### Mathematics and Geometry Age 3 to 12

Often, students learn math by memorizing facts and solutions, with little true understanding or ability to use mathematics in everyday life. Math is a series of abstract concepts for most children, and learning tends to come much more easily when they have hands on experience with concrete educational materials that show what is taking place in a given mathematical process.

**Montessori’s famous hands-on learning math materials make abstract concepts clear and concrete.** Students can literally see and explore what is going on in math. Our approach offers a clear and logical strategy for helping students understand and develop a sound foundation in math and geometry.

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 the size 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-centimeter beads; a unit of ten is made up of a bar of ten beads strung together; hundreds are squares made up of ten 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 child 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 that most adults wouldn’t recognize. The study of volume, area and precise measurement in everyday applications around the school is introduced in the early years and continually reinforced and expanded.

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, making economic comparisons, or in gathering data and making a statistical analysis.

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

Precise measurement and comparison is a crucial application of mathematics, and our students engage in all sorts of calculations: determining the amount of gas used by the family car, the electricity burned when our lights are left on overnight, and the perimeter of the buildings.

**Our students are typically introduced to numbers at age 3:** learning the numbers and number symbols one to ten: the red and blue rods, sand-paper 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 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 Frame (a color–coded abacus).

**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.)

Development of more abstract understanding of “short” division through more advanced manipulative materials (Division Board); movement to paper and pencil problems, and memorization of basic division facts (Normally 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 ad-danced manipulative materials (Test Tube Division apparatus); (Typically by age 7-8).

Solving problems involving parentheses, such as (3 X 4) – (2 + 9) = ?

**Missing sign problems:** In a given situation, should you add, divide, multiply or subtract ?

Introduction to problems involving tens of thousands, hundreds of thousands, and millions (Normally by age 7).

**Study of fractions:** Normally begins when children using the short division materials who find that they have a “remainder” of one and ask whether or not the single unit can be divided further. The study of fractions begins with very concrete materials (the fraction circles), and involves learning names, symbols, equivalencies common denominators, and simple addition, subtraction, division, and multiplication of fractions up to “tenths” (Normally by age 7-8).

**Study of decimal fractions:** 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.

**Money:** units, history, equivalent sums, foreign currencies (units and exchange). (Begins as part of social studies and applied math by age 6.)

Interest: Concrete to abstract; real life problems involving credit cards and loans; principal, rate, time.

**Computing the squares and cubes of numbers:** Cubes and squares of binomials and trinomials (Normally by age 10).

**Calculating square and cube roots:** From concrete to abstract (Normally by age 10 or 11).

The history of mathematics and its application in science, engineering, technology & economics.

Reinforcing application of all mathematical skills to practical problems around the school and in everyday life.

Basic data gathering, graph reading and preparation, and statistical analysis.

### Geometry

Sensorial exploration of plane and solid figures at the Primary level (Ages 3 to 6): the children learn to recognize the names and basic shapes of plane and solid geometry through manipulation of special wooden geometric insets. They then learn to order them by size or degree.

**Stage I:** Basic geometric shapes. (Age 3-4)

**Stage II:** More advanced plane geometric shapes-triangles, polygons, various rectangles and irregular forms. (Age 3-5)

**Stage III:** Introduction to solid geometric forms and their relationship to plane geometric shapes. (Age 2-5)

Study of the basic properties and definitions of the geometric shapes. This is essentially as much a reading exercise as mathematics since the definitions are part of the early language materials.

More advanced study of the nomenclature, characteristics, measurement and drawing of the geometric shapes and concepts such as points, line, angle, surface, solid, properties of triangles, circles, etc. (Continues through age 12 in repeated cycles.)

Congruence, similarity, equality, and equivalence.

The history of applications of geometry.

The theorem of Pythagoras.

The calculation of area and volume.