Algebra

     Classical education depends on a three-part process of training the mind. The early years of school are spent in absorbing facts, systematically laying the foundations for advanced study. In the middle grades, students learn to think through arguments. In the high school years, they learn to express themselves. This classical pattern is called the trivium.
    The first years of schooling are called the "grammar stage" -- not because you spend four years doing English, but because these are the years in which the building blocks for all other learning are laid, just as grammar is the foundation for language. In the elementary school years -- what we commonly think of as grades one through four -- the mind is ready to absorb information. Children at this age actually find memorization fun. So during this period, education involves not self-expression and self-discovery, but rather the learning of facts. Rules of phonics and spelling, rules of grammar, poems, the vocabulary of foreign languages, the stories of history and literature, descriptions of plants and animals and the human body, the facts of mathematics -- the list goes on. This information makes up the "grammar," or the basic building blocks, for the second stage of education.
    By fifth grade, a child's mind begins to think more analytically. Middle-school students are less interested in finding out facts than in asking "Why?" The second phase of the classical education, the "Logic Stage," is a time when the child begins to pay attention to cause and effect, to the relationships between different fields of knowledge relate, to the way facts fit together into a logical framework.
    A student is ready for the Logic Stage when the capacity for abstract thought begins to mature. During these years, the student begins algebra and the study of logic, and begins to apply logic to all academic subjects. The logic of writing, for example, includes paragraph construction and learning to support a thesis; the logic of reading involves the criticism and analysis of texts, not simple absorption of information; the logic of history demands that the student find out why the War of 1812 was fought, rather than simply reading its story; the logic of science requires that the child learn the scientific method.
    The final phase of a classical education, the "Rhetoric Stage," builds on the first two. At this point, the high school student learns to write and speak with force and originality. The student of rhetoric applies the rules of logic learned in middle school to the foundational information learned in the early grades and expresses his conclusions in clear, forceful, elegant language. Students also begin to specialize in whatever branch of knowledge attracts them; these are the years for art camps, college courses, foreign travel, apprenticeships, and other forms of specialized training.
    A classical education is more than simply a pattern of learning, though. Classical education is language-focused; learning is accomplished through words, written and spoken, rather than through images (pictures, videos, and television).
    Why is this important? Language-learning and image-learning require very different habits of thought. Language requires the mind to work harder; in reading, the brain is forced to translate a symbol (words on the page) into a concept. Images, such as those on videos and television, allow the mind to be passive. In front of a video screen, the brain can "sit back" and relax; faced with the written page, the mind is required to roll its sleeves up and get back to work.
    A classical education, then, has two important aspects. It is language-focused. And it follows a specific three-part pattern: the mind must be first supplied with facts and images, then given the logical tools for organization of facts, and finally equipped to express conclusions.
    But that isn't all. To the classical mind, all knowledge is interrelated. Astronomy (for example) isn't studied in isolation; it's learned along with the history of scientific discovery, which leads into the church's relationship to science and from there to the intricacies of medieval church history. The reading of the Odyssey leads the student into the consideration of Greek history, the nature of heroism, the development of the epic, and man's understanding of the divine.
    This is easier said than done. The world is full of knowledge, and finding the links between fields of study can be a mind-twisting task. A classical education meets this challenge by taking history as its organizing outline -- beginning with the ancients and progressing forward to the moderns in history, science, literature, art and music.
    We suggest that the twelve years of education consist of three repetitions of the same four-year pattern: Ancients, Middle Ages, Renaissance and Reformation, and Modern Times. The child studies these four time periods at varying levels -- simple for grades 1-4, more difficult in grades 5-8 (when the student begins to read original sources), and taking an even more complex approach in grades 9-12, when the student works through these time periods using original sources (from Homer to Hitler) and also has the opportunity to pursue a particular interest (music, dance, technology, medicine, biology, creative writing) in depth.
    The other subject areas of the curriculum are linked to history studies. The student who is working on ancient history will read Greek and Roman mythology, the tales of the Iliad and Odyssey, early medievial writings, Chinese and Japanese fairy tales, and (for the older student) the classical texts of Plato, Herodutus, Virgil, Aristotle. She'll read Beowulf, Dante, Chaucer, Shakespeare the following year, when she's studying medieval and early Renaissance history. When the eighteenth and nineteenth centuries are studied, she starts with Swift (Gulliver's Travels) and ends with Dickens; finally, she reads modern literature as she is studying modern history.
    The sciences are studied in a four-year pattern that roughly corresponds to the periods of scientific discovery: biology, classification and the human body (subjects known to the ancients); earth science and basic astronomy (which flowered during the early Renaissance); chemistry (which came into its own during the early modern period); and then basic physics and computer science (very modern subjects).
    This pattern lends coherence to the study of history, science, and literature -- subjects that are too often fragmented and confusing. The pattern widens and deepens as the student progresses in maturity and learning. For example, a first grader listens to you read the story of the Iliad from one of the picture book versions available at any public library. Four years later, the fifth grader reads one of the popular middle-grade adaptations -- Olivia Coolidge's The Trojan War, or Roger Lancelyn Greene's Tales of Troy. Four more years go by, and the ninth grader -- faced with the Iliad itself -- plunges right in, undaunted.
     The classical education is, above all, systematic -- in direct contrast to the scattered, unorganized nature of so much secondary education. This systematic, rigorous study has two purposes.
    Rigorous study develops virtue in the student. Aristotle defined virtue as the ability to act in accordance to what one knows to be right. The virtuous man (or woman) can force himself to do what he knows to be right, even when it runs against his inclinations. The classical education continually asks a student to work against his baser inclinations (laziness, or the desire to watch another half hour of TV) in order to reach a goal -- mastery of a subject.
    Systematic study also allows the student to join what Mortimer Adler calls the "Great Conversation" -- the ongoing conversation of great minds down through the ages. Much modern education is so eclectic that the student has little opportunity to make connections between past events and the flood of current information. "The beauty of the classical curriculum," writes classical schoolmaster David Hicks, "is that it dwells on one problem, one author, or one epoch long enough to allow even the youngest student a chance to exercise his mind in a scholarly way: to make connections and to trace developments, lines of reasoning, patterns of action, recurring symbolisms, plots, and motifs."

Historic & Modern Application

Classical vs. Modern Education

Modern public education does not look like the trivium. But in order to understand why, we have to understand from whence modern education has come. For, if one takes a cursory glance through Classical America, we find that even 200 years ago things were very different in American education than they are today. How did we get so far away from the classical tradition in education?

In order to approach an answer, we must understand the rise of government- sponsored education, and with it, the public school nightmare. Richard Mitchell, professor of classics at Glassboro State College, deftly details the not-so-subtle shift in educational philosophy that accompanied that change from classical education to modern education in The Graves of Academe. You must read the whole work, it is a thoughtful and considered piece of scholarship, elegantly written with humor and insight. In it, he follows the rise of the pseudo-science of “educationism,” and its rejection of academia in favor of the unmeasurable and untestable nebulous principles of behavior modification and “humanism:”

Over and against the overweening demands of scholarly intellectualism, the teacher-trainers have set the presumably unquestionable virtues of what they call “humanism.” They use this term in so many different contexts and to characterize so many different kinds of acts and ideologies that I will not attempt to discuss it fully here. It will just have to grow on you. It does not, as you might think, denote as usual a particular school of thought or slant of philosophical or religious speculation connected especially but not exclusively with the Renaissance, although many who use the term have heard of the Renaissance. This is something closer to “humaneness,” as that word is used by what used to be called the “Humane Society,” an organization that publically deplored the cruel treatment of horses. One of the aims of “humanistic” educationism is to deplore the cruel treatment of children subjected to the overbearing demands of knowledge, scholarship, and logic by the traditional powers of authoritarian intellectualism.

And so here we are today, as far away from traditional scholarship and training in the rigorous demands of the trivium that underpins it, as one can get. It is a far cry from the tested and proven methods of nurturing the intellect, drawn from the classical tradition, required by historic American education:

[The] NEA task force that had been made up largely of scholars, the Committee of Ten, [was] called together in 1892 and chaired by Charles W. Eliot, then president of Harvard University. That committee had come out in favor of traditional academic study in the public schools, which they fancied should be devoted to the pursuit of knowledge and the training of the intellect. But what can you expect from a bunch of intellectuals? The Eliot Report of 1893 was given to things like this:

As studies in language and in the natural sciences are best adapted to cultivate the habits of observation; as mathematics are the traditional training of the reasoning faculties; so history and its allied branches are better adapted than any other studies to promote the invaluable mental power which we call judgment.

Obviously, the Eliot committee did its work in the lost, dark days before the world of education had discovered the power of the bold innovative thrust. All they asked of the high schools was the pursuit of knowledge and the exercise of the mind in the cause of judgment.

Classical Education in History

However, throughout history education meant “the pursuit of knowledge and the exercise of the mind in the cause of judgment.” Grammar has been the first study of children from antiquity, and the term “grammar school,” commonly used in the past, still means a child’s first school. In the classical world, a child’s first school was where he learned the grammar of Greek and Latin, the foundation upon which all other education was built. Aristotle, who lived at roughly the same time as Alexander the Great, compiled the system of formal Logic in use today and wrote a treatise on Rhetoric used in universities for millenia. The occupation of rhetorician, or public-speaker, was a lofty one in Roman society, with Cicero being the most reknown. Quintilian, who was influenced by Cicero, was a famous teacher of rhetoric in Rome and wrote Institutio Oratoria, a work detailing the instruction of children and the training of orators; it was likewise used for centuries.

During the Middle Ages the mastery of the Seven Liberal Arts became solidly fixed as the sole educational curriculum, with no question as to its authority, little experimentation with its processes, and few new works added to the ancient texts used in schools. Everyone knew that seven was the number of perfection anyway, so nothing was added or taken away. These seven liberal arts are explained in the old couplet quoted by C.S. Lewis in The Discarded Image:

Gram loquitur, Dia verba docet, Rhet verba colorat,
Mus canit, Ar numerat, Geo ponderat, Ast colit astra.

Grammar talks, Dialectic teaches words, Rhetoric colors words,
Music sings, Arithmetic numbers, Geometry weighs, Astronomy tends the stars.

The first three -- Grammar, Dialectic, and Rhetoric -- formed the trivium, the three-fold way. The remaining four -- Music, Arithmetic, Geometry, and Astronomy -- formed the quadrivium, the four-fold way. The trivium was first mastered as a foundation to further study in the quadrivium. The medieval quadrivium was the fore-runner of our university, and is the reason today that earning an undergraduate degree still requires a four-year course of study.

In looking at the medieval concept of the trivium, then, C.S. Lewis states that “Grammar talks” means that Grammar teaches us Latin, which was the “living Esperanto of the western world and great works were still being written in it. It was the language par excellence, so that the very word Latin came to mean language.” (Greek was not studied during the Middle Ages, for the most part. All the Greek texts had been translated into Latin.) The phrase “Dialectic teaches words,” really means that “having learned from grammar how to talk, we must learn from Dialectic how to talk sense, to argue, to prove and disprove.” “Rhetoric colors words” -- it gives what we have to say, built by grammar and dialectic, “structure and style.”

During the Renaissance, and the revival of all things classical, several changes were brought about in education, most notably due to the influence of Desiderius Erasmus. He encouraged the learning of Greek as well as of Latin, so that the New Testament could be read in its original language. With the invention of the printing press, more school books and instruction books for teachers became available. Erasmus himself wrote several on the teaching of rhetoric. The tumultous years of the Reformation followed, which brought a fresh emphasis on Biblical study and exegesis. The classical emphasis on language study was brought to the New World by the Puritan colonists, who determined that every child be able to read the Bible, and every pastor be a scholar of Latin and Greek (grammar), able to determine truth from error in their doctrine (dialectic), and able to eloquently expound on the Scriptures (rhetoric). Thus the training in the trivium, underpinned by a Biblical worldview, remained the standard of academic excellence.

The classical concept of the trivium in education survived as the sole educational model for two millenia because it worked -- it consistently produced educated men, given to “the pursuit of knowledge and the exercise of the mind in the cause of judgment.”

Encyclopaedia Britannica on the History of Education
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1911 Encyclopaedia Britannica on the History of Education

The Single-Subject School

Classical, medieval, and colonial education look very different from our modern education ideal. Where were all the classes in science, arithmetic, history (or should I say, social studies), literature, art, and ...? We learned from The Public School Nightmare that the Prussians instituted the system of schooling where many subjects taught in isolation were crammed into each school day, necessitating that each one receive a small block of time--an hour or less--ending with a bell. They didn’t do this so that students could become masters of science, arithmetic, history, and so on, but so that they would learn to accept what they were told without question, (since they didn’t have time to study much of anything in depth) and become conditioned to going where they were told at the sound of a bell. It was a system designed to rob them of the power of their mind--and it worked. World War I and II are the results of the Prussian model of schooling.

Before the Prussian model, students were given large chunks of time to study a single subject: Greek and Latin grammar. This was the foundation for further study in logic and rhetoric as taught by the classics written in Greek and Latin. Students learned history and literature by way of Greek and Latin grammar, as they read for practice things like Caesar’s De Bello Gallico (Concering the Gallic War) and Homer’s Iliad and Odyssey, and other works. They learned English grammar incidentally as a result of learning Greek and Latin. Historically, science was not a part of a child’s education, and sometimes only the rudiments of arithmetic were learned. Thomas Jefferson learned his arithmetic, not from his grammar school, but from a private instructor hired by his father.

This immersion solely in the tools of learning -- grammar, logic, and rhetoric -- produced the men of colonial America who were then able to educate themselves further about any subject they desired to pursue (as Benjamin Franklin did with science). But more importantly, it produced free men, able to think for themselves, as is the aim of a liberal education (liber means “free” in Latin). It fit them for the task of forming the first free government on the earth in modern times. When faced with their own tyrant, in the form of George III, they did not mindlessly do what they were told. Their education enabled them to evaluate what they were told to do, and decide whether it was in fact the right thing to do. Having decided that it was not, their education equipped them to search history and make applications to their present situation, so that they might discover the right course of action. And all their work of the mind was guided by the Biblical worldview into which they had been immersed.

Is it any wonder that modern education, sponsored by the government, with attendance there made compulsory by law, shuns large blocks of time devoted to the Bible, Greek, Latin, Logic, and Rhetoric? That would produce free men, thinkers; and, as Richard Mitchell points out, “the free are quirky” -- hard to control. Why do you think our modern system of schooling is modeled after Prussia and not after colonial America?

The Trivium for Today

But, as homeschoolers, we have a unique and God-given window of opportunity to rectify matters in education, at least for our own children. As we study how to apply the trivium to education today, there are at least two schools of thought that have arisen. One is inspired by Dorothy Sayers, from her famous speech The Lost Tools of Learning. She points out the major deficiencies with modern educationism, and advocates a return to the classical model of education. In it, she suggests that each subject in the curriculum be divided into its grammar, dialectic, and rhetoric stages; and it is true, since the trivium describes the laws of learning, that each subject has a grammar, dialectic, and rhetoric aspect. This website attempts to detail those aspects, and meld them together into a practical and workable course of study for homeschoolers.

The other school of thought advocates limiting the subjects in the curriculum and returning to the single subject school: Greek and Latin grammar in the elementary grades, logic in the middle grades, and rhetoric in the high school grades, as was the case with traditional classical education.

My personal belief is that, in an ideal world, the latter approach is desirable with modification. First of all, in its pure form it is unworkable for most of us who homeschool because of state regulations concerning our curriculum. And in this day and age, neither arithmetic and higher mathematics, nor science can be optional subjects in the curriculum. I believe a blending of the two approaches to the trivium will give us the desired result, which remains, after all, young adults devoted to “the pursuit of knowledge and the exercise of the mind in the cause of judgment;” men and women made free by their education -- i.e. able to think, reason, and apply accurately and for themselves.

We must remember that the central subject of the grammar stage remains Greek and Latin grammar; the central subject of the dialectic stage remains logic, and the central subject of the rhetoric stage remains rhetoric. We provide the maximum time in the school day for their study. In this way, the tools of learning -- grammar, logic, and rhetoric-- become the servants to the other subjects we undertake. Grammar gives us precision and proficiency in language, which enables us to read with comprehension history, literature, science, philosophy, and anything else. Logic gives us the ability to think rightly and arrive at valid conclusions, which develops our reason used in mathematics, science, and in evaluating what we read in history, literature, etc. Rhetoric teaches us to communicate, and so becomes the handmaiden to writing, oratory, and in contributing to the ideas fostered by history, literature, science, etc.

We are not trying to perpetuate the schooling atrocity institited by the Prussians, so let us not becomes slaves to the subjects. Rather, let them serve us in inculcating the tools of learning in our children.

ABOUT THE LIBERAL ARTS

Based on the types of studies that were pursued in the Classical world, the Seven Liberal Arts became codified in late antiquity by such writers as Varro and Martianus Capella. In medieval times, the Seven Liberal Arts offered a canonical way of depicting the realms of higher learning.

The Liberal Arts were divided into the Trivium ("the three roads") and the Quadrivium ("the four roads").

The medival Quadrivium thus followed the division of mathematics made by the Pythagoreans. Recently, mathematics has been defined as "the study of patterns in space and time," which very much resembles the ancient Pythagorean understanding of mathematics.

There were other important studies in medieval times. For example, philosophy was often envisioned as a metastudy that united all branches of knowledge. For this reason, Philosophia is depicted in the illustration below as nourishing the Seven Liberal Arts.

“Whom will He teach knowledge? And whom will He make to understand the message? Those just weaned from milk? Those just drawn from the breasts? For precept must be upon precept, precept upon precept, line upon line, line upon line, here a little, there a little.”

The Three Laws of Learning

The word “trivium” comes from the Latin prefix “tri” meaning “three,” and the Latin root “via” meaning “way,” or “road.” The word literally means “the three-fold way or road.” The trivium refers to the three stages, or ways, of learning that coincide with a child’s cognitive development as he matures. We should begin an in-depth look at the trivium--the three stages of learning--by reminding ourselves that the trivium is not some arbitrary theory of teaching methodology or new fad of learning philosophy. Rather, the trivium was developed by long trial and error, through the observation of the ancients in the way children learn during the whole course of their instruction from young child to young adult. They realized that time after time, they followed three stages in the learning process. They simply pointed out what was obviously there; what God had designed: that there are three stages, which they named Grammar, Dialectic, and Rhetoric; and they progress in that order. Think of Sir Isaac Newton. He didn’t invent the three laws of motion (God did that when He created the universe), but after careful observation, he defined them by stating what was already there. So it is with the trivium. We might even call the trivium the three laws of learning.

How We Learn & the Trivium

There are two ways to look at the trivium. The first has to do with the affinity that children have for thinking about things, for seeing things in one light or another depending on their stage of brain development. It is a well-known fact that when a baby is born, his nervous system isn’t fully developed yet. There is a process of myelinization of the nervous system that must occur, and this process takes years. First he is able to hold his head up, then roll over, crawl, walk, run, jump, and finally do things like ballet or basketball or synchronized swimming with style and grace. Just as a child’s physical coordination and motor skills develop and become more refined over time, following a definable progression; his cognitive coordination and thinking skills also develop and become more refined over time, following a definable progression.

Therefore a child will first embark on the stage of brain development that classicists term the grammar stage. In this stage he has a natural affinity for storing up a tremendous amount of information on any number of things, from nursery rhymes to math facts, and recalling that information at will. He will then progress to the stage termed the dialectic, where his abilities to reason are honed and sharpened, and everything is turned into an exercise in argumentation. And lastly he will advance to the rhetoric stage, where self-discovery and expression are the paramount concerns, and where cognitive abilities come into their full flower of maturity.

The second way to look at the Trivium is just this: that anyone learning something new goes through these three stages as well. A baby learning his native language starts with vocabulary first (the grammar stage), advancing on to stringing that vocabulary together in meaningful ways (the dialectic stage), and ending with finally becoming proficient in completely expressing his thought in the common standard of language usage (the rhetoric stage.) A teenager learning to drive or an adult learning to operate a personal computer does the same thing: commit the vocabulary, the rules, the basics of the subject to memory (grammar), string the isolated parts together to make a meaningful whole (dialectic), then become proficient in the operation of the car or the computer or whatever the subject happens to be (rhetoric.)

The Grammar Stage

With that under our belt, let us look at the grammar stage more completely. Simply defined, it is the learning of the body of knowledge of a subject, and most classicists would agree that this is best done by memorization. Most of us have been trained to have an aversion to rote memorization, but it is not harmful, and neither does it have to be dull. I would venture to say that no baby had to be forced to learn to talk, but rather he enjoyed the process immensely. In reality, a child begins learning the grammar of things when he is born and continues from there, but in formal education the grammar stage coincides with the elementary years. In terms of cognitive ability, children at this age automatically zero in on the concrete facts. Therefore it is fine at this stage to concentrate on the concrete and leave the analytical and the abstract out of it.

There is a big push in modern educational theory to introduce abstract concepts to elementary children, and while there is some overlap of the stages with individual children maturing in their thinking individually, for the most part they are not developmentally able to grasp abstracts at this age. Resist the pressure to have young children wrestling with underlying abstract mathematical concepts; feel free to be the only one in miles that does not emphasize self-expression to the detriment of all else in writing class. Ideally, an understanding of anything is not the goal at this age, but rather: have they memorized their math facts and demonstrated that by being able to do computations; have they memorized their phonics and spelling rules and demonstrated that by being able to read and spell correctly; and so on. The problem with attempting to teach abstracts at this age is that children do not yet have the ability to connect relationships between factors, nor do they have the ability to question and reason out the validity of what they have been told, but they have the ability to believe that what they have been told is the truth. That is the beauty of the grammar stage.

The Dialectic Stage

The dialectic stage is defined as learning to reason, and the body of knowledge learned in the grammar stage is the stuff learning to reason is practiced on. In the grammar stage children learned facts; in the dialectic stage children try to understand the facts they have learned, and begin to relate those facts to one another in a significant way. This stage coincides with middle or junior high school, although it may actually begin for individual children earlier than that, in 5th or 6th grade. It is in the dialectic that the emphasis in cognitive skills shifts from the concrete to the analytical. This is where children are naturally inclined to ask the question “Why?” This is where they question what they have learned in the grammar stage to see if it is in fact true. Truth holds up very well under examination, and only proves its nature by this process. While not advocating children question the things they were taught, if what they were taught is true, we need have no fear of it being questioned, even if that questioning runs to things such as the existence of God or the veracity of the Word. Therefore teaching the science of Logic is critical at this stage. It gives children the tools they need to question accurately and arrive at valid conclusions. We might be conditioned to react with shock or discipline, even, when children at this age question, argue, or want to know why. If we can understand that going through this process is the necessary step to arrive at the next one and therefore on to maturity, perhaps we can temper our response and help children learn to question and reason while maintaining an attitude of honor and respect.

The Rhetoric Stage

The last stage is the rhetoric stage, which focuses on learning the science of communication and the art of expression. In the grammar stage children learned facts; in the dialectic stage children began to understand those facts, and in the rhetoric stage children learn to express what they now understand in the most compelling manner possible. This stage roughly coincides with high school. Cognitively speaking, this stage is where abstract thought reaches its zenith. In this stage, the unknown can be explored because the known is understood; the hypothetical can be introduced and grasped with the mind. The mental jump can be made from the natural to the spiritual, from the practical to the theoretical. Self-expression finally comes into its own in the language arts; “hard” sciences and advanced mathematics are more easily mastered; history can be applied to economics and political science; and Bible study can turn to apologetics.

The Stages & the Subjects

From this explanation of the stages of the trivium, we can see that each instructional stage corresponds to a related stage in cognitive development: the grammar stage to concrete thinking, the dialectic stage to analytical thinking, and the rhetoric stage to abstract thinking. The stages themselves also bear the name of individual subjects which are at the heart of that stage. The subject of grammar, which is the science of correct language usage, is best learned in the grammar stage; the subject of logic, which is the science of right thinking, is best learned in the dialectic stage; and the subject of rhetoric, which is the science of expression, is best learned in the rhetoric stage. Each of these subjects give our children valuable tools of learning which enables them to grasp, understand, and act on any other subject, area of study, or problem encountered in college and in adult life.

Some confusion in understanding the trivium has resulted from the names of the stages themselves sharing the names of these individual subjects. We must remember that each stage represents a distinct way of teaching and learning and thinking about each subject in the curriculum, while at the same time zeroing in on a particular field of study uniquely relevant to that stage. In discussing the trivium it is helpful to distinguish between the stages of grammar, dialectic, and rhetoric; and the subjects of grammar, logic, and rhetoric.

The Trivium in Summary

The Quadrivium

I: The Seven Liberal Arts

What are the "liberal arts"? Nowadays, we usually define a liberal-arts education in the negative -- "impractical", the opposite of technical or professional training; rarely leads to a job, usually due for a cutback next year. It is a vague phrase used to lump together all the academic disciplines that we perceive as soft, subjective, in distinction from disciplines like law, business, the sciences, which are supposedly based on a hard, objective body of facts. In these money-dominated days, in which a college degree is seen merely as a ticket to a higher-paying job, we increasingly forget that a "liberal arts" education has always been meant, not to train one to perform specialized labor -- which is merely "vocational" teaching --, but to liberate the mind. The word comes from Latin liber, a "free man", one who is not a slave. Little wonder that, the more we become slaves to an omnipotent and all-pervading economic order, the less we remember the origin of the liberal arts as the curriculum of philosophers, priests, and magicians.

Originally the liberal arts were seven in number. They were divided into the three-fold Trivium of Grammar, Logic, and Rhetoric, and the four-fold Quadrivium of Arithmetic, Geometry, Music, and Astronomy. These words mean, respectively, a three-way and a four-way crossroads, implying that these paths of knowledge are fundamentally interconnected -- and, by extension, that all other paths can be found to intersect here, as well. The T. was the basis of elementary education (whence we probably get the word "trivial"): Grammar taught the craft of reading and writing; Logic, of careful reasoning; and Rhetoric, of effective communication. The Q. was the basis of advanced education: Arithmetic taught the science of number; Geometry, of form; Music, of sound (and of "harmony" in the most general sense of the word -- "number in motion", as it was often put); Astronomy, of time (of "form in motion"). Moreover, from the very beginning, whether openly acknowledged or carefully alluded to, each of the Quadrivial sciences was accompanied by its complementary metaphysical art. Each dealt not only with the outer structures, but also with the inner meanings of its discipline. Thus, Arithmetic included Arithmology, the understanding that numbers were not merely quantities, but also qualities (that "two", for instance, is also "duality, polarity"); Geometry included what is nowadays called Geomancy, the understanding (in, for example, the design of temples or cathedrals, or in the graphic arts) that the spirit and the emotions can be affected in particular ways by particular forms; Astronomy included Astrology, the divination of the meanings of cycles of time; and Music included not only the study of "practical theory", of nomenclature and technique (e.g. "this is a minor third", "this is the Mixolydian mode"), but also the study of "speculative theory", of the meanings and influences of tones and intervals and scales.

II: History of the Quadrivium

The choice of these particular disciplines was by no means arbitrary. Plato, who learned the arts of the Quadrivium from the school of Pythagoras (who, in turn, probably learned them from the priestly schools of Egypt or Babylon), exalted them in the Republic and the Laws as the essential education for the philosopher -- for the study of these art/sciences awakens the mind to the intrinsic order of the cosmos, freeing it from its bondage to mere "shadows on the cave wall". By studying the links and intersections among these disciplines, one learns to recognize analogies, patterns, correspondences, through which the archetypal Ideas that underlie and unite the cosmos manifest themselves in the world of time and space. Little wonder that, a thousand years after Pythagoras, as the Roman Empire was collapsing and the ancient libraries and academies were being burned and outlawed by religious fanatics, the seven liberal arts were the one essential seed of classical wisdom that was rescued and preserved by the monasteries through the so-called Dark Ages, to bloom in the Carolingian renaissance and yet again in the age of the cathedrals as the curriculum of the first universities. The Quadrivium flourished uninterruptedly in the philosophy, art, and science of medieval Islam; in fact, as interconnected sacred canons of measurement, of spatial orientation and architectural and artistic proportion, of musical scales and modes, and of calendric cycles, the four arts the West knew as the Quadrivium were also the basis of priestly and shamanic education and practice in ancient Egypt, Babylon, India, China, Meso-america -- indeed, in practically every pre-modern culture one can examine.

In the West, up through the Renaissance, the seven liberal arts retained an important role in education, and the influence of the Q. deeply pervaded the visual arts, architecture, music, and philosophy. Composers wrote musical works that were based on arithmetical patterns or, in at least one instance, on the geometry of the cathedral at whose dedication the work was performed (1). The proportions of cathedrals and palaces were in turn often borrowed from the "consonant" musical intervals. The astronomer (and astrologer) Kepler was led to his discovery of the mathematical laws of planetary motion by his investigation of the "music of the spheres" and his discovery that the five Platonic solids could be nested within the orbits of the known planets. Ficino's and Agrippa's magickal philosophies were based on the metaphysical arts of the Quadrivium. Such historical examples could be multiplied ad infinitum.

But they end rather suddenly in the 1600's, with the combined onslaught of the Christian "witch craze" -- which branded the metaphysical arts of the Q. as "satanic" and "occult" -- and the Scientific Revolution, which inaugurated a dogmatically materialistic view of the cosmos (originally, scholars such as Frances Yates (The Rosicrucian Enlightenment) and Carolyn Merchant (The Death of Nature) argue, as a protection against religious persecution, but which soon hardened into an ideology that fit conveniently with the increasing domination of society by commercial and economic interests). It was as if a heavy iron gate had abruptly slammed shut between the outer and the inner, between the left brain and the right, between the physical sciences and the metaphysical arts. After Descartes, Newton, and the Royal Academy, numerology was permanently factored out of arithmetic; numbers henceforth were allowed to signify only quantities, not qualities. Astronomy deliberately eclipsed astrology; time was decreed to be measurable only by the uniform ticks of the clock, not the variegated images of the zodiac. Geometry was circumscribed to exclude geomancy; shape and proportion were deprived of symbolism by engineer and artist both, as utilitarian and aesthete increasingly diverged from their once-shared perspective. Music was silenced from singing of any art but its own -- the scientists who commandeered its study of acoustics could only sneer at the "music of the spheres" and the "harmonies of heaven and earth", and the artists who inherited its practice of harmony and rhythm were eventually left with a mere technical argot of chord-names, scale-intervals, and key-signatures. Although the four Quadrivial sciences survived into the Age of Reason in the new materialist priestcraft of "physics", the four arts were largely abandoned (numerology), ridiculed (astrology), forgotten (geomancy), or isolated (music). The Crossroads was buried and soon forgotten, and the link it provided between the material and the spiritual order was severed. In the new world order, knowledge and truth could no longer be uncovered through the traditional reasoning of analogy and correspondence, but only through the revelation of the Bible or the proof of the test-tube -- that is, either religious faith or cause-and-effect materialism. Once the habit of "pattern"-thinking was replaced by "straight-line" thinking, knowledge lost its unity and interconnectedness, and began to fragment into ever smaller specialties, each with its own jargon, each dominated by its own elite of "experts".

III: A Universal Crossroads

Today, the process of specialization of knowledge has reached extreme proportions. We live in a Tower of Babble, bombarded by information and facts and expertise, little of which is expressed in a common language, even less of which is seen to interrelate or form a comprehensible overall pattern. Yet, on the other hand, we increasingly hear physicists, ecologists, and religious figures speak about holism, interconnectedness, the web of all being. We sense that this is true, that we and the cosmos do participate in an organic unity; but we have as yet found few ways to practice it, or even to understand it on a practical, meaningful level. Above all else, how do we teach it? Academics and educators are at long last beginning to show serious interest in "interdisciplinary" research and teaching, and the ability to reason analogically, to recognize patterns in common among seemingly diverse areas of knowledge and experience, is coming to be recognized increasingly (even, for example, by such a committed scientific materialist as Marvin Minsky, in a recent interview (2)) as a highly practical skill in a world where people must constantly adapt to ever more swiftly changing technologies, careers, and cultural and economic conditions. Perhaps it is time to re-connect ourselves with our past as well, and to excavate and rebuild the ancient pedagogy of the Crossroads -- but in a new way, appropriate to the environment of the Computer Age.

A brief testimonial:

When you begin to follow a theme such as "cycles" or "proportions" -- or "resonance", "twelve", "symmetry", to name some others -- that runs through all four of these fields, you quickly find yourself learning about geometry and math, for example, while you thought you were simply studying music. Moreover, because the kinds of knowledge that the Q. teaches truly do intersect with nearly every other field of study, you find it far more easy and enjoyable to read and understand the underlying theories of texts on physics, electronics, ecology, computer science, philosophy, and many other technical specialties than it had ever been before. The Q. acts as a kind of "universal translating device", which -- because it represents all-pervasive patterns of order -- helps one to identify and build upon the most significant points of commonality between what one already knows and an unfamiliar new field or skill, and to penetrate the fog of jargon and terminology that so often obscures those points. It teaches you to recognize commonality and similarity, rather than the constant focus on distinctions and divisions and particularities, on learning as a linear march through a series of isolated compartments, that is enforced by conventional educational methods. In a larger sense, analogical understanding -- "that" is similar to (but not identical with) "this", "you" are similar to (but not identical with) "us" -- is an essential skill that seems to be largely missing today from a world increasingly dominated by religious separatism, factional warfare, and economic selfishness. One might even say that it is the intellect's counterpart to the emotion of compassion.

I think of the Q. as a kind of compass for navigating the Information Flood (a more accurate metaphor, it seems, than "superhighway"!). Just as you use a compass to orient you to the point on the earth where all lines of latitude and longitude unite, so you use the Q. to find the points where seemingly disparate areas of information converge. This is a twelve-fold compass, in which the arts of the Trivium are not added to the Q. as in the "seven liberal arts", but multiplied with it. Each Quadrivial discipline can be divided according to its "logic", its "grammar", and its "rhetoric". The logical branch is its scientific, "left-brain" side, the investigation into the laws by which it works; the rhetorical branch is its aesthetic, "right-brain" side, its expression of meaning and emotion, as well as its relationship to the spirit and soul; and the grammatical branch is the terminology and conventions it uses, a neutral middle ground which draws on both the logical and the rhetorical branches. For example, in Music, Grammar describes certain combinations of tones as "major triads", others as "minor triads". Logic observes that major triads are formed "upward", by frequency multiplication, with the smaller and more complex interval above the larger, simpler one, whereas the minor triad is formed "downward", by frequency division, exactly inverting the two intervals. Rhetoric uses the upward-tending major triad to express "yang" feelings like joy or courage, and the downward-tending minor triad to express "yin" feelings like sorrow or fear. I would go so far as to say that every field is usefully divisible into these three branches, and note that, by and large, the Logic and Grammar branches are nowadays given most of the educational and practical attention, whereas the Rhetorical branch is written off as arbitrary and random (because it is "non-quantifiable"), and ignored or left to individual whim. (If only architects considered the rhetoric of their geometry and not merely its logic, for example, we might have fewer harsh rectilinear surfaces dominating our landscapes and contributing to the anomie of modern life.)

Send e-mail to Steven C. Rasmussen at SRasmus@aol.com. Tell me something about your own interests and background, too.

A website for teaching and learning music theory in innovative ways (currently under construction) is the first phase of a proposed Quadrivial website, called Liberal7Arts. (In The Goodly Spellbook, you can find a more complete version of the information on the 7 Modes that this prototype website demonstrates.)

FOOTNOTES:

(1) Guillaume Dufay, in the 15th century, structured the motet "Nuper rosarum flores" according to the architectural proportions of the Duomo in Florence, at whose dedication the work was performed. Return to text.
(2) In Mira Balaban et al., eds., Understanding Music with Artificial Intelligence: Perspectives on Music Cognition, 1992 MIT Press, p. xi-xii. I cite this at length because Minsky arrives at the same conclusion as me from the opposite end of the philosophical spectrum:"So you think computer science -- or at least AI -- needs other, richer kinds of descriptive ideas and terms [than binary logic]? MM: Yes indeed. I want AI researchers to appreciate that there is no one 'best' way to represent knowledge. Each kind of problem requires appropriate types of thinking and reasoning ... For example, logical methods are based on using rigid rules of inference to make deductions. This works well inside formal domains -- those artificial worlds that we imagine for ourselves. But to cope with the unknowns and uncertainties of reality, we must base our actions on experience -- and that requires us to reason by analogy, because no two situations are ever quite the same. In turn, this means that we have to be able to recollect similar experiences and understand which differences are relevant. But in the logical world, ideas like 'similar' and 'relevant' are alien because logic can only answer questions like 'what is this an instance of?' or 'what is this a generalization of?' The trouble is that concepts like instance and generalization apply only to ideas -- because no actual object or event can be an instance of anything else. However, real things can be seen as related -- at least in an observer's mind -- by apparent similarities of structures, effects, or useful applications. Certainly this is the case in music."Return to text.

"COMPASS" SYMBOLS:

Motivation in Education

Fritz Hinrichs

Education must transfer from generation to generation the core of our culture’s accumulated body of knowledge. In our day, many think that to believe in an accepted body of knowledge that prioritizes what is important to learn and what is not smacks of elitism and exclusivity. In part, this charge cannot be denied because discernment often demands that we play the role of intellectual hatchet-men; however, if you will reject the notion of a “canon” of knowledge, you are faced with the task of creating a rational for your own curriculum that can give a convincing answer to that most awkward but ubiquitous question, “Why do we have to learn this?”

Having cut themselves free of the constraints that guide classical education, our large educational institutions have resorted to an ever increasingly frantic attempt to construct a convincing rational for their methodology out of the vacuum of their own errant psyches. Yet, whether they resort to ethnicity, technology, gender or the deviancy of pop culture, their attempts to give meaning to their teaching end up being mere exercises in personal assertiveness.

Even though our secular school system abandoned any true foundation for absolutes by rejecting God’s authority over all of life, academia has been able to keep a coherent system together by merely coasting on the inertia built up by our culture’s rich intellectual tradition. This has been especially true in Mathematics where a healthy regard for disciplined rigorous thinking and applied mental struggle were thought to guarantee mental fruit, but now, even this field seems vulnerable to the prevailing epistemological meltdown. The “new new math” curricula currently being promoted by the National Council of Teachers of Mathematics seems bent on destroying mathematical rigor in hopes of instilling students with such dubiously mathematical abilities as “group learning skills”, “guess and check techniques” and “awareness of diverse cultural approaches to mathematics”. By attempting to make math more “interesting” and “accessible”, they have removed precisely what makes Mathematics a joy to learn. No longer will students be able to savor the hard won pleasure of successfully working complex mathematical procedures. Even in the field of Mathematics where you think it would be nearly impossible for the mind to be led astray, our schools are abandoning rigor and sense for mere gimmicks and social agendas . What is the philosophical lesson from all this? The closer and closer man comes to pure epistemological autonomy (that is, the ability to accept only those ideas he finds within himself) the closer he comes to resigning himself to complete nonsense.

For students to be motivated in their studies, they need to know that what they are studying is indeed of real significance. They need to know that they are not being feed some new-sprung agenda or half-baked innovation that will simply go the way of the faddish educational chaff that, once having gleaned its profits, goes to the winds never to be seen or thought of again. Students need to know that they are being feed the best that our civilization has to offer- that they are studying something that is much, much larger than themselves. As I guide students through the study of the proofs in Euclid’s Elements, it is always a pleasure to point out to them the fact that their geometry book is the same that was used by Thomas Jefferson, James Garfield, Lewis Carroll and a host of other intellectual witnesses going back before 200 BC. Even though Euclid makes absolutely no attempt to show you how his system is “practical”, I do not find my students asking, “what will I ever us this for?” Because the Element’s is truly a classic work, the students come to see why mathematicians have admire Euclid’s beautifully constructed proofs through the ages.

When we climb out of the broad stream that comprises the wisdom of the ages, it is very easy to lose our educational bearings, being blown to and fro be the winds of opinion. Furthermore, without a good rational for our curriculum, we will be unable to resist the student’s desire to find the path of least exertion between now and breaktime. To be motivated to work, we need to instill in our children first godly character and then the conviction that their studies are indeed significant. Despite the mantras that are continually chanted around us, the motivation for pursuing an education does not come from looking at charts of the average salary levels of various degree recipients, or from following the educational atomism that reduces all educational accomplishment to a single GPA, or by explaining all labors as just steps in the great “ordu salutis” culminating in acceptance by that Ivy-league dream college. By putting before students these poor reasons for getting an education we are drumming into them the idea that education is a means and not an end. Until they understand that education is an end in itself, that indeed, the creation in which we dwell and the historical saga in which we take part are truly worthy of our interest and concentrated study, we will only see them labor with a slave’s reluctance.

Why Good Grammar?

by Richard Mitchell

I HAVE been given this assignment: To write on the question, Why good grammar? I have not been explicitly asked to answer the question, however, and for that I am grateful. It is a strange question, after all, something like Why clean hands? And its best answer is really, Well, why not? If there is anything to be proved here, it ought to be left to those who support the cause of "bad grammar."

The fact that a collection of English teachers can put that question forth as worthy of serious consideration is far more interesting than the question itself, for it suggests that we need desperately to defend the continuance of an enterprise in which we have regularly failed to bring about any noticeable public devotion to what we call "good grammar." And it conjures up a disquieting image of a colloquium of mining engineers which devotes one of its afternoon sessions to considering the topic: Why would English teachers sit around asking themselves "Why Good Grammar?" Surely they would find more fascination in that question than in "Why Good Arithmetic?"

Behind the current murmuring about the teaching of grammar, and the supposedly related teaching of writing, there seems to be a strange, but not at all unaccountable, misunderstanding. When I hear the plans and pleas of the curriculum coordinators and the language arts facilitators, I get the impression, which I am sure they intend to convey, and which they may actually believe a correct one, that the grammar of the English language is inordinately difficult to learn. They have such elaborate (and expensive) plans for the teaching of it, that one might think them engaged in training a whole nation of children as master architects or violin virtuosi. But the learning of grammar is not a difficult and improbable achievement. In fact, there is some sense in pointing out that no one really needs to learn it.

While I often do claim, out of exasperation, to have students who have no native tongue at all, the simple truth is that there is no person on the face of the earth who doesn't know the grammar of his language, for who doesn't would have no language at all. He would be unable to utter anything that we would recognize. Grammar is a strange and as yet unfathomable power to utter, and without any deliberate thought at all, any sentence or any infinite number of possible sentences.

But that power, of course, is not what the projectors of comprehensive nation-wide intercurricular grammatical renovation have in mind. They must be thinking about things like misplaced modifiers and "between she and I," and even of pronoun agreement and the difference between restrictive and non-restrictive clauses. And I think they may be wasting their time, and lots of money, some part of which is mine.

Even the "grammar" of conventional rules and regulations is not, in our language, either a large body of information or a particularly difficult one to learn. While serious meditation on the difference between restrictive and non-restrictive clauses will lead to important insights about the philosophy of meaning, the student who would simply like to recognize which is which needs neither the meditation nor the insights. And the same is true of the agreement of subject and verb and the sequence of the tenses, and true, as well, of all those things that so many students seem never to have heard of, after having heard of them over and over again for twenty long years. We ought to be asking ourselves not whether we should persist in the teaching of "good grammar," and seeking, as it must often seem to the rest of the world, some excuse to abandon the whole futile enterprise, but why it is that we can't seem to teach such a small body of principles so easy to grasp.

I have encountered (who hasn't?) governors and college presidents and deans and even guidance counselors, not to mention a few English teachers, who can not consistently make their pronouns agree with their antecedents or punctuate non-restrictive clauses. Their defense of such lapses is an interesting one; on the one hand it has some merit, but only if we grant the implicit assumption that "language is communication," which we ought to be careful not to do. On the other hand, it is an inadvertent self-condemnation, admitting to a lapse somewhat more serious than an ill-chosen pronoun. They say, "Well, you knew what I meant, didn't you?" And I did. But I also know where they had been, or, to put it more accurately, where they hadn't been.

In any mouth or pen, grammatical habits, especially those more important to writing than to speech, present a miniature character sketch or autobiography. Many of the attributes therein revealed are personal, and some collective, social, or even ethnic, and, strictly speaking, none of a teacher's business, and that is so whether the unwitting autobiographer is the student I am supposed to teach or the academic vice-president who is supposed to be more academic than I. Most of those little gaucheries and solecisms that make us think that we ought to teach more grammar are only the habits of a mind in a certain condition, a condition to whose cure all of us have supposedly chosen to devote our lives.

I know of no specific name for the condition I have in mind, and I find it hard to describe. Although it is a kind of thoughtlessness, or unknowing, it is not truly what we usually mean by ignorance--the simple lack of information. Nor is it dullness of mind, or stupidity, whatever we might mean by that. It is that condition in which the mind takes but little, or takes but rarely, the grasp of itself. Although it isn't necessarily caused, or perpetuated by a lack of reading, or cured, either, by a glut of reading, it is nevertheless the condition to which we seem vaguely to point when we sit around and complain that our students seem never to have read anything.

While I would not want to make a habit of it, I will for now call that condition "inducation," the state of being led into something, rather than out of it, as "education" suggests. But even that idea is not quite enough, for the condition I have in mind is not accurately described as one into which we are led, but one in which we are left, and out of which we might be led. In its purest form, it is the condition in which the mind operates like an organ of the senses, thinking what it must think in response to the suggestions of its environment, as the ear hears whatever it can hear and the open eye must see whatever lies before it.

As Holmes often remarked to Watson, it is not at all uncommon for the eye to see without noticing, and when the mind works like an organ of the senses, it is to be expected that it will do the same, which can perhaps be described as thinking without thinking about, without considering, reflecting, comparing, weighing, or judging. It is the condition in which the mind serves as a registry, a perpetual catalog of whatever presents itself. That condition is not only one in which we are born, but one into which we fall continually, and into which, it must be admitted, we had better fall in the ordinary course of daily life, lest we find ourselves walking into closed doors and driving our cars on the other side of the road.

Consider now the opposite condition, the one in which the mind, still the hapless receiver of the world about it, can nevertheless withdraw far enough so that it can, and will, consider, reflect, compare, weigh, and judge--comment, as it were, on the items in its register. How are those acts performed? What evidence have we, and what evidence can we provide, either of their existence or their conclusions? The form in which they exist, and the only form in which they can exist, is language, nothing but language. They are acts of language, which are to the mind what motion is to the body.

One who considers is in fact talking in himself and with himself, and one who does not consider but receives only is reciting what is really a kind of list, a list of the names of the items that appear in the world. To that list he may well add the names of items associated with those presently visible, but the act of the mind that he does not commit is the act we call, in what is truly a grammatical distinction, "predication." And thus for the reciter, any kind of language will do, for the outer reality that it names is what it is no matter what the reciter calls it, and that he calls it something in its presence is sufficient. For the considerer, the outer reality engenders a statement that is not the reality itself, but a statement about it, a predication. And that statement brings forth statements about the statement.

There is another way to understand the reciter's condition: in his case, language really is communication. His inward words correspond to the world outside, whether accurately or not, it doesn't matter. It is he alone who needs to be satisfied with the correspondence. And should he, someday, find himself not satisfied and impelled to be satisfied, he will have to move to another floor of the mind, as it were, and make statements about his statements. Then, grammar will become important to him, although he is very unlikely to say, "Aha! I now see that grammar is important."

An example: as a child in school, I was not baffled by subordinating conjunctions, but that was only because I was totally indifferent to subordinating conjunctions. I knew what the book said about them, and I could answer the teacher's questions about them in the terms of the book, and that was all I needed to do. It satisfied the teacher. But to be baffled, one must be interested, and to be interested in subordinating conjunctions, a condition that seems especially dismal but, fortunately, remarkably unlikely, requires first an interest in subordination itself.

But an interest in subordination is not unlikely, not, at least, in a mind that has discovered certain of its powers. Subordination is the root of logic, which is itself a grammatical art, the consideration of the just relation of one statement to another, and logical fallacies are errors of grammar--a confusion, for example, as to whether two statements can be related as "if" implies or as "because" implies. Some minds, at some point, discover that they can not make sense of their own predications without attention to grammar, although they do not ordinarily think of what they are doing as an exercise in grammar.

If they don't, that is our fault. We teach grammar, depending on our factions, either as a ticket of admission to "culture," or a marketable skill serving to provide industry with plenty of communicators. Some of us would rather not teach it at all, seeing in it just another penchant of the "dominant class." And in almost all cases, we are vainly trying to teach it as what it is not, a catalog of rules and regulations, to the wrong people at the wrong time, to unawakened children, of many ages, whose minds have not begun to want any grasp of themselves. For them, and their registers, communication is enough, and they are, for their own purposes, quite right in saying, just as all those unawakened deans and presidents and guidance counselors say, "Well, you knew what I meant, didn't you?"

And, just like the deans, and presidents, and guidance counselors, the children in our classes are as they are because their minds have not yet wandered or lingered in the right place, in that place where the mind does its most important work through the consideration and manipulation of language. Thus it is that even the most elementary facts of English grammar seem to them arbitrary. It is for the sake of the work of the mind that a pronoun "agrees" with its antecedent, however far away; it echoes and reasserts one relationship of ideas rather than another. But for him for whom the naming of ideas is enough, that curious "agreement" seems finicky, a strange notion of schoolteachers, and an unnecessary complication of a very simple and straightforward process--communication.

We are at least partly right when we sense that students, just like the deans and presidents and guidance counselors, seem never to have read anything. That isn't true, of course, they have read a lot, but they are in the habit of reading (in both cases) texts carefully chosen not to disrupt the mind as a sense organ. Ambiguity, irony, and wit, to say nothing of deliberate pondering and metaphoric analogy, are not common either in the texts exchanged by managers or in those by which children are taught to "read." For it is not truly reading that they are taught; it is the receiving of communication. We do this in the strange belief that they ought not to have to suffer perplexity, but it is only as the mind notices its perplexity, and suffers--for the noticing is not by itself enough--that it begins to move from recitation to consideration, to taking some grasp of itself.

We have formed the habit of teaching "good grammar" as though it were a prerequisite for other powers, especially for good writing and clear thinking. In fact, though, those powers are really one--the power of the mind to consider itself and its own works. Writers do not write grammar any more than readers read grammar. Both, unless they are mere reciters and receivers of communications, do the work of the mind in grammar, for that work can be done in no other medium. And the knowledge and understanding of the rules and conventions of grammar come most readily to one whose mind is already using them. Before we can bring our students to remember, and enforce upon themselves, those rules and conventions, we will have to bring them to need them, and to know that they need them.

"Good grammar," in the fullest sense of the term, is neither an embellishment nor an accessory to anything else. It is the Law by which meaning is found and made. It may be, of course, that a good "education" ought to provide something more, but it is preposterous, perhaps even wicked, to suggest that it can be had with anything less.