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Another article online I read -- and I highly recommend this one -- is called The Teaching of Arithmetic. It's a three part series originally published in 1935 by a superintendent of a school district in New Hampshire in which he describes the deplorable condition of arithmetic and English in his schools, and how he improved both of those by eliminating formal arithmetic from the curriculum in five classrooms and replacing it with his "new Three R's" -- reading, reasoning, and recitation. By "recitation" he meant the same thing Charlotte Mason meant by "narration." The author says, "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."
This does not mean that the children were not taught any math at all. They learned to count, to read numbers, to tell time, to understand and use money They learned to use comparatives, such as more, less, half, and double. They learned to measure distances and weights and so on, and were given lots of practice in estimation. They learned skip-counting and multiplication and fractions.
In the first two grades, most of this work was done as it came up in class. For example, reading numbers was learned "incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text." The rest was taught by doing things, like counting money, or using a ruler or yardstick to measure things.
All of the problems they were given at this stage were done either mentally or using objects, and the answers were given orally. Any problem "which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older."
The problems they solved were pretty complex. He describes a test he gave a group of fourth and fifth graders involving a map and dates and distances, and the children were perfectly able to work out in their heads that Niagara Falls had first been discovered by Europeans 250 years earlier, that the falls had been retreating upstream at a rate of one mile every 500 years, and that it would take 10,000 years for the falls to retreat all the way to Buffalo.
That is exactly what I found most interesting about his method -- the way he taught his students to reason their way through problems. In the other classrooms, the students would get hung up on which mathematical function they ought to be doing and would make things really hard on themselves and sometimes make it impossible to find the answer.
He describes their inability to do the map work mentioned above. Here's another example.
I drew on the board a little diagram and spoke as follows: "Here is a wooden pole that is stuck in the mud at the bottom of a pond. There is some water above the mud and part of the pole sticks up into the air. One-half of the pole is in the mud; 2/3 of the rest is in the water; and one foot is sticking out into the air. Now, how long is the pole?"He describes the lengthy conversation he had with this traditionally taught classroom, trying to get the children to reason their way through the problem. When he gave this problem to his experimental classrooms the students saw the point immediately.
First child: "You multiply 1/2 by 2/3 and then you add one foot to that."You will note that not one child saw the essential point, that 1/2 the pole was buried in the mud and the other half of it was above the mud and that 1/3 of this half equaled one foot. Their only thought was to manipulate the numbers, hoping that somehow they would get the right answer.
Second child: "Add one foot and 2/3 and 1/2."
Third child: "Add the 2/3 and 1/2 first and then add the one foot."
Fourth: "Add all of them and see how long the pole is."
Next child: "One foot equals 1/3. Two thirds divided into 6 equals 3 times 2 equals 6. Six and 4 equals 10. Ten and 3 equals 13 feet."
After trying this experiment successfully in a few classrooms, he convinced his principals to draw up a course of study for all their schools based on his ideas. He describes this course clearly and it's detailed enough that it could serve as a scope and sequence for the home school, but it's a compromise position -- his principals insisted on introducing formal arithmetic in the sixth grade. He mentions the textbook they used and describes which sections they taught and which they skipped or put off till the next semester or year. He covers every grade from 1st to 8th.
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From our library I checked out a book called Mathematical Experiences for Young Children, by Louise Binder Scott and Jewell Garner. While I was reading it, I started a new page, Commonplaces on Math, which is linked in the sidebar, so I could have a place to keep the things I would have underlined in the book if I owned it. In fact, this book is almost wasted as a library book. It's full of activities for teaching various concepts, so it's really practical, but it's the kind of thing you need to have on hand, unless you're the kind of creative, crafty mom who just needs a few examples and then can come up with plenty of ideas on your own. I am not that mom.