Sophia Institute online Waldorf Certificate Studies Program

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 

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.
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, conceptbuilding 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 imaginationbound 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 selfconfidence. 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 onesided 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 classteacher 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 mainlessons 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.
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 imaginationbound 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 selfconfidence. 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 onesided 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 classteacher 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 mainlessons 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 mainlesson)
* 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 mainlesson)
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
* 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 coordinates (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
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 coordinates (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