Language and the Learning of Mathematics

We think in terms of words. Yet today anyone who identifies lexical reasoning, based on gradually formalized natural language, as the key to the learning of mathematics, is filing a minority report. Only seven of the 515 sections in this program deal specifically with the importance of reading and writing in the study of school mathematics. The tendency to downplay this important idea has been dominant in mathematics education circles for over 20 years. Certainly there is, in current texts, much less emphasis on linguistic logic and the careful use of language in exposition, than there was in the early sixties when Max Beberman's University of Illinois program and the SMSG programs were flourishing.

A somewhat parallel situation exists in secondary school English. Nowadays, English teachers who seek to clarify meaning and facilitate intelligible communication by insisting on the enforcement of the accepted rules of grammar and syntax are fighting a losing battle and, worse, are patronized and ridiculed for being "uptight". Those of us who insist on the crucial importance of lexical reasoning and proof in the teaching and learning of secondary school mathematics are in the same boat even though this was once the dominant viewpoint.

We live in an age of ambiguity where it is far more profitable to be fluent and verbose than it is to be articulate. At such a time the learning of mathematics, which requires understanding of carefully worded expositions as well as appreciation of carefully drawn distinctions, is bound to suffer, and it is suffering. We are constantly being bombarded with evidence indicating that school mathematics in the United States operates at a level far below those found in most other industrialized nations (2) (3) (4) (5). Judging by the performance levels of the early sixties, school mathematics in the United States is in regression. Perhaps we should reassess some of the highly controversial doctrines which have, in recent years, gained a large measure of uncritical acceptance.

At any given time there are certain widely accepted viewpoints that are difficult to challenge. Unpopular ideas that question this prevailing orthodoxy are silenced or ignored even though their logic is simple and their validity clear.

In the present context the widely accepted viewpoints to which I refer are essentially these: If we emphasize the applications of school mathematics in a wide variety of problem solving situations two good things will happen; (1) the student will somehow learn the mathematics needed to solve these problems, and (2) seeing that mathematics is useful, he will be motivated to learn more mathematics. Surely this is the message of the Agenda for Action's recommendation 1.1 which recommends that the mathematics curriculum be organized, not around its own internal structure, but around problem solving. (1) To be sure this recommendation is carefully hedged and qualified so as to provide the writers with a measure of "plausible deniability". But the overall thrust toward problem solving and applications is unmistakable.

I come out of retirement to challenge both of these viewpoints -- because they run counter to my experience during 46 years in the mathematics classrooms of Illinois, first at the secondary and later at the college level -- and because I love school mathematics and still care about what is happening to it.

According to my experience, students must know the mathematics before they can apply it. Or to say it differently, they cannot apply mathematics they do not know. To expect them to learn mathematics in the process of applying it is preposterous. It is like trying to teach people to play water polo before they know how to swim.

Nor do I believe that students are necessarily motivated to study mathematics because it is useful. Most will concede that it is useful -- so are castor oil and other revolting forms of medication. Ultimate usefulness will not motivate study by a teenager -- and it should not. They subconsciously realize that the most miserable people in the world are those who are doing distasteful things just because they can make a living at it. Many of these are spending their lives as adjuncts to a computer.

The great mathematician, Marshall Stone, says, "I hold that utility alone is not a proper measure of value, and would even go so far as to say that it is, when strictly and shortsightedly applied, a dangerously false measure of value." (6)

Does the current emphasis on the application and problem solving aspects of school mathematics indicate that the utility criteria are being "strictly and shortsightedly applied"? I think it does. I think that utility is being overemphasized at the expense of certain intrinsic values of school mathematics which would serve the student better in the long run. We never hear anymore about the beauty of mathematics or about its structure and internal consistency -- or about mathematics as an ideal arena for the application of logic to the thinking process -- or indeed about any of the cultural values of mathematics that have been cherished by the race for generations.

All this is ignored by the sweeping recommendation that the mathematics curriculum should be organized around problem solving. In fact, this recommendation almost denies that school mathematics exists as a separate entity apart from its applications. It implies, moreover, that mathematics in and of itself is pretty dismal stuff which can interest no one unless it is attached to some supposedly interesting situation in the "real world". If we make this attachment maybe some of the interest derived from these external situations will "rub off" on the mathematics and thus render it more palatable to the student. This, I submit, is an example of the "strict and shortsighted application" of the utility criteria which Professor Stone deplored. It is moreover a bizarre position for an organization of mathematics teachers to take.

It is strange, indeed, when those of us who profess to be fascinated by the values that inhere in the subject matter of school mathematics find ourselves in the minority and on the defensive in a convocation of mathematics teachers.

You might as well know that was and still is my condition. I loved the structure of algebra and the classical theorems of geometry. My enthusiasm for these great ideas never flagged. It never occurred to me to be bored with them as I taught them to class after class over the years -- any more than it would occur to a musician to become bored with the great masterpieces of Brahms, Beethoven, and Bach.

Look back with me to what attracted us to mathematics in the first place. Was it not its structured character - - its internal consistency as a body of subject matter where answers can be checked and where results (theorems) can be established by applying the rules of logic without appeals to feelings, authority, or faith?

If we found these qualities to be attractive, challenging, and motivational why wouldn't our students find them so? I think they would -- if we still felt that way. Do we? Or are we overwhelmed by the prevailing doctrines that sweep all this aside?

If your enthusiasm for the inherent qualities of mathematics has withered under the flood of dogma emanating from the AASA and the ASCD (Association for Supervision and Curriculum Development) -- and, oh yes, the NCTM, let me try to rekindle it. These inherent qualities can be appreciated only when they are understood.

This brings me to my major thesis that natural language, gradually expanded to include symbolism and logic, is the key to both the learning of mathematics and its effective application to problem situations. And above all, the use of appropriate language is the key to making mathematics intelligible. Indeed, in a very real sense, mathematics is a language. Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof.

Due to the current overemphasis on problem solving and applications, the student of school mathematics does not get nearly enough experience with the various aspects of proof.

There also seems to be a widespread belief that even the most elementary fundamentals of logic needed for mathematical discussion are too difficult for secondary school students. (Actually it is their omission that renders mathematics unintelligible.) This belief may stem from the fact that the abrupt introduction of proof in tenth grade geometry, without the language needed to render it understandable has led to frustrating and even traumatic experiences for both students and teachers.

Whatever the reason, proof in the applications-problem centered domain of school mathematics is postponed - - suppressed - - downgraded. (We won't have any proof questions on the exam, will we?) There is even strong support for the idea that we should not presume to do much with proof at the secondary level. In Professor Usiskin's new U of C math series, formal proof does not appear until the second semester of the tenth grade. This will sell a lot of books. And the progressive emasculation of American texts in school mathematics, which began in the late sixties, will continue apace! (Algebra without structure -- geometry with little or no proof!)

I ask you this question: Can we suppress proof without distorting school mathematics and seriously impairing the students ability to understand it? Mathematics is essentially a structured hierarchy of propositions forged by logic on a postulational base. For how long do we protect our students from the pedagogical consequences of this fact? And indeed, whom are we protecting? Those who advocate this language of logical discourse are seeking clarity of exposition rather than pretentious rigor in proof.

Also, due to our preoccupation with applications, there is not nearly enough time spent in deriving key propositions and theorems. When you and I say that we understand the theorem "the determinant of a square matrix is zero if and only if its rows are linearly dependent", we are saying that we understand how this theorem fits into a hierarchy of propositions -- and could, given time, derive it from first principles.

Why should our students be any different? To be sure, they are working at a more elementary level. But this nagging question remains: What basis do they have for understanding anything without seeing how it fits into a structure based on something?

Yet when high school students say they understand the formula for the cosine of a difference, they generally mean that they have memorized the formula and can apply it. They generally do not mean that they can derive it from first principles. Very often they do not realize that this essential dimension of understanding even exists; i.e., their understanding is deficient. So they get by on memory and facility until the cumulative effects of these deficiencies ultimately overwhelm them, and they leave mathematics in frustration and despair.

We can prevent this by equipping the student with the essentials of the language needed to understand mathematical reasoning. What are they? My answers are contained in the handout "The Language of Mathematical Exposition" which is summarized below. These are the distillation of the last 25 of my 46 years as a teacher of school and college mathematics during which time I literally lived with them and applied them daily in the classroom. They were the salient characteristics of three textbooks I co-authored which were published in 1964, 1966 and 1973 respectively.

Summary of the Language of Mathematical Exposition: Set language including set builder notation. Quantification, including "Some", "All", "None" and "there exists". Negation and contradiction. Conditions under which conjunctions, disjunctions, and implications are true or false. Equivalent statements. The statement "P implies Q" is equivalent to its contrapositive "Not Q implies not P", whereas its converse "Q implies P" is only a conjecture whose truth value (T or F) must be investigated. A theorem having n premises in its hypothesis and a single conclusion has n (partial) contrapositives, each of which is obtained by exchanging a contradiction of the conclusion with a contradiction of one of the n premises. Each of these contrapositives is equivalent to the theorem. This theorem also has n (partial) converses, each of which is obtained by exchanging the conclusion with one of the n premises. Each of these converses is a conjecture whose truth value must be determined. Tautologies. Proof patterns. Valid arguments. Proof.⁽¹⁾

I will spend little time on this list which is already well-known to you. Indeed, that may be one of our problems. As math majors, we know these so well that we tend to assume that our students know them too. I respectfully suggest that this assumption is unwarranted. Our students do not know them, and they will not unless they are given the necessary instruction. It is my position that understanding of this Language of Mathematical Exposition is absolutely essential for the successful study of school mathematics -- or any other subject where deductive reasoning is employed.

Understanding this language and facility in its use emerges over a 4 to 6 year period. It should not be abruptly introduced in the middle of 10th grade geometry. Some of it should be introduced in grades 6, 7 and 8, where very little is going on now. (In fact in some series the amount of duplication between texts for these grade levels is so great that it is actually difficult to arrange them in the proper order.)

I take issue with the idea that a problem has to be "practical" or look practical in order to be motivating and worthwhile. This idea, when taken as the basis for extensive classroom activities, is often found to be misleading, pretentious, and generally counter-productive. It is also time-consuming and often leaves little measurable residue in terms of the students understanding of basic ideas. Moreover it is simply not true. Some practical problems are a bore; some fanciful problems are not only more fun but are far more instructive.

There are those who believe that the most important use of problem situations in school mathematics is to ensure that the students can employ the formalized language of mathematics to demonstrate their understanding of the underlying theory. It is this ability and this understanding that will enable them to succeed in more advanced courses in mathematics and science. It is this ability and this understanding that will confer upon them the power to formulate and solve the problems they will encounter in these encounter advanced courses and in the real world.

There are essentially three proof formats available to us: essay, ledger (two column) and flow. Beginning students have great difficulty with the essay form and, worse, I have great difficulty analyzing their essay proofs in ways that will be helpful to them. For this reason it is generally agreed that, for the beginners, we need some kind of a pattern which will enable us to point out such errors as omissions, unwarranted assumptions and non-sequiturs. This accounts for the prevalence of ledger proofs in our geometry texts. But, as you supposed, I believe that the flow proof is far superior to the ledger proof in the all important matter of delineating the structure and thrust of a mathematical argument.

I also believe that the study of proof should begin in algebra rather than in geometry where troublesome incidence relations raise their ugly heads. (How much can we assume from the drawing?)

I conclude my remarks on flow proofs with some caveats and observations.

Don't confront the class with a completed flow proof except where you:

(1) ask for reasons; (2) ask for scrutiny to find mistakes; (3) ask for translation to the essay form. Otherwise build the proof with them in class. They will gradually learn with you to work forward from the premises (the search for necessary conclusions) and backward from the conclusion (the search for sufficient conditions) until a seamless sequence of implications extends from the hypothesis to the conclusion. This can be an exciting and challenging task - - but it is not easy. But to paraphrase the great teacher, R. H. Moore, "Mathematics properly taught is difficult." Flow proofs plow up unexpected difficulties. These difficult situations where misunderstandings lurk were there all the time. The flow proof merely exposes them so that they can be resolved.

I do not expect my students to go through life using flow proofs any more than English teachers expect their students to go through life diagraming sentences. My ultimate objective is to develop the students' ability to read and write essay proofs of the kind they will encounter in more advanced courses in mathematics.

Now that I have presented my language of exposition as essential to reasoning and proof in school mathematics, let us consider the consequences of neglecting lexical reasoning and proof.

These are catastrophic: alienation of our students (later the public); isolation of our subject; and failure to prepare our students for advanced study.

Due to our reluctance to use a handful of universally valid and easily understood logical principles, we are denying our students the opportunity to understand.

For too long have we sheltered our students from explanations deemed to be too difficult for them. When explanations are inadequate or missing entirely, students attribute their resulting lack of understanding to the idea that there is something abstruse and forbidding about mathematics which they will never be able to understand. The result may be permanent alienation from both mathematics and the mathematical sciences. This alienation not only cripples thousands of students, it cripples our economy and our defense posture as well.

Examples of public alienation to mathematics abound. It is the only subject I know where intelligent people will openly flaunt their ignorance. "So you're a math professor" they say; and then almost inevitably I wince when they add, "Well, I never was very good at mathematics."

The public's misapprehension of the nature of mathematics would be amusing if it were not so tragic. They think that we have a curious predilection for carrying out complicated numerical algorithms, and they are pretty sure that the advent of the computer has rendered us obsolete. In bridge playing groups they say, "You're the mathematician, you can keep score." This is like saying, "You are the archeologist. You can dig the post hole." We've all had to suffer through stories about absent-minded and wacky mathematicians -- stories whose impact is that to do the kind of reasoning required in mathematics, you don't have to be insulated from reality -- but it helps. My only weak rejoinder is that we don't know how flaky these characters would have been if they had not studied mathematics.

All of this indicates that mathematics is the object of widespread misunderstanding and hostility on the part of the public. Our PR is PU. [Dysmetria means aversion to or fear of mathematics. We have all seen evidence indicating that many high school students suffer from Acquired Incurable Dysmetria Syndrome.]

The notion, already too widely held, that one must have a "mathematical mind" in order to deal with the peculiar thinking required in mathematics has served to isolate our subject from the real world to an extent that we cannot possibly counteract by our current determined (and somewhat trendy) overemphasis on applications and problem solving. This "mathematical mind" syndrome has hurt us enormously. It leads to the mistaken notion that mathematics is easy for those with this mysterious native ability and beyond the reach of those who lack it, no matter how hard they work.

Using this language of exposition to develop an interplay between logic and mathematics, as suggested here, will clearly indicate to the student that the same logical principles apply in every field; that there is nothing esoteric or arcane about the thinking required in mathematics.

Another dire consequence of our downgrading of proof is that it distorts our subject in such a way that the early courses, as currently presented, give the student no idea of what mathematics is really like.

Problem solving, important as it is, does not provide adequate preparation. Many students who enjoy working the problems in high school math and even in calculus, and hence decide that they like mathematics and want to major in it, find that the nature of the subject changes abruptly when they encounter the proof courses which follow calculus. They are bewildered and dismayed in courses such as Abstract Algebra (Algebra Structures), Linear Algebra, Advanced Calculus, and Topology where proof is the name of the game. Their previous problem solving courses did not prepare them for this abrupt change in emphasis. Their rude awakening often comes so late that it is difficult to change majors. This is tragic. Is it not our function to prepare students to deal with proof at the college level?

My own experience as Professor of Math at Elmhurst College abundantly confirms the fact that the student's first collision with proof can be traumatic.

Our math majors were strong students who were required to complete four years beyond calculus from these one semester courses:

Since they had virtually no experience with proof, some had great difficulty when they encountered Linear Algebra. The combination of new subject matter and the proof requirement staggered them. Since I had the same experience myself, I was sympathetic and by extra effort was able to help them pass the "proof barrier". I occasionally resorted to flow proofs. I am happy to say that most of them went on to become strong majors. But this is often not the case.

We must face up to the fact that serious deficiencies in the area of lexical reasoning are causing a general lowering of the levels of math competency among beginning college students. The remedial (precalculus) mathematics that must be offered in college now accounts for almost two-thirds of the college math enrollment in the United States (7). For emphasis, I reiterate that many potential math and science majors flounder in the higher level math classes because of their continuing inability to think in terms of the language of mathematics. Recognizing this, some colleges have tried to devise remedial courses dealing with the basic techniques of mathematical proof. Having worked on developing such a course for about eight years, Professor Susanna Epp of DePaul University (Chicago) has this to say. "During this period I came to realize that many of my students difficulties were much more profound than I had anticipated. Quite simply, my students and I spoke different languages. ...Very few of my students had an intuitive feel for the equivalence between a statement and its contrapositive or realized that a statement can be true and its converse false. Most students did not understand what it means for an if-then statement to be false, and many also were inconsistent about taking negations of "and" and "or" statements." (8) She goes on to list other deficiencies of the type which could and should have been remedied in high school by teaching the material on the handout.

Let us, therefore, resolve to realign our efforts in the teaching of school mathematics, even though this realignment is at variance with the recommendations of prestigious committees. We must not dissipate our energies in dealing with a bewildering array of specific problems. This prepares the student, at best, to deal with similar problems. No. We must focus on building proficiency in the formalized natural language required to apply the thinking process in mathematics. Only then can we open the channels of communication which will enable our students to profit by more advanced instruction. Only then can we prepare our students to cope effectively with problems which are today unforeseen and unforeseeable.

Now let me present an Agenda for Understanding which is intended for college capable(intending?) students of secondary school mathematics (grades 6-12).

1. The school mathematics curriculum must be organized around the internal structure of school mathematics. The idea that mathematics is a hierarchy of propositions forged by logic on a postulational base should begin to form in the student's mind about grade 9 and should be thoroughly established by grade 12. Then our students will know what mathematics is really like and they will be in a position to decide whether or not they want to continue. We will have been honest with them.

Will students be repelled by what may appear to some to be a rather austere and abstract portrayal of mathematics? No. On the contrary. The able ones are more likely to be repelled by a formless, flaccid curriculum in which no structure is discernible and which, as a consequence, provides no basis for understanding anything.

2. The teaching of mathematics should be regarded initially as an extension of the teaching of language. Our efforts to develop an awareness of the intimate relationship which exists between grammar, mathematics and logic should begin with games of " How do we know?" in the early grades (12) continue with the introduction of formal proof not later than grade 9 (perhaps with the aid of flow-diagrams) and culminate in the ability to read and write lucid essay proofs by grade 12.

3. We must try to take a more balanced view of the role of problem solving in school mathematics lest our preoccupation with it causes us to neglect the very mathematics that makes problem solving possible. On Page 153 of the NCTM publication "Curriculum Evaluation and Standards for School Mathematics" (9), now in draft form, the following edifying statement appears, "Students success in mathematical problem solving requires knowledge of mathematics." I was relieved to see this. Reading current literature I had begun to believe that all one needed to do was follow Polya's four problem solving stages, which as Jeremy Kilpatrick notes (10) have recently been rediscovered by trendy mathematics educators, most of whom are not problem solvers. At least I do not see them listed as either solvers or proposers in the problem sections of any of the several mathematical journals that I peruse each month. Sometimes I am almost overwhelmed by the amount of mathematics one must know in order to cope with these problems. If we do not know the definitions and theorems invoved in a given problem, Polya's four stages will not help us find a solution.

Don't misunderstand. I am not opposed to problem solving. I had large collections of problems that I used in the classroom. "Problems of rare charm and distinction" I called them. These, I thought, were challenging, ingenious, inherently fascinating and instructive -- and best of all I could work them. Such problems are the life blood of mathematics. But let us not fail to convey to our students that the body of mathematics is given structure and coherence by the bones and sinews supplied by definitions, postulates and proof.

We are mature, experienced and mathematically trained professionals and it is time for us to stop overreacting to the plaintive question "What good is all this?" Experience has shown that students lose interest in this question when they understand what is going on. The antidote to shortsighted, pragmatic demands for immediate utility is understanding of the kind that can be imparted by the linguistic approach here advocated -- not a fragmented curriculum organized around problem solving.

4. We must restore the emphasis on proof which was once a major feature of the college preparatory sequence. Formal proof, founded on this language of mathematical exposition, should begin in 9th grade algebra and should be the dominant theme in 10th grade geometry -- and thereafter. I do not object to an electronically activated, laboratory approach to the informal geometry of grades 6-8. But such experimental methods must not be allowed to supercede proof in the geometry of grade 10. Proof must remain the central theme of this geometry, as it has been for centuries. We must somehow cope with this surging aversion to proof which not only drastically reduces the proof content of geometry, but actually jeopardizes its position in the curriculum. Geometry at this level is not a laboratory subject.

Geometry with proof is the keystone of the secondary mathematics sequence. Geometry without a strong emphasis on proof offers little more than an unstructured collection of geomerric facts including many the student has already encountered in junior high school. Such a course is not worth the student's time.

Will not all this emphasis on proof make learning more difficult for the student? You bet. But education results from overcoming difficulties not from evading them. Urge your students to climb the mountain. Don't pretend that there is no mountain to climb. Besides, adolescents are not repelled by difficulty. Our students will find proof to be challenging, rewarding and even exciting provided we find it so.

5. It is time for us to reconsider our policies with respect to a national curriculum and nationwide testing. For a language to be effective it must be well-nigh universally applied. Accordingly, there is an urgent need for a mathematics sequence based on lexical reasoning, which can be (a) used universally for college capable students in grades 6-12 and (b) validated by a nationwide testing program. Not all college capable students would complete this sequence but all would take four years of mathematics in high school. For the best students the 12th grade course would be calculus, and the valuable parts of so-called discrete mathematics, implemented by matrices, vectors and computer techniques, would be in the algebra courses which precede calculus.

Our failure to provide such a program for college capable students makes us unique among the industrialized nations of the world and undoubtedly accounts for many of our failures, including the fact that foreign-born students are dominating our graduate level programs in mathematics and engineering (13).

Recommendation 6 in the Agenda for Action should not apply to college capable students. These students most emphatically do not need "A flexible curriculum with a greater range of options --- designed to accommodate their diverse needs." Diversity will come soon enough. In secondary school they need a well-defined program which develops their ability to read, write and think in the formalized, symbolic language of mathematics.

The benefits of such a program would be enormous. College instructors would know what to expect from students, depending on how far they had progressed in the sequence. But most important, it would clear the language channels that must be open if students are to learn or apply mathematics at the college level.

As we convene here in Chicago today a wave of dissatisfaction with education is sweeping over our country -- and this dissatisfaction applies with special force to school mathematics. Are we going to ignore this -- arrogantly continuing with policies that have been thoroughly discredited? I prefer to take this as a warning.

Because of some ill-considered turns we have made we are now on a collision course with disaster. It is time for us to get back on course. Let us strive for renewed confidence in the inherent values of our subject and in ourselves as expositors. We should be encouraged by the experience at Potsdam College, Potsdam, NY, where a traditional pure mathematics department contributes 20% of each graduating class as mathematics majors (about 200). (11) Recruiters from industry are literally standing in line to hire these math majors who they are convinced, have learned how to think in the language of mathematics.

(1) The Agenda for Action - - Appendix to Changing School Mathematics - - A Responsive Process, National Council of Teachers of Mathematics, 1906 Association Drive, Reston, Virginia 22091, Copyright 1981.

(2) Steen, Lynn Arthur, Smokestack Classrooms, FOCUS (Newsletter of the Mathematical Association of America), March-April 1987.

(3) The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective. Curtis C. McKnight, et al. Stipes Publications, 10-12 Chester Street, Champaign, IL 61820, 1987.

(4) Beck, Joan, "What can we learn from the successes of Japanese kids?", Chicago Tribune, February 9, 1987.

(5) "Math Instruction Doesn't Make the Grade", Sheldon L. Glashow, Virginia Pilot, April 7, 1986.

(6) Stone, M. H., "Mathematics and the Future of Science", Bulletin of the American Mathematical Society, Vol. 63, No. 2, March 1957, pp. 61-76.

(7) Steen, Lynn Arthur, Undergraduate Mathematics in China, FOCUS (Newsletter of the Mathematical Association of America), September-October, 1983.

(8) Epp, Susanna, The Logic of Teaching Calculus, Paper written for the Tulane/Sloan Foundation Conference on Calculus, 1986, subsequently published (1987) in Toward a Lean and Lively Calculus, Ronald G. Douglas, Editor, MAA 1529 Eighteenth St., NW Washington, D.C. 20036

(9) "Curriculum and Evaluation Standards for School Mathematics" -- Romberg et al -- NCTM 1987.

(10) Kilpatrick, Jeremy, "George Polya's Influence on Mathematics Education." Mathematics Magazine, Vol. 60, No. 5, December 1987. pp. 299-300.

(11) Poland, John, "A Modern Fairy Tale?" The American Mathematical Monthly1 March 1987, pp. 291-95.

(12) Lipman, Sharp, Oscanyan, "Philosophy of the Classroom", Universal Diversified Services, West Caldwell, NY, 07006, 1977.

(13) Study cited in an editorial in Chicago Tribune for February 17, 1988.

Note 1. A full version of "The language of Exposition" is available to anyone having access to Acrobat Reader by sending an E-Mail to franka@elmhurst.edu