Sophia Institute online Waldorf Certificate Studies Program

Waldorf Methods/Reading and Math 2
Introduction
Language is our most important means of mutual understanding and is therefore the primary medium of education. It is also a highly significant formative influence in the child’s psychological and spiritual development and its cultivation is central to the educational tasks of Steiner/Waldorf education. It is the aim of the curriculum to cultivate language skills and awareness in all subjects and teaching settings. Clearly the teaching of the mother tongue has a pivotal role within the whole education.
Mathematics in the Waldorf school is divided into stages. In the first stage, which covers the first five classes, mathematics is developed as an activity intimately connected to the life process of the child, and progresses from the internal towards the external. In the second stage, covering classes 6 to 8, the main emphasis is on the practical. Course Outlines
Waldorf Methods/Reading and Math 1
Lesson 1: Introduction Lesson 2: Reading/1st Grade Lesson 3: Reading/2nd Grade Lesson 4: Reading/3rd Grade Lesson 5: Math/1st Grade Waldorf Methods/Reading and Math 2 Lesson 1: Math/2nd Grade Lesson 2: Math/3rd Grade Lesson 3: Reading/4th Grade Lesson 4: Reading/5th Grade Lesson 5: Reading/6th Grade Waldorf Methods/Reading and Math 3 Lesson 1: Math/4th Grade Lesson 2: Math/5th Grade Lesson 3: Math/6th Grade Lesson 4: Reading/7th and 8th Grade Lesson 5: Math/7th and 8th Grade 

Tasks and Assignments for Waldorf Methods/Reading and Math 2.2.
Please study and work with the study material provided for this lesson. Then please turn to the following tasks and assignments listed below.
1. Study the material provided and look up other resources as needed and appropriate.
2. Create examples of curriculum that addresses the learning method and content appropriate for the age group in question as follows. Curriculum examples should include outlines and goals, activities, circle/games, stories, and illustrations/drawings:
Create 2 examples for this age group.
3. Additionally submit comments and questions, if any.
Please send your completed assignment via the online form or via email.
1. Study the material provided and look up other resources as needed and appropriate.
2. Create examples of curriculum that addresses the learning method and content appropriate for the age group in question as follows. Curriculum examples should include outlines and goals, activities, circle/games, stories, and illustrations/drawings:
Create 2 examples for this age group.
3. Additionally submit comments and questions, if any.
Please send your completed assignment via the online form or via email.
Study Material for Waldorf Methods/Reading and Math Lesson 2.2.
Arithmetic and Mathematics/Class 3
General Observations and Guidelines
Mathematics in the Waldorf School is
divided into three stages. In
the first stage, which covers the first five classes, mathematics
is developed as an activity
intimately connected to the life process
of the child, and progresses from the
internal towards the external. In the second stage,
covering Classes 6 to 8, the main emphasis is on the practical ... The ninth year onwards is characterised by a transfer towards a rational point of view.
Thus writes H. von Baravalle, the first mathematics teacher at the Waldorf School in Stuttgart.
Classes 1 to 5
Here two questions must be answered:
1. How should the first mathematical concepts be tackled?
2. What is the psychological basis on which to build them?
In answer to question one, careful scrutiny shows that the teaching of arithmetical and geometric concepts is connected to the consciousness and activity of the child's movement organism. Counting is inner movement by which outer movement can be observed. E. Schuberth calls this 'the sensory content of mathematics teaching" The results of Piaget's research on the development of intelligence in children also point in this direction: in the 'concrete operations stage' (twelve or thirteen years old), children still carry out movements when they want to connect one thing to another. Anyway, these movements are connected to the physical objects from which the children can as yet barely free themselves, if at all.
This leads on to question two: if the development of mathematical concepts occurs in the static, concrete phase, our purpose must not be to 'generalise and abstract', but to 'make concrete and look at individual cases: 3 This defines the means whereby it is possible to avoid confronting children with abstract logical structures, but rather to immerse their whole capacity for experience in mathematics. We can now refer to form drawing where the consciousness necessary for using mathematics is nurtured and practised. This physical experience is a basis and assumption for a healthy immersion in the 'formal operations stage' (Piaget). The rule 'from hand, through heart to head' (which is meant by the child's above mentioned 'whole capacity for experience') makes it possible for the children to bring their own capacities into play.
It is evident that the best, most fruitful questions about concepts and explanations come from pupils who do not pose questions using a quick intellectuality, but a capacity for involved feeling, which allows clarity to come into thinking:'
To this concrete approach in mathematics at primary school level we should add something further which does not depend on the element of movement. This is the quality, one might say the identity, of the individual numbers.
As the accent above is on a quantitative approach to numbers, as the result of a brief pause in movement or even based on the movement itself, we need to place our introduction to qualitative number concepts beside those of quantity. We approach these qualities when we examine many examples where the number in question is really active in the world, as for example the number five in the flowers of a rose. Here we use the child's desire to question what lies behind the world and human creations, that is, to seek what lies behind the phenomena. The nuclear scientist W Heider referred to this when he said in a lecture:
One directs one's attention to qualitative phenomena, to characteristics which have something to do with the totality of the objects observed.
Steiner recommended taking this as the starting point for an introduction to number concepts:
We have gradually come to the point in the course of civilisation where we can work with numbers in a synthetic manner. We have a unity, a second unity, a third unity and we struggle while counting in an additive manner to join the one to the other, so that one lies beside the other when we count. One can become convinced that children do not bring an inner understanding towards this. Primitive human beings did not develop counting in this way. Counting began from unity. Two was not an external repetition of unity but it lay within unity. One gave us two, and two is contained in the one. One, divided, gave us three, and three is contained within one. If we wrote the number 1, translated into modern terms, one could not get away from the 1,for instance to 2. It was an inner organic picture where two came out of unity and this two was contained in the one, likewise three and so on. Unity encompassed everything and the numbers were organic divisions of unity. 5
This 'real' way of looking at numbers leads on to written numbers, to symbols. It is not a picture like that which should be used for introducing letters, but pictures of number qualities. The picture belongs to the being of the number, not to the outer symbolic form. At this point we indicate a further point of this 'qualityoriented' teaching: today, especially, when we face the results of a quantitative world view in ecological catastrophe and destruction, it is increasingly important to make a beginning of this kind in teaching mathematics.
By beginning with the concrete qualities of number and by working with the properties of movement in counting and calculating, children develop a kind of intelligence which seeks and finds the way to reality.
This brings us to the second stage: the approach to mathematics teaching mentioned earlier. Here we must deal with the practical use of calculation.
If calculating has been practised thoroughly enough during the first stage in the manner indicated, then applied calculation also gets a qualitative colouring. The forces of intelligence which are served by business mathematics as well as percentages and interest, are not 'valuefree', but can retain a colouring for balanced testing and judging. The human significance of what is thought out can and should be made clear. In this connection one should point to the suggestion by Steiner that elements of bookkeeping should also be taken up in mathematics lessons. To see what the general idea behind such an indication is, one should ask what skills may be developed through bookkeeping. There you will see how above all a moral method of trading can be decisively supported by these means.
All these themes can lead us on to other educational aims: inner mobility leads to imaginative ability in solving mathematical problems. Through the experience of number qualities the children experience trust and security: Number, world and human being belong together. The children can experience further security through the correctness of solutions to problems By this means they win some independence. For this reason mathematics is an activity suited to freeing children from the fetters of authority even if, at first, they depend on the help of the teacher. A final educational aim which should not be undervalued and which is connected with the latter is calculation. Calculation is not possible without regular practice, which also makes it an excellent medium for schooling the will. An explicit presentation of the third stage will be given under the heading Classes 9 to 12 and is therefore omitted here. Geometry as part of mathematics teaching begins in Classes 5 and 6 and is taught in separate main lessons. One of the principal intentions in this subject is to develop and nurture the ability to visualise space. The controlled security of directed movement and the estimation of proportions and relationships, practised in freehand geometry, is well prepared for by form drawing in Classes 1 to 4.
The establishment of skills, knowledge and techniques partly in connection with subjects is taught with increasing complexity related to age.
* Pupils should gradually learn to discover, to grasp mentally and to apply geometrical properties, and the practical, drawn, solutions representing them.
* Work with geometric drawing instruments should lead to clear and exact construction.
* Patience, care and precision should be developed, as well as independent creative work through enjoyment of drawing.
Thus writes H. von Baravalle, the first mathematics teacher at the Waldorf School in Stuttgart.
Classes 1 to 5
Here two questions must be answered:
1. How should the first mathematical concepts be tackled?
2. What is the psychological basis on which to build them?
In answer to question one, careful scrutiny shows that the teaching of arithmetical and geometric concepts is connected to the consciousness and activity of the child's movement organism. Counting is inner movement by which outer movement can be observed. E. Schuberth calls this 'the sensory content of mathematics teaching" The results of Piaget's research on the development of intelligence in children also point in this direction: in the 'concrete operations stage' (twelve or thirteen years old), children still carry out movements when they want to connect one thing to another. Anyway, these movements are connected to the physical objects from which the children can as yet barely free themselves, if at all.
This leads on to question two: if the development of mathematical concepts occurs in the static, concrete phase, our purpose must not be to 'generalise and abstract', but to 'make concrete and look at individual cases: 3 This defines the means whereby it is possible to avoid confronting children with abstract logical structures, but rather to immerse their whole capacity for experience in mathematics. We can now refer to form drawing where the consciousness necessary for using mathematics is nurtured and practised. This physical experience is a basis and assumption for a healthy immersion in the 'formal operations stage' (Piaget). The rule 'from hand, through heart to head' (which is meant by the child's above mentioned 'whole capacity for experience') makes it possible for the children to bring their own capacities into play.
It is evident that the best, most fruitful questions about concepts and explanations come from pupils who do not pose questions using a quick intellectuality, but a capacity for involved feeling, which allows clarity to come into thinking:'
To this concrete approach in mathematics at primary school level we should add something further which does not depend on the element of movement. This is the quality, one might say the identity, of the individual numbers.
As the accent above is on a quantitative approach to numbers, as the result of a brief pause in movement or even based on the movement itself, we need to place our introduction to qualitative number concepts beside those of quantity. We approach these qualities when we examine many examples where the number in question is really active in the world, as for example the number five in the flowers of a rose. Here we use the child's desire to question what lies behind the world and human creations, that is, to seek what lies behind the phenomena. The nuclear scientist W Heider referred to this when he said in a lecture:
One directs one's attention to qualitative phenomena, to characteristics which have something to do with the totality of the objects observed.
Steiner recommended taking this as the starting point for an introduction to number concepts:
We have gradually come to the point in the course of civilisation where we can work with numbers in a synthetic manner. We have a unity, a second unity, a third unity and we struggle while counting in an additive manner to join the one to the other, so that one lies beside the other when we count. One can become convinced that children do not bring an inner understanding towards this. Primitive human beings did not develop counting in this way. Counting began from unity. Two was not an external repetition of unity but it lay within unity. One gave us two, and two is contained in the one. One, divided, gave us three, and three is contained within one. If we wrote the number 1, translated into modern terms, one could not get away from the 1,for instance to 2. It was an inner organic picture where two came out of unity and this two was contained in the one, likewise three and so on. Unity encompassed everything and the numbers were organic divisions of unity. 5
This 'real' way of looking at numbers leads on to written numbers, to symbols. It is not a picture like that which should be used for introducing letters, but pictures of number qualities. The picture belongs to the being of the number, not to the outer symbolic form. At this point we indicate a further point of this 'qualityoriented' teaching: today, especially, when we face the results of a quantitative world view in ecological catastrophe and destruction, it is increasingly important to make a beginning of this kind in teaching mathematics.
By beginning with the concrete qualities of number and by working with the properties of movement in counting and calculating, children develop a kind of intelligence which seeks and finds the way to reality.
This brings us to the second stage: the approach to mathematics teaching mentioned earlier. Here we must deal with the practical use of calculation.
If calculating has been practised thoroughly enough during the first stage in the manner indicated, then applied calculation also gets a qualitative colouring. The forces of intelligence which are served by business mathematics as well as percentages and interest, are not 'valuefree', but can retain a colouring for balanced testing and judging. The human significance of what is thought out can and should be made clear. In this connection one should point to the suggestion by Steiner that elements of bookkeeping should also be taken up in mathematics lessons. To see what the general idea behind such an indication is, one should ask what skills may be developed through bookkeeping. There you will see how above all a moral method of trading can be decisively supported by these means.
All these themes can lead us on to other educational aims: inner mobility leads to imaginative ability in solving mathematical problems. Through the experience of number qualities the children experience trust and security: Number, world and human being belong together. The children can experience further security through the correctness of solutions to problems By this means they win some independence. For this reason mathematics is an activity suited to freeing children from the fetters of authority even if, at first, they depend on the help of the teacher. A final educational aim which should not be undervalued and which is connected with the latter is calculation. Calculation is not possible without regular practice, which also makes it an excellent medium for schooling the will. An explicit presentation of the third stage will be given under the heading Classes 9 to 12 and is therefore omitted here. Geometry as part of mathematics teaching begins in Classes 5 and 6 and is taught in separate main lessons. One of the principal intentions in this subject is to develop and nurture the ability to visualise space. The controlled security of directed movement and the estimation of proportions and relationships, practised in freehand geometry, is well prepared for by form drawing in Classes 1 to 4.
The establishment of skills, knowledge and techniques partly in connection with subjects is taught with increasing complexity related to age.
* Pupils should gradually learn to discover, to grasp mentally and to apply geometrical properties, and the practical, drawn, solutions representing them.
* Work with geometric drawing instruments should lead to clear and exact construction.
* Patience, care and precision should be developed, as well as independent creative work through enjoyment of drawing.
Classes 1  3
The dynamics of will activity should be internalised by the experience of countability. Motivation should be awakened through pictorial description of number qualities. This dual aspect is important: on the one hand it educates the bodily senses through experience of movement, unfolding of movement possibilities (both coarse and fine), and coordination exercises. On the other hand internalising of the expressed activities in soul activity (Le. calculation). Here the main medium for achieving this is the use of pictures. Through pictures, children can grasp internally what is intended. Pure symbolic, logical presentation can never achieve this. (Nevertheless one should always be conscious that calculation is aiming at a pictureless world, in contrast to the introduction of letters of the alphabet.) In order to be able to handle quantitative numbers freely, an inner numerical space needs to be created, in which one learns to move, rhythmically at first, with varied number patterns. This is achieved, amongst other means, by a memory developed by learning the timestables through rhythmic movement e.g. through clapping, passing bean bags or skipping. It appears important initially to approach actual calculation as concretely and visually as possible and to keep in mind the principle 'from the whole to the parts: This means that the right connection between analytical and synthetic thinking is produced. Work with the temperaments should be formed in the sense given in the ath discussion in Discussions with Teachers? At the end of Class 3 the pupils should have a confident grasp and clear view of numbers up to at least 1,020. This does not mean just the quantity or extent but equally the quality of the numbers.
Class 3
In mathematics lessons the approach is analytical in the sense of the reference by Steiner given above. Starting with the number 1 as unity, numbers (symbols) from 1 to 10 should be produced in a qualitative manner (see above), which are contained as manifold in unity. With written numbers one can begin with Roman numerals which are less abstract than the Arabic. Alternatively, the Arabic numbers can be introduced in pictures as with the letters of the alphabet.
* Mental arithmetic
* Sums using numbers up to 1,020 or 1,100
* Written addition and subtraction using several places (place value)
* Written multiplication using two place values
* Written division using units as divisor
* Up to 15 times tables; 10 times table up to 900
* Square numbers by heart as a sequence
Weights and measures (practical subject) and calculations with simple practical problems
* Mental arithmetic
* Sums using numbers up to 1,020 or 1,100
* Written addition and subtraction using several places (place value)
* Written multiplication using two place values
* Written division using units as divisor
* Up to 15 times tables; 10 times table up to 900
* Square numbers by heart as a sequence
Weights and measures (practical subject) and calculations with simple practical problems
Numeracy Checklist for Classes 1 to 3
Most children within the normal range of ability will be able to:
Number
1 have working knowledge of four processes and their symbols +  x ..; (including processes in verbal and written sentence form)
1 appreciate number qualities 112
1 understand Roman numerals IX and Arabic numerals 1100
1 count from 1100
1 know number bonds up to 10
1/2 understand difference between odd and even numbers
1 3 have work knowledge of the multiplication tables 112
1/3 apply simple mental arithmetic in narrative form using above listed skills
2 know number bonds up to 20
2/3 recognise, analyse and count to numbers up to 1,000
2/3 work with tables as division (24 shared between 6 is 4)
2/3 know patterns in multiplication tables 10, 9, 5,4, 11
2/3 use place value to four places (ten, H, T and U)
2/3 carry numbers across columns e.g.
19 74
+2 x2
3 be able to recite tables 1  12 in chorus and individually
Form Drawing
1/2 draw straight line, curves, linear forms, symmetry on vertical axis
1 3 draw common geometric forms freehand
3/4 draw symmetrical reflections: about horizontal and diagonal axis
Measurement
2/3 use money for simple bills and calculating change
3 tell time using hours, half hours, quarter hours on 12 hour clock
3 calculate simple practical sums, e.g. how many milk bottles in a crate holding six by six, bricks in a wall, floor boards, etc.
3 calculate simple sums in measurement of length, capacity and weight
Number
1 have working knowledge of four processes and their symbols +  x ..; (including processes in verbal and written sentence form)
1 appreciate number qualities 112
1 understand Roman numerals IX and Arabic numerals 1100
1 count from 1100
1 know number bonds up to 10
1/2 understand difference between odd and even numbers
1 3 have work knowledge of the multiplication tables 112
1/3 apply simple mental arithmetic in narrative form using above listed skills
2 know number bonds up to 20
2/3 recognise, analyse and count to numbers up to 1,000
2/3 work with tables as division (24 shared between 6 is 4)
2/3 know patterns in multiplication tables 10, 9, 5,4, 11
2/3 use place value to four places (ten, H, T and U)
2/3 carry numbers across columns e.g.
19 74
+2 x2
3 be able to recite tables 1  12 in chorus and individually
Form Drawing
1/2 draw straight line, curves, linear forms, symmetry on vertical axis
1 3 draw common geometric forms freehand
3/4 draw symmetrical reflections: about horizontal and diagonal axis
Measurement
2/3 use money for simple bills and calculating change
3 tell time using hours, half hours, quarter hours on 12 hour clock
3 calculate simple practical sums, e.g. how many milk bottles in a crate holding six by six, bricks in a wall, floor boards, etc.
3 calculate simple sums in measurement of length, capacity and weight