"The next point is to demonstrate everything demonstrable." This part seemed straightforward enough that in my original analysis this was the only part of the entire section that I noted. Demonstrating everything demonstrable means using counters of some sort to actually show the problem while working it until the child has internalized the concept.
"A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate."
No sheets of problems until the same problems have been successfully, and repeatedly, worked with counters and without written numbers.
"He may arrange an addition table with his beans, thus––
0 0 0 = 3 beans
0 0 0 0 = 4 "
0 0 0 0 0 = 5 "
and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9, etc."
Addition Table
We need to create that table, but do it with counters, one line at a time, working with that line until it is known without needing the counters.
"Thus with 3, 4, 5,––each of the digits: as he learns each line of his addition table he is exercised upon imaginary objects, ’4 apples and 9 apples,’ ’4 nuts and 6 nuts’ etc.; and lastly, with abstract numbers––6+5, 6+8."
After we have learned one line from the table, we then repeat the exercise using imaginary objects instead of counters. After that is learned well, we do the same process with actual numbers, but orally rather than written. (She does mention that the child might write out the problems on his slate after each line of the table is finished, but only if he is already writing figures.)
"A subtraction table is worked out simultaneously with the addition table. As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5?"
After we have learned a line from the addition table, presumably all the way through the abstract numbers step, we then repeat the process using the same line from the subtraction table.
"When the child can add and subtract numbers pretty freely up to twenty, the multiplication and division tables may be worked out with beans, as far as 6×12; that is, ‘twice six are 12′ will be ascertained by means of two rows of beans, six beans in a row."
When we have learned the addition and subtraction tables with digits, we move on to multiplication, but it appears that here we just learn each line of the table using counters and don’t continue on to the mental work yet.
"When the child can say readily, without even a glance at his beans, 2×8=16, 2×7=14, etc. , he will take 4, 6, 8, 10, 12 beans, and divide them into groups of two: then, how many twos in 10, in 12, in 20? And so on, with each line of the multiplication table that he works out."
Once the line from the multiplication table has been learned, we do the same line from the division table.
"Now he is ready for more ambitious problems: thus, ‘A boy had twice ten apples; how many heaps of 4 could he make?’ He will be able to work with promiscuous numbers, as 7+5-3. If he must use beans to get his answer, let him; but encourage him to work with imaginary beans, as a step towards working with abstract numbers."
Then it looks like we start working with mental word problems, followed by actual numeric problems, using counters when needed by trying not to.
"Carefully graduated teaching and daily mental effort on the child’s part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and effort of mind."
This part does seem key. We have to take it slow and steady, and keep working at it a little at a time.
All my Charlotte Mason math posts.
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