Steiner's reversed number teaching: write the total first and let the child take it apart, so analysing precedes adding instead of following it.
What is Arithmetic from the Sum?
Arithmetic from the Sum in Anthroposophy is Rudolf Steiner's reversal of the direction in which number is taught, set out in the tenth lecture of The Renewal of Education (GA 301), given at Basel on 5 May 1920 and returned to in the closing lecture of 11 May 1920. Instead of placing addends on the left and asking what they make, the teacher writes the total on the left and asks in how many ways it can be separated. Ten becomes five and five, or three and three and three and one. Multiplication starts from the product and hunts for its factors. Subtraction asks from which number five must be taken so that eight remains. Steiner held counting strictly apart from arithmetic, calling counting a naming of quantities, and he tied the reversal to the child's analytic force, which he paired with waking as he paired synthesis with falling asleep. Waldorf classrooms have worked this way since 1919.
Arithmetic from the Sum names the practice of beginning each operation at its result. A child handed ten and asked to break it apart is choosing freely; a child handed two, five, and three has one lawful answer already waiting. Steiner read that difference as a matter of will rather than of arithmetical convenience, and built the reversal into the first Waldorf curriculum on those grounds.
In Steiner's Own Words
In the beginning, we need to attempt, for instance, to begin with the number ten and then divide it in various ways. We need to show the children how ten can be separated into five and five, or into three and three and three and one. We can achieve an enormous amount in supporting what human nature actually strives for out of its inner forces when we do not teach addition by saying that the addends are on the left and the sum is on the right, but by saying that we have the sum on the left and the addends on the right. We should begin with analyzing the sum and then work backwards toward addition.
What it Means Today
Hans Freudenthal, the Dutch mathematician whose Utrecht institute now carries his name, gave the fault Steiner was describing a precise name. In Mathematics as an Educational Task (1973) Freudenthal called it the anti-didactical inversion: handing learners the finished, tidied result of mathematical work and asking them to run it forwards, when the work itself ran the other way. Mathematicians do not meet sums waiting to be totalled; they meet quantities that have to be taken apart. Freudenthal's counter-principle, guided reinvention, puts the learner back where the structure was still open and lets the answer be arrived at rather than received. Realistic Mathematics Education, built at the Freudenthal Institute and carried into Dutch primary classrooms, works this way in its splitting and number-bond exercises.
Thalira synthesis: Steiner reached the same diagnosis fifty-three years earlier by a different road, because he was asking not what mathematics is but what a nine-year-old's will becomes when it is never given a free choice, which makes the anti-didactical inversion a moral problem before it is a pedagogical one.
The practice itself is plain. Write 12 on the board and collect the ways down. Ask which number loses five and keeps eight. Let the partitions stay various, because the variousness is the point.
Where to Read More