Mathematics and Teacher Training

Mathematics in Montessori’s method follows the same pattern as the rest of her work: concrete materials before abstract concepts, the child engaging directly with objects rather than with symbols on paper. The article works through the specific materials she developed for early mathematics and the global teacher-training programme that made the method spread.

Mathematics with concrete materials

Montessori placed great emphasis on mathematics within her curriculum, but mathematics in her hands does not look like the mathematics of conventional schools. The starting observation: typical children very easily learn numeration, which consists of counting objects. The counting is concrete, not abstract; the child counts real things they can see and touch, not symbols on paper.

This insight reshapes the early mathematics curriculum. Conventional schools introduce mathematics through worksheets, addition tables, and abstract written numerals. The child is asked to manipulate symbols before they have a clear grasp of what the symbols mean. Montessori inverts the sequence: concrete objects first, abstract symbols second.

She focused on developing materials specifically for the introduction of counting and number operations. Her materials are now widespread in early-childhood classrooms even beyond Montessori-branded schools. Each material has a specific role.

Number rods. Rods painted in alternating red and blue divisions of 10 cm each, in lengths from 1 (single 10 cm rod) to 10 (rod of 10 segments). The child handles the rods, compares their lengths, and develops a concrete grasp of the magnitudes 1 through 10 before any written numerals appear. The visual and physical contrast lets the child see that 5 is bigger than 3 and smaller than 7 without having to manipulate symbolic numerals.

Spindle box. A box with ten compartments numbered 0 through 9. The child places the correct number of spindles (small wooden dowels) in each compartment. The compartment labelled 5 holds 5 spindles; the compartment labelled 0 holds no spindles, demonstrating the concept of zero in concrete form. The exercise links the numeral to the actual quantity.

Small cubes and number cards. Small wooden cubes paired with cards marked 1 to 10, set out in orderly arrangement. The child practices matching cards to quantities and learning the conventional symbol-to-number associations.

Sandpaper numerals. The same principle as the sandpaper letters from the previous chapter, applied to numerals. The child traces the numerals with a finger, getting tactile and kinaesthetic input alongside the visual input. The numerals are introduced in preparation for written work and a brief explanation of how to use the number rods.

Mathematics in daily life offers many opportunities to work with numbers in practical ways. The directress uses these opportunities for addition, subtraction, multiplication, division, and other mathematical operations. The mathematics is not confined to a separate subject hour; it is woven through the whole day. A child counting plates for lunch, measuring ingredients for cooking, or distributing materials to the group is doing mathematics in a way that is connected to real practical purpose.

Montessori also introduced cardboard substitutes of money so children could learn to make change. Practical money handling, again with concrete materials, prepared children for the economic competence they would need later in life.

Flashcard

What materials did Montessori develop for introducing children to numbers 0 through 10?

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Answer

Number rods, spindle box, small cubes and number cards, and sandpaper numerals

Number rods: red and blue divisions of 10 cm, from 1 to 10, for concrete magnitude. Spindle box: ten compartments 0-9, child places correct number of spindles in each. Small cubes and number cards: matching quantities to symbols. Sandpaper numerals: traced with the finger for tactile and kinaesthetic input. The materials provide concrete objects before abstract symbols. Daily life offers further mathematics opportunities (counting plates, measuring ingredients) that the directress uses for operations across the day.

The need for teacher training

The success of the first Casa dei Bambini created an immediate problem. Montessori was able to repeat the successful results in a second Casa by providing exactly the same learning conditions. She quickly realised that she could not be in every Casa personally; the method needed teachers who could be trained to provide the same conditions Montessori herself created.

The training requirement was substantial. Teachers needed to be trained as directresses rather than as traditional instructors. The shift in role required more than a change in title. It required a different way of thinking about teaching, a different relationship to the children, a different daily practice. A traditionally trained teacher could not simply be relabelled as a directress and continue doing what they had been doing; the work itself had to change.

The first Montessori teacher training course was planned in 1909 for the fifth Casa, established in a prior Italian-Swiss orphanage. About 100 potential Montessori directresses attended the course. The number is striking: there was a real population of educators interested in learning the method, almost immediately after the method was published.

The training course content covered Montessori’s full educational philosophy. Lectures included her overall framework, the conception of teachers as observers and researchers of free children, the idea of the classroom as a prepared environment functioning as a laboratory, the principles of scientific pedagogy with individual children as case studies, demonstrations of how children taught themselves (with local children used as the demonstrating cases), spontaneous self-development of individuals, the establishment of scientific child psychology, and demonstrations by Montessori herself with local children, including the famous Three Period Lessons.

The combination of theory and demonstration was distinctive. A pure-theory course would have left the trainees with abstract principles they could not apply. A pure-demonstration course would have left them able to mimic specific lessons without understanding why those lessons worked. The Montessori courses combined both: theory grounded the practice, and the practice illustrated and refined the theory.

Flashcard

What made Montessori's teacher training distinctive?

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Answer

Combination of theory and live demonstration: the principles were grounded in observed practice with real children

A pure-theory course would have left trainees with abstract principles they could not apply. A pure-demonstration course would have left them able to mimic specific lessons without understanding why they worked. The Montessori courses combined both. Lectures covered the philosophical framework, scientific pedagogy, and the classroom as a laboratory. Demonstrations with local children showed the principles in action, including the Three Period Lessons. About 100 directresses attended the first course in 1909.

Global teacher training

The success of the early Montessori schools created a global demand for teacher training that Montessori personally tried to meet. Reports of countries opening Montessori schools came in from across the world, and everywhere there was a desperate need for trained teachers. Montessori travelled extensively, giving lectures, speaking at conferences, writing articles and books, while conducting the teacher training courses herself all over the globe. The pace was punishing, but she was committed to maintaining the quality of the training that the method required.

Switzerland was among the very first nations to manifest interest in the Montessori method, largely through the endeavours of Teresina Bontempi and the University of Geneva. The University of Geneva invited Montessori to demonstrate her method in person with a group of young children, in the presence of its teaching staff. The demonstration was meant to show the method working in real time, with real children, in front of academic observers who could assess what they were seeing. The format became standard for many of Montessori’s later demonstrations.

In Rome, Montessori planned two teacher training courses in 1910, giving up everything else so she could do all the training herself. The reason was practical. One of the greatest difficulties she faced was how to encourage new teachers to put her principles into practice, to follow her experiment, and to procure the same results that the original Casa had achieved. She found that other people teaching her method tended to drift away from the original principles in subtle ways that compromised the results. Doing the training personally let her maintain fidelity to the method.

The methods Montessori developed for training teachers globally include three specific features. Presentation of materials: each training course gave full explanation of the didactic materials and their demonstration. The trainees needed to handle the materials themselves, in the way they would later present them to children, so that the demonstration was correct in every detail.

The three-period lesson: a specific teaching procedure used to introduce sensorial materials. The classic example uses the rough and smooth boards or the colour tablets. The three periods are: naming (the directress names the quality, such as rough or smooth), recognition (the child finds the named quality among examples), and recall (the child names the quality when presented with examples). The procedure trains the child’s sensory discrimination and the related vocabulary together.

Characteristics of the three-period lesson: the lesson is brief (the teacher uses few words), simple (the teacher speaks only the truth without elaboration), and objective (the child concentrates on the materials being demonstrated rather than on the teacher’s personality). The three characteristics produce a teaching style very different from the verbose, ornate teaching of conventional schools. The brevity, simplicity, and objectivity together make the three-period lesson distinctively Montessorian.

Flashcard

What is the *three-period lesson*, and what are its three characteristics?

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Answer

A specific teaching procedure for sensorial materials, with naming, recognition, and recall as the three periods; brevity, simplicity, and objectivity as its characteristics

The classic example uses the rough and smooth boards or the colour tablets. Period 1, naming: the directress names the quality (rough or smooth). Period 2, recognition: the child finds the named quality among examples. Period 3, recall: the child names the quality when shown examples. The lesson is brief (few words), simple (only the truth), and objective (child focuses on the materials, not the teacher’s personality). The combination produces a teaching style very different from the verbose ornate teaching of conventional schools.