Early tasks:
Learning the numbers and number symbols one to ten.
Introduction to the decimal system: 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.
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. Work with this material over a long period is critical to the full understanding of abstract 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.)
Elementary tasks:
 Memorization of the basic math facts: adding and subtracting numbers under 10 without the aid of the concrete materials.
 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. 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.
 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.
 Development of still more abstract understanding of “long” multiplication through highly advanced and manipulative materials (the Multiplication Checkerboard)
 Development of still more abstract understanding of “long division” through highly advanced manipulative materials (Test Tube Division apparatus)
 Solving problems involving parentheses, such as (3 X 4)  (2 + 9) = ?
 Introduction to problems involving tens of thousands, hundreds of thousands, and millions
 Study of fractions: 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”
 Study of decimal fractions: All four mathematical operations.
 Practical application problems, solving word problems, and determining arithmetic procedures in real situations becomes a major focus.
 Money: units, equivalent sums, counting, performing mathematical operations
 Reinforcing application of all mathematical skills to practical problems around the school and in everyday life.
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.
Early tasks:
 Basic geometric shapes.
 More advanced plane geometric shapestriangles, polygons, various rectangles and irregular forms.
 Introduction to solid geometric forms and their relationship to plane geometric shapes.
Elementary tasks:
 Study of the basic properties and definitions of the geometric shapes.
 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.
 Congruence, similarity, equality, and equivalence.
 The history of applications of geometry.
 The theorem of Pythagoras.
 The calculation of area and volume.
