For the past twenty years, when I explain how mathematics are taught in Montessori schools, I am always surprised and amused by the reaction of students’ parents : « if only I had been taught fractions this way, I would have understood immediately … », « substractions seem so obvious now ! », « with all this material, mathematics seem so easy and obvious, etc… »

The same way, when I taught maths to young students arriving from other schools, I would be told : « I don’t understand anything in maths, it’s too hard, I’ll never make it ! ». All these negative reactions are the result of a misadapted teaching to a young child’s needs. After a few months of class and after mastering a few concepts with Montessori material, those same children began to consider mathematical concepts like a game and understood it more easily. Where does this miracle receipe come from ?

It comes from the fact that in Montessori teaching, **all mathematical concepts are presented with adequate and concrete material that the child can manipulate.**. Contrary to what is done in traditional schools, a notion will never be taught in an abstract way based on « things » that you must apply in automatic without understanding them.

In the Montessori training that I followed, the President of the American Montessori Association, who came from Canada, told me that as often as possible, **the child should be able to find the reasoning on his own, that he should understand how to get to the result on his own.**

Thanks to concrete material used in Montessori schools, **a child always has a notion of what he is doing.**. For example, in exercices involving the notion of unity, of the dozen, of the thousand, the teacher gives the student material that he can hold in his hands. He understands that a unit is very small. On the contrary, he has difficulty holding the cube used for one thousand in his little hand and understands that a thousand is a lot, an important quantity.

Thanks to this concrete notion of quantities, the child **will be less tempted to write an absurd result to a math problem.**. Thanks to the manipulation of quantities, he will be chocked by an incoherent result.

In traditional schools, children are often asked **to memorize only**. This is the case with multiplication tables for example that children learn by heart. It is of course quite useful, but why not teach children what these tables are made of?

In a Montessori class, we use material made with colored beads bars (see illustration). From kindergarden, a child knows that bead bar 1 is red, beads bar 2 is green, beads bar 3 is pink, etc… (see photo of the scale of beads) ; then we teach him that multiplication is an addition of the same quantity a certain number of times.

Therefore, when the child calculates 4 times 3, he takes the pink beads bar 4 times and counts how many multiple of 3 there are. When he calculates 6 times 8, he counts all the multiples of 8, 6 times. That way, if one day he does not remember the result, he will find it by counting 8 by 8, 6 times. A child who will have learned the tables by heart without understanding how they are elaborated will be penalised in case he forgets a result (for example because of stress during an exam).

In a Montessori class, since all children are different and have different ways of learning, they can choose from a wide variety of different material. For this particular notion of multiplication, there are beads bars but also a multiplication board (see photo), a stamps’ box, an abacus, etc… This other material enables the child to realize that, to make multiplication tables, we put the same quantity the number of times we are told.

**Manipulation is essential. Always keep in mind the five senses. Everything is learned in an efficient way thanks to the senses. **

A child who has extracted the beads with his hands while doing a substraction is quite conscious that he withdrew something and that the result can only be an inferior quantity to the original one.

**Visual sight is also essential. **. When a child sees Montessori material for fractions, he notices immediately that in a half, he can put two quarters. He will not forget this, while abstractly, everything is a question of « things » and « memory ». Don’t we often hear people say : « I don’t see what you mean… » ? Don’t we all need to see to understand ? ».

Same thing with geometry. A child who has manipulated geometry sticks and constructed squares, rectangles, obtuse and acute angles with his own hands will not have difficulty remembering that to make a square, he took 4 same colored sticks, all equal and that to make a rectangle, he used 2 different colored sticks twice.

At the age of 3, when a child has worked visually with the geometric cabinet, when he has put the right forms in the right embeddings, he will be familiar with geometrical forms, their names etc… and will have overcome difficulties compared to the child who abstractly learned the lesson.

In Montessori schools, we never teach « things » or rules to memorize without first trying to understand them, like it is often the case in traditional schools.

Like for example, when a child is told to write a little « one » at the top to resolve the operation without explaining to the child the meaning of that number.

Never has an adult or a child been able to tell me why they did this!

In a Montessori class, we concretely explain what we do. For example, when you want to substract a quantity and that quantity is insufficient in unity, dozen and hundred, we proceed to changes (see lesson on the substraction with changes above and at the end of this article).

This is a considerable help to understand substractions with hourly notions where the use of « little numbers » does not work.

Same thing with the notions of measures, **everything is presented concretely and linked to the child’s everyday life. **. He is handed a booklet where he writes down the measures of tables, shelves, his booklet, the corridor of his school, etc…

During these measurements, he notices that a centimeter is small and that it is used to measure small objects while a meter is bigger and used to measure larger things in the classroom… Same thing with weight measures. We introduce a scale in the class and children are asked to weigh themselves. With a smaller scale, they weigh smaller objects. Children are often taken to the fruit and vegetable stand of the local supermarket to weigh things. They therefore understand the notion and necessity of weighing things.

**It is important to link mathematical concepts to a child’s life so that he knows that ** he uses these concepts on everyday basis. During math classes, when he is learning how to write numbers with letters, he is usually thrilled to use a cheque book edited with his own name ; he understands the purpose of writing numbers with letters…

In a Montessori class, **we insist on vocabulary. **We always keep in mind that we are dealing with a number of notions presented to very young children (between the age of 3 and 6) who are in a sensitive period to language. Even for older children, **words are essential**.

We use the word « sum » for the result of an addition, « difference » for that of a substraction, « product » for that of the multiplication and « share » for the division. This way, when resolving a problem, a child will read the question « find the difference… », and his brain will immediately understand that it needs to substract, etc…

It is also important to learn mathematics on an individual basis. The child goes through different sensitive periods during which he will be more attracted to certain acquisitions. During these periods, he will learn certain concepts more easily and durably. I worked with children who showed interest to divisions for several weeks. They would spend entire days resolving them. Then one day, the passion would end and they would go on to new things.

We should then at all times be careful with the rhythm of our teaching. The teacher must keep in mind what the student must learn during the year but the order of acquisition should not matter. Maths should be taught in such a way that they respect the rhythm and evolution of all.

A child needs to **understand and to reason**. For example, when he is told to measure the circumference of the circle by multiplying its diameter by the number « pi » without ever being told what that number correponds to. The child is also told to memorize that pi equals 3,14. But why ? In a Montessori school, he will be told what pi is and why it is equal to 3,14.

In traditional schools, children must do what they are told, accept things because they are taught by an adult and try not to disturb the class by asking questions. How can they learn to like mathematics ? How can they grow up to be adults who think, who reason, who do not accept things as fatality and who keep their faculty of acceptance or refusal, and still have the courage to take initiative ?

A child who will have learned the abstract concepts using his senses, on his own, who will have understood the whys and wherefores by the handling of things : that child will be active during his adult life ! He will know how to think, how to create, how to construct and he will do it by using his brain but also all of his senses.

For the gifted child, the constraint of learning without understanding is even more unbearable. He wants to find the solutions on his own. This is why some gifted children find themselves in total failure in maths because they are taught in a way that everything seems incomprehensible. These children often complicate things even more and just cannot make it. The repetition of notions during primary classes also causes them unbearable boredom..

One of the characteristics of the child gifted in maths is that he understands everything on his own and remembers it forever. It is therefore important to let him learn at his own rhythm and to let him use his own reasoning. This way, at the International Montessori School, children would find themselves four or five years ahead in mathematics without any problem.

These gifted children are often blamed because they find the result without using intermediate steps. This is true but can be overcome by explaining to them that during certain steps, the teacher wants them to reason, to express this reasoning and that the result is not essential. They usually understand this concept even if it is not how their brain would like to work. It is even more important when they are young to let them have fun, to progress and play with mathematics…

For the child with calculus problems, learning maths in an abstract way like in traditional schools, is a great source of suffering. It is impossible for these children to learn maths in an abstract way and they rapidly find themselves in great difficulty.

The manipulation of Montessori material helps them progress, understand a certain number of notions and even if they will never be geniuses in maths, they will not be in difficulty and will be able to progress at their own rhythm, which is essential for these children.

These children generally show a great will and never totally earn the fruit of all their work. It is therefore important to help them obtain a certain success.

For the dyslexic child, life is not easy. All decoding of texts and all subjects using language are a difficulty. It is important to find subjects in which he can excel. Maths can be a solution but at the condition that the child is not left alone infront of his book, infront of instructions, of problem statements which will only be a source of difficulty. Let’s not mix everything ! The important thing is to understand math concepts, to be able to resolve operations, to find the right reasoning, to know how to draw perpendicular lines…

**It is important for the teacher to read the instructions and the statements** to allow the child to focus on reasoning in which he is usually very capable. He can therefore be very good in maths. And how great it is for him to finally have success ! It can even help him with other topics because he will gain a certain self estime ! This is essential for the dyslexic child who tends to think that he is not intelligent which is absolutely not the case!

I will regularly publish articles on my blog on the teaching of mathematics. I started with small children with the teaching of numbering and I will write about relative numbers and the resolution of equations with Montessori type material.

**Do not leave your child in difficulty with mathematics!**

*****

**Lesson on the substraction with changes**

**For example: 53 – 17 =**

I have 5 dozens and 3 unities. I have to take away 7 unities when I have only 3. How can I do that ? I take one dozen out of the 5 because I know that in one dozen, there are 10 unities. I can therefore exchange this dozen with 10 unities and find myself with 13 unities. I can now take away 7 and I am left with 6. Now with the dozens, I now have 4 and have to take away 1, which is possible and I am left with 3. The result is 36.

1^{st} step: 2^{nd} step: Last step:

d u d u ** Remains: 36**

5 3 5 4 **1**3

– 1 7 – 1 7

**2 ^{nd} example : 304 – 186 =**

1^{st} step: 2^{nd} step: 3^{rd} step:

c d u c d u c d u

3 0 4 32 10 4 32 109 14

– 1 8 6 – 1 8 6 – 1 8 6

Last step:

c d u

32 109 14

– 1 8 6

1 1 8 ** Remains: 118**

**Sylvie Rousseau d’Esclaibes**