How to introduce numbers 1-12 to young children for the first time!
HOW WILL CHILDREN BENEFIT FROM THE ALBUM
THE QUALITY OF NUMBERS
- The content on this Album is from a recognized Waldorf Teacher Resource which has been widely used internationally.
- The Album was produced and created by an experienced Waldorf teacher according to Waldorf educational principles.
- The stories and songs give children an understanding and reference point of where numbers came from and how it originated
- Children learn that numbers are not just dead and abstract concepts but something that is living and intimately related to themselves and the universe around them.
- Children learn about the oneness of the universe for instance there is only one earth and one me. They learn about the duality of twoness for instance dark and light, day and night, hate and love and so on.
- Many studies have shown that musical activity improves mathematical ability in children.
- Many studies have shown that rhythmical activity develops the will of the child and their ability to persevere in the face of obstacles.
- Artistic activity is deeply embedded in human consciousness and is imperative for developing the emotional and feeling life of the child. The stories and songs appeal to this aspect of children’s psyche and establish a healthy emotional relationship towards numbers. It cultivates a natural curiosity and love towards numbers that will stay with them as they grow older.
- Children become aware of the mystery of numbers and it becomes part of their unending journey of discovery. This imbues the study of numbers with a sense of meaning and purpose.
- The singing and reciting of the verses and stories will enhance the children’s language and oral abilities.
- Through the music, singing, reciting and rhythmical activity numbers become part of the children’s bodies and beings.
- Working artistically with the content on the album for instance creating pictures of the stories using various artistic media as well as dancing, walking and clapping rhythmically along, develops gross and fine motor skills.
- Although professionally done and beautiful to listen to, the Album is organic and authentic. Children relates to it immediately, knowing instinctively that the teacher and children singing on the CD could have been just as well their teacher, they themselves or their class mates singing.
- The songs are simple and easy to learn because they are all pentatonic and in harmony with the children’s current developmental phase. Adult harmonies and tunes are much too complex for the the young child’s mind and understanding and should only be introduced around the age of nine.
- Although every effort was made to compose the arrangements and harmonic accompaniments as beautiful as possible, it was done simply for the important reason that it doesn’t distract children from the actual purpose of the album which is the learning of the songs.
- The stories although imaginative do not contain too much information and are easy to remember and learn.
WHAT PARENTS AND TEACHERS ARE SAYING ABOUT
THE QUALITY OF NUMBERS – CD AND SONGBOOK
“This is a wonderful CD and a great help in the classroom. The songs and poems are really easy to learn and the children love it. I found it very helpful when I taught Class 1 a couple of years ago, and still use some of the verses and songs. I can really recommend this CD for all primary school teachers!” Elizabeth Swanepoel
“The songs and the stories are really beautiful, meaningful and easy to learn. The material is deeply grounded in Waldorf pedagogy and makes an excellent resource and teaching aid.” Monica Robinson
“What a fresh and original presentation! It catches the imagination in a miraculous way.” Hes van Vuuren
“Thank you so much!! I want to say that my children LOVE your materials, best investment I have made so far in our home school. Beautiful!! Will you publish / come out with more materials.” Marie Löfstedt
“As a typical right brain, at school I believed that I had no affinity or liking for maths, ending up with a complete emotional block against it. What a pity I had no exposure to Waldorf Education! And holding this CD in my hand, I could not help but think: “If only…!” Stephan Spies
“We absolutely love this. The story of each number is beautiful and the songs are as well. Thanks for putting this together!” Liliana Varela-Tello
“The songs on the CD are catchy and fun. A must for all homeschooling moms and kids! Trust me… maths CAN be fun!!” Ronel Kemp Wright
“The moving power of mathematical invention is not reasoning but imagination.” — Augustus de Morgan
READ MORE ABOUT THE QUALITY OF NUMBERS
In Waldorf Education grade one children are introduced to the quality of numbers in their very first arithmetic block. What does the ‘quality of numbers’ mean and why is it the first thing Waldorf Education teaches children on the outset of their mathematical journey? There is a profound significance and brilliance to this method and to understand it we need to have knowledge of the following:
First we need to look at the historic development of numbers and counting from the earliest times. Secondly we will look briefly at the development of the philosophy of numbers and how symbolic meanings of numbers came about – especially from the point of view of the early Greek mathematicians and philosophers. Thirdly we will reflect on how Rudolf Steiner viewed mathematics and his instructions on the correct way of teaching numbers to children. Once you have read it you will see how it all forms a beautiful unity and why it is simply a brilliant way of introducing numbers to children for the first time. It is organic, beautiful and in harmony with human development.
A Brief History of Numbers and Counting
Do we need numbers to count?
What is the human being’s ability to grasp numbers without counting? For instance if we look at a group of identical objects will we be able to tell how many there are at a glance or do we need to count them first.
Through observation and experimentation it has been concluded that the ability of animals to grasp quantities is limited. It has been established that certain animals or bird species can distinguish up to four objects but get confused when there are more than four.
Without the ability to count according to a number system, the ability of human beings to grasp quantities, is more or less on the same level. The largest number the human brain can comprehend without counting or estimating is four. Most people can identify five elements in a group by quickly counting them. However everything beyond five can only be an estimate or a guess and if we would like to know how many elements there are, we will have to count it.
When people were still in the hunter-gatherer stage they lived in a societal structure which had little need for counting. As a result they only invented words for the numbers they could grasp at a glance. So they had words for one and two (some tribes would go up to three and four) and quantities beyond that were refer to as many. This still holds true in the hunter-gatherer tribes that managed to survive and still exist today. They are (among others) the Aranda of Central Australia, the people of the Murray Islands in Torres Strait north of Australia, the Indians of Brazil and in Tierra del Fuego, the Abipón of Paraguay and the Bushmen of Africa.
The need for counting and number systems came with the idea of ownership. Hunter-gatherers encountered large numbers of animals but as long as there were many – enough for food and clothing – they saw no need for counting them. However when the idea of ownership and trading came into being, the owner had to know the exact amount of animals in the herd.
Still they didn’t invent words for counting as we do today. They used a pebble system. A flock of forty sheep would be represented by a sealed container of forty pebbles ensuring a safe delivery during a transaction for instance. For very large numbers pebbles of a different shape were introduced. This system of counting was used in many early societies of all continents for many millennia. Interesting to note that Calculus is Latin for “pebble”. So to calculate means “to move pebbles.”
With the introduction of pebbles of a different shape, number systems came into being. Our decimal system is a number system and it uses the base ten. So we would use pebbles of different shapes for units, tens, hundreds, thousands etc.
All evidence shows that humans used their fingers when they started counting. Our decimal system’s number base is ten and based on our ten fingers. Using body parts as a base for a number system comes naturally. People in China use only one hand to indicate numbers. They can communicate any number from one to ten, using the five fingers of one hand. They form different shapes representing the different numbers. This is very useful in the stock exchange when the broker is talking on the phone and having to give instructions to buy and sell at the same time.
In the Middle East they use a method which is still in practice today. They use the thumb to point to the three different parts of each finger. So four fingers on one hand cover the numbers one to twelve. From this followed the duodecimal system which uses twelve as an alternative base for a natural number system derived from our bodies.
A logical extension of the duodecimal system is the sexagesimal system, which uses the base 60. It uses one hand to count from 1 to 12 and the other hand to indicate the multiples of 12.
In some regions of temperate climate where people do not wear shoes and therefore can also use their toes for counting, a number system with the base twenty developed. This number system is called the vigesimal system.
The Philosophy of Numbers
What exactly is the nature of numbers and mathematics? There are many obvious applications and uses for numbers and math in an advanced society but do numbers have a deeper mystical significance? Do they symbolically point to universal truths that can only to be expressed eloquently in numbers and mathematical expressions. These questions have been asked since ancient times and have led to many different branches of thought trying to find the elusive answers to the pertinent questions regarding the true nature of numbers.
The Early Greeks
The early Greeks (Pythagoras, Socrates,Plato, Aristotle etc) saw numbers as more than just mere numbers. They believed that there was an intrinsic spiritual quality to numbers – a symbolic value that pointed to a deeper existence more perfect than this one. Individual numbers contained certain universal spiritual truths specific to each number and was related to the creation of the cosmos.
Pythagoras who is perhaps most famous for his Pythagoras theorem, made many great contributions to the history of mathematics, sciences, music theory and astronomy. To him the universe was ruled by numbers and numbers were the substance of all things. His famous quote says: “All is number”. He also believed geometry is the knowledge of the eternally existent. Plato and Aristotle were influenced by his teachings. Plato who was a student of Socrates, believed that numbers were actual things that existed in a perfect dimension. To him numbers were the highest degree of knowledge. In fact he believed it was knowledge itself. Plato had a particular passion for Geometry and believed that it contained the secrets of the universe. He felt so strongly about the study of geometry that there was an inscription at the door of his Academy in Athens saying: “Let no man ignorant of geometry enter here.” Plato’s perception of numbers and mathematics had a deep influence on how people view mathematics over the centuries and some people adhere to Plato’s view of numbers up to today. Aristotle studied at Plato’s academy in Athens and also believed all the mathematical sciences were about order, symmetry and limitations and that these were the greatest forms of the beautiful.
All major religions reflected this belief of an intrinsic spiritual symbolic quality inherent to a specific number and still reflects it today especially in their esoteric teachings. In all the major religions Christianity, Islam, Buddhism, Hinduism and Judaism etc – you will find that numbers still hold special symbolic and spiritual meanings.
When you compare the meanings of numbers given by the different traditions you will see that some numbers have more or less the same symbolic meaning across the spectrum. Other numbers have different symbolic meanings in different traditions.
A brief summary of the most common symbolic meanings attributed to numbers zero to twelve in major religious and esoteric traditions:
Zero is an empty circle. Its outer represents that which comes before birth and after death. Before the one – symbolic of the one from which all earthly creation sprung – there is that which comes before. It can be referred to as the mystery, void, nothingness or the eternal. However at the same time zero also points to the totality of life contained within the circle.
Zero is a very powerful number and symbolic of the Cosmic Egg occurring in many traditions and mythological creation stories. It is said that it can bring great transformation about.
People counted and used numbers for hundreds of years before they figured out the use of a sign that depicted the value of nothing. Zero appeared and disappeared throughout the ages until it was finally adopted in formal mathematics. Today zero is used in two different ways. Firstly it serves as a empty place indicator in our number value system. For instance in the number 1705, the zero holds the place of the tens in its absence and ensure that the rest of the numbers are placed in their correct positions. The second use is as the number itself – meaning nothing.
One is seen as a noble number. It refers to the whole. It is also seen as representing primordial unity. In most traditions one refers to the all inclusive one namely the God-head or the Source.
One is divisible only by itself. From the one comes the many. In Taoism it is said Tao begets One, One begets Two, Two begets Three and Three begets all things. Pythagoreans saw one as the very essence of things and called it the Monad.
Two is mostly seen as a symbol for duality, opposition, separation and antagonism. It is the first number which points to the separation of the One and therefore symbolizes the corruptibility of things. It is also a symbol of love and generosity referring to the duality of the giver and the receiver. It refers to sexuality – male and female. It refers to opposites, day and night, hot and cold, white and black etc. Two is about balance. The balance between the two points of extreme. It is also about division.
Three represents the Holy Trinity in the Bible – God, the Son and the Holy Spirit. It also symbolizes a return to God. The duality which splitted from the one can now in three, relate to the One (God-head) again. It represents mother, father and child. It refers to beginning, middle and end as well as birth, life and death. The sun, moon and earth are also three.
Four is associated with the earth. It is symbolic of order and rationality. It also is associated with the concept of foundation. There are the four elements: earth, fire, air and water. One thinks of the four seasons – winter, summer, autumn and spring. There are four directions (cardinal points) – north, south, west and east. On the fourth day of the week of creation the material world was created according to the Bible. The Bible also refers to the four corners of the earth. There were four rivers flowing out of Eden – Genesis 2:10.
In Christianity five is the symbol of God’s grace, favor and goodness towards men. The Ten Commandments contains two sets of five. The first five commandments set the rules of conduct in our relationship with God, and the last five set the rules of conduct in our relationship to others.
It is also generally accepted that there are five senses: sight, hearing, smell, taste and touch.
Five is often seen as the number of the human being. It symbolizes a human figure who is stretching out its four limbs and head in the same way as the pentagram or pentagon in geometry.
Throughout the ages the pentagram or the pentagon had special significance for people. Many movements have attached different symbolic values to this geometrical figure. I will mention a few of them.
According to many the pentagram is a symbol of the magical and the sacred. According to popular belief Eve gave Adam the apple of knowledge. If you cut an apple in half, you will find that the seeds in each half has the shape of a pentagram. Hence the the pentagram became a symbol of the wisdom and power to make choices but also the fall of man.
To the Pythagoreans the pentagram was a sacred symbol and it symbolized health and light. It represented higher knowledge and the five points symbolized spirit, air, fire, water and earth.
The Freemasonry uses the pentagram as a symbol called “Blazing Star”.
For Christians it symbolizes the five wounds of Christ. They see it as a symbol of Christ, as “Alpha and Omega”. It symbolizes the star of Bethlehem – a symbol of guidance. It is important that the star has one point upwards.
Turned around with two points up the pentagram symbolizes the devil. Satanists use the pentagram with two points up as their symbol. They use the inverted pentagram with two circles around it and the head of a goat inside it.
The Pythagoreans saw six as the first perfect number. In mathematics a perfect number is when all the numbers a number can be divided by (called the divisors) – excluding the number itself – are added together. The sum of the divisors must be equal to the number itself. If we take six as an example it will be 1+2+3=6. Another property of a perfect number is that when you add the number itself to the sum of the divisors for instance 1+2+3+6 = 12, and divide the answer by two, 12÷2=6 the result will be the number itself. In this case 6.
In the Bible humans were created on the sixth day. It is the number of the human being and its imperfections. Six is also associated with other created creatures. It represents human systems of government.
In other traditions six is related to earthly matters. This is because the number is often found in nature for instance the six legs of an insect, six-sided snowflake, some water crystals, the cells of a honey comb are shaped in the form of six-sided hexagons.
The star of David is the ancient Jewish symbol of a six pointed star. The six points are frequent symbols of health and healing. It also represents equilibrium, harmony and balance.
Six is a powerful number which reminds us to keep our thoughts and actions as positive as possible.
There is no doubt that there is something very special about the number seven. It is widely revered as a holy number. The number seven occurs countless times in mythology, stories, legends, world religions, alchemy, astrology, philosophy of mathematics and history. Its symbolism is always more or less the same. It refers to completion, totality, perfection, plenty, security and synthesis.
In the Bible for instance the number seven is symbolic of completeness, divine perfection or something that is finished. It is symbolic of the creation week. The earth and everything in it was created in six days. The seventh day is a day of rest and communion with God. The six (six days of physical creation) plus one (the God-head) equals seven, The seven is then symbolic of the fact that completeness and perfection are only possible when in relationship and communion with God.
In Buddhist thought seven is the number of ascent and of the ascension to the highest. It is about attaining the center. The seven steps of Buddha symbolize the ascent of the seven cosmic stages transcending time and space in order to reach enlightenment.
Pythagoras revered seven as the most spiritual number of all. Seven is a cosmic number with three of heaven and four of the world.
There are seven notes in a major and minor scale. The different frequencies of sound is closely related to the formation and creation of the universe.
In the Kabbalah and the Bible the number eight symbolizes that which is higher than the natural order – one step higher than nature and its limitations. Eight is seven plus one. In seven days God completed the creation. Eight is what comes after the completion of the earthly creation. This is why it is often connected to miracles. Eight also signifies a new beginning or an era.
Eight lying down is the sign of infinity and represents everything good that is infinite for instance infinite love, infinite supply, infinite energy, infinite time etc. In other words the number eight is symbolic of infinite abundance and its underlying causes.
Nine is the last unit before ten. It is a sacred number. In the Bible nine is symbolic of divine completion or finality. Christ died at the ninth hour of the day in order to bring salvation to mankind. There are nine fruits of the Holy Spirit: faithfulness, gentleness, goodness, joy, kindness, long-suffering, peace, love and self-control. It is associated with heaven.
The Pythagoreans called the number nine the “Ennead”. They thought of nine as unlucky as it is one short of ten. The Pythagoreans considered ten to be Perfect.
In the Kabbalah nine is associated with beneficence. It takes nine months for a human baby to be born. It is painful to give birth but necessary because it is a new life that enters the world.
To the Pythagoreans the holiest number of all was the ten or the “tetractys”. They saw it as a triangular number composed of the sum of one, two, three and four.
The following prayer from the Pythagorean order illustrates how revered the number ten was:
“Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.”
In the Bible ten is also seen as a complete and perfect number in the same way as three and seven and twelve. There are ten God given commandments and therefore ten is associated with responsibility, law and order.
Ten marks the completion of a round in our decimal system. We count in tens. A hundred consists of ten tens. A thousand consists of ten hundreds and so on. It represents the completeness of order. It implies that nothing is wanting and that the cycle is complete.
The Bible views eleven as follows: Eleven comes just after ten. Because it is one more than ten it is seen as the first number that unbalances ten. Therefore it is associated with rebellion, chaos, disorder and judgement.
In other traditions it is viewed quite differently. In numerology it is one of the master numbers because it is the double digit of the same number. One represents the Source. So one doubled represents double the power of one.
Twelve represents a complete time cycle. Because eleven comes just before twelve it contains a sense of urgency to complete things before the time runs out e.g. the eleventh hour.
In other traditions eleven is associated with balance. It is symbolizes male and female equality and the perfect balance between opposite forces.
Twelve is one of the perfect numbers, as are three, seven and ten. It is associated with completion and wholeness.
The Bible see it as a perfect number and associate it with God’s power and authority. It is seen as the perfect governmental foundation.
Twelve is the number of space and time. Time is measured by two groups of twelve. There are 12 x 2 hours in a day and night cycle. There are twelve months in a year. Therefore it is commonly viewed as the symbol of cosmic order.
A brief mention of the development of the philosophy of mathematics in the nineteenth and twentieth century.
During the nineteenth century there was a general shift in scientific circles towards a more empirical approach. The Platonian view of mathematics was rapidly losing support and in light of this there arose a need to formulate a philosophical theory of mathematics free from Platonian influence. Four theories arose from this movement early in the twentieth century. I will just briefly mention them here: logicism, formalism, intuitionism and predicativism. However in the years before the second world war it became clear that they fell short in many aspects. As a result there was a movement to return to Platonistic views. This brings us to how Rudolf Steiner views mathematics and its place in human knowledge.
Mathematics and its Spiritual Place in Human Knowledge
Steiner adhered to the historic Greek mathematician Plato’s understanding of the place of mathematics in human knowledge. According to Plato there is a spiritual realm that exists beyond physical existence. This spiritual realm is called the Realm of Forms (also called the Realm of Ideas or Realm of Ideals) and the physical world and its forms are only shadows or images of this true reality. Plato believed that human beings could know nothing of this true reality if their thoughts were still permeated by what the senses transmit. Plato asked how can man escape or move beyond sensory perception? This question played a vital role in the education of the spiritual life of his students.
With a little reflection one will soon realize how difficult it is for any person to move beyond material perceptions. Even if a person had to withdrew into themselves – allowing no sense perceptions from the outside – residues of sensuous perception still lingers in his mind. Also another problem is that a person with no proper spiritual training would face the ‘void’ or ‘nothingness’ if they managed to move completely beyond the senses. He would face the absolute annihilation of consciousness. However just as the human being has certain organs to perceive sensory stimuli, it also has certain organs – if developed – that can perceive spiritual truths. If these organs which can perceive spiritual truths are developed and are functioning, a human being can move beyond the sensory realm into the spiritual world. It would not face nothingness. It was such a mind that Plato was striving for. A trained mind that was capable of leaving this sensory world behind and could move into the spiritual world.
Plato looked at mathematics as a way to train the mind for existence in the spiritual Realm of Forms. The mathematical images hover over the border-line between the material and the purely spiritual world. Take a circle for instance. If we think of a circle we do not think of a circle that has been drawn on paper. We think of a circle that we have come across in the material world for example the sun, the moon etc. It does not specifically relate to one sensory experience. Instead it relates to many occurrences. Steiner says and I quote:
“When I think mathematically, I do indeed think about something my senses can perceive; but at the same time I do not think in terms of sense-perception. It is not the material circle which teaches me the laws of the circle; it is the ideal circle existing only in my mind and of which the concrete form is a mere representation. I could learn the identical truths from any other sensible image. The essential property of mathematical perception is this: that a single sense-perceptible form leads me beyond itself; it can only be for me a representation of a comprehensive spiritual fact.”
In the way Steiner described above is it possible to learn about super-sensible concepts by way of the sensory world. This was very important to Plato. In order to truly know the essence of an idea we need to visualize it in a purely spiritual manner.
“Learn to emancipate thyself from the senses by mathematics, then mayest thou hope to rise to the comprehension of ideas independently of the senses” – Plato
How to teach children about numbers and counting for the first time.
Instructions and suggestions by Rudolf Steiner
The development of the child from infancy until adulthood follows the development of humanity from the earliest ages up to today. People first used their body parts (hands, fingers, toes, eyes etc.) as well as the concrete environment to learn how to count and to develop number systems. In the same way young children need to follow this instinctual way of learning and discovering numbers. They first need to link the numbers to their body and the concrete world around them. Numbers need to be concrete and be a part of their very humanity. As the mathematical journey continues it will evolve naturally to the abstract and the thinking realm. During the early years the approach needs to be concrete and part of their physical existence.
Abstract thinking of teachers causes difficulties in teaching
Steiner noticed that many teachers live in an abstract world and are removed from life itself and that this abstraction is the cause of many difficulties in education and teaching. How many of us remember how we learned to count things for instance? Most of us have no idea what we do when we are counting. We just do it automatically without much thought.
Many theories on how to teach numbers and counting to children in this abstract manner have been formulated. Teachers act on those theories and get external results. The children seem to be able to count just fine. However the problem is that the child’s being is not touched with this kind of counting. There is no connection to real life. It is too abstract. Just think of the inventions of the abacus and the bead-frame. It might has its applications in later life but to a seven year old these inventions are just too abstract. It is not applicable to the developmental phase of a child that age. Calculating machines are even worse and shouldn’t be brought near a young child learning to count for the first time. They prevent children from dealing with numbers according to their nature.
Counting and numbers should come from life itself
Counting and numbers should come from life itself. If we think at how humans historically started to count. They used their body parts – their fingers and toes etc. Counting and numbers have symbolic values and have been connected to happenings in the natural world since the earliest times. For instance one is related to the whole that can contain many. It is connected to that aspect in the world that exists as a one or a unit. Two is related to that aspect in the world that has a natural two-fold quality. It refers to the duality of life for instance day and night, hot and cold, you and me, male and female etc. Three is related to the three-fold quality in the world for instance mother, father and child, the divine Trinity etc. So it continues. This relationship of numbers to certain qualitative aspects of creation is refer to as the quality of numbers.
Wouldn’t a young child find the quality of numbers hard to understand you may ask. Steiner is very clear on this point. He says and I quote “…it is supremely important to know from the beginning that you should never expect a child to understand every single thing you teach. Children must take a great deal on authority, but they must take it in a natural, practical way.” When people in adulthood finally come to an understanding of what they have been taught as a child, it awakens new life. This is of great significance. Things should never be trivialized in order to “bring it nearer to a child’s understanding”.
Applying the pedagogy of Rudolf Steiner in the classroom.
Lets look at what we need to keep in mind in order to prepare a lesson according to Waldorf educational principles.
a) The Threefold Approach to Teaching in Waldorf Education: Thinking, Feeling and Willing.
b) The Developmental Phase of the Child
c) Living Concepts
Teach living concepts. Living concepts are capable of growth and not finished. The soul should be constantly called to work on a concept again and again. Concepts which are finished in themselves, are dead and no food for the soul. Living concepts always have a sense that they are not finished. There is still more to be done, more to learn, more to figure out and more to discover.
d) The Seven Lively Arts
In ancient times scholars would study the seven Liberal Arts in order to become educated. It developed late in the medieval times and became central to university education.
From the Liberal Arts the Seven Lively Arts developed. There are many applications but in Waldorf education it is understood that they refer to the following: Drama, Drawing, Movement, Music, Modeling, Painting, Speech.
Rudolf Steiner placed a high priority on artistic activities. To him the arts were crucial to human development. Not only does artistic activities serve to strengthen the will but it teaches the mind to think imaginatively and engages the feeling life. Artistic activities are used and incorporated extensively in the Waldorf classroom. Please see my blog on Rudolf Steiner and the Arts.
e) Other Things to Keep in Mind
- Although there is a new awareness in the process of waking up, the child is still imitative in many ways.
- The child will follow and trust healthy authority naturally.
- Important to remember learning takes place in the body. The head is a mere spectator. Everything needs to be related to the body. The natural world should be presented as an extension of the human being.
- One should always teach from the whole to the parts. The child should have a sense of the whole before moving onto the parts.
- Everything you teach should be permeated with the qualities of truth, beauty, goodness and wonder.
- The teacher should teach the lesson with a natural authority that flows from kindness and love. A lesson taught with kindness and love will have a much deeper impact on the child than a lesson taught with impartiality and distance.
Practical example of a teaching lesson introducing the quality of numbers according to Waldorf educational principles, following the Three Day Rhythm:
Teaching the child about the number One.
The child should discover and derive numbers from real life and the concrete – much in the same way as our early ancestors discovered numbers through their ten fingers, ten toes and the occurrences of unity, duality, threefoldness etc that they witnessed in the natural world around them.
The children are standing or sitting in front of you and everyone is waiting for the lesson to begin. The important part is to bring the concept of a unit or a whole across. A one that forms a unit and cannot be divided. Examples of this would be the sun, the moon and of course the child itself. You can ask them: What is there only one of in the world? Let them wonder and guess about it for a while. Then you take a stick and cut or break it in half. You can then ask: Can I do this to you? Of course you can’t. In this way the children will start waking up to the idea of a unit. You can go ahead and encourage the children to find other examples in the world that can only exist as a unity in themselves for instance the sun and the moon etc. Perhaps they can go for a walk outside to see if they can find anything that is a unit in itself. Do not worry if they do not get it at once. This is all about teaching living concepts that will keep them wondering and growing.
Then you can go on and make the children aware that one can contain many. For instance Class One contains 22 children. Class One is a unit but contains 22 children within its unity. It is important to remember to always work from the whole to its parts. There is first the unit or the whole and then the parts. The parts function within the unit. Rudolf Steiner said the following: “This is something that is ONE. Now you divide it like this, and you have something that is TWO. It is not two ONEs put together but the two come out of the ONE. And so on with three and four. Thus you can awaken the thought that the ONE is really the comprehensive thing that contains within itself the TWO, the THREE, the FOUR. By teaching the child in this manner the child will have concepts that are living and thereby come to experience something of what it is to be inwardly permeated with the element of number.”
Important to remember to guide the child to discover these qualities and occurrences of one in the concrete world themselves.
In the next part of the lesson you can start incorporating the seven lively arts. Most of the knowledge in Waldorf education is brought to children through narrative rather than instruction. So you can tell the children a story about number one. You can also incorporate the singing of a song into the storytelling. Children are essentially musical beings and the song will reach deep into their souls. This will engage their imagination and feeling life. The story will place the unity of one in context within the broader world and the child will discover that one is part of his everyday life and surroundings. Let them take the story, the activity and the song into their sleeping night.
On day two you should first let the children do a recall on what they did the day before. In this case they should recall the story and the activity about the number one. After the recall you can proceed to introduce the number two in the same concrete manner as you did number one the day before. We derive two from the body first using the two hands. We show the difference between the two hands and their interaction as opposed to the body that can only move as a unit. Then we move on to the different occurrences of duality in the natural world for instance black and white, night and day, dark and light, you and me etc. One can be imaginative. We can then tell the story and sing the song about two. Afterwards the children can draw pictures about the unity of one they were introduced to the previous day.
On day three you will first do a recall on the previous days work. The children must remember the activity and story about two. Then we can proceed with three. First an activity for instance we can have two friends and then call a third child over to join them. This is something the two hands cannot do. Then you can tell them the story about three and sing the song. Then you can proceed to show the child the sign for one. We first teach them the Roman numerals. So the sign for one will be I. These signs comes naturally to children. From the earliest ages numbers were depicted with strokes. So they instantly understand I, II, III and IIII. One can derive V from one hand with its five fingers. You can count to four – the four fingers stretched out but the thumb tucked away. When you reach five you stretch the thumb out. It makes a natural V with the hand. In this way you can derive the form from the body itself. So on day three they can draw the Roman sign for one – I – beautifully in their first main book for math and draw a picture to illustrate its unity.
And so the teacher can continue with the three day cycle until the numbers 1-12 have been introduced. The time it takes to do this will differ from class to class and from learner to learner. If there is still some time left in the block, one can continue and introduce the quality of some significant numbers beyond ten.
Confirming and strengthening what has been taught in main lesson in lessons during the school day where the seven lively arts are presented as subjects onto themselves:
In circle time the children can recite verses about the unity of one rhythmically. They can learn to sing the song about one in the music class or they can play the melody on the recorder. In a drama activity they can act out the images and concepts in the verses. During beeswax modelling they can model a sun or the moon, themselves etc. In the painting class, they can paint the moon, the sun and the earth for example. The possibilities are endless. As learning is done with the body, expressing what has been learned through artistic activities is a powerful way of reaffirming and instilling knowledge in the child’s being.
To introduce children to the quality of numbers in the right way is crucial. It sets the foundation from where the young mind will view numbers in the years to come. When introduced correctly children will develop an intuitive understanding of what one, two, three etc are. They will wake up to the element of number in the universe. Teaching them with living spiritual concepts will result in a sense of awe, reverence and wonder. There will be a sense of meaning and purpose and a connection to the universe.
The actual memorizing of counting in sequence from a one to a hundred and further, is another process and should be approached differently. It is done with rhythmic clapping and movement. It follows after or alongside the introduction of numbers in this qualitative way. More about this later.
Rudolf Steiner. 1982. The Kingdom of Childhood. [ONLINE] Available at: http://wn.rsarchive.org/Lectures/GA311/English/AP1982/KinChi_index.htm.
Rudolf Steiner. 2002. Mathematics and Occultism. [ONLINE] Available at: http://wn.rsarchive.org/Lectures/19040621p01.html. [
Rudolf Steiner. 1971. Theosophy. [ONLINE] Available at: http://wn.rsarchive.org/Books/GA009/English/AP1971/GA009_index.html.
George Ifrah. 2000. The Universal History of Numbers. [ONLINE] Available at: https://archive.org/stream/TheUniversalHistoryOfNumbers/212027005-The-Universal-History-of-Numbers#page/n1/mode/2up.
Horsten, Leon, “Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2016/entries/philosophy-mathematics/
J.J. O’Connor, E.F. Robertson. 2000. A History of Zero. [ONLINE] Available at: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html. [Accessed 20 April 2017].
FAMOUS MATH QUOTES
“The moving power of mathematical invention is not reasoning but imagination.” — Augustus de Morgan
“All mathematicians share … a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics. ” — Martin Gardner
“Beauty is the first test; there is no permanent place in the world for ugly mathematics.” — Godfrey Harold Hardy, A Mathematician’s Apology (1940)
“Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God. ” — from The Man Who Counted by Malba Tahan
“Mathematics is the music of reason.” — from A Mathematician’s Lament by Paul Lockhart
“The book of nature is written in the language of mathematics.” — Galileo