"But when she was asked to solve the problem 12 3 5, she "brought down"
first the 2, then the 5, and then the 1 and came up with an answer of 152."
should be 12 * 5
and under "mathematical notation" another confusing plus sign:
"Common mistakes are to add numbers in the order in which they
are said--from left to right--and to line up numbers from the left.
Misunderstandings of this kind produce such errors as
writing 123 1 1 as
123
1100
223
Children who produce such answers are behaving as they are expected to do
in school in one important respect: they are applying previously acquired
knowledge to solve new problems. As long as they don't become discouraged,
they eventually catch on to the logic underlying the written number system
their teachers present to them."
=2E.. which should be 123 + 1
and, I suppose:
123
+100
----
223
-- which all (the way there's sense in children's common errors) still
holds good as useful reminders for young people who are going to work with
kids.
The (lack of) shortcut treatment of 4 + 9 - 9 and the other inversion
problems (and other similar experiments) always makes me think there's an
"authority component" in arithmetic -- where plus and minus serve as
commands to be followed without question and in the order given.
It takes some time figuring out the game is played differently, that it's
the numerical relations and notihng else that matter.
Eva