"When the child is able to work pretty freely with small numbers, a serious difficulty must be faced, upon his thorough mastery of which will depend his appreciation of arithmetic as a science; in other words, will depend the educational value of all the sums he may henceforth do. He must be made to understand our system of notation. Here, as before, it is best to begin with the concrete: let the child get the idea of ten units in one ten after he has mastered the more easily demonstrable idea of twelve pence in one shilling."
So after we work with basic arithmetic and achieve mastery of the four operations with small numbers, we move to working with money for a time to introduce the concept of place value. Two skills are drilled during this process: converting a quantity of one coin into larger coins, and noting on paper the value of the whole.
"Let him have a heap of pennies, say fifty: point out the inconvenience of carrying such weighty money to shops. Lighter money is used––shillings. How many pennies is a shilling worth? How many shillings, then, might he have for his fifty pennies? He divides them into heaps of twelve, and finds that he has four such heaps, and two pennies over; that is to say, fifty pence are (or are worth) four shillings and two pence. I buy ten pounds of biscuits at fivepence a pound; they cost fifty pence, but the shopman gives me a bill for 4s. 2d.; show the child how to put down: the pennies, which are worth least, to the right; the shillings, which are worth more, to the left."
Then we introduce place value.
"When the child is able to work freely with shillings and pence, and to understand that 2 in the right-hand column of figures is pence, 2 in the left-hand column, shillings, introduce him to the notion of tens and units, being content to work very gradually."
"We have but nine figures and a nought: we take the first figure and the nought to express another number, ten; but after that we must begin again until we get two tens, then, again, till we reach three tens, and so on. We call two tens, twenty, three tens, thirty, because ‘ty’ (tig) means ten. But if I see figure 4, how am I to know whether it means four tens or four ones? By a very simple plan. The tens have a place of their own; if you see figure 6 in the ten-place, you know it means sixty. The tens are always put behind the units: when you see two figures standing side by side, thus, ’55,’ the left-hand figure stands for so many tens; that is, the second 5 stands for ten times as many as the first."
We must drill with this concept, just using the tens and ones, for a time until the child is completely comfortable with the idea.
"Let the child work with tens and units only until he has mastered the idea of the tenfold value of the second figure to the left, and would laugh at the folly of writing 7 in the second column of figures, knowing that thereby it becomes seventy. Then he is ready for the same sort of drill in hundreds, and picks up the new idea readily if the principle have been made clear to him, that each remove to the left means a tenfold increase in the value of a number."
Then we move on to larger units, and drill again. However, we do not work any problems with large numbers until the concept of place value for that number has been mastered.
"Meantime, ‘set’ him no sums. Let him never work with figures the notation of which is beyond him, and when he comes to ‘carry’ in an addition or multiplication sum, let him not say he carries ‘two,’ or ‘three,’ but ‘two tens,’ or ‘three hundreds,’ as the case may be."
All my Charlotte Mason math posts.