Method
At school the method is explained first and similar examples follow. We do it the other way round: the child gets a problem for which the method has not been handed out yet.
Children try things, argue, make mistakes out loud and check one another's guesses. The teacher does not supply the answer — they ask the next question. The rule is put into words at the end, by the children, and so it is remembered as a conclusion rather than a formula.
Up to ten — any more and there is no time to hear out every child's reasoning. In Family Club up to sixteen, because there children come with a parent and work in pairs.
The tradition of mathematical circles is pedagogy, not geography. It grew recognisable outside the language it was invented in: in the United States, schools working by this method became a notable phenomenon, The Atlantic and NPR wrote about them, and their students place regularly at olympiads.
We work in that tradition in Espoo and Helsinki, and we teach in Russian, Finnish and English, because the method does not depend on the language.
Sixteen topics, each one its own lesson. The programme differs by age; below is the one for grades 2–3.