Montessori believed that you should introduce math to young children as "materialized abstractions." This was accomplished with hands on apparatus for math. Montessori believed that a strong foundation of math at a young age (preschool) prevented children from failing in math later on. "Math phobias" are a common symptom of children who have had little or no introduction to math manipulitives during the sensitive period between 3 and 6 yrs. old.

Here is an interesting part of a lecture from Ms. Child concerning math for preschool children ---

"You will find in the Children's House there is a great deal of material for learning Arithmetic. This is in order that the children may acquire a real familiarity with numbers at an age when they have natural interest in sensorial work. And what they learn through the senses at this age is absorbed into the unconscious, it is known perfectly, it is not uncertain and hard to recall as knowledge acquired unwillingly in a later period tends to be.

The fundamental feature of the system of numbers in our civilization is that we count in tens. We have a decimal system. This should be made apparent from the very first. All the early sensorial material for dimensions, the cubes and stairs are in sets of ten. The long stair [red rod] consists of rods 1,2,3, etc. decimetres long, and the tenth is 10 decimetres, or one metre.

When the child has mastered the arrangement of these rods he has a sensorial conscious. First the number rods are given. These rods are identical with the long stair [red rods], except that the different sections of one decimetre are coloured red and blue alternately to draw attention to the significance of these gradually sizes. Now the arrangement of the rods is very easy with these guiding marks, so that the next step can be taken.

The names of the numbers are taught in connection with the appropriate rods. Then the names are taught in connection with the sandpaper figures so that finally the rods can be laid out and the figures placed on them.

Other simple counting exercises are also done with the spindle box, and the loose counters and number cards.

All this work perfects the knowledge of the nature of number and the numbers up to ten, but the important thing to notice is that the introductions given with number rods where the units are fixed, so that there is no possibility of confusion. As the units are fixed the child can work them and absorb them into the unconscious, in a way that would not be possible with the moveable counters. With these it is possible to teach the names of the numbers before the child has grasped the abstract conception.

Until a thorough knowledge of the units is achieved the material is limited to ten. In none of these first three exercises is it possible to go beyond ten. This is an important point to avoid confusion.

But once the units are mastered, the next step is the whole decimal system. The method that is sometimes followed of studying next the numbers from 10 to 100 is very dull indeed and any interest there may have been is exhausted long before any conception of large numbers can possibly be arrived at.

In fact may people seem terrified at a sum that includes numbers with commas. The idea seems to be "How terribly I suffered with those sums of two or three figures. Certainly so many figures would be such torture I could never survive." But this is quite a mistake, as the size of the number makes very little difference to the difficulty of the sum. The same nine figures are employed in all the hierarchies so that operating with millions cannot be very different from operating with units.

This simple fact often strikes even adults with surprise, and once it is realized it can help a good deal to give courage. But it should never be necessary to point this out to anyone who ever properly master arithmetic.

In fact children seem to be fascinated by thousands and millions and prefer working with large figures if they are introduced early.

In the Montessori method the knowledge of numbers and counting is given with the bead material. The units are single gold beads, the tens are bars of ten gold beads, the hundred, of course, are squares made up of ten 10-based bars, and the thousands cubes made up of ten hundred-squares. There are also hundred chains and thousand chains made up of tens linked together. The child naturally learns the names of the most conspicuous objects first, the hundreds and the thousands, then with the aid of number cards he learns how these figures are written. In all this he never learns a name that has not got a meaning to him, he associates it with a certain quantity of beads.

This may be criticized as clumsy or materialistic but for most people it is necessary step in the achievement of the understanding of numbers.

However late the child may start arithmetic in the ordinary school, he cannot get a clear idea of number from abstraction alone; it is better to give the material to the young child who can work with it patiently and with interest for many years before it is necessary for him to attempt to work sums on paper.

……… And this real value of the material rests more in the difference it makes to the ordinary child.

…..Children are not really interested in blocks and beads for long without the real intellectual interest of associating them with sizes and numbers. For when they have achieved the manual dexterity to build them up securely they want to go on to some new difficulty. And this is the time to give the numbers. We greatly underrate the intelligence of the four or five year old if we think that this learning is too much. If we do this we are also likely to make the opposite mistake later and expect the child of five and half or six to grasp the numbers in a few short lesson with counters. The units may seem obvious to us after years of practice, but they are not so to the child.

If his initiation is hasty and unprepared, he will learn the names off by rote and it may be years before he gets a real conception of the different numbers and their relations to each other. The names tend to be associated with a figure only, and his arithmetic is a succession of verbal formulae connected extremely precariously by counting on the fingers.

Seeing we have a decimal system it seems possible that there is a good precedent for this in pre-history but, whether this is so or not, we cannot avoid the conclusion that, as a matter of fact and experience in conditions today, counting on the fingers is not a success. Counting with the fingers, definably yes, but on the fingers, no.

Now all this finger counting, that lets us down so often and causes so many disappointments when we find that the laboriously achieved answer is wrong, is only necessary for the combinations of the nine units-adding and subtracting the numbers from one to nine. If we know these combinations and know them correctly we need not interrupt our sums to count on the fingers.

If we start with a real conception of the numbers, gained from the use of number rods, that is a great help. We learn from this which pairs of numbers make ten. Then later we can do all kinds of exercises with the small bead bars and the addition board that will teach us may more little additions and subtractions, until finally all the combinations of the nine units are perfectly familiar, as familiar as the linear counting itself, so that it becomes easier and more natural to say at once sixteen and three are nineteen because we have a mental conceptions of six and three, than to embark on the precarious operation of adding 3 (or four!) live fingers to a misty conception of the numerical sixteen in the imagination.

So life becomes easier and happier for the child and the Primary school teacher. They are spared the fog of misunderstanding, the slough of despair in which they often stock when the child is seven. This is not necessary if the foundations are laid by work with sensorial material in the Nursery school."

It is important to introduce the fundamentals of math to preschool children while it is so easy for them to absorb

concrete information. This mathematical foundation given to preschoolers will have a life long effect on their learning ability.

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