CM Multiplication and Division Question

I’m trying to go through Volume 1′s arithmetic section and make an outline of the steps recommended.  I can get through the addition and subtraction parts just fine (I think – see this post for my analysis), but I have a question about the multiplication and division parts from pages 256-257.

For addition and subtraction, there’s a three-step process for each line of the addition table, followed by the same three-step process for the same line of the subtraction table.  First work the whole line with counters, then with word problems, then with mental numbers.

For multiplication and division (page 257), there appears to be just a one-step process for each line of the multiplication table, followed by a one-step process for the same line of the division table.  It seems we’re just supposed to work out the line using counters and then go on to the next one.  But after working out both tables all the way through, with counters, then she recommends moving to complex word problems that involve both multiplication and division within one problem, without any mention of ever having done the simpler word problems in the course of working through the multiplication and division tables.

Do you think this is what she meant or did she mean us to do the three-step process here as well before moving on and just omitted mention of it?

Modifying Ray's Arithmetic

***Update:  I no longer use the Beechick guide.  Instead I use the original teacher guide published in the Eclectic Manual of Methods.  As a result, our early math lessons look much different from what I planned here.  And the Eclectic Manual meshes well with Charlotte Mason! ***

I have for years now planned to use Ray’s Arithmetic when my dd was ready for formal math.  That time is now, and I’m finding as I look closely at both Ray’s New Primary Arithmetic (this links to a copy of the actual text) and Charlotte Mason’s math recommendations (page 253 at the link) that the two are not exactly in sync.  I prefer to follow CM’s recommendations, but I’m hoping I can modify Ray’s to fit so that I don’t have to create the whole shebang from scratch.

I think we can follow this course for the first several lessons.  My lessons are numbered with Arabic numerals; Ray’s are numbered with Roman numerals.

Lesson 1 - Lesson XI – work out the table at the top with counters and drill over that, with counters

Lesson 2 – Lesson XI – drill on the word problems, orally

Lesson 3 – Lesson XI – drill on the word problems, orally, but phrase them as arithmetic problems (2+1 instead of using the word problem format)

Lessons 4-6 – Lesson XXV – repeat three steps above

Lessons 7-60 – Repeat this process for each of the next arithmetic lessons, alternating addition and subtraction lessons.  Optionally skip the last arithmetic lesson since it works with 10 and the implication from CM is that we would stop at 9.

Lesson 61 – Lesson XXXIX – work out the table at the top with counters and drill over that, with counters

Here I’m at a loss because I’m not sure if we should drill with word problems, as above, or continue straight to addition.  Any thoughts are welcome.

More Charlotte Mason Math, Volume 1 pp. 253-264

As I prepare to begin formal math instruction next week, I’m looking at my curriculum choice in more detail, and I’m finding that I need to look more closely at CM’s actual recommendations for the beginnings of math instruction.

"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.

Bird Songs

I just wanted to share a birdsong CD we purchased in preparation for starting Year 1.  We love it!  Instead of being simply a list of bird names followed by the song, this one groups the birds by similar sounds, then a narrator explains how to tell the birds in the group apart.  This is critical for us, because once the leaves appear on the trees (and we have leaves for the vast majority of the year) we see very few birds, although we can hear a great variety.  Once we go through this CD thoroughly, I think we’ll be ready for one that follows the list format, but for beginners like us, this seems perfect.


Birding by Ear: Eastern and Central North America (Peterson Field Guides(R))

Beating the Heat

Educating the Charlotte Mason way means spending lots of time outside.  I’m 5-1/2 months pregnant,
so I’m anticipating a hot summer.  My last baby, two years ago, was born in early September so we did the whole pregnant-all-summer thing then too.  Having a tiny baby in the summer is harder than being pregnant all summer, imo, because although I’m uncomfortable I’m not in danger in a reasonable amount of heat, but with a baby, particularly keeping it protected from mosquitoes, you do have to be careful.

We go out early, and then again late.  There is one local park that has some shade for the play equipment, so we can play there.  There’s another park that has a water feature the kids can run through, and I think the last two summers (since we’ve lived here) we spent time almost every week at that park.  There’s a local walking trail that feels like we’re in the woods, and last time around we spent many mornings walking there, slowly, the kids riding their scooters/bikes and looking for flora and fauna.  We sometimes went to one of the area state parks, where we could walk in the woods (so it was shaded) or swim in the lake.  Some swimming areas are not suited for little ones, but one local lake has a beach and doesn’t really ever get deep in the swimming area.  They looked for shells, watched vultures, saw a turtle on the road and rescued it, etc.

We have a big yard, with a swingset that’s shaded almost all day and parts of the yard that are shaded throughout the day, so that’s a big help.  But with that previous pregnancy, we were in a rental house and didn’t have that.  I do think that in the really, really hot parts of the summer (which haven’t arrived yet–yay!), we stayed inside except early mornings and evenings.  It’s similar to the way people in really cold places have to handle winter weather.  But with a little conditioning, you’ll find you can tolerate hotter temperatures than you would have expected.  I kept the house really cold all winter, and the kids were comfortable, but now that it’s hot outside (sort of) they’ve adjusted and tell me they’re freezing when it’s 78 in the house!

We live in east Texas.  When we started with CM, we lived in the Houston area.  So much of the year is extremely hot that if we don’t adjust, we’ll end up spending most of the year inside.  It probably won’t get cooler until November, and it’s going to get a lot hotter before then!

Charlotte Mason Math: Vol 1 Pgs 253-264

It seems like it might be worthwhile to try to summarize what CM values in a mathematical approach for the early years, meaning ages 6-9 or so. That will help anyone who wants to evaluate a curriculum to decide what to use when the school years approach, or who wants to create their own plan of instruction.  This is just a rough draft.

The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.

In the early years, she sees arithmetic as training the mind just as the work with language (hearing great stories, learning to read with accuracy and attention, etc.) trains the mind.

So this list she gives us helps us see at least some of the qualities we should be able to find in any math program we want to use.

She then talks about how important it is not to put the child in a situation where he does not know the right way to work the problem he is given. He must always know the right approach to take before he begins the problem.

Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.

Another point to watch for when looking at a curriculum.

The next point is to demonstrate everything demonstrable.

The child should be shown how each new concept works, using manipulatives whenever possible.

She then gives us an outline of the order in which concepts should be introduced:
*  addition and subtraction together
*  multiplication
*  division
*  word problems (done mentally and without counters–or with them at first then without them later)
*  notation, introduced first with money to get the idea of place value in a concrete way

Let him never work with figures the notation of which is beyond him, and when he comes to ‘carry’ in an addition or multiplication sum, let him not say he carries ‘two,’ or ‘three,’ but ‘two tens,’ or ‘three hundreds,’ as the case may be.

She discusses the importance of working with weights and measures, but doing it in the context of the real world–actually measuring and weighing real objects.

Therefore I incline to think that an elaborate system of staves, cubes, etc., instead of tens, hundreds, thousands, errs by embarrassing the child’s mind with too much teaching, and by making the illustration occupy a more prominent place than the thing illustrated.

Perhaps a warning against too much emphasis on props like cuisenaire rods and the like?

I do not think that any direct preparation for mathematics is desirable. The child, who has been allowed to think and not compelled to cram, hails the new study with delight when the due time for it arrives. The reason why mathematics are a great study is because there exists in the normal mind an affinity and capacity for this study; and too great an elaboration, whether of teaching or of preparation, has, I think, a tendency to take the edge off this manner of intellectual interest.

An interesting observation, and worth considering.

OK, this was my shot at outlining what CM wanted in basic arithmetic instruction in the early years (in this case meaning about ages 6 through 9).

All my Charlotte Mason math posts.

Why Wait for Year 1?

I think most of us at one time or another feel the itch to go ahead and jump into Year 1, since after all our little ones can already handle the reading and many if not most of the other activities.  Why wait?  And we get lots of outside pressure, in many cases, to be doing formal school rather than this "whatever-it-is" that we call Year 0. 

I don’t say this to imply that anyone is wrong for pressing on with Year 1 early, but just to let you know that there are good reasons for waiting.  You may decide that despite those reasons it is worth starting in, and that’s ok.

Year 0 is a lot more than just read-alouds.  (In fact, read-alouds are really a very small part of it. CM didn’t recommend spending huge amounts of time reading to the children in the early years anyway.  She wanted them to be up and around.)  There’s so much more to focus on, including habit training and nature study (and lots and lots of time spent outside just running around), and those things are *extremely important*, not just nice-to-haves or something to kill time before the kids are ready for "real school".  If you start doing Year 1, you may find that you are keeping the kids inside much of the time to "do school" and robbing them of the outside time they need.  There isn’t really a Year 0 booklist; those books are just some suggestions to get you started.  The Year 0 Yahoo group has some booklists, or you could use FIAR as many do.  But I’ve found that as we’ve emphasized outside time more (the goal is 4-6 hours *a day* remember!) we don’t need so many books to read because we’re not inside reading.

One of the benefits to waiting until age 6 (or thereabouts) to start Year 1 is having time to get the extras firmly entrenched in your family schedule before you have the pressure of school.  For instance, this is a great time to start doing art study, composer study, poetry, tea time, physical education, foreign language, hymns, folk music, etc.  If you get those subjects going smoothly, then you can add in the Year 1 subjects without so much stress.  Habit training makes life easier for everyone, and this is a great time to work on it intensively.  You can still teach many of the concepts you want to teach without making it school.  Math is easily covered through games, cooking, and other real life situations.  Pre-reading and even early reading lessons can be done very casually, but you could even do a formal reading program without having to start Year 1.

Starting Year 1 early may lead you to have to slow things down later, when the readings get much more intense.  That’s another factor to consider.  In a few years the readings take a big jump in difficulty and in the maturity of the themes, so you may find that you need to do one of those years for two years to give your kids time to mature before moving on. 

CM very strongly encouraged parents to delay formal education until approximately age 6, for a whole host of reasons.  Children need time to really play, to imagine, to run and jump and breathe fresh air, to explore the world around them without interference, to let their brains mature without a lot of strain.  She’s not alone in recommending this, and research supports her.  So I guess I am just suggesting that it might be worth looking at this time not as "wasted" but as valuable preparation for the later years, even if the preparation doesn’t look like school.