Math begins with the recognition that things can be enumerated and counted, so this is where we begin our discovery of mathematics.
Rote counting:
Children are in a sensitive period around age 3 where they begin to become passionate about counting. They quickly pick up 1, 2, 3, 4, and 5. In the classroom we count whenever possible: we count the days of the year, the beads in the spooning work, the squares on the carpet, and much more.
Visual Discrimination:
Children ages 3–4 begin to recognize amounts of things, that numbers mean something. They are able to internalize the differences in the amount of objects assigned to a given number (8 means more objects than 2). To isolate this concept in the classroom we use materials that have physical objects for the children to count and match to the corresponding number, such as spindle boxes and cards and counters. To be successful in these works, children must, at the same time, learn to establish one-to-one correspondence, only counting one object per number spoken when counting a group of things. This is done with activities such as playing simple two-person board games, setting the table with one fork and spoon for each plate, and understanding that when you play outside there are only eight swings, so only eight can swing.
Symbol Naming:
While children are visually discriminating objects and the numbers we use to represent them, they are also beginning to name the symbols when they are isolated away from a quantity. They are able to see the number 7 on a sign and know that that reads the spoken word seven. They also begin to recognize that the symbol and word five always mark the same amount of things. Children age 3–4 love this! In the classroom we introduce the children to sandpaper numbers (rough number cutouts glued to a hard board). We show the children how to trace the number as they would write it and to say the word it represents. They can then trace the number free-hand in a small box of sand. This uses their desire to touch to internalize the names of numbers and the way they can be reproduced.
Once rote counting has begun, the symbols can be identified, and the numbers and quantities can be correctly sequenced, children are ready for beginning addition.
Simple Addition and Beginning Place Value:
We introduce simple set addition by showing the children how to physically add objects: adding a set of 5 stones to a set of 4 stones, resulting in one big group of 9. Once this simple set addition begins, it naturally flows into keeping track of all those numbers, and place value can be explored. We begin with the teen board: one set of 10 and one unit equals 11, one set of 10 and 2 units equal 12, etc.
Once place value has begun, we work on numbers 1–100, both quantity and number recognition. Until place value is understood, numbers such as 17 and 71 are confusing to the children. We stress the differences in these place values by having the children build numbers with unit beads, ten chains, hundred squares, and thousand cubes. These numbers are then written in a corresponding color so that the child knows that units are green, tens are blue, hundreds are red, and the pattern repeats with thousands. This utilizes the child’s desire to use both touch and sight to learn the concept of place value. As these materials are presented, children are also sequencing the numbers 1–100. As these numbers are mastered, children will continue to practice the concept of place value until it is fully internalized. It is the most important concept in mathematics. Everything in math is built upon it.
As children succeed in physically adding and building numbers, problems are then worked on paper. They transfer the concrete to symbolic as a transition towards abstraction. We learn about even and odd numbers, equal and not equal, greater than and less than, counting by tens (helps with place value) and fives (helps with time and money) and twos.
As the children master single- and double-digit addition, they are introduced to subtraction and multiplication (which is repeated addition), and lastly if their time in class allows, division. These functions are all taught in the same way, giving concrete experiences then moving to the abstract.
Geometry
We begin the study of geometry with 3-year olds with flat or, as we say in class, plane shapes: circle, square, triangle, rectangle, pentagon, hexagon, trapezoid, rhombus, oval, ellipse, quatrefoil, and curvilinear triangle. We use metal insets that the children trace and color to help them form these shapes (this also strengthens their hands for writing). Sandpaper shapes, like the sandpaper numerals above, are used to provide the language for the shapes. When children have learned to identify these shapes, they are ready to see how they relate to one another with our constructive triangle, rectangle, and polyhedral boxes. Within these materials, the children take apart a square and find that it is two triangles, that two halves of a rectangle can be resituated to make a parallelogram, and that triangles are the root of all sided shapes. To complete this understanding, children trace and cut these shapes and construct them on a poster, labeling them as they go.
After the plane shapes are identified, the children begin their study of solid geometry with the cylinder, cube, cone, and sphere because many of the Montessori materials are made with these shapes. The earth is a sphere, half of the earth is a hemisphere, and boxes are rectangular prisms. We add ellipsoid, ovoid, triangular prism, and square-based pyramid to complete this introduction to solid geometry.