Montessori wrote: "Human intelligence today is no longer a natural intelligence but a mathematical intelligence. Without a mathematical education it is impossible to understand the progress of our time or to participate in it. In our time, a mind without mathematical culture is comparable to that of a [person] ignorant of the alphabet... In its natural state the human mind is already mathematical: it tends toward exactness, measure and comparison."
Current research continues to reveal the importance of the environment for both life in the womb and the first three years of life. The child in utero has a life rich in sensory stimulation. The unborn child senses the hustle and bustle of everyday life, the patterns of the mother, the cycles of night and day, of activity and inactivity. This child hears the voice of the mother and the father, the voices of the family. If this child has a parent who loves classical music or believes in the "Mozart effect," this child might hear classical music. The Mozart effect theory states that because the neural pathways for music learning are very similar for those for mathematics, listening to music enhances the wiring and myelinization of the neural connections in the brain. Thus, listening to classical music, both inside and outside the womb may enhance the development of the logical mathematical mind.
At birth, a baby's brain contains 100 billion neurons, as many as stars in the Milky Way. Continued neural development depends on direct sensory experience with the environment. Newborns must be free to move and explore their new environment sensorially. When the child is free to move and interact with the environment, neural development, or learning, takes place.
The crucial role for the parent at this time, beyond loving, bonding to and attending to the basic needs of the newborn, is to provide order within the environment. When there is consistency and order in basic activities such as feeding, changing, putting to sleep, bathing, and playing, the newborn will not only develop a sense of security but a sense of the sequence of an activity. This sense of sequence and order helps the child think logically and know what to expect.
Maria Montessori observed that throughout their lives humans have tendencies that attract them to both order and mathematics.
This tendency toward mathematics moves the child to observe, to contrast and compare, and to classify. The child at a very early age will notice when another child has more than they do. The child from birth to three also uses language to create order and logic within the environment. Clear and consistent names for things help the child make sense of the world and give words to their needs. This very young child also has experience with mathematical concepts such as big, small, more, one, few, many.
This child from birth to three is in a sensitive period for order. The child has a great interest and need for consistency, sequencing, and recurring patterns. This is illustrated by how upset an infant can become when a routine is altered, or how a toddler reacts when you try to read a beloved book in a different manner. The new interpretation is lost on the toddler; they want the familiar, the comfortable.
Montessori observed sensitive periods in children in the early years of this century. Today these periods of sensitivity to intense learning are labeled "touchpoints" or "windows of opportunity." Current research indicates that the window of opportunity for math and logical learning is between birth and four years of age. This is a critical time for experience in routines, sequence of activity, in sizes and shapes, in simple mathematical concepts. The child should have activities with a beginning, a middle, and an end, should sort and classify, and do activities that support one to one correspondence. All of the activities of practical life or daily living provide the experience for the learning that this child needs. Setting the table is a perfect activity for learning one to one correspondence. The child carries and sets out one plate, one spoon, one glass for each person.
The child from birth to three is a sponge. They will absorb whatever is in their environment, whatever we give to them, each in their own unique manner. As we observe this child interact, absorb, and unfold like a beautiful mysterious flower we can appreciate Maria Montessori's assertion that "The greatness of the human personality begins at birth."
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Mathematics is one of the most useful and fascinating divisions of human knowledge. Montessori speaks of the mathematical mind and says it is "a natural human tendency to think in mathematical terms; to count, to measure, to recognize shape and symmetry." In the scientifically prepared materials with an understanding and analysis of the child's difficulties, Montessori compares the math materials to a mental gymnasium in which the child's mind can be exercised.
When a child enters Children's House, she has already had a great deal of experience with numbers. The child knows her age, the number of brothers or sisters she has and will often ask a teacher, "How many more minutes until my Mommy comes?"
Working with the math materials does not begin right away. The child is prepared indirectly for the math materials through his work with the Sensorial Materials and the Exercises of Practical Life. The child exhibits his natural tendency for calculation in practical life through pouring water, polishing and washing. All the child's movements require calculation in order to be precise. For example, the adult marks pitchers and cups with a red dot to assist the child in pouring activities. The orderly manner in which the adult arranges the environment assists in ordering the child's mind.
The five fundamental abilities the child needs in order to understand and carry out basic concepts of math operations are discrimination, recognizing similarities and differences, constructing and comparing a series, finding relationships and understanding exact terminology.
Through repeated work with the sensorial materials the child absorbs the properties of the physical objects in the environment. These impressions are then abstracted and related to the environment through their application. Several materials that refine the visual sense help the child experience the concept of 10, the basis of our decimal system.
Montessori realized that children of four years could count numbers up to 1000 after mastering only numbers 1-10. With the decimal system, we give the child a visual sense of unit, ten, hundred and thousand. The child sees these numbers in geometric form; a point for the unit, a line of units for ten, a square of hundred and a three-dimensional cube for thousand. The children exhibit such joy when they hold a thousand in their hands for the first time. Next, we show the children how to put numbers together with addition and multiplication to form a larger quantity. The quantities are sub-divided into smaller quantities with subtraction and division, and the children begin to understand the true nature of the operation.
Through repeated work with the materials, the child readily and gracefully makes the passage to abstraction. Dr. Montessori's genius and keen observation enabled her to see the children's powers, needs and potential in the realm of mathematics.
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The mathematical mind is the defining characteristic of human intelligence. Thought itself is almost impossible to envision without the counting, ordering, measuring, observation of pattern, and classification that are key elements of mathematical thinking.
The goal of all our work in a Montessori classroom is the full development of the human personality. The study of mathematics plays a crucial role in this development. Stanislaw Dehaene, a cognitive scientist at the National Institute of Health and Medical Research in Paris, suggests that mathematics is "engraved in the very architecture of our brains." Work in mathematics is profoundly satisfying to the children because mathematical work corresponds with the way in which their minds naturally work. As Montessori guides we try to ensure that the child is presented with materials that correspond to the child's interests and developmental needs.
This lofty goal is achieved in the classroom through the use of the Montessori materials that the child is free to manipulate and that represent abstract concepts, such as fractions or multiplication. As the child continues to use and explore the possibilities of the material, the mathematical abstraction is revealed to the child's mind. Unlike the Children's House child, the elementary age child is not usually drawn to repetition for its own sake. The materials are carefully constructed to provide repetition through variety. A new lesson on "multiples" for example can draw in the child who is tired of practicing multiplication facts. Still the multiples lesson is essentially the practicing of multiplication facts in a format that is new and therefore intriguing. This principle of repetition through variety also comes as various small but interesting refinements are introduced with a familiar material.
The moment I wait for as a Montessori guide is when the child, dutifully manipulating the fraction pieces, finally says, "You know, we could just add the numerators without using the pieces." At that moment the material has done its job of supporting the child's intellect with a concrete referent for the imagination. The child is then able to infer the mathematical process. Children who learn math in this way know it is not just with their memories, but with their whole being. It may be difficult for the younger children to communicate their understanding, but when they do reach an abstract level of understanding a mathematical concept, it does not easily leave them.
The elementary age child is a powerful learner. The universe is barely big enough to satisfy the wide-ranging curiosity of these children. Two primary characteristics of children at this age are the imagination and the reasoning mind. These are both powerfully engaged in the study of mathematics. In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic. This marriage of logic and imagination makes the study of mathematics a compelling pursuit for any learner. The Montessori materials ensure that this engaging study is open to all children as they work to internalize concepts and master skills.
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Behind the wall the gods are playing. They play with numbers of which the universe is made.
The human mind is a mathematical mind. It continually creates mathematical ideas and applies them to the real world. Right now in class we are:
- reading about the fifth dimension in A Wrinkle in Time,
- discovering how the pattern in a pine cone or sunflower reveals Fibonacci's sequence,
- tripling a recipe to serve 29,
- measuring the material to sew pillow covers for the loft,
- making a job chart dividing a 18x24 poster board to fit 28 uniformly measured orange pockets,
- drawing the orb of a spider's web,
- learning that 97% of the virgin forests in the continental USA are gone,
- realizing that the earth is 70% water,
- pondering that the earth is moving at a million miles a day,
- and asking, "How many words is the essay supposed to be?"
Where you might least expect math, there it is! Mathematics is the language of science, technology, engineering, medicine, and art.
Our basic "curriculum" is cosmic education. The child is offered the wonder of the universe to explore in all ways. All areas of investigation are interconnected. Wonder, awe and gratitude are inspired through touching the imagination and exciting the child to make discoveries. Mathematics is not only a tool for this exploration but the imagination is touched by the stories of math. The students are inspired by images of:
- Erastothenes, using the idea of a simple household tool to create THE SIEVE of Erastothenes to reveal the prime numbers,
- The 120 primes between 3000-4000. What about between 4000-5000, and what about Euclid and the Unending Primes?
- The Babylonians calculating the square root of two and recording it on a clay tablet instead of a computer,
- The astonished man as the hieroglyphic for 1,000,000,
- Two brothers, building a special computer in their apartment in New York that simply tried to find repeating pattern in the digits of pi and doing that all day,
- or of Pythagorus who, with his secret brotherhood, discovered the dodecahedron and made it the symbol for the universe - but kept it secret!
Perhaps another Archimedes will appear who will decide to calculate the number of grains of sand on the beaches of the world!
Cube root is a rite of passage in Class F. It is the grandest in material presentation and the most profound work. It is a reverent moment when the cubing material comes out. And the onlookers look in awe upon the material. What a brilliant piece of material designed to show the principles in a concrete manner. The student is ready! The preparation has been set - it is time! The discovery of cube root brings the "ah-hah" of enlightenment to this complicated principle. The child remarks "I worked with that in Children's house and EI. It was a puzzle to put together, I never knew it could help you learn cube root!"
With all the wonder and excitement there is also fear particularly around calculating. Fear of mathematics is like a fear of dancing, both are overcome by a little practice. Practice comes in many forms at Lake Country, the lessons don't stop and the new lessons inspire practicing the old skills. Calculations in all operations are practiced in EII and individual standards of performance are kept. Children are prepared in the practical aspects of math and are given the opportunity to take responsibility for practicing what is needed and expected.
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Adolescents are in transition from childhood - the person who lives in the family - to adulthood - the person who lives in society. Mathematics is one of the vehicles by which they will find their place in society.
Their interest in mathematics is connected to their powerful thinking minds and parallels their interest in logic and philosophy. They embrace math for the intellectual challenge it offers. They enjoy the fun of mathematics because they are great problem solvers and come to know the range and power of their own minds through grappling with mathematical situations. Adolescents are ready to move into a fairly sophisticated understanding of mathematical ideas. They love algebra and the idea of the unknown takes them to a heady mathematical high.
Secondly, the adolescent enjoys the challenge of mathematics and revels in the excitement of not only getting it right, but of knowing if he is right. Mathematics offers that sudden understanding of how things work, that ah-hah reaction, and generates frantic hand waving and explosive answers.
Thirdly, in spite of their intense enjoyment of the competitive chase of problem solving, the adolescent works collaboratively with great effectiveness in mathematics. This is the age of strong social identity with the peer group and they can be brilliant working together, especially around an open-ended problem. In these collaborations they are intense, passionate, argumentative, and persistent.
Fourthly, the adolescent likes technology. Montessori says that the adolescent must be accustomed to using machines. "Civilization has given [humans], by means of the machine, power much greater than [their] own." Using machines well is part of the adolescent's entree into society. They love their calculators and the challenge is to help them find the balance between the calculator as an appropriate tool and a crutch that they use for the simplest addition problems. As Montessori says, there is an important moral lesson learned in working with any machine - who is in charge, the person or the machine? The power machines have given us ought also "to create new duties, always a higher morality." As the student embraces the use of the calculator and computer for doing mathematics, he is also engaged in finding the balance between using the machine as a tool and using the machine as an end in itself.
Fifthly, the adolescent is practical, in so far as the practical is focused on the self. This is an intensely introspective, self-absorbed age. We know that the adolescent is an eager consumer, but that is a surface view that masks the adolescent's genuine and fundamental interest in economics. Understanding economics is one way that the adolescent finds and understands his place in society. That is why the work at the Land School is so important and why the Junior High marketplace is a high point of the year for so many students. Developmentally, these young people are interested in the exchange of money and goods because those transactions represent the bridge into adult society. Economic participation signals acceptance and power in the wider society.
Montessori called the adolescent a "social newborn" who has experienced a birth to a new life. For the young person mathematics is a powerful tool to show the way in this new life.
This article appeared in the Fall/Winter 1999 issue of the Lake Country School Courier
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