Learning mathematical concepts in a Montessori classroom begins concretely and progresses towards the abstract. They are developed from simple to complex. Process is taught first and facts come later. Order, coordination, concentration, and independence are experienced by the child using these materials. The math activities are organized into five groups.
Group one introduces sets of one through ten which prepares the child for counting and teaches the value of quantity. Children begin to associate numeral and quantity with number rods and number cards. A child will gain a growing understanding of sequence. Spindle boxes, cards and counters, the short bead stair, and other 1- 10 additional counting activities a teacher may add, reinforce the one through ten numeral concept.
Group two involves the decimal system using the golden bead material. The child will become familiar with the names of the decimal categories; units, tens, hundreds, and thousands. A concrete experience with each category is represented by beads. Quantity will be followed by symbol and association.
Group three deals with the operations using the golden bead material. The concept and process of addition, subtraction, multiplication, and division are presented. Children work with each other and benefit from these exercises using the bank game. Progression then continues using operations with the stamp game.
Group four consists of linear counting. Quantity is presented using the teen and ten boards followed by symbol and association. The one-hundred board and bead chains develop number concepts and recognition of numbers one through one-hundred. The bead chains also introduce the child to skip counting; five, ten, fifteen, twenty, etc.
Group five contains activities such as strip boards, snake game, and memorization of facts. Fractions are also a part of this group. Fraction skittles and insets serve this purpose.
The activities in the Math area are not to be implemented at a set pace. Providing the child with the materials at precisely the right challenge level will enable the child to demonstrate his development to the teacher through his progress. A child that is able to grasp such math concepts as addition and subtraction demonstrates the successful use of the math materials. The materials are so beautifully designed and appropriate for each child during his sensitive periods of learning math. Mathematical apparatus provides the necessary stimulation for the child to learn math concepts more readily.
Number Rods - Ten rods, grated in length, the shortest rod being ten centimeters in length, and each succeeding rod increasing by the length of the first, with the longest rod being one meter. These rods prepare the child for 1-10 counting.
Number Rod Variation - Using the same object children can count out the objects onto each section of the rods.
Sandpaper Numerals - Numeral symbols 0-9 in sandpaper. These help the child with recognition of numerals 0-9.
Association of Rods and Numerals - Number rods and numeral cards 1-10. Children associate the number rod quantity with the number symbol. Children gain a growing understanding of sequence as they work with association exercises.
Association of Rods and Numerals
Small Number Rods - These can also be used for simple addition along with Bead Stair Addition when the child is ready to work with simple addition. I placed the photo here as an extension work wih the Small Number Rods.
Small Number Rods
Addition with Small Number Rods
Spindle Boxes - A child counts out the quantity of spindles in their hand and places each set of spindles into the appropriate rectangular section. The Spindle Box indirectly helps to develop the idea that each quantity can be made up of loose units taken together as one set. They also begin to learn the concept of zero as an empty set.
Extensions 1-10 - These are additional exercises that can be added to the Math shelves to provide practice in associating quantity and symbol. I change these monthly according to unit studies to enhance those as well.
Cards and Counters - A set of numerals 1-10 and 55 plastic discs. The layout of the counters is what makes this counting work different. Children will count out the discs in pairs under the numeral. When all the discs are counted out, point out that some of the numerals have a disc that doesn't have a partner. Those numbers are called odd numbers. The discs which all have partners are called even numbers.
Cards and Counters
Memory Game - A small basket with slips of paper each with a numeral from 0-10 written on it. 55 small similar objects such as buttons, seashells, or tiny pebbles. This exercise can be done with two, three, or four children. Each child chooses a number. Explain that they are to keep their number a secret. Ask them to look at their number, remember it, and bring that many objects from the basket of objects that is on the shelf. You can also provide small trays in which the children can use to count out their objects. When all the children have returned to the mat have them take turns to read their number and count out their objects.
Short Bead Stair - A set of small numeral cards from 1-9 and one set of bead stair beads. These beads provide exercise in counting. The bead stair will be used in later math exercises such as the teen board and the snake game.
Short Bead Stair
Short Bead Stair
Introduction Tray - Golden Beads - Two felt green squares, one red felt square, and one blue felt square are needed along with one unit bead, one ten bar, one one hundred square, and one thousand cube. This work helps the children to learn the language of the decimal system and progresses with association of symbol and quantities.
Introduction Tray - Golden Bead Material
Introduction Tray - Quantity and Symbol
Tens Tray - Using the golden beads in amounts of ten units, ten 10's, ten 100's, and one thousand, children are helped to see that ten of one category builds one of the next category (ten units make one ten bar).
Exchange Game - Manipulation of beads between catergories
Nine's Tray - Tray containing: nine units, nine 10's, nine 100's, nine 1,000's, and four squares of felt. Nine units beads are placed onto a green felt square. Say to the child, "If I had one more unit, that would make one ten." Place blue felt square on the mat and put one ten on it. Remove units and green felt. Continue with remaining bead material.
Association of Quantity and Numeral
Composition of Numbers - Need a set of small numerals, Nine's Tray or the Bank, unit container, empty tray, place value paper pages and colored pencils.
Association of Symbol and Quantity - Using the golden bead material, this is a sample of one of the presentations combining quantity and symbol.
Association of Quantity and Symbol
Composition of Numbers
Operations with the Golden Bead Bead Materials - Using the golden beads, the concept and process of addition, subtraction, multiplication, and division are taught.
Operations with the Stamp Game - The Stamp Game leads the child into a more abstract way of performing the basic operations of addition, subtraction, multiplication, and division.
Teen Board - Helps children to recognize numbers 11-19 in quantity and symbol as well as reinforces the concepts of place value. Prior learning to this work is the Short Bead Stair. This can be presented after the Introduction Tray.
Ten Board - Presents to the child the names of the decades and counting from 1-99.
100 Board - Aids the child in the development of number concepts and logical thought, recognition of numbers 1-100, and provides exercise in counting to 100 with the symbols. Prior Learning: Teen Board, Ten Board, and the 100 Chain, all of which can be presented simultaneously with 100 Board.
Addition with Golden Beads
Addition with Golden Beads (completed)
Stamp Game - Addition
Stamp Game - Addition (continued)
Stamp Game - Addition (completed)
Addition and Subtraction Cards
Ten Board (mixed numbers)
Bead Chain Cabinet - Beautiful cabinet of color coded chains, squares, cubes, and number cooresponding number tiles.
Squares of Numbers - Children are introduced to skip counting. The five ssquare chainis made up of all fives and when you fold it, it makes a five square. Children count out each set adding a number tile to each. They roll out the register tape and write the numbers along the paper.
Cubes of Numbers - Same as the Squares of Numbers, but now children find out about the cubes of numbers withthe colored chains.
Bead Chain Cabinet
Squares of Numbers
Squares of Numbers
Cubes of Numbers
Memorization of Facts - Learned through mateirals such as the Snake Game, Short bead Stair, Addition and Subtraction Strip Boards, and Charts.
Multiplication and Division - Learned through Bead Bar Exercises, Boards, and Charts.
Fraction Skittles (I make a set of paper ones that the children can cut and make a chart of fractions to take home)
Money (There are many ways you can introduce and teach simple money concepts to the children, just be sure to have real money available.)