Today I found out that Sam has been paying attention. I still have no idea what she understands, but at least I now know that her mind is engaged and she is taking things in in some form.

Sam finished math with the problem you see at the top of the page in the picture below. She started doodling so I told her to let me know when she was done. As I did other things at my desk, I heard her making up a story, while drawing the pictures and words to go with it. She did two of them. Her stories were like a summary of the past nine months of school. She included math, astronomy, physics, literature, history, grammar, penmanship, spelling, and wow, just everything. It’s like all of her knowledge came pouring out of her in this form, and it was completely unplanned. It was definitely a stream of consciousness thing – not any kind of true integration – but I was impressed that she actually had all of these thing in her consciousness at all.

Here is the best I can do at summarizing her stories. It’s hard to see the drawings, I know.

Once upon a time, there was a Mr. Sun Robot and a Mrs. Moon Robot. Mr. Sun Robot made it light, and Mrs. Moon Robot made it dark. [you can see the robots labeled with their names on the left, but the interesting part is that she started the drawings first by drawing the squares and rectangles on the graph paper, which is something straight from math lessons. The shapes must have made her think of robots. I don't know why she thought of the sun and moon, but you can see that she has drawn those objects as the robots' heads.]

…Mr. Sun Robot would always make it light and Mrs. Moon Robot would always make it dark, so they were always fighting. [This is directly inspired by all of the myths we've been reading...she is beginning to get a sense of the structure of these stories!] And there were a lot of people on the earth [see the people she drew all in a row at the bottom right?], and some of them liked it dark and some liked it light, so they were always fighting. [There was much more about all of this, but most of it was repetitive. There was no real resolution to the fighting, though.]

And there was the earth [see drawing above the row of people...the sun and moon made her think about the earth], and there were many people on it [the little circles along the edge of the big circle are the people] and the light shined down on them, straight down, down down down on the earth [she is recalling her lessons about gravity now...a little bit misapplied, but we've talked about the earth's gravity a lot lately so I know that is what she was thinking about] and the people liked it, but then it was cloudy [see all the clouds around the earth?] but that’s okay, the people still liked it–it was still light. And the earth went around the sun and the sun went around the moon [I wasn't about to correct her at this moment!] And the sun shined on the earth [see the ball on the curved stick...that is the sun. And there was much more about all of this that I can't recall.]

The end. [written above]

[Name and date at the bottom.]

I managed to keep my mouth shut the whole time, except when she asked me to remind her how to write Mr. and Mrs. (We’ve been working on abbreviations!) When she was done, I told her that she had just made up her own myth. I told her that when she could write words faster, she could write down all the words of the story (she said, “just like YOU do!”), along with the pictures, and that way she would never ever forget it. I told her I would love to read her stories, but right now, I just loved to hear them. She said she wanted to do another.

Once upon a time, there was the sun and the moon. The sun was lite [sic] and the moon was dark. And the moon went all the way up to the sun [see the arrow?] and it got BURNED [one of our upcoming science lessons involves burning things. Also, the fact that she knows the sun gives light and heat is not something I could have guaranteed Sam knew before today - we've talked about it and read about it a million times, but I bet if I had asked if the sun was hot or cold, she would have said, "cold" just to piss me off or keep me guessing, or whatever her unfathomable reason is for never allowing anyone to see into her mind].

[Skip down to the pictures under the word "crash".]  And there was a road, and there were people going this way [she drew lines showing motion from left to right at the bottom of the road] and there were people going the other way [she drew lines showing motion from right to left at the top of the road... she even got the sides we drive on correct!] and they were going FAST and they went CRASH into each other [she drew the vertical line in the road to represent where the cars met, then she wrote "crash" which she sounded out phonetically in a very deliberate manner.]

[Then she basically repeated this story and drew another road.]

And then there was a circus [the big rectangle near the top middle, which remained empty for this part of the story]. See, it’s a rectangle. And there were animals in the circus [she did not draw the animals or any of this part]. They were wild animals, cats and dogs and wolves, and they were big animals. But the people kept them in their homes and made them their pets [Egyptian history lessons...domesticating cats] so it was okay. [This part went  on for a while.]

And there were houses. [The three squares below the circus]. They were connected together like this [drawing]. [My only interjection happened now, I said "like townhouses?"]  Yes, townhouses. And there were windows for the people to see out, and stairs [and more and more about houses]. And the people went to the circus, and there were three tents [the three black spots at the bottom of the circus], and you see, the people went from here to there [the lines with arrows connecting the houses to the tents...she is drawing a lot of lines connecting things in her workbooks right now...I'm making her practice workbooks so she can take standardized tests and I know the lines come straight from that skill because she has never drawn lines connecting things like that before]. And the people stayed in the tents!

And then the person found…[doodling]…the person found [draws a circle]…the person found…a dollar [draws a dollar sign inside the circle; we've been working on money lately]. And he put it in his pocket. And then he found another dollar and another and another. And he put them all in his pocket [good girl!].

And then there was a car [left, middle]. And he put all the dollars in the trunk of the car [the thing that looks like a flame on the left side of the car was the trunk and the lines were the dollars he put in]. And the car looked like a rocket ship! [Yes, it did end up looking like a rocket, didn't it? I don't remember what the circles with crosses in them were.]

And then he found…[back to the circus...the big circle with dollar sign] a really, really, really big dollar! And it was so heavy, he could not pick it up. He needed more force to move it [yes, she said that, and it is straight from our science lessons]. He was thinking and he decided to build a box on top of the dollar [see the square on top of the big dollar?]. But that didn’t work.

So, he invented [yes, she used that word] a giant to pick up the dollar. [See the smiling figure with a dollar at the bottom left?] The giant was nice. He was a good giant.

The end.

I have never seen Sammy do anything quite like this before. It was just beautiful. And I’m so proud that I was able to mostly keep my mouth shut. The whole process must have taken about a half hour. (There was much more talking that I could possibly write here.) When you have a child like Sam, who rebels against giving any feedback at all about what she understands, this kind of thing is a miracle.

Here is the built Decanomial Square:

Since she didn’t do the greatest job making things neat and square, I was a little lost about how I’d talk about squares. And she was getting kind of tired of doing problems and counting the beads. I ended up pulling out a sheet of graph paper and drawing squares and rectangles to remind her of the difference between them. Again, I couldn’t really tell if she was getting anything out of it. I let her color in the shapes. I counted the length of the sides. Nothing seemed to be getting through. Then, I tried counting the graph paper boxes within the shapes (which we called “units”), just like we’d count the beads from the Decanomial Square to get the area of each shape (although I didn’t introduce the word “area”). I wrote the numbers in the boxes, left to right, top to bottom (consistent with how we were working the Decanomial Square). Then we counted the appropriate beads and got the same answer. Then she did the multiplication problem on her calculator and got the same answer. Finally, I got my “aha” moment. Somehow, that made it all clear to her and Sammy was thrilled! Again, it took doing two exercises at the same time, one just a level more abstract than the other, to get the concept through.

Well, we had gone back to First Language Lessons after all my complaining about memorization. It turns out that when Sam parroted back things to me and it seemed like memorization, it wasn’t so much that she had no understanding, but that she was rebelling against the question/answer method. It’s true that she doesn’t have a great understanding of common and proper nouns yet, but she does have some grasp of it. In later lessons, she showed me so. I’ve found that there are times when she just won’t answer an oral question from me, even when she knows the answer. I have not yet figured out why. Sometimes it seems like she won’t answer when it is too easy, sometimes when it is too hard. But I have a suspicion she just doesn’t like it when she is made to feel like I am the authority and I am testing her. I try to avoid that atmosphere, but in any Q&A method, it’s going to feel a bit that way.

I know she knows more than she lets on because sometimes I can trick her into revealing what she knows by making a game out of it or being silly. That doesn’t work every time because she sees through it, but occasionally I’ll get amazing answers out of her. Once I asked her to summarize a short story and her first response was, “Kengeng went to Fengbang and they went doodlie-doodlie-doo,” and then she started flopping around on the couch like a fish and giggling like a maniac. Then I did something or other that removed the tension, and she stated the essential from the story like it was nothing. There is much more in her head than she is willing to reveal.

Anyway, that’s why we are back to FLL. We’ve moved on through pronouns (which Sam loved) and are on to verbs. She likes to identify these words in all of her reading. There are more and more writing opportunities in the FLL lessons. We’ve been using those for printing.

But all of that is nothing compared to the most exciting development – Sam is beginning to write her own thoughts! She has always loved making up stories and role-playing. She does it orally for hours with her dolls and figurines, but her printing has never been good enough to keep up with her thoughts. Now, she can write a few sentences such as, “Sammy wint [sic] to the playground. Hayden wint [sic] to the playground. Hayden played on the swings.” I know it sounds like nothing, but it is a huge accomplishment for her to put together her thinking, printing, spelling, grammar skills to say something on paper. That is the purpose of writing, after all, right? I suspect that I won’t have to struggle  to find content to use for writing skills anymore. Sam is finally on the path towards writing in all of her subjects. (We’ll probably start cursive soon, before her thinking-writing connection becomes wedded only to printing.)

Right along with this achievement has come another explosion in Sam’s conceptual development. She understands all the science we’re doing and she is enjoying the story of history also. She isn’t very interested in math lately, but when she works on something, her understanding is much better. In fact, I think that she is less interested in math because she has gotten bored with the easy stuff we are doing. Sometimes she leaps forward so quickly, it’s hard to keep up.

I now consider Sammy to be in first-grade. Not that grade-levels matter at all in our homeschool. But I think of preschool and kindergarten as preparation for a conceptual education, and that’s mostly what we’ve been doing up until now. This writing is what marks the transition!

Just stunning. I know she can interpret music, but I didn’t know she could create the right music to suit a situation or an emotion. And she did it without any real thought or planning. She just sat down and went into the zone and played it. It didn’t even bother her that her little brother was pecking away at the high end of the keyboard at the same time. She was into her own little world. I told her that there are jobs for people to do this work. She could be a writer of movie soundtracks. Wish I could have captured this on video – it was a thrill.

Sam came out of Montessori without understanding any of the operations – not even that addition means putting together. I freaked out a little bit, but I’ve been able to observe her for many months now, and there is no real problem with her or with the Montessori method. Sam is on the slow side for math, for sure. And I’m peeved at her teacher for not being able to observe what I have about her lack of understanding. But we’ve been plugging away at all the Montessori math work, and Sam is beginning to get it. If she had continued for another year in Montessori Primary as her teacher recommended, she might have gotten up to speed. But maybe not, because I found a better way to get through to her: combining and integrating the math exercises.

First of all, here are the things that she has done for the past five months:

Exchange game with a die: we play a game together where we take turns rolling a die – 1 through 6. Using single “unit” beads, you take the amount shown on the die on put it in your pile. When you reach ten, you trade in, or exchange, ten unit beads for a “ten bar.” Whomever gets to 100 first wins. (And there is a “one hundred square”.)

Exchange game with a deck of cards – aces through tens, four each. Same thing as above but with cards, and the person with the most at the end of the deck wins, instead of playing to 100.

Exchange game on paper – Same as above but write down the number on paper and sum it up each turn.

War – the standard card game. The concept to learn here is “higher or lower” which Sam was terrible at, coming out of Montessori.

The Bring Me game – I write down a four digit number and Sam has to bring it to me on a tray in the form of beads. (Unit beads, ten-bars, hundred-squares, and thousand-cubes.)

The Bring Me game in reverse – Sam writes the numbers and I bring them to her.

The Stamp Game – we use an iPad app for this. A stamp is just a rectangle with a number on it: 1, 10, 100, or 1000. There are boxes full of each at the top of four columns. Then you get an addition or subtraction problem. Say 1,111 plus 2,222. You start with units. You bring down one “1″ stamp, and then you bring down two “1″ stamps. You do the same with tens, hundreds, thousands. You “exchange” for higher quantities when you reach ten of anything. You have an answer and write it down.

Coin counting – pour out a random pile of coins. Group them by type. Count each pile. How much is each worth? Put the beads for each value next to the pile of each type: unit bead, five-bar, ten-bar, and I have beads tied together to represent a quarter. Mommy writes it down, but multiply to get the total value like this:

• pennies: 6 X 1 = 6
• nickels: 4 X 5 = 20
• dimes: 14 X 10 = 140
• quarters: 3 X 25 = 75

Mommy helps as necessary with the multiplication (many ways to help without telling or using a calculator and that is part of the value here). Then put the coins in a coin-counting machine to check your work. Fun!

Multiplication bead board – times tables using beads in columns and rows to count up the totals.

Girl Scout cookies: use the beads to understand what “four dollars per box” means and how to do the math.

Finger charts for addition and subtraction – these are just charts where you use your fingers to find an answer to a simple math problem. There are control charts to check your work. This is the beginning of memorization, not a tool for conceptual understanding, except that you can see patterns on the chart and that helps.

A skip-counting iPad game – not very useful for us.

Adding and subtracting on our fingers – we do this every day, all the time.

So a lot of that is pure Montessori. But Sam can do any of this by rote, and she gets it down so quickly. I thought that changing things up would help engage her mind more. That was how I decided to do the exchange game with cards after doing it with the die. I felt like she was in a rut with the die and getting to 100. So we used the cards, and she had to integrate that we were doing the same thing. And playing until we ran out of cards forced us to count up and compare our totals at the end (although we would do that most every turn anyway because Sam really loves to WIN right now.) Also, just the idea of turning up a ten card and getting to pull a ten-bar directly – that was different. At first, she couldn’t understand because she was used to taking units – just one through six.

Once I saw the success of this, I decided to do other things to change things up. Instead of Bring Me with Sam always bringing the beads, I had her make up and write the numbers for me to bring. I could see in her responses that this added a new level of understanding to quantities for her. It’s hard to just think of a random four-digit number. I had to help her realize that all she had to do was think of a number between 0-9 for her units, then do the same for each column. (We used four-column paper with headings of “units” “tens” etc. for this). Then, she still has trouble with the idea of not writing down a zero on the higher orders…so if her number is 456, she will want to write “0456.” She also had a lot of trouble reading the numbers. This is something I don’t see Montessori addressing: how we say the numbers is different. “Fifty four” is a weird thing to say rather than “four units and five tens.” What’s the best way to connect how we say the numbers with the decimal understanding of them? I don’t know, but this reverse game helped.

The most awesome thing I did was to make a two-person game out of the Stamp Game and the beads. We use the iPad app to generate some problems. Then one of us finds the answer with the Stamp Game while the other finds the answer with the beads. When we both come up with the same answer, Sam is thrilled. The Stamp Game is just one level more abstract than the beads, and one level less abstract than doing written problems with standard carrying. Doing them together links them.

Then I realized that this was how I would introduce carrying on paper. We played the Exchange Game. She used the beads and I used the beads and paper, summing my running total each turn. Whenever I had to carry, she would see me exchange ten unit beads for a ten bar, and she would see me write a tiny “1″ on top of my tens column. She got it immediately.

One day, we went crazy with the Girl Scout cookie game. One of us would sell the other a random number of boxes. Then we’d get out the “four” bead bars, since each box costs four dollars. We added all the fours. Then we skip-counted by four. Then we multiplied by four. We had huge piles of beads on the floor and did the problems every way we could think of. I could practically see the light bulb go off over Sammy’s head that day.

The moral of the story is that all the Montessori materials are great, but Sam needs more integration between them. What she needs now is more word problems like the cookies. I don’t have a source for those, except the random things that come up during the day. But usually we aren’t around our beads when those things come up. Finding a source for word problems, along with more memorization work, is the next order of business.