[titter]

> Oh dear. When a student answers that 7/12 is 1.5 and doesn't immediately see why this couldn't possibly be true you know that the problem is rote learning of algorithms.

Not really, no.

1. Students have to have a good concept of division, as you suggest.

2. Students have to know a method of division that accurately gets a precise answer. (whether this is via an algorithmic method or via a calculator)

3. Students have to know that their answer to (2) should correspond to their concept in (1)

4. Students have to recognise that they can check their answer using (3), and then remember to actually check this.

A good teacher will teach all four parts of this process. However, it's not possible to teach part 4 without first teaching parts 1 and 2. Every single child makes this same mistake at some point in their learning. This is not really a "problem" and it doesn't indicate a failure of teaching - in fact, the opposite here: the important thing is that the teacher has identified it and can advise the child on how to improve their understanding.

That's what teaching is. As you say, many adults have not consistently achieved part 4 - in fact, it's not immediate at all: it has to be learned.