Te Au Pāngarau | Mathematics
Class
1
Statement of Intent
The children are introduced to the magical world of number through story, rhyme, song and practical activity. The initial focus is on the quality of the numbers one to twelve, fostering a sense of feeling connected with the 'being' of each number.
Continuous opportunity to practise counting, sequencing one to one, with practical objects is given.
The four operations are introduced in a way that characterises their essential purpose to foster an emotional-affective connection.
Number patterns are experienced through rhythm, movement music and imagination. Practical experience that engages all the faculties is striven for.
Learning to count as a rhythm, using the heart as well as the head will enable a true relationship with number work that will enrich the thinking processes, as the child grows older.
Initially all processes work from the whole to the parts, and particular attention is given to the moral qualities of giving, sharing and helping rather than amassing large amounts for oneself.
Lessons
Introduction to Numbers
Having learned to form alphabet letters, the children are now ready to form numbers. Number sequencing, games, songs and poems will have been introduced orally and through movement in Morning Circle time, allowing the teacher to ascertain the level of prior knowledge of individual children, and this will now be used to further the children's abilities.
Possible Lesson Content
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Quality of numbers, e.g., number 1 can be very small (1 child) or very large (1 country, 1 sun)
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Roman numerals to establish a personal connection (e.g., through corresponding number of fingers) to the quantity represented
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Arabic numerals
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Forming numerals in different mediums
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Sequencing and sorting exercises using materials
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Exploring ordinal numbers and their suffixes
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Identifying odd and even numbers through movement and games
Introduction to the Four Operations
Having learned to sequence numbers and identify odd and even numbers, the class will now be imaginatively introduced to working with the operations of addition, subtraction, multiplication and division.
Working from the whole to the parts that make up the whole (analysis) will be experienced first. Further experience of moving from the parts to the whole (synthesis) will be worked with later.
Possible Lesson Content
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Exploration of the relationships between the operations through story and practical work with materials
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Learning to name the operations’ signs and their meaning (Continuing emphasis is placed on the moral significance of working with arithmetic. For example, the children learning to give' or 'share' a piece of apple allows for a healthier soul and moral development than seeing how many pieces they can amass in front of themselves.)
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Exploring the inter-relationship between the operations: addition and subtraction are particular 'friends', and multiplication and division are helpful to each other.
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Practising the 2-, 3-, and 5-times tables rhythmically
Further Work with the Four Operations
Building upon prior experience, the class will now have the opportunity to practise writing number sentences from a variety of sources, including stories, and, using the four process signs and the equals [=] sign will be able to read and analyse their sentences.
Possible Lesson Content
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Working through number stories, then writing numbers or number sentences (operations)
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Rote counting and times tables exercises, in rhythmical movement or games
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Mental arithmetic, also in the Morning Circle/rhythmical part
Waldorf Achievement Objectives
Soul and Emotional Development: The children will be led towards an experience of
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The ‘quality' of numbers up to at least 12
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A feeling for the inherent morality of numbers
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Community building through giving and sharing
Within the expected range of Class 1, the children will be able to:
Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Number quality in myths (e.g., Papatuanuku and Ranginui); counting in te reo
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Form Drawing: Lines; shapes; movement in space; directions and distances
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English Language: Speaking; listening; directionality of sentences; presenting number quality
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Science: Counting objects found on walks etc. – investigation; days and seasons
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Art: Painting and sculpting numbers – quality and quantity; pictorial representations of numbers; Main Lesson Book illustrations
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Eurythmy: Rhythmical patterns; counting
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Handwork: Fine-motor skills; directionality; working with simple patterns; counting; simple adding and subtracting
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Languages: Simple counting rhymes and verses
Class
2
Statement of Intent
In Class 2 the children further immerse themselves into the world of number. From the foundations built in Class 1 they expand their number knowledge and become nimbler with mental Maths. Will-based, kinetic learning is especially useful to help the children memorize arithmetic facts.
Frequent and regular rhythmic and speech activities will ensure that the children’s awareness of, and engagement with, number and arithmetic remains high, and help them to grow their confidence. Story, rhyme and song still have a place in Class 2 Maths as they allow the children to access new learning through their feeling and own experience.
The four operations and their inter-relationships are explored. Understanding the value of a number in its place will allow the children to venture into the higher number realms and work there with increasing confidence.
The four operations and place value are the themes of carefully sequenced Main Lessons which are planned to consolidate foundations built in Class 1 and then expand knowledge and understanding through the year.
Lessons
The Four Operations
In Class 2 story and real-life contexts remain an important element of Maths learning. A variety of materials (including fingers) is used to illustrate mathematical questions and investigations, and the children will work with them using these materials. As the class becomes more confident, they will be guided towards imaging when working with number mentally and when writing number sentences.
Addition and subtraction, multiplication and division are all explored at the same time.
Possible Lesson Content
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Sequencing, orally and practically
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Counting to 1000 and back, individually, in groups, as a class
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Mental arithmetic, with basic facts in particular
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Times tables practice, with movement, speaking, writing
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Investigating addition and subtraction as reverse operations
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Exploring the nature of multiplication as repeated addition, of division as equal sharing
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Investigations in small groups
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Learning to write number sentences and horizontal algorithms
Place Value
When the class is confident with writing out simple sums as number sentences, it will be time to introduce the concept of place value. Again using stories and other imaginative contexts, the children will practice working with sums and numbers of no more than four columns. Learning to read these numbers will take time for some. The children will learn to name the columns as units (ones), tens, hundreds and thousands; they will practice setting the correct digit in the correct column to form the correct number. They will practice the use of columns practically and in written work and learn to recognize and analyse any 4-digit number. An introduction to 'carrying' will also happen in this Main Lesson. More complex addition and subtraction sums may be given as the Main Lesson proceeds. It is very important for the moral development of the children to bring an imaginative picture of 'giving' rather than 'borrowing' when practicing subtraction.
Waldorf Achievement Objectives
Soul and Emotional Development: The children will be led towards
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An inner relationship to the significance of place and value of numbers
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A deepening connection with the analytical process leading from unity to multiplicity
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A more conscious sense of the relationship between processes
Within the expected range of Class 2, the children will be able to:
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Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Te reo – cardinal numbers to 100, ordinal numbers to 10; context for number stories
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Form Drawing: Table and number patterns (e.g., “Table Spider”); geometry
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Science: Measuring (e.g., quantifying birds, plants, clouds); probability in “Living World” contexts
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Art: Conceptual drawing (e.g., place value towers); Main Lesson Book illustrations
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Eurythmy: Rhythmical patterns; counting
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Handwork: Fine-motor skills; directionality; working with simple patterns; counting; simple adding and subtracting
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Languages: Simple counting rhymes and verses
Class
3
Statement of Intent
The world of mathematics assists the nineyearold in taking hold of their environment and learning to understand it.
The historical perspective of the Main Lessons beginning with the human being as the origin of measurement will give the children a basis out of which to develop an understanding of the modern use of measurement and money.
Learning happens through doing: measuring, weighing, counting, adding and subtracting; multiplying and dividing are all applied to everyday tasks, needs and queries.
Opportunities for the children to develop their skills of estimation will be given as part of the process of finding answers.
The mathematical questions and applications will spill over into the practical Main Lessons of House Building, Farming and People at Work, offering the children more opportunities to practice these skills.
Lessons
Number and Measurement are the focus for Class 3. The nine yearold wishes to take hold of the environment and understand it. Bringing to consciousness the various ways in which historically and pictorially humankind came to the abstract forms of time, measurement and weight will aid the children’s connection to the environment and give them a basis for future development of mathematical work. The traditional Māori measurements can provide a relevant starting point, and also link to the House-Building Main Lesson later in the year. Practical application and development of the four operations is continued, and the challenge of stepping out of the safety of group work and into the realm of solo presentation of tables is placed before the soul of the nine-year-old.
Measuring Length and Weight
From an experiential beginning, the children can be led in a careful sequence to an understanding of the need for a normed measurement system such as the metric system. Their own activity is central to this understanding. The children will at various times work individually, in pairs or in groups.
Possible Lesson Content
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Using traditional Māori measures to measure various lengths
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Exploring scaling (e.g., 10 mārō = 1 kumi [approx. 18 metres]) to measure longer distances, e.g. outside
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Making measuring instruments for traditional measures and a 1-metre rule
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Comparing measuring results; learning to construct simple charts
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Some discussion about effectiveness and reliability of various measures
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Brief histories of traditional measures (Māori, imperial) and the metric system
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Exploring the graduation on the 1-metre rule through practical measuring tasks
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Concepts of weight, e.g. the difference in “a handful of…”
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Making and /or using sets of scales
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Comparing weights in both traditional/imperial and metric systems
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Exploring the best ways to weigh or otherwise measure liquids
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Experimenting with measurement in authentic contexts such as baking, cooking, making cheese
Time and Measurement
Tātai arorangi (Māori astronomy)
is an ideal starting point for this lesson.
From a Māori perspective, the teacher could introduce the concepts of tracking the rhythm of the day, a week, a month and a year. The division of the year into its four seasons, and festivals such as ‘Matariki’. The modern division of the day into hours, minutes and seconds will also be traced and the corresponding sentence structures learned.Telling the time using Te reo Māori.
Measurement - traditional Māori forms of measurement.
Possible Lesson Content
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Tracking the rhythm of a day, a week, a month and a year, including seasons and festivals
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Exploring the traditional Māori lunar calendar Maramataka and its months
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Parts of the day and night, in reference to the children’s experience and context
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Exploring the division of the day into hours, minutes, seconds
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Constructing a sundial and tracking the position of the sun at specific times
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Learning to read analogue and digital timepieces
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Constructing clock faces
Money and the Four Operations
Using and understanding money will be introduced in a practical way with the children having opportunities to work out payment for items and how to make change. Assessing 'value for money' exercises will be given and opportunities to practice the four operations in relation to money will be given, with problems presented verbally, mathematically and in written form.
The concept that ‘money is work’ can be explored through story and the concept and language around the concept of ‘utu’ brought to the children through role play.
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Form drawing using mirror images of NZ coins - money ML and form drawing
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Stories of Creation - Ranginui and Papatūānuku
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Planting and cultivating traditional Māori kai - farming ML
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Looking at the structure of wharenui? Dwellings ML
Waldorf Achievement Objectives
Soul and Emotional Development:
The children will be led towards
An inner connection with the abstract processes of time, measurement and weight
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An inner connection with measuring time
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A sense for the need to have common standard forms of measurement
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An experience of the traditional ways of measuring
Within the expected range of Class 3, the children will be able to:
Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
Kaupapa Māori: Te reo – directions, times of the day, clock times; traditional measurement methods; astronomy – seasons and the passage of time
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Form Drawing: Geometry
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Science/Technology: Measuring materials, time (farming)
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Art: Illustrations (e.g., maps); Main Lesson Book illustrations
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Eurythmy: Rhythmical patterns; measuring space, movement and tempo
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Handwork: Measuring patterns; counting; applying four operations in patterns
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Languages: Currencies; vocabulary
Class
4
Statement of Intent
The Class 4 children will be led to an experience of deepening and extending basic skills in their Maths work. Opportunities will be provided for them to experience patterns and to discover the mysteries of the number world.
Ongoing work will ensure the practice of tables is carried through to a fluent level, and that individual children are enabled to use this knowledge in a practical way.
Children are expected to be competent at working with problems involving up to six-digit numbers in all four processes by the end of Class 4. Fractions are introduced and the children should attain a level of competency with solving fractional problems by the end of the year.
The soul wish of the child in Class 4 is to penetrate and understand the world of numbers. The teacher must strive to keep the enjoyment level high at both the investigative and practical levels, ensuring all problems can have a practical application. Equating fractions with their own experience of being a part of a community will appeal to the soul of the ten-year old.
Lessons
Number Patterns
It is now time to introduce the wonders of the mathematical world through the experience of patterns, puzzles, games and forms. The teacher can use existing table patterns or invent their own. A table chart can be made by the children; magic squares can be investigated, and the children can invent their own. there's the Magic Number Nine to investigate, the prime numbers to discover and the secret of 'Best Friends' to unearth (checking answers to problems: how and where to add to check a subtraction sum is correct; how subtraction will check an addition sum and how division and multiplication can assist in self checking work).
Introducing the Magic Circle of Twelve at this time will serve as an introduction to more in-depth work with freehand geometry in a later lesson.
The children should be given as many opportunities as possible to 'play' with numbers; groups could be given the challenge to create new puzzles or games and individuals should face the challenge of proving their table knowledge and their ability with 'mental arithmetic’ problems. The creative teacher will find many ways of weaving math problems into the stories that are being told.
The Four Operations
Work with the four processes is continued, to ensure that each child is competent with the longer forms of addition, division and multiplication. Continued practice of selfchecking answers to problems using the ‘friends’ system should be encouraged. Numbers of six digits should be worked with and children should be able to read and write numbers up to six digits long; clarity of columns should be maintained while problems are being copied down and solved. The children should have the continuing opportunity to relate their work to practical experience and to enjoy a variety of mediums as appropriate in this number work.
For the child who knows they are progressing well in this Main Lesson, there is a sense of satisfaction and a new relationship can be made with Maths work. This is a wonderful opportunity to raise enthusiasm and delight in number work for those children who have been challenged by it, and it is important that the teacher monitors closely all skill levels. It is possible to have a visual aid to record individual progress with tables for example that will reward the achievers and encourage the slower workers. Peer teaching can be implemented to extend the able child and assist the slower worker.
Fractions
Through practical and pictorial themes, the children will begin to investigate the world of fractions. Beginning with themselves, they could look into their social world of class community and family. Here they can perceive the class and family as the whole and themselves as a part of that whole. Moral lessons can be learnt alongside the skills: here the child can see that no matter the size of an older sibling, they are both an equal fraction of the family. Here they can also investigate that even the least significant member of the class is an equal fraction of that class. Many different ways of making 'the whole' can be chosen, and the children can investigate what fraction they may be of that entity.
A lot of the work at the introductory stage should be of a haptic nature. The children’s feeling and will is easily engaged here, and the concept of “the more pieces, the smaller the size” can be illustrated visually in many ways.
Class 4 sees the formal introduction to common fractions. A lot of time should be spent on developing the understanding that fraction pieces are equal. The children will learn the language of fractions, find fractions in many different contexts and learn to add and subtract fractions with the same denominators. It is important that time is given to a thorough grounding of these basic concepts in Class 4; calculating with common fractions, renaming improper fractions etc. will come in Class 5.
Waldorf Achievement Objectives
Soul and Emotional Development:
The children will be led towards
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The inner satisfaction of discovering number patterns
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Sensing themselves as part of a community
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Sensing themselves as an equal part of their family
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Sensing themselves as an equal part of the class
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Sensing others as an equal part of the class
Within the expected range of Class 4, the children will be able to progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Te reo – numbers; telling the time in minutes (Specialist Kaiako)
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Form Drawing: Double symmetries (fractions); Geometry
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Social Studies: Mapping local geography; family trees (fractions)
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Science/Technology: Measurement
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Art: Illustrations (e.g., maps); Main Lesson Book illustrations
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Eurythmy: Rhythmical patterns; moving fractions
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Handwork: Number patterns in projects; counting and multiples; applying four operations in patterns
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Languages: Vocabulary
Class
5
Statement of Intent
As the child moves into a state of balance, the soulwish to be accurate in movement and skills can be deepened in the mathematics programme. The truth of numbers, the discovery of a variety of ways to view the world, or a specific problem, can be explored.
The children will experience the many faces of life in many ways and extend their view of the world, looking at it from different places, moving towards an understanding that just as we may have differing viewpoints so there will be differing ways to work with the same problem.
The interface between common fractions and decimal fractions, Form drawing and geometry offers the opportunity to further explore these concepts. Deepening and consolidating the work that has gone before with common fractions will lead into an initial exploration of decimal fractions; leading on from skills gained in Formdrawing lessons, plane geometry will be explored. The accuracy demanded in creating a freehand geometrical form will both strengthen and demonstrate the inner balance of the children and can also prove to be a source of deep soul satisfaction to the eleven year olds.
Throughout the year, the children will continue to practice the four processes, develop their ability to use the twelve tables and extend their ability to recognize prime numbers. As new processes are introduced through Main Lessons, the Maths practice lessons will enable continued exposure to the subject and practice of the skill.
Lessons
Freehand Geometry
Formdrawing now leads into geometry. Initially working freehand and as accurately as possible, the children will draw and describe common geometrical shapes: triangle, square, circle, ellipse etc.
They will have the experience of dividing a circle into twelve parts and describing the patterns that arise from numerical forms. The many different ways colour can be used to highlight a form will be part of the children’s experience in this lesson.
Babylonian astronomy and mathematics can be used as a context for introducing the concept of angles, while Pythagoras’ theorem can be experienced using cut-outs and creating knotted-rope constructions.
The children will be given many opportunities to work with freehand forms, practicing accuracy and with the emphasis on sharp pencil tips and precise ruling, to aid their inner development and handeye coordination before introducing the geometrical instruments. The lesson has a particular relationship with the inner sense of 'balance' that may be achieved at this time. This stage of the children’s development may bring the perfection of handeye coordination, the balance of breath and heartbeat, and this will show in the children’s freely drawn patterns. We reflect upon: 'as within, so without'.
Introducing and/or extending the children to Māori, Celtic, Egyptian and Greek geometric motifs, patterns and designs in Form Drawing lessons will refine their abilities further and successful designs could be incorporated artistically into Main Lesson books.
Common Fractions
Continuing from the work of Class 4, the children will extend and develop their ability with common fractions. They will now investigate improper fractions and convert them to mixed numbers. Finding fractions of a given whole can still be experiential. The children will explore calculating with common fractions, albeit still at basic levels e.g., addition and subtraction where the denominators are the same.
Common fractions are now placed in relationship with decimal fractions. The children should have opportunities to compare and relate the two types of fractions through manifold practical, visual activities.
Decimal Fractions
Introducing decimals to the class as another form of fraction will widen and deepen the children’s world view; just as there are many ways of looking at an event in the English language, so there are many ways to look at Maths problems!
Reminding the class of how we use money and revisiting the metric system could introduce the subject. Olympic games are also a wonderful context.
Opportunities will be given to solve decimal fraction problems using the four processes from stories and life situations. Comparing and investigating the similarities and differences between fractions and decimal numbers will consolidate the learning process and enliven the children's interest in the different ways problems can be solved.
Waldorf Achievement Objectives
Soul and Emotional Development: The children will be led towards
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A growing awareness of, and appreciation of, that there are often different ways to solve a problem
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A developing sense that mathematics and geometry are an essentially human achievement whose beginnings rest in curiosity, thinking, observation and experimentation
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An experience of the balance and eternal truths that are inherent in mathematics and geometry
Within the expected range of Class 5, the children will be able to:
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Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Geometric elements of ti rakau and kapa haka
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Formdrawing: Geometry; geometric designs and motfis
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Social Studies: Mathematics and geometry in ancient civilizations – achievement and effects
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Science/Technology: Number patterns in nature; 3D design and construction
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Art: Illustrations; Main Lesson Book illustrations; colour work in geometry
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Eurythmy: Movement of geometric shapes
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Handwork/Woodwork/Gardening/Cooking: Number patterns in projects; geometric designs; measurements
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Languages: Vocabulary
Class
6
Statement of Intent
The accurate construction of given geometrical forms using instruments accompanies the children’s developmental stage where they are increasingly able to experience the world through rational thinking.
Constructing complex forms with accuracy can be a challenge for even the most gifted of twelveyearolds. Learning through experience of the practical construction of given forms, they will now expand their understanding of geometry to include the laws that govern relationships.
The introduction of percentage fractions is a further ‘earthly’ aspect of mathematics. The many practical applications touch directly on the children’s lives - percentage fractions are an integral part of the modern financial structure. Discussion of the three uses of money, of interest and of the deeper moral and social aspects of money will stimulate the children’s deeper questions about causality, effect and of their own agency.
A first experience of algebra expands the children’s sense of how problems can be solved through first finding a structure which can then be applied in diverse contexts.
Throughout the year the children will continue to practice those skills that have been introduced in previous Main Lessons. They should feel confident in using all the processes that have been introduced and practiced over the past five years; they should be competent in the times tables (some children may have learnt tables higher than the twelve) and basic facts, and able to identify and use appropriate strategies when solving problems.
Lessons
Geometry
All that has previously been practiced freehand will now be taken up with compass, protractor, ruler and set square. The children will benefit from carefully considered steps in the introduction of these instruments, including setting the historical context of the discipline of geometry. The children could explore working with the straight-edge, string and shadow, emulating perhaps the Greek geometers such as Thales. Reference can also be made to the achievements and understandings of the Babylonians and Egyptians in the field of geometry.
The work to be undertaken will initially be modelled accurately, step by step, so that the children may see and understand what is required. Later, as the children become more confident, verbal instructions for the construction may suffice.
Forms should be large-scale and colourful. The children will be challenged to work accurately and have the opportunity to compare forms made of accurate and inaccurate measurements. Simple geometrical constructions will lead to forms that are more complex.
Line and angle bisection will be practiced; the construction of triangles, parallel lines and right angles will follow. The children will construct regular polygons using compass and protractor or trial and error methods. This leads to the beginning of exact deductive geometry, especially that of the angles of a triangle adding up to 180 degrees.
To give the children a good grounding, geometry may be taught in a second Main Lesson.
Simple Interest, Percentages and Graphs
The Ancient Rome Main Lesson offers itself as a springboard for this lesson: the soldiers returning home must borrow to replant their fields and rebuild their homes. There are then many areas that the children could enter and practise working with simple interest, including any fundraising they are undertaking.
A visit to the bank may be arranged and the children could have the opportunity to make deposits to the class account. The teacher may also introduce the concept of discount at this time and the children may practice working out more complex problems that include concepts of both interest and discount. Percentages are introduced as part of the simple interest exercises and understanding can be extended in the math practice lessons through their conversion into fractions and decimals and viceversa.
Graphs may be introduced at this time and the children may practice tracking future earnings and showing them as a graph. Graphing skills can be attached to many other lessons and practiced throughout the year. An overview of the history of money may be told, from sharing to bartering, to its use in the Crusades and the invention of cheques by the Knights Templar, and the invention of paper money in China. The precious metal content of a coin in Roman times can be compared with the content in modern coinage; the phenomenon of the cashless society and its use of EFTPOS, credit and online banking should also be touched on.
The three uses of money, purchase, loan and gift should be introduced now and the children should be led to a sense of the morality inherent in the right use of money.
Waldorf Achievement Objectives
Soul and Emotional Development: The children will be led towards an experience of
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The ‘quality' of numbers up to at least 12
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A feeling for the inherent morality of numbers
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Community building through giving and sharing
Within the expected range of Class 6, the children will be able to:
Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Te reo; geometry in Pacific navigation
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Form Drawing: Geometry; geometric designs and motifs
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Social Studies: Mathematics and geometry in ancient civilizations – achievement and effects; the history of money and its role in societies
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Science/Technology: Geometric patterns in nature (e.g., geology); 3D design and construction
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Art: Illustrations; Main Lesson Book illustrations; colour work in geometry
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Eurythmy: Movement of geometric shapes; angles
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Handwork/Woodwork/Gardening/Cooking: Geometric designs; measurements; cost of materials
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Languages: Vocabulary
Class
7
Statement of Intent
The children are increasingly able to comprehend through their own logical faculties that which was previously acquired through experiential thinking. From a foundation of number knowledge and operations they can now being to recognize the general rules that occur, for example, as algebraic formulae. Here then the foundation to conceptual thinking is built, and the transition should be practiced in many ways.
In mathematics the children now need to learn to pay attention to their own thinking. This allows them to transcend their expanding emotive-affective inner experience and build trust in their own thinking, and their ability to comprehend the world.
When we calculate mentally, our will is strengthened, and for the children in Class 7 this is important work. Hence the use of calculators should still be avoided.
In geometry the aesthetic quality is now correlated to the quality of thinking and will. Achieving precise and beautiful constructions can motivate the children to inquire into the rules and relationships that cause them. Finding geometric proofs further develops and also tests the children’s causal thinking. The necessity to learn the correct language to formulate them takes them another along their developmental path as they must become proficient in using language that describes what is rather than what is desired.
Lessons
Number, Higher Operations and Financial Literacy
The children will be introduced to negative and positive integers. The need to define negative numbers in various situations (e.g., temperature, altitude or sea level measures, bank balances and credit) can be discussed prior to abstract representation using number lines etc. The four operations with negative and positive integers are introduced and practiced. Here it can be useful to illustrate the “direction” of a calculation with movement.
Study of the concept of profit and loss can be part of this lesson, building on the “Simple Interest” lesson in Class 6. The concept of 'borrowing' may be explored and the accruing of compound interest should be investigated from both the lender’s point of view and from that of the borrower. A class trip could provide a useful platform for investigating this subject. The children could, for example, cost out the venture and explore ways of raising funds, investing the money raised and also investigate hypothetical problems that may arise.
The three uses of money, purchase, loan and gift should be explored further now and the children could practice working with budgets to meet their personal needs.
The children will be familiar with the 'Square Times Table' and will now be introduced to, practice and extend their knowledge of squaring and cubing powers. Whole numbers, fractions and decimals can all be explored. The process will be reversed so that square roots may be found, and the children should become competent at recognising, naming and finding them. The students will explore the concept of higher powers and have practice at working them out.
The BEDMAS rules are now introduced and practiced.
Algebra
This Main Lesson may well be introduced through the history of Algebra, and particularly the work of Muhammed Al-Khwarizmi (780 – 850 CE) whose book gave the discipline its name and was the main reason it reached the western cultural realm just before the time of the Renaissance.
Previously used formulae such as that for simple interest (I = PRT/100) may be recapitulated, and the children will be led to the understanding that they have been working with algebra all along (e.g., Class 1 maths problems, or V = LxWxH), and that now they will begin to apply the known rules and learn some new rules and applications.
They should have many opportunities, in practical contexts at first, of working with the four rules, like and unlike terms and the processes involved in working with brackets. They may also have some practice of showing algebraic problems as graphs. The children should have ample opportunities to work in groups, networking and sharing their problem-solving techniques.
Geometry
Extending the work done in Class 6, the children will now be taken as far as proving and applying the Theorem of Pythagoras, and practice will be undertaken in all aspects of recognising, naming and measuring angles.
The class will have opportunities to experience finding differing types of quadrilaterals and their symmetries, leading to simple set theory and the intersection of sets. They will also practice constructing triangles with the same area as polygons and the transformation of plane figures, especially from square to general quadrilateral, which will lead to simple perspective drawing.
Care should be taken in all aspects of this work and practical contexts or stories help to give them meaning. For instance, a look at the change in spatial and depth representation as seen in the transition from medieval to Renaissance art will help give the exercises a practical base and encourage the children to discover and experiment with aspects of the geometrical principles themselves.
Waldorf Achievement Objectives
Soul and Emotional Development: The children will be led towards experience of
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A sense that they are beginning to take hold of the world through their ordered thinking
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The difference between rules that arise from human-made conventions (e.g., the formula for interest) and those that exist independent of humans (e.g., the sum of angles in a triangle)
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Financial literacy as an essential aspect of the independence of the ‘captain’
Within the expected range of Class 7, the children will be able to:
Progress within the SEANZ Learning Steps and Signposts Framework (Mathematics).
The Integrated Curriculum – Mathematics in the Other Learning Areas
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Kaupapa Māori: Te reo; geometry in Polynesian navigation
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English Language: Speaking; listening; interpreting number stories; language of Mathematics – subject-specific vocabulary
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Social Studies: The history of mathematics and geometry; development and spread of ideas n and across societies; the role of money in society, and the social contract
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Science/Technology: Geometry and mathematics in navigation and construction, and technological developments
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Art: Main Lesson Book and other illustrations; colour work in geometry
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Eurythmy: Movement of geometric shapes; angles
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Handwork/Woodwork/Gardening/Cooking: Geometric designs; measurements; cost of materials
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Languages: Vocabulary