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The audio you are about to hear is an appendix from the book, "Teaching the Trivium, Christian Homeschooling in a Classical Style." It's article 11 in the appendix, entitled "History and Research on the Teaching of Math." This is one of the more well-researched articles I've ever found on this subject.

And the authors of this book have brought together their sources and their research here and formed the opinion that formalized workbook-based instruction in mathematics should not begin for children until about the age of 10. While I myself do not strictly follow this practice, it is something that I think should be widely debated and should be widely discussed and understood.

Because while we are working hard to develop our children's brains by catalyzing a very high level of numeracy in them, it's important that we do that at the proper time. And mathematics is something that I'm not sure that it should be learned early for most people. As parents, we need to exercise a great deal of discretion and good judgment as to what is working well with our children and what is not working well with our children, that's the goal.

And if your child is not succeeding with math at the age of five, then you should supplement something that will be better for this child's age of study and simply push off the study of math to a time when the child's brain is more prepared to accept it. But in today's world, where we understand the importance of mathematics and science and technology, et cetera, to our generalized culture and success in academia, parents often feel this intense pressure to make sure that a child is succeeding at a very young age.

And I fear that that can many times cause us to pressure our children in a wrong way, a counterproductive way, a way that destroys their confidence and their ability to be excellent at mathematics. And it would be helpful and useful if we had a little bit of context to inform our opinions.

And so while my wife and I in our homeschool have indeed been using some workbooks and doing some workbook math prior to the age of 10, I am very confident that it would be perfectly okay to delay until the age of 10. And I have suggested this to people when a child is struggling.

And I think it's one of those areas that I wanna reinforce you in so that you can exercise good judgment. This appendix is full of historical quotes and especially in the beginning, it's just a little bit slow. So if you don't wish to follow all of the historical quotes, I'm going to read them for the sake of completeness.

But if you don't wish to listen to all of the quotes from their research and sources, just skip forward, say 15, 20 minutes. And at the end of this article, I will again sum up some of the reasons why I think in many cases people should wait until the age of 10.

One more comment, in the previous episode of the podcast called "How to Catalyze a High Level of Numeracy in Your Children," I mentioned the stories of Art Robinson and his family, as well as citing this particular author and their experience. And in short, waiting until age 10 to do workbook math does not mean that you wait until age 10 to do math.

There are lots and lots of things that you can teach your child, mathematical concepts, numbers, counting, telling time, all of the that you can use in daily life when you're cooking with your child or playing with your child, et cetera, so that your child will have a strong, generalized understanding of mathematics.

And in addition to that, if you will work diligently at teaching your child math facts, all of his addition, subtraction, multiplication, and division tables, then your child will be very well prepared to succeed in mathematics. And what Harvey and Laurie Bludhorn, the authors of this book called "Teaching the Trivium, Christian Homeschooling in a Classical Style," say with their children, I think they had five children, and what Art Robinson's experience has been with his six children and with the many acolytes of his who have used his course of study and methodology, is that children are very able to come in and basically just start with fifth grade math and succeed.

And if indeed any makeup work is necessary, at that point in time, you can cover in just a few weeks what previously you may have spent years wrestling over. And if you have to tie your child to a chair and move his hand for him to get him to do his math, it's a pretty good sign that maybe workbook math is not what he should be focusing on right now, and you can put that time to other productive use.

So without further ado, let's begin the article with the actual formal research on this topic. "History and Research on the Teaching of Math" by Harvey and Lori Bludhorn. Formal arithmetic at age 10, hurried or delayed? Provincialism is the word which we use to describe an opinion which is narrow and self-centered in perspective.

Because the common practice in our culture in our day is to begin formal instruction in arithmetic as early as age four or five, many have questioned the suggestion that one may wait until age 10 before beginning formal instruction in arithmetic. Waiting until age 10 for formal instruction in arithmetic is often misnomered "late start" or "delayed academics." A broader perspective would examine more than what is simply the prevailing practice of a particular culture at a particular time, especially if that practice is a policy largely imposed by the government.

We don't claim to have the last word on the subject, but we have examined the matter more broadly. And in this article, we will present some of the things which we have discovered. We will quote only a small selection from authorities we have found, and we will allow you to form your own opinion before we comment.

The historical perspective, what the ancients did. Quote, "Strange though it may seem at first, it is nevertheless quite clear that addition, subtraction, multiplication, and division, comparatively simple operations, which we inflict on our children while they are still quite young, were in antiquity far beyond the horizon of any primary school.

The widespread use of calculating tables and counting machines," abacus, "shows that not many people could add up. And this goes on being true to a much later date, even in educated circles." That quote is from "A History of Education in Antiquity" by Henry Maru, published 1956. What the medievals and later did.

These are various quotes from several different pages. Quote, "Before the Reformation, there was little or nothing accomplished in the way of public education in England. In the monastery, some instruction was given by monks, but we have no evidence that any branch of mathematics was taught to the youth. Some idea of the state of arithmetical knowledge may be gathered from an ancient custom at Shrewsbury, where a person was deemed of age when he knew how to count up to 12 pence." See, Tyler's "Primitive Culture," published 1889.

"In the 16th century, on the suppression of the monasteries, schools were founded in considerable numbers, and for centuries have served for the education of the sons, mainly of the nobility and gentry. In these schools, the ancient classics were the almost exclusive subjects of study. Mathematical teaching was unknown there.

Perhaps the demands of everyday life forced upon the boy is a knowledge of counting and of the very simplest computations. But we are safe in saying that, before the close of the last century, the ordinary boy of England's famous public schools could not divide 2,021 by 43, though such problems had been performed centuries before, according to the teaching of Brahmagupta and Bhaskara, by boys brought up on the far off banks of the Ganges." By the way, just to insert, always remember that when you hear the term public schools in the context of England, that means what I use the term private schools to mean.

And when you hear private schools in the context of the country of England, that means government schools. So one of the reasons why, when I refer to schooling, I use the term government school and private school, because it helps to solve this international issue with the English language. "All the information we could find respecting the education of the upper classes points to the conclusion that arithmetic was neglected, and that De Morgan was right in his statement that as late as the 18th century, there could have been no such thing as a teacher of arithmetic in schools like Eton.

In 1750, Warren Hastings, who had been attending Westminster, was put into a commercial school that he might study arithmetic and bookkeeping before sailing for Bengal. At the universities, little was done in mathematics before the middle of the 17th century. During the reign of Queen Elizabeth, fresh statutes were given, excluding all mathematics from the course of undergraduates, presumably because this study pertained to practical life and could, therefore, have no claim to attention in a university.

This scorn and ignorance of the art of computation by all but commercial classes is seen in Germany as well as England. It was not before the present century that arithmetic and other branches of mathematics found admission into England's public schools. At Harrow, vulgar fractions, Euclid and geography and modern history were first studied in 1829.

At the Merchant-Taylor School, mathematics, writing and arithmetic were added in 1829. At Eton, mathematics was not compulsory till 1851. Since the art of calculation was no more considered a part of a liberal education than was the art of shoemaking, it is natural to find the study of arithmetic relegated to the commercial schools.

The poor boy sometimes studied it, the rich boy did not need it. In the Latin schools, it was unknown, but in schools for the poor, it was sometimes taught. The first arithmetics used in the American colonies were English works. The earliest arithmetic written and printed in America appeared anonymously in Boston in 1729.

Though a work of considerable merit, it seems to have been used very little. In 1788, appeared at Newburyport, the New and Complete System of Arithmetic by Nicholas Pike. It was intended for advanced schools. Reform in arithmetical teaching in the United States did not begin until the publication of Warren Colburn in 1829, 1821, excuse me, of the intellectual arithmetic.

This was the first fruit of Pestalozian ideas on American soil. The success of this little book was extraordinary, but American teachers in Colburn's time and long after never quite succeeded in successfully in grafting Pestalozian principles on written arithmetic. Those quotes all from a history of elementary mathematics with hints on methods of teaching by Florian Cajori, PhD, published 1917.

The American, quote, "The American Calculator," first published in 1828, "is reasonably typical of the colonial period. "The text was used with older students, "beginning at about the age of 11. "It was complete in itself, not one book of a series "such as the text that evolved in later years." Quoting from "Readings in the History "of Mathematics Education," edited by James Bidwell and Robert Clayson, published, I don't have a published date, quote, "The study of it," meaning arithmetic, "used to be put off to a very late period.

"Scholars under 12 or 13 years of age "were not considered capable of learning it, "and generally they were not capable. "Many persons were obliged to leave school "before they were old enough to commence the study of it." From "Readings in the History of Mathematics Education," taken from the text of an address delivered by Warren Colburn before the American Institute of Instruction in Boston, 1830.

"In the early 17th century, "the grammar school curriculum "was almost certainly confined to Latin grammar, "the catechism, and Bible study. "Pupils arriving at Oxford and Cambridge "frequently did not have any knowledge of Arabic numerals, "to say nothing of the elementary arithmetical operations." From "A History of Mathematics Education in England," by Geoffrey Howson, published 1981.

So, when it changed, quote, "Pestalozzi, "it is to the latter," 1803, "that we owe the greatest impetus "in the rational teaching of arithmetic to young children. "The essential features of his reform are as follows. "One, he taught arithmetic to children "as soon as they entered school, "basing his work on perception.

"Two, he insisted upon a knowledge of numbers "and the simplest operations, "using objects before the figures were taught. "Three, he approached the subject of fractions "in the same way. "Four, he made arithmetic the most prominent subject "in the school, and it is to his influence "that its present prominence is due.

"Five, he emphasized oral arithmetic, "a movement that led to the great success "of Warren Colburn in the United States. "Tuerck," 1816, "did not wish arithmetic "in what we call the first grade, "nor before the child reached the age of 10 years. "And of this idea, there is just at present "a temporary revival as if it were a new discovery, "although it was practically universal before Pestalozzi." That is a quote that's quoting from "Acyclopedia of Education," edited by Paul Monroe, published 1919.

Summary of the historical evidence. "The material which we have read indicates "that the formally teaching of arithmetic "to young children was not practiced by the ancients, "the medievals, nor up to modern times. "In fact, it was common to withhold "formal instruction in arithmetic "until somewhere between the ages of 15 and 18.

"It was not until the 16th century "that arithmetic began to be taught to children "as young as age 12 or even 10. "It was Pestalozzi at the beginning of the 19th century "who began to teach arithmetic to children "as young as age six or seven, "though the practice of waiting until age 10 "persisted well into the 20th century." So to wait until age 10 to teach arithmetic is actually from a historical perspective to advocate an early start.

It is only from a decidedly modern perspective, a provincial perspective, that waiting until age 10 would appear to be a late start. We have not discovered any material which might indicate the contrary. The research perspective. The research is quoted here to demonstrate the point at hand. We recognize that studies are open to various interpretations.

Inefficient period. Quote, "Early childhood may simply be an inefficient period "in which to try to teach skills "that can be relatively quickly learned in adolescence." This comes from "Prime Time for Education, "Early Childhood or Adolescence" by William Rower in the Harvard Educational Review published 1971. I'm gonna interrupt here my reading of this appendix to comment on this.

In addition, and forgive me, I'm repeating a tiny bit of what I said yesterday, but I came to thinking about some of these topics quite a lot when I started thinking about the efficiency of learning at different ages. And if we know that we need to learn something, for example, we need to teach our children arithmetic and help them to be excellent in mathematics, we know that that needs to be done.

We need to think about by what age that needs to be done. So let's say we say it needs to be done by age 18. The question is, we have a limited number of hours that we can use for formal instruction. How are we best to use those hours?

Commonly, if you do the math on the amount of time children spend in school, it's usually something like 15,000 hours, 15 to 18,000 hours, depending on how we do the math, that an American child in the American system will spend in school. So if we have 15,000 hours, how should we dedicate those hours?

Well, imagine for a moment that we could successfully teach mathematics in say 3,000 hours by waiting to start until say 10 or 11. But if we start before that, we use say 1,000 or 2,000 hours more than is necessary. And I'm just making these numbers up. I have no basis in evidence or even calculation in them, just to drive the example.

Well, now that's 2,000 hours that we can't dedicate to other things. And as I'll talk about, or I think they mentioned it in a moment in this appendix, those early years are really beautifully suited to other things. I myself became convinced that language acquisition is what a five-year-old brain is wired to do.

So is a seven-year-old brain and a nine-year-old brain, et cetera. And so what if instead of teaching mathematics from the age of five to the age of 10, we could do it much more efficiently by starting at the age of 10. And instead we could teach a second or a third or a second, third, fourth, and fifth language in those same 2,000 hours.

Unless you think I'm being excessive, I think those numbers are just about right. If we assume a 180-day school year and we take 180 days out over five years, we wind up with basically 900 hours. Well, to learn Spanish or French for a native English speaker adult learner is a 675-hour project.

I would say that with more efficient methods and with childhood and just a few extra hours in here and there, the hours that the child might be spending doing homework and other work, that it should be no problem for a 10-year-old to have his native language as well as two foreign languages and then start math at age 10.

And since we have abundant evidence that this can be done more efficiently at that age, and let's say the 10-year-old is completely caught up with his peers by the age of 11, probably faster, but would you rather have an 11-year-old who's caught up with his peers and speaks one language or would you rather have an 11-year-old who is current with his peers and speaks three languages?

What's gonna be in the better interest of the child? So that's what that quote by Rohwer published in the Harvard Educational Review is so important. He said, "Early childhood may simply be "an inefficient period in which to try to teach skills "that can be relatively quickly learned in adolescence." Children don't do better in math, but they do learn to hate school.

Quote, "In a cross-national study "of mathematics achievement, Hewson, 1967, "stratified samples were drawn from the total population "of all students enrolled in the modal grade "for 13-year-olds in 12 different nations, "Finland, Germany, Japan, Sweden, Belgium, "France, Israel, Netherlands, Australia, "England, Scotland, and the United States. "Among other observations, a score was obtained "for each student on a standardized test "of mathematics achievement, "and in an attitude inventory on a scale designed "to reflect the degree of positive attitude towards school.

"For each national sample, information was also obtained "yielding the median age of school entry. "Thus, it is possible to rank the samples "in terms of age of school entry, "and to obtain rank correlation coefficients "between this variable and those "of ranked mean mathematics test scores, "and ranked mean attitude towards school scores.

"The results reveal a negligible negative correlation "between age of school entry and mathematics achievement." Rho equals negative 0.06, P is greater than 0.05. "And a strong negative correlation "between entry age and attitude towards school." Rho of negative 0.72, P is greater than 0.01. "The average performance of students "on the mathematics test did not improve significantly "as a function of additional years of schooling, "despite the fact the extremes of the nation sampled "were separated by nearly two years of formal academic work.

"More alarming is the suggestion inherent "in the high negative correlation "between entry age and attitudes towards school, "that the longer the student was enrolled prior to testing, "the more negative his attitudes towards school itself. "Clearly, there is no indication in these results "that revising the mandatory age of school entry "to younger levels would improve the students' chances "of subsequent school success." Of course, it might be argued that these results do not confront the issue directly, since the argument in favor of earlier school entry is most persuasive for low-income children.

The Hewson, 1967 report, however, speaks directly to this point as well. "For all of the students tested, "information was obtained to permit a categorization "of father's occupation. "Accordingly, correlation coefficients "can be computed separately for two large groupings "within each national sample, "that is, for those occupations falling "in the higher SES, social economic spectrum, "higher SES categories, clerical through professional, "and for those falling in lower "socioeconomic spectrum categories, SES, "skilled through unskilled manual.

"The correlations between entry age "and mathematics achievement test scores "are not significantly different from zero in either case, "but it is interesting to note "that the coefficient for the higher SES categories "is positive, rho is plus .19, P is greater than .05, "while that for the lower SES categories is negative, "rho equals negative .39, P greater than .05.

"Thus, even in its qualified version, "the presumption that early school entry "promotes school success in children "from lower-income families finds no support "in the results of the Hewson study. "Indeed, these data appear to contradict the presumption. "In these examples, support can be found "for the assertion that legitimizing curricular demands "in terms of later extra-school success "is vulnerable with respect "to the typically rigid timing of those demands." Quoting from "Prime Time for Education, "Early Childhood, or Adolescents." Again, same article by William Rower in Harvard Educational Review.

"Fractional Reserve. "Research in grade placement and readiness "has had two effects on the arithmetic curriculum. "They are commonly known as the stepped-up curriculum "and the stretched-out curriculum. "The stepped-up curriculum is largely due "to the study of the Committee of Seven. "Over a period of a few years and in hundreds of cities, "the Committee sought to determine the mental age level "at which various topics could be taught to, "quote, completion.

"Typically, they found that addition of like fractions "required a mental age of 10 to 11 years, "and unlike fractions, 14 to 15 years. "Two-figure division required a mental age "of 12 to 13 years. "As a result, many courses of study and textbooks "move selected topics to higher grade levels.

"Hence the name stepped-up curriculum. "Bennett Sett in Manchester, New Hampshire, "carried out a study from which he concluded, "quote, if I had my way, I would omit arithmetic "from the first six grades. "The whole subject could be postponed until the seventh year "and mastered in two years' study. "This led many people to conclude erroneously "that all arithmetic could be deferred "until the seventh grade.

"However, closer observations show "that there was much arithmetic taught in grades one to six. "Teal visited the Manchester, New Hampshire schools "and said, quote, first-hand observation "leads me to conclude that Bennett Sett "did not prove that arithmetic "can be taught incidentally. "Instead, he provided conclusive evidence "that children profit greatly "from an organized arithmetic program, "which stresses number concepts, relations, and meaning.

"Buswell found that Bennett Sett "had only deferred, quote, formal arithmetic, "and that all other aspects "of a desirable arithmetic curriculum were present. "Of the formal arithmetic, "Buswell said, I should like to eliminate it altogether. "On the same topic, deferred arithmetic, "Bruckner says, quote, from these studies, "the conclusion should be drawn "not that arithmetic should be postponed, "but that the introduction of social arithmetic "in the first few grades does not result "in any loss in efficiency "when the formal computational aspect of the work "is introduced later on, say, in grade three." That's from an article entitled, "What Does Research Say About Arithmetic?" by Vincent Glennon and C.W.

Hunnicutt, National Education Association, Washington, D.C., published in 1952. By the way, inserting one of Joshua's comments here, there are a couple of components, and let me add a little bit of context to help those of you who haven't studied this. Number one, there is a strong movement among many schooling experts to introduce arithmetic using lots of tangible items, using lots of physical, manipulative items, et cetera.

And I think that that emphasis that many alternative educational models have and have discovered is along this line, that just expecting a young child to do formalized, book-based arithmetic is probably not the best expectation. So using manipulatives, et cetera, can help with those concepts. Separately, there is a strong background of arithmetic being taught in a social way.

I have a complete set of the classic American educational texts called Ray's Arithmetic. McGuffey readers and Ray's Arithmetic were very widely used in American school systems one to two centuries ago. And a huge portion of that was done as oral participation from the whole class. And then finally, I think this is good reason to consider a narrative approach.

And so I mentioned in the previous episode that I'm a big fan of the Life of Fred books. My children enjoy them, et cetera. So I think that if you wish to avoid formal workbook-based math, there's abundant evidence that using manipulatives, teaching arithmetic informally, drilling and memorizing math tables, especially in a social setting, doing it at the breakfast table with all your children, et cetera, teaching, learning even formulas and things that are verbal that'll be used later, and then doing something like reading aloud from Life of Fred or allowing your children to read Life of Fred without too much hardcore pencil-on-paper calculations is an abundantly effective math curriculum.

And then when they are ready, developmentally ready, then introducing the workbooks is appropriate. Drive them to abstraction. Quote, "Harris has pointed out that in the first stages of the development of the mind, the mathematical process is decidedly more complex than the other mental processes which are taking place at that time." Quoting, "The reason why it requires a higher activity of thought to think quantity," which is an abstract number, "and understand mathematics than it does to perceive quality or things and environments," which are physical objects, "lies right in this point.

The thought of quantity is a double thought. It first thinks quality, object, and then negates it or thinks it away." In other words, it abstracts from quality. It first thinks thing and environment, quality, and then thinks both as the same in kind or as repetitions of the same. A thing becomes a unit, number, when it is repeated so that it is within an environment of duplicates itself, number among numbers.

Several very important consequences for the practical teaching of mathematics can be drawn from the fact formulated. The mathematical process may not be introduced before there is a considerable stock of qualitative facts in the child's mind on which to work, and not until the child's mental powers are sufficiently developed to take the steps implied in even the simplest mathematical concept.

It is a question whether we are not tending to introduce the abstractions of mathematics too early. The German boy who enters the gymnasium at the age of nine is expected to know only the four fundamental operations on integers, and in his first year, corresponding to our fourth grade, he learns further only the German weights and measures, decimal system, and the simplest operation with decimals.

By this time, our children are introduced to the complexities of fractions, common and decimal, to our system of weights and measures, far more complicated than the international decimal system used in Germany, and even sometimes to percentage and some use of generalized, literal numbers. And yet the German boy does not come out behind at the end of the race 10 years later.

It has even been urged that no formal study of mathematics is needed at all, but that pre-collegiate mathematics at least could be developed incidentally in the study of natural phenomena. Though this proposal is extreme, it contains much good. Yet the time must come when the child sees that he will save himself much trouble if he makes a mathematical tool and practices with it enough to have a fair amount of skill in its use.

The concrete application gives zest to the work, but there must be occasions when the mathematical process itself is a center of interest. It's an article or a book entitled, "That Comes From the Teaching of Mathematics in the Elementary and the Secondary School," by J.W.A. Young, Longmans, Green & Co, New York, published 1919.

Math class postponed, quote, "Several groups of important investigations on the teaching of arithmetic have contributed findings that have led schools to make changes in the organization of the curriculum. One group of studies dealt with the effect of postponing or deferring the teaching of arithmetic in the primary grades. Included in this group are the studies by Ballard in 1912, Taylor in 1916, Wilson 1930, and Benezet in 1935 and '36.

In these studies, formal arithmetic instruction was withheld in one group and administered as usual in another group. At the end of the experimental period, the comparative achievements of the two groups were measured. In each case, the experimenter recommended the postponement of formal arithmetic. Ballard for two years or the age of seven, Taylor for one year, Wilson for two years, and Benezet until grade five.

On the basis of these and other studies, the plan of eliminating formal arithmetic instruction from grades one and two, sometimes also grade three, has been adopted by a considerable number of school systems. In some systems, there is not even an approved plan of informal or incidental arithmetic. Such procedure fails to recognize certain very important facts about the studies referred to above.

A careful reading of the reports of these four experiments shows that, while formal practice on computational processes was postponed in the experimental groups, there was a great deal of use made in these classes of various kinds of activities, games, projects, and social situations through which the child was brought into contact with numbers and given the opportunity to use them informally in meaningful ways.

It is especially clear in the studies by Wilson and Benezet that arithmetic was not in fact postponed at all. It is evident that what happened in these two studies was that computational arithmetic was replaced by what I called earlier in this paper social arithmetic. In each study, the plan was to emphasize number meanings, to develop an understanding of the ways in which number functions in the daily lives of children, both in and out of school, and to develop what is called number readiness for the more formal work to follow.

That comes from an article entitled Deferred Arithmetic, published in Mathematics Teacher, volume 31, October 1938, by Leo Bruckner from a paper read at the annual meeting of the National Council of Teachers of Mathematics in Atlantic City, New Jersey. Two years before the math. Quote, "Preliminary to any useful discussion of the topic, "it is wise to clarify the issue.

"Although the proposal to defer parts of arithmetic "has been made periodically for a number of decades, "the present rather widespread interest "was no doubt stimulated very directly "by the series of three articles "written by Superintendent Benezet "and published in the Journal "of the National Education Association in 1935 and 1936.

"In the first of these articles, "Mr. Benezet expressed his belief as follows. "Quote, 'If I had my way, I would omit arithmetic "'from the first six grades. "'The whole subject of arithmetic could be postponed "'until the seventh year of school, "'and it could be mastered in two-year study "'by any normal child.' "From the Mathematics Teacher, "article entitled Deferred Arithmetic by G.T.

Buswell, "published May 1938. "The Korean War," Korean spelled C-H-O-R-E-A-N, "The Korean War. "Quote, 'The exaggerated ideas "'of the efficacy of arithmetic "'and the cultivation of the mind, "'and the resulting overpressure and premature training "'are strongly condemned by studies in mental hygiene. "'As English physician, Dr. Sturgis "'has studied Korea in children, "'and many of the cases he has found do, "'as he thinks, to causes connected "'with the schoolwork and arithmetic, "'he deems an especial factor in producing the disorder.

"'In the case of a nervous child, "'he maintains that working sums is liable to cause chorea. "'Chorea is an irregular nervous twitching of a muscle "'or group of muscles accompanied by irritability, "'forgetfulness, sleep disturbance, visual difficulties. "'The majority of cases begin between ages five and 10, "'and usually go away after the child is removed "'from classroom and schoolwork for three months.

"'In the case of some children, "'as pointed out by General Walker, "'work in arithmetic is a frequent cause "'of worry and interference with sleep. "'When children do sums in their dreams, "'this is a danger signal. "'Certain habits of interference of association, "'certain arrests, as they have been called by Dr.

Triplett, "'illustrate very well these secondary effects "'of certain methods and processes of learning. "'Number forms sometimes illustrate "'the secondary effects of instruction. "'Such habits represent not only so much "'mental ballast, but usually also interference "'of association and often the germs "'of pathological neuroses. "'They are probably pretty common. "'The counting habit, arithmomania, so-called, "'is likely to have several representatives in each class, "'according to Triplett's investigations.

"'This is a real handicap, "'filling the mind with quantitative ideas "'to the exclusion of causal relations. "'Hygiene is especially concerned "'with the problem of the age "'when work in arithmetic should be begun. "'In order to answer this question, "'it is necessary to consider briefly "'the mental operations involved in arithmetical work.

"'In the simpler study of number and number relations, "'in addition, subtraction, and the rest, "'the process of learning is chiefly one "'of acquiring habitual associations. "'What hygiene demands here "'is that these should be formed naturally, "'and that interference of association "'or mental confusion shall be avoided. "'Again, in teaching arithmetic to very young children, "'all sorts of objective methods "'and devices have been developed, "'and these are deemed necessary in such instruction.

"'Still further, it appears that the number forms "'and the like which are common in adults "'are developed in the early years of instruction. "'From these are likely to develop "'artificial and grotesque habits of thought, "'as illustrated by Dr. Triplett's so-called arrests, "'and by some of the number forms. "'The problem of the proper age for beginning arithmetic "'is then something like this.

"'At what age can a child be drilled "'in arithmetical processes "'without the aid of artificial devices, "'and the like, which are likely to persist "'as arrests or habits of interference of association? "'And at what age should the study of logic be begun? "'At what age does the child have a nascent interest "'for arithmetical work?' "We have at present no adequate data "for answering these questions, "but until further investigations have been made, "the verdict of hygiene "is that ordinarily formal instruction in arithmetic "should be postponed until at least the age of eight or 10.

"The Italian physiologist, Masso, "President G. Stanley Hall, Professor Patrick, "and others agree in condemning formal instruction "in this subject before this age. "Quote, 'Mathematics in every form,' "writes Professor Patrick, "is a subject conspicuously ill-fitted to the child mind. "It deals not with real things, but with abstractions. "When referred to concrete objects, "it concerns not the objects themselves, "but their relations to each other.

"It involves comparison, analysis, abstraction. "The grotesque number forms which so many children have "and which originate in this period "are evidence of the necessity which the child feels "of giving some kind of bodily shape to these abstractions, "which he is compelled to study. "The practical teachings of the hygiene of instruction "as regards arithmetic may be summed up "in the light of our present knowledge somewhat as follows.

"The formal instruction in this subject "should not be begun before the age of eight or 10. "Arithmetical work before this "should be spontaneous activity on the part of the child. "By postponing arithmetic until this age, "it is possible to do away for the most part "with artificial devices and methods "which may lead to arrests or interference "of association later on.

"The work in arithmetic should be simple, "and the complex examples in logic and the like "should be eliminated. "In the case of nervous children, "special care should be taken to avoid worry "and the development of neuroses like chorea. "And in general, special attention "should be given in this subject to the secondary effects "which are important from the point of view "of mental hygiene." That's from "Cyclopedia of Education," article by William Burnham, PhD, professor of pedagogy and school hygiene, Clark University, Worchester, Massachusetts.

Moore says less, quote, "In 1972, under the auspices "of the Hewitt Research Foundation, "we conducted a broad investigation "of approximately 3,000 sources "in early childhood education research "and other literature. "The Hewitt investigation traces "the single idea of school readiness. "We then carefully checked the bibliographies "of relevant items for further sources.

"These various sources yielded "more than 7,000 studies and papers. "About 1,000 items were closely analyzed "and categorized, of which 700 or so "have been included here." Scott cautions wisely that much research and education fails to produce new information that can be used beyond the situation in which it is acquired.

But when the findings of such studies, in concert with the findings of many other studies, all point in the same direction, the implications deserve examination. It is obviously unscholarly, unethical, and unwise to wave aside a possible truth because it does not agree with presently accepted knowledge or conventional practice.

Some of the trends here identified in early childhood literature are provocative in this respect. Here is a challenge to early childhood scholars to reexamine the early childhood dilemma. Rower shows that the effects of early instruction in mathematics, noted by Austin, were not statistically significant. What was significant was a strong negative correlation between school entry age and attitudes toward school.

Additional years in school did not contribute significantly to average performance in mathematics, but the earlier children had started school, the more negative their attitudes toward school. Developmental readiness, however, was still the most important factor for doing arithmetic and understanding paragraph meaning. A number of studies verify, the younger a child is when he starts to school, the more chronological age appears to affect this process throughout his school life.

Cumulative records over a period of six years revealed a continued disadvantage, even though as a group, they had a slightly higher IQ than those who entered school from six to nine months later. Children in the younger group were also more likely to repeat a grade. Fayberg's results showed that successful school achievement in areas requiring use of concepts, such as numbers, classes, and spatial and causal relationships, correlated highly with mental age.

Developing these concepts was especially associated with success in arithmetic, problem solving, and spelling. Strom observed that the excessive value attached to academic achievement and the pressures to grow up and achieve earlier could be damaging to personal development. If, as neurophysiologists suggest, brain structure and function move along together, requiring a child to undertake tasks for which he is not fully prepared is risking damage to the central nervous system.

It may also risk potential difficulties in the effective and motivational aspects of learning due to frustration, because the learning tools simply are not yet ready. Recent findings confirm this. If we expect reading and arithmetic based on understanding rather than on rote learning, delay of formal training in these areas appears wise, although informal education through warm parental responses is desirable.

Some scholars and clinicians conclude that formal education should wait until ages 10 to 14. Strong clinical and research evidence indicates that early exposure to the so-called stimulation of school often destroys childhood motivation for learning. By grade three or four, many children become stranded on a motivational plateau, never recovering their early excitement for learning.

Most primary teachers agree. That comes from a book called "School Can Wait" by Raymond S. and Dorothy N. Moore, Brigham Young University Press, Provo, Utah, published 1982. The brain, its plane is sprained if it is strained. Quote, "The axons or output parts of brain neurons "gradually develop a coating of a waxy substance "called myelin, which insulates the wiring "and facilitates rapid and clear transmission.

"At birth, only the most primitive systems, "such as those needed for sucking, "have been coated with myelin. "The process of myelination in human brains "is not completed at least until most of us are in our 20s. "While animal studies have shown "that total myelin may reflect levels of stimulation, "scientists believe its order of development "is mainly predetermined by a genetic program.

"While the system overall is remarkably responsive "to stimulation from the environment, "the schedule of myelination appears to put some boundaries "around appropriate forms of learning at any given age. "We should stop for a moment to discuss "some potential hazards in trying too hard "to make intelligence or learning happen.

"Some of the skill deficits of today's school children, "in fact, may have resulted from academic demands "that were wrong, either in content "or in mode of presentation, "for their level of development. "The same mentality that attempts to engineer stimulation "for baby brains also tries to push learning "into school children, much like stuffing sausages.

"For example, some parents now wonder "if their schools are any good "if they don't start formal reading instruction, "complete with worksheets in preschool. "Before brain regions are myelinated, "they do not operate efficiently. "For this reason, trying to make children "master academic skills for which they do not have "the requisite maturation may result "in mixed up patterns of learning.

"As we have seen, the essence of functional plasticity "is that any kind of learning, "reading, math, spelling, handwriting, et cetera, "may be accomplished by any of several brain systems. "Naturally, we want children to plug each piece of learning "into the best system for that particular job. "If the right one isn't yet available "or working smoothly, however, "forcing may create a functional organization "in which less adaptive, lower systems "are trained to do the work." As an example, let's take the kind of reasoning needed for understanding, not just memorizing one's way through higher level math.

Perhaps some readers of this book shared a common experience when they took algebra. Many of us functioned adequately until we reached Chicago, where two planes insisted on passing each other every day in class. When it wasn't planes, it was trains, people digging wells, or other situations that did not seem in any way related to graphs and equations of X, Y, and Z.

Personally, I found the more I struggled, the more confused I became, until soon I was learning more confusion than algebra. Moreover, I began to believe I was pretty dumb. Was I developing what Herman Epstein calls negative neural networks, resistant circuitry, toward this worthy subject? Having fled from math courses at the first available opportunity, I have since talked to other adults who confided that, after a similar experience, they also avoided math until forced years later to take a required course in graduate school.

At this point, their grownup brains discovered that they actually liked this sort of reasoning. In this personal example, it is very possible that the necessary neural equipment for algebra, taught in this particular manner, may not yet have been automatically available in my early adolescent brain. The areas to receive the last dose of myelin are the association areas responsible for manipulating highly abstract concepts, such as symbols, X, Y, Z, graphs, that stand for other symbols, numerical relationships, that stand for real things, planes, trains, wells.

Such learning is highly experience-dependent, and thus there are many potential neural routes by which it can be performed. Trying to drill higher-level learning into immature brains may force them to perform with lower-level systems and thus impair the skill in question. I would contend that much of today's school failure results from academic expectations for which students' brains were not prepared, but which were bulldozed into them anyway.

The brain grows best when it is challenged, so high standards for children's learning are important. Nevertheless, curriculum needs to be considered in terms of brain-appropriate challenge. Reorganizing synapses is much more difficult than having the patience to help them get arranged properly the first time around. Abstract rule systems for grammar and usage should be taught when most students are in high school.

Then, if previously prepared, they may even enjoy the challenges of this kind of abstract, logical reasoning, only, however, if the circuits are not already too cluttered up by bungled rule teaching. One ninth-grade student who came to me last year for help with grammar was hopelessly confused about the simplest parts of speech.

Although she was intelligent and could, at her current age, have mastered this material in a week, she had been a victim of meaningless grammar drills since second grade. As Michelle and I struggled on the simple difference between adjectives and adverbs, I often wished I could take a neurological vacuum cleaner and just suck out all those mixed-up synapses that kept getting in our way.

It took us six months, but finally, one day, the light dawned. "This is easy," she exclaimed. "It is." When brains are primed for the learning, and the student has a reason to use it, we're left with real literary models. Immersing children in good language from books and tapes, modeling patterns for their own speech and writing, and letting them enjoy their proficiency in using words to manipulate ideas are valid ways to embed grammar in growing brains.

No amount of worksheets or rule learning will ever make up for deficits resulting from lack of experience with the structure of real, meaningful sentences. It is folly to ignore the importance of oral storytelling, oral history, and public speaking in a world that will communicate increasingly without the mediation of print.

These skills build language competence in grammar, memory, attention, and visualization, among many other abilities. I personally believe that helping students at all grade levels memorize some pieces of good writing, narrative, expository, and poetic on a regular basis would provide good practice for language, listening, and attention. I do not mean reverting to a rote-level curriculum, but simply taking a little time each week to celebrate the sounds of literate thought.

At the same time, schools must get into the business of teaching children to listen effectively, because no one else seems to be doing it. That is from a book entitled "Endangered Minds, Why Children Don't Think, and What We Can Do About It" by Jane Healy, Simon & Schuster, published 1990.

What should we then do? Historically, the age for instruction in arithmetic and mathematics seems to have slowly shifted from age 15 or later down to age 10. Then, about a century ago, this was shifted again to about age seven or six. In very recent times, it has shifted again to age five or four, but recorded history may not be the place to go in order to find substantive support for the practice of beginning formal instruction in arithmetic at any age, five, 10, or 15.

There is more material in arithmetic and mathematics to learn and to use today than the ancients studied. Some may argue that starting earlier allows more time to learn more material. It seems obvious to them that if a child learns to do multiplication and division at age 10, then he is five years ahead of the child who learns to do multiplication and division at age 15, right?

Perhaps. So if we teach him to multiply and divide at age five, he would be 10 years ahead. At birth, 15 years ahead. Get the point? This is more than merely an issue of enough time. This is an issue of development. How much math there is to learn and how early children may have been forced to learn some math.

These considerations do not give us data to define the time when it is most effective and most efficient to begin teaching arithmetic and mathematics. Most obviously, there is a time when it is too early. Those who advocate formal arithmetic at age five appear to have ignored this developmental issue.

And when the results are not like they want, they patch them up with experimental classroom methods which try to emulate informal experiences in arithmetic, a tacit witness to informal instruction before age 10. In our culture, we erroneously perceive that the only way anyone anywhere at any time can learn arithmetic is from early formal instruction, usually in a classroom school.

But young children have learned the basic concepts of number in every culture without any formal instruction. Games, measurements, and commercial activity have been the primary childhood instructors. They are still the best instructors of young children. Withholding formal instruction until age 10 will by no means guarantee failure. Depending on what arithmetic activities are done, it may actually guarantee the child's success.

What we suggest is, one, formal textbook or workbook instruction in arithmetic may begin at age 10. It is about age 10 that the developmental light bulb goes on and the child becomes capable of a great deal more mental and physical skill. Of course, that's not an absolute rule. With a few children, it is as early as eight.

We call them bright children because the developmental light bulb goes on early. Waiting until the child is developmentally prepared to handle the concepts makes instruction in arithmetic very easy because the child learns very quickly. Two, there is no necessity for formal teaching in arithmetic before age 10. Once all of the developmental parts are there, most children can learn in a few weeks everything which they might have spent six years learning, kindergarten through fifth grade.

That is, if they haven't already learned it through questions and experiences and working things out on their own, which is generally the case. Three, depending upon the child, upon the method, and upon the subject matter covered, there exists the potential for developmental harm from the formal teaching of arithmetic before age 10.

Small children cannot understand many arithmetic concepts at an early age. We can teach them to perform the process, but we cannot make them understand the concepts. The child learns to hate learning. The child's understanding develops along the wrong lines. He may actually develop mental blocks to arithmetic, actual physiological blocks in the brain.

This may give new meaning for the term blockhead. Four, not formally teaching arithmetic before age 10 frees up a lot of time for other activities which will build the vocabulary of the child. Vocabulary is the number one index of intelligence. Developing vocabulary was one of the deliberate foci of ancient education.

We waste valuable time for developing vocabulary and verbal language skills if we instead spend those hours teaching a five-year-old to count by fives. He'll know it intuitively by age 10 anyway without ever being taught. Instead, we ought to spend those hours reading to him. We only have so much time in the day.

Do we wanna spend it trying to force math skills into a child who developmentally is not optimally prepared or spend it doing what is developmentally natural to a young child, learning new words and associating them with new ideas and experiences? Stretch the child's vocabulary during the formative years and when he's developmentally ready to do some deeper thinking, he'll have a mind prepared to take on the task and he'll take off like a rocket.

Please note, we are not saying that no child should ever utter the name of a number before age 10. Not at all. About age four, most children discover money and there is no hiding numbers from them after that. They encounter numbers all of the time. If we encourage learning, then they'll be asking lots of questions and we'll be full of opportunities to teach numbers and measurement.

But we would not encourage using a formal workbook before age 10 unless the child has a genuine desire to do so. He shows that he is competent to handle the work and it does not take away time from other valuable activities We are not going to ruin the child if we wait until age 10 before beginning the formal teaching of arithmetic.

Thus concludes the appendix of teaching the trivium, Christian homeschooling in a classical style by Harvey and Laurie Bludhorn. Again, I wanted to hasten to add this after focusing so heavily on the value of using mathematics as a way to grow our children's brains. And I want you to be aware because as parents, all of us feel this intense pressure from other parents around us, perhaps our parents, perhaps school administrators and authorities who in some cases are quite literally looking over our shoulder, inspecting our work, et cetera.

And we read stories about incredible prodigy children who can do all these incredible things. And we all say, well, I just gotta get after it. Gotta push, push, push, push, push, push, push. But we need to not be influenced by those things and recognize that there is a time and a place to do everything.

And that just because you can do something at an early age doesn't mean that you should do something at an early age. There are three basic concepts that I think are important for learning. And let me use a physical example, right? Because we can look at physical development. I'm tempted to use an example from the sexual development of children, but I'll let you consider that on your own.

And instead I'll substitute something like running. I would bet that we could teach and train a five-year-old to run marathons. I'm not aware of five-year-olds who have run marathons, although I'm certain that there are some that could walk run them. But I would bet you that we could train our five-year-old children to run a marathon.

Question is, does that mean that we should train our five-year-old children to run marathons? First, we say, what is the purpose of running a marathon in the first place? Why is that necessary? Why are we doing it at all? And you can see that running a marathon could be a useful expression of development, but that doesn't mean that age five is the age to start.

It seems much more natural to have, say, a 13-year-old or a 15-year-old, somebody who's approaching that adult space, start to engage in marathon training. And again, I don't know anything about teaching five-year-olds to run long distances, but it would just seem to me very inefficient to try to get a five-year-old who has physically small legs and physically small lungs, et cetera, to try to become an elite long-distance runner.

And while we might encourage running, we may not choose a formalized training system. Number two, if we want our five-year-olds to enjoy running, we probably ought not to inflict upon them the kind of marathon training that an elite Kenyan marathon runner engages in, the hours and the hours and the hours of grinding.

That would probably be a pretty decent way to destroy our child's love of running. So even if we did have a five-year-old who was spectacularly capable at running, we don't want to inflict upon the five-year-old a professional marathoner's training schedule. The professional marathoner, while perhaps he doesn't love every minute of his training, on the whole is going to enjoy his training.

Otherwise he's not gonna do it. But the child is not ready for such a grueling pace and does not get the same sense of fulfillment by doing hard things that the adult does. And then third and finally, let's assume that we have a five-year-old who is skilled at running and we want to encourage running.

Doesn't it seem that there are better ways to do that? Taking our child and putting him on a soccer team or just playing pickup games of soccer or some other running-based sport is a much more natural way of indulging that love of learning. And we can encourage him to develop his heart, develop his lungs, develop his legs, but we can do it in the format that is more appropriate to a child.

And then we can wait and see in the fullness of time. Well, by way of analogy, we ought to look at mathematics the same way. Mathematical ability is wonderful, it's great. And in the fullness of time, perhaps we will want to press and press and press our children to achieve their personal mathematical potential.

But let's not do it in a way that is inappropriate for young children. Let's bring in the mathematics in a simpler, more informal way. Let's wait for the child to be developmentally ready before we really go after it. And let's try to nurture the child's love of learning. If you want your child to be a lifelong mathematical expert, then you will need to nurture his love of mathematics, not grind it out of him at an excessively young age.

One final comment. I mentioned that I was not myself specifically following this rule of no formalized mathematics until age 10. It's not because I disagree with the advice, but rather it's that I didn't come across the advice until very recently. I didn't read this book until last year. And what sparked my interest in this was studying mathematical curricula and thinking about what curriculum I'm interested in using for my children in high school ages.

And I was reading essays by math teachers and one of them said very clearly, don't put your children in algebra until they have hair on their armpits. And I thought that's an interesting and quite provocative, vivid image to think about. And then when I found this other book that I just got last year and read, it so impressed me that I thought, I think there's something here and I think more of us should pay attention to it.

But I'd already started on the traditional workbook program. My children have been using math workbook texts since first grade. Now, they're fairly light and I haven't identified them to be causing problems, but I was already kind of on what is now the traditional path. And I've continued on it.

What I have noticed is that my eldest children who are using workbook maths, I think are mentally a little older than a lot of their peers. I've noticed quite a bit more kind of mental age. And so I've tried to be careful and I definitely don't want to overwhelm them.

This point, I'm not stopping intentionally the workbook math, but I am very free in my mind that if math is causing problems, then I'm going to release it and use this concept of waiting on the formalized stuff until a later age. And I think that it's important that we talk about this because there are a lot of tears shed in many children's bedrooms over math and it ought not to be so.

The other thing I have spent a lot of time thinking about is, okay, what can we do with the time instead? Now, when we get to training the soul or the character, I would point out there's a lot that a child needs that doesn't fit into an academic rubric.

I believe children need hours and hours playing outside with their friends or with their siblings, just complete unencumbered free play. That's really important to me. And I don't want to sit around and do workbook math. I mean, some of the Asian cultures, I admire them for their mathematical prowess.

And sometimes when I have been there physically or spoken to people about some of the intensity, I cry for the lack of childhood development that can happen with just playing outside in the woods with your friends. And so that's an option. I think the thing that you can definitely press at a young age is language development.

I can't see any downside to language development. I see downsides to other things. So if we think about the three Rs, which I've stolen from Naval, has now five points, that what is education? Well, it's reading, writing, arithmetic, coding, computer coding, and persuasion. Well, persuasion is something that can clearly wait till a later age, right?

We can talk about that in terms of rhetoric at a later stage in adolescence and beyond stage where we can formally study persuasion. Coding, I don't see any benefit to starting at a young age. I'm open to research, to the contrary, but I haven't found any computer science expert, any programmer, et cetera, who has said, "No, Joshua, absolutely, you have to start coding at the age of five in order to be a world-class coder." I don't know the specific age, but I think at least age 10, more likely age 13 or so, and is probably fine for coding, even though I have looked at some of the curricula that have been developed for bringing coding into first grade, et cetera, and I'll talk about that in a separate show.

Well, reading, writing, arithmetic. We talked about arithmetic today. What about writing? Well, for writing, a child needs a certain amount of physically writing, right, a certain amount of coordination, muscular skill, hand-eye coordination, et cetera. The child needs something to say, and so I think that formal writing should be delayed.

Now, what we do is we practice what's called narration, which is preparation for writing. Narration is something that is the verbal equivalent of writing that allows the child to practice composing thoughts and basically composing small essays, but without the limiting effect of writing, 'cause writing needs time to develop that strength and that muscular fluidity.

Well, reading is also important, but we don't wanna try to force our three-year-olds to read. We wanna encourage them to read when they're capable, but we wanna encourage them, but we wanna be thoughtful. So at its core, what is the thing that we can do that's not going to harm our child?

Well, number one is, again, play, relationships, character development, a huge thing, intentionally trained habits in the character, but we can do language development. A child who is three is in a mode of rapid language acquisition. So it was a child who's five, who's seven, who's eight, and who's 10, et cetera.

So this is the age at which the child is wired for language acquisition. So we wanna emphasize lots and lots of language. And even if that's not physically reading until the child is ready, then two hours of reading aloud to the child, say two hours of audio books as well, you're gonna have lots and lots of language acquisition.

And so help your child to develop that broad vocabulary that we talked about previously. Also, what is a subset of language development? Well, other languages. A child's young brain, while I'll discuss this separately, I do not think that a child is necessarily a better learner of languages than an adult's.

That is certainly an interest, and the child's brain is very open to languages. And that's where we will go next. If you want to help your children to be smarter, then help them to learn multiple languages at an early age, even if you do this somewhat intensively. That same aspect of normal life can be a component of basically intensive language acquisition.

If you pick your family out from Atlanta to Georgia and move to Hong Kong and put your child into a local play group, or let's choose a different place with a little bit more nature, into a rural China or somewhere, then your child is gonna be outwardly playing and running and playing in the woods, but is going to be in a very intense period of language acquisition.

And the same thing can be done, of course, right at home. And so that's where we go next, is let's talk about multilingualism as a way to enhance your child's intelligence. Hope this essay helps you to consider what is right for you and yours. Always remember, you're the parent, you know your child.

It is your job to make sure that the teachers that you have brought to help your child are not harming him, but are helping him. And thus, that's why I want you to be equipped with this information and background.