Edited to give a more detailed description.
Here is part of my math album that discusses the math theory
and some of the materials we use.
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
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.
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
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
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
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.
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.
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.
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
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: