“To start with, the “I” grapples intensively with the task, then this is looked at and commented by some “YOUS”. Finally, the teacher sums up all the interesting insights into a “WE” position.” (Gallin, 2010)
The principle of I – YOU – WE requires a balance between teachers' and pupils' activities. Prof. Dr. Gallin calls this “Dialogic Learning”. He is convinced that by offering choices and the possibility of pupils' collaboration in correcting their work, the danger of traumatising children through maths can be avoided.
Write down what you are thinking and how you approached the task!
Rather than a traditional notebook, the author suggests the use of a so called “journal”, an ongoing logbook or record. There, the pupils note their thoughts and the steps they took towards a solution. Thus, the pupils' thoughts become visible, possible solutions to the task are described and some self-correction is used.
Genesis of numeracy and implications for the notation system
For Gallin, successful mathematical thinking takes place in the balance between functional thinking (such as rhythms, stepping forward, processes) and predictive thinking (such as size, length, statics), but the structured, rhythmical counting process is often neglected. The author suggests the use of a counter; this is an illustrative device which is often overlooked. A counter is easily made and allows the children to independently investigate adding and subtracting.
"Human beings only play when in the full meaning of the word they are human, and they are only completely human when they play." (Schiller 1795)
This quote by Schiller points to other playful devices illustrated in the text, such as the “multiplication glasses”. The children are encouraged to “look for a place with a lot of countable things, for example a skyscraper with many windows, a roof with numerous roof tiles....” Further, the author explains the “multiplication table” or the “multiplication landscape”. These illustrative devices activate three dimensional thinking, they do not just emphasis factorization but also support motor skills, accurate measuring, imaginative thinking and much more.
Complicated calculations lose their horror if you can play with mathematical terms.
Finally, the author describes examples for an imaginative approach to the conversion of mathematical terms. The aim is to find as many calculations as possible relating to a specific number. Numerous examples by pupils illustrate the various approaches.
The readers gain not only some theoretical insight into Gallin's approach but also plenty of illustrative, practical examples by pupils. There is also a list of further publications by Gallin.
Unfortunately, Peter Gallin's excellent paper is only available in German. (Download) We are looking for volunteers to translate it into English. Please get in touch with the editors at Waldorf Resources if you are able to provide an English translation. Thank you.
Forum group "Mathematics" see here