Everybody's mountain; 1; The University of Illinois School Mathematics Program

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 black blood 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 on one of those mountaintops the University of Illinois school mathematics program. Negative 1 times negative one time. Negative one might approach. I disagree with Dan I believe in a negative light negative one but see the negative one time how much is this negative one times negative one. Janice.

And positive one times negative one is right the University of Illinois High School in our bene harks back architecturally to the stone clusters of the Middle Ages. For Professor Max b been in seventh grade as However the architecture of twentieth century mathematics was taking shape swiftly at a level of perception ordinarily not exhibited if at all until college years. I'm getting tired of writing all day so I just talk about OK now what if I had six of these negative ones all strung out with time science between them to come out to be a very positive one. Well great at that. Really I do. OK suppose I had 21 of them. Twenty one negative ones all strung out with time signs that they couldn't find the negative one. Suppose I had two hundred fifteen negative ones all strung out with time science but that work out to be negative why negative Why suppose I had one thousand four hundred and twenty

six of those negative ones all strung out to five positive ones. OK in junior highs elsewhere in Illinois and in the other states seventh graders might be continuing the arithmetic to fourth place multiplies rounding off decimal fractions reviewing addition and subtraction and Roman numerals. A teacher might be telling them that five multiplied by three cannot be written as five to the third power. Here at the University of Illinois school mathematics program they were about to be irritated to discover a rule for themselves. Now what I want to write down 1400 twenty six of these make it once. How long would it take about ten minutes to take back. That's too long. Can anybody suggest a short way in which we can like this battery. The number you said times a negative one. Let's see one thousand four hundred and twenty six times negative one. Like this. Yes

this looks like a multiplication problem which I'm multiplying 1426 by negative 1. They might somebody might even think that was positive one thousand point twenty six times negative Why would that work out to be positive or negative one thousand four hundred twenty six all so that wouldn't work. What I'm looking for is a quick way of expressing this idea where you could write one hundred one thousand four hundred twenty six negative ones times negative one to see negative 1 like that. Well that might do it. OK let me tell you now what what mathematicians actually do with this. They take the symbol negative one but still put some parentheses around it and then write up in the corner. There I won four to six when one looks at bat.

You know he's looking at an abbreviation for negative one times negative one times negative one type of negative one all the way out. OK that's the mediation. Now tell me what they did and won two of the five thousand eight hundred fifty five brutal negative line they could have won. How about negative 1 to the 4 8 7 times negative 1 to the 5 21. Think carefully about possible probably how many think a positive one. OK that's pretty good. That's a good one. Professor Baden is a former math teacher at the campus high school. He is now the director of the University of Illinois project for the improvement of school in mathematics. Professor Beeman What were you up to with this class of seventh graders. Well I was trying to illustrate one of our typical teaching methods that in which we make the student

aware of the need for a new symbol. If we want to get the students to use the exponent symbol like six of The Fourth Power eight to the tenth power what we would do traditionally would be to say that if you have an expression like six times six times six times six you could write it in a shorter way by writing a six with a little four in the upper right position. The student's natural reaction to such a method of presentation is who cares. Sol So what so you can abbreviate it. What we were trying to do in this class was first to get the student involved in problems in which he had to make use of these expressions. And so we gave him fantastic examples like. Multiply may get it 1 times negative 1 times negative one for one thousand four hundred twenty six of these now will quite clearly get the answers to these problems though when faced with the problem of actually writing such an expression on paper on the blackboard he would rebel. And so it's quite

natural to say Well is there a shorter way of writing such an expression. Then they're quite willing to work with the abbreviated symbol. Moreover they feel the symbol makes sense because it's directly connected with the kind of problem they have just been involved with. They don't ask the question why would anybody do this. They know the answer to it because they've just finished doing something with expressions of this type. A corner of our banners University High School overlooks West Springfield Avenue. Number Twelve hundred eight is a few steps down the avenue a small three story frame dwelling which houses Professor Bieber and his staff of 15 people who comprise the mathematics project the physics labs on American campuses are generally new shiny and expensive. For some quaint not too upskilling reason. The breakthrough areas in educational research seem always to be emblazoned in the university's most dilapidated architecture. The professor's office is up a ship galley staircase. Mountains of mimeographed text materials loom up everywhere in the groaning cubicles

but the spirit of mathematics is nonetheless triumphant. And by the sound of workmen in the street below heard through the open window. Professor be them and this project was born in 1051. What were its objectives. Well our primary objective in the early days was to. We are going to EIS reshuffle the traditional content of my school mathematics and try to put it into a more economical package as far as time is concerned. The professor is a young man who likes to get to work at 3 o'clock in the morning. He related how the project got its start with the horrendous realisation of the line I engineering profs that the students coming to them from the state's high schools were ill prepared in mathematics. A committee of educators mathematicians and engineers was formed the committee discovered that high school math was so compartmentalize that students generally forgot what they had learned in the early grades and wasted precious time in the higher grades reviewing what they had forgotten. The committee agreed to demolish the partition separating Algebra Geometry

Trigonometry and calculus. They decided to try teaching the subject in a half dozen basic threads to preserve the essential unity of mathematics. No textbooks were extent so the project wrote its own text to a road tested found successful in classrooms and then the project ran into its first major roadblock. What was it Professor Beeman And what did you do about it. Three years after we had to do this thing we discovered that the students were just not exhibiting the kind of basic understanding of say the use of variables that we expected them to have. What did you suppose was wrong. Well I suppose we had hit upon one of the things that was wrong with with conventional mathematics that the imprecision of of language the kinds of incorrect descriptions that are given in textbooks and we were guilty of the same of the same failing. We went to Professor von on our campus. Is what is called a foundation is a specialist in mathematics and he seemed to

have some kinds of answers that we thought were workable. Professor Herbert even on his own a just a philosopher of mathematics. His displacement from the recondite towers of blue sky theory to the blue jean classrooms of high school freshman still wet behind the ears with the identification and drawing of polygons was as radical an innovation in Upper academia as it was a welcome one. Professor Ron you have been teaching mathematics students at the University of Illinois for 20 years. What were your impressions of the students. Well too many of them. Were very poorly prepared. They. Made. Elementary mistakes. They didn't know how to apply mathematics to new situations. They were afraid of mathematics. The rather typical situation I think over the country. One of the reasons for this failure. One answer is that the students. Just have no way of knowing what mathematics is about. From reading conventional texts there is no

distinction made between when you are talking about the symbols that are used when you're using them. The student has no way of telling what it's about to distinguish for example between when you are talking about numbers and when you are talking about the symbols that you use to name numbers. I had no idea that there was a distinction between a number and a numeral. Is this commonly expressed in textbooks for mathematics. No not a top. Nowadays the distinction is completely overlooked in the student is quite at a loss. One writes a numeral four on the board and then writes another expression three in a plus sign in numeral 1. These things that he's writing are merely names names for a certain thing a number. If these are called Numbers students naturally confused here are two numbers of the same number. What does this mean. Let me go into law instead and we lose a good mathematician. Yes it's the brighter students actually who are most confused by this sort of thing.

Bright students mystified by the esoteric ritual of conventionally taught math. Afraid I'm prepared repellant. Our young minds unable to breathe in the stratosphere of abstractions. Not at all asserts professedly even when you and I were 13 it goes on to say we were enchanted by the mysteries of life. We weren't concerned with practicalities. The more abstract a question was the hotter We argued about it. Angel's absolutely delighted us. How many of them could dance on the head of a pin numbers the professor continues are as abstract as angels. Mathematics is as creative as music painting or sculpture. The high school freshman will revel in it if we let him play with it as an obstruction. But insisting that he pin numbers down is like asking him to catch a butterfly to explain the sheen on its wings. The magical glint of the sun rubs off on his fingers and the fluttering thing in his hands can never lift into the air again to renew his wonder. Prue and Barry were two seventh graders who were chasing mathematical

butterflies with Professor B the man and his number thoughts pro you came here from another junior high. Did you take arithmetic there. Yes we did. It was alright but it was kind of easy. What about you harder and you have to challenge it to class before you didn't have to make a challenge on it. Well this is a challenge to make it more interesting for you to prove what were your impressions when you came to this course. I found myself talking different ways. Well I thought it was just a list. Think about the problem for you just like. Yes well girls aren't supposed to be interested in mathematics but about the most girls don't know that this kind of mathematics tell us about a boy not in his class to think of mathematics taught in this way. They get more interest in it if they could learn the. First few

pages and see how it works I think. Boy the proof if you want to describe what you learned by this method of teaching math what would you say. I think I've learned a lot so far I have to spank. I could you know just everybody speak up at school and you'd more competition you know what you're doing. Heidi there are 13 units in the Abana program they begin with the arithmetic of real numbers and pro numerals with generalizations and algebraic manipulation. They range three questions in equations ordered pairs complex numbers circular functions postulation all systems and analytic geometry. That teachers are not drill sergeants rules once discovered are not required to be verbalized. Freedom in problem solving techniques is encouraged. So is intuition estimating ounces. Even a wild guess and then adjusting the gas in the light of subsequent verification.

Professor but even that has broken the fun barrier in mathematics 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 had positive 10 to it and I come up 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 81. What number was I think three body three are negative three make up your mind what you eat or drink. You can write a good three. OK. I'm thinking of a number I multiply by itself. And that answer is negative 36. You know I wasn't thinking right.

But mathematics professor Biederman isn't supposed to be fun for children. How do you justify this heresy in mathematical tradition. I think we need a bit of comic relief. The end of the hour. It also gives them just the sheer practice of doing computations and it gives them a feeling of excitement as they leave the mathematics classroom and I think if they come back to the mathematics classroom the next day and disobey the same kind of excitement and stimulation there is a pragmatism in contemporary American education which relates content directly to utility. Of thought uses it the student is supposed to ask when learning something. So conventional math curricula have him figuring family budgets reading radio and TV logs using numbers in installment buying things even investing in Wall Street. Professor B Timmons emphasis is not on the what for but on the why for. Habits of thinking are involved in which the details are forgotten. But a mental discipline is retained which can be useful in such advanced modern math techniques as public opinion sampling

programming of high speed computers or calculating the orbit of a manmade moon from the position of the beeps in the sky. To youngsters outside Illinois I respond in the same fashion. Or is this merely a Midwestern mathematics miracle Miss Eleanor McCoy is a teacher coordinator at the project. She helps observe the program which numbers 12 pilot schools in Illinois and 37 participating schools in 15 other states 1 participating schools in Hawaii. Miss McCall What are some of the things you have been observing about the program. It's very interesting to see the enthusiasm that the students have when they're working with this material. They seem to bubble over with ideas and it's very difficult to keep them quiet and so many conventional classes a teacher has to sort of pry information out of them. And it's. Rather dull and dry. And in these classes there is much more response from the students and I have observed that and then I've had other teachers make that comment themselves how do this reflect their interest in homework. It makes their homework. A

real challenge and something that they want to do. In fact I visited one of the pilot school teachers just this weekend and she said that she has a freshman boy who is a son of a veterinarian and his father had said. You know my son never did want to do is homework before in fact it worried me because he never brought any home and now he brings this math home he does at the very first thing he just can't wait to work on that math course he likes it so well. You were trained in the conventional way as a high school math teacher in a liberal arts college. What were your impressions when you were first exposed to the project here and about it. Well it seemed very strange and very different to me and I'll be frank to say that I was quite skeptical at first as to whether some of the ideas could be taught and whether some of the ideas which they expected to teach would be taught in the way they were presenting them because I had thought from a conventional book for a long time and I just thought that it couldn't be done the way they said it could. The converts a minute. For one hundred twenty five high schools and one hundred twenty eight colleges have purchased individual sets of the project's mimeographed teaching materials. There many are because

they keep changing all the time in response to classroom testing. Even adults have caught the contagion. Employees of the Polaroid corporation in Cambridge Massachusetts have constituted themselves a participating school. All teachers who participate must be especially trained. I watched an in-service training class of some 30 math teachers from all over the United States. They took the same test as professor of the women's ninth graders. They broke even in the results. I listened as the teachers wrestled with the concept of sets and probability. They made subtle distinctions between the word order and the word end in mathematical verbalizing. One teacher looked at a formulation on the blackboard and asked. Is it a verb or a noun or a sentence. Another answered. It's just the name of something a description of a situation. I felt like an aboriginal at a count down at Cape Canaveral. My high school mathematics stopped with Euclid. I remember vaguely that it was a model to prove a theory in any other way than by dividing the page into two

columns and placing the statements in one column and the reasons in another. Well we feel it as one of the democracies that day. We've got to make mathematics hit the spot for our youngsters. We feel that curriculum changes are inevitable and we feel that a number of the answers here are ready made. You may have placed the accent. It's Australia. Alastair McMullan teaches math the Teachers College in Sydney. The problem of developing enough qualified mathematicians for the Space Age is a severe down under as it is everywhere. Mr McMullan is spending a year with the Illinois project. Then he will return to Australia. An apostle of mathematical reform Mr McMullan How do you teach math in Australia today. Well we've got a lot of pressure on the youngsters academically and we also teach in rather a formal way and what is the effect of its method of teaching on your youngsters. Well some very bright children stay with mathematics more or less despite the system and I think other very good mathematicians are probably lost as a result of this system.

Well you've been over here one month but you've actually been teaching already. What are your observations on your experience of teaching. Well I've been very surprised to see the spontaneous enjoyment in the youngsters response. Is this good for girls as well as boys. Well as far as I can see girls seem to enjoy mathematics at least as much as the boys do. And when you have them in the 12th grade do you have you been able to observe any progress. While I've been extremely surprised to see the creative attitude of these youngsters in 12th grade have to watch mathematics they will attack any problem on its own merits rather than looking for some pattern that they have had before. This sort of mathematics is intrinsically interesting to children it hits the spot. They don't need any added visual motivation from the adult world they don't need to be shown that this process is necessary because it is useful in the adult world because I feel that they will say well now this is a challenge to me intellectually. This is fine and I'm quite sure that with we won't need anything beyond that I think it will.

Getting all those people whom we have we're losing kind of the conventional system the notion of cartesian products is a fairly contemporary development in mathematics. It requires a good deal of sophistication in dealing with abstract matters. Listen now to a class of ninth graders freshman. Only one year removed in growth from the positive and negative contingent you heard at the start of the broadcast. OK let's say we start out with the set that's different from the set B and let's say there's an element in B which is not a. OK now show me why the cartesian product of A on B is different from the cartesian product of B on the way. Let's talk about that element in B which is not in a class with elements in B but it's not. That means that element of the first component of an ordered pair and if it's not in a Y then anyone have it to be a first component therefore you will have

ordered pairs which are therefore the two sets of ordered pairs will be different. OK if you got these exactly to you. Well not quite how she explained it. How would you explain if you got another element in set B. It will only be used as. Component in the ordered pairs it won't be used as a first component so that you couldn't use it instead. Therefore you can't change it just because it wouldn't be the same ordered pair. Let's see if suppose I have this element say let's give it a name. That's a that's a nice name. 7 7 because I have 7 which is in B but not e and you know that means in a cross B I'm going to have an ordered pair with a second component is 7 7 in B Cross they are going to have an ordered pair whose first I'm called on to 77. But you wouldn't have A's first component 7 you would not want a crosspiece 7 isn't an 8. OK anybody else like to talk

about why Cartesian multiplication is not communicative. Anyone. It began you remember with the inadequate preparation of high school students for the engineering class. Professor Vaughn do you expect that the graduates of the 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. Well 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. And I think it's also important because the. More kinds of aesthetic experience that a person is able to

perceive. The better the life. They are about a revolution has been handsomely fired by a grant from the Carnegie Corporation in New York. Your experiment Professor Beaven then will eventually come to an end here in Illinois. What do you expect the universal fruits will be. I suppose so will the university go and I will publish the textbooks that we have been developing these past years and make them available to the profession. Is yours the only group in the country reshaping math teaching in the high schools. Unfortunately we have some company in this. The Commission on mathematics which represents the College Entrance Examination Board is in the process of issuing a report which will be a recommended syllabus for students preparing for for college. There's another group which was started last summer at Yale University called the school mathematics Study Group. This group is just getting underway and within the next two or three years will bring forth a series of what they call prototype textbooks the adoption of these

prototype texts by American high schools and the training of teachers who will be able to use them will constitute an important challenge to the educators. At our banner they are already extending the method into the primary grades kindergarten not accepted. What about the public profess to be them in the great mass of adults for whom mathematics is always meant sudden intellectual death. We need to have a new attitude on the part of of the lay population that it is no longer respectable to admit to illiteracy in mathematics that the time has passed when a person can say with a certain amount of pride that he can no longer add three and five will never know how to add three in five. I don't think this is a good carry over for students to recognize that parents are proud of this fact but generally speaking how shall a layman think about mathematics in the new age. They they have to regard it as as a subject which is constantly growing. They will find their 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 and taught mathematics professor Bateman a final personal question. Why are you interested in mathematics. Well I think it's fun. The origin of the University of Illinois Mathematics program was a failure in the state's educational machine. Educators mathematicians and engineers began to ask questions and the questioning led to the provision of funds for experimentation. It led also to the uncovering of the university's own doorstep of the creative talent required to staff and to direct the program. How actually is mathematics being taught in your community. Are your children being challenge stirred awake into the exciting is static vistas in a traditionally double subject.

The answer will not be found until responsible people begin asking questions. The stink is not only survival for our way of life in a perilous time of scientific and political change. It is an important aspect of the very quality of that life itself. If we are not concerned with the release of our children's imaginative powers then our children's imaginative powers will never be released for us. Education is your mountain. Your children's Mountain your neighbor's mountain. Education is everybody's mountain. The 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 NEA E.B. Radio Network.