I think if I could ever have understood the difficulty, I would've had a chance of working around it. The problem seemed to be that they could place things on the number line, see that this was to the right of that, say that 0.4 was to the right of 0.35, say that hence 0.4 was bigger than 0.35, and then go right back to doing what they had always done.
My model of what was happening in the student's head is illustrated by this. A survey was conducted asking people:
A. Which is true:
(a) The Sun goes round the Earth.
or
(b) The Earth goes round the Sun.
?
B. How long does it take?
Of those people who gave the answer to A as (b), over 80% answered B with "a day" or "24 hours" or something equivalent. In short, they got the "right" answer to A, and then a completely disconnected and obviously "wrong" answer to B.
In this case I think people are answering A by reciting a fact they learned when young. We are told that the Earth goes round the Sun, so that's what people recount. By they don't connect that with their personal experience. Their personal experience is seeing the Sun going overhead. They "know" that the Sun goes round the Earth, and that it takes a day. They're told the Earth goes round the Sun, and that's what they parrot for question A, but when asked B, they answer from their own personal experience of seeing the Sun go overhead.
And that takes a day.
So I think the student could follow the things they had been told, and they could recite, parrot fashion, all the "right" things that would get them the marks on the test designed specifically to assess their knowledge of decimals places, and then when they had to put it into action, they fell back on what they felt things looked like, and were driven by their intuition and experience.
Which was completely wrong.
So that's my "theory" of where the disconnect lay. But maybe I'm wrong, because I certainly never fixed it.
I know that when I'm debugging a piece of code, and I get an idea in my head about how it works or should work, and that idea is shown to be wrong by clear and incontrovertible evidence, it can still take between a few hours and a few days for me to stop thinking it.
My model of what was happening in the student's head is illustrated by this. A survey was conducted asking people:
Of those people who gave the answer to A as (b), over 80% answered B with "a day" or "24 hours" or something equivalent. In short, they got the "right" answer to A, and then a completely disconnected and obviously "wrong" answer to B.In this case I think people are answering A by reciting a fact they learned when young. We are told that the Earth goes round the Sun, so that's what people recount. By they don't connect that with their personal experience. Their personal experience is seeing the Sun going overhead. They "know" that the Sun goes round the Earth, and that it takes a day. They're told the Earth goes round the Sun, and that's what they parrot for question A, but when asked B, they answer from their own personal experience of seeing the Sun go overhead.
And that takes a day.
So I think the student could follow the things they had been told, and they could recite, parrot fashion, all the "right" things that would get them the marks on the test designed specifically to assess their knowledge of decimals places, and then when they had to put it into action, they fell back on what they felt things looked like, and were driven by their intuition and experience.
Which was completely wrong.
So that's my "theory" of where the disconnect lay. But maybe I'm wrong, because I certainly never fixed it.