Trivium Pursuit

On Early Academics

I. Our Opinion of Unschooling

How Misunderstandings Arise

Then Peter, turning about, seeth the disciple whom Jesus loved following . . . . Peter seeing him saith to Jesus, "Lord, and what shall this man do?" Jesus saith unto him, "If I will that he tarry till I come, what is that to thee? follow thou me." Then went this saying abroad among the brethren, that that disciple should not die: yet Jesus said not unto him, He shall not die; but, "If I will that he tarry till I come, what is that to thee?" This is the disciple which testifieth of these things, and wrote these things: and we know that his testimony is true. — John 21:20-24

The Word is Misinterpreted

The Lord Jesus, the Word, the perfect expression of God incarnate, was continually plagued with strange rumors which came from misinterpretations of His very words. An example is cited above. Though the Apostle John was still alive and could be asked at any time directly about this rumor, nevertheless Christian disciples continued to spread the false rumor. So John published this denial of the false interpretation and had his denial attested by other witnesses ("WE know that his testimony is true").

Facts First

Too much imagination was added to too little information. No doubt, everyone reading these words has been the victim of such. Hopefully, this has led you to be more careful in handling what others say. Critical speaking skills [rhetoric] should not express the results of critical thinking skills [logic] unless the critical listening skills [grammar] have first been engaged. Or, as they used to say long, long ago, in a far away distant galaxy, "Facts first."

Rumors of Our Dearth Have Been Greatly Exaggerated

The Wealth of Notions

At any rate, we would like to make a statement, for the record, regarding the false but persistent notion that we advocate some form of unschooling before age ten. From the way some people talk, you could get the impression that we believe children should be left to wander about aimlessly, foraging in the forests for food, suffering from a dearth of discipline, until their tenth birthday.

Hear Goes Nothing

We only suggest that there are better ways than workbooks to accomplish the same academic ends for children before age ten. Anybody who suggests we say differently hasn’t listened to what we say. If thousands of parents have heard what we say, but have not extrapolated our words into a whole world of undisciplined disorderly unschooling, then the fault cannot be in our words. Perhaps when we say "maybe no workbooks" some hear "absolutely no schooling" because they know no other way of schooling. And perhaps they transform "ten" from an average final cut-off date to a razor sharp division, because they think only in even integers. Or perhaps, they simply do not know what "unschooling" is, and they apply the term to whatever does not agree with their experience or notions about schooling. We cannot discern all of the reasons, and we don’t want to. We just want to set the record straight.

Unschooling, In Their Own Words

The Unschooling Method (John Holt) seeks to provide an unstructured and unguided environment of books and resources. Parents 1) provide a model of interest in learning, 2) involve their children in their own adult experiences, 3) surround them with a rich environment of resources, 4) make themselves available to answer questions and suggest things to help the children to explore their own interests. Here, in the words of the proponents of unschooling, are definitions for the term unschooling:

SOURCE #1: Unschooling Undefined, by Eric Anderson http://www.olin.wustl.edu/Staff/bradford/unschool.phpl

. . .Where unschoolers differ from other homeschoolers is the extent to which we let children be responsible for their own education. . .Unschoolers believe that the natural curiosity of a healthy child, given access to a rich environment, will lead the child to learn what he or she needs to know. . .Child-driven learning is fundamentally active. Children are doing things because they have taken responsibility for carrying out the actions needed to fulfill their desires. Unschooling is centered around the idea of learning, with the student as the center of action and the source of activity, rather than on the idea of teaching (with the teacher as the center of action and the source of activity). . . .The child learns that if he wants something to happen, he has to make it happen. . .Unschooling families do not set up miniature classrooms, with time set aside for studying, a parent playing the role of teacher, formal lesson plans and imposed curricula. Beyond that limit, we differ in how much order we try to lend to the learning process. "Radical" unschoolers impose little or no structure, though books and such are available to act as guides. Others allow children to learn what they wish, but provide strong organizational assistance to help the children reach their goals. (Assistance can take the form of lessons, or workbooks, or even assigned projects.) Some families use curricula for some subjects (often math) but are freer with others. Most try to squeeze learning out of the activities of everyday life.

SOURCE #2: Defining Unschooling, by Pam Sorooshian

http://www.comenius.org/chn/netnews/unschool.php

. . .A better way to state the unschooling position is that children should not be forced to learn something against their will. . . .Unschoolers trust children to choose to learn and they recognize that the deepest and most satisfying learning comes about when someone is at least fascinated with, if not passionate about, a subject. . . .Unschooling does mean that parents have consciously decided to leave many, most, or all decisions about what and how to learn up to the learner. . .Unschoolers believe that there is so much that is worth learning and so many ways to learn anything that there is no good reason to force anyone to learn certain materials at certain times or in certain ways. Across this spectrum, unschoolers are distinguished by their high degree of confidence that children will benefit from being allowed tremendous amounts of freedom to choose their own educational paths.

I believe that we learn best when we, not others, are deciding what we are going to try to learn, and when, and how, and for that reasons or purposes; when we, not others, are in the end choosing the people, materials, and experiences from which and with which we will be learning; when we, not others, are judging how easily or quickly or how well we are learning, and when we have learned enough; and above all when we feel the wholeness and openness of the world around us, and our own freedom and power and competence in it. What then do we do about it? How can we create or help create these conditions for learning? – John Holt, from What Do I Do Monday

John Holt's ending in the book "How Children Learn" is a good definition of unschooling. "Birds fly, fish swim, man thinks and learns. Therefore, we do not need to motivate children into learning by wheedling, bribing or bullying. We do not need to keep picking away at their minds to make sure they are learning. What we need to do, and all we need to do, is bring as much of the world as we can into the school and classroom (in our case, into their lives); give children as much help and guidance as they ask for; listen respectfully when they feel like talking; and then get out of the way. We can trust them to do the rest."

SOURCE #3: What is Unschooling?, by Earl Stevens

Then what is unschooling? I can't speak for every person who uses the term, but I can talk about my own experiences. Our son has never had an academic lesson, has never been told to read or to learn mathematics, science, or history. Nobody has told him about phonics. He has never taken a test or has been asked to study or memorize anything. When people ask, "What do you do?" My answer is that we follow our interests - and our interests inevitably lead to science, literature, history, mathematics, music - all the things that have interested people before anybody thought of them as "subjects"

What God Thinks of Unschooling

Notions at War

The primary tenet of the advocates of unschooling is that the child be left to determine his own direction. The whole tenor of the Scriptures militate against such a notion. We have often cited the following text:

The rod and reproof give wisdom: but a child left to himself bringeth his mother to shame. — Proverbs 29:15

The first half of this verse tells us that a child must be disciplined with the rod and reproved for his misbehavior. Misbehavior may be manifested in a multitude of ways, including intellectual misbehavior.

Train up [/catechize] a child in the way he should go: and when he is old, he will not depart from it. — Proverbs 22:6

We are to intellectually train up our children. They are not to be left to determine their own direction.

. . . A wise son maketh a glad father: but a foolish son is the heaviness of his mother. He that begetteth a fool doeth it to his sorrow: and the father of a fool hath no joy. A foolish son is a grief to his father, and bitterness to her that bare him. — Proverbs 10:1; 17:21,25

We are not to raise fools, and fools are those who are left to themselves, not trained in the ways of the Lord. We would not limit this wisdom and foolishness to outward behavior and misbehavior. It includes inward and intellectual behavior.

. . . out of the abundance of the heart the mouth speaketh — Matthew 12:34

What We Think of Unschooling

Little Train, Off Track

We should focus on the second half of Proverbs 29:15 – "a child left to himself bringeth his mother to shame." Children are not to be left "unschooled." They are to be trained and taught. We have often said that "a young child needs more training than he needs teaching." Notice, we did not say "a child needs only training and no teaching." Training and teaching go hand in hand, but with younger children, the training predominates, and this need for training gradually gives way to more teaching as the child matures. If you’ve done your training at an early age, then an older child will be prepared for plenty of teaching and will need very little training. But in neither the training nor the teaching do we advocate leaving the child alone to explore his own interests. Children are not little adults.

Train on Time

On the basis of Scripture, we thoroughly reject the notion of unschooling – in principle and in practice – especially before age ten. The Scripture doctrines of original sin and total depravity should forever end the discussion. We affirm disciplined age-appropriate schooling for all ages. A child must be trained in self-control and self-discipline, and he must acquire the tools and equipment which he needs before he can embark upon a guided exploration of the world around him. He also needs some academic training in order to be able to use these resources to the best advantage.

Unschooling Versus Discipline

Free Access To Discipline

Once we have instilled such disciplines in our child, then providing an open environment rich with resources may be the best thing we could do for him. As a child matures, especially at the Wisdom Level, he should gradually be allowed more freedom in what he pursues. But even students approaching adulthood – age twenty – need some structure in their environment. So we propose, in strong contrast to unschooling, a Resource-Rich Environment for Disciplined Students. The more disciplined the student, the freer his access to resources.

II. Some of Our Views on Early Academics

Formal versus Non-Formal Early Academics

The Waiter Delays

There are some who believe we should introduce every child to formal structured classroom-like instruction from as early an age as possible – by four years of age, or even earlier. They believe time is wasted if we wait. The early years are academically very important. Others believe we should delay such formal academic instruction until eight to ten years of age.

Research, and You Will Find

Research indicates that if we start such formal instruction too early, then it causes developmental problems. Stress is placed on the child’s systems – such as vision, hearing, nerves, and coordination – which are not yet fully developed. In the early years, the brain is not yet formed to handle complex abstract thought. It is better adapted to receive lineal instruction. If it is strained to go beyond its development, it will lack proper comprehension, and it will store the information in less accessible places than if it is taught these same things when the brain has properly developed and is prepared to receive it. (For more discussion of this, see History and Research on the Teaching of Math.)

Much More is Involved Than Mere Academics.

Poles Apart

In order to train in proper behavior and to build moral character, the early years are well spent in doing projects, learning household management chores, and doing service for others. If we lay a good foundation of self-discipline and moral character, the academic instruction of later years will be more productive. The more we read to children, the larger their mind will be – larger vocabulary, larger store of categories of thought, and a greater love of learning. We are talking about the polar opposite of "unschooling."

Train up [/catechize] a child in the way he should go: and when he is old, he will not depart from it. — Proverbs 22:6 KJV

Against Indulgences

Because some, such as we, do not advocate early formal structured academics, others have speculated that we must recommend an indulgent, non-demanding, even negligent approach to schooling before age ten. Would anyone say that doing such things as reading and narrating Charles Dickens, memorizing Greek and Hebrew Alphabets, memorizing and reciting passages of Biblical and Classical literature, learning to speak, read, and write foreign languages – that doing such things before age ten fits into the category of indulgent or non-demanding? Definitely not! Yet these are all things done by those who take a later formal-academics approach. We agree wholeheartedly that early instruction is important. We disagree only with the notion that it is necessary to follow the modern methodology of early formal academics. Instead, we advocate the Classical method of instruction which prevailed until the twentieth century, which emphasized informal instruction until an age when formal instruction is developmentally comprehendible. Why attempt to stuff things into a child’s mind at a time or in a manner which renders the material less digestible?

Missing Connections

The last century of education has been an ever-expanding experiment in earlier and earlier formal academics, accompanied by ever-multiplying "learning disabilities" and such like things as were never heard of before. The evidence suggests a direct connection between early formal academics and learning disabilities, and this warrants considerable caution regarding this modern methodology. Again, we recommend something closer to the classical methodology of delayed formal academics as a general rule, but there are many and varied exceptions, and the parent is most qualified to discern what is best for his own child.

Two Extremes

Imagine these two extremes: On the one hand, we have a strict, constrained, starched, rigid, formal, military academy classroom setting. On the other hand we have the casual, loose, offhand, perfunctory, come-as-you-are-and-do-as-you-please-and-go-as-you-may hippy-dippy setting. Now we tell you that you must choose between these two settings. What have we done? We have suggested that there are only two possibilities. This is known as the Either/Or Fallacy. It is like saying, "If you are not a fascist, then you must be an anarchist." Well, there are numerous other possibilities than just fascist or anarchist. The early formal-academics people are not necessarily like the military academy, and the later formal-academics people are not necessarily like the hippy-dippy setting. We believe forcing young children into too rigid a structure is counter-productive, but too little structure is also counter-productive. We are not talking about rigid opposites, but different points on a scale, and the best point is different for different families and different children at different ages and in different circumstances.

Brain Strain

We emphasize that the physical properties of the brain should be fully developed before engaging in those activities which place stress on those properties. When this rule is followed, learning progresses very rapidly and more satisfactorily. If there were some simple but accurate diagnostic tests to assess whether the mind is physically ready, we might recommend them. As it is, the best test we know of on the market is the parental assessment – the homeschool hunch – a carefully designed method whereby Mom and Dad, after working with the child, come to agree, "Yep, he’s ready for this," or "Nope, we’re going a bit fast, we’ll hold back a little here." Please notice that the parent is being trained through this process as much as is the child. We suspect this was the original design.

Compulsive Parents

Some parents are a bit compulsive. They want to drive their children like Hebrew slaves without supplying straw for their bricks. (That may be the child’s view of it, anyway.) These parents may be driven by their desire to show off their children’s progress, or they may expect their children to perform at adult levels of ability. Or they may simply be trying to find ways to keep the curious little rascals plenty busy and out of trouble. But children are rather resilient creatures, and as long as we are in tune with what’s happening, and we don’t try to force things too strongly before their time, then a little too much here or there won’t burn the little tykes' brains out, and they’ll progress along at a nice pace.

Indulgent Parents

There is, however, the opposite difficulty of doing too little or being too lax. This can be due to laziness or an indulgent spirit on the part of parents. Or it may be weariness with the work of raising lots of rambunctious little rug rats. Guess what? Children have resiliency in this direction as well. As long as we don’t soften things too far, then a little laxity here or there won’t turn the little lads and lassies’ brains into cornmeal mush.

They’ll survive the minor flood or the minor drought.

Guilty Parents

Now, name the parents who are not guilty of both of these tendencies? (We are not asking you to confess your faults openly. Just admit them to yourself and to God.) What keeps us in balance? Our love for God and for our children. What gets us out of balance? Our lust for the world and our flesh. You see, this is a spiritual problem at its core. We cannot teach our children discipline if we do not have it ourselves. One of the reasons God has given us children is to goad us into self-discipline. Everything rests on us doing our jobs as parents.

The Downside-Up Solution

Upside Down Culture

Our culture is turned upside down. We are taught to believe that everything rests on the government, or on the economy, or on the church. We become dependent upon those things which are, in fact, dependent upon us. Those who will not rule themselves will be ruled by others, and treated like children. Once we break the pattern of being conformed to the external structure which this world imposes upon us, then the internal structure and self-discipline, which have been built into us through the operation of the Spirit of Christ, will begin to work out of us to transform the things around us. When we restore the Biblical jurisdiction and order to the family, we will find parents behaving much more responsibly – like mature adults should. Then order should grow outward from the family into all of society. We will then be able to restore Christian order and the rule of law.

Cultural Slide-Backing

We lament the cultural slide which has made it necessary for parents to rely upon external structures to keep order in their own families – whether it is rigid programmed curricula, or classroom schools, or other. The solution to this situation is not to give them more outside structure. The solution is to build structure within them, which will make the outside structure less necessary. The solution is to promote the self-discipline of the parents. In other words, the solution is to get the parents to grow up. Then the structure inside the parents will generate the outside structure which the children need while they develop their own internal structure under the nurture and admonition of the Lord administered by their parents. This will make children more greatly honor and admire their parents, which means they will love them more. Proper order will be re-established. God’s order. And God will be glorified.

Buckets of Fire

Burning Light Bulbs

Chris Davis of Elijah Company is fond of quoting W. B. Yeats, "Education is not the filling of a bucket, but the lighting of a fire." (Or maybe it was Plutarch.) The light bulb for academic learning does not go on until about age ten. Before this time we can waste much time teaching simple things, often to the exasperation of both pupil and parent. Instead of teaching a five year old to count by fives – and take five days to do it, we can wait until the child is age nine or ten and teach him in ten seconds – if he has not already figured it out on his own.

Promise Broken

Pushing formal-academics at an early age may destroy more promise than it creates. Those children who have been pushed in very early years often experience a jet lag of academic burnout in later years. Of course we should never discourage a child from learning, but neither should we over press academics to the point of strain or exasperation. We should stretch and challenge our child in order to strengthen him. When your child is ready to learn something, then teach him. When he has a question, then answer it. Encourage questions – and teach him the proper manners for asking questions.

Spoiled Eggs

Every child is different, and there are precocious little prodigies who take to academics and go farther and faster than others. But these children still need more training than teaching in their early years. A bright young child who is undisciplined, ill-mannered, and amoral can be a genuine grief, especially if he is also academically learned. Formal academic teaching is not the same thing as child training. What we most want to do with children during their early years is to train them to function in the Scripture-controlled Christian culture of our own home.

Training Which Enslaves

The goal of the socialist-state is much the same – to socialize children to function in the government-controlled humanist culture of the public school. They train children to look to the government to care for them, to provide for them, to educate them, to solve their problems for them. And they do a good job of training children for humanist culture. They have taught a whole generation to think that the solution to every problem is a new government program, and never to consider whether a government program may have created or exaggerated the problem to begin with. The government is not seriously concerned about early academics, but only about early socialization. Modern Education has invented early education: Kindergarten, then Head Start, then Preschool, then Daycare. The newest fad is Educational Preparedness – for six month-olds! Their ultimate goal seems to be prenatal indoctrination of the fetus, or perhaps genetic engineering which creates a predisposition towards collectivism.

Preparation for Academic Progress

God invented the family to train up and enculturate the child with godly principles, values, and goals. There is no legitimate substitute for the family. The early child training and development of the physical and mental faculties are preparation for more formal academics. At about age ten, after the child has learned to read, and his vocabulary and categories of thought have been well-developed through informal academics, the formal subjects of the Trivium can easily and readily be taught, and academic progress is a joy to the child.

Early Language Skills First

So our focus before age ten is building language skills – speaking, reading and writing – especially vocabulary. This is the primary index of intelligence throughout life. Do things in a concrete way. At age ten, when the brain physically changes, and begins to make the complex connections, you begin the more complex and abstract learning. With this emphasis in the early years, we lay a proper foundation for a full academic load later.

Early Informal Math

Our Research

We have often been asked about our suggestion that math before age ten is best taught "informally." This seems most uncustomary to many. At the end of this article we have placed our article History and Research on the Teaching of Math. This information also appears in the Appendix of our booklet series. We very much want to learn if there is any contrary research or historical evidence. Everything which we encounter on the question continues to confirm this common sense view on the matter. We continually receive positive and enthusiastic feedback from families which have followed these suggestions – though many at first followed somewhat apprehensively. We are still waiting for our first negative feedback. We’re somewhat surprised. We at least expected there would be some families which were generally lax or unschooling in their approach, and would try to blame their math troubles on our recommendations. But we recommend no lax learning. Lax learning lacks learning.

What We Recommend for Math (and Grammar Too)

It’s the Method, ______

What we and others recommend regarding math is basically what was practiced with outstanding success until the twentieth century, when formal math before age ten was largely introduced into the world. Cultural math failure coincides with the innovation of early formal "workbook" math. We argue that it’s the method. We believe in math before age ten. But we believe the evidence is against workbook math before age ten. The developmental evidence appears very supportive of that view. The same is true with grammar – not language, but grammar. It is best to learn to speak and read and write a language before age ten. But grammar – identifying Gerunds and Participles – is best left until age ten. (Approximately age ten.)

The File Drawer Analogy

Math and Grammar can be "learned" – and "learned" – well before this time, but it’s not the kind of learning we want. We compare it to putting information in the wrong file cabinet – you have trouble later finding it and using it. At age ten, the information is literally stored in a different part of the brain than before age ten. (Again, approximately age ten.) Learning math in an abstract workbook fashion before age ten literally causes the brain to be structured differently. If the child depends upon his early math learning drawer, and does not develop a new file draw for later math learning, he runs into a brick wall when he encounters algebra. (We like to mix our metaphors.) Now, if he learns abstract workbook math before age ten, then he will either develop a second math memory after age ten (and, hopefully, not have a cross-indexing problem), or else he will begin to fail in upper math. But if he learns math in a concrete – not abstract – way before age ten, and he begins to learn abstract workbook math at age ten, then the brain will develop properly, the right connections will be made, and – assuming normal abilities and developments elsewhere – he will advance in math at a regular pace without unusual difficulties. The same is true with grammar.

The Computer Analogy

Or, to put it in computer terms, some word processors can handle some simple calculations. You can type in the data, and it will work with numbers on a simple level. But if you want to do complex calculations, you have to load a much more complex program on the hard drive. Until about age ten, children only have word processors. About age ten, the more complex spread sheet program begins to be loaded up on the hard drive. If you enter all of your math information in the word processor, then it is likely that when the child switches to the spread sheet program, the data will not be compatible. Formatting errors will abound. You’ll have to re-enter the data. Why not do something more profitable until the spread sheet program is up and running?

Time Better Spent

We are satisfied that the time spent studying math – which the young child is not yet developmentally equipped for – could better be spent developing verbal skills – which the child is a sponge for at these early ages. Deal with numbers in a concrete and verbal way until age ten. Use actual objects when you can, and when you can’t, then use words and names for actual objects. Our culture is so full of numbers and measurements, that we let them pass without notice. Teach the names for numerical values with dominoes. Teach counting with cards or Rummikub. Teach addition with checkers or chess. Teach base ten and place value with money or Cuisenaire rods, or other manipulative math programs. Teach measuring systems with tape measures, measuring cups, weight scales, odometers. Teach fractions with pies and cakes and cooking. Teach area by garden plotting and room arranging. One mom who had struggled with waiting in math wrote us that her son wrote down on a Sunday school form that math was his favorite subject. Since they didn’t do math, she was surprised and puzzled. When she asked her son why he wrote down math, he said, "What do you mean, Mom? We talk about numbers all the time." When she sat down with her son and looked through a math program, she discovered that he already knew it all. This may be a little more intuitive and less structured than we have in mind, but it demonstrates well how these things are taught as part of life.

The Ideal and the Real

In our opinion, the ideal would be to learn to speak and write several languages and to become familiar with a wide scope of literature before age ten, which lays a wide and solid foundation for formal math and grammar beginning around age ten. Everything seems to point to this as the best course to take. But we have never said "don't ever teach math before age ten." The whole idea is as ridiculous as it sounds. You cannot avoid exposing your child to arithmetic concepts. They will discover it on their own at a very early age. Teach them what they are ready to learn. But teach them in a concrete way, not in an abstract way. That’s what informal math is. It is not leaving the child to discover what he wants. Also, we have never said, "don't ever teach formal math before age ten." We have always said that that was a judgement call to be made by the parent, and if you should have a precocious little tyke who wants to learn math and works well with workbooks, then you would probably be mistaken if you were to hold him back. But if you force him beyond his developmental capabilities, then you are more prone to cause developmental abnormalities. In other words

And O ye fathers, do not aggravate [/exasperate] your children, rather, nurture them to full maturity in the correction and counsel of the Lord. — Ephesians 6:4, Very literal translation.

O ye fathers, do not overstimulate [/provoke too far] your children, in order that they should not be broken in spirit [/disheartened]. — Colossians 3:21 Very literal translation.

Stimulation and Aggravation

The Greek words for aggravate [parorizete] and stimulate [erethisete] convey the idea of pushing the child too far, too fast, beyond his capacity, to the point of justifiable anger and broken desperation. An exaggerated example of this in English literature would be Paul Dombey, a Charles Dickens’ character. Fathers are to nurture their children to full maturity, not drive them there. The process is different with each and every child, requiring plenty of micro management. We cannot just run our children down an academic conveyor belt and expect them to emerge at the end like well manufactured machines.

Don’t Bring the Classroom Home

The Apostle Paul gives these warnings in Ephesians and Colossians precisely because it is necessary and ought to be heeded. Just because something can be characterized as "disciplined" does not mean it is the most profitable in the end. We discourage the formal abstract workbook type of rigor as an "aggravation" on the child developmental level. We encourage other avenues and other time schedules which reach precisely the same goals with as much or more discipline, but with less time commitment and with more satisfactory results. In other words, don’t bring the classroom school home, but tutor your child instead. This frees up time for other good and important things, such as reading aloud. Nevertheless, as we have always said, the parents are the best judges of what is best for their own family, and what may be too much for one family may not be enough for another. The socialist one-size-fits-all academics is incompatible with homeschooling!

III. Summary, and A Final Note on the Views of Others

Five Point Summary

  1. We thoroughly reject unschooling in principle and practice. "Lax learning lacks learning."
  2. We affirm disciplined age-appropriate schooling for all ages. No, we insist on it. But disciplined does not necessarily mean formal narrow abstract academics.
  3. Before age ten, we encourage – as a general rule – building language skills, general knowledge, and concrete perspectives, while training the character in godly attitudes and personal discipline. This builds a broad and sound foundation for later formal academics.
  4. Before age ten, we discourage – as a general rule, but not as an absolute rule – formal narrow abstract academics. This is ordinarily not the best use of one’s time, and it may cause trouble in the long run.
  5. We say that the parents are the best judge of their own family’s needs. There is no "one-size-fits-all" plan.

Please feel free to quote us.

A Final Note on the Views of Others

Now, one more thing before class is dismissed.

Measuring the Middle

Other persons have other emphases. That’s fine. We wish them well, and we encourage you to read them as well. Our ideas may appear to rub against theirs at some points, but we don't see the difference as near the clash as some people try to make it. We simply approach the same problems from different perspectives. One says, "not too little of this." Another says, "not too much of that." When you evaluate the two, you will find that we are only trying to measure how wide the middle is. Some people see it more clearly from one side. Others see it more clearly from the other side. Hurray for seeing it more clearly.

The Same Difference

You might compare our differences to the differences between Strong’s Concordance and Young’s Concordance – same material, different arrangement.

Gentle Correction

If we all understood each other perfectly, that would be heaven. We can tolerate that, and we hope you can also. In the mean time, we do not presume to speak for anyone else, and if we ever misrepresent anyone else’s views – even slightly, please feel obligated to correct us – gently.

Also read Research on the Teaching of Math.

Appendix

Why Smart Teenagers Can Still Be Very Silly

On the outside, teen-agers appear to be nearly grown up. But inside the skull a vital part of their brain is closer to a child's than an adult's.

New findings in neuroscience and paediatric psychiatry link brain immaturity to teens making foolish judgments and reckless decisions. . . . [O]ne of the last parts of the brain to mature is the prefrontal cortex the part responsible for self-control, judgment, emotional regulation, organization and planning. "The teenage brain is a work in progress," said neuroscientist Sandra Witelson, of McMaster University in Ontario.

The old belief was that by the time a child reached the age of puberty and pimples, his brain's hardware was completely connected.

But by using magnetic resonance imaging, researchers showed that the brain had a lot of developing to do well beyond the start of adolescence. The brain reaches about 95 per cent of its maturation by the age of five years. But the corpus callosum, a cable of nerves connecting the right and left halves of the brain, continues growing beyond 20-something. The corpus callosum is linked to intelligence and self-awareness.

The prefrontal cortex matures most between the ages of 12 and 20. Add to this brew of disconnected neurons a dose of active hormones spiked with the power of peer pressure and a need for autonomy. That is a recipe for sometimes risky teenage behavior.

The brain research suggests that teenagers must be trained to handle the freedoms they demand.

Researchers say that, after puberty, a pruning process takes place in the prefrontal cortex. About the age of 10, the prefrontal cortex goes through a growth spurt when neurons grow new connections. But these die off if they are not used.