The teaching of mathematics in Waldorf schools is based upon a form of realism, which Steiner also designated as monism (Steiner GA 1, chapter 6 ‘Goethe’s way of cognition’ (German edition 1987, p. 129)- and Steiner GA 4, chapter 7 ‘Are there limits of recognition?’ (German edition 1995, p. 124)). The import of this is that concepts, active in the mind, are directly involved in the construction of reality, as opposed to being generalities abstracted from multiple instances in the manner of nominalism. In the recent philosophy of mathematics such positions are being aired once again in a number of variants (see Wilholt 2004)

According to Steiner, knowledge arises through combining appropriate concepts with a particular sensory or psychological experience. This necessarily implies that the process of forming mathematical concepts also involves imbuing some element of experience with conceptual content and thus lifting it into the realm of pure thinking. Insofar as mathematics can be based upon sensory experience, in relation to the formation of concepts the primary area in question is that of the activities of the senses of balance and movement (the kinaesthetic sense). These are primarily concerned with ordering the body’s movements in relation to space, and as such are self-activating rather than involved in conveying information from outside (Steiner GA 21, ‘5. About the real basis of intentional relationships’ (German edition 1983, p. 143ff.)); Schuberth 2012). Following the course of child development, then, the path gradually leads from a cultivation of the senses in connection with numbers to pure thinking in mathematical concepts, as vividly described in exemplary fashion by Alexander Israel Wittenberg (Wittenberg 1968, 1990). Mathematical concepts are thus not acquired through abstraction from external sense experience. In terms of content they are supra-empirical. Even though they can be applied to the outside world, their beauty and flexibility go beyond such practical application. In pursuing mathematics, the human being inhabits an objective world of ideas, beyond sympathies, antipathies and mere opinions. All who take hold of a particular concept with their thinking enter the same spiritual sphere. (Locher-Ernst 1954).

Given all this, mathematics cannot just be a matter of learning definitions, theorems, proofs and problem-solving strategies, nor of studying the products of nominalist thinking (cf. e.g. Bourbaki 1957: p. 8; Field 1980). Any mathematics teaching worth its salt should provide the students with authentic encounters with mathematical phenomena – in the primary school, of course, in association with concrete actions and appropriate types of question. Mathematical “objects” cannot, of course, be presented in the same way as, say, experiments in physics. They arise in the mind when students create a mental picture through an active process, often through movement or a transformation of that movement (e.g. Bernhard 1984; 1999: Ulin 1987, and more recently, for instance, Weber 2009). The experience of mathematical phenomena, therefore, goes hand in hand with productive inner activity, enabling students to engage in the dynamics of developing mathematical concepts.

Only then are the phenomena put in systematic order, and finally, in class discussions, their wider implications and consequences for the formation of other concepts considered (Steiner GA 302, 14.06.1921 (German edition 1986, p 42ff.); Sigler 2010)). This allows students to learn to practise mathematical communication and argumentation. In this way, doing mathematics remains an open-ended process, which stimulates imagination, the aesthetic sense and the desire to find out more. The aim is to arrive at a genuine understanding of the phenomena and their inter-relationships, and not merely a formal resort to secure knowledge. This is achieved through adapting the content and methods to the students’ gradually evolving capacity to form judgements, so that the lessons foster a healthy development of their overall ability to judge.

This is a systematic, long-term plan, which supports the process outlined above, in which the development of mathematical concepts is organised on different levels. As each one unfolds, its content is set in a new context, and in this way its implications extended. A striking example of such a cumulative process of skills development is the theorem of Pythagoras:

Class/grade 5: first hint of Pythagoras’ theorem through the derivation of two congruent squares from one square

Class/grade 6: practically oriented approach to Pythagoras’ theorem; especially in the isosceles right-angled triangle

Class/grade 7: intuitive proof of Pythagoras’ theorem by adding and dividing areas; alternative visual proofs

Class/grade 8: proofs by means of shear transformations

Class/grade 9: Pythagorean triples; square roots

Class/grade 10: geometric interpretation of the cosine theorem as a generalisation of Pythagoras’ theorem

Further steps could be found in the geometry of spheres or – more generally – in the geometry of curved surfaces (Steiner GA 300c, 30.04.1924 (German edition 1975, p. 155)).

Such sequences form thematic groupings (Wittenberg 1990: p. 122ff.), which eventually meet and intermingle such that the “processes of concept formation characteristic of each of them” (Wiegand 2014: p. 114) lead to the creation of “living (flexible) concepts” (Steiner GA 293, 30.08.1919 (German edition 1992, p. -140), which are susceptible to extension and growth.

A further important perspective is the relation of mathematics to every-day living. Time and again opportunities will be taken to demonstrate practical relevance, as, for instance, in the high school main lesson on land-surveying, where trigonometry finds its practical application, or in the geometric methods of technical drawing used in architecture and machine design, or in the application of algebra and functions theory to calculations in the world of business and finance.

Already in the first primary school years, however, numbers and operations are set in relation to simple every-day situations or imaginative descriptions. In grade 3 simple arithmetic is introduced in connection with measurement in the building and farming main lessons. Fractions involve addressing a variety of senses in a wide range of activities. Algebra begins by working with commercial formulae (calculating interest), and the approach to equations also entails focusing on many useful, practical examples. “All teaching must give instruction in the art of living.” (Steiner GA 192, 11.05.1919 (German edition 1991, p. 98)). Nowadays this injunction of Steiner’s is increasingly regarded as relevant for all kinds of mathematics teaching (cf. Maass 2011). It is important, however, to distinguish between what is called “modelling” in current educational discussions, and what Steiner meant by “instruction in the art of living”. The former is based on the assumption that mathematical forms of thinking can only ever be models of reality. This being the case, mathematical models may legitimately be applied to all areas of life equally. Their degree of relevance in each area is discussed later. The latter, by contrast, represents the view that mathematics is only applicable to those areas which are either inherently mathematical anyway, or where its practical relevance to every-day life is readily demonstrable to the students.

All this in no way implies, however, that there are no purely mathematical reasons for opening up new topics. Number theory, and more particularly, geometry generate ample motivation to pursue mathematics out of love for its intrinsic clarity and objective lawfulness.

In all schools, in addition to the normally-covered subjects of arithmetic, algebra, analysis and statistics, geometry forms a second main branch of the curriculum. Elementary geometry, solid geometry – both of which are often an element of art lessons in the high school – and projective geometry together constitute the central pillars of what is in effect a curriculum in its own right. They are treated independently of the ultimately more algebraic tendency of the topics on offer. Only in class 11, in analytical geometry, do these two aspects of the curriculum come together. This occurs mainly with a view to creating opportunities to practise these two ways of thinking and to assess their relative merits.

The method employed here is that of targeting the development of different modes of thinking. Accordingly, from the first year of school onward – for instance, with the very first introduction of numbers – a beginning is made in approaching things from the whole to the parts (analytical thinking). This is subsequently paralleled, as Steiner says, by synthetic thinking, in which the whole is understood as the co-operative interaction of the parts. This can be continued in all succeeding classes and subjects (Steiner GA 294, 21.08.1919 (German edition 1974, p. 13)); Schubert 2012; p. 38ff.). It is confidently assumed that the mutual effects of such modes of thinking, intrinsic as they are to certain subjects, will rub off on other realms of knowledge and also upon the students’ life of feeling and will (GA 301, 05.05.1920 (German edition 1977, p. 152 ff.)); Schubert 1990).

Mathematics is given in the form of main lesson blocks all the way through the school. From around class 6 two running lessons are added in. During the class-teacher period (from classes 1 to 8) the organisation and timing of the main lessons is basically at the discretion of the class-teacher. The original intention was – and this was the practice of the first Waldorf school in Stuttgart – that there should be 12 weeks of mathematics per year in the first five years, 10 in each of the next three years (classes 6 – 8), and then 8 weeks in each of the high school years (9 – 12). Of course, these times are by no means fixed, and can be varied according to the particular school’s requirements.

The running lessons are mainly there to give opportunities for acquiring and practising certain abilities and skills, so that the necessary mathematical procedures and techniques can be mastered. To this end the work is centred around problems, involving practice in the application of heuristic strategies and in this way finding approaches to a solution (Götte, Loebell and Maurer 2009; p. 280; Ulin 1987: p. 26ff.). This requires a store of graded exercises, capable of being fitted to individual needs, which enables the students to observe, document and gradually take responsibility for their own progress in learning.

The guiding principle for the use of electronic aids such as various kinds of calculators or of “dynamic geometry” software is the extent to which they encourage active engagement with the mathematic phenomena concerned and thus promote genuine skills development. As a rule this leads to computer aids being used in very consciously controlled doses from class 9 or 10 onwards.

Mathematics teaching for grades 1 to 8 falls into three main stages:

• Up to grade 3: becoming alive to the idea of number; working with natural numbers and elementary arithmetical operations; first steps in applied arithmetic; first steps in arithmetic writing; geometry as a branch of form-drawing
• Grade 4 and 5: concentration on fractions; geometry as the comparative study of forms
• Grades 6 to 8: introduction to algebra starting from the calculation of interest (its use in commerce and the economic issues it raises); equations, negative numbers and powers; beginning of geometric proofs.

The following two sections have been written in terms of the above 3 stages. It should be noted that not all the topics named here can be covered in the lessons. The teacher is called upon to make a selection that is felt to be pedagogically appropriate for the particular class.

Methodological considerations   

Initially the important thing is that children develop a feeling for numbers. This is done by incorporating them in songs and games involving rhythmic movement, a dynamic will activity, that should ideally be part of every (main) lesson. The benefits of this are two-fold. On the one hand, it is a concerted focusing on the body-centred senses, schooling them through the experience of movement in all its facets of basic and fine-motor co-ordination exercises. On the other, it creates the possibility of making the bodily actions more conscious in the form of mental arithmetic which is not dependent on finger counting or the use of other objects. To this end the main means used are descriptions of simple situations. This enables the child to grasp what is meant inwardly. At this age approaching things through pure logic will not have the required effect. It must nevertheless be kept in mind that the ultimate destination of arithmetic is the abstract world of numbers, which is a different realm entirely from the one the introduction of the alphabet is leading to.

First learning numbers in combination with movement (which will, of course, be appropriately varied) is the vehicle by which the child internalises them. The multiplication tables committed to memory in this rhythmical manner can then begin to be used and the child gradually come to feel completely at home with numbers. Through all this the child will also experience the fact that number has both a temporal (ordinal) and a spatial (cardinal) aspect.

An important principle has already been mentioned: “First from the whole to the parts and only then from the parts to the whole”. In other words, the correct relationship between analytical and synthetic thinking must be established. If, as recommended in chapter 4 of “Discussions with Teachers”, the temperaments of the children have been brought into relationship with the four arithmetic operations (Steiner GA 295, 25.08.1919 (German edition 1984, p 40 ff.)), principle mentioned above will have been almost automatically taken into account (Schuberth 2012: p. 42ff.).

By the end of class 3 the children should have a reasonable overview of the numbers up to 1000, by which is meant not just a quantity, but to an equal extent the quality of numbers, as revealed, for instance, in the process of division. More detail is given in what follows.

Suggested lesson content

Class/grade 1

Arithmetic

•  “1 is the largest number”. Subsequent numbers, up to 10 or 12 to begin with, are developed by subdividing 1.  In writing the numbers, it is best to start with the more pictorial Roman numerals, which can later be replaced by their Arabian equivalents. Having these two ways of writing makes it easier to distinguish between the concept of number and its graphic representation in numerals
• Counting (from one) up to 120
• Rhythmic practice and memorisation of multiplication tables, possibly as far as the 7-times table
• In conjunction with the temperaments the four basic types of arithmetic operation are introduced within a range of, say, 1 to 20. Starting from the body, the transition is made to simple graphic representation, and thence, having introduced the operational signs, to doing actual sums. Here also beginning from the whole plays a decisive role, and the experience of doing the sums inwardly takes priority over writing them down.
• Practice in all four arithmetic operations, both analytically and synthetically
• Appealing to various senses through guessing games

Geometry

• This is done as “geometry in action”, in other words form-drawing. It involves not simply having the children imitate outward forms, but goes more in the direction of developing their artistic feeling for the qualities of movement expressed in the forms they draw. These exercises also support the development of fine-motor coordination. (® fine art, drawing/graphics)

Class/grade 2

Arithmetic

• Further practice in mental arithmetic
• Extension of the range of numbers used in practising the 4 operations, up to about 120
• Memorising multiplication tables up to 12 x 12
• Displaying the multiplication tables in both movement and drawing
• 1 x 10 to 1 x 90
• the ordinal and cardinal aspects of numbers, which already figured in class 1, are now distinguished more clearly and practised accordingly

Geometry

• Geometry is continued as form-drawing. The focus is on free-hand drawings of symmetrical forms divided by an axis (® fine art, drawing/graphics).

Class/grade 3

Arithmetic

• Mental arithmetic
• Sums in a range up to 1000 (or more)
• Addition and subtraction sums (i.e. written) involving multiple-digit numbers
• Multiplication sums involving a two-digit number
• Division sums involving a one-digit number
• Simple divisibility rules
• Multiplication tables up to the 15-times table
• Memorisation of the square numbers up to 25 x 25
• Increasingly complex rhythm exercises
• First explorations into numbers and multiples (numbers with many or few multiples, perfect numbers, prime numbers)
• Applied arithmetic in relation to the main lessons on housebuilding, trades and farming

Geometry

• Geometry as form-drawing continues. The focus is now on mirror images in relation to curved axes, and exercises relating to the inside-outside problem (Steiner GA 307, 14.08.1923 (German edition 1986, p. 178 f)); Jünemann 1992; Schuberth 2008, p. 69ff.; ®fine art, drawing/graphics).

Methodological considerations

Around the age of ten the child’s soul-life goes through a major change, which brings about an increase in self-awareness, together with an associated distancing in relation to the surrounding world (® The child between the ages of nine and twelve).

This turn-around in the quality of the child’s experience is reflected in the arithmetic curriculum insofar as it recommends introducing fractions in grade 4. Here the child finds something in the presented material that corresponds to his or her own life-experience. The main point of this is not that the children learn very quickly how to work with fractions. Much more important is that the way a fraction comes about becomes a profound experience for them, resting upon a foundation of sound, concrete ideas. In this connection, the fact that the mathematics of fractions originated in the culture of ancient Egypt supplies the teacher with a host of perspectives that could feed into his or her teaching (Bindel 1982: p. 64ff.).

Here once again there is ample opportunity to exercise both analytical and synthetic procedures: from whole to part and from part to whole. The important thing is not to fix the concept of the fraction to one or too few images (cake, pizza), but to bring it into relationship with as many phenomena of magnitude as possible (distance, area, volume, time etc.). Once reduction and expansion have been covered, the four basic operations can be introduced step by step in connection with fractions. In grade 5 reducing the denominators to prime factors in order to determine the common denominator can be introduced.

Once ordinary fractions have been introduced, working with decimal fractions follows on quite naturally. Steiner formulated the objective as follows: “In grade 5, then, we carry on with fractions and decimals, equipping the children with everything they need to be able to move around at will within the realm where numbers express themselves in such forms.” (Steiner GA 295, 06.09.1919 (German edition 1984, p. 168))

From grade four on form-drawing should be taken in the direction of elementary geometry. So that the children have as powerful an experience of the geometric forms as possible, it is advisable that they draw them at first without compasses or ruler, in other words, free-hand. “Although with our first steps in geometry we are working at the most elementary of levels, it is important that the children gain a sense of that dimension beyond all practical utility that has to do with the solution of ultimate questions. This is made all the simpler to the extent that we succeed in conveying, besides its effective laws, a feeling for the beauty of its forms and the strictly regulated interplay of their complementary relationships” (Bühler in Kranich/Jünemann 2015; p. 190).

In grade 5, then, the introduction of ruler and compasses is recommended. The pupils can be given their first acquaintance with Pythagoras’ theorem, beginning with the right-angled isosceles triangle, e.g. as derived from a square in relation to two smaller squares of equal area. In connection with stories from the history of ancient Egypt the Pythagorean “rope trick” can also be introduced.

Suggested lesson content

Class/grade 4

Arithmetic

• Mental arithmetic
• Written sums using large numbers
• Introduction of fractions; expansion and reduction; fractions and ratios
• Addition, subtraction, multiplication and division with fractions having both similar and different denominators; transforming improper fractions into mixed numbers, and vice-versa
• Introduction and first application of decimal fractions in connection with practical situations
• How to do long multiplication and long division sums

Geometry

• Transition from artistically enhanced form-drawing (“active geometry”) to juxtaposing simple geometric forms – circle, ellipse, various quadrangles etc. – for purposes of comparison (Schuberth 1998).

Class/grade 5

Arithmetic

• Regular exercise in mental arithmetic
• Revision: the four basic operations with natural numbers and decimal fractions
• For determining the common denominator of fractions, reduction to prime factors can be introduced
• Illustration and comparison of the magnitudes of common fractions
• Continuation of decimal notation
• Verbally formulated problems
• Simple problems involving direct and indirect proportions
• Possible introduction of percentages

Geometry

• Triangle, square, circle, isosceles and right-angled triangles
• Pythagoras’ theorem in terms of the right-angled, isosceles triangle
• Description of the complementary relationships between fundamental geometric forms; transformations; no general proofs as yet
• Towards the end of class 5 use of compasses and rulers can be introduced; constructions of circles and circle-based patterns; discovering geometric figures, such as the triangle, hexagon, square, rhombus, parallelogram, trapeze (these should all be familiar from free-hand drawing); in addition, a study of symmetry
• Different types of angles: acute, right-, obtuse etc.; measuring angles
• Introduction to some basic constructions (bisecting a line in the middle with a perpendicular, finding the midpoint of a line, dropping a perpendicular etc.); these are used to practise composing descriptions of constructional procedures.

Methodological considerations

Around the age of twelve, the concepts that have by now been internalised - the children having acquired them in concrete situations in earlier grades - will have worked  on under the surface, and now emerge  as a stronger capacity to penetrate and order things with the power of logic. In algebra this step is made particularly clear: it leads from working with numbers and operations in an elementary way to the perception of lawful relationships, which take the form of algebraic formulae. This is the beginning of purely conceptual thinking. Making this transition should be practised in many different ways (Schuberth 1995: p. 166). Of particular note here is Steiner’s suggestion that algebra should be developed from the world of commerce – the calculation of interest. He points out that at this phase of development the child has an instinctive relationship to profit and loss in the context of financial transactions and to the economic and social issues associated with them. This should, he said, be taken up in mathematics lessons, and be brought into the light of informed judgment (Steiner GA 294, 05.09.1919 (German edition 1974: p. 191)); and GA 295, 06.09.1919 (German edition 1984: p. 168f.)).

This is also the time to make clear the distinction between laws that have arisen as a result of convention (e.g. the formulae regulation interest) and those that have not (such as the sum of the angles of a triangle).

As the child approaches adolescence, his or her feeling life expands in all directions. Mathematics can be a great help at this time. The subjective opinions and ideas of adults are not necessarily welcome! Mathematics demands attention to one’s own thinking. In the process of discovering and applying mathematical laws the young adolescent can come to trust a kind of thinking that is not just a subjective experience, but a part of understanding the world as it is; in thinking we can inhabit the orderliness of the world.

Mixed-ability classes present special methodological challenges, which have been compounded with the introduction of multi-cultural inclusiveness. These challenges must be met by setting problems that incorporate many degrees of difficulty, and by varying the ways of working.

Mathematics is the education of the will in the realm of thinking. For this reason we should still forego the use of pocket calculators. Repeatedly using the mind as the calculator trains the will. This means that every opportunity should be taken to exercise the mind’s mathematical faculties, and not just the main lessons that are directly focused on them. One of the ways to provide extra opportunities for this to happen is to institute (from around class 6 onwards) running lessons in mathematics.

In geometry the aesthetic quality of the drawings now arises primarily from their being placed within a context of ordered thinking. The accurate use of compasses, ruler and set-square in connection with the reasoned description of the construction in hand leads to precision and beauty, which in turn can become the authorities motivating the students to undertake their own search for lawful connections (Fuhrer 2010). Causal thinking is in process of awakening, and working out geometric proofs independently, or approaching them through discussion, does not merely encourage it, but feeds and ignites it. Formulating proofs and laws demands a form of language that exactly matches what it seeks to describe. Experiencing and learning to use such a language, which is free of all emotionality and relates only to how things actually are, and not to how we would like them to be, is important for young people on the way to finding their own individual modes of expression (Schuberth 2001).

Suggested lesson content

Class/grade 6

Arithmetic and algebra

• Mental arithmetic continues here and in subsequent years in ever more demanding ways
• Consolidation of all the techniques of working with natural numbers, positive decimals and fractions
• More complex problems involving direct and indirect proportions
• Continuation of percentages into the realm of commerce: calculating interest, profit and loss, discounts, value added tax (VAT) – these topics should be included within a wider context looking at the question “what is money?” In this connection an introduction to modern economic structures involving a division of labour can be included (Schuberth 1995).
• First encounter with variables in the development of percentage and interest formulae
• Introduction to thinking the various operations in algebraic terms

Geometry

• Working with the flexibility of geometric figures and incorporating it in proofs
• Consolidating or extending the basic constructions in the geometry of point and line: bisecting equally with a perpendicular, halving an angle, erecting and dropping a perpendicular, parallel shift
• Various kinds of angles
• Experience and recognition of the universal validity of geometric laws as an introduction of the process of geometric proof
• The sum of the angles in a triangle and other many-sided figures
• Thales’ theorem
• Pythagoras’ theorem: from the isosceles to the general right-angled triangle
• Angles and magnitudes (see also class 3)
• Constructions with triangles, and possibly the congruency theorems

Classes/grades 7 and 8

We are putting classes 7 and 8 together here, because their agendas are more or less identical. Much is put in place in class 7, which is then consolidated and extended in class 8. It must be left to the class teacher to decide how he or she is going to organise the suggested themes over the two years. Basically it is advisable to introduce all the suggested topics in class 7, and then take them up again in class 8.

Arithmetic and algebra

• Introduction of negative numbers as an outcome of “surplus subtractive power” (“less than nothing”), and not of the linear sequence of numbers; opportunities to practise thinking in terms of balance sheets
• The four operations in connection with whole and negative rational numbers
• How to use brackets
• Introduction of the higher operations: powers and roots; it is possible to take the first steps in understanding logarithms (Schuberth 2008)
• Binomial formulae
• Exercises in factorisation and polynomic division
• Linear equations with one unknown factor; a variety of applications
• Extension of business mathematics introducing some elements of book-keeping; distinction between liquidity and capital in daily monetary transactions; continuation of the consideration of social questions – also in connection with history
• The root algorithm as an application of binomial formulae
• Calculations associated with the Pythagorean group of theorems
• Surface and volume calculation on the cube, parallelepiped and prism
• The mathematics of the circle, the constant p, circumference and area

Geometry

• Triangle constructions and their accompanying descriptions – also from bisectors of altitudes, sides and vertex angles
• Shears as form transformations of equal area
• Pythagoras’ theorem in connection with proofs involving shear and dissection transformations, which lead on to the theorems of Euclid
• Special lines and points in connection with triangles; in-, circum- and ex-circles; the medians and the centroid or centre of gravity (physics main lesson), altitudes and the orthocentre
• Simple drawings in perspective (in connection with modern history; but it is recommended that colour perspective be introduced before linear perspective (®fine art, painting)
• Elementary geometry of the circle: tangent constructions, periphery and centre-angle theorem proof

Possible further topics

• Examples of different kinds of proof in geometry – those that use congruence, and those that use symmetrical reflections
• The golden section and the pentagon
• Symmetrical and asymmetrical quadrangles and their properties
• Circles inside and around four-sided figures
• Elementary theory of curves (Locher-Ernst 1988)
• The astonishing polarities of the platonic solids

Methodological considerations

On the one hand, class nine students are characterised by high idealism, which often shows itself in impulsive ways, and display a relish for the world of ideas. The other side of what they reveal of their inner life is a strong tendency to see their own needs as primary and to pursue them with a will. Mathematics at the beginning of high school picks up on this motif by taking concrete situations or human intentions as objects for thought. It thus sets an example of how the mind caught up in its own desires can expand out of itself, and how initially subjective and selfish intentions can be applied in the world in a more appropriate way. This can provide a firm foundation for the idealism typical of adolescence.

Real-life situations are the starting point for students to exercise and sharpen their power of judgment. Mathematical thinking is thus linked to practical outcome, and the students feel its inherent power. This becomes especially clear when we enter the topic of combinatorics: from the innumerable abundance of individual cases super-ordinate patterns and structures arise, which bring order to the chaos. A complementary train of thinking arises from Steiner’s suggestion to work with “approximations” based on practical examples (Steiner GA 300b, 17.01.1923 (German edition 1975, p. 222)). This brings us back to the approach described above, in which reason and practical intelligence show themselves to be useful and serviceable. It culminates, then, in constructions leading to the discovery of irrational numbers, which can be approached by various methods (e.g. continued fractions, “guess and check” (trial and error) etc.)

Further examples of the practical application of mathematical thinking are found in algebraic equation theory and in the realms of plane and solid geometry. The exact construction of bodies in space in geometrical (technical) drawing, which exercises the faculty of spatial visualisation, is also part of this, and can be of special relevance in connection with art and craft lessons such as woodwork (® Handcrafts, woodwork). Above all, the work on the conic sections lends mathematical thinking an aesthetic nuance, through which the sense of truth also acquires a dimension of feeling. This process of aesthetic deepening continues, moreover, in class 11, when this subject expands into analytical and projective geometry.

Besides the development of technically and practically oriented understanding, it is of paramount importance to empower the will so that idealism will have a sound basis on which to express itself. This entails offering, mostly in the running lessons, a wealth of individualised and differentiated exercises which encourage reflection and lead not to a mere mechanical facility for certain techniques, but to abilities compatible with heuristic strategies, and thus capable of being transferred to other areas. At the same time the method employed is geared towards the students gradually learning to assess their own progress.

Mathematics at the beginning of the high school should not simply continue the work of the middle school. It needs to be placed on a completely new footing: all that has been learnt so far must be understood in a new way, incorporated into new contexts, expanded. The experience of mathematics as an act of thinking now occupies centre stage.

Suggested lesson content

Arithmetic/number theory

• Smallest common multiple, highest common factor, possibly in connection with the Euclidean algorithm, divisibility rules and their rationale
• Prime numbers and the question of how many there are
• Figurate numbers
• In case it has not been covered in class 8: simplification of the processes of squaring and finding roots without electronic aids, building upon the binomial theorem
• Distinguishing the number ranges N, Z, Q with consideration of their particular features
• Rational numbers developed from their periodic decimal fractions; characteristic properties of the period numerals
• Extension into the realm of irrational numbers and the new number range R

• Derivation of irrational roots from continued fractions, preparing the ground for the concepts of sequences and limits
• The Golden Section (application in art, architecture, nature and the human body)
• Pythagorean triples
• The guess and check (trial and error) method
• If possible, Euclidean algorithm for determining incommensurability
• Recognising incommensurability in geometric examples
• squaring and the square root of 2
• equilateral triangles and the square root of 3
• regular pentagon and the square root of 5 in connection with the Golden Section

-counting systems with different bases, especially the binary system in connection with computers

Algebra

• using terms: addition, subtraction, multiplication, division of polynomials and fractions
• linear equations with two or three unknowns, applied to relevant situations, especially in the sphere of economics
• quadratic equations, quadratic complement, Viëta’s theorem, solution formulae (possibly not until grade 10)
• equations involving fractions, also using direct and indirect proportionality applied to various areas of practical life (calculating percentages, interest, compound interest)
• possibly linear and quadratic inequations

Combinatorics

• problems of selection and sorting, permutations, combinations, variations
• binomial coefficients, general binomial theorem, Pascal’s triangle
• possibly basic elements of probability theory, arising from combinatorial problems; gambling games, their social issues and economic ramifications

Geometry

• similar figures, proportionality theorems derived from practical problems; can also be done in connection with the trigonometry main lesson in class 10
• group of Pythagorean theorems: altitude and cathetus theorem
• if possible, theorem of centre-angle and angle at the circumference
• further steps in calculating area (triangle, quadrangle, square, rhombus, parallelogram, trapeze, deltoid)
• circumference and area of the circle and of circle segments
• calculating volume (cube, parallelepiped, prism; perhaps also, pyramid, cylinder, cone, sphere, cf. class 10 topics)
• the conic sections and some of their properties, e.g. various types of curves (locus curves, envelopes, Cassini ovals etc.)

Geometrical drawing

• diagram of planar bodies in horizontal and vertical perspective
• drawings of the Platonic solids
• transformation of the simple Platonic solids into Archimedian solids through progressive “smoothing” of the corners or edges
• discovering the duality of Platonic solids
• visualisation exercises in connection with the above tasks
• descriptions of the spatial relationships involved
• exact descriptions of the construction procedures
• various ways of making diagrams of planar bodies

Methodological considerations

In class 10 young people are losing their easy-going frame of mind and becoming able to take things more seriously. Their thought life is no longer so caught up in individual details and the gratification of their immediate personal needs. Their thinking becomes a more conscious and inward activity and shows an increased ability to see the general picture; at the same time, thinking can become more abstract and self-absorbed. This tendency can lead them to the questions: How can it be that concepts and methods developed in a purely mathematical way fit together so well with the world’s phenomena? How come they are so effective and powerful? These are questions that form an undercurrent in maths lessons and are taken up in a very similar way in physics (® Physics). Teaching takes full advantage of this increase in the power of thinking: from purely mathematical beginnings, trigonometry proceeds to open up completely new territories, which follow on from the earlier triangle constructions and congruency theorems, widening their frame of reference in such a way that they become directly applicable to practical situations. This usually culminates in a two-week land-surveying practical. This yields hands-on experience of just how efficacious trigonometry is. On the one hand, it gives precise ways of checking and double-checking measured distances and angles, and on the other its use of triangulation makes calculating unknown distances so much easier. At the same time, however, it also exposes the gulf between the living landscape and the one-sided mentality that seeks only to measure it, turning it into a grid of surveying poles and sight-lines. It is worth pointing out and discussing the fact that this latter way of looking only pays attention to what is accessible to rationality and leads to a feeling of self-aggrandisement, resulting from (apparent) mastery over nature. Historically this has manifested in the fact that the existence of exact maps has made it politically possible to assert territorial claims. Technology also has often been an expression of this kind of mentality.

Logarithms and exponentials are likewise suited to modelling natural processes, but they can work equally well for social contexts. Here again the students have the experience of there being a good fit between mental constructions and living phenomena. At the same time, the narrow one-sidedness of this approach to the world should also become the subject of critical discussion (see esp. Köhler 1992)    

Suggested lesson content

Powers with whole-number and rational exponents, Logarithms

• arithmetical and geometric number sequences (this could also be done in grade 11)
• extension of the range of natural exponents to the ranges Z, Q and R
• the concept of the logarithm
• if possible, calculation of approximate values of logarithms with the help of continued fractions and other suitable methods
• logarithmic laws, base transformations
• historical context and working with logarithmic tables; possibly also, biographies of L. Eulers, J. Bürgis, J. Nepers
• exponential equations; if possible logarithmic equations
• if possible, logarithmic scales in science, Archimedian and logarithmic spirals with morphological examples from nature; the evolute
• modelling various types of object
• if possible, numerical relationships within music (harmonic series, intervals, tempered tuning etc.)
• if possible, elements of business mathematics (interest, compound interest, principles of credit etc.) in connection with social issues

Algebra

• more complex equations involving fractions
• higher order equations, bi-quadratic equations; if possible, root equations
• equation systems with several unknowns, if possible, the Gauss algorithm

Plane trigonometry

• the concept of similarity in connection with the interceptor theorem and centric elongation
• sine, cosine, tangent
• solving basic problems in conjunction with right-angled triangles, and carrying out calculations involving planes and figures in space
• extension of calculations in connection with angles larger than 90°, relationships between angles of the unit circle, radian measure
• the sine theorem
• the cosine theorem, and in its geometric form as the Pythagorean theorem extended to apply to triangles in general
• a variety of practical applications

Land-surveying practical

• familiarity with the normal measuring devices (especially the theodolite, optical level and angle prism)
• methods for measuring angles, elevations and distances
• if possible, tachymetry
• the use of error estimation, mean value determination and interpolation
• the uses of trigonometry in surveying, especially triangulation
• constructing an – as far as possible – accurate map

Area and volume calculations

• working out the value of p, for example through using the guess and check method on a circle inscribed within a regular polygon
• circle, circle sector, circle segment, annulus
• calculating the volume of pyramids, cylinders, cones and spheres

Geometrical drawing

• continuation of the drawing of figures of equal area by various methods
• exercises in axonometry
• curvilinear figures
• shadow constructions, limiting case of infinitely distant points in the casting of shadows
• transition from the dodecahedron to the icosahedron; the stages of their interpenetration
• screw, spiral, helix
• technical drawing: sketches and detailed design drawings of a carpentry project see Handcrafts – woodwork)
• central projection; function of vanishing points and the horizon

Methodological considerations

In class 11 a capacity for judgment develops that takes its lead from individual acts of thinking. As such it represents a different experience of the nature of thinking, no longer accepting the fact that a thought structure fits well with the external world or is practically efficacious as the sole measure of its truth. Thinking thus attains a more existential significance: the activity of the individual mind becomes responsible for bringing forth its own thoughts, which thus constitute a self-dependent cognitive reality. The challenge here is no longer to rely upon any kind of sense-based outlook, but to look to one’s own act of thinking as the means of arriving at a correct view of the matter in hand. From this cognitive vantage point the student now has the task of gradually developing his or her own philosophical stance.

In this process the main lesson on projective geometry – which adopts a synthetic approach – plays a central role. This subject was very close to Steiner’s heart; he repeatedly drew attention to its connection both to purely conceptual and to super-sensible knowledge (Steiner GA 324, 21.03.1921 (German edition 1991: p. 84ff.). The fact that it requires working with the concept of infinity means that the subject cannot be grasped in a naïvely naturalistic way. Rather, it involves taking up a position by means of an individual effort of will that can no longer lean on any kind of sensory correlative, and thus is beyond anything that can be easily imagined. It can and must, nonetheless, be taken hold of by concrete thinking. In this main lesson it becomes especially clear that to think about mathematics is actually to think about one’s own identity: in how far can we be sure there exists something beyond the senses? And does this non-sensory element have any meaning in the phenomenal world? What does this imply for my own mental life?

The theme of the analytical geometry main lesson stands in marked contrast to all this: whereas projective geometry investigates phenomena in relation to the (infinitely distant) periphery, analytical geometry brings everything into relation with a fixed system, the coordinate axis with its point of origin, which in pictorial terms can be seen as a point of reference projected out of the human being. But other possible coordinate systems should be described as well, in order to make clear that a functional relationship defined by an equation can be expressed in a variety of geometrical ways. Analytical geometry sets the experience of space on a firm footing, in that it combines geometry with arithmetic and algebra, whereas its projective counterpart opens up the experience of space to such an extent that other modes of space are conceivable. With regard to this widening of perspective, aspects of the geometry of spheres can be included. These in turn can be connected with astronomy.

In the realm of numbers, then, the concept of limits can be tackled in exact terms in connection with the theme of sequences and series. Here it also becomes clear that in thinking overstepping limits is possible, for the limit value does not lie in the (finite) individual components of a sequence, but in the law governing its overall behaviour.

Suggested lesson content

Sequences and series

• arithmetical as well as finite and infinite geometrical sequences and series (so far as not already covered in class 10)
• derivation of geometric series formulae and their application in natural science and economics
• development of the limit concept
• examples of how to display geometric series graphically
• periodic decimals as limits of geometric series
• compound interest as a special case of a geometric series: derivation of the formulae and their various uses in the contexts of nature and economics (cf. the lesson content of class 10); following on from that Eulers’ number e

Analytical geometry

• historical background
• the Cartesian and polar coordinate systems and their relationship
• the linear equation in its various forms and its graphical representation in the planar coordinate system
• application of mathematical knowledge to geometric problems (intersection of two lines, calculating the special points associated with triangles …)
• derivation of the circular equation
• investigation of the relationship between straight lines and circles
• correspondence between algebraic and geometric form, between equation and curve
• introduction to the concept of the function; relationship between curve and equation especially in whole-rational and simple-rational functions, zero points, symmetry, behaviour at the limit of definability. Connection with the equation theory work of classes 9 and 10
• possible further topics

• angle of intersection of circle and straight line
• polarities
• derivations of the equations for ellipse, hyperbola and parabola in initial normal position, of its tangents or asymptotes
• other higher order curves and their analysis
• derivation of the condition of contact and the tangent equation; significance of the discriminant and introduction to complex numbers as solutions to quadratic equations
• fundamental law of algebra
• introduction of the vector concept, and, if possible, developing it into vector geometry in observational space, and thence to a variety of section problems
• interpenetration of geometric figures

Projective geometry

• central projection, vanishing points and horizon (if not done in class 10)
• the point, line and plane at infinity
• central co-lineation; in particular, the significance of the position of centre and axis in relation to the line at infinity can be worked out in connection with elementary geometrical constructions, such as the parallel shift, centric elongation and shear etc.
• the concept of duality
• Desargues’ theorem
• Point and line domains
• Theorem of Pascal and Brianchon
• Harmonic quadrangle, harmonic position
• Polarities
• Continuity of form transformations in conjunction with the line at infinity and duality
• If possible, circle inversion or expansion into space

If possible, the geometry of the sphere

Introduction to non-Euclidean geometry via the sphere

• The “axiom on parallels”
• Diagrammatic representation of large and small circles on the sphere
• The spherical lune
• The construction of the spherical triangle by means of three determinant dimensions (congruency theorems); sum of the angles in a spherical triangle
• Construction of tangents with respect to a sphere at the points of its triangle
• Further examples, taken from mathematical geography and astronomy (® Astronomy):

• Navigational calculations, such as distance, azimuth etc.
• The pole triangle
• The sine and cosine theorems in sphere trigonometry
• Graphic representation of the horizon system, also diagram of determination of the position of a star
• The equator system
• Construction of a nautical triangle; the annual course of the sun; time determination (local time, time zones, sidereal time); calendar calculations; Platonic world year; moon and sun rhythms

Methodological considerations

In grade 12 the kind of thinking that in the previous year was experienced as an inwardly directed, self-dependent act now begins once again to connect itself to the world. In mathematical terms the processes by which phenomena become actualised can be shown, and only in such terms can their characteristic features be understood. This can be seen as relating both to the phenomena of the world and to individual human beings, insofar as they are also, as shapers of their own biographies, in process of becoming. In this connection, the subject of analysis offers a rich field of potential experience. Here Steiner strongly urged that differential quotients not be derived from tangential increments, i.e. from the field of geometry, but from the realm of pure number (Steiner GA 300c, 30.04.1924 (German edition 1975, p. 154)), as is indeed increasingly the case in current discussions on the teaching of mathematics at university level (cf. Danckwerts/Vogel 2006). With this it becomes clear what form of thinking is required. In order to judge how two processes in the form of the quotient of two differential sequences both tending towards zero behave in relation to each other, the focus must be on the qualities of the two processes and their interaction in the quotient. With the breaching of the limit, something new forms out of the features of these processes: the limit value. This method demands a higher degree of inner activity than does the graphical display of a function in a geometrical approach, which mostly conceals the actual problems. Looking at the world of numbers from this perspective, every number can be understood as a limit, i.e. as the product of a process, as, strictly speaking, is already evident with periodic decimal fractions. Since, then, analysis provides us with tools, which are exceptionally well-suited to being applied to processes of all kinds, it is safe to assume that this form of thinking is an inherent aspect of the world: “Applying the infinitesimal calculus to natural processes in mechanics and physics, we in fact accomplish nothing other than the calculation of the sensible from the super-sensible. We comprehend the sensible by means of its super-sensible beginning or origin.” (Steiner GA 35, ‘mathematics and occultism’ (German edition 1984, p. 12f))

These more narrowly mathematical matters should be followed up by philosophical reflection upon the foundations of mathematics. This could involve consideration of the relative merits of, e.g. formalism, Platonism, constructivism, intuitionism. It could also address the question of the continuum in relation to the potentially or actually infinite. In these deliberations everything that has been learnt all the way through school can be included, so that high school work can be rounded off with an experience of mathematics as a whole. Only now can it become really clear to the students, what an important position their own acts of thinking and their products have in the world, and, indeed, in the shaping of their own lives.

In the realm of geometry it would also be worthwhile to continue paying special attention to the qualitative aspect of the subject, for instance, in form transformations, so that here also the groundwork can be laid for a way of thinking that learns to acknowledge a spiritual principle at work in the processes by which the objects of this world come to be.

The order of the main lessons will follow the sequence of points given below, but the demands of state examinations also need to be taken into account.

Suggested lesson content

Differential calculus

• Revision of the concept of functions
• The range of definitions and values; graph of a function; introduction to inverse functions
• Working out the relationship between function and graph on the basis of elementary functions (cf. class 11)
• Short look at the history of infinitesimal calculus using the examples of Newton and Leibniz
• The difference quotient and the differential quotient
• The numerical continuum, the complete set of real numbers
• Investigating the differentiation rules for polynomial functions
• Possible interpretations of the differential quotient as the gradient in one point of the function graph, as local alteration in the rate of change, as instantaneous speed, as marginal tax rate etc.
• Graphic representation of the relationship between the indefinite integral (antiderivative) and its derivative functions
• Curve sketching, particularly in connection with polynomial and simple rational functions
• Application in various areas of life and of technology
• e-function, natural logarithms (possibly deepening what has been covered before)
• determining the term of the function from graphical evidence
• maxima and minima problems with examples from stereometry, economics, optics (the Fermat principle) etc.
• if possible, complex numbers

Integral calculus

• differentiation and integration of the indefinite integral (antiderivative) as inverse mathematical processes
• derivation of the integration method for polynomial functions, possibly starting from the method of exhaustion (of the parabola) as used by Archimedes and as applied more generally by his successors
• Integration as the calculation of the area under the function graph: from the rate of change to the integral
• the integral function as the function of the upper limit of a definite integral
• the concept of the antiderivative and the indefinite integral
• if possible, directional fields
• some integration rules (basic integrals)
• fundamental theorem of differential and integral calculus
• calculation of area of surfaces bounded by curves, and other applications
• if possible, calculation of the volume of rotated figures

Geometry (selected topics)

• Projective and affine descriptive geometry, both constructive and analytical:

• diagrammatic representation of points and lines
• fixed elements
• diagrams of conic sections (analytical, only in convenient positions with respect to the coordinate system)
• invariants
• introduction to the interpretation of the diagrams in terms of set theory
• Geometry of curvilinear surfaces:
• singularities of curves: thorn-points, pointed beaks, turning points, double points, double tangents and their possible positions relative to the line at infinity
• dual forms
• reciprocal polar transformations
• curve metamorphoses

• Non-Euclidean geometry
• Outline of vector geometry in observational space:

• scalar product, cross product, scalar triple (parallelepipedial) product
• area and volume calculations
• distance and intersection calculations of points, lines and planes
• spheres

History and philosophy of mathematics

• Personalities who shaped the development of mathematics
• Georg Cantor on questions surrounding the limits of the numerical size of sets, in conjunction with infinity construed as either actual or potential
• The philosophical schools of formalism, Platonism, constructivism, intuitionism
• The use of axioms
• The crisis as regards the foundations of mathematics; the formalist paradigm of David Hilbert; antinomials in set theory as presented by Paul Finsler
• Inter-relationships between mathematics, astronomy, botany, embryology and geometry

Possibly probability and statistics

• Concept of probability; binomial distributions
• Laplacian and non-Laplacian distributions; testing hypotheses
• Rules of addition and multiplication

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