My dear Greaves,

You are aware of the nature of those exercises which were adopted at my suggestion, as calculated to employ the mind usefully and to prepare it for further pursuits, by eliciting thought and forming the intellect.

I would call them preparatory exercises in more than one respect. They embrace the elements of number, form, and language; and whatever ideas we may have to acquire in the course of our life, they are all introduced through the medium of one of those three departments.

The relations and proportions of number and form constitute the natural measure of all those impressions which the mind receives from without. They are the measures, and comprehend the qualities, of the material world; form being the measure of space, and number the measure of time. Two or more objects distinguished from each other as existing separately in space, pre-suppose an idea of their forms, or in other words, of the exact space which they occupy; distinguished from each other as existing at different times, they come under the denomination of number.

The reason why I would so early call the attention of children to the elements of number and form is, besides their general usefulness, that they admit of a most perspicuous treatment - a treatment, of course, far different from that in which they are but too often involved and rendered utterly unpalatable to those who are by no means deficient in abilities. The elements of number, or preparatory exercises of calculation, should always be taught by submitting to the eye of the child certain objects representing the units. A child can conceive the idea of two balls, two roses, two books; but it cannot conceive the idea of "two" in the abstract. How would you make the child understand that two and two make four, unless you show it to him first in reality? To begin by abstract notions is absurd and detrimental instead of being conducive. The result is, at best, that the child can do the thing by rote without understanding it; a fact which does not reflect on the child but on the teacher who knows not a higher character of instruction than mere mechanical training.

If the elements are thus clearly and intelligibly taught, it will always be easy to go on to more difficult parts, remembering always that the whole should be done by (questions). As soon as you have given to the child a knowledge of the names by which the numbers are distinguished you may appeal to it to answer any question of simple addition, or subtraction, or multiplication, or division, performing the operation in reality by means of a certain number of objects, balls for instance, which will serve in the place of units.

It has been objected that children who had been used to a constant and palpable exemplification of the units by which they were enabled to execute the solution of arithmetical questions, would never be able afterwards to follow the problems of calculation in the abstract, their balls, or other representatives being taken from them.

Now, experience has shown that those very children who had acquired the first elements in the palpable and familiar method described, had two great advantages over others. First, they were perfectly aware, not only what they were doing but also of the reason why. They were acquainted with the principle on which the solution depended; they were not merely following a formula by rote; the state of the question changed, they were not puzzled, as those are who see only as far as their mechanical rule goes, and not farther. This, while it produced confidence and a feeling of safety, gave them also much delight a difficulty overcome, with a consciousness of a felicitous effort, always prompts to the undertaking of a new one.

The second advantage was that children well versed in those illustrative elementary exercises, afterwards displayed great skill in (head-calculation) (calcul de tête). Without repairing to their slate, or paper, without saking any memorandum of figures, they not only performed operations with large numbers but they arranged and solved questions which at first might have appeared involved, even had the assistance of memorandums, or an execution on paper, been allowed.

Of the numerous travellers of your nation who did me the honour to visit my establishment there was none, however little he might be disposed or qualified to enter into a consideration of the whole of my plan, who did not express his astonishment at the perfect ease and the quickness with which arithmetical problems, such as the visitors used to propose, were solved. I do not mention this, and I did not then feel any peculiar satisfaction, on account of the display with which it was connected, though the acknowledgment of strangers can by no means be indifferent to one who wishes to see his plan judged by its results. But the reason why I felt much interested and gratified by the impression which that department of the school invariably produced was that it singularly confirmed the fitness and utility of our elementary course. It went a great way, at least with me, to make me hold fast the principle that the infant mind should be acted upon by illustrations taken from reality, not by rules taken from abstraction; that we ought to teach by things more than by words.

In the exercises concerning the elements of form, my friends have most successfully revived and extended what the ancients called the (analytical method) -the mode of eliciting facts by problems, instead of stating them in theories; of elucidating the origin of them instead of merely commenting on their existence; of leading the mind to invent instead of resting satisfied with the inventions of others. So truly beneficial, so stimulating is that employment to the mind that we have learned fully to appreciate the principle of Plato, that whoever wished to apply [himself] with success to metaphysics ought to prepare himself by the study of Geometry. It is not the acquaintance with certain qualities or proportions of certain forms and figures (though, for many purposes, this is applicable in practical life and conducive to the advancement of science) but it is the precision of reasoning and the ingenuity of invention, which, springing as it does from a familiarity with those exercises, qualifies the intellect for exertion of every kind.

In exercises of number and form less abstraction is at first required than in similar ones in language. But I would insist on the necessity of a careful instruction in the maternal language. Of foreign tongues, or of the dead languages, I think that they ought to be studied, by all means, by those to whom a knowledge of them may become useful, or who are so circumstanced that they may indulge a predilection for them if their taste or habits lead that way. But I know not of one single exception that I would make of the principle that, as early as possible, a child should be led to contract an intimate acquaintance with, and make himself perfectly master of, his native tongue. Charles the Fifth used to say that as many languages as a man possessed, so often was he man. How far this may be true I will not now inquire: but this much I know to be a fact that the mind is deprived of its first instrument or organ, as it were, that its functions are interrupted and its ideas confused, when there is a want of perfect acquaintance and mastery of at least (one language). The friends of oppression, of darkness, of prejudice, cannot do better nor have they at any time neglected the point, than to stifle the power and facility of free, manly, and well-practised speaking; nor can the friends of light and liberty do better, and it were desirable that they were more assiduous in the cause, than to procure to every one, to the poorest as well as to the richest, a facility, if not of elegance, at least of frankness and energy of speech - a facility which would enable them to collect and clear up their vague ideas, to embody those which are distinct, and which would awaken a thousand new ones. (PSW 26, p. 127-131)