Mathematics For The Montessori

Sensory and daily life activities offer many practical opportunities to use numbers and quantities. For example, when setting the table, we usually tell a child “Bring me two napkins” or “We need four spoons. Let’s see: one, two, three..”. Such situations take place dozens of times each day, which helps children internalize the existence and meaning of numbers from a very young age.

1. Teaching quantities before numerals

We always teach children quantities by showing them groups of physical objects, and not numerals. Written numerals will be learnt later, when children are already able to understand the meaning of "three flowers" or "five bears". Being able to associate the symbol 3 with the word "three" won’t be of much help if they are not able to bring you three pencils when you ask them to.

1. One-to-one correspondence

Little children tend to start showing interest in numbers quite early, but they usually lack understanding of one-to-one correspondence, which is the ability to assign a single number to a single object while counting. Most toddlers will vaguely point at a group of objects and say: “One, two, three, four, five…" just repeating a sequence of words they have heard before, but not really understanding their meaning. For them it’s similar to singing a song or reciting a poem: they are just trying to mimic adult behaviour, and adults tend to count very fast, and without pointing clearly at each object.

Understanding one-to-one correspondence is key in order to learn to count properly. In order to achieve this, count things all the time in front of your child, and do it slowly: say a number and touch an object, pause, then say the next number and touch the next object, etc. The pauses are very important, as it is actually touching each object. This way, children understand that each number corresponds to one object only. Start with small groups (three or four objects will suffice) until you are sure the child is able to match one number word with one object only. It will be easier if you start counting groups of identical objects, to avoid confusion.

1. The number 10 and the decimal system

Learning to count and recognize groups of ten objects is a critical skill which needs to be mastered before moving forward to other, more complex mathematical concepts. In everyday life, we unknowingly use the decimal numbering system all the time. The decimal system contains ten numeric characters (0-9) and the number ten as its numeric base. This means that each time you add a zero to the right, you increase the magnitude of the previous number by ten times: for example, 70 is ten times 7. To strengthen the concept of ten, the majority of Montessori sensory materials have exactly ten pieces: for example, the knobbed and knobless cylinders, the brown stairs, the pink tower, etc.

Counting to 10

You can easily teach little children to count to ten without using any Montessori materials. Just show them groups of ten (or less) identical objects, and count them slowly. Do it as often as you can, and take advantage of every opportunity. Be creative and use jelly beans, crackers, plums, apples… anything will do! Remember to count while doing sensory and daily life activities: count the steps while you go up, count the eggs in an egg carton, etc.

Number rods:

The number rods are ten wooden bars of gradually increasing length, formed by alternating red and blue rectangles. They are meant to:

• Help children to count,
• Understand the sequence from one to ten,
• Understand how the number ten contains ten units, the number nine contains nine units, etc.

You can build your own number rods quite easily. Just go to a home improvement or DIY store and ask them to cut the rods for you: you will need ten pieces, the shortest being 10 cm long (approx. 3,97 inches) and the longest 100 cm. Each piece will be 10 cm longer than the previous one. Once you have the ten rods, divide them into 10 cm long segments and paint them in red and blue, like in the illustration.

The number rods have a maximum length of 1 meter (3,28 feet).

If you can’t get hold of wooden rods just make them out of cardboard or even building blocks, like in the picture below these lines:

Sandpaper numerals:

Sandpaper numerals are used in a similar manner to sandpaper letters, which were discussed in the reading and writing section of this manual. Their purpose is to teach children the symbols which represent quantities, once they already understand the latter. We start with number one and present the numerals in groups of three, using a three-part lesson (first 1-2-3, then 4-5-6 and finally 7-8-9).

ACTIVITIES TO REINFORCE THE UNDERSTANDING OF QUANTITIES AND NUMERALS:

Once the child can recognize groups of ten objects and has learnt the numerals from one to ten, we will strengthen these skills with related activities.

Cards and counters:

Make ten white cardboard cards and write on them the numbers from one to ten. Get 55 small, identical objects (counters or tokens) and put them in a basket. You can use beads, candy, dry beans, toothpicks, pompoms, etc.

Cards and counters.

Place the cards in order on a placemat. Then start putting the right number of counters under each numeral card. Place them in orderly rows and count aloud while you do it. When you finish, pick up all the materials and offer the child the opportunity to repeat the activity if he wishes to.

Counting from 11 to 19

Once you have mastered the numbers from one to ten, it is not advisable to jump directly to counting up to one hundred without explaining 11-19 before. This is a very particular set of numbers (the “teen” numbers) because their names can’t be formed in an intuitive way, unlike the following (from 20 on) which can be easily named once you know the rules.

Montessori beads or pearls are chains of one to ten beads. The most used ones in preschool environments are:

• The golden beads: these are chains of ten yellow beads, and you will need at least ten of them,
• The bead stair: these are nine bead chains of different colours, which contain one to nine beads.

This is a fairly affordable material and it’s easier to buy it than to make it at home. You can get good quality Montessori beads online, as they are small and light and shipping shouldn’t be expensive. However, it is also possible to create a homemade replica by threading plastic beads on a piece of wire. Another possible option, less resistant but faster to make, is printing or drawing them on cardboard strips.

In order to learn the numbers from 11 to 19, we will need the bead bars from one to nine, at least one ten-bead bar and cards with numerals from 1 to 19.

To begin, show the child the bead bars from one to nine, placing each bar next to the card with the corresponding numeral.

Then add a ten-bead bar next to each of the teen bars, in order to show the child how to form the numbers 11, 12, 13, etc. This way of presenting the numbers is very graphic for the child because it clearly shows how the number ten is the basis for creating the numbers 11-19.

Once you have shown the child how to do it, you can slightly change this activity and place the groups of beads in a random order. Let the child count the beads and decide which numeral card goes where.

Counting from 1 to 100

The golden beads are bars of ten yellow beads which are used to learn the numbers from ten to one hundred. You will need at least ten of them, so that you have enough to count up to one hundred.

The golden beads allow us to teach the concepts of ten, twenty, thirty, etc. As usual, use a three-period lesson, such as:

1. Show the child two golden bars. Count the beads one by one: “This is twenty”.
2. Afterwards, do the same with the numbers thirty and forty. “Which one is twenty?”
3. When you are sure that the child is able to recognize twenty, thirty and forty, check if he can also name them on his own: “What is this?”

Seguin’s boards

Seguin’s boards are a learning material which is used to form numerals from 10 to 99. There are two versions: the teen boards (from 10 to 19) and the tens boards (from 10 to 99). They are just a rectangular wooden frame with the numbers 10, 20, 30 etc. up to 90, printed on the base. On top of the printed numbers we can place wooden slats with the numbers one, two, three, etc. which then cover the number zero and allow us to write whatever number we want (for example, we put the number 7 on the 40 in order write 47).

I made our own Seguin’s boards with sixty simple, handwritten cardboard cards and they work perfectly. If you want to make your own, too, just copy my card set from picture

To begin with, we will form the numbers 11-19 using bead bars. Then we will try to write those numbers with our tens tables and match them (see picture 81 in order to see how to work with the tens/teens boards).

The next step is to form the numerals 10, 20, 30, 40… up to 90: first with bead bars and then with numerals (using the tens boards).

Finally, we will form the numbers from 21 to 29, from 31 to 39… etc., until we get to number 99.

After a while, you can prepare a few groups of beads and let the child attempt to write the correct number next to them, using Seguin’s boards (or homemade cardboard cards).

The Hundred Board

The hundred board is one more on the list of Montessori materials which are worth buying. It consists of a square frame with a hundred small wooden tiles which contain the numerals from one to one hundred.

If you can’t get hold of one you can still try and make your own with a square of thick cardboard or foam board (foam board is a material mostly used to build models, made out of two pieces of thin cardboard and a layer of foam inside. Its thickness should be 3-5 mm/0.15 inches). You will have to cut 100 equal squares of 1 x 1 inches (2.5 x 2.5 cm), which will be used for the tiles from 1 to 100, and a square frame measuring 10 x 10 inches to fit them in.

How to use the hundred board:

First of all, put all the tiles on a mat on the floor or on a big table. Start by sorting the numerals by tens, that is: put all the twenties together, all the thirties together, etc. At this time, it is not yet necessary to put the tiles in the correct order: just sort them by tens, like in picture 83.

Once you finish sorting, begin to place the tiles on the hundred board one by one, while you name them: "one, two, three…".

This is a job which requires a good amount of time and patience. Some children are able to complete it in one session, while others will need to take a break from time to time.

The first few times you can use just a few tiles (for example, from one to thirty) instead of all the tiles at the same time. This will prevent the child from losing interest too soon and leaving the job half-done. When the child starts to get familiar with the hundred board you can gradually increase the number of tiles.

Units, tens, hundreds

Montessori beads can be used to introduce the concepts of units (with loose pearls), tens and hundreds. We can also practice skip counting by tens if we combine the golden bead bars (ten-bead bars) and the hundred board.

Counting to one thousand

If your preschooler is interested you can also count up to one thousand. Print ten squares of 100 beads (see the previous illustration) and show the child how to make the numbers 101, 102, 103, etc. Then try to skip count by the hundred until 1000.

You can represent the number 1000 with ten squares of 100 golden pearls. Create a tangible number 1000 with a cube-shaped box (all the squares should have 10 beads/circles of side).

You can use small items (such as dry beans or pompoms) to prepare this effective addition activity. Get two small bowls: two for the addends and one for the result (sum).

87: Addition with bowls and pompoms.

• Put three pompoms in one of the addend bowls and two in the other.
• Count each group of pompoms carefully and place a card with the corresponding numeral (3 and 2) below each bowl.
• Ask the child how many pompoms we will have if we put them all together. Add the card with the plus (+) symbol and explain to the child its name and what it means.

88 : A nice (not strictly Montessori) educational toy: addition with bee-shaped magnets.

• Explain that the third bowl is where the result goes. Put all the pompoms together in the third bowl and count them.
• Put the equals sign (=) in the right place. Explain that this symbol is called "equal" and it means "result" or “is the same as”.
• Repeat the exercise with different amounts, letting the child help or work on his own.

The addition snake is a game in which the child must find combinations of numbers which add up to 10. You can substitute Montessori bead bars with bead bars drawn on paper or made at home with loose beads, as we saw a few pages before.

We start with a snake of many colours who wants to turn golden. The game consists in helping the snake become golden, that is, exchange her multi-coloured bead bars for yellow (10-bead) bead bars.

What we need:

Note: in Montessori schools they also use black and white bead bar stairs for this exercise, but at home I have never used them, for the sake of economy.

How to play:

• Make a snake shape by putting a few bead bars together, in a zig-zag pattern (see illustration above). Start by using bead bar combinations which always add up to exactly 10 as you go from the tail to the head of the snake, such as:

3 + 7 + 4 + 6

• Tell the child: “Now we are going to exchange these coloured beads with the same number of golden beads”
• Start counting beads. When you count up to ten, remove the coloured bead bars and replace them with a ten-bead bar:

(3 + 7) + 4 + 6

10 + 4 + 6

• Start counting from number one at the point where you left off. Again, when you come up to ten, replace the coloured beads with yellow beads.

10 + (4 + 6)

10 + 10

• Finally, put the resulting golden bead bars together and say: “This is twenty, so the snake was 20 beads long. Now it has become golden!”

You can play again, adding numeral cards and symbol cards (+, =), creating the following equation:

3 + 7 + 4 + 6 = 20

Finally, you can make this game harder if you use coloured bead bars which don’t add up exactly to ten. In this case, you only remove whole bead bars when you can replace them with golden beads. What happens if you have a 7-bead bar next to an 8-bead bar? What can you do then?

1. a) You simply count up to ten and put a marker at this point (a pencil, a small piece of paper). Keep counting until you end up having an exact number of tens, then replace with golden bars.
2. b) If you own black and white bead stairs, you can use them until you get to the next ten (black and white colour helps to distinguish them from the coloured ones). In this case, you would remove your 7- and 8-bead bars, and replace them with a golden, 10-bead bar, and a black and white, 5-bead bar.

Subtraction

Learning subtraction with bowls

You can explain the concept of subtraction very much like you did with addition. You will need three bowls on a tray or mat, numeral cards, and symbol cards (-, =). Use the same small counters you used to teach addition (jelly beans, almonds, etc.).

Explain your child you are going learn subtraction today. Use an example, such as: “Subtracting is a way to find out how many walnuts will be left after I eat a few of them”.

First decide the equation you are going to solve. You can write it on a card and leave it aside (face down) to use it at the end as control of error.

Let’s calculate: 8 – 2 = 7.

Count eight walnuts and put them in the first bowl: “I have eight walnuts”.

Take out two walnuts, counting aloud, and place them in the second bowl: “Imagine I eat two of them”.

Finally, take the walnuts which were left in the first bowl and put them in the third one (the “result” bowl). “How many walnuts are left after my snack?” I count them. “There are six walnuts left”. “Eight minus two equals six”.

90: Subtraction (1): first step.

91: Subtraction (2): remove the subtracted amount and place it in the second bowl.

92: Subtraction (3): the leftover pompoms are the result of our equation.

Multiplication

In Montessori, we learn to multiply with the multiplication board, which is just a square material with the numbers 1-10 written from left to right and from top to bottom. This creates a grid of 100 spots, which can be filled with counters to multiply any amount from 1x1 to 10x10 (see illustration).

A printed or drawn multiplication board will be enough. Of course, it is nice to have the original Montessori material, but it’s not strictly necessary. You will need 100 small counters, such as dry beans, almonds, one-cent coins, etc.

Print the multiplication tables and keep them as a control of error.

How to use the multiplication board:

First, explain to the child you are going to do multiplication: “Multiplying means adding a number (multiplicand) to itself as many times as another number (multiplicator) tells us”.

Explain that, on our board, we will mark the multiplicand on the vertical columns and the multiplicator on the horizontal rows.

Let’s imagine you are learning the table of four. You would, of course, start with 4x1, 4x2…

In this example we will do 4x3, using a homemade multiplication board and some dry beans.

4 x 3 = 12

Multiplicand x multiplicator = product

I like to tell children that I am going to give the same amount of beans to a few friends and use multiplication to see how many I should buy.

• Mark the number 4 on the left column, and the number 3 on the top row.
• Now put beans on the first, second, third and fourth rows, like in the picture below (I would count them as I go, and say something like “Three beans for my first friend, three beans for my second friend, etc.”).

94: 4x3 with beans.

• Finally, count how many beans there are on your board (12). This is the result. Now check your printed multiplication tables and make sure you calculated it correctly.

Division

Once the child has practiced addition, subtraction and multiplication, and is capable of working on them without help, you can introduce the concept of division.

Explain that dividing means to distribute or share equally.

In Montessori, we work with a division board. Just like the multiplication board, we can make our own with a square piece of paper and some small items to count.

95: Drawing your own division board is fairly easy.

Draw a division board with nine rows and nine columns, like the one in the illustration.

Get 81 counters (in the example I used almonds) and 9 tokens (chess pawns are useful) to represent the divisor number.

Write a few equations on cards. Choose equations which result in a whole number. Some examples you can use:

4 ÷ 2 = 2

9 ÷ 3 = 3

15 ÷ 3 = 5

In this example we will solve the last one:

15 ÷ 3 = 5

Dividend ÷ divisor = quotient

• You can start with an example, such as distributing 15 almonds equally between 3 children. “Let’s divide 15 by 3”.
• Count 15 almonds and put it in the first (dividend) bowl.
• Mark the divisor on the top row of your division board (the “children” you are going to give your almonds to, in my example). Put a chess pawn on each number, to represent your children or divisor.
• Then start distributing the almonds. Give one to each child (that is, put one under the numbers 1, 2, 3, then go to the next row and do the same, then to the next row, etc., until you run out of almonds).
• In order to read the result, you just have to count how many rows you managed to fill (5).
• Check your written equation to make sure you calculated the division correctly.

96: If we distribute 15 almonds between 3 children, each child will get 5 almonds.

Fractions

Daily life activities offer hundreds of opportunities to talk about fractions with your children. Fractions can be learnt before division because we encounter them extremely often in our day-to-day lives (especially in the kitchen).

Fractions with food:

Whenever you have to cut a pizza, cheese or pie, take advantage of the situation to casually explain simple fractions to your child.

97: Fractions with cottage cheese: 1, 1/2, 1/3 y 1/4.

Say to your child: “Let’s share this cheese between you and me”. Then you cut it in half. “This is one half. And this is the other half. The cheese has two halves, and they are equal. One is for you, and the other is for me”.

98: Fractions with apples.

You can do the same in order to divide a pizza between four family members. “Let’s divide our pizza so that there is the same amount for each of us. We are four, so let’s divide it into four equal pieces”. Cut it in half (say it), then in four. “This is one quarter. We have divided one pizza into four quarters.”

Fraction cards

Once the child understands the meaning of a half, a third and a fourth (of a cheese, of a donut, of an apple…) you can take one step forward and use abstract materials such as fraction cards or diagrams. Afterwards, you can also teach him the numeric representation of fractions (1/2, 1/3, 1/4).

99: Fraction cards (1/2, 1/3, 1/4).

100: Comparing fractions and looking for equivalences.