Montessori Sunday: Golden Beads (Part 3)

Ok, so I'm a day late but the Golden Beads are just so darn useful that I can't short them now. Trust me, they do justify having 3 separate posts for themselves.

After the child finally has a solid foundation and understanding of the beads themselves, and how they relate to on another it's time to begin the basic math operations. This same material will be used to teacher addition. It'll be used to show how subtraction works. The children will multiply, they'll divide--all with the same golden beads. Now that is a multipurpose activity.

I do want to point out that the golden bead material includes both the physical beads and their matching cards. The beads are present to give the children a concrete material to manipulate, while the cards are there to show the abstract symbols that society uses. Both parts are important to help prepare the children for further mathematics work at the Elementary level. While doing the operations with the golden beads we're not focusing on whether the children are getting the right answers or not. The process of how things are done is what matters in the children's house. Having the children physically experience adding the beads together to make a larger pile matters more than what 8,932 + 6,247 equals. They can see that putting two smaller quantities together makes a larger one, so they can really understand what the process of addition is.

The idea of subtraction is similar. The children get to be a part of actually taking away a certain number of beads from a larger quantity. They can see that this process results in a smaller number then they started with.

Multiplication usually puts a grin on their faces because they have a 'secret' when they go to collect their beads and cards. (Hint: The secret is that they're all getting the same number). The children who have done multiplication before think it's hilarious and the new ones are properly shocked when the secret is revealed. It's definitely a more fun way to show that multiplication is really just the adding of the same number a certain amount of times.

Finally the children learn the process of division, which usually ends up becoming their favorite way to use the golden beads. I think it's because everything ends up 'fair' and these older children are just beginning to build up a strong sense of right and wrong. The beads are handed out from largest to smallest, with each person getting the same amount and any extras 'remaining' with the teacher.

That's what a remainder means! I remember thinking during my Montessori training. Somehow I made it all the way through high school calculus and all of college without having a real understanding what was happening with division. And I could have learned it at 5 years old.....I kind of feel like I missed out...

Montessori Sunday: Golden Beads Part 2

Last week I introduced the Golden Bead material, one of the key Montessori materials for teaching math. This work allows young children to work with numbers in the thousands while only needing to be able to count up to 10. Pretty neat.

When Dr. Maria Montessori first came up with the idea of the golden beads she thought they would be used at the elementary level, that the younger students in the Children's House would not be interested or able to understand the idea of them. What she found though was that it was children around 4 or 5 years old who were truly interested in using these beads to convey the idea of numbers, not the older elementary children! It's pretty amazing what children can comprehend if given the right context to experience something with.

Once the child has become comfortable with identifying all the beads and cards by their names (1, 10, 100 & 1000), they can begin learning the change game. Learning this as adult even made an impression on me. I remember needing to carry over numbers in school, but didn't really have a good understanding of why. Learning to do so with the golden beads finally made it click exactly why and what was happening. Now I wish I would have just learned it this way in the first place!

The directress will have the child bring a large tray over to the supply of golden beads on the shelf and begin loading it up in a haphazard manner. Handfuls of 10's, fistfuls of units and piles of hundreds are added to the tray without much thought as to their order. At a mat the directress informs the child that they are going to count ALL these beads, usually sparking a shocked response since the messy pile looks unconquerable. To make it a little less intimidating the beads are sorted into categories.

"Let's start with the units. 1, 2, 3..." The directress shows how to count the beads one by one into a small cup. "10! Wait, 10 units? That's the same as 1 ten bar. Let's go back and exchange these." Together child and adult go and exchange the units for one of the bars of 10. And so the counting goes until there aren't enough unit beads to make into a ten bar. The ten bars are counted and exchanged in to hundreds, then the hundreds counted and made in to thousands. Finally there's only a small piles of beads left and the child is sent to fetch the matching cards to form the written number.

"That's our answer, we counted all the beads we had on this tray. Want to do it again?"

The act of simplifying this messy mountain down is a favorite with children. It seems so intimidating until they begin to actually do the process and find out how simple it is. It's also extremely important that they fully understand this concept because they will need to be able to exchange beads freely using the four basic math operations.

Montessori Sunday: Golden Beads (Part 1)

The Golden beads...I'm not sure if there is a more versatile material in the classroom. This lovely material introduces the child to the decimal system (including numbers up to the thousands), the idea of carrying over from one category to another (such as 10 tens turn in to a hundred), and also gives the child a concrete way to preform the four basic math operation (addition, subtraction, multiplication and division).

Wow, that's a lot to get from one material, so to make this a more complete post I'm splitting it into two parts. Needless to say learning this material is a long process; children may end up using the golden beads for both their second and third year of children's house. A classroom that I student taught in did a great job as building a solid foundation with the golden beads and it was very apparent that the children had a good understanding when they progressed to later materials.

The initial presentation to the child is simply naming each of the different categories (units, tens, hundreds and thousands). Ideally the beads for this material are made out of glass, so when the child is introduced to the thousand cube there is a very distinct impression of how heavy and large a number this is. It is also convey to the children that there are 10 units in one of the ten bars, and 10 ten bars in a hundred square, and then 10 hundred squares in a thousand cube. This idea will be reinforced later with a special activity with the beads.

 From there the children begin fetching certain quantities of the numbers. For example: "Bring me 5 10's.No that's not a typo, I said "5 10's." The child may not have mastered the formal names of numbers yet (such as twenty, fifteen, etc.) but they can count to 10, so why should we hold them back? Yet another reason why the English language makes education so much harder!

As the child get competent they are asked to get more and more numbers until finally they are fetching from all four categories (ex. Get me 3 thousands, 4 hundreds, 5 tens, and 6 units). How's your memory feeling? I know that I sometimes forget what number I've sent them for!

Around this same time the child is introduced to the decimal cards, which are introduced the exact same way as the beads. The children also learned to recognize the cards by fetching them, and eventually work up to getting four cards at once.

Finally the beads and cards are combined and the child's memory exercised as they are asked to get numbers such as 6,592 with both the cards and beads. This can take children weeks to months to master, but it's a vital step to make sure they understand and can remember these large numbers because the next step is doing math problems with them. 

Yep, we teach children how to add, subtract, multiply and divide using numbers in the thousands. None of this silly single or double digit stuff for Montessori children....ok, ok so it sounds like bragging, but is it really all that tricky when all you need to do is be able to count to 10 and remember the names of the categories?

Presentations, presentations and more presentations!

A presentation is what a Montessori directress calls the process of showing a new material to a child. Most children love getting presentations, even if they're not particularly excited about what the activity is they love the one on one time with the adult. I normally don't do a lot of new presentations in our classroom because we want to limit how many new things the child is introduced to at a time so they go back and practice what they have already been shown. So my fellow teacher and I have decided that it's best if she does the majority of the presenting and I focus more on showing the children extensions and reinforcing what they learned the first time. Today though it just worked out that I had a lot of opportunities to give new presentations.

I caught L right when she came in and said that magic words "I haven't worked with you in a while, what would you like to do?" The first thing she blurted out was the color wheel. I grinned and reminded her that she already knew that work so well that she didn't need me to help her. Instead I offered to show her the leaf cards and had her set out the mat. She laid out one set of the cards and I read off the names of the leaves before handing her the card to match.

Unfortunately I was interrupted a few times during the presentation. Ideally this should be a time where the teacher is completely focused on the one child, but with one of the other teachers busy greeting children at the door I had to step away from L to remind other children to stay in the classroom. Luckily L was patient and I did try to leave her with a card to match when I stepped away. Overall I think L enjoyed matching the cards and she was really quick even though there are a lot of different leaves to look through, and since they're real leaves they aren't truly identical.

My next presentation actually started before I even finished doing leaves with L. A came in and the first thing she asked me was if I was going to give her the presentation I had given her friend M the day before. I told her sure, surprised she had remembered that I wanted to show her subtraction and seemed pretty excited about it. So I had her start setting up the cards for the golden beads. That's one of the really nice things about working with the older children, they can really take direction, and set up even new works independently.

A and I flashed through the process of doing subtraction. She knew exactly how to count the beads, remembered the numbers she needed to get and eagerly followed the steps. In fact she had such a good time that as soon as we finished reciting the problem she asked if we could do it again! I apologized that I needed to work with some other children and suggested a few other friends who knew subtraction that she could ask to work with her. Unfortunately they all declined and she just cleaned it up instead.

Next I reviewed a counting work with one of the four year old boys, helping him recognize his numbers and double check his work. I got another boy started on the sound book he was supposed to be making, helping him find a table where he wouldn't get distracted. I also tried to engage one of the four year old girls in to doing a fetching work since she wasn't working, but rather being silly and distracting her friends. She didn't like that idea though and turned me down. Ouch. Though she was polite about it. Just asking though had desired affect of getting her to buckle down and choose a work to do.

As the day was drawing to a close I saw N running around the classroom, trying to play chase with some other children. I got him to stop running, asked if he had a work out and then told him to come with me and choose one of the chains. This was the first presentation I gave today that wasn't optional. Usually it should be optional if the child wants to come to the presentation, but occasionally, like in this situation, it's necessary to divert them to something more productive.

So N chose the short 7 chain and I introduced him to the arrows that went with it. He quickly caught on to the idea of counting and placing the arrows. Since it was so easy for him I also introduced him to writing the numbers down on special graph paper. Why hold him back when he's obviously ready for more challenging work? It took right up until the end of the work cycle, but N finished writing his number for the chain and I'm sure he's excited to bring that paper home.

Whew, a lot of new materials today. Now hopefully they all liked the work enough to try it again tomorrow.

In over their heads

Usually there's a lot of success stories over the course of the day, but coupled with those has to be some stumbling.

Story 1: Too much work

M and M decided that they wanted to do "Big" addition. Guess they were really inspired by the one we did in the afternoon last week. They spent a good twenty minutes trying to recruit another friend to work with them since doing addition with the golden beads works best with three people; two to add their numbers together and a third to be sort of in charge and help get the counting done. After failing to find a third they were still inspired enough to set it up for just two children with me giving them the numbers to collect. They gathered their cards and beads with just a little difficulty and then got stuck. This whole thing took about half an hour and when I noticed both of them just wandering around, occasionally returning to their mat I had to step in and ask what step they were on. I reminded them to set out the big set of cards and left them at it, hoping it would spur their memory of how to finish up the activity.

No such luck. By this point they were still a little fuzzy on what to do, add that to the fact that they were tired from working on this one project for about 45 minutes But they pulled it together, finished the work and without a second thought began cleaning up. I wish they could have done it completely on their own, but on the other hand I didn't want them to get frustrated or bored so much they didn't want to ever do it again. Oh well, better luck next time.

Story 2: Toy temptations

This story happened in the afternoon, when only the oldest children are in the classroom. E and A had the farm out, one of the most coveted and anticipated materials in the classroom. The farm is just what it sounds like, a play farm. It is used in the classroom to teach the parts of speech (noun, adjective, etc.) and isn't introduced until children can read fairly well. Of course all the younger children recognize the fun animals that go with it and want to play with them. Even the older children can get distracted from the reading by the temptation to play with the animals.

So when I heard E telling A about how 'super pig needs to be here to he can tell all the other pigs what to do,' I went over and reminded them they needed to finish setting up the farm animals so they could label them. They buckled down for a little bit, but not long after I noticed they were simply arranging the animals in a tight circle. I asked E to come and read some of his book to me, telling A she could arrange the animals while waiting for her turn to read. Usually things get less silly if there's only one child but still they couldn't get those animals arranged. They were just too tempting. I could tell that E and A both knew they weren't using the farm like they should have though, because as soon as I said they needed to put it away since they weren't using the labels they complied without a single complaint. Sigh, well at least they knew what they were supposed to be doing.

Story 3: You can't read with hands in your eyes

Reading is hard. And it's even harder when you're struggling with it and you really want to do it, but you can't, and all your friends have read the books you're still struggling through, but you still can't make out the words, and you just can't stand to look at the words on the page anymore and your eyes hurt from it.

Yeah, I think that sums up the frustration A was feeling this afternoon. She has such a hard time remembering what sounds the letters make that reading is nearly impossible for her. Sounding out 'it' or 'rag' is almost impossible when you can't remember what sound a 'g' or a 't' makes. In a Montessori classroom we really focus on helping children recognize the sounds of letters so they can read and write phonetically, then as they progress forward we introduce 'puzzle words' (things like 'the' 'two' and 'these' that can't be sounded out and just have to be memorized). Poor A has such a hard time recalling sounds that she treats ever word like a puzzle word, staring at it and reading it over and over again until she can recognize it on site.

It's a long and laborious process. I could tell how hard it was as she struggled to read the short book, constantly rubbing her hands over her eyes and looking away from the book. We spent nearly 20 minutes on  a book with maybe 25 phonetic words in it. Most of that was focusing on getting her to keep her hands away from her face so she could actually see the words, rather than just trying to guess what is on the page. A did agree with me that it's even harder to read when you can't see the letters and kept trying to stop herself rubbing her eyes. And we did finish. We're going to have to read that book again tomorrow, but that will be ok. All I can say is that I hope it gets easier, because I know she really wants to read like her big sisters.