Everyone who learns a little bit about educational history in the U.S. learns about John Dewey. His published ideas, which would seem commonplace and hardly revolutionary today, caused a change in the system of education throughout American public schools which seemed at the time quite momentous. Ordinary people noticed it and referred to “the new schools.”
I get these things from reading old books.
A few points about all this before moving on.
- We are still living with Dewey. I dare say that my own essay, below, will be accidentally littered with terms and idea unknowingly inherited from him. It’s almost impossible to avoid. One might think that as he dwindles into the chronological distance, other important figures intervening, he would become less important. In truth, as far as I can tell, his thought is the overarching category under which all intervening figures of importance and power work. The wandering, lost, constantly multiplying and proliferating quality of educational philosophies since he published first in 1893 – these are all the results of people trying to flesh out his basic ideas. Those who try to work outside his concepts are pushed to the side and looked at as oddballs and non-participants.
- Dewey reformed not just certain aspects of education – but our conception of the whole purpose of education. Concomitant with this new conviction came a desire to revolutionize the entire practice and structure of education. He might as well have borrowed Ezra Pound’s motto in regard to poetry – “Make It New.”
- As is usually the case with more-or-less reasonable men who bring in revolutionary ideas, his followers went much farther with it all than he ever meant them to.
- Nevertheless, Dewey explicitly meant his revolution of education in the U.S. to turn public schools into primarily sociological, instead of educational, institutions. And the purpose of sociology, in his intent, was to produce liberal and progressivist citizens. In other words, he imagined an institution to which all parents, of every stripe of conviction, would send their children – and from which those same children would emerge uniformly liberal and progressivist. It was a brilliant maneuver to stamp out differences of opinion – or rather, to make differences of opinion feel unsocial. In the process, of course, it became inevitable that education itself would suffer within that very institution supposedly dedicated to it. We noticed these effects even at a Christian college. The focus of the administration was on creating good Christians – a mere christianization of Dewey’s ideas. Learning itself suffered. The more virtuous and sensitive the student, the more it suffered.
- Dewey’s active principles are clothed and presented in all sorts of phrases which indicate kindness to, and attention toward, the child. Of course, any mother would feel. (And it is the mothers who really have to be convinced in a matter like this.) Of course, the child’s potentialities should be attended to in the construction of the curriculum. It’s no good teaching him all sorts of wonderful things if he isn’t capable of learning them.
I have always been a proponent of kindness to children. I will be the first to recognize that when conservatives become reactionary, their response to such intellectual subterfuge as John Dewey employed is often to become cruel and harsh and inattentive to children, to show that they are unaffected by liberal thinking.
As I mentioned in the Fascism/Collusion post, every revolutionary movement has two parts – the ostensible philosophy on which the whole thing is based (which is inevitably built up around the recognition of some real virtue to make it more plausible) and the real intentions, which are always quite practical and would horrify the same people who accept the ostensible philosophy.
To refute revolutionary thinking, then, the conservative who would not be a mere reactionary must both refute the ostensible philosophy, and pull back the curtain to show the real tendency of the practical plans.
In order to do the first, he must delicately separate the genuine virtue from the strands of euphemism, false argumentation, and insinuation in which it has been wound, must show how that true virtue has been bloated to force out other virtues, and must demonstrate a better use of that virtue.
The second is far simpler, especially when 100 years of practice or so makes evaluation supremely possible.
For the first, one might say: Yes, education should be kind to children. It should, if possible, take their natures into account. But let’s not forget that to our forefathers, one aim of education was to improve on nature. They believed a liberal education – meaning, the education which would produce the virtues of a free man rather than of a slave – had to curb certain tendencies in boys (and sometimes girls) and encourage others. All our important cultural figures were produced by such methods. Why, John Dewey himself was produced by it. Oughtn’t we to think a moment about whether their schoolmasters knew a thing or two?
For the second, one might continue by asking – for what have we given all this up? For what have we given up the possibility of producing another Robert Frost – who was educated in the old manner? Has anything greater arisen as a result of all this? Or have we simply bogged down?
Dewey, who believed that schooling could produce civic virtues but who was also a utilitarian, might find grievously ironic the fact that education now, having been through several ideological revolutions, is resolving into a simple, middle-class economic affair of preparing boys and girls to earn wages. Utilitarian, yes. Bourgeois, beyond a doubt. Fascist – more than a little. Ideology? No thank you – it has proved exhausting, impossible to live by, and more than a little confusing. Let’s just stick to getting children ready for jobs in the tech sector.
But of course, the real purpose has not failed him. Public Schools are still churning out dutiful little liberals who go on to live miserable, inhuman, and servile lives of uncritical rage at the system.
Is it possible – isn’t it, in fact, likely – that in abandoning a freeman’s education, American public schools have simply thrashed around a bit before settling down to the natural level of a slave’s education? Obedience, conformity, thoughtless agreement, self-control even at the level of belief, uniforms, and ceaseless labor that does one no apparent good – this is the education of Dewey’s future and our present.
A great part of the virtue of homeschooling is that you really can individualize your educational methods and pace to the needs of the individual child – an ideal which is actually impossible in any affordable school system.
In practice, it turns out that most copyrighted and trademarked educational methods are mere gimmickry, designed by inspiration of the devils (or so it seems) to distract people from the fact that the simplest methods of conveying knowledge are really quite inexpensive.
I’ve plunged through – and sometimes skimmed through – a few of the more promising child development, intelligence-theory, and educational philosophy books in the past few years. I’ve tried quite a few things out. Here’s what I’ve settled on – the easy, no-frills, no-nonsense educational methods that are inexpensive, easy, and most of all, beloved by children.
For it turns out that tuning the educational methods to the child is not that hard, once you get over the dewey-eyed confusion about a freeman’s education being somehow cruel. Despite learning style differences, of which much can be made, most children fall into an average which responds to a few basic methods. There are universals on which education can depend, and methods which can be used over and over again to the best effect.
Dewey was, I think, forgetting (or ignoring) one thing. Boys hate school. (Many girls of spirit do as well, but with girls you are more likely to get a suicidal conformity.) They always have and always will – with the partial exception of a few boys whose intellectual hunger is so great their natural and quite proper dislike of being treated like a prisoner is overcome.
As they grow, they learn that it is good for them to bear the yoke in their youth. (It makes them, you see, determined that they shall never bear it in their maturity – and that is a salutary determination.) They begin to understand that it is necessary for them to acquire this learning in order to do well in their lives and in the world. But still, the basic attitude is persistent – please don’t make me spend anymore time on this than necessary.
That is the true nature of the child to which education must be shaped. It is why modern education, with its roundabout methods, extraneous subjects, and endless hours is really and truly cruel. It is why modern education triggers mass-murders.
And that is why nearly all the educational models that have sprung up over the last century are largely useless. They are really just “tricks” – little roundabout ways of making students “get it,” which really have the effect of forcing kids to spend hours on something most of them could get in minutes if it were presented directly.
Here they are then – educational principles for the simple life, the free life, the traditional life.
- There are two kinds of things which must be learned. The first are processes. The second is a body of knowledge. Everything, as far as I can tell, falls into those two categories.
- Not all processes and a not all knowledge can be taught at school. The most important and the most basic must be taught there, and the students left with the capacity to teach themselves the rest.
- Accordingly, there are two ways of learning – an instinctive one and a cultivated one. The instinctive manner of learning is through imitation. The cultivated manner of learning is through symbol.
- Processes are learned primarily by imitation. Knowledge is acquired primarily through symbol.
- However, in fact, the acquisition of knowledge through symbol must be approached first by the acquisition of process through imitation. In other words, the student must first learn how to learn – and he must do that by imitative actions which will get him started by sparking the activity of learning.
- Ultimately, the development of the student’s own potential is not at odds with artificial, conventional, and demanding learning methods. Reading and writing are highly artificial and non-instinctive activities, and yet they are the core of education. It is, you might say, human nature to do extra-natural things.
Now for the discussion and explanation.
Children begin life primed to learn by imitation. This is the common path for both free people and slavish people. By the time they are five years old, children’s minds have been shaped by their surroundings and family to a degree and in a way that is never fully reversible. The purpose of this is merciful. It is to condition the child to the way he will have to live.
Early childhood education must focus on imitation. For babies, this works two ways. First, the mother (and her helpers) imitates the baby’s random actions and noises. This delights the baby – which encourages everyone to keep going – but the effect is also ontological. It helps the baby feel real, and like a real person. It helps her understand that she is an individual in relation – or rather, a person in communion. After that, the result is educational. Imitation is the first thing the baby must imitate. After the mother has imitated the baby for while, the baby imitates that imitation. She begins to imitate the mother in return. Now real learning can begin.
Walking, talking, and all the “developmental milestones” will be learned when older people hold babies while they do things, and when they directly elicit imitation from babies through babyish games.
When the baby becomes bored by an activity (not an easy stage to reach) that means she has learned what she can learn from it. It’s time to progress, make more complex, or extend the game – or move on to something else.
Until the child learns how to read, which is the beginning of the second stage of education, learning is dependent first, on the child’s having things to imitate, and second, on the child’s having things to experience. Of course, the natural environment itself is quite rich in potential experiences. But if a child is to be pointed toward, and readied for, that education which will make her free of herself and of the whole human traditional and of all life’s possibilities, she has to be primed for academic attainment, through imitation of human behavior. The best human behavior, of course.
This is why young children should be with their parents while they work, as much as possible. It is why adults should talk to one another around children, and talk to the children themselves.
There is one practice in early childhood which will do more for a child’s intellectual readiness to learn than anything else I know of. It is a practice which can feel quite artificial, but which children adore. That is the practice of giving very young children phrases, sentences, chants, and rhymes to repeat.
Mother: “The ball is behind the block. Say it.”
Child: “The ball is behind the block.”
Mother: “Where is the ball?”
Child: (Giggles. Has no idea.)
Mother: “It’s behind the block. Give Mommy a kiss and go play.”
Two Months Later
Mother: “Sweetie, did you see where Mama put her earrings?”
Child: “Behind the lamp.”
Learning to speak is the first introduction to the symbology which will overtake imitation as the primary vehicle of learning by the time the student is 14 years old or so. Of course, speaking itself is a process and must be learned by imitation. Most of the mistakes of revolutionary educational methods come from confusion about this. It is assumed that as soon as the child can speak, it should be encouraged to speak creatively, spontaneously, and from its own thoughts. This is basically like asking a car to run without first putting fuel in. The gas-tank of language should be filled to the brim before the trip to the far fields of human thought begins.
Beginning Formal Education
Imitation is still the primary method to be used until about 11 years of age. However, this imitation branches out into diverse forms around the age of 7.
The most basic, and most historically effective, methods involving imitation are recitation and copywork. These methods, be warned, are anathema to post-Dewey educators. Mention them and such educators will immediately react to mental images of boys being flogged for forgetting a word in their recitation, or girls having their hands beaten with a ruler for blotting their copywork. The conversion, you see, was really accomplished emotionally.
Of course, the dreadful and real effect of no longer requiring recitation in school is that the American people has lost its voice. Shrieks from the left about “voiceless populations” is, of course, meant to cover up the leftist intention that the average person, himself, should become voiceless. The average person isn’t easily indoctrinated. Only people who have been outraged and destabilized by trauma can have impractical ideologies implanted in them, can be persuaded to spend their days in howling for change.
No one can learn to speak well who has not first imitated those who speak well.
So, newsflash: you don’t have to flog you children if you set them recitation and copywork. (And you shouldn’t.) Simply point out the errors, ask the child to correct them in good time, and praise the success so far as it is real.
(A note: never praise the child herself. This gives rise to vanity, which sets in motion a cycle of self-overestimation and self-doubt that is quite paralyzing in the long run. Praise her actions in so far as they are well-meant, or successful. Praise them for what they are, specifically. Praise honestly. Never ever say “Good job,” for that will become a meaningless, anxious habit, and convey to your children that you are really afraid they are not doing a good job, and don’t want them to know because they might feel badly. “Well done,” will do very well when it is really merited. “I’m impressed,” is for when they have genuinely surprised you with some initiative or extra effort. Noisy and excited praise, combined with some treat or reward, can be used to encourage a child to try even harder. However, exhaustion can quickly set in and the child may regress to a former level before coming around and trying again. Better use it sparingly.)
The marvelous thing about recitation and copywork (besides the fact that it is direct, to-the-point, and doesn’t take up much time) is that these methods get in the beginnings of all the other subjects while the student is still learning how to speak and how to write. History, literature, logic, science, religion – it can all be introduced by the simple expedient of setting a related sentence to copy or a paragraph to short-term memorize and recite.
It is important that these sentences and paragraphs should be borrowed from high-quality sources. A child knows the difference when you set him “The quick brown fox jumps over the lazy dog,” as opposed to a Bible verse or a line from Shakespeare or paragraph from Winston Churchill’s speeches. He knows whether or not you are taking him seriously as a potential full-statured man.
Imitation gives rise to process – but in this unexamined, rote interaction with the cherished and hoarded works of humanity throughout history, the first inklings of thought arise. They do so spontaneously. They are not put there by a teacher and therefore they are the seeds of intellectual freedom.
Mathematics involves the acquisition of a certain body of knowledge – the numbers and their basic relationships to one another. However, in itself it is basically a process, and must therefore be learned by imitation. Here is how to teach a child to add.
Mother (holding up one finger) “How many fingers am I holding up?”
Mother: “Correct. How many am I holding up in my other hand?”
Mother: “Well done. Now watch. My hands are coming together and meeting in the middle. How many fingers am I holding up?”
Child: (Giggles, doesn’t know)
Mother: “Repeat after me. Mother is holding up two fingers.”
Child: “Mother is holding up two fingers.”
Mother: “Can you hold up two fingers?”
Child: (Holds up two fingers, one on each hand.)
Mother: (Separating her hands and then bringing them together again): “Repeat after me. One… plus one… is two.”
Child: (After a few tries, gets it right, along with the motion of separating the hands and bringing them together.) “One… plus one… is two.”
Mother: (Doesn’t bother with deep explanations.) “You did it. Well-done. Go play now.”
Play time is essential. While the child is playing, the background mind is working through what has just happened. A few days later, the process will be repeated, and the child will be much closer to understanding.
Later, mathematical problems can be set the child in the ordinary manner. Avoid modern math textbooks like the plague. They are becoming more and more convoluted in the attempt to make mathematics, a decidedly unpleasant subject for most children, conform to the Dewey principles.
In the home, math is best taught through conversation. You didn’t think it could be done? It can! Up to a certain point, arithmetic can be demonstrated in everyday activities. This grounds mathematical concepts in concrete reality. That is where they originated in the history of mankind, and that is where they should originate for the child.
In our family, we often use car time to set one another riddles, play twenty questions or solve tricky syllogisms. Once it’s established that this is fun, math questions can easily and naturally be introduced. The point here is to play with numbers, not to get assignments done.
Once the child’s mind has sufficiently abstracted mathematical concepts from the rote, concrete knowledge of numbered things and what we can do with them, then he can be set mathematical problems on paper.
Imitation is still critical here. When I taught my son long division, I sat down beside him on the couch with plenty of paper and a pencil. I wrote out the four steps: divide, multiply, subtract, bring down. I knew there were further little sub-steps, but I did not require him to memorize them. I knew he would automatically subsume them under the main headings. I did require him to recite the four steps several times, at different points in the process.
I began by writing out long division problems, and solving them myself. As I worked, I talked my way through what I was doing, while he watched and listened.
When he began to have a hard time concentrating, I knew he had gotten as much out of the demonstration as possible. It was time to move on to a new stage.
I still held the pencil, but I stopped talking. I now required him to tell me what to do next. At first he simply recited the four steps. “You should be dividing now. Now you have to multiply.”
“Multiply which numbers?” I would ask.
“I don’t know.”
“The four and the two. You see? That allows us to set up the subtraction and get the remainder.”
“It’s so hard!”
“I’ll do another one.”
Finally, when he was correctly instructing me to do the problems, I passed the pencil to him. At first, he felt lost and couldn’t transfer his understanding of the problem from a situation where he was telling me what to do, to self-direction.
(This aim of self-direction, by the way, is the secondary aim of home-schooling, and probably of liberal education, as well. The home-schooled child becomes a self-teaching adult. He has learned how to learn, and he will never be dependent on someone else to fill his mind.)
I had learned by now never to become upset when a child can’t give the answer. The most expedient way for the child to learn the answer is for the teacher to give it to him. It’s not cheating; it’s imitation. Give the child the answer as many times as he needs; eventually he will start giving it to you. In the process, you will bypass that anxiety which usually accrues to question-and-answer style teaching, in which the child must answer or fail.
Whenever my son didn’t know what to do next, I told him what to do, quite simply. Sometimes I added in a “because,” connecting the step to the whole process. “Because if you don’t subtract, you don’t know the remainder, and you can’t use the remainder to create a new, temporary dividend.”
Success came at last. He was sitting there working through long division problems in silence, with hardly any mistakes. Throughout that time period, I had been gradually increasing the size of the numbers we were working with.
At about this time I noticed signs of fatigue. I decided enough was enough. We didn’t pick up division again until the following week. At that point, I set him review problems of many, many short division problems. At the rate of 20 problems per day, it took him several days to work through.
Then he reached a page in his text book where he was supposed to do long division again.
“Do you remember everything?” I asked. “All the steps? You don’t need my help?”
“Mom,” he said scornfully, “I’m not an idiot!”
So that was how a boy learned long division in two hours flat. No tricks. No breaking things down into endlessly smaller steps. Just minimal explanation, lots of imitation, and repetition leading to mastery.
This “primary education” time between learning how to read and entering the second phase of education requires that facts be learned. Facts about history, geography, science, and religion are now to be driven in, like pegs in the wall of the mind. Later on, these facts must bear the weight of whatever complex ideas the student is able to acquire.
Again, this is really to the child’s liking, far more than most adults might suspect. Long-winded explorations of trade-routes are pointless when they have no context. The names of nations, both ancient and modern, must be memorized. (Maps and pictures can be looked at, but as they are not essential they don’t need to be required as a permanent acquisition.) Capitals, rulers, wars and battles must be learned. The names of plants and animals, both common and scientific, should be acquired. The continents and the nations should eventually be as familiar as a child’s own street address.
Unfortunately, because we are cut off from the past, it may be quite confusing for an adult to figure out which facts the child should be learning. I have adopted the expedient of creating a notebook where I jot down lists of things I think my children will need. These fall into three categories:
- Common knowledge, which they will be embarrassed by not knowing
- Prerequisite knowledge, without which they will be unable to remember and understand the complexities of history, science, literature, language, and advanced thought, and
- Cultural knowledge, which is basically religious and poetical
When I’ve accumulated enough of these lists, I print them out in large type, punch holes in them, and put them in the child’s notebook. The child draws on the notebook for copywork and recitation.
This notebook, a math text from the 1940’s, spontaneous conversations, and an enormous, every-growing pile of books for reading, is my curriculum. The notebook has cost me less than $20.00 so far. The textbook I picked up for fifteen dollars in an antique shop. The pile of books is fed by thrift store and Amazon purchases.
In short, anything you wish your child to understand completely later on must be introduced in skeletal form now, by making him memorize the “what.” Other wise he will not know “what” you are talking about when you try to discuss things at a fuller level. Names, places, events, dates, and objects – drive in those pegs. Make sure they are the biggest and the most important.
Imitation is still key at this stage. These facts should be recited aloud. As the child’s physical voice develops, he needs continual practice using it to produce all the words he knows. It is a tendency of homeschooling, and a wrong one, that students typically know far more words than they can easily call up in speech.
Because I believe it is essential that every person of sufficient intelligence should learn formal logic, children at this stage of education should memorize, copy, and recite syllogisms. You will not have any trouble explaining what logic is to a student who has already been using it.
At the same time as all this learning-by-imitation (or rote, if you will) is going on, something else must be happening as well. The child must be acquiring experience. Concrete experience is the foundation of all abstraction. Some experiences can be imitative.
For instance, John Holt details many delightful experiments in which he takes some machine or artifact into a classroom, and then just sits there using it or playing with it. The children naturally crowd around and soon begin asking, “Can I try?” In the process, they end up learning, not the theory of a pendulum or some other simple machine, but the sense-experience of it. Later, when they must begin theorizing about such machines, they have something to go on.
The experience gained in such activities is not directly academic; it feeds the intellect by introducing ideas to it in the only way ideas can be introduced: through sensual data. It also feeds freedom because thoughts are allowed to rise spontaneously in the child.
It might seem odd to some that I could promote both the ideas of John Holt, the father of unschooling (“just leave your children alone and let them discover for themselves”) and traditional, highly formal, rather austere methods borrowed from traditional education.
However, it is Aristotle, and the whole Aristotelian tradition of formal logic and Reason, which insist that all ideas begin as sense experience – must, and do. (Whether idea-shaped slots in the mind are there waiting receptively for those experiences, ready to form the accompanying ideas, is another question, and one about which a Platonist will have more to say.)
Therefore the complete education is complementary – sensual experience must complement rote reproduction of facts; imitation and acquisition of process must complement the secret, inner freedom that only arises when a boy, a girl, a man, or a woman has in some quiet moment an original thought or a private discovery.
It does not matter whether someone else somewhere has also had that thought or made that discovery. It matters that someone is beginning to really think.
Siegfried Engelman, from whose ideas of Direct Instruction I have borrowed here, also talked about enrichment of the environment. He unified the two ideas by pointing out that the teacher is also part of the environment. The adult who converses with the child, shows him interesting artifacts and creatures, gives him the names of everything in his surroundings, teaches him the moral code, and sings nursery rhymes to him is part of the reality against which the child wants to dash himself again and again to find out its shape. The parent or teacher is the human part of that environment, and therefore no less important than the toys, books, rocks and grass which the child must explore for himself.
This experience of being set loose to explore is something joyous and productive in ways that no teacher can predict or replicate in a controlled situation. He can, as John Holt did, cunningly participate in and provoke such experiences. If he can, perhaps he should.
At any rate, all through the primary education, we want to see children who spend as little time as possible acquiring the most important facts and processes, but are then set free in the richest possible environment to make discoveries, have thoughts, and appreciate the freedom which so starkly contrasts to the school hours.
The Second Phase of Formal Education
This phase comprises all the preparation for University or career. It begins at about the age of 11 or 12, when the student’s rational faculties are blossoming at a fairly furious rate. He has already become competent and quick at reading, speaking, writing and all the basic mathematical functions. (If he’s behind in some facet of these prerequisites, you can continue the same methods you were using before; he will catch up quickly now that his mind is growing up.) He’s also picked up lot and lots of facts – pegs in the wall of the mind on which more complicated ideas and understandings can be securely hung.
Now the second kind of learning, and the second object of learning, begins to take over (although imitation is still available to use whenever it is needed.)
Now recitation and copywork give way gradually to composition and speech-presentations, both in the form of extemporaneous argumentation and in prepared speeches. Most students will naturally feel some embarrassment at having to use their voices in this manner. The way to overcome this embarrassment to is to require them to do it any ways, but in a situation where everyone else is doing so as well.
For homeschoolers, this means that family nights should include recitations when the children are young, and speech nights when they are older. For a homeschooling family, intellectual pursuits have to be woven into family life as enjoyable activities. It’s only in this way that the schooling, in the formal sense, can be prevented from taking over home life. Actual instruction is minimal when mental activity is pursued for fun.
However, I have to emphasize here that the embarrassment about public speaking will be rightful, and probably harder to deal with, if the student has not been prepared for making his own speeches by reciting the words of others first.
Some students will simply be naturally diffident and will never really get used to this aspect of schooling. For them, it can be truncated and minimized on the assumption that they will spend their lives in a different manner. Bad experiences, of course, should be avoided or they may make someone’s shyness worse. In the home, embarrassment is usually never acute unless something extraordinarily traumatic occurs, or unless criticism is harsh. When the child reaches adolescence, he must be treated with an even more sensitive dignity than an adult requires, for a time. He should never be criticized, scolded, or reproached in front of others – least of all, in front of his siblings.
It may seem as if high-school education is terribly complex and really beyond the ordinary parent. Actually, true learning is just as easy, if not easier, at this level as at the first level. It is the intellect which becomes complex – and that makes the access to academic success easier.
Here is the immemorial method for learning most subjects at this level.
The student reads an assigned reading. He is then examined on his reading by being required to write a specific number of paragraphs in answer to a question which is put to him about the content of the reading. He should not know ahead of time what the question will be – this will make him read the whole assignment attentively. However, he should also have the book open in front of him as he is examined. If he has a general idea of what is in the reading, he should be free to locate and quote it exactly in his paragraph(s).
This method, which seems so simple, is actually the face of an extremely deep and delicate intellectual process which is going on in the student. He is using symbols (words) at an advanced level, increasingly so as he practices, and he is functioning intellectually, not just academically. Actually, the intellect is seated in the academy and is becoming at home there.
This method can encompass the acquisition of all knowledge except the processual. Since mathematics is essentially processual, it may seem that this subject must continue to be learned by imitation. Sometimes, however, if the intellect is strong, the student will make leaps and solve a problem himself, without being shown how. This student may have the makings of a mathematician.
How can such leaps be encouraged in the ordinary student?
First of all, the foundation of such original thinking is of course, the earlier imitative thinking. The intermediate stage has already been mentioned – conversations that involve speculations and observations about mathematics, and mathematical puzzles set in fun.
Moving on, in my opinion, geometry should be taught before algebra. First, because it appeared earlier in the history of man. Second, because it is founded on the concrete art of building and design, and can be taught while pursuing design projects. And finally, because geometry is inextricably wound up in logic, and so it insists that students think about numbers – which is different from merely solving problems, as recitation is different from conversation.
True geometry, traditional geometry, involves the construction of “proofs.” A proof is a logically accurate demonstration, in correctly used technical terms, that a geometric proposition is true. A great deal of rote memorization will be initially required, in the elementary manner, to acquire that terminology. At 11 or 12 one may begin. Demonstration of existing proofs may use shapes and objects. And then, finally, one leaves the student to make his own proof – to demonstrate the truth of some geometric proposition. He has reality – in the experience of which he is ideally rich – and those knowledge-pegs to help him. He also has rudimentary logic which has awakened in his mind by all those syllogistic games. Now to see whether the spark will arise in the mind.
The maturity of the intellect, mathematically speaking, is reached when the student can reason confidently and accurately about numerical relationships. Most of the time, students never get anywhere near this level.
Ideally, for most students, two years would be spent on geometry, and formal logic would be begun just before, so that the two might complement and support one another. For both of these disciplines, old textbooks should be acquired. New ones are usually gimmicky, and therefore frustrating and time-wasting.
Algebra is a detestable and slavish discipline which bears very little relationship to reality. It is essentially a math-game – an occupation invented for leisured slave-owners. Where it is used as a game, as a diversion, it can be profitable to the agility of the mind. Insisted on too much, it makes one cruel. A half-year of algebra, and the acquisition of what is currently considered a 6th-grade understanding of it, is sufficient for most people.
Calculus is interesting and is a prerequisite for computer programming. (For some kinds of programming, trigonometry, a branch of geometry, is also needed.) A year of Calculus can do no harm. It is best treated as a mental game, however, especially if one wishes to avoid the eternal question – to which there is no good answer in most cases – of “how will I use this in the future?”
Ultimately, the high-school graduate should have learned:
- A modern and an ancient language (again, beginning with rote acquisition of vocabulary and grammatical structures, and proceeding to a more symbolistic, theoretical, and intellectually complete usage)
- Intermediate-level formal logic
- Intermediate-level geometry
- First-level calculus (with whatever algebra is necessary to understand the conventions of calculus, if indeed any are)
- The rudiments of music and drawing (or more, if so inclined)
- A grasp of world history and of the history of his own nation, founded on facts and the recitable acquisition of the main points of the timeline, and proceeding to ideas
- An understanding of the history of science, of the scientific method, and of the important scientific people, laws, discoveries, and beliefs that matter to our society – again, beginning with the memorization and recitation of facts, and proceeding to a more complex understanding
- A deep, at-home, affectionate familiarity with the literature and literary conventions of his own culture and language over a large period of time, along with a survey-level knowledge of the characteristics of other cultures’ literatures; also, the ability to discuss and write about this literature according to a basic scheme of criticism
- The main personages, stories, requirements and doctrines of his religion – at first, their names, outlines, main points, and titles, and later on, the theoretical framework of doctrine. This must of course be supported in the home by such level of observance as the family finds practical and proper.
- That indefinable and indescribable collection of random facts called “common knowledge” – the main usefulness of which is to prove to his fellow-citizens his bona-fides as an educated person.
It don’t believe that the content and method of this education scheme needs to be different for different children. One child will naturally emphasize and pursue different aspects of it more than other children will. Permissiveness is all that is needed for this kind of tailoring to take hold. Not too much permissiveness, or something important will be missed.
The structure of liberal education is founded upon the assumption that there is something Universal about Man – some core of design and being which transcends and surmounts all the individual differences. The Universal core is traditionally what is being treated in primary education. Specialization may begin quite early in the case of extremely and specifically talented children; most of them, however, will want to try many different things before settling on a specialization later in the educational process. Specialization is the development of the individual, rather than the Universal nature, and ought not be put first.