Here’s round 1, I hope it helps. -R
1. Is the main difference in grades 2 and 3 that grade 2 is learning the concept and beginning mastery. The goal of grade 3 is to master the tables through more practice? Would the lessons in grade 3 consist mostly of working through problems? And lesson time during grades 2 & 3 consist of working on the rules and tables?
The difference would be that the values worked with become larger i.e. Add numbers within 1,000 in one year and add numbers within 10,000 the following year, master the multiplication table up to 5×10 one year and the next year master up to 10×10. The way the work was done is consistent throughout with proficiency shown before proceeding .
2. Page 19 does not have anything listed on the geometry row for grade 4. Should that be a continuation of outdoor geography or move into practical geometry?
Geography continues but along a different vein and practical geometry won’t start for another year. During Elementary Geography in Form I, students worked more in their own vicinity, making observations during outdoor outings, learned pacing, plan drawing along with learning the larger picture of the earth and sun, seasons and direction finding…all the things mentioned in the chapter on Geography that would later aid them in the study of geometry. After the first form, Geography went further afield to foreign countries, readers, travel books and books of a more literary nature were read along with map work, geography associated with current events (following explorer or army routes) was also studied. Diagramming, map drawing, etc. were still used but I believe outdoor geography became more of a study of geology and scouting skills. Does that make sense?
3. On the grade 4 S&S, it lists multiplication by a product and division of one quantity by another. I thought the answer of a multiplication problem was the product. And the definition of division is division of one quantity by another. Are these just another way of listing these concepts?
In the S&S, Long Multiplication (in this case, multiplying by more than 12) preceded Multiplication by a Product. Multiplication by a Product would be a shorter way of doing long compound multiplication. For instance, if you have to multiply a number (say by 35)which is a product of two or more numbers (5 and 7), you could multiply by one of the smaller numbers, and then multiply the result by the other. Students could work a problem both ways to prove which is the shorter method.
Division of One Quantity by Another – These problems require reducing given quantities to obtain the answer. For example, “2ft. ÷ 8 in.” or “$1 ÷ 5¢.”
4. Is there a place that lists the rules to help get multiplication and division sums right? And why would this not be listed until grade 4? To make sure they understand what to do and how to do it and not take a short cut?
Q.4 In the textbook used, the rules were given immediately (in this case, A New Junior Arithmetic, by H.Bompas Smith) but Charlotte by-passed these pages and did not present them until the end of study. One of my favorite parts during the research for the book was seeing CM’s philosophy in action while comparing a chosen textbook’s sequence with the PNEU’s sequence of study. Charlotte gave students the time and ability to obtain rules for themselves before introducing them formally.
For example, one rule might simply be, “After you have got the answer, ask yourself whether it is a likely one.” Another would be, “In multiplication you can check the answer by dividing it by either the multiplier (the number you have multiplied by) or the multiplicand (the number you have multiplied) and the answer to the division sum will be the other of the two numbers.”
An example of this with my own children. We used a popular math curriculum with my eldest and he was told of the “commutative property” right away. My second child, using a CM-approach, realized the idea on his own, i.e. that you can swap the numbers and still get the same answer such as 3+2 = 2+3. He’ll be formally introduced to the law later.