Mathematics: An Instrument For Living Teaching

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.

Well, I started on the living math journey a year and a half ago. We ditched the math program. We were using Right Start. It is a good program and my dd was doing well with it. But I didn’t like having to follow their plan (we are pretty structured and follow a plan, but it’s rarely someone else’s). I had read the Benezet articles (http://www.inference.phy.cam.ac.uk/sanjoy/benezet/) and decided (from the articles and personal observation of my dd) that the program (and others out there) covered more than we needed. I saw that some of the concepts covered in first grade (congruent shapes for example) were ones that were not needed at that age. Similar to grammar in the Charlotte Mason method, we would be covering the same topic multiple times in multiple years without real life practice or enough practice in the curriculum to remember what we learned. But if we wait until later, we can cover it once when she is ready for it and she will learn it faster and remember it. So while she was learning the material it was only the four rules and tables that she was remembering long term – because we worked on it at other times too. And Life of Fred came out with the Elementary Series. I don’t think the Elementary Series is stand alone because there is not enough practice. But it is good for showing how you use math in daily life and in helping with logical thinking. We read Fred, other living math books, work on word problems and problems on paper, and play math games. (I will be posting a blog post soon about what we use for math. We are on a trip right now, but once I get it done I’ll add it to this thread for you.) The book from SCM has confirmed for me that all we need in the first three grades is working on the basics and working on mental math skills and add measuring and weighing in the fourth grade. In fifth grade I plan to begin fractions and decimals with Life of Fred. I’ve heard from both sides of the fence that it is stand alone and some say it isn’t. So it probably depends on the child. Regarding money, we have found it to be a great way to teach the concepts. I’m not sure if this helps or not, but I think it boils down to finding what works for your family/child and sticking with it. Regardless of what you use, you can make it more CM friendly with the suggestions Richele gave in the book.

Yes, it was all worked out basically the same way.  Maybe some answers to your first questions will help:

1. Addition and subtraction tables

Right, the first year the children are beginning to feel at home with numbers and “unraveling the[ir] mysteries.” This is their beginning acquaintance with the science of number, which includes the ideas of symbols representing quantities, an order of things and the idea of magnitude.   The formal memorization of tables takes place in Year 2 -though your child will most probably know his addition and subtraction facts by the end of Year 1 through all the work he or she has accomplished.

1. Rapid mental math

This was the wording from the schedules and programs. The tables were worked out using rapid mental math as well, primarily using money sums for five minutes each day at the end of the lesson.

Construction of tables:

First the child is introduced to the idea of times or multiplication as an extension of addition through the concrete (ie three rows of four beans, five rows of two beans, etc) then the symbol “x” is introduced.  After this they work some simple problems to reinforce this idea and then write them down using the symbol “x.”  Tables are then introduced.

Charlotte’s students constructed their tables up to 12 due to the pre-decimalized currency system. Today our children construct up to 10.

Working with one number at a time (in order), the child constructs a table of ten “rows” of the number being worked using beans, beads or pennies and exercised upon in order. You can use money problems as well as pure number. The child will prove the facts using concrete objects. Now comes the time to write out a multiplication table and then commit it to memory, using the steps outlined on pp. 34-36. A multiplication table would be pointless unless repeated groups of objects added together so reliably and consistently. Using CM’s methods, the child sees the rationale of the table, that we have a reliable God-given tool that will save us time and the properties of multiplication will point to that beauty and truth found in mathematics, that there are fixed laws and properties designed and upheld by God that “exist without our concurrence.” 

In reply to the second part of this question, each table is the three lines as shown on p. 35. In our home, I have my child construct a written table in his squared multiplication notebook so we can date its construction and subsequent memorization for our records. I don’t see evidence for or against filling out an entire multiplication table in Charlotte’s writings.  As my son masters each table, he adds to large one as well.