Click here for home, brianrude.com
Experimentation is held in high regard in science, as indeed it should be. However not all experiments are equally valuable. If an experiment is based on misconceptions, or is loosely controlled, the results may add more confusion than clarity. There is a great deal written about education, some of it professing to be rigorously scientific. But the general state of education as a science seems to be very low. Students of education may be given ideas, and they may adopt those ideas with considerable zeal, but those ideas don't seem to get much respect in many situations. Any parent whose child is having trouble in some subject feels justified in dismissing as an educational fad the ideas that seem to be the source of the child's trouble. The "math wars" of the last few decades offer evidence of this. Indeed educational ideas seem not to have advanced much in the last century. "Constructivism" as a case in point, seems to be primarily a new term for a set of ideas that gained a following seventy or eighty years ago, and at that time went under the name of "progressivism".
Many educational ideas claim to be based on research, which implies some sort of scientific experimentation. I am not very familiar with educational research, but at times in the past I have made a concerted effort to see what is there. I might find a study, perhaps on the internet or in an educational journal, and try to give it enough attention and effort to understand it. The study may claim to use scientific methods. There are control subjects as well as experimental subjects, and detailed statistical analysis of the results is done. But for all of this effort on the part of the researchers I have never found anything memorable, anything that captures my imagination, anything that really seems to establish some fact, idea, or perspective. Everything I find seems eminently forgettable.
One reason for this sad state of affairs, in my opinion, is that the field of education has skipped a very important first step, and without that first step not much can be expected. Without that first step experiments, no matter how rigorous, will be based on misconceptions and faulty premises. My argument is that any science must start with simple accurate description of reality, of what actually is, not what should be. For the past century the influential education writing does not seem to be this at all. Rather the most influential writing is usually about what should be.
One very influential essay in the field of education was "The project Method" written by William H. Kirkpatrick in 1918. It seems entirely fanciful, as I think a few quotes will show. His central point is that it is better if a child acts in accordance to his own purposes rather than being coerced by the teacher. When a person has a purpose in mind, that purpose will not only be the link that connects together a long series of actions, but also, since those actions are meaningful to the person, will facilitate learning, understanding, and retention. He gives the following example:
"Suppose as extreme cases two boys making kites, the one with wholeheartedness of purpose as we have just described, the other under direct compulsion as a most unwelcome task."He goes on to assert that the actions will be quite different for the two boys, since one is acting on his own but the other is acting under the direction of the teacher. For the first boy, according to Kilpatrick:
"The unified set of wholeheartedness will render available all the pertinent connected inner resources. A wealth of marginal responses will be ready to come forward at every opportunity. Thoughts will be turned over and over, and each step will be connected in many ways with other experiences. Alluring leads in various allied directions will open before the boy, which only the dominant present purpose could suffice to postpone. The element of satisfaction will attend connections seen, so that the complex of allied thinking will the longer remain as a mental possession."But for the other boy, according to Kilpatrick, things are different:
"All of this is exactly not so with the other boy. The forbidden ‘set’ so long as it persists will pretty effectually quench the glow of thought. Unreadiness will rather characterize his attitude. Responses accessory to the work at hand will be few in number, and the few that come will lack the element of satisfaction to fix them." "Where the one boy has a wealth of accessory ideas, the other has poverty. What abides with the one, is fleeting with the other. Even more pronounced is the difference in the by-products or concomitants from these contrasted activities. The one boy looks upon his school activity with joy and confidence and plans yet other projects; the other counts his school a bore and begins to look elsewhere for the expression there denied. To the one the teacher is a friend and comrade; to the other, a taskmaster and enemy. The one easily feels himself on the side of the school and other social agencies, the other with equal ease considers them all instruments of suppression. 1Kilpatrick presents ideas, and he makes arguments for these ideas. But he makes no attempt to describe any particular bit of reality. He has no anecdotes, no history, no data, no examples from his experience or others'. He paints pictures, and these pictures are believable. But there are many opposing pictures which could equally well be painted, and which are equally believable. One can indeed imagine a boy making a kite and diligently working out the problems that arise One can imagine ultimate success for the boy and his satisfaction of accomplishment. But one can equally imagine the boy getting discouraged and losing interest, or being distracted, or deciding that it's a lot easier to buy a kite than to build one, or badgering his dad to buy him a kite, or simply losing interest and moving on to something else. Yet the whole argument for the project method depends on the reader imagining things the way Kilpatrick imagines them.
Here is some similar imaginings from much more recent times. The following is from an article I found on the internet. I don't think it's an important article. Rather I present it as a typical article, one among many that I wade through while browsing the internet, perhaps exceptional only in the fancifulness of its imaginings.
. . . . . . . The theme, or educational philosophy, underlying our approach is one we call ‘communal constructivism’, by which we mean an approach to learning in which students not only construct their own knowledge (constructivism) as a result of interacting with their environment (social constructivism), but are also actively engaged in the process of constructing knowledge for their learning community.People who want to become researchers in a field, genuine researchers, typically get a Ph.D. degree in that field. Yet according to these authors, apparently, that is unnecessary. Any kid can do it. But how, one may ask, are all these advances in a field to be recorded and learned by others, if every boy and girl is doing original research? Such thoughts do not seem to bother the authors. Like Kilpatrick in the first example, they put forth ideas and argue for these ideas, but it all seems far removed from reality.
. . . . . . . In this model students will not simply pass through a course like water through a pipe but instead, riverlike, leave their own imprint in the development of the course, their school or university, and ideally the discipline. This will result in a gain for the institutions or course, but more importantly the students themselves will benefit.
We argue that a diverse range of techniques can, and should, be used to enrich this type of learning environment within which the focus is on learning with and for others. Peer tutoring and project-based learning are obvious techniques but we also advocate the ideas of cognitive apprenticeship, the publishing of information, flexibility in the time table, a radical look at the way in which assessment is done, and so forth.
. . . . . One of the most interesting effects observed in the first year of the course was, therefore, the extent to which students on the course began to replicate the learning environment they themselves were part of in the classes that they in turn were teaching. The most dramatic example was where one of the authors, herself a primary teacher, employed both peer- tutoring and Web-based publication with her class of 10-year olds. The class were taught how to use a simple story-authoring package and gained experience in its use. They then were given the task of going into the classes of 9-year-olds in the school and teaching not just the students how to use the package but also the teachers of the other classes. The whole process was documented on a Web page that the students took great delight in showing their parents. The teacher in question would have described herself as barely IT literate at the outset of the course and what is interesting is to see how she felt empowered to apply her own experiences from the Masters course to her very different classroom situation.2
And here is yet a third example of what we might consider as description. This is the "vision statement" of the 2000 "Standards" of the NCTM. I used this same quote as a springboard for discussion in my article "Some Disagreements With The Standards".
"They draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers.” 3This is about as close to a description as there is in the standards. But again it is a description of what might be, not what is. It is not hard to envision what is described, though I would argue that it is not typical of any real group of students. But one can envision other scenarios equally as well. One can envision students being frustrated because their teacher expects them to act like students in this vision statement and therefore gives little effective instruction. And, in a totally contrary scenario, one can envision students being enthusiastic because they are accomplishing substantial learning in a traditional classroom, and it makes them feel good. If we are only imagining then we can imagine whatever we want.
Of course it can be argued that none of the three examples I have given are meant to be descriptions of reality, and therefore not subject to criticism as being poor descriptions. A presentation and development of ideas does not have to include description. Indeed I do this all the time. Without describing anything real I present ideas, and put ideas together in ways that I think are logical and beneficial. Sometimes no examples are needed. Sometimes an imaginary example is best to make a particular point. However I try to bring in real examples as much as possible. Other things being equal, a real example will carry more weight than an imaginary one. My point in this article is that plain, simple, accurate, and extensive description is rare in educational literature, but is needed in any science.
Now I will present a few examples of what I consider genuine description of classroom situations. Each example I give can certainly be subject to many criticisms. To say that an example is a good description is not to say that it is accurate in every detail, or that it is not slanted by one agenda or another, or that it is typical. But these descriptions, in my opinion, can be called actual descriptions, not idealistic imaginings.
My next two examples come from the book The Schools, by Martin Mayer. This is a rather old book. I picked it up in the early 60's. At the time I thought it was a good book, but then over the years I pretty well forgot what was in it. A few years ago, for some reason or another, decided to look at it again. After rereading the entire book over a period of a few weeks I realized that what I like about it is its descriptions. Here are a few:
From the classroom:
At the Sir James Baine School, in Battersea, one of the worst of the London slums. It is a “Junior school,” which takes children from age seven to age eleven; an “infant school” in another wing of the same building cares for children from age five to age seven. Altogether, there are some four hundred children in the two wings. The school was built in 1953 and is as attractive as an essentially prefabricated structure can be; its bright colors have held up moderately well under the Battersea smog. It stands in the middle of an island of grass — real green grass, between railroad tracks and factories, in the heart of Battersea. There is also a school garden, used in the science program and just for recreation; and among the plants in it is corn, put down a few years ago when Headmaster C. J. R. Grubb (B.Sc.) found after a Harvest Festival that only three of the three hundred children in his school had ever seen an ear of corn. The children made their own sandpit in the playing area for broad-jumping and high-jumping.
Mr. Grubb is “taking assembly.” A thin young man with long fair hair, wearing a tweed jacket and a big if not always easy smile, he sits in solitary grandeur, his legs crossed, on the stage of the small school gymnasium-auditorium. Before him the junior school is assembled, the younger children cross-legged on the floor, the eleven-year olds on chairs at the rear. The teachers sit along the walls, removed from their charges; two members of the staff are West Indians, though there are no Negro children in the school. Religious services are over (“council schools” follow an “Agreed Syllabus” of Protestantism) and the children are all intent on Grubb, who has the gift of holding them. He is handing out “oranges” (compliments) and lemons” (complaints) about their conduct. The big orange is congratulations for their behavior the day before, when a troupe of actors had visited the school and put on a show. “They told me you were the best audience they’d ever seen,” Grubb says in that soft North Country accent which is one of the world’s most pleasant ways of speaking English. “I must admit I was very proud of you.
“Now” — with a little wince and a stern look — “I have a rather sour little lemon. I know squirting water is good fun, because I used to do it myself — but I was never surprised when I got spanked. I’m not going to spank anybody. I’m going to leave it up to you. We can stop this squirting water. Do you know how?”
A few hands are raised.
A nine-year old boy says, from his place on the floor, “Turn down the taps.”
“Turn . . .” says Grubb in a tone of suggestion.
A chorus rises from the floor: “Off!”
“That’s right,” Grubb says, and smiles. “If you don’t want to lose your drink, don’t squirt water — and if you see somebody tempted to squirt water, stop him.”
The halls are full of animals (mostly assorted rodents and birds) in wire-mesh cages, part of a Pet Club which is intensively used to help establish interest in reading, arithmetic and natural science. The school resounds to squeaks and tweets. In the library alcove (there is no separate room for a library), a tall young lady teacher, sitting uncomfortably on a child-sized chair, is working with six eight-year-olds around a table. Remedial reading.
“Dive—drive—driver,” says the teacher, showing the words on cards. “Now, what do they have in common?”
“Dive,” says a girl, troubled.
“Ay vee ee,” says a boy, more confident.
“Chir-rup!” says a bird.
“That’s right. Now, how do you make ‘dive’ from ‘ive’?”
“Add the ‘d,’” says the little boy
“Brrr-eep!” says another bird.
“And drive? . . .”
A chubby eleven-year-old boy stays behind while the others go out for prelunch games, to show an electric motor he made by himself, whittling the ends of a stick, winding the wire around it, emplacing the poles and constructing a switch to the six-volt battery. He explains what he has done in a strong cockney accent.
“That lad will get a grammar school,” Grubb says after the boy runs off to join his companions. “Very few of my children will. In London the standard for a grammar school is higher than it is in the surrounding areas. Most of my children go to a single-sex secondary modern. There is a comprehensive school — but it’s very, very picky and choosy, I’m afraid. They don’t take a fair sample.
“This is a neighborhood to get out of. All the most intelligent parents move out before their children are eleven. But the children here are as good as we are. In ten years, some of them will have more brains—if that’s what we’re talking of—than we have.”
In another classroom, nine-year-olds are singing, following a BBC broadcast with open scores, supplied by the BBC schools service. Some of them can read staff notation, some cannot. Another group of the same age are painting, houses and green fields. Ten-year-olds are writing in blue notebooks, some finishing an arithmetic assignment, others working on a book report or an individual project connected with the school’s forthcoming excursion to the Isle of Wight — which offers Portsmouth Navy Yard, Nelson’s Victory , a trip over the waves, Roman ruins, chalk cliffs. But the children will have to pay part of their own expenses, and only a quarter of them will be able to go.
. . . . .
It is a good school. The children come to it clean and willing, and truancy runs less than 3 per cent. The classrooms are quiet - quieter than almost any American classroom, because as an ordinary matter nobody - especially not the teacher - is talking. The children are learning what they have to learn, with a little over, though their writing will always be better than their reading, which does not get quite to the mark. There is nothing nasty about the place: no rowdyism, no deliberate incivility, little vandalism, few attempts to interfere with the orderly progress of the school day. The children are not beaten down - indeed, the school atmosphere is much freer than they are used to at home. Few American slums have such cheerful elementary schools. Yet there is something terribly shabby about the whole operation, for Battersea is a place without a future; and the children, too, have no future. 4
If one wants to criticize this description, there is indeed much to criticize - and many questions to be asked. What does the author mean “their writing will always be better than their reading”, for example? And this is obviously not meant to be an objective description. The author interjects much of his own opinion, such as the last sentence in the quote. But for all its faults, I would argue, it is still description. The author wrote the book for a general audience, not for professional educators. It is written to inform, but not to establish any general or specific educational principles. There is no pretense that it is a scientific study. But, unlike many educational studies, I think it has value for teachers and prospective teachers. It may be an imperfect representation of reality, but it is much better than the whimsical imaginings of the first three quotes I gave.
Here is another quote from the same book:
At a district junior high school in Denver, an “accelerated” class of thirteen-year-olds is in the middle of the health unit of the Holt eighth-grade science textbook. They are sitting on stools at the counters in the laboratory, arranged in a U, and the young teacher at the front is droning on, following the Holt manual. “Now,” he says, “sunburn. If all of us pick up a hot pot from the stove, all of us will get burned. If we go out into the sun this isn’t so. Some will burn easily,” and so forth. Commercial preparations are mentioned for joke purposes — Golden Glow, Man Tan — and a recommendation follows. “So we just have to use a little common sense. We have a lot of concoctions you can buy and smear all over yourself. If you do get a burn, rub on cold cream, salad oil or shortening. I’d recommend cold cream, because it smells better.”
A girl interrupts. “Doesn’t salad oil have salt in it?”
“Not the salad oil you’d use for this purpose. But remember, don’t use butter or margarine, because they have salt in them. Then there are chemical burns, if you touch fresh see-meat you can get a burn. All right, your first aid for chemical burns is . . . . ”
This explanation completed, the teacher writes on the blackboard the words
“Heat exhaustion,” he says, “is one thing that can occur. The prevention would be not to overdo it in exceedingly warm places. Now, heat stroke. For some physiological reason the body temperature goes up, that’s what’s serious.”
A boy asks, “How come such a small change in temperature — five or six degrees — can do so much damage to the human body?”
The teacher says, “Well, it’s more than that.”
“All right,” the boy says. “Ten degrees, then.”
“We are so designed,” the teacher says, “that we maintain a constant temperature. if you’re out playing softball on a hot day, you’re perspiring. You think you’re very hot, but if you pop a thermometer in your mouth, you’ll find you’re about the same. If temperature goes up for some reason, we’re in trouble.”
Near the visitor, a rather handsome boy mutters to the girl beside him, “What that means is, ‘I don’t know the answer, but I won’t admit it.’” The girl, who wears glasses, looks at the boy adoringly
“Heat cramps,” the teacher says, proceeding industriously — one cannot blame him for the material, he is worse bored than the children — ”heat cramps you’ve never heard of, probably. I’ve never known of it myself. But it’s muscle cramps from being in a hot environment, usually in the limbs or abdominal muscles.” He turns and writes on the board again, “FROSTBITE.” Then he asks the class, “What is frostbite?”
A boy says, “When the tissue is frozen?”
“When part of the tissue is frozen. Remember the tissue is composed of cells, and the cells are full of liquid and that liquid freezes just like water. To prevent frostbite, you keep warm . . . . . . 5
This does not seem to be a picture of good teaching. I think the author considered it mediocre teaching, at best. I think perhaps the author also thinks there is something wrong with health education in general, as I have thought most of my life. But my interest here is not whether it is good teaching or bad teaching. I would argue that as description it succeeds. It seems real. After reading it I can feel that I got a real picture of that situation. I have no way of independently confirming that it is an accurate description, of course, but even if it is inaccurate it opens the door for others to try to describe such things, and try to be more accurate. Then we have something real to work with, not idle imaginings.
The phrase "good description" is not a very definite standard. The next example I will give might not at all be described as good description in that it doesn't give much detail. It is written by a mathematician observing a math class:This article is written for mathematicians, not the general public, and for mathematicians the sentence that begins with "The teacher then called on another student . . ." hits like a freight train. The student hit the nail on the head. The teacher should have seized on that student's statement and not let it go until that central point was understood and appreciated by the whole class.
In 2001 I was invited to observe some of these workshop model classes at a magnet high school in lower Manhattan. In one of the ninth grade classes, the lesson problem of the day was (in mathematical terms) to determine the equation of a line through the origin that does not intersect any additional points on the integer Cartesian lattice in R 2 . The students began the exercise working unsupervised in groups of four. Then the class convened as a whole to discuss their findings with the teacher serving as moderator. The tenth (or so) student to speak observed that if the line were to intersect another lattice point, then it would have a rational slope. The teacher then called on another student, and this key observation was soon lost. The discussion devolved into an unsuccessful effort to understand the difference between rational numbers with finite decimal representations and those with repeating decimal expansions, and the math period ended with no solution to either question. 6
I don't understand this situation, as described. A "magnet school in lower Manhattan" surely would have a teacher who understands the math. Was the situation described accurately? I don't know. Perhaps another observer, with similar mathematical qualifications as the auther of this quote, would see the same situation quite differently, though I cannot imagine how.
It is not at all clear to me why this problem was chosen by the teacher in this example. If the class is just learning about graphing, slope, linear equations, and so on, I can't see that any purpose is served by bringing in number theory. Perhaps graphing of linear equations is well understood and the subject they are studying is number theory. Or was this problem chosen by the teacher? Perhaps he was just stuck with it for some reason. This example is described as a "model class" at a magnet high school. I can well imagine that this problem was chosen for some reason other than being an appropriate part of a coherent curriculum.
But my purpose in this article is to talk about description of teaching, not mathematics or curriculum coherence. This example describes the essential happenings. It is a sparse description, and it does not include much information about the setting. But it very well serves its purpose. I think this description is most meaningful in contrast to how the situation would be described by educators who have an ideological commitment to the teaching methods that are being used here. Such a description, I believe, would sound like the communal constructivism example I gave.
I have not given many examples of descriptions of teaching and learning. Perhaps someone more extensively acquainted with educational literature could find more. Surely there must be more. Of the examples I have given only the two descriptions by Martin Mayer seem to qualify as descriptions of everyday life in the classroom.
How do teachers teach today? What goes on in second grade classrooms? What goes on in high school classrooms? What is the rationale for these actions that actually occur? What went on in one room rural schools in 1910? What went on in big city junior high schools in 1920? Will we ever know? A few years ago in my reading I came across a mention of a book, "How Teachers Taught" by Larry Cuban. This sounded very interesting. I assumed the author would try to construct, from historical sources and whatever else, a picture of teaching practices of earlier times. I assumed, in other words, it would be a book of descriptions of teaching. I assumed it would also be a book of analysis of those descriptions. After a few months, or perhaps years, I got a copy of the book, and was very disappointed that it was not what I had hoped for at all. Rather it was a history of reform attempts. It described, as I recall, the educational fads of different times and recounted the efforts of various big city school systems to implement them. It made no attempt to describe real classroom practices at various times and places in the past.
I have in the past had a few books that give real experiences of first year teachers. These are anecdotes or case histories, but, as I recall, the descriptions were rather brief and shallow, and do not give a very good picture of the larger setting in which these incidents take place.
I have made attempts to describe my own teaching. As a teacher of college freshman math courses in recent years I mostly lectured. I felt, and still feel, that a careful explanation of concepts and related details is the essence of college teaching. The basic framework of a good course, I would argue, is a carefully selected sequence of topics, careful explanation of those topics, a sequence of well choosen homework assignments that are regularly graded and promptly returned to the students, and a set of tests that require the students to demonstrate a certain level of understanding and fluency in those topics. Thus my everyday routine is to spend a few minutes answering questions on the current day's homework, followed by a careful explanation of the current day's topic, and ending with the assignment of the next day's homework. I have elaborated on this basic description in several articles that I have put on my web site. 7 There is nothing dramatic or exciting in these articles, but I tried to be complete enough and accurate enough so that others can get a picture of what I do.
But what do others do? I would expect most math teachers do much as I do. But how can I know? I can remember math classes that I have taken, and these memories mostly confirm that my pattern of teaching is very common. But a student's perspective and a teacher's perspective can be quite different. And memories may be very imperfect and colored in ways that are had to identify and appreciate. I would like to know what other math teachers do by reading their first hand accounts, or by reading accounts written by observers, but if these descriptions exist, they are hard to find..
I think there should be a lot more description in educational literature. However I won't claim that more description will work any magic. I have argued that description is a basic first step in any science, but taking a first step does not guarantee that a second step will be taken. My argument is rather that if the first step is not taken the second step can be expected to be faulty. The quality of educational writing that we observe today is, in my humble opinion, consistent with this expectation.
1. “The Project Method,” Teachers College Record 19 (September 1918): 319–334, available on the internet at http://www.historymatters.gmu.edu/d/4954/
2. Communal Constructivism: Students constructing learning for as well as with others. Bryn Holmes, Brendan Tangney, Ann FitzGibbon, Tim Savage, Siobhan Mehan. https://www.cs.tcd.ie/publications/tech-reports/reports.01/TCD-CS-2001-04.pdf
3. Principles And Standards For School Mathematics, National Council Of Teachers of Mathematics, Chapter 1, A Vision for School Mathematics, nctm.org
4. The Schools, Martin Mayer, 1961, Anchor Books, Doubleday & Company, Inc., page 153
5. The Schools, Martin Mayer, 1961, Anchor Books, Doubleday & Company, Inc., page 307
6. Alan Siegel, Understanding and Misunderstanding the Third International Mathematics and Science Study, http://www.cs.nyu.edu/faculty/siegel/siegel.pdf
7. See “Thoughts On My Teaching At SCSU” and “Thoughts On My Teaching At NDSU” on my website, brianrude.com.