# Mathematics: An Instrument For Living Teaching

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• Crystal Wagner
Participant

I think this is the last post for awhile. What is meant by having the child make a table of be weights and measures. I understand having them weigh and measure different amounts and lengths, but what should they put in their table?

Richele Baburina
Participant

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.

MamaSnow
Participant

I finally read it too and found it helpful as well..thanks Richele for your hard work. I do want to jump in with a question of my own. One of the things that jumped out to me was the way that CM introduced practical use problems – such as money – right away from the very beginning, helping to connect math to the child’s own experience. This is one area where I kind of feel like our current program is lacking. I realize I could bring some of this stuff in on my own, or ditch a math program entirely, but I know myself well enough to know that I if I don’t have something written out for me I won’t be able to keep up with it. So ditching a formal math curriculum isn’t a option for me, at least not in this season. So I guess what I want to know is how do other people approach this…IS there a math program out there that introduces this practical connection from the beginning? Or do others of you have resources or activities that you use to supplement your current program in terms of practical application? I’d be intersted to hear about it. I’m trying to decide if we want to stick with what we’re doing, stick with it with a bit of supplementation, or change to something else entirely.

Thanks!

Crystal Wagner
Participant

Crystal Wagner
Participant

To follow up on a couple of items –

~ Should the grade 4 weights and measures tables that are listed be like 2 pints in 1 quart, 4 quarts in 1 gallon?  They figure this out through experience of course, not copying a table.  Or does this refer to something different?

~ In grade 1, the child explores numbers.  I understand exploring 1-20 and introducing place value with the bundles.  We will also work on the addition and subtraction tables with manipulatives.  Do I understand correctly that we will work on the addition tables all year.  If they have them memorized it’s great, if not they will continue that in grade 2.  I’m a little confused about numbers 30-100.  They are supposed to explore those numbers.  We talk about them in terms of the bundles/units.  I assume we would also work on money sums with those numbers, reducing to the smallest number of coins.  Addition isn’t formally introduced until grade 2, but was it worked on in simple terms with concrete examples?

Richele Baburina
Participant

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

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.

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