Montessori Mathematics – Why it is the BEST!

This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child's special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations. ~The Discovery of the Child, Maria Montessori.

The Montessori Mathematics curriculum is an amazing system that is highly successful and incredibly built. This curriculum consists of incredible hands-on learning materials, detailed lessons, one-on-one instruction, life application and deep levels of understanding process, not memorizing products. As you read through this article, please take your time to read and re-read each section. This article is only a summary of how the Montessori Math curriculum works and why it is the best. Please take the time to visit with your teachers to understand how each material works and how the entire system works together.

Montessori Math: Ages 0-6 At Lighthouse Academy we introduce Math to very young children for many reasons. One of the most basic reasons is because these young children are in a sensitive period for order – and the study of math helps to satisfy the needs and interests that go along with this developmental stage. Children are exposed to materials with which they can explore mathematical principles as early as 2 ½ or 3 years of age, and the curriculum follows them as they develop skills and understanding.

Maria Montessori was fond of quoting the French philosopher and mathematician, Pascal, who said that all humans have a “mathematical mind”. Montessori believed this to be especially true of children. They explore the world by organizing & categorizing what they find there; they love to find the patterns in the world around them and this is the beginning of their mathematical education.

You can find activities that introduce and allow the exploration of math-related concepts in each area of the Montessori classroom. The concept of one-to-one correspondences, or pairing, is taught in the Practical Life area with such activities as: using keys and locks, matching cups to saucers, and buttoning or snapping on the dressing frames – to name just a few. In the Sensorial area the same concept is at work in the Knobbed Cylinders activity, as well as in matching pairs of colors with the Color Tablets; while in Language the children might match pictures to concrete objects or play Lotto games. These are just a few of many examples. Seriation is practiced in the Sensorial area with activities like grading the Sound Cylinders from loud to soft, arranging the Touch Tablets roughest to smoothest or fabric tablets from lightest to darkest. Back in Practical Life we find clothes- dish- or table-washing activities involving a long series of steps to accomplish.

We may also find excellent demonstrations of the principles of conservation here, as the children pursue such activities as pouring dry or liquid substances from one container to another, or to multiple containers. Other classic mathematical concepts demonstrated and practiced throughout the Montessori classroom include: similarity and difference, combination (adding), spatial relationships and forms; and temporal relationships.

Children in the Montessori classroom have the opportunity to learn many mathematical concepts before they are six years old. They’ll begin with very concrete materials and gradually progress to abstract concepts. They learn to recognize, recite and eventually to write the numerals one through one thousand, and to associate those numerals with the appropriate quantities (number). At three or four years of age they might work with the Number Rods, Spindle Boxes, Sandpaper Numerals or Numerals and Counters as they absorb information about the concept of zero and even & odd numbers; while at five they’ll enjoy the Tens and Teens Boards, the Hundred Board and the :Long Bead Chains. Threes ands fours can explore the concept of sets of numbers, again with the Spindle Boxes or the Numerals and Counters. Skip Counting can be demonstrated with the Short Bead Chains, while formation of number (place value) and exchanging (“carrying” in arithmetic problems) are concepts that children pick up with an activity like the Bank Game. Here as well are good opportunities for learning addition, multiplication, subtraction and division. Children between four and five will also pick up the concept of squares and cubes of numbers while working with, for example, the Bead Chains and the Hundred Board. When they’re ready, the older fives and sixes can be introduced to the memorization of basic number facts (i.e. addition and multiplication tables) with materials like the Table Rods, Addition Strip Board, Multiplication Bead Bars, and several more. Six-year-old children also often embark on the “path to abstraction” with variations on the Stamp Game that include addition, multiplication, subtraction and division. They also love the Dot Game and the Small Bead Frame.

The Montessori math curriculum is varied, never boring, and challenging – but never forced upon the children. They love these materials! They’re colorful and fun to work with, intricate and beautiful. At Lighthouse Academy, Math is fun, and has little to do with memorizing formulas without understanding the logic behind them. It’s a terrific, positive and effective method to introduce Math to our young children!

Preparation for mathematics a) One-to-One Correspondence

b) Books - Counting c) Songs – 3 Little Speckled Frogs d) Sorting/Classification e)Sequence/Patterns – Do As I Do f)Sequence/Seriation – Bead After Bead

Numeration Sets Children 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.

Number Rods

Sandpaper Numerals (0-9)

Number Rods and Numerals

Spindle Rods

Numerals and Counters

Linear Counting Children learn 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. 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. Short Bead Bars

Snake Game – Search for Ten

Teen Boards

Squaring Chains – 1-10

Cubing Chains – 1-10

Base 10–Decimal System 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. Decimal System-Quantity (Golden Beads)

Here are some of the ways the Montessori Math program has been specifically designed to insure the children’s success with early Math concepts:

· Concrete, manipulative materials (i.e. Number Rods, Spindle Boxes) are presented and explored thoroughly before the introduction of related abstract activities and concepts. · Materials are designed to isolate one area of difficulty so that children are not overwhelmed. · There is a built-in control of error in the materials so that a child may independently recognize and correct their own errors

· The materials are attractive and fun – so children want to use them · Montessori math is presented in a logical way with interesting materials that have a high possibility for success. · The emphasis is always on the process of mathematical operations – not only on getting the correct answer – which works to remove the fear and mystique often associated with Math.

Montessori Math: Lower & Upper Elementary

The inquisitiveness of the lower and upper elementary Lighthouse Academy student is astounding. The beauty of the advanced squaring and cubing materials beckons like beacons, inviting the students to come explore and learn with them. They dive into the study of fractions and decimals, eager to move beyond to more complex mathematics, geometry, and algebra. While the concrete materials are still in place, the need for repetition is gone. “Show me. Then, show me more” is the litany of the upper elementary Montessori math students. Upper elementary students move quickly from the concrete experience to abstract thought. They are eager to test their knowledge with pencil and paper and need, at times, a gentle reminder to return to the materials as a way of building neurological pathways.

Geometry, math, and invention are languages used to explore and manipulate, to theorize and create, real objects in a real world. At this age children continue to enjoy exploring math and geometry concepts if they are related to real life, and if they are presented with materials which can be handled, manipulated, used to create. We must keep sight of this fact when teaching children. We give manipulative materials in all areas of math and leave it to each child to decide when she is ready to work without materials—in the abstract—on paper with pencil.

We encourage children to make up their own problems—especially story problems related to their lives and the subjects they are studying—for themselves and for their friends, in order to come to a very practical and clear understanding of geometry and math. Children enjoy making up problems for each other, and examples that stump their teachers. This process of math concepts makes them stick in the child's mind.

With higher math, geometry and algebra, we give many practical examples and help the children come up with their own formulae after much experience. For example, if a child measures all of the rectangles in the room—tables, windows, books, etc. for figuring surface area, he will easily create, and even better understand, the formula "A=lw."

For each grade level, from 1st through 6th, the children learn to plan and schedule their work. It is left to each child to decide the best system and schedule, through trial and error, and with adult help, depending on learning styles, and interests.

This teaches the math of planning, scheduling, allotting sufficient time, and it teaches responsibility.

When children are given this solid, material foundation, and see the relationship of geometry and math to the real world, it makes it easier for them, in later years, to spend long periods of time working on paper.

This is because they know that these steps are just that—steps which will take them to a new level of understanding in the exciting world of math and science.

Montessori Math: Middle School

Abstract work is a higher mental level of work, which comes naturally after the child has learned to picture the object being measured or related to other objects in her mind.

In the elementary class stories are told and experiments carried out to show children how humans used their imaginations in the past, and how they are using them today, to solve problems and come up with great inventions—the use of fire, measuring the earth, compasses, boats, and many others. They see how inventions, geometry and math came about as the result of human progress, to meet specific needs.

Geometry, for example, arose from the practical need to reestablish planting boundaries after the annual flooding of the Nile in Egypt. In "geometry," geo stands for earth, and metry for measure.

Middle school students love to reach back into history with their imaginations and reconstruct these needs and solutions and the creation of systems of learning. The Hindus introduced the use of "0." Let the child try to do math without it! Where did algebra, calculus, trigonometry come from? They want to know!

Children are inspired by these stories, and by examples and pictures, to find out more. Children come to realize that mathematics has evolved and is still evolving from a practical need. Math, graphing, fractions, all become logical tools for recording and measuring, and algebra a short cut for recording.

A main thrust at this level is pre-algebra, binomials and trinomials, at ages 10 and 11 and then algebra at age 12. The materials are incredible and continue the idea of didactic materials using the senses to process higher level mathematics as opposed to a continuation of abstract computations without deep understanding.

ALGEBRA

2 dimensions, base x

Algebra involves the study of numbers in terms of their functions and relations. In

mathematics, a formal language, we use numerals to represent known or specified numbers, and letters to represent unknown or unspecified numbers.

In English, a natural language, we use names or proper nouns to represent known or specific individuals, and pronouns to represent unknown or unspecified individuals. The Montessori student learns proper use of the English language while interacting with

physical objects represented by the proper nouns and pronouns of English, not by employing some other natural language as a metalanguage in which to discuss the English language. (It is generally poor practice to use a natural language, which the student has substantially mastered, as a metalanguage in which to study a second natural language. Such practice usually results in the student translating expressions from the second language into the first language for processing, rather than learning to think and speak fluently in the second language. A similar practice, using English as a metalanguage in which to study the language of mathematics, is even more costly to the student.)

The following is an explanation of how the student learns algebra at Lighthouse Academy while interacting with manipulatives, physical objects, represented by the numerals and variables of mathematics. For simplicity here, the explanation is given in two dimensions. Of course, the youngest students interact with perceptual three-dimensional objects, concrete rectangular prisims, rather than with imaginary two-dimensional objects, abstract rectangles.

1. The proper interpretation of a numeral by answering three questions:

• How many objects? • What base? • In how many dimensions?

Here we have a unit square, it is one unit in two dimensions, in width and in length. We use numerals to name specific numbers in mathematics (just as we use proper names to name specific objects in English). *1 N.B. We can name this object “one unit-square”, “(1 × 12)”, “one unit”, or simply “1”. How many? ... 1. What base? ... 1. In how many dimensions? ... 2. (1 × 12) = 1 Here we have five unit-squares. We can name these objects “five unit-squares”, “(5 × 12)”, “five units”, or simply “5”. Alternatively, we have one bar consisting of five unit-squares, i.e., a bar five units in width and one unit in length. Thus, we can name this object “one five-one-way”, “(1 × 51)”, “one five-bar”, or simply “5”. How many? ... 5. What base? ... 1. In how many dimensions? ... 2. (5 × 12) = 5 How many? ... 1. What base? 5. In how many dimensions? ... 1. (1 × 51) = 5

Here we have another bar. It is one unit in length, but its width is not divided into units, so it doesn't have a specified width. Since the name of the width does not represent a specified number, the interpretation or value given to that name may change or vary; so that name is a variable. We use variables to name unspecified numbers in mathematics (just as we use pronouns to name unspecified objects in English). We use letters as variables in mathematics. We name the width of this object “x” because we can't count it as any specific number of units. Thus, we have a bar x units in width and one unit in length. We name this object “one x-one-way”, “(1 × x1)”, “one x-bar”, or simply “x”.

How many? ... 1. What base? ... x. In how many dimensions? ... 1. (1 × x1) = x We can now regard our unit square as a bar, in base x, that measures x in no (zero) dimensions. We can now name this object “one x-no-way”, “(1 × x0)”, “one unit”, or simply “1”. How many? ... 1. What base? ... x. In how many dimensions? ... 0. (1 × x0) = 1 Likewise, we can rename this object, in base x, as “five x-no-way”, “(5 × x0)”, “five units”, or simply “5”. How many? ... 5. What base? ... x. In how many dimensions? ... 0. (5 × x0) = 5 Here we have three bars in base x. We name this object “three x-one-way”, “(3 × x1)”, “three x-bars”, or simply “3x”.

How many? ... 3. What base? ... x. In how many dimensions? ... 1. (3 × x1) = 3x We can arrange nine units in two dimensions so that we have a rectangle that is three wide and three long, or a square. We name this object “three-units squared”, “(3 × 3)”, “one three-two-

ways”, “(1 × 32)”, “one three-square”, or simply “32”.

How many? ... 1. What base? ... 3. In how many dimensions? ... 2. (1 × 32) = 32 Here we have a rectangle that is x wide and x long. It is x in two dimensions, and we name it “x-squared”, “(x × x)”, “one x-two-ways”, “(1 × x2)”, “one x-square”, or simply “x2”.

How many? ... 1. What base? ... x. In how many dimensions? ... 2. (1 × x2) = x2 2. The construction of compound (polynomial) numbers, and the composition of compound numerals (polynomials): Here is two x-bars plus four units, named “(2x + 4)”:

How many? ... (2, 4). What base? ... x. In how many dimensions? ... (1, 0). (2x1 + 4x0) = (2x + 4) Here is one x-square plus three x-bars plus two units, written “(x2 + 3x + 2)”:

How many? ... (1, 3, 2). What base? ... x. In how many dimensions? ... (2, 1, 0). (1x2 + 3x1 + 2x0 ) = (x2 + 3x + 2) Here is two x-squares plus five x-bars plus four units, which we name “(2x2 + 5x + 4)”:

How many? ... (2, 5, 4). What base? ... x. In how many dimensions? ... (2, 1, 0). (2x2 + 5x1 + 4x0) = (2x2 + 5x + 4) 3. The factoring of polynomial numbers: In arithmetic and algebra we often need to find the factors of a number to solve a problem. The factors of a number are two or more numbers which, multiplied together, are that number. For example, 3 and 2 are factors of 6, because 3 times 2 is 6. Using manipulatives we can find the factors of a number by reconstructing a bar of unit squares into a rectangle of unit squares. Here are the factors of 6:

We can represent this fact with the expression, “6 = (3 × 2)”. Since the trinomial “(x2 + 3x + 2)” names a compound number constructed of three groups of

rectangular parts, we can find the factors of this number by reconstructing the rectangular parts into a single rectangle. The factors are the resulting rectangle's width and length, (x + 2) and (x + 1).

To represent this fact we write “(x2 + 3x + 2) = (x + 2) (x + 1)”. Let's factor the number, (x2 + 5x + 6). First we group the blocks so that each group is represented by a part of the trinomial. Then we reconstruct this row of grouped blocks into a single rectangle, and find the width and length of the rectangle, which are the factors of the number.

The sentence that represents this fact is “(x2 + 5x + 6) = (x + 3) (x + 2)”. 4. Higher dimensions and definite bases: After the Montessori student learns to factor polynomial numbers and perform basic operations in two dimensions, such as addition, subtraction, multiplication, division, and square roots, using manipulatives, he may then use three-dimensional blocks, in base x, to perform basic operations, such as addition, subtraction, multiplication, division, and cube roots. Having mastered these algebraic manipulations and notations, the student may substitute a definite base (such as base ten) for base x. He then discovers how to perform these same operations using multidigit, composed numerals, in a number system incorporating place value to facilitate working with large numbers. Such a student does not require “lessons” or

“demonstrations” of traditional Montessori math equipment. He discovers for himself how to do long division or how to calculate a square root. The joys of such discoveries are even available to primary-level students who have not yet acquired the perceptual ability to distinguish a numeral “2” from a numeral “5”, or a numeral “6” from a numeral “9”. Such students can master basic operations, such as long division and square roots, using only the digits (“0” “1” “2” “3”) of the base-quad number system, before they can reliably decode the ten digits (“0” “1” “2” “3” “4” “5” “6” “7” “8” “9”) required to operate in the base-ten number system. By having their learning environments enriched with base-quad Montessori math materials, students will not be prevented by limitations of traditional, base-ten materials from developing mathematical aptitudes during earlier and more fruitful “sensitive periods” for the acquisition of those aptitudes. 5. Multiple bases – working with more than one variable: As the student employs this knowledge of algebraic operations using expressions with a single variable to the task of mastering arithmetic, he also extends his study of algebra to numbers incorporating several variables. To build these numbers he requires additional blocks with distinct lengths and widths representing additional unspecified numbers (y, z, etc.) which may vary in value from both x and 1. Now let's factor the number (x2 + 2x + 2xy + y2 + 2y + 1). First construct the blocks into groups, each of which is represented by a part of the polynomial. Then we reconstruct the row of grouped blocks into a rectangle, and find the width and length of the rectangle, which are the factors of the number. In this case we get a square, bearing an unmistakable similarity to the face of the trinomial cube found in many Montessori classrooms.

The sentence that specifies this fact is “(x2 + 2x + 2xy + y2 + 2y + 1) = (x + y + 1)2”. In order to derive the maximum benefit from interaction with a specially prepared Montessori environment, it is crucial that the student not confuse the concept of a numeral – ‘a name of a number’ (which is a word from the language of mathematical discourse), with the concept of a

number – ‘a quantity or measure’ (which is an object from the universe of mathematical discourse). Confusion of word with object is a more serious problem for the young student of mathematics than for the young student of English. Despite our imprecise usage of English, the student is not very apt to confuse a tree, which he can learn to climb, with “tree”, which he can learn to pronounce, spell, write, and read. If we mistakenly ask him to put mom on the blackboard, he will probably reach for the chalk rather than reach for his mother, but we had better not ask him to put paint on the blackboard unless we are prepared to learn an expensive, albeit valuable and well deserved, lesson! However, he is more apt to confuse 1 and 2, from which he can construct 3, with “1” and “2”, from which he can compose “21” or “12”. (He may learn to avoid this latter confusion by unconscious attention to context, but even world-renowned mathematicians have been known to confuse a material conditional with an implication.) When working with a very young mathematician, who has not yet developed the level of auditory discrimination necessary to reliably distinguish between the sounds of the spoken words “number” and “numeral”, it is preferable to use language which will not contribute to the student's confusion: Vocalize the expression “construct (x + 2)” as “construct the number, x plus two”. Vocalize the expression “compose “(x + 2)” ” as “compose the name of x plus two”. Do not vocalize “compose “(x + 2)” ” as “compose the numeral for x plus two”. In order to derive the maximum benefit from interaction with a specially prepared Montessori environment, it is crucial that the student not confuse the concept of similarity or equivalence – ‘the same in some respect(s)’ (which constitutes joint membership in a set or class) – with the concept of identity – ‘the same in all respects’ (which constitutes being one and the same object). “(3 × 2)” and “6” are two different names (numerals) for the same object (the number 6). These two names are similar names, in that they each name the same object; they are equivalent names, in that they each take the same object as their semantic value. However, they are not identical names. We can represent such facts with metamathematical equivalency statements about the numerals (the names) that we use to refer to numbers: “ “(3 × 2)” names the same number that “6” names ”, “ “(3 × 2)” is equivalent to “6” ”, “ “(3 × 2)” ≡ “6” ”, or “ “(3 × 2)” equals “6” ”. Usually, however, we simply represent such facts with mathematical identity statements about the actual numbers (the objects) themselves:

“ (3 × 2) is the same number as 6 ”, “ (3 × 2) is identical to 6 ”, “ (3 × 2) = 6 ”, or “ (3 × 2) is 6 ”. Therefore, whenever vocalizing mathematical expressions (identity statements or equations), it is preferable to use language which will not contribute to the student's confusion: Vocalize the expression “1 + 1 = 2” as “one plus one is two”. Do not vocalize “1 + 1 = 2” as “one plus one equals two”.