• "It is possible to introduce a religious element into every subject, even into math lessons." • "Four is the sign of the cosmos or of creation ... Seven is the number of perfection ... [etc.]" • "Adding is related to the phlegmatic temperament, subtracting to the melancholic, multiplying to the...." • “Basic geometric concepts awaken clairvoyant abilities.” — Rudolf Steiner
MYSTIC MATH The Waldorf Curriculum: Arithmetic and Beyond
Two plus two equals four. The rules and truths of mathematics do not change. Math is math, and therefore kids who learn math in a Waldorf school learn the same things that kids everywhere else are taught. Right? Well, not exactly. As with most other subjects and activities, the study of math in Waldorf schools has an occult purpose. Waldorf math is not so much about learning how to handle numbers as learning to clairvoyantly perceive the gods' divine cosmic plan. That is, it is about becoming an occultist. Here are statements made on the subject by Steiner and his followers. You'll find some truth in these statements — mathematics does indeed open doors to an understanding and appreciation of the structure of the universe. Whether any parts of mathematics ought to be considered sacred, however — and whether math, properly comprehended, leads us to occultism — may perhaps be debatable. The quotations take us somewhat far afield, but the ride is worth taking if you want to grasp the Waldorf approach to math. As much as it may strain our credulity, all of the following comes within the context of mathematics as Steiner, using his marvelous clairvoyance, understood it.
◊ “Those who steep themselves in the right way in what, in the Pythagorean sense, we may call the 'study of numbers,' will learn to understand life and the world in this number symbolism.” [4]
◊ “[N]umbers and numerical proportions have a certain meaning for the cosmos and the world. It is in numbers, we might say, that the harmony that wells through space is expressed.” [5] ◊ “[N]umbers can give you a clue to what is called meditation if you have the key to plunge deeply enough.” [6] ◊ “Those who draw attention to this remarkable law of numbers explain it in an altogether materialistic way; but the weight of the facts themselves is already compelling people today once again to recognise the spiritual, mathematical law prevailing in the things of the world. We see how deeply true it is that everything which comes to expression in personal form in the later course of human evolution is a shadow-image of what was present formerly in elemental, original grandeur, because the connection with the spiritual world was still intact.” [7]
Waldorf math teachers may not usually state their occult views openly in class. But sometimes they do state them. These are the views they embrace, the "meanings" they find behind their subject — and naturally, on occasion, they voice such meanings. As we will see presently, Waldorf math teachers may lead students to the view that mathematics proves the truth of occult or at least spiritualistic beliefs. And to lead students in this direction, Waldorf math teachers may not always be above twisting the numbers. For Steiner and his followers, studying numbers comes close to affirming numerology. They see mystical significance in numbers, and they affirm the ancient concept of "sacred numbers" — the notion that some numbers have special occult significance, being imbued with sacred power. Here's one example, taken from a lecture by Rudolf Steiner: “[T]here is in Berlin an interesting doctor who has made remarkable observations ... I will indicate it on the blackboard ... [S]uppose that this point represents the date of a woman's death ... The woman is the grandmother of a family. A certain number of days before her death a grandchild is born, the number of days being 1,428. Strange to say, 1,428 days after the grandmother's death another grandchild is born, and a great-granddaughter 9,996 days after her death. Divide 9,996 by 1,428, and you have 7. [Steiner taught that 7 is the number of perfection.] After a period, therefore, seven times the length of the period between the birth of the first grandchild and the death of the grandmother, a great grandchild is born. And now the same doctor shows that this is not an isolated case, but that one may investigate a number of families and invariably find that in respect of death and birth absolutely definite numerical relationships are in evidence ... In short, the very facts compel people today to rediscover in the succession of outer events certain regularities, certain periodicities, which are connected with the old sacred numbers.” [13] Essentially, in the Waldorf belief system, math entails playing with numbers and geometric shapes in order to "discover" ideas that Anthroposophists already believe and are absolutely determined to affirm. Thus, they decide that 7 is the number of perfection. Then they make calculations (based in this case on unsubstantiated anecdotes) that enable them to arrive at the marvelous result: 7. So, golly! Seven! But once we have recovered from our astonishment (if any) occasioned by such elaborate occult reasoning, we might ask whether anything has actually been proven. Have we learned, for instance, that 7 is really a sacred number? Have we learned that 7 actually represents perfection? Have we learned that there are "periodicities" related to "the old sacred numbers"? Have we, in fact, learned anything? Or have we just watched some people fool around with numbers in order to beguile themselves? Here's another example:
" 333 BC [+] 333 AD 666
It is always possible to get the numerical answer you want if you are willing to play with the numbers that "lead" to it. Note that Steiner says Ahriman was at his peak "around" 333 BC. Perhaps. But the math gets mighty fuzzy — especially if the calculation also depends on the idea that 333 AD was exactly the year when humans needed to start consciously reentering "the realm of the higher hierarchies." All that Steiner has done is to get the answer he wanted, 666, by arbitrarily pinpointing two nebulous dates. Granted, 333 + 333 equals 666. But then so does 200 + 466, or 665 + 1, or... On the other hand, 330 BC + approximately 330-340 AD equals approximately 660-670, or thereabouts, give or take, more or less... The more we consider Steiner's numbers, the less impressive they become. And we haven't even asked ourselves whether Ahriman's power really peaked around 333 BC, or whether Ahriman even exists, or whether human beings actually began using logic in the year 333 BC (i.e., they hadn't used logic before and didn't begin in some year other than 333 BC), or whether mankind really turned a corner and needed to "strive to re-enter" the "higher hierarchies" precisely during the year 333 AD, or whether the "higher hierarchies" really exist, and so on. The more we consider Steiner's numbers, the less impressive they become. In fact, the more we examine Steiner's numbers, the more convinced we should be that his numbers mean nothing.` I may have mentioned that I attended a Waldorf school. The math teacher there spoke, from time to time, about the Platonic nature of phenomena. All "real" phenomena are actually just shadows of their ideal prototypes that exist in a supersensible realm, he taught us. That realm is real, and indeed the ideal phenomena there are more real than the "real" things we see around us. A chair, for instance, or a geometric form, or indeed a mathematical result, is only a poor manifestation of the ideal chair, form, or result existing out there in the wild blue ether. (I'm not using my teacher's exact words, which I can no longer recall.) The lesson I took away was that the "real" world is unreal, while the "ideal" or supernal world is reality. I didn't learn much math, but I absorbed this occult lesson, and I took it very much to heart. Indeed, this was the lesson we were given in all sorts of classes, not just math. It was one of our central tenets. The unreal is real, and it is all around us. When I saw Jimmy Stewart in the play "Harvey" — about a man who believes he is accompanied everywhere by an invisible six-foot-tall rabbit — I took it seriously. Sure, invisible realities are all around us. You say you have a six-foot-tall spirit-rabbit pal? Who am I to gainsay it? I left Waldorf a seriously befuddled lad. We find mathematical hocus-pocus in Waldorf instructional materials published quite recently — teachers' guides for Waldorf math courses. For instance, drawing on indications given by Steiner, Waldorf educator John Blackwood recommends the following math exercise. Determine the average respiration rate for children in a class. The result should be approximately 18 breaths per minute. Multiply this by 60 to get the number of breaths per hour, then multiply that result by 24 to get the number of breaths per day. The result should be 25,920. [15] Now, the “Platonic Cosmic Year” (also called the “Great Year”) is the length of time required for the Sun to cycle through all 12 signs of the zodiac. (That is, the Great Year is the period of the precession of the equinoxes.) The length of this “Year” is 25,920 regular Earth years, more or less. Now, calculate the number of days in a human life. Multiply 72 (years) by 360 (days) and what is the result? 25,920! Wow! This surely indicates the great design of the universe, no? The number of breaths we take in a day (25,920) is equal to the number of days we live (25,920), which in turn is equal to number of years in a “Great Year” (25,920). Bingo. From this, we can plainly see that human beings are microcosmic representatives of the great macrocosm, the vast universe presided over by the divine powers! Here’s how Waldorf teacher Blackwood puts this: “Put these [results] side by side and it all gets interesting — the human being is surely the microcosm in the macrocosm. Man is made in the image of God.” [16] Perhaps we were made in the image of God, but this mathematical sleight of hand does not prove anything of the sort. Break it down: ◊ How many breaths do we take per minute? This varies widely, depending on all sorts of factors. The results can be anything from about 10 to about 30. The average, in other words, is about 20. Let’s try the calculation using 20 breaths per minute instead of 18. Multiply 20 x 60 x 24, and what do you get? 28,800. If we accept this new result, our nice little paradigm wobbles: We want an answer of 25,920, not 28,800. We're off by 2,880 (28,880 - 25,920). So let’s ignore the new result. Disregarding the facts, let's cling to the idea that we breathe 25,920 times a day. ◊ How long is the “Platonic Cosmic Year” or “Great Year”? According to many sources, it is 26,000 years, more or less. Other sources, aiming for greater precision, say it is 25,800 years, and some specify 25,765 years. [17] Sadly, none of these numbers is precisely what we want (25,920). We're off by anywhere from 80 to 155 years (approximately one to one-and-a-half centuries). So let’s ignore these alternate results. Disregarding the facts, let's cling to the idea that a Great Year is 25,920 years long. ◊ How many days do we will live? The Biblical span of three score years and ten is 70 years. A year is 365 days long. So, the average life should be 70 x 365 = 25,550 days. [19] To be more precise, a year is really 365.256 days long (remember leap year), so the result is 70 x 365.256 = 25,568.2 days. Neither answer, sadly, is 25,920. We're off by 370 or perhaps 352.08 days — in either case, approximately a whole year. So let’s ignore these results, too. Disregarding the facts, let's cling to the idea that a human life is 25,920 days long. Not all Waldorf math instruction is suffused with occultism. Some of it is merely superficial. For example, H. v. Baravalle’s GEOMETRY [21] sidesteps logical proofs of the kind that are central to geometry as taught in most schools. Instead, the book emphasizes looking at and creating pretty geometric designs. This is fun for kids, and it may have some educational value — but it is intellectually lightweight. The same holds for the same author’s TEACHING ARITHMETIC AND THE WALDORF SCHOOL PLAN [22], which dwells on such matters as “magic squares” (in essence, what today we might call simplified sudoku) — fun, but trivial.
If you subscribe to the religious doctrines of Anthroposophy, you may approve. If not, you may want to look for a different sort of school for your children. You may find some benefit in considering what passes for reasoning among Steiner and his followers. They explicitly downplay "mere intellect, mere logic." Their opposition of rationalism leads them to the sorts of propositions we've seen, above. Intellect and logic are, of course, absolutely essential to math. The science called mathematics is really nothing more than the use of logic as applied to numbers. The Anthroposophical aversion to logic helps explain why math in Waldorf schools is so often shallow. What do Steiner and his follower turn to instead of rationality? Write down the answer and pass your paper to your neighbor. Hint: Consider the titles of Steiner texts I've quoted here: “Mathematics and Occultism”, OCCULT SIGNS AND SYMBOLS, OCCULT HISTORY, and THE FOURTH DIMENSION: Sacred Geometry, Alchemy, and Mathematics. In brief: the answer is occultism. Did you get the right answer? Excellent, A+. Now, for extra credit... — Roger Rawlings Above I referred to Platonic nature and Platonic years. Plato pops up pretty often in Waldorf schools. Plato said many things that are more or less mystical, so he is popular among Steiner's followers. Thus, for instance, Anthroposophists accept as reality the myths Plato spun about an ancient land: Atlantis. [See "Atlantis and the Aryan".] But Plato is big in Waldorf schools for another reason, as well. Waldorf teachers try not to talk about Steiner too much in front of their students — they know they are not supposed to "teach Anthroposophy" to the kids (not openly, anyway). So, instead, they often make use of Plato, Goethe, Wordsworth, Emerson, or any other famous figure whose words might be twisted to seem to agree with Steiner's. They use these proxies to stand in for Steiner. Why? In order to teach Anthroposophy to the kids. AFTERWORD: That Big Result, Again Steiner repeatedly stressed the importance of the wonderful number 25,920. His account was somewhat more sophisticated than Blackwood's, but not much. Here is one version (Steiner explained this wonderful matter more than once): By the way, on other occasions Steiner said that the earth breathes much more slowly than he indicates here.
Waldorfish math is even more mystical t han I've let on in this essay. To gaze upon other aspects of this subject, see "Magic Numbers" and "Temperaments".
[Illustration from p. 40; I have added color to the b&w image.] "THE PROBLEM OF DIVERSITY. "From Kircher's
some non-Anthroposophical images, such as the one above, to encourage everyone to compare and contrast Anthroposophy with other systems that claim to embody occult, arcane, esoteric, or mystery wisdom. [Design from Wil Stegenga's PICTORIAL ARCHIVE OF GEOMETRIC DESIGNS (Dover Publications, 1992). I have added colors and spooky shadings — R.R., 2010.] Because Steiner said that geometry helps foster clairvoyance, this branch of math receives special emphasis in Waldorf schools. Students are often led to create dramatic geometric designs. Here is one I did recently, more or less capturing the feel of designs I remember creating as a Waldorf student. (But. if I recall, all of them consisted of straight, intersecting lines connecting points on a circle, and the choice of color was more programmatic. Still, the result was similar: swirling, concentric rings.) [R. R., 2010.] Clearly, such designs are related to mandalas, which are traditional aids to religious meditation. [Design by Alberta Hutchinson; I have tweaked it and altered the colors — R.R., 2010.] Anthroposophists show considerable interest in mandalas. Not long ago, for instance, a Rudolf Steiner Institute summer course bore the title "The Mandala: An Archetype of Self and World." [Anthroposophic Press, 2001.] From the back cover: Steiner discusses, among other things, "The relationship between geometric studies and developing direct perception of spiritual realities." For "direct perception" read "clairvoyant sight." (As for alchemy...)
Some people's math is other people's moonshine. Here is Steiner explaining, as he often did, that the planets move along geometric lines — but not ellipses or circles, since they do not orbit the Sun.
[R.R. sketch, 2010.] [OLD-TIME CUTS AND ORNAMENTS (Dover, 2001), p. 32.] The mathematics teacher at the Waldorf school I attended often said that if you enter a city from one direction, it will be a different city than if you enter it from another direction. He did not simply mean that you would see the city differently; he meant that the city would literally, truly be different. This concept is consistent with Anthroposophy, which teaches that thoughts exist as real beings in the spirit realm; what we think comes to pass, literally, because we think it. The universe is malleable — our subjective states make the universe different from what it might be if we had different subjective states. Even the import of mathematics would be different if we had a different mental attitude. This is the radical subjectivity promoted by Waldorf education, and it is clearly wrong. Facts are facts; they do not bend to our preferences. Truth is truth; we do not make something true by thinking it. (You do not alter the nature of New York City, for instance, by perceiving it from the east instead of from the west.) We Steiner was speaking, here, about the mystical meaning of mathematics and geometry (or what is sometimes called sacred geometry). He found occult meaning in numbers, in geometrical design, and indeed in all orderly phenomena. This is what "a mature soul-condition" may find. But is it truly an apprehension, or merely a subjective desire? Is it found in phenomena, or is it read into them? Our subjective states are, of course, important. How we feel about things is, of course, important. The spirit in which we act is, of course, important. But recognizing the importance of such things should not muddle us. Our inner states are important, but they are separate from — and do not control — outer, objective reality. Steiner's teachings result in such concepts as the following:
What we believe certainly may shape reality if we act on our beliefs — but Each of us is a microcosm. A pretty idea, perhaps. But... [R.R., 2010.] A non-Anthroposophical version of the micro/macro concept [Lewis Spence, AN ENCYCLOPEDIA OF OCCULTISM (Dover, 2003), facing p. 256]. The occultism found behind and in Waldorf schools differs from other forms of occultism is many ways. But there are also many links between Anthroposophy and competing types of occultism. Cosmic impression, by a Waldorf grad [R. R., 2013.] To visit other pages in this section of Waldorf Watch, use the underlined links, below. ◊◊◊ 5. THE WALDORF APPROACH ◊◊◊
Some illustrations on the various pages here at Waldorf Watch are closely connected to the contents of those pages; others are not — they provide general context. ENDNOTES
Manvantara is a period of manifestation. Buddhi is an advanced stage of spiritual consciousness, the transformed etheric body. Kama is desire, which is "colorless" and either good or bad, depending. [2] “Mathematics and Occultism”. [3] Robert Trostli, RHYTHMS OF LEARNING: What Waldorf Education Offers Children, Parents & Teachers (SteinerBooks, 1998), p. 123. [4] Rudolf Steiner, OCCULT SIGNS AND SYMBOLS (Anthroposophic Press, 1972), lecture 3, GA 101. [5] Ibid. [6] Ibid. [7] Rudolf Steiner, OCCULT HISTORY (Rudolf Steiner Press, 1982), p. 77. [8] Ibid., p. 75. Steiner spoke of sacred numbers as well as sacred geometry. [For more on the occult significance of numbers, see "Magic Numbers".] [9] Rudolf Steiner, THE FOURTH DIMENSION: Sacred Geometry, Alchemy, and Mathematics (Anthroposophic Press, 2001), p. 92. [10] Ibid., p. 74. [11] Ibid., pp. 24-25. For more on mediums, see "seances". [12] Ibid., pp. 39-40. [13] OCCULT HISTORY, p. 75. [14] Rudolf Steiner, GUARDIAN ANGELS (Rudolf Steiner Press, 2000), pp. 96-97. In popular belief, 666 is the mark of the Antichrist. In Revelation 13:18 (King James version), we find
[15]
Both Steiner and Blackwood acknowledge that there is some imprecision in the calculations, so the number 25,920 is an approximation. Whether this is helpful in math class is questionable (is 2 + 2 = 4.1 approximately correct or simply wrong?), and it does not remove the central problem: The astrological and other occult premises in Waldorf math are baseless. (A carpenter who needs to cut a board 4 feet long but cuts it 4.1 feet long will be off by almost an inch and a quarter; the board won't fit.) [16] MATHEMATICS IN SPACE AND TIME, p. 102. [18] To put this another way: We can always get the results we want if we play fast and loose with numbers. So, if we say that an average life is 72 years long instead of 70, and that a year is 360 days long instead of 365, we can get the result we wanted from the beginning: 25,920. But this is not a real result, it is merely the number we were determined to get, come hell or high water. There’s a larger point, too. Let’s say that we don’t fiddle with any numbers — let's say that scrupulous calculation really shows that the number of breaths in a day is equal to the number of days in a lifetime, and this in turn is equal to the length of a Great Year. Have we proven anything? Or have we simply found a coincidence? Specifically, have we proven that we are microcosms of the universe? Or have we made a huge, illogical leap? Consider the following, for instance. What if instead of using breaths per minute we use the average number of heartbeats per minute? (Like breaths, heartbeats are extremely variable, but let’s pretend that they aren’t.) Some sources give an average of 50 beats per minute. 50 x 60 x 24 = 72,000. This is nowhere close to 25,920, so our nice little paradigm is knocked to pieces. For this reason, a Waldorf teacher will insist on breaths per minute, not heartbeats per minute. To get a predetermined answer, you must take care to select only the data that will produce that answer. In other words, you must fudge. Steiner fudged with numbers incessantly. For occult reasons, he was determined to categorize phenomena in groups of seven and twelve, for instance. He very often succeeded, and this impresses some people. But forcing phenomena into preselected, arbitrarily delimited brackets proves nothing. You can always get the results you are determined to get, if you are willing to cut, trim, and paste to suit. [19] Actually, in Germany today, the average lifespan is about 80 years. (Remember that Steiner delivered most of his lectures in Germany.) In Great Britain, it is about 80.4 years, in France about 81.4 years, in the USA about 78.2, in Japan about 82.9. [See www.google.com/publicdata.] Thus, the average Japanese lives about 30,280 days, or 4,360 days longer than the magic number, 25,920. Of course, lifespans were shorter in Steiner's day; but the point is that the wonderful pattern he claimed to spot has no basis in reality, and any apparent plausibility in Steiner's words has only declined over time as lifespans have lengthened. [20] The exercise suggested in MATHEMATICS IN SPACE AND TIME amounts to Anthroposophical indoctrination or at least softening. At first blush, members of mainstream Western religions may think the exercise seems okay: We are created in God's image; who wants to deny this? But consider whether your faith includes astrology, which is so important in Anthroposophical thinking. [See, e.g., "Astrology".] Christianity, Islam, and Judaism do not embrace astrology. Anthroposophy does (albeit an odd astrology reworked by Steiner). [21] H. v. Baravalle, GEOMETRY (Publications of the Waldorf School, Adelphi College, 1948). The book is a junior high school teachers' guide, so we should cut it some slack. However, most guides and textbooks for junior high geometry in regular schools are considerably more substantial. [22] H. v. Baravalle, TEACHING ARITHMETIC AND THE WALDORF SCHOOL PLAN, (Publications of the Waldorf School, Adelphi College, 1950). [23] Christians, Hindus, and Zoroastrians may be attracted to Anthroposophy, since it contains elements of their faiths. But they should realize that Anthroposophy diverges far from their faiths in many of its others doctrines. Meanwhile, of course, Muslims, Jews, secularists, and most others should realize that Anthroposophy explicitly rejects their viewpoints. [24] Rudolf Steiner, THE CHILD's CHANGING CONSCIOUSNESS AS THE BASIS OF PEDAGOGICAL PRACTICE (Anthroposophic Press, 1996), p. 94. [25] Rudolf Steiner, “Man’s Position in the Cosmic Whole, the Platonic World-Year” (ANTHROPOSOPHIC NEWS SHEET, Jan. 8, 1940, No. 1-2). The formatting at Waldorf Watch aims for visual variety, seeking to ease the process of reading lengthy texts on a computer screen. I often generalize about Waldorf schools. There are fundamental similarities among Waldorf schools; I describe the schools based on the evidence concerning their structure and operations in the past and — more importantly — in the present. But not all Waldorf schools, Waldorf charter schools, and Waldorf-inspired schools are wholly alike. To evaluate an individual school, you should carefully examine its stated purposes, its practices (which may or may not be consistent with its stated purposes), and the composition of its faculty. — R. R. |