Really Big (and Really Small) Numbers

A couple of months ago, I realized that fractions and decimals were coming up in the third grade textbooks. This would be the first introduction to decimals for most of my students and a far more in depth look at fractions on top of that. So before we got to that point, I would need to make sure they had a handle on place value.

How would Montessori do it? I asked myself. I somewhat regret my decision to drop out of Montessori training. No matter. I could handle it on my own.

So we all got out pieces of paper and some crayons, I went to the white board with my dry-erase crayons, and we accomplished this: (click on the images to view them full sized)

They caught onto the pattern very easily, as I was hoping they would. I started with Ones, Tens, Hundreds, Thousands, Ten Thousands, and Hundred Thousands, noting all of the colors aloud, some of them catching on before the others and verbally prompting me. Then I let them do most of the work. "What comes next?" I asked, and they all replied, "Millions!" "And what color is it?" "Green!" "How many zeroes does it have?" "Six!" We continued in this fashion, hesitating when we reached the next unit. "What comes after Hundred Millions?" I asked. "What comes after Hundred Billions?"

(There was a moment after I asked one of these questions when the class became silent, considering. One student blurted out a concept he vaguely knew as a very large number, "A Google! ... And after that comes a Google Chrome!" I lost it. The entire class burst out laughing. It's a wonderful, carefree memory that I cherish.)

We had to turn to the internet to help us after Hundred Trillions, because I honestly didn't know what came next, and we were able to authentically search for knowledge together.

I ran out of space on the board after Hundred Quintillion, but many of them were eager to go on. I let them go as far as they were able, though the color pattern became a bit skewed for some.

A few days later, we did the same going backwards, which was much more difficult. We didn't go very far, but having done the previous exercise was definitely a benefit.

From there we just followed the textbook, but they were definitely far better off.

Multiplication Tables

Third grade is the year when children are typically asked to master basic multiplication facts. I haven't let myself think too hard about this because I'm not sure what conclusion I would come up with. I know that by the end of the school year, my students will be expected to have reached mastery in this skill, and that's the end of it. No way around this one, really. Besides, some things just have to be memorized, right? Simple multiplication must be one of them. Yes, we've talked at length about what multiplication means, and my students are all aware that 4x5 means taking four groups of five items. But when it comes to working through problems, it's good to have the facts memorized so that one doesn't have to spend a great length of time finding the answer. At least that's what I tell myself.

I, for one, never finished memorizing my times tables as a child. When I multiply, I struggle when I get to points such as 7x8 or 6x9. And of course, I'd like to spare my students from that struggle.

So we studied our multiplication tables.

I began with the assumption that my students, who were blessed with a wonderful teacher last year, were already proficient with 2x (which was correct). So our proper start began with 3x. I stood at the white board and asked the students to count slowly by threes. (I amended this for all later sessions to, "Give me the multiples of [x].") I wrote them down as the students counted, stopping at 30. Then I circled the multiples into groups of three, leaving 30 by itself, to give more easily remembered chunks. We said them aloud with a little rhythm, "Three, six, nine. Twelve, fifteen, eighteen. Twenty one, twenty four, twenty seven. Thirty!" We did this little chant three times, then repeated the process every day of the week. On Friday, while they were working on another assignment, I called them individually to my desk to recite. If they stumbled, I asked them to go back to their seats and practice a little more, then called them back at the end of the rotation.

I didn't take grades on it. I didn't put their names on the board. I just kept track of it in a little notepad and gave only the reward of a small praise. Hopefully it kept the intrinsic motivation going.

The next week, once the multiples of four were on the board on Monday, the students commented on an easily follow-able pattern. On Tuesday, they kept going past forty. By Thursday, they recited multiples of four up to 100. On Friday, when I called one boy up, he kept going past forty. I remained quiet, merely listening. Other students came up to ask questions about an assignment, but I signaled for them to wait. Around 120, some groaned with boredom. Closer to 500, an excited crowd had gathered around him. By 800, nearly the entire class had gathered, and at 1000, the crowd burst into applause. The boy grinned and went sheepishly back to sit down.

It wasn't repeated, however, because whatever excitement they found in the game somehow became lost after the first initial run.

At any rate, we DID manage to make it through our 9x tables. They wondered if I would have them continue on through 10x, 11x, 12x,... where would we end? I didn't see the need to go on, though, because while it might have been an interesting venture, I wanted to spend more time making sure that they were all completely confident with 3x through 9x.

The next week, we started our timed tests. I found this useful website that generates multiplication problems, however many of them you want and with whatever maximum and minimum numbers you want. We started with 30 problems, minimum of 0, maximum of 9, then graduated to minimum of 3. I gave them one minute to complete as many problems as they could. Again, I took no grades, just kept track of progress in a notebook.

When they began asking for more time on the clock, I became a little concerned. After a week and a half of no progress, I decided that we needed to have a conversation about what was happening. They told me that the timer was putting too much stress on them and they could hardly focus on the problems for the want to finish on time. So we stopped the timed tests.

"But what should we do?" I asked them. "You all need to master these facts before you go to fourth grade so that you don't struggle."

We decided to give flash cards a try. So I guess we'll see how that goes in a few weeks.

On Autism and a Book Review: The Reason I Jump

I have a secret to admit.

I am a discriminatory person.

I don't do well with people with special needs.

One of my best friends is a special education teacher, and I don't know how she does it. The stories she tells me about her work... It takes a certain kind of wonderful person to be a special education teacher, and I definitely am not one of those people.

My husband and I have decided that if we find through in utero testing any disabilities in our future children, we will terminate the pregnancy. Having a special needs child is a stressful situation that would put a strain on our marriage we know we wouldn't be able to handle.

But autism is different. It is not detected in utero, and is rarely even detected until after one year of age. Believe me, I understand the hype. If I had a healthy, lovable, regularly developing baby for one year that suddenly seemed to regress and turn into something I hadn't known my child to be, as some anecdotes suggest, I'd look for a cause, too, and the vaccine I just recently gave him might look like a pretty easy excuse. But the science just isn't there.

There are plenty of sources I could choose to link to here. Let's go with a recent National Geographic article that's easy to read and sums up the situation pretty nicely, a CDC article saying that there's nothing to worry about, a CDC faq, and, for good measure, a Skeptical Raptor post that perhaps conservatives wouldn't accept as legit but links to many important studies and gives valid reasoning for why myths are incorrect.

So. The science isn't there. Let's move on, shall we? We know where autism doesn't come from, and we know that we don't yet know where it does come from.  Causality aside, what about dealing with already existing autism?

No matter how many times I tell it not to, my brain discriminates, specifically, against those that seem to have the incapacity to comprehend the world around them and social order as we, as a society, have come to create. People who can't seem to understand what is going on around them or make sense of anything. People who act without reason against societal norms.

Can they think logically? Do they even have thoughts? Are they even self-aware? "No," my brain tells me, "obviously they can't. And I can't handle people without intelligence." And for the most part, I left it at that. I put it out of my head as much as possible, because it's so difficult to think about.

But when Jon Stewart had David Mitchell on The Daily Show to speak about translating the book, I listened. I was struck by how sincerely Jon recommended the book. It seemed as though he did not have enough words to praise it as much as he wanted to. "The Reason I Jump is one of the most remarkable books I think I've ever read. It's truly moving, eye-opening, incredibly vivid... It is the most illuminating book I think I've ever read on the syndrome... I don't normally urge you," he told his audience. "The Reason I Jump is on the bookshelves now. Please. If you get a chance, please pick it up. It is remarkable." With that kind of authenticity, I could hardly say no. I swallowed my pride and went to the bookstore.

The Reason I Jump is a series of question and answers from then-13-year-old Naoki Higashida. The questioner asks things that anyone observing an autistic individual would want to know. For instance, "Why do you speak in that peculiar way?" "Why do you echo questions back at the asker?" "Why don't you make eye contact when you're talking?" "When you're on one of your highs, what's going through your mind?" "Why do you flap your fingers and hands in front of your face?"

And Naoki answers the questions, all of them, with sincerity and honesty. And it's abundantly clear that he is way more self-aware than I ever would have given him credit for. He can articulate most of the reasons why he does the things that he does, and in the few instances that he can't, he can even articulate that he doesn't have the exact reasoning, it's just pleasant to him in some way. Furthermore, he has the comprehension to acknowledge both what's going on around him and what other people must be feeling because of him.

Naoki is at no loss for intelligence, but he feels as though he is trapped within a body that he cannot control, one that doesn't follow the directions his brain gives.

And realizing this, truly, was a redefining moment for me.

The most remarkable question for me was 39, Why do you like being in the water? Here's the response Naoki gives:

We just want to go back. To the distant, distant past. To a primeval era, in fact, before human beings even existed. All people with autism feel the same about this one, I reckon. Aquatic life-forms came into existence and evolved, but why did they then have to emerge onto dry land, and turn into human beings who chose to lead lives ruled by time? These are real mysteries to me. 

In the water it's so quiet and I'm so free and happy there. Nobody hassles us in the water, and it's as if we've got all the time in the world. Whether we stay in one place or whether we're swimming about, when we're in the water we can really be at one with the pulse of time. Outside of the water there's always too much stimulation for our eyes and our ears, and it's impossible for us to guess how long one second is or how long an hour takes. 

People with autism have no freedom. The reason is that we are a different kind of human, born with primeval senses. We are outside the normal flow of time, we can't express ourselves, and our bodies are hurtling us through life. If only we could go back to that distant, distant, watery past--then we'd all be able to live as contentedly and as freely as you lot!

Deep, deep, powerful stuff, man. Naoki Higashida, this 13-year-old Japanese autistic boy who screams, doesn't speak or look anyone in the eye, who runs away from home and has a panic attack when he spills a drop of milk from the pitcher, is an intellectual. Is a creative author who has a wonderful sense of language and a powerful sense of empathy, particularly for those he loves. He's just trapped inside of a body that can't express it in a way the rest of us are familiar with.

It's still quite hard to think about, but Naoki mentioned frequently how just having patience was helpful. And that's all we can try to do with all children, I suppose. *sigh*

The Reason I Jump can be purchased from Amazon here. (I am not affiliated and am no way compensated from sales.) I couldn't find anything else from him translated, but if you want a screwy half-intelligible Google translation of his blog, it can be found here. He apparently has a new book that just came out in Japan.

A few months ago, my special education teacher friend posted a video of Carly Fleischmann, a teenage autistic girl that communicates with a computer. Her story is similar to Naoki's, but she's American, so if you want to learn more, she might be the better bet. She keeps a website, a Facebook, and a Twitter.

Project Time

In August, I found myself, finally, in a classroom of my own. There I was, two days before students, sitting alone in the room, faced with the task of development. In a public school, everything is thrust upon you--scope and sequence of curriculum already created and, in some cases, scripted. But not here. I had next to nothing to go on. All I had was a list of 15 names, some desks, a few shelves of reading books, a cupboard of junk left over from last year's teacher, and some text books I could use as curriculum. There was no foundation upon which to build--I had to develop it. It was perfect. Everything I could ever hope for. Mine to create anything I liked with. A block of freshly cut clay waiting to be molded by my hands.

My only hindrance was the knowledge that last year's teacher's contract hadn't been renewed because she let the children have too much freedom. 

Wow, that's... well, that's... interesting knowledge to have. Because I chose to work at a private school, I knew I wouldn't have the same troubles I had seen in the district with administrators micromanaging what I taught, but this left me with an entirely different problem--how was I, as a first year teacher, going to maintain enough classroom management to not follow in that teacher's footsteps and yet still create a classroom that allowed children to thrive?

The fearful first year teacher side of me won, I'm sorry to say, and I veered on the side of caution. I moved all the desks into traditional rows, three of five, all facing the white board. I started diagramming a curriculum map based on the textbooks provided. I created a short list of classroom rules, rather than having the students generate their own. I set everything up nice and neat like a good little school teacher. I prepped my stern voice. No one was going to tell me that I had no classroom management. I can learn from others' mistakes.

There was one thing I couldn't resist putting a tiny twist on. When I wrote up my schedule (which I was amazed to find that I could actually do. That's when I knew I was in a private school), I left a period towards the end of the day open for "Project Work." I think I may have done it subconsciously. I definitely wasn't sure what it meant at the time. I just wrote it down without understanding why.

The first few weeks of school was chaotic, anyway, what with both my students' acclimation to change as well as my own. We didn't follow the schedule until nearly September. But all the while, the kids kept asking, "When are we going to do a project?" The way they said it made it sound like an important work. Apparently their second grade teacher had done a lot of hands on projects and science experiments with them, and they were craving that type of work again. But I was caught up in the stress of a new job at that time and couldn't think that far ahead. All I could tell them was, "Soon."

When my mentor came in to observe me, she noted that I didn't do any sort of intervention. "Yeah, that's not really my style," I hinted as politely as possible. I hadn't seen any beneficial results from my experience with intervention in the district. I knew that maybe I just hadn't seen the right methods, but again, I was still trying to figure out how to do the basic functions of my job and couldn't handle that much on my plate yet. Still, there were those students that I had been meaning to work one-on-one with. To pull some students aside, though, I'd need an extended period of time in which the rest of the class could be working independently. Then somehow it just clicked. If I gave the class some time to work on anything they were curious or passionate about, I could pull those students that needed extra help. The Montessorian in me grew giddy. I couldn't wait to begin.

I don't really do much with my intervention time now that I have it, to be fair. Even though I made a big show about how it's a privilege to come to intervention because you get the chance to practice something you're struggling with and learn more, the students are still resistant to coming. And I hate to force them. They're not receptive learners when they're frustrated or upset. They just dislike missing project work, so for the most part, I leave them to it. Sometimes I pull students one at a time to check for comprehension or fluency or the like, or sometimes I pull students that I know didn't understand a lesson, but most of them get a minimally-interrupted 40 minutes of project time a day.

My only rule for Project Time is that all projects must be written down in the form of a proposal and approved by me before they can be begun. There are expectations, like "You must work quietly enough that the students I'm working with can focus," but they weren't really stated in the form of a rule. (Perhaps they should have been. Occasionally when that expectation isn't met, everywhere except for my group becomes a No Talking Zone, in which all communication must be carried out non-verbally. We switch back to whispering after a few minutes, but I still have to go through with the process every other day or so. They get so excited for project time. But that's to be expected, I suppose. We just have to have rules because classmates are distract-able. But I digress.) No rules about what types of projects are available, grades for projects, or even that one must do a project were ever stated. We spent a day or so working on proposal writing, and then I just set them free.

And they've completed some pretty cool projects, only two months in. I'll take this opportunity to throw in some pictures. All thumbnails open into full-sized images.

First, an example of a proposal. Most don't go into as much detail as this one, unfortunately. These girls went all out with the proposal, spent nearly two months working, and just recently gave a presentation on their accomplishments.

Working hard. The girls in front are working on the solar system project. The ones in the back are working on a wildlife diorama. The boy in the back is, of his own initiative, grabbing some wipes to clean up a mess he made.

Unfortunately, the only computer I have available during project time is the one the school provided for me. I can't send students to work in the computer lab unsupervised, and I can't ask everyone to come to the computer lab every time someone needs to use the computer, so I just let the students use mine. This student is typing a story he wrote to publish on the class website.

Finished wildlife diorama featuring plastic figurines and easter grass. They decided against gluing anything down so that they could scrap the cardboard when they were done and use the figurines for something else.

An exploded baking soda and vinegar volcano. One of the many. This one features a clay model that took three days to create and lava rocks recycled from another presentation.

Elephant's toothpaste (hydrogen peroxide and yeast). This student painstakingly copied down the instructions from a YouTube video in order to do this. Word for word. It took three days of writing and a lot of patient listening.

Finished solar system diorama. They had trouble connecting the styrofoam balls to the yarn. It was difficult first to make a hole in the ball (which they did using scissors), and then to get the string successfully through the hole. The latter took a lot of trial and error, but they finally realized that they could tie the string around a pencil, push the pencil through, and grab the string on the other side.

Borax-and-glue slime, illuminated with glowsticks. A big hit for all of the boys. He even shared some of the final product with anyone that wanted to take some home.

And now for some observations.

-First, this group of students seems to gravitate towards science experiments. Since they had experience with it last year, it's safe ground. We had a long run of volcanoes and baking-soda-vinegar-based projects, probably five or six presentations. Currently we're having a run of sensory projects--slime and silly putty.

-The students often inspire each other. The volcano thing started with a project from one boy at the beginning of September, and after that, the rest of the boys wanted to do their own. I'd like to encourage them to make minor changes to projects when they do similar projects and record the differences, but for now, I know they just like having the experience of doing the work for themselves.

-I do have a few students that can rarely ever "think of anything to do." At first, I told them to sit and plan something, but after a while, the started getting up, wandering around the room, watching, and occasionally helping others. Now I have a few students that regularly spend all project time assisting others.

-And I do have some students that occasionally prefer to read at this time or use it to do their homework, like a study hall.

-When doing a project, though, especially an involved one, the students rarely like to work on their own. I specifically didn't set a limit on group size, so occasionally on longer projects, there are normally three to five students working together, though participants can come and go. If participation in a project drops to one sole member, it is usually abandoned.

-Sometimes students have several simultaneous projects occurring at once, spreading across multiple groups. The one they choose to work on each day varies.

-Occasionally if I assign a longer assignment during the day, or one that the students get deep into, they will choose to continue the assignment during project time.

-And watching the natural leadership roles play out is certainly very interesting.

A couple weeks ago, this Edutopia article, titled Just Ask: Harnessing the Power of Student Curiosity, came across my radar. Turns out that my Project Time is already a thing. Employees of Google are given the opportunity to use 20% of their time working on personal creative ideas. This has led to many innovative programs, such as Gmail and Google News. The initiative has spread to education, becoming known as Genius Hour or 20% Time, and some books have been written on the subject, including Passion-Driven Classroom by Angela Maiers and Amy Sandvold, which is now on my To Read List.

This is all well and good, and I'm happy for these people, but honestly, I'm already looking towards my next big venture: a Montessori-inspired modern classroom. I think Project Time was necessary only as a transitional stage until I can get the ball rolling on my Menu-based Modern Montessori-inspired classroom.