Our kids are typically introduced to numbers at age 3: learning the numbers and number symbols one to ten: the red and blue rods, sandpaper numerals, association of number rods and numerals, spindle boxes, cards and counters, counting, sight recognition, concept of odd and even, and the famous Montessori Golden Bead materials. Furthermore, children sees and feels quantities and numbers and their relationships. The child’s “hands-on” work with the didactic manipulative materials brings the “abstraction” of mathematics to the “concrete” laying a firm foundation of real learning. Children gain a lifelong love of advanced work in arithmetic, geometry, algebra, trigonometry, and calculus.
Introduction to the decimal system typically begins at age 3 or 4. Units, tens, hundreds, thousands are represented by specially prepared concrete learning materials that show the decimal hierarchy in three dimensional form: units = single beads, tens = a bar of 10 units, hundreds = 10 ten bars fastened together into a square, thousands = a cube ten units long ten units wide and ten units high. The children learn to first recognize the quantities, then to form numbers with the bead or cube materials through 9,999 and to read them back, to read and write numerals up to 9,999, and to exchange equivalent quantities of units for tens, tens for hundreds, etc.
Linear Counting: learning the number facts to ten (what numbers make ten, basic addition up to ten); learning the teens (11 = one ten + one unit), counting by tens (34 = three tens + four units) to one hundred.
Development of the concept of the four basic mathematical operations: addition, subtraction, division, and multiplication through work with the Montessori Golden Bead Material. The child builds numbers with the bead material and performs mathematical operations concretely. (This process normally begins by age 4 and extends over the next two or three years.) Work with this material over a long period is critical to the full understanding of abstract mathematics for all but a few exceptional children. This process tends to develop in the child a much deeper understanding of mathematics.
Development of the concept of “dynamic” addition and subtraction through the manipulation of the concrete math materials. (Addition and subtraction where exchanging and regrouping of numbers is necessary.)
Memorization of the basic math facts: adding and subtracting numbers under 10 without the aid of the concrete materials. (Typically begins at age 5 and is normally completed by age 7.)
Development of further abstract understanding of addition, subtraction, division, and multiplication with large numbers through the Stamp Game (a manipulative system that represents the decimal system as color-keyed “stamps”) and the Small and Large Bead Frames (color-coded abacuses).
Skip counting with the chains of the squares of the numbers from zero to ten: i.e., counting to 25 by 5’s, to 36 by 6’s, etc. (Age 5-6) Developing first understanding of the concept of the “square” of a number. Skip counting with the chains of the cubes of the numbers zero to ten: i.e., counting to 1,000 by ones or tens. Developing the first understanding of the concept of a “cube” of a number.
Beginning the “passage to abstraction,” the child begins to solve problems with paper and pencil while working with the concrete materials. Eventually, the materials are no longer needed.
Development of the concept of long multiplication and division through concrete work with the bead and cube materials. (The child is typically 6 or younger, and cannot yet do such problems on paper without the concrete materials. The objective is to develop the concept first.)