Mathematics

"In general, like everybody else, I assumed that arithmetic was very difficult and that it would be foolish to expect more than what had been attained at so early an age."
The Discovery of the Child p 276, Chap 19

Early on in her work Montessori realised that what young children needed was to be given very precise information about the world in which they were living.

She saw that they had a natural sense of order and an innate tendency towards categorisation. In fact she felt that the term the 'mathematical mind' was particularly appropriate.

She therefore developed the sensorial materials to satisfy this need and then went on to look at number operations. She developed existing materials such as the long rods and introduced new ones, such as the spindles and cards and counters, that continued to give the child a concrete representation of number.

At first she assumed that these exercises would provide sufficient material for such young children to work with, but one day she found that a group of four-year-olds had taken the new golden bead apparatus that she had designed for older children in elementary school, and that they were copying what they had seen the older children doing. They were not only able to complete the exercises, but showed great interest and excitement at what they had discovered.

She then realised that the children were capable of much more complex thinking than had previously been expected, but that success relied upon their being provided with concrete materials that only then led to abstract thinking.

"This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child's special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculation."
Ibid p 279, Chap 19

Quotations

"In our work, therefore, we have given a name to this part of the mind which is built up with exactitude, and we call it 'the mathematical mind'. I take this term from Pascal, the French philosopher, physicist and mathematician, who said that man's mind was mathematical by nature, and that knowledge and progress came from accurate observation."
The Absorbent Mind p 169, Chap 17

"Just as the form of a language is given by its alphabetical sounds and by the rules for arranging its words, so the form of man's mind, the warp into which can be woven all the riches of perception and imagination, is fundamentally a matter of order."
Ibid p 169, Chap 17

"In our tiny children the evidence of a mathematical bent shows itself in many striking and spontaneous ways."
Ibid p 169, Chap 17

"Order and precision, we found, were the keys to spontaneous work in the school."
Ibid p 169, Chap 17

"The results we obtain with our little ones contrasts oddly with the fact that mathematics is so often held to be a scourge rather than a pleasure in school programmes. Most people have developed 'mental barriers' against it. Yet all is easy if only its roots can be implanted in the absorbent mind."
Ibid p 169, Chap 17

"The first material used for counting is the series of ten rods already used in the education of the senses for teaching lengths."
The Discovery of the Child, 263, Chap 18

"The fact that a group is enlarged through the addition of a new unit and that this increasing whole must be considered constitutes the chief obstacle for children of three-and-a-half to four in learning how to count."
Ibid p 263, Chap 18

"The rods correspond to numbers and gradually increase in length unit by unit. They therefore provide not only an absolute but also a relative concept of numbers. Their proportions have already been studied in the sense exercises. Here they are judged mathematically, and this provides an invitation to arithmetic."
Ibid p 264, Chap 18

"Although the rods are the principal help given to a child so that he can begin arithmetic, two other objects are also used. One of these leads to the counting of separate units and gives a child a concept of numerical groups... This material, which we have called the tray of spindles, has compartments, each marked with one of the ten figures placed in sequence... The other material ... consists in a group of cards in a box containing different objects (coloured markers) ...This is all the material that we have deemed to be necessary for laying the foundations of counting and arithmetical operations."
Ibid p 265, Chap 18

"It seemed important to us that the children should be able to count up to one hundred and to carry out the exercises connected with this operation, which unites a rational study of the primary numbers with simple reckoning... For more than twenty years we limited our teachings to these exercises."
Ibid p 276, Chap 19

"In the meantime I had prepared for older children in elementary schools... material which represented numbers through geometrical forms and moveable objects which enabled a child to make various combinations of numbers. We have called this splendid material 'the material of the beads'... Now it happened that some children about four years old were attracted by these brilliant objects, so easy to carry and handle; and, to our great surprise, they began to use them as they had seen the older children doing... The result of this was such a great increase in enthusiasm for work with numbers, and especially with the decimal system, that arithmetic actually became one of their favourite exercises."
Ibid p 277, Chap 19

"The obvious pleasure which the children had in these exercises and their ability to handle the small geometrical solids prompted me to make objects resembling Froebel's famous 'gifts' of cubes and prisms arranged into a square box."
Ibid p 277, Chap 19

"All this teaching of arithmetic and of the principles of algebra by means of cards which assist the memory and by other materials produces results which might seem to be fantastic. They indicate that the teaching of arithmetic should be completely transformed. It should start with sense perceptions and be based on a knowledge of concrete objects."
Ibid p 278, Chap 19

Study guide

The Absorbent Mind - Chapter 17

The Discovery of the Child - Chapters 18, 19

Journal articles

Baker, K (1996) 'The Mathematical Intelligence seen through the lens of the Montessori Theory of Human Tendencies', NAMTA Journa,l v21, n2, p98-107

Bausch J, Hsu J (1988) 'Montessori: Right or Wrong about Number Concepts?', Arithmetic Teacher, v35, n6, p8-11, February

Chattin-McNichols, J (1980) 'Algebra in the Montessori Elementary Classroom', AMS Constructive Triangle VII, 5

Hellenberg, H (1995) 'Mathematics in History', Montessori Life, v7, n2, p30-32, Spring

Luckenbill, L (1995) 'Biological Superiority in Math: Calvin or Susie? Spotlight:Gender differences', Montessori Life, v7, n4, p28-32, Fall

McGhee, J (1995) 'Mathematics and Language Experience', Montessori Life, v7, n2, p34-36, Spring

McNamara, J (1994) 'Montessori Mathematics: A Model Curriculum for the Twenty-First Century', NAMTA Journal, v19, n1, p3-9, Winter

Scott, J (1995) 'The Development of the mathematical Mind', Montessori Life, v7, n2, p25 Spring

Sirgo, L (1995) 'New Life for Early Childhood Math', Montessori Life, v7, n2, p34-36, Spring

Thompson, D (1995) 'Preschool Math and the Didactic Materials', Montessori Life, v7, n2, p20-21, Spring

Turner, J (1995) 'Math Education and Piaget's Theory: A conversation with Constance Kamii', Montessori Life, v7, n2, p26-28 Spring

Woessner, R (1995) 'Mathematics: Montessori or Traditional?', Montessori Life, v7, n2, p40, Spring

Zener, R (1994) 'Extensions in the Mathematics Area of the Children's House', NAMTA Journal, v19, n1, p11-23, Winter

Archive resources

Boyd, W (1917) From Locke to Montessori, George Harrap & Co London.

Culverwell, E (1913) The Montessori Principles and Practice, G.Bell & Sons, London.

Kilpatrick, W (1915) Montessori Examined, Constable, London.