies were in philosophy before he switched to mathematics. He worked for the Office of Naval Research taught at Cambridge England and has been a Fulbright scholar. Mrs. Mary Lou Wilson the first great teacher took our ABN political science at Stanford.

She has two young children of our own. Mrs. Wilson What prompted you to talk to Dr. Sue PS after he spoke at the curriculum meeting you both attended. Well I knew his daughter very well and I wanted to understand how he felt about these things. And I also wanted to get here to tell you need to understand just what we were doing in the classroom. I got my own mathematical back it was very limited it had been one of my weak areas. And perhaps he could enrich our first great experience in mathematics and I would like him to help us in any way he can. Why geometry Dr. So peace for enriching the curriculum. Why not accelerate arithmetic or introduce the elements of Sept theory. I think that the geometry has a particular appeal to it and the vividness of the of the drawing the figures the concreteness of the figures is the kind of thing that exactly can be made to appeal to students at this age level. Holly and Sue piece talked informally to Mrs. Wilson's class about mathematics. The

children were provided with adjustable thumb screw compasses and straight edges. The experimenters were curious about the degree of pupil interest they would arouse about attention spans difficulties and vocabulary and about comprehension of abstract concepts. They were even curious about the children's ability to handle compasses accurately and safely. The class worked individually with pencil and paper at their desks and also as a group following the university professors at the blackboard who used to string in pieces of chalk. The atmosphere was often artificially charged with the presence of observers a visiting sixth grade class one day couldn't restrain its enthusiasm and the older children raised their hands in answer to questions addressed to the first graders. This was not a formal educational experiment. It was a cooperative personalized probing by an informal team of university scholars and then elementary grade teacher into the mysterious realm of the yet unknown capacities of children.

Last time he taught you reviewed some definitions didn't you. Let's go on to constructions. Professor Hartley was at the blackboard now his objective to discover how much the children had remembered about the fundamental geometrical constructions. Let's see if we can find out how to construct an equal arrow triangle. Brian first you would just a straight line with a pencil and a straight it and then you would take your can compass. And measure that last 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 end of the yes. And then you would take take it up to the top there and swing a little arc have drawn an arc around the border. Then you take your compass in put the point on the other side and said it's still the same length.

Now bring it up. Give it another. Are all right 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 up from the top. Top this point of intersection. Then you take use storage over to the other side. And place it on the other line and draw a line to see where the two arcs owners set each other in. Is this an equal lateral triangle. Yes it is because all. Three sides are the same. The children were asked to draw the six lines determined by two circles and they sent us. They were asked to draw a perpendicular from given points to a given line segment to

bisect a given angle to find the midpoint of a given line segment. The Diary of the experiment kept by Holly in soupy is not the delight experienced by the children in making discoveries. Interest and excitement ran high the diary notes one day there was an initial break through it records and then the discoveries came quite easily and rapidly. Conceptual difficulties lay in the way the students were often not yet aware of the meaning of new words such as bisect. Diary notation reads. They have not yet grasped the fact that we must define certain properties before we can state that figures have these properties. Nevertheless Dr Sue PS despite the irrelevancies and the shaky ability of the children to reason as precisely as you wish. What did you and Dr Holly find. What we found was that with a few emendations we could and we could follow the classical programme 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 about geometrical problems what about the so-called compass Beriah the student's ability to handle compasses accurately. We found in point of fact that they could to our delight and to some extent to our surprise were all the children in the class able to handle the mathematical concepts. 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 Stanford elementary as in the Palo Alto Unified School District. John Caffrey director of research for the district first heard about the Holy See peace project several weeks after it had been gotten by way of a telephone call from Dr. Winfield questions on the school's principal. He thought Christiansen was joking when he announced that Euclid was now in the daily curriculum of Mrs. Wilson's first graders. He was amazed when he came over to find the class doing so well. Mr. Cafferty

isn't it true that geometry is generally not taught in the primary grades because of certain assumptions made by educators concerning children's readiness for learning the subject. Assumptions based frankly on the fact that we don't have enough information about some of these things for example we don't know what readiness for the Study of geometry consists and we know that children have to be ready to undertake the study of any subject and we don't know enough yet about for example in the area of geometry exactly how old a child has to be either physically or mentally before he can begin to study it. So there was obviously a challenge here to the notion that first grade is too early for the Study of geometry because we normally don't undertake it until about the tenth grade. There is no data in any case then either to prove or disprove what was being done here. It isn't that there is no data there's just not enough data. Well there have been some experiments but we don't really know for sure. Exactly what age children should reach before they're introduced to a variety of topics of us once again. Mathematics is only one area.

Dr. Holly did the Greeks teach geometry to their children and so were only in age to my knowledge this wasn't done. However I'm no historian of the Greek pedagogy. However I can say that Euclid's Elements were certainly addressed to the tour audience. There were however of course more elementary geometry texts than Euclid's Elements. These are in general lost to us nowadays through the centuries. Apparently the age at which we have assumed geometry can be taught seems to decline even in the standard curriculum we teach geometry at an earlier age than was attempted several hundred years ago. Do you believe the children have an unsuspected mathematical maturity. I think they have unsuspected maturity and many domains. Mathematics merely happens to be my specialty. Dr Sue PS Why do you suppose this notion of geometry to first grade youngsters has never been thought of or at least tried to practically have at the first grade level before.

From a purely practical standpoint in terms of the everyday affairs of a living it is absolutely necessary that the students be taught arithmetic. Consequently this subject of the subject of mathematics has dominated the curriculum in the elementary schools. But on a deeper level there are undoubtedly questions of a rigidity of love and willingness to experiment. I would have led to the exclusion of topics like geometry from the elementary school curriculum particularly in the primary grades the geometry instruction remember was carried on as only a part of the children's regular school day which included arithmetic and spelling as well as the children's periods of rest and play. To appreciate the uniqueness of the experiment. Contrast for example the first graders attention to Euclid's propositions with their dedication to the one little duck with the feathers on his back. Seat. The.

Game. Was weak. Flat of the musical mathematicians were Brian Jones Patricia superior's charity Scott and Mike little. Brian do you like geometry. I know I like bisecting a line in line. How about you. Oh my God. Because I'm well I'm not. Do you actually like it Mike. Yes. Why do you like it. I think that this last idea by like it is arithmetic is interesting in geometry. You know why is geometry more interesting. It's more complicated a deal I would talk about studying geometry

here in the first grade outside of class or do you just forget it when you leave the class. Charlie how about you Brian. I don't really forget you remember geology because I like an awful lot. Brian wants to be a scientist when he grows up. How about you Mike what do you want to be when you grow up. As time are finite game game low buy do you think you guys are lucky. What do you think is more interesting to do to make construction clients want to talk about it. Brian and I do make them better. Would you like to go on with studying geometry. Yes I I would love a lie. How about you treasure and Charlie and you Mike. Yes. Dr. Christiansen the school's principal observed that the children could follow the relatively difficult vocabulary used by holy and superiors with ease and interest. Mrs Weston did you find that this war cabinet was sometimes too adult for them.

Yes particularly in the discussions that were conducted by Dr. Lee and Dr. Sophy's. Not so much as the words such as Isosceles or equal lateral but words. Like when a line is determined by the parents with a child of this age does not know what the word determined means and such words as this had to be explained and modified. What about the stresses and strains emotionally on the learners in this situation did any of the children need relief Mrs. Wilson. Yes very definitely that they found times when Dr. Holly or Dr. cities would be involved in a discussion and would become so interested in there that they would not notice that the children were becoming nervous. And I felt that there were times when these discussions were carried on too long and the vocabulary was over their heads. Dr. Christiansen Were there any occasions when there were extreme manifestations of failure in the class. Yes sometimes children were asked to do more than they could do with comfort or assurance. Sometimes there were tears sometimes

evidence of boredom and inattention. Did you point these trains out to the university professors Mrs Wilson. Yes I did and they were very responsive to suggestions because after all they had not dealt with children this age other than their own two children. They had it out with them in a classroom situation and they were very open to suggestion and use of certain little techniques that first grade teachers use that they were not familiar with they found that they had a great deal more success in the years more children and not didn't concentrate on one when they would change the pace of activity when we'd have a few minutes of discussion and then allow the children to go back to their seats and be involved in some physical activity where they were constructing or just cutting a discussion shorter. You found no irreconcilable conflict between the content people and the method people. No there wasn't. Doctor wrote seven descriptions of constructions on the board which the children had worked on. He asked Terry to read the first one.

A sick girl with giving. Yes very good. Let's try the second one. Go ahead Melissa that's jogged a little triangle was given for Selig can you read the third one. Find the midpoint of the of. A law and find who can read the fourth one. Mark draw ng cute angle. All right what about the fifth one. So you can draw an obtuse angle. Fine. What about sex. Pat. By Step eg a very fine by me angle right. Finally what about seven Buddha three line segments. Construct a triangle Botho was called Buddha because she sat like a Buddha when she was a baby.

All right now these are seven constructions which one do you think the hardest Can you remember. Bill no triangle were given the why do you think it's the hardest. Me looks no I don't look the hardest. Let's quickly take a vote on which one is the hardest. How many think the last one is the heart. 3 How many think by taking a given angle is the hardest too. What about constructing an ego out of a tribe. How many think that's the hardest. No but I know what I know I said. What about finding the midpoint of a line. Oh I see a lot of people wired two three four five six seven eight. Dr. Holly you want professors who presented into your experiment with the belief that children could deal successfully with geometry in the first grade. Where do you stand with respect to this belief now.

Our beliefs have if anything been considerably reinforced by our experience and we are firmly convinced that one should explore the possibilities of determining to what depths the reasoning abilities of children can be carried. What plans do you undock the Supremes have now with respect to the continuation of this experiment. At present the details of the future of this experiment are somewhat uncertain. However we certainly will continue with the experiment in some form or other. Mr. Cafferty your school district is seeking a grant to support further research along the lines of this experiment. What do you want to explore. Unless our research shows that the experienced classroom teacher one without however special I think. Background can do what has been done here. The experience will not be of general value to us or to other school districts. We want to try this out only in first grade but all the way through the sixth grade. Mrs. Wilson if you get this graphic. That has been applied for how would you modify the actual classroom teaching experience for the

children. Well I would start earlier with and then we did last year and I would give them a great deal more opportunity to handle and actually quit and add to the compasses in the straight edges and they would make circles and become used to the equipment before they would go on into the deeper concepts of the program. And after teaching them how to manipulate what would you what would you expose them to. Then I would expose them just to the shapes the Eternals and the circles and that served them and then get to the conceptual material. Yes. What would you do about grouping. I would very definitely group as I found it necessary it would probably be at least two perhaps three groups. We would always take into consideration in individual towns ability. But these modifications would in no way spoil the essential driving purposes conceived by the university professors do you think. You know I don't believe Dr Sue PS A further research should indicate that geometry can be taught as early as the first grade. Wouldn't that require changes in the training of elementary teachers.

I think that by changes in the present teacher training programs the elementary school teachers can be more adequately prepared to teach this kind of material. I think it's worth noting that the same kind of problem faces us and any attempt to introduce at a serious level the teaching of foreign languages in the elementary schools the majority of elementary school teachers are not prepared to teach a foreign language and to some extent the same problem faces us in the in the teaching of any substantial body of geometry. We need to offer it in the College Program of the elementary school teachers the kind of material that would prepare them for teaching geometry and additional portions of mathematics in addition there is the need to prepare workbooks or textbooks that would be available to the students to the classroom. For covering this kind of material but it doesn't seem to me that any of these problems are insurmountable are really extremely difficult in terms of finding a solution. Dr. Hawley Why has this experiment of significance to the nation as a whole.

There seems to be in our curriculum elementary and secondary and appalling lack of confronting the student with problems about which they much must reason frequently the problem with which the student is confronted is of a very routine nature. He is given a problem. He is not made to understand the significance of the problem and he is told how to solve the problem and merely learns to repeat this solution at this level we're trying to teach the student how to reason about his problem and produce his own solution in a rational manner. There is in geometry an intrinsic beauty and intrinsic intellectual interest which one can find a deep satisfaction from having met unusual problems understanding what the significance of the problem is and of 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 the even the educated public that the creative aspects of science are a genuinely and is deeply creative as those aspects of the art it is recognised by almost everyone that 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. Individuality can be expressed as much in the domain of reasoning and in solving a problem with an original outlook. Even in geometry a subject over 2000 years old as it can be in painting a picture or or drawing in or any of the other plastic arts that might be considered. There is a strong and general pressure today to sink shafts of learning traditionally reserved for more adult plateaus of experience ever more deeply into the bedrock of the educational process. One reason for this pressure is the rapidly

accelerating closing of the gap between the university and the elementary school worlds. Another is the realisation based on a century of research that growing children pace themselves dynamically and differently some faster some slower in their learning activity. A misguided reason is that all children should learn tough subjects because tough subjects are good for all children. In a democratic society at least this conclusion is untenable however incomplete our knowledge is of the act of learning. It is certainly worth exploring what are the commonalities and the peculiarities of children with respect to mathematics in the elementary grades. A further broadening rather than a narrowing of educational opportunity seems to be the consensus of opinion among experienced educators. We cannot keep any child educationally productive However if we insist on his achieving goals that are beyond his reach.

The mountaintop in Palo Alto. Geometry in the first grade. Is elevated because the enthusiasm of the probers into the mysteries of mathematical maturation was tempered by the compassion which recognizes individual differences and respects the infinitely variable skills insights and contributions waiting to be awakened in all children. Education is your mountain to. Your children's Mountain your neighbor's mountain. Education is everybody's mountain. They recorded series everybody's mountain was written and produced by Robert Louis Shea on for the Educational Television and Radio Center. The programs are distributed by the National Association of educational broadcasters. The series consultant was Dr. Ernest O MELBY professor of education at Michigan State University and former dean of education at New York University.

This is the end E.B. Radio Network.