Te Ao Pāngarau | Maths

The teaching of math involves instructing students in mathematical concepts, theories, and problem-solving techniques, including algebra, geometry, and calculus. It aims to develop logical reasoning, analytical skills, and the ability to apply mathematical knowledge to real-world situations.

Math Class 8

Math class 8

The generative impulse in class 8 is one of identity, progress, and power. During this important threshold year and transition into High School, students are looking for authentic explanations that have meaning for the world they live in. The class 8 experience builds self-sufficiency through students developing skills to address fears and anxieties associated with identity, community, and personal challenge.

Working with practical problems offers a rich fund of activities for students and can also be formed into a life skill, which might open various avenues to the real world of work. Working by means of mathematical exercises fosters an active connection to these areas. The practical component brings students towards life and reality. Calculations help with the area of thinking. The beauty of geometrical figures and shapes bring the students to a greater awareness of the real world. Geometrical rules and proofs are important for them as they develop their individual forms of speech and expression.

The study of platonic solids seeks to bring to the students both a Mathematical understanding and inner experiences in their relation to the geometry of form. At a time where their bodies and emotional lives are undergoing significant change, it is supportive to encounter Mathematical laws that are established and timeless.

An understanding of the historical significance of the Platonic Solids over millennia of human culture is both engaging and rewarding, and gives students an appreciation of the elegance, value, and beauty intrinsic to Mathematics.

The study of Algebra is revised, consolidated, and extended, with focus on the development of strong foundations in the topic. The students should be led to an appreciation of the power and versatility of Algebra as a Mathematical tool and develop confidence in its application in a wide variety of contexts.

The middle school years are an important period of learning, in which the foundations of knowledge of many fundamental disciplines are established. Algebra is a significant Mathematical tool that finds application in the further study of Mathematics, but also appears in the Sciences, Engineering, Technology, and Economics, amongst others. A thorough grounding in Algebra is indispensable in many tertiary courses and provides skills that create a pathway towards continued success in further education, training or employment.

The study of Statistics is another foundational discipline which gets revisited. The various applications are not only pervasive in modern life but can also be immediately relevant to students at this age. With the right data sets, exploring this data can be a new way for students to explore their individuality, and the differences in our society, and can easily link to questions of belonging or fairness.

UNITS OF LEARNING

In some schools these could be stand alone Main Lessons and in some schools they could be taken as ongoing practice lessons or a combination.

Geometric/Platonic Solids

ESSENTIAL

Possible Content:

Historical significance, relevance, and pervasiveness of the five Platonic solids.
Precision drawing, through accurate net drawings using compass and straightedge.
Creating accurate 3D models out of card, or heavier paper, using drawn nets.
Creating 3D models out of other tactile materials such as straw, reed, or other wireframe models.
Creating 3D models out of clay, which is an excellent way to demonstrate how duals are connected.
Other regular geometric solids such as the Archimedean solids, Catalan Solids, Johnson solids, Kepler–Poinsot polyhedra.
Naming and counting vertex, edge, and plane.
An exploration of duals in geometric solids, and the duals in the Platonic solids.
Euler’s formula.

Statistics

Possible Content:

Statistical Investigation Cycle (PPDAC)
Sampling
Appropriate displays for discrete and continuous data, including box-and-whisker graphs.
Informal comparisons using box-and-whisker graphs.
Relevant data: measuring things around school or in student’s every day life.
Relevant data: taking class surveys of all kinds of things.
Relevant data: Census at school.
Relevant data: Hans Rosling’s Gapminder.
Relevant data: ourworldindata.org.

Relevant Pedagogical Aims:

To polish skills for learning.
To re-discover notions of beauty and goodness in light of one’s own responses to the world.
To support the individual’s developing sense of uniqueness.
To encourage courage.

Relevant Pedagogical Aims:

To foster a sense of connectedness to the world.
To polish skills for learning.
To support the individual’s developing sense of uniqueness.

Math Class 9

The impulse in class 9 is polarity and contrast. Curriculum content, concepts and ideas are grounded in authentic contexts to support students to make the necessary connections, and to become more adept at putting their thoughts into action. As thinking is an expression of our ego activity, Mathematics affords quite special opportunities for the pupils’ inner development and self-knowledge. The practice of creative problem solving helps in the development of the students’ thinking sphere.

Working out of Steiner’s indications for the students of this age group, the study of Number and Algebra brings an experience of the manipulation of abstract concepts regarding number. Beginning with the concrete and practical, the students deal with increasingly complex algebra of polynomials, simultaneous linear equations, the factorisation of quadratics, and surds arising out of squares, triangles, and pentagons.

The extension of concepts from concrete and perceptible to abstract serves to nourish cognitive processes within the students that allow them to develop confidence in the power of their thinking. This provides students with the impetus to become enterprising individuals, who show initiative, explore ideas, and use their creative abilities to make discoveries about the worlds around and within them.

Statistics and Probability offer students good experience and practical skills in formal logical thinking. Statistics has become a vital tool in our modern society, and students will have increasingly encountered statistical facts and figures in newspapers and other media as their awareness of the world has developed. The concepts of randomness, choice and fairness challenge the students at a time where their developing ability to form judgments is still largely based on emotional response.

The artistic educational quality of Steiner education is especially evident through the teaching of Mathematics.

Study of Geometry reviews and extends the student’s experience of working with form. Geometry theorems provide a vehicle for the development of logical thought processes in building sequential proofs. Accurate construction and measurement are skills that will be drawn upon in many other future topics, and in areas as diverse as art, surveying and cartography, and projective geometry. Trigonometry reviews and extends the student’s experience of working with triangles. Trigonometry provides a link between the side lengths and angle measure, which in the student’s prior experience, appeared to be two seemingly disparate features of the triangle. This topic lays the foundations for the practical experience of land surveying in class 10.

Teaching of Conic Sections enables students to approach the problem of infinity as one did before with parallels and helps them understand polarity and contrast. This topic provides an experience of the creation of form in space through the polarities of the radial and peripheral growth processes. These polar processes find their expression in the construction of the conic sections, firstly by the intersections of concentric circles, and then through the construction of their envelopes. Through the process of construction it becomes clear that the conic sections are all merely metamorphosed forms of the circle.

The study of Conic Sections employs the experience of the planar sectioning of the cone as a vehicle to develop students’ two and three dimensional spatial visualisation ability. The students experience form as dynamic and able to undergo metamorphosis. The conic sections are shown to arise in the natural and built world. They begin to gain an inward sense of the influence of forces working inwards from the infinite periphery, and this prepares them to meet the concepts of infinity encountered in their study of Projective Geometry.

UNITS OF LEARNING

In some schools these could be stand alone Main Lessons and in some schools they could be taken as ongoing practice lessons or a combination.

Geometry of Conic Sections

ESSENTIAL

Possible Content:

Conic Sections as planar sections of a double cone
Constructing the Conic Sections using physical restraints such as string around a peg.
Constructing the Conic Sections using their mathematical invariants – distances from lines using ruler, and distances from points using compass.
Constructing the Conic Sections using envelopes
The Cartesian equations of the Conic Sections, this is a beautiful link between Geometry, Graphing, and Algebra.
The three so called degenerate cases of the Conic Sections. Note, These are actually the more fundamental geometric elements of point, line, and plane.
Integrating the Conic Sections into Māori art or patterns.
Explore applications: parabolic reflectors, arched bridges, elliptical whisper chambers, caustics of light in a cup, orbits of celestial objects, where they occur in nature and architecture.
Explore polarities and the grey areas in between by morphing one conic section to another – this involves pushing some process to infinity and leads nicely into Projective Geometry in the later years.

Relevant Pedagogical Aims:

To awaken to the polarities of subject knowledge through both heart and will.
To show how causes, issues, “facts”, phenomena, opinions, etc. can pull in two directions, have two sides.
To work with accurate observation, objectivity and detail.
To bring idealism to the fore, to push ideas to the limit.
To work with the hands and soil.

Probability

ESSENTIAL

Possible Content:

As Probability deals with the interplay between certainty and impossibility, and the grey area between where most probability happens, this topic is suitable for either Class 9 or Class 10.
Definitions of probability.
Theoretical vs Experimental Probability, and the Law of Large Numbers.
Games, simulations, or repeated trials.
Coding simple simulations, for example in Scratch.
Probability trees.
Arrangements, Selection, Permutations, Combinations.
Monty Hall.
Benford’s Law
Analysing games or gambling problems.
Applications of Probability

Relevant Pedagogical Aims:

To awaken to the polarities of subject knowledge through both heart and will.
To develop the reasoning power of the student.
To show how causes, issues, “facts”, phenomena, opinions, etc. can pull in two directions, have two sides.

Math Class 10

As students progress through this year, they develop an increased capacity for self-control and balance. The curriculum provides opportunities for students to come to a deeper understanding of basic and underlying laws, structures, and processes that relate to the place of human beings in the wider world. At this stage students become more aware of commercial processes and the way businesses develop, which means that they can now reflect on these things.

The study of financial mathematics helps students to deal with real life problems such as taking a student loan, mortgage, and how to operate a small business.

Through the study of Number and Algebra students develop various solutions, methods, and formulae. They learn to think outside the box. They consolidate and strengthen their understanding of Algebra and discover areas in which Mathematical disciplines which previously appeared separate begin to overlap and merge. Students are also exposed to different number bases and their applications.

The study of Trigonometry offers a wide field of activity where students discover a new structure of relationships and their applications. Surveying allows students to measure and draw a small piece of land and by doing so precision is learnt. Trigonometry focuses on the use and understanding of Trigonometry and its applications to areas as diverse as surveying, mechanics, navigation, engineering, physics, astronomy, mapping, military operations and construction. A thorough picture is presented of the historical significance and development of Trigonometry and Surveying, with emphasis on practical work, applications, mathematical theory, and worked examples.

The increased exposure to concepts that move out of the practical and into the abstract serves to nourish cognitive processes within the students that allow them to develop confidence in the power of their thinking. This provides students with the impetus to become confident, creative individuals, who are enterprising, show initiative, explore ideas, and use their creative abilities to make discoveries about the worlds around and within them.

In some schools these could be stand alone Main Lessons and in some schools they could be taken as ongoing practice lessons or a combination.

UNITS OF LEARNING

Financial Mathematics

Relevant Pedagogical Aims:

To help begin recognizing the individual’s life-path toward adulthood.
To foster awareness of one’s own actions.

Possible Content:

GST
Income Tax
Compound Interest, Loans, Mortgages, Investments.
Budgeting
Student loans
Banqer.co has useful resources
Easily links to NZ CSE assessment LO 1033.

Surveying and Trigonometry

ESSENTIAL

Possible Content:

This unit often incorporates a camp or other extended measuring task where a group of students work together towards a single shared outcome.
Learning to use technical equipment with a high degree of precision, such as Theodolites.
Recording and processing large amounts of measurement data.
Creating an accurate map of a small area, by working together as a whole class.
Using cartographical conventions to label a created map.
Right angled trigonometry.
Coordinate systems, and using trigonometry to convert measurement data to coordinates.
Law of Sines and Law of Cosines.
Can link to other investigations of local area, flora and fauna, environmental health, history, etc.
Easily links to NZ CSE assessments LO 1024 and LO 1026.

Relevant Pedagogical Aims:

To bring more consciously the aesthetic sense.
To recognise relationships between the inner and outer worlds.
To broaden the powers of perception.

Math Class 11

Class 11 students engage their analytical and reasoning skills as they apply their capacity for greater objectivity and explore the world in increasing detail.

In areas of pure Mathematics students are now confronted by infinite processes on all sides. A new stage in thinking meets the pupils as they work with series towards the sum of an infinite series, adding up ever smaller parts to arrive at a concrete whole. In percentages, a new process is discovered as steps tend towards zero. Non-linear graphs can now be examined in closer detail, including asymptotic behaviour. Graphs are also the introduction to Calculus, as an infinite limiting process is now required to find gradients and areas.

The areas of Algebra and Geometry which were dealt with separately until now are brought together as analytical Geometry. The laws of Euclidean geometry are raised to a new stage in the teaching of Projective Geometry. This asks students to use their imaginative reasoning to grasp the concept of infinity by working through the elements of infinity such as infinite point, line, and plane. This enables students to acquire an extension of their thought space.

Trigonometry is brought to motion giving a basis for the wave theory[GU1] [MF2] . Physical and Geometrical processes can thus be modelled by the graphs of trigonometric functions. With spherical trigonometry the student can experience an enhancement of plane trigonometry.

[GU1]Does this mean 'wave theory' as in physics (light, etc) or theory of waves as in modelling them using trig (amplitude, frequency, horiztonal/vertical displacement) with respect to larger scale things like tides?

(Jane)

[MF2]Both! Trigonometry is everywhere

UNITS OF LEARNING

In some schools these could be stand alone Main Lessons and in some schools they could be taken as ongoing practice lessons or a combination.

Sequences and Series

Possible Content:

Arithmetic, Geometric, Harmonic, Musical Series, and the Fibonacci sequences.
Artistic or visual representations of these different modes of progression.
Arithmetic and Geometric Series.
Infinite Geometric Series.
Solving problems leads to a natural place to introduce or teach Logarithms.
Some of the formulas can be found by students’ explorations of carefully crafted examples.
Applications of theses sequences and series in everyday life.
Easily links to NZ CSE assessment LO 2018.

Relevant Pedagogical Aims:

To develop and guide capacities for critical thinking.
To develop a reflective questioning attitude to the world of phenomena and opinions.
To develop an awareness of the internal processes of phenomena

Projective Geometry

ESSENTIAL

Possible Content:

This unit is suitable for either Class 11 or Class 12.
Point, Line, and Plane, in two or three dimensions.
Point, Line, and Plane situated at infinity.
The methods and results of pushing point, line or plane (where possible) to infinity when using the following theorems.
Dualities of shapes, concepts, and theories, in two or three dimensions.
Desargues’ Two-triangle theorem.
Brianchon’s Theorem.
Pascal’s Theorem.
Pappus’ Theorem.
The study of Path Curves as more true representations of natural forms than Algebraic Curves.
The extensive amount of drawing in this topic lends itself to producing artistic representations.
Links to perspective drawing in Art.
Easily links to NZ CSE assessments LO 2017 and LO 3005.

Relevant Pedagogical Aims:

To develop a reflective questioning attitude to the world of phenomena and opinions.
To be given time to reflect on philosophical issues as they arise in relation to their studies.
To develop a personal sense of aesthetics and style.
To develop an awareness of the internal processes of phenomena.

Math Class 12

This is the students’ final year, and it is the culmination of an education which seeks to produce individuals who will work with a sound understanding of both themselves and the world. Mathematics in class 12 forms a synthesis of what they have learnt over the course of their school life. In doing so, the focus of study narrows to allow for a more in-depth, and interrelated view of the subject. Therefore, the discipline is split into two separate areas of study: Mathematics with Calculus, and Statistics.

In Calculus, students focus on topics such as Trigonometric models and proofs, Conic Sections, Differentiation and Integration, and algebra with Complex numbers and systems of equations. In Calculus, students move from purely numerical into an experience of differentiation and integration. The study of Mathematics, while still ultimately used to model real processes, is now almost entirely abstract in its study.

In Statistics, students focus on topics such as Probability and Distribution models, Linear Programming, Forecasting from a Time Series, and Confidence Intervals. The study of statistics now requires taking the understanding of the model and integrating that into a wider context.

Despite its narrower focus this year, Mathematics still has a few topics suitable for whole class instruction.

In some schools these could be stand alone Main Lessons and in some schools they could be taken as ongoing practice lessons or a combination.

UNITS OF LEARNING

Budget.
Kiwisaver.
Comparing investment schemes.
Student Loans, and paying for Tertiary Study.
Taxes if you’re self-employed.

Life Skills

Relevant Pedagogical Aims:

To be aware of themselves as members of a world community.
To feel in command of their future direction/destiny.

Contemporary Mathematics

Modern Maths
Various applications of Maths in modern life
Chaos Theory
Data Science
AI

Relevant Pedagogical Aims:

To perceive the inter-dependence of phenomena, processes and human endeavours.
To have well-developed moral, ethical and personal standards.