Research on the Teaching of Math

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Formal Arithmetic at Age Ten, 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 ten before beginning formal instruction in arithmetic. Waiting until age ten 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

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. — A History of Education in Antiquity, by Henri I. Marrou, translated by George Lamb, Sheed and Ward, London, 1956, page 158.

What the Medievals and Later Did

[page 204] Before the Reformation there was little or nothing accomplished in the way of public education in England. In the monasteries 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 twelve pence. See Tylor’s Primitive Culture, New York, 1889, Vol. I., p. 242.) . . . In the sixteenth 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 every-day life forced upon the boys 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 [page 205] could not divide 2021 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 . . . . 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, [page 206] who had been attending Westminster, was put into a commercial school, that he might study arithmetic and book-keeping 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 . . . .

[page 207] 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, geography, and modern history were first studied” in 1829. At the Merchant Taylors’ 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 shoe-making, 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 . . .

[page 217] 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 . . . . [page 218] Reform in arithmetical teaching in the United States did not begin until the publication of Warren Colburn, in 1821, [page 219] of the Intellectual Arithmetic. This was the first fruit of Pestalozzian 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 engrafting Pestalozzian principles on written arithmetic . . . .

A History of Elementary Mathematics with Hints on Methods of Teaching, by Florian Cajori, Ph. D., London: Macmillan Company, 1917, pages 204-207, 217-219.

. . . The American Calculator, first published in 1828 . . . is reasonably typical of the colonial period. This text was used with older students (beginning at about the age of eleven). It was complete in itself, not one book of a series such as the texts that evolved in later years. — Readings in the History of Mathematics Education, edited by James K. Bidwell and Robert G. Clason, National Council of Teachers of Mathematics, Washington, D. C., n. d., page 2.

. . . The study of it [arithmetic] used to be put off to a very late period. Scholars under twelve or thirteen 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. — Readings in the History of Mathematics Education, page 25, taken from “Teaching of Arithmetic,” text of an address delivered by Warren Colburn before the American Institute of Instruction in Boston, August, 1830, reprinted from the Elementary School Teacher 12 (June 1912): 463-480.

[In the early seventeenth century, the grammar school curriculum was] almost certainly confined to Latin grammar, the Catechism and Bible study . . . . [P]upils arriving at Oxford and Cambridge frequently did not have any knowledge of Arabic numerals, to say nothing of the elementary arithmetical operations. — A History of Mathematics Education in England, by Goeffrey Howson, Cambridge University Press, Cambridge, 1981, page 30.

When It Changed

. . . 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: (1) He taught arithmetic to children as soon as they entered school, basing his work on perception. (2) He insisted upon a knowledge of number and the simplest operations, using objects, before the figures were taught. (3) He approached the subject of fractions in the same way. (4) He made arithmetic the most prominent subject in the school, and it is to his influence that its present prominence is due. (5) He emphasized oral arithmetic, a movement that led to the great success of Warren Colburn in the United States . . . . Tuerk (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 . . . . — A Cyclopedia of Education, Edited by Paul Monroe, New York: The Macmillan Co., 1919, Volume I, Article on The History of Teaching Arithmetic, by David Eugene Smith, Ph. D., LL. D., Professor of Mathematic, Teachers College, Columbia University, New York City, page 206.

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 fifteen and eighteen.

It was not until the sixteenth century that arithmetic began to be taught to children as young as age twelve, or even ten.

It was Pestalozzi, at the beginning of the nineteenth century, who began to teach arithmetic to children as young as age six or seven, though the practice of waiting until age ten persisted well into the twentieth century.

So to wait until age ten to teach arithmetic is actually, from an historical perspective, to advocate an “early start.” It is only from a decidedly modern perspective – a provincial perspective – that waiting until age ten 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

. . . early childhood may simply be an inefficient period in which to try to teach skills that can be relatively quickly learned in adolescence. — Prime Time for Education: Early Childhood or Adolescence? by William D. Rohwer, Jr., Harvard Educational Review, Vol. 41, No. 3, August 1971, page 316, from the summary.

Children Don’t Do Better in Math, They Just Learn To Hate School

In a cross-national study of mathematics achievement (Husen, 1967), stratified samples were drawn from the total population of all students enrolled in the modal grade for thirteen-year-olds in twelve 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 toward 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-toward-school scores. The results reveal a negligible negative correlation between age of school entry and mathematics achievement (rho = -. 06, p>. 05) and a strong negative correlation between entry age and attitude toward school (rho = -. 72, p<. 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 that the extremes of the nations 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 toward school that the longer the student was enrolled prior to testing the more negative his attitudes toward school itself. Clearly, there is no indication in these results that revising the mandatory age of school entry to younger levels would improve the student’s 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 Husen (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 categories (clerical through professional) and for those falling in lower-SES categories (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 = +. 19, p>. 05) while that for the lower-SES categories is negative (rho = -. 39, p>. 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 Husen (1967) 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. — Prime Time for Education: Early Childhood or Adolescence? page 322.

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 “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 moved selected topics to higher grade levels. Hence, the name “stepped-up” curriculum . . . .

. . . Benezet in Manchester, N. H., carried out a study from which he concluded “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 observation showed that there was much arithmetic taught in grades I to VI. Thiele visited the Manchester, N. H., schools and said: “Firsthand observation leads me to conclude that Benezet 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 Benezet had only deferred “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,” Brueckner says, “From these studies the conclusion should be drawn not that arithmetic should be postponed, [page 18] 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.” — What does Research say about Arithmetic? By Vincent J. Glennon and C. W. Hunnicutt, National Education Association, Washington D. C., 1952, page 17.

Drive Them to Abstraction

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.

“The reason why it requires a higher activity of thought to think quantity [abstract number] and understand mathematics than it does to perceive quality (or things and environments) [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 [page 50] 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 ten years later.

[page 100] 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 centre of interest.

The Teaching of Mathematics in the Elementary and the Secondary School, by J. W. A. Young, Longmans, Green and Co. New York, 1919, page 49, 50, 100.

Math Class Postponed

[page 288] 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 in 1930, and Benezet in 1935-36. 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 5.

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 a 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 . . . .

Mathematics Teacher, Volume 31, October, 1938, pages 287-292, article “Deferred Arithmetic” by Leo J. Brueckner, from a paper read at the annual meeting of the National Council of Teachers of Mathematics in Atlantic City, J. J., Feb. 26, 1938.

Two Years Before The Math

[page 195] 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 expresses his belief as follow: “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 years’ study by any normal child.” — The Mathematics Teacher, Volume 31, Number 5, May 1938, article “Deferred Arithmetic” by G. T. Buswell, pages 195-200.

The Chorean War

. . . The exaggerated ideas of the efficacy of arithmetic in the cultivation of the mind and the resulting over-pressure and premature training are strongly condemned by [studies in mental] hygiene . . . .

. . . An English physician, Dr. Sturgis, has studied chorea in children, and many of these cases he has found due, as he thinks, to causes connected with the school work, and arithmetic he deems an especial factor in producing the disorder. In case of a nervous child he maintains that working sums is liable to cause chorea. [page 643, 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 ten, 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 8 or 10. The Italian physiologist, Mosso, President G. Stanley Hall, Professor Patrick, and others agree in condemning formal instruction in this subject before this age. “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 8 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.

Cyclopedia of Education, p. 208, article by William H. Burnham, Ph. D., Professor of Pedagogy and School Hygiene, Clark University, Worcester, Mass.

Moore Says Less

[Page 10] 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.

[Page 11] Scott cautions wisely that much “research” in 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.

[page 140] . . . [Rohwer] showed 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 attitude 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.

[page 141] . . . 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 progress 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.

. . . Feyberg’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.

[page 142] . . . 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 . . . .

[page 228] 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 affective and motivational aspects of learning due to frustration, because the learning “tools” simply are not yet ready. [Our emphasis] 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 ten to fourteen . . . . 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.

School Can Wait, by Raymond S. And Dorothy N. Moore et. al., Brigham Young University Press, Provo, Utah, Hewitt Research Foundation, Berrien Springs, Michigan, 1982.

The Brain, It’s Plain, Is Sprained If It Is Strained

[Page 66] [T]he 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 [page 67] coated with myelin . . . . The process of myelination in human brains is not completed at least until most of us are in our twenties. 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 . . . . [W]e 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 schoolchildren, 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 schoolchildren 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, etc. – 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.

[page 68] . . . 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 or people digging wells or other situations that did not seem in any way related to graphs and [page 69] equations of X, Y, and Z. Personally, I found that 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 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!

[page 289] 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 with real literary models.

[page 290] 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.

Endangered Minds: Why Children Don’t Think And What We Can Do About It, by Jane M. Healy, Simon and Schuster, New York, 1990.

What Should We Then Do?

Historically, the age for instruction in arithmetic and mathematics seems to have slowly shifted from age fifteen or later down to age ten. 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, ten, or fifteen.

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 ten, then he is five years ahead of the child who learns to do multiplication and division at age fifteen. Right? Perhaps. So if we teach him to multiply and divide at age five he would be ten years ahead. At birth, fifteen 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 ten.

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 ten 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:

  1. Formal textbook or workbook instruction in arithmetic may begin at age ten. It is about age ten 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.
  2. There is no necessity for formal teaching in arithmetic before age ten. 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.
  3. 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 ten. 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.”)
  4. Not formally teaching arithmetic before age ten 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 ten 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 want to 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 ten. 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 ten, 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 ten before beginning formal teaching of arithmetic.



  1. What I Have Learned About Mathematics - Unembellished - […] Well-researched article on the timing of teaching math […]

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