Edited to give a more detailed description.

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Here is part of my math album that discusses the math theory

and some of the materials we use.

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General areas in the Montessori Math Curriculum

The Montessori math curriculum in 3-6 moves from the concrete

understanding of math to an abstract understanding. A clear

example of this can be seen in the Montessori materials of the

Golden Beads and the Stamp Game.

The Golden Beads are physical representations of the numbers

1, 10, 100, and 1000. One is signified by one bead. Ten is

shown with ten beads put together in a long line by a wire.

100 has ten of the ten bars strung together to make a square.

1000 has ten of the hundred squares put together. When a

child holds the one bead in one hand and the 1000 cube in

another hand, the cube is literally 1000 times the size, 1000

times the weight, and 1000 times the visual representation.

By holding these sensorial objects, the child has a very

concrete representation of the difference between one and one

thousand.

A material known as the stamp game takes this to another level

of abstraction. It has small wooden “stamps” that are painted

various colors. On the stamps are painted the numbers 1, 10,

100, or 1000.

A child begins addition of four digit numbers with the

concrete beads, which give a very concrete representation of

the numbers through various senses, then progresses into the

stamp game, which provide information primarily through the

visual sense. This helps the child move from a concrete

understanding to a more abstract understanding.

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Golden Beads

Here, the golden beads are shown in the demonstration tray and

the building tray. The thousands are on the left and the

units are on the right. The Stamp Game

The stamp game is used to help the child move from the

concrete ideas of the beads into a more abstract concept of

math. The colors are painted green, blue, and red to match

other materials in the classroom that represent the same concepts.

Materials have built into them what we call “control of

error.” This is something that helps the child understand if

they did the material correctly or if they made a mistake and

need to double check their work. One of the most common

examples of the control of error in the early math materials

involves the quantity that the child is provided. In many of

the materials, the child will be given 45 of something – such

as beads or wooden spindles.

If the child counts everything out from 1-9 and realizes he

does not have enough, or has too many, he notices the mistake

and has to count it again. The realization and the correction

is immediate, as opposed to the next day or week when the

child is given back a graded test and no longer cares; nor

feels like they can do anything about it. Since children seek

to perfect their work, this provides the child with the

opportunity to do so.

The Montessori math curriculum also follows a certain

sequence, with certain materials to help the child discover

these concepts.

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Introduction of numbers.

This is where the Montessori math curriculum begins. There

are several materials that help to teach the child this

concept, such as:

Number Rods: The number rods are wooden rods, varying in

length from 10 cm to 1 meter long. Every 10 cm is painted a

varied color of red and blue. The rods help a child overcome

one of math’s greatest challenges. As Maria Montessori

stated, “The advantage of this material is that it represents,

united together, although distinct and countable, the units

comprising each of the represented numbers. The rod of 5, for

example, is all one piece representing the number 5, but the

five units are distinguishable by the different colors on it.”

Sandpaper Numerals: The sandpaper numerals are numerals cut

out of sandpaper and glued onto a smooth board. The child can

trace the numeral and have a sensorial understanding of what

the numeral is. This also helps the child with a pre-writing

skill.

Spindle Boxes: The Spindle boxes consists of a set of 2

boxes with different compartments. Painted over each

compartment are the numbers 0-4 on the first box and 5-9 on

the second. Included are wooden spindles that the child can

hold and place into the box. These materials “leads to the

counting of separate units and gives a child a concept of

numerical groups, and at the same time, it fixes before his

eyes the succession of the following signs: 0, 1, 2, 3, 4, 5,

6, 7, 8, 9.” This also helps to teach the concept of zero.

“In the box containing the pegs there is one compartment over

which 0 is printed. Inside this compartment ‘nothing must be

put’”

The child does not necessarily need to fully understand the

order of numbers at this point. These materials are designed

to help the child form the concept of the numbers. The

spindle boxes begin to help the child see the order, but it is

not fully necessary at this point whether or not the child

notices this particular order. That will come with time and

with other materials in this section, such as the number line

– where children lay out the numerals 0-9 in successive order.

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OVERVIEW AND FUNCTIONING OF THE DECIMAL SYSTEM

The next step in understanding numbers in the Montessori

curriculum is an understanding of the decimal system.

This begins with the introduction tray, shown and described

above. The child begins by learning the names of the beads –

units, tens, hundreds, and thousands. The child then learns

the names of the symbols that correspond to those. The final

step is learning to put them together. The building tray is a

set of 10 of each of the decimal places which the child can

actually build and develop these numbers himself.

Working with the golden beads enables the child many

opportunities for learning the decimal system. One of the

most interesting works children do is called the 45 bead

layout, where they lay out the entire decimal system and match

up the numeral cards with it. Here, they connect the

understanding of each decimal place by putting the unit beads

down in 1-9 on the right side of the rug then placing the unit

bead cards next to it. They do the same with the tens,

hundreds, and thousands. This layout is first used to help

the child name the place values (5723 is 5 thousands, 7

hundreds, 2 tens, and 3 units) then later to give the proper

name to the number.

The exchange game involves learning that we can exchange

these objects for objects. Since the child already

understands on some level that ten 10 bars makes up 100, there

are various activities that help him to understand that we can

exchange 10 tens for 1 hundred and switch it around to

exchange 1 hundred for 10 tens.

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Linear Counting

Understanding the decimal system is a step that many programs

do not focus on, for whatever reason. Many skip ahead into

linear counting without making sure the child has some concept

of our decimal system.

To teach linear counting, we use several materials. The

first of which is the small bead stair, where the child counts

from 1-10 using beads similar to the golden beads. The one

bead, with a single bead on a wire, is painted red. The two

bead, with 2 beads strung together, is green. 3 is pink, 4 is

yellow, 5 is light blue, 6 is purple, 7 is white, 8 is brown,

9 is dark blue, and 10 is the familiar golden bead color. The

child counts the beads and places a numeral on the last bead

to signify its quantity.

The teen boards and the ten boards are used to count numbers

past 10 and up to 99. The teen boards have the number 10

painted on it 9 times. There is a slot where the child can

slide in another number, from 1-9, and create a new number.

For example, they might take the number one and slide it in,

covering the number 0 on the 10. This then becomes 11. The

child places a ten bar (because of the first numeral) and a 1

bead bar next to each other. The child can then count the

beads and discover that there are 11 beads. The teen boards

are similar, except they have the numbers 10, 20, 30...all the

way up to 90. There are 45 ten bars to go with the ten boards.

The hundred board is a square board divided up into 100

different sections. Included with the board are 100 different

tiles, numbered 1-100. The child places the numbers on the

board in successive order from 1-100.

One of the signature Montessori math works is the bead

cabinet. The bead cabinet helps the child learn to skip count

and linear count. Each chain is made by taking multiples of

the small bead chain and linking them together. For example,

the small two chain has two of the two bars. The large two

chain has two of the small two chains linked together. What

we get is the result of cubes and squares of numbers. A child

leaving the 3-6 program of a Montessori school will likely

have a good understanding of what squares and cubes are of

certain numbers because they counted them out and actually

made the squares and cubes out of the numbers.

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THE PROCESSES OF MATH

Once a child has a clear understanding of counting, decimal

places, numerals, and can properly understand them in a

variety of activities, we begin moving the child into the four

operations of math: addition, multiplication, subtraction, and

division.

Many of the materials the child is already familiar with are

used. The golden beads serve are used as a very concrete

material to help the child understand not simply addition, but

the importance of place value in addition. Many times, in

more traditional educational systems, children might learn to add: