A Geometry Course That Changed Their Lives:
The Guinea Pigs After 60 Years
Annual Conference of The National Council of Teachers of Mathematics
April 6, 2001: Orlando, Florida
Dr. Frederick Flener
Northeastern Illinois University
Chicago, Illinois 60625
This project was supported by a grant from the Committee on Organized Research of Northeastern Illinois University.
Imagine yourself living in 1932. The United States, as well as most of the world, is in the middle of a major economic depression. You live in Columbus Ohio, and the Ohio State University is about to open it doors to a new experimental high school. Not a good time for experimentation in education. They will be admitting kids from all over the Columbus area and charge them about $65 a year for tuition. Not much money by today's standards, but in 1932 ...?
Your 11 year old child says that all his friends will be going and he would like to go too. Ohio State claims that students will be selected from the applicants on a lottery basis, but you are a bit suspicious. The school is advertising that it will be a so-called, "progressive school" -- following the guidelines of the Progressive Education Association (PEA). This means the school will have a "democratic" curriculum -- there will be no grades, and students will learn by engaging in so-called life activities." You ask yourself, "Why would any parent who values a college education send a child to this school?" This is a school at which your child might not learn the appropriate precollege curriculum, the school would not be able to send a transcript detailing the courses taken and grades received, and your child might even avoid college if the "life experiences" were too inviting? Shouldn't parents make sure the kids understand the basics, to prepare them for college or for employment? Weren't these reasonable goals for schools -- and aren't they still today? This is why I was surprised that in 1938 when a group of 57 graduates wrote a book describing their experiences at this experimental school, they gave glowing praise to the school, and they wrote about the exceptional regard for their education and especially to a course called The Nature of Proof. That is what this presentation is about.
(OVERHEAD 1: Title of Guinea Pig Book)
The book they wrote, Were We Guinea Pigs?, was sort of a replacement for the typical senior yearbook produced in most high schools. This was the inspiration of one of the English teachers at the school, Louise LaBrandt ("Lou LaBrandt"). First, because the school had this so-called democratic tradition, she had to convince the faculty and administration at the school that the students were capable of completing such a task. She then cajoled the Holt Publishing Company into publishing the book, convincing them that there was a commercial market for the book. Finally, she had to ride herd over the 57 students for the greater part of their senior year while they compiled chapter after chapter until the book was finished. It was a successful effort (although not necessarily a financial success), and the students were very proud of their product, so much so that they referred to themselves as the "Guinea Pigs," and 60 years later they still use the same name.
Who were these guinea pigs, and why did they go to this experimental school? Most of them began at the school on the first day it opened in 1932. They started in sixth grade and attended the school for six years. Several were children of faculty, a few attended the very small experimental school which previewed the University School, and a few students went because the alternatives were the Columbus School for Boys, and the one for Girls which were much more expensive, or the Columbus public schools which were very crowded. But, there doesn't seem to be any clear reasons why these students were enrolled. Robert Butche, is a 1954 graduate of the University School and is writing a historical book about the school. When I talked with him he said that many of the parents were either insane, or more likely, they had tremendous faith in the faculty. The official claim was that the selection process was by using a random lottery of those who applied, but I am a bit skeptical of this. The demographics of the class were not typical. Most of the guinea pigs were from well educated families; 88% of the fathers and 45% of the mothers attended college which was a much higher percentage than the norm in 1932. The $65 may have excluded many children from applying, but a large number of students had scholarships. Were the scholarships based only on financial need? Possibly, but maybe not. What were the criteria for offering a scholarship? It may have been the same as it is today, potential for success academically. However the students were selected, one thing is quite clear. This was probably not a randomly selected cross section of the adolescent population in 1932 Columbus. We know the guinea pigs weren't necessarily the most intelligent kids in Columbus, but they were probably an above average group academically.
Regardless of why the kids were enrolled, the parents concerns about the potential effect on their child's future had to be alleviated. The Progressive Education Association was pushing to establish a research program to see if progressive schools could be successful and the "30 School Experiment" was established. These schools were identified as "progressive" and the head of the PEA, Dr. Wilford Aikin, got over 200 elite universities to agree to admit students from the 30 schools on the basis of the schools' recommendations and a couple of comprehensive exams to convince the universities that the students could at least read and write. No minimum SAT or, ACT scores, nor even a class rank, was needed to be admitted to schools such as Princeton, Stanford or the University of Chicago. That was a carrot which may have appealed to some of the parents. Or, as Butche suggests, it may simply have been that the teachers at the University School were going to be exceptionally good. Whatever the circumstances, about 150 students, in grades 7 to 10, began in September of 1932, sixty of whom formed the nucleus of the guinea pigs.
The Carnegie Foundation funded an Eight-Year study to examine about 2,000 of the first college bound students in the 30 School Experiment, and compared them with an equal number of conventional high-school graduates. Here are some of the results of that study as reported in a 1940 Time Magazine article:
(OVERHEAD #2A, B: Time Mag. Information)
The Progressives had slightly better grades in standard college subjects. The more extremely progressive the high school, the better its graduates did in college. (The University was one of the most "progressive.")
A group of 46 who deliberately avoided mathematics in high school surpassed their classmates in every college subject, including math. Conventional students joined more social, religious and service clubs, attended more movies.
Progressive students went in for more extracurricular activities (except athletics), took more active interest in politics and art, talked more, wrote more, listened to more speeches, and music, read more books, went to more dances, had more dates.
Progressive students did just as much worrying, had as many personal troubles as their fellows. (Time Magazine, November 25, 1940)
In general, one could call the experiment a definite success. What happened? Why aren't all schools today like those schools? Are there any? There were probably research design problems which prevented such a program from gaining universal implementation. Supposedly, the students were not selected on the basis of academic or intellectual criteria, but one might suspect that families who applied to these 30 schools like the University School were not typical. The teachers in the programs were probably some of the best in the nation, and much of this presentation will be devoted to one such teacher. Sure, these universities modified their admission criteria for graduates of 30 schools, but could they have done so if many more schools, say 100 or 200 participated in the program? In other words, by it very nature the "30 School Experiment with Progressive Education" in all probability could have only be applied to a very limited number of schools.
This gives you a bit of a background about the general conditions that existed when Harold P. Fawcett's began his experiment with a course in high school geometry, The Nature of Proof. Fawcett taught this course at the University School at Ohio State. It was a unique course, and probably the most popular course at the school. In fact, one might argue that it was the best course ever taught in a secondary school -- at least I will try to convince you of this today.
When I began this investigation, I fully expected to determine what went on in that course which made it so successful. What I discovered was a course that was intertwined with the story of the school. This was a unique program, and this was a unique time in our nation's history. The guinea pigs found themselves in this progressive school, with a gifted teacher, and all of this tied together through a clever approach to teaching geometry. I can only share a small part of the story today -- an abstract, or a peek into a much larger story. I have talked a bit about the school, let me now turn to the teacher and his students -- the guinea pigs.
Who was Fawcett? Who were the students, the guinea pigs, and why did they (and still do today) think so highly of him? What did they learn? What evidence is there that this was that good of a course? These are questions I hope to answer in the next 45 minutes.
First, I would like to take a couple of minutes to explain what piqued my interest in this project. About two years ago I was asked by the Illinois Council of Teachers of Mathematics to make a presentation in support of "proof" in geometry. As many of you know, there seems to be less emphasis on the importance of proof as part of the geometry curriculum, with much more time given to geometric investigations. I have always been a proponent of teaching critical thinking in a geometry course because as an undergraduate I was influenced by my professors, especially Robert Pingry and Ken Henderson, who extolled the virtues of Fawcett's 1938 NCTM Yearbook, The Nature of Proof, which you can buy from NCTM for about $15. (HOLD UP BOOK) In preparing for that talk, I re-examined that book, but I also recalled that I had heard of some book called, Were We Guinea Pigs?, but I knew little about it. So I went to our library and there was a copy of the book in our library, along with another book called, The Guinea Pigs After 20 Years, written by Margaret Willis in 1959. In this she described the guinea pigs after they had lived in the "real world" for 20 years.
What did these books have to do with Fawcett's course? The first book was one written by the graduates of the first class of students to complete the full six years at the University School. The second book was a glimpse into the lives of the Guinea Pigs as they approached "middle age." Both books gave high regard to The Nature of Proof course. First, graduating seniors praised the course, and 20 years later they spoke with their enthusiasm about the influence the course had on their lives. It was close to a consensus among the students that this was an outstanding course, and several thought it was the best course they ever had -- elementary school, high school, or college. When I made my presentation, I gave the audience all of the positive information from the three books. But, like so many sermons the talk had very little influence on them. They said, "Nice talk Fred, but we are going to do more with technology, investigations, etc., and there is still going to be less emphasis on proof in our classes." So-be-it, but I felt I gave it my best shot.
It was afterwards that I began to wonder if any of these graduates are still living, and if they are what do they now think about the course? I did a little math -- after all, it is my subject -- and decided that most of the students would have been born in 1920 or 21, and many are probably retired and living in Florida. However, I had no way of contacting them. I learned that the University School had closed in mid-60's and records of graduates were not well updated. All I had to go on was the list of names from the Guinea Pig book. (OVERHEAD 3: Names). I wrote to the Alumni Association at OSU with the list and asked if there may have been any 1942 graduates from the list. They came up with 11 who might have been students from the University School. The Association then forwarded a letter from me to these 11, and slowly my contact grew. I now have either located or have learned of the death of 50 of the 57 students from that "Guinea Pig" class. Over the next two years I interviewed 21 of 27 whom I knew were still living. I also interviewed two of Fawcett's daughters, Dorothy Zechiel and Winifred Evans (Winnie), some of their children, as well as others who knew Fawcett personally and professionally. What you are hearing is a story of Harold Fawcett and the Nature of Proof course, through the eyes of the Guinea Pigs and others who knew him. (Incidentally, whenever I refer to someone as a "guinea pig," I do so as they themselves do, affectionately.)
One of the first letters I received was from Tom Bowen, a (retired? still active?) CPA living in the Cleveland area. He wrote a nice letter describing his memory of the course. In the letter he said, "In any discussion of Dr. Harold Fawcett's presentation of geometry, The Nature of Poof, one caveat is required: One cannot separate the person and character of the man from his message." (OVERHEAD 4: BOWEN QUOTE) Therefore, you need to know who this person, Fawcett, was, and what do we know about his "character."
Here are two different photographs of Fawcett. (OVERHEAD 5: SIDE BY SIDE) . This top one was taken in 1964 when he was President of NCTM. The bottom one was taken in 1938, from a TIME magazine article, when people were discussing the book written by the group of high school students. I don't know about all of you, but my image of a scholar is the top one. When I showed the photos to the guinea pigs their memory was of this one.
But, in many ways, the next photo of Fawcett and his wife, Muriel, is a more meaningful photograph. (OVERHEAD 6: H.P. AND MURIEL) During my travels I learned much about Fawcett, the "person and his character." Let me share a few of the insights I have learned about Harold Pascoe Fawcett. In addition to being a highly respected mathematics educator, Fawcett was a kind and caring person. I learned this from a few of his colleagues like Eugene Smith who taught The Nature of Proof course at the University School from about 1945 to 1956. From two former students turned professors, Ken Cummins who died last year and Ken Henderson, my former professor, and from the writings of others such as Joe Crosswhite, a colleague at Ohio State, or William van Til, who was a fellow teacher at the University School. In almost every interview, letter or article, usually the first thing they talked about or wrote about was his personality. They did this before they even mentioned his teaching ability or his scholarship. I also learned about Fawcett through some of his poetry, much of which he wrote to his wife and children. I have permission from his daughters, Dorothy and Winnie, to show you a couple of his poems. ("doggerel" is what he used to call his poetry) The first is one which he wrote for his wife Muriel. He often wrote the short poems and left them for her. His daughter Dorothy found a few of these in an old book. When he wrote this one, he was simply going for a walk. He just wanted Muriel to know he was thinking of her.
(OVERHEAD 7: H.P TO MURIEL POEM)
my darling ~ 3:45 p.m.
Since sleep has gripped your body
And closed your tired eyes;
I've gone into the open air
To take my exercise.
Since you are nice and drowsy
I do not wish to talk;
And so this verse is written
To say I've gone to walk.
My love for you always
I don't know about other males in the room, but clearly, this makes me think I might be a bit more insensitive than I feel I am. Maybe my wife would argue with me on that point, but ... Well, let's just say that the probability of my writing such a poem is slightly less than winning the lottery.
He also cared deeply about his children and his friends. His daughter, Dorothy shared a collection of poems he wrote and gave to his family after his wife died in 1966. Here's one he wrote for his daughter, Dorothy, when she was a small child and experienced -- at least for her -- a very traumatic experience. She lost her boots. The actual poem is 9 stanzas long and this is only four of the stanza, but you should recognize the principles he was trying to teach Dorothy.
(OVERHEAD 8: DOROTHY POEM)
THE LESSON OF THE LOST RUBBERS
You've lost a pair of rubbers
And I wish it were not so,
But I'd rather lose my rubbers
Than some other things I know.
You haven't lost your eyesight,
And you still have power to see
The glory of the sunshine
Which is shared by you and me.
You haven't lost your power
To bring beauty into life,
And I hope you'll always use it
To end bickering and strife.
You've lost a pair of rubbers
That cost a paltry fee.
Just go down town and buy again
And charge them up to me.
I met with Dorothy and her husband Leon. As you might guess, Fawcett's commitment to his family was passed on to his children. When I visited with Dorothy and Leon, not only was I impressed with their strong family ties, but a few other things struck me. Much of their home was hand made. That is, it was not loaded with Rembrandts or Picassos but with things that measured their life together -- a quilt, candlestick holders passed down in the family. If you visit them you know who they are and where they have been.
I also met a couple of times with Fawcett's youngest daughter, Winnie and her husband Tom, and they told me I should interview Winnie's son, Harold. When we were going to meet, Harold asked if his friend Joe Hillsman could come along, and I said "sure," not knowing why he wanted his friend to be there. When we met it was very evident. During the last seven years of his life Fawcett lived with his grandson, and Joe was a frequent visitor. Joe referred to Fawcett as "Grandpa" and in the interview contributed more to the discussion about Grandpa than grandson Harold did. Joe told one story of when the three were walking at the airport, Fawcett tripped on some loose carpeting. As he fell he simply did a forward roll, got back on his feet and continued walking. Not bad for a man in his 60's.
We also met with a couple of old friends of the Fawcetts, both over 90, who talked warmly about their friendship throughout the years. He was fondly remembered by young and old alike.
There is much more I can tell you about Fawcett, the "person and his character," but I just don't have time in this presentation. I would like to move to the course, "The Nature of Proof," possibly the best course ever taught, and its impact on the guinea pigs.
What was this course? What went on? What do the Guinea Pigs remember about the course? Is the impact of this course independent of the school? That is, from our point of view, could the course be taught in other regular schools?
As I mentioned earlier, I have interviewed many of the guinea pigs and have compiled more than 50 hours of audio and video tapes and cannot possibly hope to discuss all of the conversations in this presentation. What I would like to do is simply select a few highlights that I hope you find interesting. I have a photo taken when Henry Holt invited the entire class to NY. Remember Holt published, Were We Guinea Pigs? (OVERHEADS 9 AND 9A) Ironically, three of the Guinea Pigs whom others considered to be the "smartest" in the class are not in the photo. Ben Burtt - who has a Ph.D in chemistry and is professor emeritus from Syracuse, Joe Levinger, who has a Ph.D in nuclear physics and is professor emeritus from Rensselar Institute of Technology, and Warren Mathews, who also has a Ph.D in physics and is retired as a vice president of Hughes Aeronautics.
My first visit was with Elizabeth Stocking Seale, who now lives in Moscow -- Idaho, that is. Her husband, Bob, is a retired professor of forestry from the University of Idaho. My wife and I spent two days with them at their cottage on a lake in northern Idaho. One of the things which triggered my interest in meeting Elizabeth was in her response to my initial letter. She said, "I went to Stevens College, Ohio State and the University of Idaho, and I never had a more influential course (than The Nature of Proof)." Powerful words. Did she really mean it? Throughout our visit, she repeated comments similar to those. One of the benefits of seeing Elizabeth first was that she was the photo editor for the Guinea Pigs book and she knew who most of the people were in the photos. I noticed that she was in several of the photos. She simply said, "Of course I'm in many of the photos -- I was the photo editor, wasn't I?" Throughout the visit, she insisted that the type of thinking she learned in The Nature of Proof course influenced her throughout her life. She also gave me my first insight into the whole University School environment. We spent a lot of time discussing the Guinea Pigs book -- especially the photos.
One of the photos in the book that impressed me was the sign over the Math/Science door. (OVERHEAD 10: Prize the Doubt.) When I first saw it, I liked the first clause, but was a bit concerned about the second. There must have been some controversy among the students as well because I heard from most of them that the "Low kinds" referred to are "lower order animals." However, when I interviewed Joe Levinger, he thought the quote was from Shakespeare, and said, "It probably referred to lower class people, because that was the way things were in Elizabethan times." Whatever the second clause meant, the first did have an impact on the students-and was also relevant to The Nature of Proof course. Most of the guinea pigs that I interviewed were very open minded, whether it was politics or social mores. In one of the first letters I received, Helen Spencer Lynch wrote the following;
(Overhead #11-Quote by Helen Spencer Lynch)
"About three weeks before your letter came, I attended a very emotional and turbulent meeting determined to listen and learn but not open my mouth and add to the conflict. Finally, I had to speak and what I said came entirely from my University School experience and was based on our "Nature of Proof" thinking. To my great astonishment the room full of people applauded. They calmed down and we were able to reach a reasoned conclusion."
Later, when I visited with her, I asked what the issue was. She said it was related to a couple of controversial issues in her church, homosexuality and female ministers. Here is a 77 year old woman speaking in support of some very liberal positions. Throughout my time spent with the guinea pigs I was continuously impressed by the clarity in their thinking. Was this the result of The Nature of Proof course? Let's turn our attention to the course itself. Fawcett begins the yearbook with the statement, (OVERHEAD 12: Quote from Yearbook) There has probably never been a time in the history of American education when the development of critical and reflective thought was not recognized as a desirable outcome of the secondary school.
In his view, the best course in the secondary school in which to teach critical and reflective thought was geometry. To support this he cites several other mathematics educators. Here are a few of those quotes; (OVERHEAD 13, 14, 15: Quotes from Yearbooks)
"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)
Fawcett was disappointed with the traditional geometry courses and did not feel we were doing a very good job at getting kids to think critically, (OVERHEADS 16, 17 AND 18) Here are a few of the others he cites to support his opinion.
"The reasons given by pupils for statements often seem to disregard entirely the thought of the situation. Often it seems that it is mere habit that dictates the response, not a thought process. Pupils have often used the various theorems as reasons and with satisfaction. They seem in some cases to have used them so often without meaning that they give them as so many memorized non-sense syllables." H. C. Christofferson (1930)
"If school children fail to get some conception of geometry and close reasoning out of the course in 'geometry' they get nothing, except possibly a permanent inability to think straight and a propensity to jump at conclusions which nothing in reason or sanity warrants." E. Bell, (1934)
"The mere memorizing of a demonstration in geometry has about the same education value as memorizing of a page from the city directory. And yet it must be admitted that a very large number of our pupils do study mathematics in just this very way. There can be no doubt that the fault lies with the teaching." J. W. Young, (1925)
So, with his views of the importance of critical thinking in schools, and his belief that the geometry course was the best place to teach this, Fawcett wanted to try a new approach. And, what better place than at the University School? Here were his thoughts about the nature of the course. (Overhead #19)
"While teachers of mathematics say they want the young people in our secondary schools to understand the nature of proof, that should not be and probably is not their total concern. What these teachers really want is not only that these young people should understand the nature of proof but that their way of life should show that they understand it. Of what value is it for a pupil to understand thoroughly what a proof means if it does not clarify his thinking and make him more "critical of new ideas presented"? The real value of this sort of training to any pupils id determined by it effect on his behavior and for purposes of this study we shall assume that if he clearly understands these aspects of the nature of proof his behavior will be marked by the following characteristics:
- He will select the significant words and phrases in any statement that is important to him and ask that they be carefully defined.
- He will require evidence in support of any conclusion he is pressed to accept.
- He will analyze the evidence and distinguish fact from assumption.
- He will recognize stated and unstated assumptions essential to the conclusion.
- He will evaluate these assumptions, accepting some and rejecting others.
- He will evaluate the argument, accepting or rejecting the conclusion.
- He will constantly re-examine the assumptions which are behind his beliefs and which guide his actions.
These principles represent Fawcett's concept of critical thinking. Fawcett also made several assumptions about the kids before beginning his study.
- Students have reasoned and reasoned accurately before beginning the study of demonstrative geometry. (Learning to reason is sort of a developmental skill.)
- Students should have the opportunity to reason about the subject matter of geometry in their own way. (Would he be called "a constructivist" today?)
- The logical processes which guide the development of the work should be those of the students, not those of the teacher. (This is a blend of 1and 2.)
- There should be opportunities for the application of postulational methods to non-mathematical material. (He believed transfer must be an overt commitment.)
His thoughts about how pupils viewed geometry relationships were drawn from five levels of activity that pupils use to gain understanding, that were attributed to W. H. Kilpatrick (1922), with the 5th being the level he hoped his pupils would reach.
- A pupil memorizes the bare words of a demonstration.
- A pupil memorizes the idea of a demonstration and can reproduce it in different words.
- A pupil makes a given demonstration his own, it becomes his thought, he can use it in a new situation.
- A pupil of himself demonstrates a proposition that has been proposed by another.
- A pupil of himself sees in a situation the mathematical relations dominating it and of himself solves the problem he has thus abstracted from the gross situation.
With these conditions to draw upon, Fawcett went about designing his course. He used the following principles in his design.
- No formal text is used. Each pupil writes his own text as the work develops and is able to express his own individuality in organization, in arrangement, in clarity of presentation and in the kind and number of implications established.
- The statement of what is to be proved is not given to the pupil. Certain properties of a figure are assumed and the pupil is given an opportunity to discover the implications of these assumed properties.
- No generalized statement is made before the pupil has had an opportunity to think about the particular properties assumed. This generalization is made by the pupil after he has discovered it.
- Through the assumptions made the attention of all pupils is directed toward the discovery of a few theorems which seem important to the teacher.
- Assumptions leading to theorems that are relatively unimportant are suggested in mimeographed material which is available to all pupils but not required of any.
- The major emphasis is not on the statement proved, but rather on the method of proof.
- The extent to which pupils profit from the guidance of the teacher varies with the pupil and the supervised study periods are particularly helpful in making it possible to care for these variations. In addition individual conferences are planned when advisable.
Fawcett was also convinced that the essential elements of the course should be built around the pupils understanding of definitions and assumptions, with a full understanding of the relationship between undefined terms and defined terms. Furthermore, he did not want the activities to be limited to levels 1 or 2 as described by Kilpatrick. Therefore, he had the following assumptions within the course:
- With respect to the undefined terms,
- The terms that were to remain undefined were selected and accepted by the pupils as clear and unambiguous.
- No attempt was made to reduce the number of undefined terms to a minimum.
- With respect to the definitions.
- The need for each definitions was recognized by the pupils through discussion. Definitions were an outgrowth of the work rather than the basis for it.
- Definitions were made by the pupils. Loose and ambiguous statements were refined and improved by criticisms and suggestions until they were tentatively accepted by all pupils.
- With respect to the assumptions.
- Propositions which seemed obvious to the pupils were accepted as assumptions when needed.
- These assumptions were made explicit by the pupils and were considered by them as a product of their own thinking.
- No attempt was made to reduce the number of assumptions to a minimum.
- The detection of implicit or tacit assumptions was encouraged and recognized as important.
- The pupils recognized that, at best, the formal list of assumptions is incomplete.
What an interesting concept of a geometry course. Could it work? When I first began this investigation I knew little about the school itself. I now realize that the success of the course was tied to the school environment itself. The guinea pigs had been at the University School for about three years and used to open ended investigations. The school was truly "student centered" so when Fawcett began the course, he had students who were different than those we see today.
On the first day of class Fawcett's comments were; (OVERHEAD 24) There is no great hurry about beginning our regular work in geometry and since the problem of awards is one which is soon to be considered by the entire school body I suggest we give some preliminary consideration to the proposition that "awards should be granted for outstanding achievement in the school."
Following that introduction, he had the student try to determine what a "school" was, so the Board could decide whether to give, say, someone who wins a mathematics competition an award. Was this "outstanding achievement in the school?" Here were the students initial self constructed definitions of "school." (OVERHEAD 24 Bottom)
12 considered "school" as a "building" set aside for certain purposes.
10 considered "school" as a "place for learning things."
3 considered "school" as "any experience from which one learns."
Following that conversation, other controversial definitions were discussed. Here are some definitional issues raised on that first day:
- Is the librarian a teacher?
- What is 100% American?
- How do I know if I am tardy?
- What is a safety in football, or a foul ball in baseball?
- What is the labor class?
- What is an obscene book?
They then went on to examine how a definition can influence a logical argument. Here's an example of how they approached the definition of a restaurant.
(OVERHEAD 25: White castle)
This was the first day of a geometry class! As you can see, his approach was not to teach geometry specifically, but to focus on elements of critical thinking. But the course was still a geometry course. Throughout the year, the pupils discussed geometry, creating their own undefined terms, definitions, assumptions and theorems. In all he lists 23 undefined terms, 91 definitions, and 109 assumptions/theorems. The difference between an assumption and a theorem is whether it was proved. Briefly, let me tell you a few of the undefined terms which the pupils understood, but were unable to define. For example, as a class they couldn't come up with "the union of two rays with a common endpoint" as the definition of an angle, so they left it as undefined. Nor could they define horizontal or vertical, or area or volume. Yet they went on to define terms like dihedral angle, and the measure of a dihedral angle-which I assume involved having rays perpendicular to the common edge. Although the coverage was probably not as thorough as a standard textbook, they proved some rather tricky theorems. A couple of these were, "If two chords intersect, product of the segments of one chord equals the product of the segments of the other chord," and the theorem that the altitude to the hypotenuse in a right triangle is equal to square root of the product of the two segments. How receptive to geometry was the class when Fawcett began? In some ways they were, remember this was the University school (open, progressive, no curriculum). But he didn't begin with a highly positive group of kids. When the year began he interviewed the students and recorded their preconceived notions of what they expected. Here is a table of their comments.
(OVERHEAD 26: Table of initial student comments.)
When the course was completed, Fawcett interviewed students, parents, observers -- and according to the guinea pigs, there were observers in the class all the time, and there are lots of comments recorded in the yearbook-most of which are very positive. Here are a few of their parents' comments.
(OVERHEAD 27: Table of parents comments.)
In general, we can say that the course was a success. From my interviews I found that the type of thinking emphasized in The Nature of Proof class seemed to carry over to the pupils lives. When I visited with Warren Mathews, who you might recall had a Ph.D. in physics and had been a vice president of Hughes Aircraft Corp. I asked him what he got from the course. His comments were, (OVERHEAD 28: Quote from Warren)
"I remember all our work with definitions. When I was a vice president at Hughes, and now in my work with my church, I realize how important definitions are. It is amazing that when we can agree on our definitions most of the conflict ends."-Warren Mathews
I thought about this. In the field of education we probably argue at cross purposes more because we don't have the same definitions in mind. For example, I don't think I have ever heard someone argue for lower standards, or lower quality in schools. However, we usually cannot agree upon what constitutes "high standards" or "quality." At one time when I was a school board member I recall some parents coming to a meeting chastising the Board for allowing "incompetent teacher" into our schools. (Obviously we didn't have high enough "standards," or we permitted teachers to have low "quality" performance). What was the criterion for their making a judgment about the quality of this teacher? This was a teacher who had given the students assignments which contained several spelling errors. (By the way, this was before "spell checkers.") Spelling was something some parents perceived as a way of evaluating teachers, and her poor spelling meant poor quality. But let's look at a different set of criteria.
This teacher was an art and photography teacher at the junior high level, and her work in teaching children photography was superb. She taught the kids the technical skills of taking photographs and developing their own. The composition of the photography was excellent, and the kids' photos were on display throughout the community. They understood the principles of enlarging, improving resolution, and even some of the chemical reactions which brought about the images during the developing process. Furthermore, the kids tended to love her as a teacher. My own children really enjoyed the course, and I must admit probably preferred this course to their mathematics class. From a curricular, pedagogical perspective this teacher was excellent -- but her spelling skills were awful. (In fact some of her spelling errors were fairly blatant.) However some of the parents differed with this view of what constitute "high" quality in teachers. What was needed was a dose of the "Nature of Proof" thinking.
In addition to many, many comments such as these from the guinea pig, there were some qualitative results from Fawcett's study. He considered the affective and cognitive benefits of the course. Because he was concerned about the transfer of logical reasoning in geometry to non-mathematical content, Fawcett gave pre- and post-tests of critical thinking to four classes, The Nature of Proof class, another geometry class at the University School, which used a traditional curriculum, and two classes from another school. The questions were based on advertisements, political platform statements, editorials and other such written information. The students were then asked to reason logically about the material. First, let me show you one of the problems given to the students.
George came home early from glee club practice at school and picked up the newspaper. In one of the advertisement he saw a picture of Bing Crosby and a package of Old Gold cigarettes. Beneath the picture was the statement,
"My Throat is My Fortune ... That's Why I Smoke Old Golds," says Bing Crosby.
What fact would have to be proved before this advertisement would influence you to smoke Old Golds? List these facts in the space below. I think you can see he is asking them to list the assumptions. Here are the results of the study: (OVERHEAD 30: Results of Logical Reasoning Test-Tab. 10, and Predictions for B-Tab.11)
Notice that only the University School classes improved from the pretest to the posttest. Isn't it interesting that both of the classes from the other school actually declined in critical thinking following a year in a geometry course. Fawcett was also concerned that the students in the Nature of Proof class may not have covered sufficient geometry content to show comparable levels of achievement with regular geometry classes. He wrote,
"While the control of geometric subject matter was not one of the major purposes to be accomplished by the pupils in Class A (The Nature of Proof class), nevertheless it seemed desirable to compare their achievement in this respect with that of pupils who had followed the usual course in geometry. The April, 1936, Ohio Every Pupil Test in plane geometry was used for this purpose. The highest score was 80, and the scores of 2,772 pupils who took the test ranged from 2.0 to 79.0, while the scores of the pupils in Class A ranged from 15.0 to 79.0. the median score of the pupils throughout the state was 36.5 while the median score of the pupils in Class A was 52, this score falling between the 80th and 90th percentiles of the state scores."
Apparently, not only did the students in The Nature of Proof course learn to reason well, and to transfer that understanding to a non-mathematical context, they also learned as much geometry content as any of their peers. In my opinion the course was a huge success.
What surprised me when I first began this study was that none of the guinea pigs knew they may have been part of Fawcett's research. None had ever seen the 13th yearbook, so I asked the NCTM to give me copies to distribute, and I have given these to the guinea pigs. They seem to be surprised that he may have used them as real guinea pigs. None of them could identify their own comments or their parent's comments in Fawcett's book. Actually, they were probably not the subjects in the Nature of Proof Study. More likely it was the class which preceded theirs. However, I don't think it makes that much of a difference, because the reactions of the guinea pigs were as positive as those reported in The Nature of Proof yearbook -- and I suspect their parents were as pleased as the parents reported in the yearbook.
Fawcett continued to teach at the University School until around 1946, (I believe he was not full time at the University School after 1940) because he was also a professor at OSU. Many of his colleagues were at one time students of his. In talking with several of them, the "person and character of the man," was evident. Eugene Smith, who served as president of NCTM was one of Fawcett's students. Gene said that whenever he went to see Fawcett during the writing of his dissertation he usually went in about "one inch high, but came out a mile high." Fawcett had the ability to build up a person's character. My own advisor, Ken Henderson did a study similar to Fawcett's Nature of Proof, and Fawcett was on his thesis committee. Henderson wrote, "Through the study Harold sustained me. ... He was a great guy. Gene Smith knew him better than I did for he spent more time in residence than I was able to do. Gene was one of Harold's favorites. (this may have been due to Gene's teaching the Nature of Proof course for 11 years.) But perhaps he [Fawcett] had no favorites for he was so supportive of all his students."
Fawcett was president of NCTM from 1958-60, in 1961 was named Ohio State's teacher of the year, and in 1988 was named to the OSU Education Hall of Fame. He had many, many publications and leadership roles in mathematics education. He was a superb teacher and scholar. Which leads me to my closing remarks.
When I began this endeavor I realized the course was a good one. In the book, The Guinea Pigs after 20 Years, Margaret Wills reported that almost every one who took the course remembered it 20 years later (OVERHEAD 32: Table from book), and she selected a few comments that the guinea pigs said about the course 20 years later. Here is one of the quotes from a guinea pig about The Nature of Proof course. (OVERHEAD 33:) I can not well remember science courses taught at the University School, and how they were taught. I can think of nothing, except for the wonderful laboratory equipment .... Contrast this reaction with the host of references to the benefits received to The Nature of Proof course.... But, if science had been presented in the manner that Nature of Proof was taught, then it is possible that several of us would have had our lives changed. (Pg. 190)
Willis herself was somewhat surprised by the reaction, she writes; (OVERHEAD 34)
"The fact that after twenty years the responses to Nature of Proof are so favorable is particularly interesting because at that time it was a very radical departure from the traditional way of teaching mathematics." (Margaret Willis, Pg. 189)
When I did get my first written responses from the guinea pigs, they were all positive comments about the course. But when I began I knew nothing about the University school nor of Fawcett, the person. Nor did I know about these graduates from the school. I am not sure where to go from here. Should I simply report about the history of the school, about Fawcett and the course, or about the guinea pigs, or should I be an advocate for another 30 school experiment?
Helen Spencer Lynch was one of the guinea pigs I mentioned before. Helen is about the most prolific readers I have met -- I don't think I know anyone who has read ALL of James Michner's books, each of which is about a zillion pages in length. She continues to send me materials which she feels is related to this investigation. I get copies of editorials from the New York Times or Wall Street Journal, but one time she shared with me a copy of her Swarthmore Alumni Bulletin. In it there was as an article by a professor, Barry Schwartz. He was arguing that grades, SAT scores and other such objective criteria have had an impact on admission to schools like Swarthmore. Although he is advocating a more open ended admission policy, with less emphasis on grades and test scores, part of the article is relevant to reviving other "30 School" type experiments. He states; (OVERHEAD 35)
Is "good enough" good enough for Swarthmore? In many high school classrooms today, experimentation is discouraged because so much is riding on the results. (September 1997, Pg 16)
His argument is that schools like Swarthmore, Harvard, and other such schools, usually accept about 10% of the applicants. He suggests that start with a broader range of possible acceptances, then once a pool is established pick students randomly. He believes that there is not that much difference between those accepted and those who are almost, but not quite "good enough" He states, (OVERHEAD 36)
With a procedure like this, the desperate efforts by high school student to climb to the top on the backs of their classmates could stop. Schools could one again be places for experimentation. Learning could once again be driven by curiosity rather than competition. ( Pg. 17)
Helen believes that there may be a place for another "30 School Experiment" to allow schools like Swarthmore to admit a few students who might attend high schools which have no grades, no fixed curriculum and can be admitted on the basis of counselors recommendations. I was fascinated that this 78 year old guinea pig was able to read an article about a more liberal admission policy at Swarthmore, and connect it to her experiences at the University school some 60 years ago.
Is it possible in 2001 to recreate this experiment? Are there any Fawcetts around to teach the students to reason critically? I don't know the answer, but it might be worth the effort. As well as I can tell each of the guinea pigs that went through the experiment 60 years ago will serve as testimony that it can't hurt. Their minds are filled with positive memories, and they certainly were not limited academically.
My last remark is a quote from a guinea pig, Elaine Bucher Lyons. (OVERHEAD 36 : Elaine Quote)
"It was the depression, my parents were divorced, I had much less money than most of the other students at the University School, but I can say without a doubt, it was the happiest time of my life."
How many of us can say that about our high school experiences? Thank you for listening. I have only shared a glimpse into a much larger story. I hope you enjoyed it.
- Necessary and Sufficient
(An attempt to draw conclusions from a remarkable experiment of more than half a century ago and some more recent ideas.)
- The Nature of Proof
(A report on the above experiment.)
- Simson Line
(A sequence of nice geometric facts with the word define emphasized. Just imagine what would happen if we did not agree on the definitions or did not use them altogether.)
Copyright © 1996-2018 Alexander Bogomolny