Just like the sensitive period to reading (see preceding article), your child will go through a sensitive period to mathematics. During that special time, as Maria Montessori said : « If a child does not obey the directives of his sensitive period, the opportunity of a natural conquest is lost, forever gone. »

Therefore, it is very important, when your child goes through that sensitive period, to be able to help him structure his first mathematical concepts : digits and numbers. I remind you that 13 (thirteen) is a number written with the two digits « 1 » and « 3 ».

If your child counts repeatingly 1, 2, 3, 4… if he asks questions about digits, or if he is interested in games involving the sorting of objects, you are facing his sensitive period to digits.

Invite him to sort out things. These sortings will be his first mathematical exercices. For example, the sorting of buttons by shape, by color (see photos), the sorting of little cars by color, the sorting of candies by type or the color of their wrapping, the sorting of fluorescent ice cubes, the sorting of colored decorative pebbles, pearls, seashells, pegs or any small object your child sees around him…

The purpose of these sortings is the grouping of elements according to a criteria ; it will enable your child to develop an organized mind.

Maria Montessori insisted on the fact that you should introduce only one difficulty at a time. Therefore, you must be careful to introduce only one criteria at a time (for example : shape or color). Later on, your child will do the same thing when dealing with a mathematical problem : unconsciously, he will sort out the data in his head before solving the problem. This is the reason why exercices involving the sorting of objects are very numerous in a Primary class at Montessori.

Maria Montessori had also noticed that when dealing with mathematics, it was necessary to teach the concrete concept before the abstract one. This is the reason why the Montessori method is the best method to teach mathematics because it uses concrete equipment for each concept, from a simple addition to an equation with multiple unknowns. The formula (a+b)3 is concretely resolved during the first year (generally at the age of 3) in Montessori schools

**Teach the concrete notion of quantity.**

You will notice that your child often counts mecanically from 1 to 10 without knowing what it represents. You must play games with him during which he will count identical objects.

In a Montessori class, we first use red bars which a child will sort out from the smallest to the biggest. This equipment is often expensive for a family budget and also needs space to store.

**1 – Show your child what quantity is.**

Use a collection of small identical objects and show your child that when he says « one », there is « one » object. When he says « two », take one object and say « one », then add one by taking a second object and tell him : now there are two cars, two buttons, two small horses, two glasses etc. Do the same thing with « 3 » : leave the quantity « 1 » and « 2 » on the table and add a third identical object and say « now, there are 1, 2, 3, there are three ».

**2 – Ask him to show you the right quantity.**

Leave the small objects on the table sorted out by quantity and ask your child : « now count two » (ask him to count « one », « two », « there are « two » here »). The same thing should be done with « one » and « three ».

Then shuffle the series of objects leaving two objects together, three objects together and one object. Move around the series several times and ask your child : « show me where there are two objects ? », « give me three objects ! », « take one object in your hand », etc. Make sure to always vary the question. Keep in mind that learning must never be boring but on the contrary attractive and fun for your child who can even use his own fingers to count !

**3 – Ask your child to give you the right figure.**

When your child completes this exercice correctly, you can go on to the next stage. Put a series of three identical objects on the table and ask him : « how many objects are there ? » or « count and tell me how many objects there are ? » ; ask your child all sorts of questions which will enable him to count the three objects and to tell you how many there are. Then ask him the same questions with only one object or two objects. Make sure these objects are seperated from the rest. Do not leave the other series on the table to avoid confusion.

When your child has assimilated the correspondence between digit and quantity, you can end the exercice : take one of the series of small objects and tell your child : « today, we learned « 1 », then « 2 », and finally « 3 » showing him the corresponding series of objects.

This is called « the three step lesson » which is used in Montessori classes to teach the majority of vocabulary and in this present case the vocabulary associated to mathematics :

Step one : « you see, this is… », you teach him the lesson,

Step two : « tell me where is… », you use the vocabulary and he must show you what it corresponds to.

Step three : « what is it ? », the child must remember the word associated to the concept.

As a conclusion : « today, we learned… ». Your child learned the concept and the vocabulary to go with it. End of the lesson.

After verifying that the preceding lesson has been assimilated, you can teach the following step. If this is not the case, you must not continue.

These exercices must be done in a quiet and orderly space, where brothers or sisters are not present and do not disturb the learning process. The child must be able to concentrate on the activity without being disturbed. It is important to have him manipulate the objects, he must touch them.

His hand must have the perception of quantities : when he holds one object in his hand, it is little. While when he holds 9, he needs his two hands. He is therefore conscious that 9 is a lot. This information is transmitted to his brain which integrates the idea of the hierarchy of numbering. Thanks to his hands and his eyes, he learns the concrete concept of numbering : to introduce the next figure, we always add an element to a group of identical elements. Later, when asked for the next number, or the last one, his brain will immediately know that it has to add or substract a number because it was integrated in a sensorial way (sight and touch).

When your child knows the quantities from 1 to 3, you can go on to 4, 5 and 6. Maria Montessori often said to teach things three by three. If you notice that this is too much for your child, teach him two by two, especially if he is very young.

**Teach the abstract notion. The symbol.**

You can simultaneously begin to teach your child the abstract notion : the name of the digit, or its symbol. The best way is to do it with the rough digits made by Maria Montessori. As for the letters of the alphabet, you can make these digits on your own with sandpaper. It is important that your child touches the digit with his hand to be conscious of its form (the driving plan to writing) while he pronounces its name. Once more, the teaching is linked to the senses.

You can then teach him the digits during a lesson in three steps as seen above (three digits or less if needed).

– Today, we are going to learn « 1 », « 2 » and « 3 » (and you show him the form at the same time).

– Touch « 2 », touch « 1 », touch « 3 »,

– Shuffle them around the table and try again several times in disorder, « draw 3 », « draw 1 », « draw 2 », etc…

– Then you take away the three digits from the table and show them to your child one by one and ask him the simple question « what is it ? », then again « what is it ? » with another digit. Repeat this several times.

– Once this step has worked, you can sum up the lesson : « today we learned « 1 » (draw it and then invite him to do it too), then « 2 » and « 3 ».

– The next day, you can teach him the names of the next digits.

Now that your child knows the concrete and abstract notion of the digit, you need to associate them. Ask him to sort out the digits (its symbols) in order (you can do it with 1, 2 and 3 at the beginning) and ask him to associate the corresponding quantity with small concrete objects.

This way, your child will learn digits from 1 to 9 and know the exact quantity of objects to which the digits correspond.

You can also have him trace the digits by following the dotted lines that you will have prepared and if you want, to reinforce his notions, you can ask us for specific booklets that I have made at the International Montessori School, booklets that were very appreciated by our Primary school students.

Once your child has assimilated the abstract and concrete notions of digits and quantities, we will be able to go on to the next lesson : the addition, the substraction, the notion of « zero », even and uneven numbers. These lessons are taught to children from 2 to 4 during the sensitive period to mathematics in Montessori schools.

See you soon for more « operations ».

Sylvie Rousseau-d’Esclaibes