Well, a problem can be diagnostic without it being a "problem solver"-question. Ie. the problem 0.4x0.3 will not show whether the student understands anything, while the problem 0.4x0.2 will show it.

About formulas. A big problem with maths today is that students are given a bunch of formulas, explained what they're for and then set to learn them by heart(like rory studied physics). Let's say you've accomplished that, what is it you've really accomplished?

Nothing more than a 24-page booklet can tell you. In other words, you have just wasted years of education.

What students should learn is why those formulas are the way it is. Why the relation between percentage and desimalnumbers has to be the way it is, why you calculate area the way you do. Really, if you understand that the area of a rectangle is side x side, I don't see why you should have much problems with any other figures.

For example.