Thanks for the article!

I've never understood why the main approach (at least in Italy) is "learn what I tell you to, one day you will find it useful". Is that so bad to tell first what we want to achieve and the start learning from the basics?

I can tell you a personal story. I am a programmer and I start programming at high school (personal projects, school didn't teach programming well enough).

I remember an interesting fact. When teachers introduce programming, they often introduce flow diagrams. Now, flow diagrams can be boring. But I was amazed when I discovered by my self that I could actually translate those diagrams into a computer language (quick basic at the time) and run them! Why, on the hearth, did the teacher not tell us that flow diagrams are actually a visual way of programming and that, with a basic knowledge of a programming language, they could be executed? Probably because for him/her flow diagrams were just a check on the to-do list of the program to follow. Something she/he had to teach, nothing more.

Besides, mathematics (and not only math) is full of anecdote. Often the life of mathematicians wasn't so boring as we erroneously think! One of them was killed in a duel for honor/love. Another sentenced to death one of his students…

Finally, I don't know if it can applied to elementary school, but sometimes a question is better than an explanation.