# Mathematics and Geometry

## Mathematics and Geometry Ages 2.9 -12

As an example, consider the very basis of mathematics: the decimal system-units, tens, hundreds, and thousands. Since quantities larger than twenty rarely have any meaning to a young child, Dr.Montessori reasoned that we should present this abstract concept graphically. Children cannot normally conceive of a hundred, thousand, or million, much less the idea that a thousand is equal to ten hundreds or one hundred tens.

Dr.Montessori overcame this obstacle by developing a concrete representation of the decimal system. Units are represented by single one-center meter beads; a unit of ten is made up of ten beads strung together; hundreds are squares made up of ten-bars; and thousands are cubes made up of ten hundred-squares. Together, they form a visually and intellectually impressive tool for learning. Great numbers can be formed by very young children: “ Please bring me three thousands, five hundreds, six tens and one unit.”

From this foundation, all of the operations in mathematics, such as the addition of quantities into the thousands, become clear and concrete, allowing the children to internalize a clear image of how the process works. We follow the same principle in introducing plane and solid geometry to very young students, using geometric insets and three-dimensional models which they learn to identify and define. Five-year-olds can commonly name geometric forms adults wouldn’t recognize.

Montessori mathematics climbs in sophistication through the higher levels. It includes a careful study of the practical application of mathematics in everyday life, such as measurement, handling finances, and gathering data.

Elementary students continue to apply math in a wide range of projects and challenges. They prepare scale drawings, calculate area and volume.

Our students are typically introduced to numbers at age 2.9: learning the numbers symbols one to ten: the red and blue rods,sand-paper numerals, association of numbers rods and numerals, association of number rods and numerals, spindle boxes, cards and counters, counting, sight recognition, concept of odd and even.

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 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 and 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.

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.

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 practices mathematical operations concretely. (This process usually begins by age 4and extends over the next two or three years). This material is practiced over a long period to help the child understand the abstraction of mathematics. 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.)

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.)

Development of more abstract understanding of “short” division through more advanced manipulative materials (Division Board); begins the memorization of basic division facts(Normally by age 7-8).

Development of more abstract understanding of “short’ division through more advanced manipulatives materials(Division Board); memorization of basic division facts( Usually by age 7-8).

Development of still more abstract understanding of “long” multiplication through highly advanced and manipulative materials (the Multiplication Checkerboard); (Usually age 7-8).

Development of still more abstract understanding of “long division” through highly advanced manipulative materials (Test Tube Division apparatus); (Typically by age 7-8).

## Solving problems involving parentheses, such as (3×4)-(2+9)=?

## Study of Decimal Fraction

All four mathematical operations. (Normally begins by age 8-9, and continues for about two years until the child totally grasps the ideas and processes).

Practical application problems, which are used to some extent from the beginning, become far more important around age 7-8 and afterward. Solving word problems, and determining arithmetic procedures in real situations becomes a major focus.