The group around her began to calculate, some of them with fingers. Suddenly someone said: “does not go…” A crowd surrounded the task, many of them working with finger-counting. Another girl said very convincingly: “8 – 9 doesn’t go, look, eight fingers minus 9 fingers.” Kaija changed a number in the chain.
This situation was a dense moment of curiosity, emergence and reflection (Allen & Bickhard, 2013). Freudenthal (1991) would say “realistic” mathematics. The teachers will work again with Kaija’s first task. The children learn to use commutative and associative laws, with addition and subtraction. They get over the actual borders of the natural numbers mathematically. They discover the integer numbers. – These kids are unplugged little mathematicians.
Kaija's '1 + 6 – 7 + 8 – 9 + 10 – 11=__' is a stepping stone for mathematical inventions and for co-operation. Math-lessons would be realistic and poetic, if teachers and psychologists could imagine mathematic (listen John Lennon, read Kierkegaard).
Allen, J. W. P., Bickhard, M.H. (2013). Stepping off the pendulum: Why only an action-based approach can transcend the nativist-empiristic debate. Cognitive Development, 28(2), 96-133.
Freudenthal, H. (1991). Revisiting Mathematics Education. China Lectures. Dordrecht: Kluwer Academic Publishers.
Kierkegaard, S. (2006). Fear and Trembling. Cambridge: Cambridge University Press.