In our Montessori classrooms, children begin dealing with numbers early and in multiple ways – (counting snack items, getting four colored pencils for the “inset work,” etc.) and soon advance to manipulating things in terms of ten (ten cylinders, ten color tablets, ten pink cubes) in preparation for learning the decimal system. They then build on this experience by using the Montessori math materials, which can take them all the way from basic counting to the four operations of arithmetic into the thousands – all by the end of their 3rd preschool year (the “Kindergarten year”.)
Counting – Number Rods and related materials. Dr. Montessori recognized that a very young child (age 3 – 4) often has trouble understanding how to count by adding one unit to another. She found that children struggle to realize that the increasing whole must be considered: often, a child would count “1, 1, 1” instead of “1, 2, 3.” Therefore, a child in a Montessori classroom is introduced to numbers with the Number Rods, a series of 10 which increase in units from 1 to 10 – each number being represented by one rod, with the units marked in alternating red and blue stripes.
The child can thus handle the Three – the rod with three alternating stripes, or the Ten, as one item. The child sees “Three” or “Ten” as a set, a unit of three or ten things, that can be broken down into three or ten individual things, and hence he acquires the basic idea of a number. The teacher helps the child count the alternating red and blue sections of each rod as he arranges them in an attractive stair formation. Later, when he has learned to write the numbers using Sandpaper Numerals, he puts number cards next to the rods to illustrate that quantity. He can also discover many mathematical facts – for instance, many children will place the One rod on top of the Nine rod, to make Ten, the Two rod on top of the Eight rod and so on, thereby demonstrating to themselves the different ways in which two smaller numbers can add up to Ten.
Once the child has mastered counting to ten, he can practice counting using other materials. For instance, he can use the Spindle box to count out 45 units into compartments labeled 0 – 9; he can use the Numerals and Counters, which introduce the concepts of odd and even; he can use the Seguin Boards, which represent first the “teen” numbers and then enable the child to construct numbers from 11 to 99.
Arithmetical Operations: Bead Stairs. The Bead Stairs are akin to the Number Rods. Various quantities of beads are strung on wire to represent the whole numbers through 10. The bars are made of different color beads, according to the numerical value of the bar: The 10-bead bars are orange; the 9-beads are dark blue; the 8-beads are brown, and so on; the bead bars can be arranged by their numeric value to form an attractive bead stair. Five- to six-year-olds enjoy manipulating these materials, and fill page upon page with math work using them. By working with color-coded chains, they quickly learn to move from counting to abstract calculations. In Dr. Montessori’s words:
Little by little the child ceases counting the beads and recognizes the numbers by their color: the dark blue he knows is 9, the yellow 4, etc. Almost without realizing it he comes now to count by colors instead of by quantities of beads, and thus performs actual operations in mental arithmetic. As soon as the child becomes conscious of this power, he joyfully announces his transition to the higher plane, exclaiming “I can count in my head and I can do it more quickly!”
Place Value & Large Quantities: Golden Bead Materials. The Golden Bead Materials introduce our student to the concept of place value – i.e., the Decimal System. This material includes individual beads or single “units”, strings of ten beads or “ten bars”, ten ten-bars combined into a “hundred square” and ten hundred squares combined into a “thousand cube”. Number cards go along with these beads – with Units printed in green, Tens in blue, Hundreds in red, and Thousands in green. With these cards and beads, children build large numbers: the teacher may, for example, make a number with the cards, such as 3,572 – and ask the child to bring her the corresponding number of beads – that is, 2 Units, 7 Ten Bars, 5 Hundred Squares and 3 Thousand Cubes.
Children continue to work with these and related materials for a long time – learning how to add, subtract, multiply and divide. They also obtain a perceptually grounded awareness of quantity: a related material, the thousand chain, consists of ten hundred-chains strung together – and measures twenty feet in length.
Further mathematical materials. As our groups are multi-age, and as children progress at different speeds, especially in mathematics, we offer a wide variety of additional mathematical materials in each classroom. Children learn about fractions with the fraction circles and skittles; they learn skip counting and multiplication with colored bead materials which expand upon the colored bead stair; they learn division with a Division Board, where division is introduced as sharing quantities between people, represented by skittles; they practice addition and subtraction with Strip Boards; they arrange the numbers to 100 in order using the Hundred Board. (Geometry, an important part of the Montessori primary classroom, is included among the Sensorial Exercises.)
All of our primary classrooms contain materials appropriate through first grade – for example, the bead frames – and our teachers are trained in their use (in contrast to some other Montessori preschools, where students generally leave before the kindergarten year, and where teachers are often unfamiliar with the more advanced math materials and therefore not able to help a mathematically gifted child advance as far as he could.)