Te Ao Pāngarau | Math

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 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 the 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.

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.

Statistics

Possible Content:

Class 8 is a good fit for statistics because it links in with the theme of individuality, and differences.
Activities could include getting a range of things to measure, then comparing measurements to see averages etc. Could turn this into a rap.
Census at school.
ourworldindata.org
Hans rosling: Gapminder
Not necessarily unique to Class 8, and Statistics will be woven throughout the Upper School Mathematics curriculum.

Relevant Pedagogical Aims:

To support the individual’s developing sense of uniqueness.

Geometric/Platonic Solids

Possible Content:

Conic Sections - links geometry with algebra.
Conic sections - polarities and the gray areas in between i.e. how one conic section when pushed to its limit turns into the next conic section.
Construct the conic sections from their definitions, as well as their envelopes.
Sows the seed for projective geometry. Can also introduce the three “degenerate cases” to get the full set of seven conic sections.
Create a Māori pattern incorporating the 4 conic sections.
Looking into nature and architecture to see where the conic sections occur.
Natural links into cartesian geometry. Can introduce cartesian equations of the conic sections, also links to pythagoras.
Parabolic reflectors and tealights are a nice practical application.
Also links into euclidean geometry.
Dipping ice cream cones into chocolate sauce 🙂

Relevant Pedagogical Aims:

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.

Math Class 9

The impulse in class 9 is polarity and contrast. This stage of adolescence is often an emotional roller coaster, and life may be experienced by students as a series of highs and lows. This is reflected in many of the themes during the year. Students at this stage are encouraged to use both their senses and intellect; they are challenged to think deeply and observe meticulously. These critical thinking skills help students begin to see beyond black and white polarities. 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.

Statistics and Probability offer students good experience and practical skills in formal logical thinking. 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. The artistic educational quality of Steiner education is especially evident through the teaching of mathematics. 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.

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. Over the course of this topic the conic sections are described both graphically and algebraically.

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.

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 and the solution of simultaneous linear equations in 2 and 3 unknowns, 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 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.

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, projective geometry etc. 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.

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.

Probability

Possible Content:

Monty Hall!!
Simulations. Dice games, card games, games in general. Good way to link theoretical vs experimental probability.
Often covered in practice lessons. Sometimes covered in Class 10 instead. Perhaps at the end of class 9.
Can easily lead into coding simulations as well. Scratch.
Main idea of the two extremes of “certainty” and “impossibility” and the grey areas in between where we find most probability.

Relevant Pedagogical Aims:

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 Algebra students develop various solutions, methods, and formulae. They learn to think outside the box. The study of number and Algebra allows students to 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.

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.

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 enables students 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. With spherical trigonometry the student can experience an enhancement of plane trigonometry. A new stage in thinking meets the pupils as they work with series towards the sum of an infinite series. 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.

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 form a synthesis of what they have learnt over the course of their school life. They focus on topics such as Probability and distribution models, system of simultaneous equations, Linear Programming, Trigonometric models and proofs, Conic Sections, Differentiation and Integration, and Complex numbers. In Calculus, students move from purely numerical into an experience of differentiation and integration.