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Sophia Institute online Waldorf Certificate Studies Program

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Waldorf Methods/Reading and Math 3

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 3.5.

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 3.5.

Arithmetic and Mathematics/Class 7 and 8

General Observations and Guidelines

Classes 6 to 8

So far, concept-building about method given  pictorially has been rooted in an approach to the  child's soul. Now, after the twelfth year, children  can increasingly create order out of what has  been gained with the strength of their ability to  experience internal logic. This step is exemplified  in algebra: it leads from the activity of calculating  to observation of the processes and from there to  the discovery of general relationships.

The purpose of an algebraic formula, of  'calculating with letters of the alphabet', is to  express the formal, intelligible processes. This is  a general step forward in the development of the  child as only the method is formulated: By this  means the transfer from an imagination-bound  thinking to a conceptual thinking is facilitated  ... The process: the delineation of a concrete  problem (interest), the solution of the problem,  the evidence of the validity of the solution  method, and finally the applicability of the  discovered rule. All this would be experienced  by the children in many situations.

As the children approach puberty, their feeling  life expands in all ways. Mathematics can offer an  important support in this stage of life. Their own  subjective opinions and ideas are not required!  Mathematics attracts their attention not. only to  the numerical material but especially to their own  thinking. If the pupils manage to become confident  and secure with mathematical laws, they learn  self-confidence. When this is achieved the young  people are on the way to the most important aim  in mathematics teaching: that of gaining trust in  thinking.

However, this thinking can now connect itself in  a one-sided selfish way to its mentor, the human  ego, and this leads to egoism. It is essential to  link thinking to world interests in practical and  necessary life situations. It is, however, important  that the attempts to solve problems do not lead  to resignation with the 'I can't do that' attitude,  because mathematics lessons then achieve exactly  what they should not. Instead of enjoyment and  confidence, they create boredom and despair. There  is hardly any other subject which is so equated to  scholarly ability and intelligence as mathematics.  To 'give up' here or to have problems means to give  up generally, and simply to be 'stupid:

For this reason, mixed ability classes make  particular demands on the teacher as regards  method or possibly even remedial measures.  During the class-teacher stage, what the pupils  have to do must be differentiated although all  of them must deal with the basic mathematical  questions. Work with practical problems offers  a rich fund of activities for the pupils and can  even be formed into a life skill, which might open  various avenues to the real world of work. Working  by means of mathematical exercises to make  thinking energetic fosters an active connection to  these areas. The practical activities bring the pupils towards life and reality and also to a description of  basic connections.

Calculation is an education of the will in the  area of thinking. For this reason practice lessons  are added to the main-lessons from about Class 6  onwards.

The precision and beauty of geometrical figures  are the teachers who will lead them to greater  awareness. What has been experienced through  amazement in geometry in Class 5 should be  worked on in thinking in Classes 6, 7 and 8.  Geometrical rules are sought and formulated. The  pupils must also experience geometrical proofs  adequately. It is important for them as they develop  their individual forms of speech and expression  that they can experience something like this, which  is quite free of emotion and concerns itself purely  with what ought to be. In Class 8 one can use the  new subject of conic sections to approach the  problem of infinity as one did before with parallels.  Infinity is still not defined specifically.

Class 7

* Continuing practise in mental arithmetic * Revision: the four rules in natural and positive  rational numbers
* Basic bookkeeping
* Introduction to negative integers (through debt calculation)
* The four rules with negative numbers 
* Extension to cover all rationals
* The four rules with rationals and their connections.
* Introduction of brackets
* Recurring decimals, later on the value of ∏ . Full understanding and comparison of decimal  places and Significant figures
* Compound interest
* Simple statistical data rendered in graphical  form and deductions therefrom

Algebra
* Simple equations, including brackets, fractions  and negative numbers. Their practical application to solving problems
* Making and transforming formulae
* Powers and roots of numbers. The exact  evaluation of square roots
* Ratio and proportion
* Calculation of the areas of figures bounded by  straight lines and circular arcs
* Types of quadrilateral and their symmetries,  leading to simple set theory

Geometry

* Areas of geometrical shapes through construction and calculation
* Area of circle, and using this to calculate the  value of
∏, by cutting the circle into pieces 
* Pythagoras; theorem; area proof
* Shapes and stretches of simple shapes  * Tangents to circles
* Further transformations of pentagons. Construction of decagon and polygons
* Perspective drawing. (Can be linked with  modern history main-lesson)

Class 8

Revision
* Fractions
* Squares and roots
* Equations
* Practical problems

Algebra
* The commutative, associative and distributive  laws in algebra. The factors of the difference  between the squares and the application of this  to practical problems
* Volumes of rectangular blocks, pyramids,  prisms, cylinders and cones. Density and  weight of solid objects
* Simultaneous linear equations and problems 
* The dissolution of complex brackets in algebraic expressions
* A brief look at balance sheets and mortgages 
* Number systems. Binary arithmetic
* Further statistical work including mean, mode  and median
* Graphs of more complicated curves. The  solution of simultaneous equations by graphs

Geometry

* Locus of line and plane
* Locus and conics defined geometrically
* Enlargements, rotation, reflection of shapes
* Angle properties of circle (angles in the same  circle, intersecting chords)
* Construction of five regular Platonic solids. Orthogonal view of them
* Exact spatial perspective drawing including  the golden section
* Discussion of general triangle sides and  altitude formulae as part of the development of  the investigation of Pythagoras' Theorem
* Optional: Internal and external angles of a  polygon
* Similar figures especially triangles

Numeracy Checklist for Class 6 to 8

Most children within the normal range of ability  will be able to:

Number
6    convert percentages to fractions and vice  versa
6    estimate results by rounding off number  prior to accurate calculation
6    business maths: balance sheets: profit and  loss, discount, commission, VAT and book-  keeping, bank accounts
6    work out averages including speed
6    read co-ordinates (e.g. for map reading)
6    use letters in formula
7    know powers of numbers
7    work out ratio and scale
7    use algebra as a general solution to specific problems
7    use negative and positive integers
7/8    know how to work with square roots
7/8    calculate compound interest, mortgage rates,  income tax
6/7    make time and speed calculations
7/8    calculate mechanical advantage in simple  machines, e.g. pulleys, levers

Data
6    present information via pictograms: use pie  charts, bar charts, linear graphs (foreign  currency exchange)
7    use algebraic graphs

Geometry
6    make precise use of compasses, ruler, set  squares to draw constructions of major  geometric figures
6    make use of freehand perspective
6/7    use protractor
6/7    draw translations, reflections, rotations
6/7    know Pythagoras Theorem and its  applications
7    use instruments to draw linear perspective
7    know properties of triangles, parallel lines  and intersecting lines
7    know and apply formulae for area of regular  geometric forms, including triangle, circle,  parallelogram, derivation and use of
7/8    calculate areas of irregular forms
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