Sunday, April 27, 2008

The Stamp Game - Addition



The stamp game is a wonderful tool for learning and reinforcing knowledge of the four operations: addition, subtraction, multiplication and division. This post will discuss and describe simple addition work with the stamp game.

I was lucky enough to attend the International Montessori Conference in Paris in 2001. It was a wonderful experience. It would take a post dedicated to the conference itself to articulate the impact that it had on me and my future as a Montessori teacher.

Many of the Montessori materials from the first Casa were on display. The originial color tablets were included. They were in fact made from embroidery floss (individual colors) wrapped on small cards that you can actually still buy at JoAnn Fabrics. The stamp game, I am sure you can now guess, was created from duplicates of postage stamps with values that represented ones, tens, hundreds and thousands. I saw some of these materials again in the ongoing Montessori Centennial museum show in San Francisco a year ago. If you ever get a chance to see any of these materials jump at it. For me, they connect me to the source; the ideas and materials behind their creation.


The stamp game has stamp sized flat, wooden squares that come in three colors: green, red and blue. The question is sometimes asked - "Why isn't there a fourth color for thousands?" I will tell you the answer here. One, ten and hundred are in the simple family. One thousand, ten thousand and hundred thousand are in the thousand family. One thousand is represented by green as it is the ones of the thousand family. The other very important point to remember is that children in the Montessori classroom learn math via the decimal system. They are counting up to ten. Quantities more than ten are converted into tens, hundreds or thousands. I will explain more later. So they learn math via the decimal system plus language. The success a child has with the math materials is dependent on their knowledge and memory of the terms ones (units), hundreds, thousands, ten thousands, hundred thousands - etc.



This is an addition problem placed by a child on the table. The green stamps in the foreground are ones (units). In the background the green stamps are the thousands. This problem reads as follows: 3,245 + 2,312 (*Note: the comma written in the quantity - ex. 2,312 is denoting the separation between the two families: simple and thousands). The child has laid out the first quantity and then placed a ruler under that first amount. Beneath the ruler she has placed the second quantity.



Now the child has removed the ruler leaving the two quantities, still separate, on the table.





The child now moves the second quantity up to join the first. Addition is two smaller quantities joining to make one larger quantity. The operation addition is based on gathering behavior - getting more - as is multiplication. Division is an equal distribution operation.


The child counts the number of stamps starting at the last stamp and moving up the vertical row towards the box. They then note the answer on their sheet. I have laminated cards with problems written on them that children independently copy onto stamp equation paper. These papers are color coded. The spaces for ones are green, tens blue, hundreds red and thousands green. Prior to the child independently copying static or dynamic math equations, I write them so as to ensure that the child 1. sees where to place the numbers for each quantity 2. that the answer does not either include a zero or requires carrying over - yet.


Above: Stamp Game paper. These fifteen boxes are cut into individual ones and kept on the shelf next to the Stamp Game itself.

Carrying over is very easy with the stamp game material if the child has worked with the golden bead materials, the teens and ten boards. Say the child is doing this problem - 2, 467 + 1,325. They place the first number on the table, then the ruler, then the second number. They remove the ruler and join the two quantities. Counting up towards the box, the child counts to ten (as there will be twelve one (unit) stamps). Once the child has counted ten ones, they slide them away from the rest of the ones and turn them into the "bank" - which is the box with the remaining stamps in it. They turn ten ones (units) in for one ten stamp. They then place this additional ten stamp below the vertical row of the already placed tens. They count the remaining ones and note the quantity on their problem paper. They continue counting the remaining quantities and noting their results until they have completed the problem. Note: if the child hesitates when they see more than ten ones (or other quantities) remind them that "ten ones equals one ten." Once in a while you will see a child write 12 in the single box denoted for ones on the problem sheet. Again, remind them to write only one number in each box and that they need to go to the bank.

There are skittles included in the stamp game. They are used for division work and I will discuss and explain this in another post.

Thursday, April 24, 2008

Practical Life Reviews - How to Sew a Pillow and Sock Rolling

How to Sew a Pillow



Above: Sewing a pillow work on the shelf with the basket of fluff - stuffing - to the right of it.

Materials on the tray include two identical sized pieces of fabric (large enough to make a small pillow). A small basket holding spools of thread. A small wooden box with lid that holds a needle. Two small clothespins. A pincushion - I made the one shown here. A pair of children's scissors.

Above: The child uses the two small clothespins to hold the two pieces of fabric together eliminating the use of pins. He has the darker sides of the fabric on the inside and the lighter (under) sides of the fabric facing out.
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The child in the photograph below has threaded the needle and is stitching the two pieces of fabric together following closely to the edge of the fabric. The running stitch is used for this work. After several stitches, he removes one of the clothespins (and later the second) as it is no longer needed.



Running stitch.

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At a point where there is just about an inch of un-sewn fabric, the child turns the pillow inside out so that the right side faces out. He then uses two fingers to stuff the pillow. I love seeing his sandpaper letter finger positioning in the photograph above.






Now, he simply stitches the small opening closed, ties off his thread, snips it and he is done.



What a nice pillow he made for his Dad.
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Sock Rolling




On the left is the sock basket. A child has taken out a pair of socks (pairing socks is also a first step to the work and is sometimes the only part done during an initial presentation with a very young child) and placed them on the table so as one sock is slightly above the other.




The child places the sock that is slightly lower on the table on top of the other, again slightly lower than the other. She uses her fingers to hold the socks in place. Next she grabs the rim of the under sock on both sides with both her hands. She uses her thumbs to push the top (lower)sock up into the under sock. Simultaneously, her fingers pull the rim of the under sock down over the body of both socks.


She continues until the socks are pretty well balled together.

Done! Three year olds love this work and so do their parents, as it is great work to do at home helping with the laundry.

Wednesday, April 23, 2008

Math - The Multiplication Board

Multiplication in the Montessori Primary classroom is a favorite work. We have many materials for the children to do multiplication with. Multiplication is done with the golden bead material, the stamp game, the multiplication bead bars box, the small and large bead frames and more. In addition there are the multiplication boards - blank boards and answer boards. These are pictured above.

As with the division work in the classroom, language is key in the initial and subsequent presentations. In division we tell the children that we divide the dividen by giving an equal amount, no more no less, to each skittle (they collectively represent the divisor) on the board and that the answer is what one gets.

In multiplication we are also careful with our words and have made a slight difference in how we explain multiplication from traditional methods. Here is the key: instead of saying multiplication is 3 times 4, we say multiplication is the same number taken many times. So we would say 4 taken 3 times.

Why that language specifically? Because it makes logical sense to the child. Many children confuse the reference to time to the the telling of time. I remember during my first year of teaching, I read the equation 3 times 4 to a child and they looked up at me with these huge eyes and said, "I don't know how to tell time." I suddenly remembered my error and corrected my language and the child immediately understood. It really is easier to comprehend. Simply take 4 three times. With the multiplication bead bars a child literally takes out three four-bead-bars and places them one under the other and counts them. "This is so easy," is usually their comment because it actually is.

Multiplication Bead Bars in their box.


The multiplication bead board has numbers 1-10 printed across the top. There is a small grooved circle in which a red marker rests when not in use. Half way down the board on the left hand side there is a side slot for a numbered card to be placed into. Above it is an open top circle which allows the number to be seen. There are white cards also with the numbers 1-10 printed on them. Some teachers have the children carefully place all of the cards on the table in order with one at the bottom of the row and ten at the top. Others have the children leave the cards in their wooden box and have the children draw one card out at a time. Both ways are fine.

The child is also given a multiplication tables booklet or, after much work with this material, a sheet of single digit problems. The picture above shows the problem 5 taken 2 times. The number 5 card is inserted into the slide slot. The red marker is placed above the number two which tells the child where they need to stop - the last row of beads will be placed beneath it. The child then places five beads under the number one printed at the top of the board and then five more under the number two. Next, they count the total beads and find their answer. 5 taken 2 times equals 10.

The greatest challenge of this work is in the counting as the answers go up to 100. The child must be able to count to 100 first and therefore should have mastered the hundred board before having a lesson on this work. The other physical challenge is that the beads need to be carefully placed in their grooves or they roll off the board and then you see children on the floor, under their table looking for red beads. Another child or two soon joins the hunt. I show up and give the helpers a few kind words for their eagerness to help and then send them on their way back to their own work.

The real problem is that there are 100 red beads and that's it. So if a child is doing the problem 10 taken 10 times and a few of the beads were lost, they come over to you announcing that there aren't enough beads. Hint: Just for these moments - go to a bead store or craft store and buy similiar looking red beads and keep them in a small jar hidden from all eyes but your own and those of your assistants. These beads, like an extra small pink cube to top the pink tower are our emergency supplies such as bandages and bug spray are.

Next week I will post pictures of the children doing this work - I took some and after examining them closely found that the child I photographed was making a few errors - time to re-present...so no photo posts yet! But soon....

Math - The Divison Board



The divison board is one of Maria Montessori's mathematical masterpieces. I say that because she has made division work so absolutely understandable that it questions any other method of initially learning this work. The board has a green colored heading. The box is divided into two sections - one is for skittle-like "people" and one is for counters. Both are also green. There are grooved spaces for up to nine skittle-like "people" and 81 counters.

Initially, during the first lesson, I introduce the child to the division board and the skittle-like "people" and the counter beads. Also, I give the child a divison tables booklet which can easily be made on the computer. Examples of these equations are as follows: 2 divided by 2, 4 divided by 2, 6 divided by 2, the last equation being 18 divided by 2. I read several equations re-introducing / reminding the child what the division sign looks like (they would have already done division with the golden bead materials and with the stamp game) and I give them the name of the division symbol - obelus.

I make it a point to tell the children that mathematical notation is read like sentences are read. I also tell them this so as to remind them to show their work so that at any time I may stop by their table and read the specific problem they are currently working on. I feel very strongly that children should learn earlier on to show their work and that it should become routine for them.

At this stage of the work there are no remainders. But, the children quickly figure out the pattern in the division tables booklet and either fill in the answers without using the board or don't want to do the work as it is either too easy or too boring. So I have blank sheets that I use for division on which I write a list of equations. I must remember not to give an equation that can not be answered on the board. Ex. 99 divided by 9.

The equations that I write have no pattern and therefore the children must be very focused on completing the one at hand and to read the next one so as to do the work correctly. I find creating these individual sheets works the best. The children will often say that the work is hard or challenging which is what older five year olds and 6 year olds want. They want the other children in the classroom to see that they are doing difficult work. This is part of the profile of a child about to graduate from the primary classroom.

After a child has worked on this material once or twice, I introduce the concept of a remainder. That may be sooner than my album states but again I find that it works the best. I make an extra box next to the one in which the child writes the answer to the equation. Notation in this box is always done with a red pencil. At the top of the page, just above the placement of this extra box I write a red r. This r is for remainder.

I really didn't catch onto the concept of a remainder when I was first learning this as a child in second or third grade. But the language is the key! In the picture below note that there is what looks like a brown circle - the limitation of this photo - this is actually a small wooden bowl like the one used with the golden beads bank game. If the child is given the equation 9 divided by 4, they place four skittle-like "people"at the top of the board in their specific places and then take nine counter beads and place them in the bowl. Carefully they give each of the "people" the same amount, no more - no less. Therefore each of the four "people" would have been given two counter beads. The answer in division is what one gets. That is what we tell the children.

Then they note that there is one bead left in the bowl - now what?! Here is the beauty of Maria Montessori's mathematical mind -she didn't just give teachers this great board material for lessons on division, but she (because she listened so carefully to the dialogue of her students and of their questions regarding math that she knew that how the method of the work is explained, the very phraseology that the teachers used, was intrinsicly important to the child's ability to master the work) gave us the very language to use for the child to master divison - so we too look in the bowl and see the one bead left there and then look at the child and say so clearly - "What remains in the bowl is the remainder - r." The child generally lets out a little laugh. They find that explaination a relief and that it simply makes sense. Next I show the child how to use the red pencil to write 1 in the box to the right of the answer box. The equation now reads 9 divided by 4 equals 2 r 1


This child is solving the equation four divided by two. The child initially placed two skittle-like people at the top in the designated area of the heading. Then they put four counter beads in the bowl. Now they give the beads to each skittle remembering the first rules: In division the answer is found by giving each "skittle" or "person" the same amount, no more no less. The answer is what one gets.

After reading several of their completed equations, I introduce the language for division - if I have not done so earlier with other materials - I take a singular problem and name the parts as I would in giving a three period lesson:

64 divided by 8 = 8

64 is the Dividen and 8 is the Divisor.

Also after several difficult equations, I show them that they may check their work using the division boards (the ones with the problems and answers listed - they are rectangle in shape and larger than the above spoken board), but that multiplication is also the opposite of division and therefore may be used to check division problem answers. The above equation of 64 divided by 8 = 8 can be easily checked using the multiplication bead bars or any material that will allow the child to take eight eight times.

I will post additional photos to give a larger overall demonstration of the work next week - we are on Spring break all this week. Good Luck!

PS - Materials purchasing note - The division board and the multiplication board are two of the least expensive math materials - My school purchased my classroom's through Appleseed Montessori Materials. Each board was under $60. Save your pennies and buy them. They are worth it!


Saturday, April 19, 2008

Photographic Album #4


Here is the fourth "page" in the photographic album capturing a sampling of the work done by the children last week.



A parent started the week with a lesson on yarn winding with an umbrella swift.




Towards the end of the week a grandparent gave a lesson on Passover and brought the ingredients for the children to make haroseth: apples (which the children cut), raisins, cranberries, cinnamon, honey and grape juice. It was delicious.
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My afternoon class used cupcake papers to make flowers. I loved seeing the variety of vases made and the detailing on the leaves.

A child's number scroll approached 1000. He passed it and continued on.



I was delighted to see one of my oldest students trace the shapes from the botany and geometry cabinets and then start filling them in. This work is an ongoing project for her. By the end of the week she was about three quarters done. She does put it away now and then to do other work such as math and language. It is really amazing to see her fill in the shapes with objects ranging from an apple to a sea shell.

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Two boys continued making geometric solids from paper. First they used the plastic hemisphere used for map making to create large circles on easel paper. Next, they illustrated the circles and cut a flower like shape with several petals from the paper.


This is how the worked looked before it was carefully folded into a sphere.

A second student worked across from the first.


This is the sphere the second student made. They are getting more exact.

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One morning this week my assistant Jill and I simply put out white paper and allowed the children to illustrate a couple of sheets each. The above is one example of student art work.


Another child drew the continent map from memory alone.

What began as a sketch of a rocket ship taking off for the moon from the Earth became a small chapbook on the solar system. To do the work the child used the solar system classification materials. He had never used this work before but his art showed he was interested so I made it available to him.



He spent most of the morning working on his solar system book. He was so pleased to take it home and show his parents.


A group lesson on the parts of a leaf left captivated one student. She finished all of the work in one sitting. What a grin she had on her face when she finished it!

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And there were lessons given on the multiplication board and the division board. Children made a variety of languge booklets and sooooo much more! On top of all of that my assistant Jill wrapped the gifts children made for our first Parent Night that was held Thursday night. It was a pretty busy week. We have a week off now for Spring Break. Happy Spring!