Introduction to Montessori Mathematics

This is the introduction I wrote for my Mathematics album for a Montessori preschool diploma course in 2010.  It is not properly referenced, although I have included a bibliography, so be warned if you want to refer to any of it for a more formal application.

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Introduction to Mathematics

Mathematical and number concepts are an essential part of everyday life. Even before starting school, young children are exposed to numbers and mathematical concepts daily (e.g. sorting, counting, estimating quantity, measuring), through television, through books, when shopping, and through many different kinds of infant toys as well as in our daily activities. For example, setting the table or putting on shoes develops the idea of one-to-one correspondence, because there must be a place for each person, or a shoe for each foot. Packing away clothes or shopping is an exercise in classification and sorting, and helping with baking gives the young child experience in measurement and estimation as well as the experience of working with volumes or mass in different types of media (e.g. flour and oil). Counting is often introduced directly in educational television programs, books for young children or number rhymes and games.

The study of mathematics, however, is more than simply the acquisition of mathematical skills. Pure mathematics is the study of patterns, structure and relationships. It is the ultimate abstraction, where the rules governing interactions are the object of study. The “things” on which the rules operate become immaterial. Maria Montessori said that it is the the mind’s power of abstraction, that, with imagination, goes beyond the simple perception of things, so that the two powers “play a mutual part in the construction of the mind’s content.” (M Montessori, The Absorbent Mind)

Maria Montessori referred to the part of the mind that deals with order and abstraction and is precisely and logically constructed as “the mathematical mind”. This part of the mind is vitally important, as it allows the child to order and thus understand his world. Because of the ordered and logical nature of mathematics, the study of mathematics provides intellectual training for precise and rigorous thought, as well as helping to develop the mathematical mind.

Before the child can construct number concepts in his mind, other concepts must be well internalised and understood. Three important principles that are essential for mathematical understanding are:

Conservation: This implies understanding that two equal things remain equal even if their appearance is altered, or their spatial arrangement changed, as long as nothing is added or taken away. This principle applies to number, length, liquid, mass or substance, weight, area and volume. The understanding of different conservation principles is achieved at different times in a child’s life, with number conservation, essential for the understanding of early mathematics, developing first, and the understanding of volume conservation usually developing last.

Reversibility: This involves understanding that an action or operation can be undone, or done in the opposite direction so as to relate back to the original situation e.g. when water is poured from a long, thin glass into a short, thick glass, it can be poured back into the original glass and then the original situation is restored. The understanding of conservation and reversibility is linked – a child who cannot imagine a situation being reversed cannot understand conservation, and a child who does not understand conservation will not be able to relate a changed situation back to the original one.

One to One Correspondence: This describes the process of pairing each member of one set to each member of another set and is an essential part of understanding number. Children need to understand that each number word must be said for one item only, and that each item must be paired with a number word. Children develop an understanding of this principle through repeated practise. They need lots of opportunities to touch or move things while counting them (e.g. plates at mealtimes, boys or girls at school) in order to prevent them from merely reciting the number words in a meaningless way. Understanding one-to-one correspondence allows the child to consider the relationships of “more than”, “less than” and “as many as” without needing to count the objects under consideration. (These are also known as ordinal relations. Ordinality refers to the relationship or order of numbers to each other, including positional relationships like 1^st, 2^nd etc.)

Other important fundamental mathematical concepts include:

Seriation and Transitivity: These concepts refer to grading (by size, colour etc), and to generalizing the idea of grading objects.

Cardinality: The cardinal numbers are the ordinary numbers (1,2,3,4,5), and the principle of cardinality is the understanding that the last number used is the total number of counted items i.e. when counting the 5 number rod, you say all five number words, and the last word you say is five, which is then the total number of lengths in the rod, and the ‘name’ of that rod.

Indirect preparation for Mathematics

The activities and materials of the Practical Life area, as well as the child’s training and development through the Sensorial Materials, give the child a concrete introduction to fundamental mathematical concepts as well as to the logical reasoning underlying them. In this way, as well as through his exposure in everyday life, the child is indirectly prepared for learning the mathematical concepts before these are presented to him through the mathematics equipment and activities.

Practical life preparation for Mathematics:

The exercises of Practical Life help the development of concentration and a sense of competence. These are essential for mathematics work as the child needs to have both the motivation and the ability to concentrate for long periods in order to complete the Maths activities. The presentation of the Practical life activities in an ordered and logical sequence e.g. the precise sequence of steps involved in polishing brass or shoe cleaning, encourages the development of mathematical thinking. All practical life materials are used in a set sequence, and the child must be precise and thorough in checking at the end of an activity to ensure that it is complete for the next child to use. The following table gives examples of how certain activities prepare the child for Mathematics.

Mathematical preparation	Practical Life Activities
Conservation, Reversibility	Pouring liquids / solids between different or the same size containers Spooning between different size containers
One to one correspondence	Matching/ pairing lids to jars or boxes Setting a table Planting seeds, one per pot
Volume, mass, measurement	Cooking, weighing out ingredients
Pattern, symmetry	Paper cutting
Geometry, fractions (½, ¼)	Napkin folding

In addition to the standard classroom activities, the child’s experience will be broadened by any activity that involves matching or pairing, such as games like dominos or snap. Games such as these and other can encourage the child to think logically.

Sensorial Materials as a preparation for Mathematics:

The Sensorial materials in the Montessori classroom provide the child with concrete sensory impressions of the basic mathematical concepts he will encounter later. This satisfies the basic principle contained in the ancient expression: “There is nothing in the intellect which was not first in some way in the senses.” (quoted in The Secret of Childhood). The sensorial material can be understood as “a system of materialized abstractions, or of basic mathematics.” (M Montessori,The Absorbent Mind), The exercises help the child to form a logical mathematical framework for future use and as such are a prerequisite for deep, positive mathematical understanding.

The sensorial materials direct the child’s attention to differences or similarities and to sequence, giving practise in classification, and seriation. The child learns about different properties such as colour, shape, texture, sound, size, temperature and weight. Indirect preparation for the decimal system is given by the dimension materials. These are all in sets of 10, using units of 1-10 in several different dimensions. Through comparing and classifying objects, as well as matching them together, the child is able to have concrete experience in discriminating between sizes, in sequencing, grading and making comparisons. Children get visual and muscular impressions of plane (flat) and 3-dimensional shapes with the geometric cabinet and the geometric solids respectively. Together with the Binomial and Trinomial cubes, these allow the child to have an early sensorial experience of materials that have a foundation in geometry and algebra. Children’s mathematical vocabulary is enriched as they learn the correct shape names, as well as the descriptive terms of measurement like narrow, long, short, wide and their comparatives (longer, bigger than) and superlatives (longest, biggest).
Maria Montessori said “As cement is to brick so is the sensorial apparatus to mathematics.” This describes the vital importance of the sensory impressions in fixing the mathematical concepts in the child’s mind. Without these impressions, the mathematical concepts encountered are not firmly established, and the more advanced concepts, that require deep understanding of the basic ideas, will not be successfully understood.

How is Mathematics learned in the Montessori classroom?

Maria Montessori’s first intellectual love was mathematics. As a young teenager, she studied mathematics at an all-boy technical school, in preparation for studying engineering. In order to facilitate the learning of mathematical and number concepts, she designed a system of precise, logically ordered, didactic equipment that is used in the preschool mathematics area. Together with the sensorial equipment, these beautiful materials form one of the more striking physical features of the Montessori preschool classroom.

The didactic material and the associated activities provide the child with a concrete and dynamic impression of the mathematical concepts presented. As with all the Montessori activities, movement is an integral part of learning. The child uses his hands and body while working with the materials. He combines, shares, counts and compares. He learns by doing, and by self-discovery. All materials contain a built-in control of error that guides the child towards doing the activity correctly, and allows the child to work independently of adult guidance, leading to true self-discovery and self-education. Through the use and manipulation of these concrete materials, the young child absorbs a visual and muscular impression of quantity and order. He discovers rules and patterns as he gradually progresses from concrete objects to the symbolic representation of number.

The mathematical exercises are presented in a logical sequence and systematic progress is made from the concrete to the abstract. The sequence of the presentations is respected and the directress does not allow herself or the child to be rushed through the exercises. The child takes one small step at a time, building on his prior knowledge. To proceed from one concept to the next, he needs to have fully established, understood and internalised the first. This allows him to develop a clear, positive understanding of numbers.

Concepts are always introduced in a concrete form, followed by the abstract symbolism. No abstractions are presented until the child has obtained a concrete impression of the concept that is being represented. He never learns a name that doesn’t have a meaning. e.g. the number cards are only introduced after the child has worked with the number rods. The early maths exercises and equipment (e.g. number rods, numerals, spindle box, cards and counters and number games) help the child to grasp the concept of quantity and the symbols 1 to 10. Once the units are understood and mastered, the child is introduced to the whole decimal system through the golden bead equipment. As the child progresses through the equipment, he works simultaneously with very large numbers while practising and honing his skills with addition, subtraction, multiplication and division of small numbers. By about 7 or 8 years old, the child will be able to work out operations in his head with true understanding and confidence.

In his biography of Maria Montessori, E.M. Standing speaks about Mathematics as a process of abstraction, and describes this process as follows:

“This process of abstraction is by its very nature an individual one, no one can do it for another, however much he may wish to do so. Abstraction is an inner illumination, and if the light does not come from within, then it does not come at all. All we can do, is to help the child by presenting them with external concrete materials. In these materials, the abstract idea of mathematical operation which we wish to teach is, as it were, latent. The child works with them for a good while, and as he does so- his mind rises eventually to a higher level.”

As explained by E.M. Standing, the process of abstraction cannot be forced, or rushed. It depends on the child’s environmental experience, as well as on the inner development of the child’s mind and mental abilities. The directress can only offer opportunities for environmental experience with the concrete materials. Each child then comes to his own understanding of the mathematical concepts in his own time.

Maria Montessori has been criticised for introducing Mathematics too early. Yet the material is only presented to a child who is observed to be ready for it. The correct and systematic introduction of Mathematical principles and concepts during the child’s absorbent mind period allows these to be naturally and effortlessly understood and internalised, avoiding the commonly found lifelong repugnance which many feel for this beautiful, elegant and fundamental subject.

Bibliography:

The Secret of Childhood, Maria Montessori

The Absorbent Mind, Maria Montessori

Maria Montessori: Her Life and Work, EM Standing accessed online at http://www.archive.org/details/mariamontessorih000209mbp

http://www.lms.ac.uk/policy/tackling/node5.html

“A Child’s World Infancy Through Adolescence”, Papalia, Olds, Feldman (10th Ed)