Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from ea… - Hans Freudenthal

Educational technique needs a philosophy, which is a matter of faith rather than of science.

No statistician present at this moment will have been in doubt about the meaning of my words when I mentioned the common statistical model. It must be a stochastic device producing random results. Tossing coins or a dice or playing at cards are not flexible enough. The most general chance instrument is the urn filled with balls of different colours or with tickets bearing some ciphers or letters. This model is continuously used in our courses as a didactic tool, and in our statistical analyses as a means of translating realistic problems into mathematical ones. In statistical language " urn model " is a standard expression.

Vertical mathematising is the most likely part of the learning process for the bonds with reality to be loosened and eventually cut.