Multiplication Board

The multiplication board was a hit with our older children this week.  They were first introduced to the idea of multiplication ("fast adding") with the golden beads.  Here we show them how to add quickly.  "Two times three is just two, three times!" and we place the golden beads down on the mat in three groups of two.  We bring them down to the bottom and add.  Voila.  They absorbed this idea so quickly, so the multiplication board was introduced.  Here it is the same idea set up in a different way.  Shown above, this child is working on "2 x 9"  He fills in the beads as he goes along and is able to count to get the answer.  This is an example of using the concrete Montessori materials to facilitate easy absorption of a more complex idea.  These children will have a solid base of multiplication tables since they have worked with this material.  They will not be afraid of math as they have so many experiences using concrete apparatus.  One of the goals in our classroom is to help facilitate internal motivation to learn.  The two children who worked with this material both told the teachers how much fun it was and requested it on their own.  Learning can be a joyful experience!

Squaring Chains

As we have mentioned before, Montessori math materials are BRILLIANT and will help the child understand so many math concepts for years to come.  The contact that these children have with these concrete materials will last a lifetime!  The squaring chains are no exception to this brilliance!  Some of our older children are very interested in the chains and excited to interact with this material. 

We carry the chain to the rug and begin by counting.  We actually have labels for each break in the chain that are not shown here.  The child counts and labels, then we "square" the chain.  Here is the six chain.  At each break, the child places the label.  So for six it is 6, 12, 18, 24, 30, and 36.  The child can see how the chain makes a square when we fold up the chain. 

This is how we hold and carry the chain.  (To avoid it being swung around!)

 Many children like to complete the worksheet by filling in the circles and labels so they can take the information home. 

Glimpses of work at AACH this past week...

Paper Weaving

 Addition with the Addition Strip Board

 More circuit work!!

Stencils!

Using the Golden Bead Material- Math!

The beauty of Montessori Math!

We were lucky enough to have a pair of students join us from out East this year.  The older child (Kindergarten age) is in his third year of Montessori.  He and one of our same age but second year children worked with the Golden Beads to complete some addition work.  Here you can see the culmination of the Montessori math work that has prepared them to this point. 

 First they laid out the large numbers on a rug.

 They brought their trays over and were each given a small number to take to the bank and gather the bead materials on their tray. 

 They both carefully place the numbers down next to the beads.  Here they see each addend in quantity and symbolic form. 

After the addends are placed, the beads are then "brought down" to the bottom.  They begin in the units and use the large numbers to "write" the sum.

 

If a child is given the opportunity to stay for several years in a Montessori environment you are able to see so many things come together.  Here they complete static addition in the thousands.  The next day these children completed dynamic addition the same way! 

Discovering the Bonds of Ten

Using the addition strip board, the children are able to discover all of the numbers (addends) that add up to another number (sum).  Here, a child is discovering the bonds of ten.  It is important to show children the relationships between numbers.  They begin to understand a whole is made up of parts.  If you know the parts, you can add them together to find the whole.  You can also take away a part to find the other part.  The children also begin to think about subtraction.  They naturally see the inverse relationship between addition and subtraction. 

We usually start with ten and work our way down.  Many children choose their "bond" work in the mornings.  They have decided to do "one bond a day!" 

Large Number Rods- Associating quantity with the symbol

When we talk about associating the quantity with the symbol in mathematics, this is what we mean!  A child can often count by rote memory but may not understand how many are in eight or five.  We use the larget number rods to help the child first understand one to one correspondence.  When they count the rods, they are to touch each red or blue area.  This demonstrates each is one.  Usually when beginning this task they count and skip over an area or two so they end up at 5 when it is the 8 rod.  We give the child a lesson with a few rods at first and then increase the number.  Once they understand this concept, we introduce the symbol which is the number.  Once the child has learned all of the numbers, we put it together.  Here the child has placed the rods randomly on the rug.  He counts slowly, "one, two, three, four!" He then finds the symbol or the number and places it on the rod.  Now he understands symbol and quantity! 

Short Bead Frame

The blend of ages in a Montessori classroom is amazing.  We have 2 1/2 to five year olds this year and it is such a lovely mix.  Since we have Kindergarten aged children, it is important to keep them challenged and progressing along the learning path!  Here this child is doing static (without change) addition in the thousands with the help of the small bead frame.  This child has progressed through many of the brilliant Montessori math materials and is now ready to add numbers in the thousands.  This apparatus helps the child add in a concrete way.  Soon he will begin to add in the thousands abstractly but for now this helps him to complete many equations without stress and the demands of jumping to abstraction.  This helps the child to truly understand how he is coming up with the sum of the two addends.  He will complete dynamic addition soon which is when he has to borrow.  The same materials will be used.

The Stamp Game

This child is moving along  through the math curriculum (she is a Kindergartner) and is ready for slightly more abstract materials.  She has used the golden beads for the operation of addition and has now been introduced to the Stamp Game.  Here she uses small tiles which represents units, tens, hundreds and thousands.  In a similar way to the golden beads, two addends are written, here in colored pencil which coordinates with the tiles.  She puts the first addend down on the table. (She likes to line up the tiles perfectly with the ruler- this is something she did on her own.)

The second addend is placed below the ruler.  Once the addends are double checked, the ruler is removed and we slide the tiles up together. 

The tiles are then counted by moving them down slightly.  We always begin in the units.  This way if we go up to ten, we change ten green unit tiles in for a blue one and put it in the appropriate column. 

The sum is then recorded on the sheet. 

Addition with Golden Beads

For the older children, they have progressed through many of the other math materials and have moved onto golden beads. Again this is an amazing Montessori material that teaches units (ones), tens, hundreds and thousands concretely. We have many exercises that begin with teaching the quantity then the symbol and finally matching quantity with symbol. Once they understand this we move onto the operations beginning with addition without change.

Once the child understands this concept, we begin addition with change or dynamic addition. Here the child realizes they have to go to the "bank" and change 10 unit beads for a ten bead bar and carry it over into the tens column. From here we actually move to multiplication as it is really just "fast adding!" Stay tuned!

Addition with the Short Bead Stair

Once the child has gone through the sequence we call the "first five" of the Montessori maths materials, we move onto other aspects of math. The first five helps the child understand the concepts of 1:1 correspondence, the idea of zero and matching the symbol to the quantity of 1-10. One of the directions to go is to introduce the Short Bead Stair. This helps to reinforce the quantities 1-9 and to prepare the child for the operations of addition and subtraction. Many works are completed to familiarize the child with the short bead stair and the children enjoy working with the beautiful colorful beads.

We place a set of beads and the child counts with their fingertip or the tip of a pencil. They then write the number in the first slot. For some, as they can be only three, they need help writing so we use a dotted stamp for them to trace. For most, they love to stamp the numeral too. We then place the second addend on the green felt. They count this as well and write the number. We then show them how to put the beads together and add up with the fingertip or a pencil and write the "sum" in the box.
This is an example of how the Montessori math materials progress from concrete to abstraction in a specific sequence. The children really enjoy this activity and ask for it when they come to school. Many tell us, "I want to do math!"

The Bring Me Game

To aid comprehension of various letters and numerals, Dr. Montessori would play a game with children called the "Bring Me" game.  This involves setting up the material in a separate location and then telling the children to "bring me the _____ ."  The child then retrieves the object.  The children are requesting this game on a daily basis lately.  We have given many lessons with our sandpaper numerals so they are familiar with the names.  Because we have played this so much, we decided to let them have a turn being the one who asks for the numerals and the one who brings them back. 

Here the child is playing with two other children.  She has asked for a specific numeral from one child and the other is waiting.

"A seven!"

This is a fun activity to try at home. It can be used with colors, numbers, sounds etc... Just make sure you have told them the name first so it is not frustrating!

Math: Matching Quantity and Symbol

Many of our children are very interested in math!  They have entered their sensitive period for numbers and all things related (see post on sensitive periods).  The Montessori materials follow the developmental sequence of the child when they become interested in math.  We have heard many voices saying the numbers which is called rote counting.  This is when a child can repeat the numbers because they have memorized them. 

We have also seen some number recognition.  This is when a child can visually identify the number which we call the symbol.  (We use the sandpaper numerals.)

We are working toward 1:1 correspondence.  This is when a child can match the quantity to the symbol, for example, the number four with four items.  Once a child understands this concept, they are able to begin the  mathmatical operation of addition. 

1-1 correspondence up to 5- this is a good place to begin with those just entering into this type of activity

another 1:1 correspondence game with a die

table top number rods

Montessori cards and counters, matching quantity and symbol.

If your child is interested in numbers, take a few pinecones or leaves and count them.  In the beginning we may need to touch the object with our finger or four can become six.  The children are used to rote counting which can go faster than the finger can move.  If we stop on each one as we count, it reinforces the 1:1 correspondence.  Use refridgerator magnets as numerals and count pieces of fruit in the kitchen.  Place the number next to the amount.  There are many different activities you can do to help your child learn this concept.  As always, we never push the child to do work if they are not ready or interested.  If they are showing you signs of readiness, count away!