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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
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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.

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 'quality-oriented' 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 'value-free',  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 co-ordination 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  picture-less 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  times-tables 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

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 1-12
1    understand Roman numerals I-X and Arabic numerals 1-100
1    count from 1-100
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 1-12
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
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