Whereas the advice seems sound, it strikes me as much more in the
Piagetian/constructivist genre than in the Vygotskyan/sociocultural
genre. Do we make any educational use of Vygotsky's insight that
learning comes about through internalization of social processes, or
is this just a general characterization of how development
progresses. When it comes time to teach specific conceptual content
(rather than broad methods) we become Piagetians.
From: email@example.com [mailto:xmca-
firstname.lastname@example.org] On Behalf Of Steve Gabosch
Sent: Sunday, June 28, 2009 8:58 AM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus
Andy, the following from Leontiev I happened to be reading tonight
spurs a thought that might pertain to kids and adults when they get
"stuck" when learning. Maybe this has some relevance to the situation
you are speaking of. Just stabbing in the dark, of course.
Interesting idea in any case.
Leontiev talks about returning students to an **earlier** stage of a
learning process when they may have not have yet fully reorganized
their use of external operations into the new kinds of internal mental
required for the next stage. One step backwards creates the
possibility for steps forward, so to speak.
These paragraphs are from your new version of Leontiev's book,
Development of Mind, pages 392-393, at the end of the last essay, The
Child's Psychological Development and Mental Deficiency. As the notes
explain, this is the text of a lecture AN Leontiev gave to a World
Health Organization seminar in Milan in 1959.
I find this to be an interesting idea, that sometimes we need to go
backwards and reorganize old methods before being able to competently
proceed to new processes. Hmmm. Come to think of it, this sounds
like something I wind up doing a lot! LOL
"To explain what I mean, let me cite a simple experiment I once
made in a school for mentally backward children.
"I drew attention to the fact that the pupils, while doing mental
addition, were using their fingers for it in a concealed way. Then I
asked for several saucers, gave two to each pupil, and told them to
hold them above the desk while they were giving their answers. In
these conditions it proved that the operation of adding numbers
broke down completely in most of them. More detailed analysis indi-
cated that these children had in fact remained at the stage, as
addition, of the external operations of 'counting by ones', and had
not passed to the next stage. They therefore could not advance in
learning arithmetic beyond actions within the first ten numbers with-
out special help. For that purpose it was necessary, not to take them
further, but on the contrary to return them first to the original
of developed external operations, to 'reduce' these operations prop-
erly and to transfer them to the oral plane, in short to build a
'to count in their head' all over again.
"Research has shown that such a reorganisation is actually possible
even when working with children of quite pronounced mental back-
wardness. It is specially important that this approach has the effect,
cases of a slight lag in mental development, of completely eliminating
"Such intervention in the forming of mental operations of some
kind or other must, of course, be prompt and timely, because other-
wise the forming of the process cannot proceed further normally be-
cause the stage of its forming has sometimes not been built up by
chance or has been built up incorrectly, with the result that an im-
pression of alleged mental incapacity in the child is created."
<end of quote>
On Jun 27, 2009, at 11:00 PM, Andy Blunden wrote:
I hope people won't mind if I continue to pick the brains of this
list on the problem of my niece's progress in maths, or lack of it.
It seems that the suggestion last time - that Marissa may have
missed important lessons while on holiday - may explain her poor
performance last year in maths, even though maths has always been
her weak subject. She has caught up a bit but she is still badly
It seems that the issue Mike has raised also applies: she is getting
homework that seem to presume she know things that in fact she
doesn't. The only other negative in her school reports is that she
doesn't participate in class discussion or ask questions when she
doesn't understand something.
I presume the hesitancy about speaking up is probably the cause of
failure to correct her maths problems and the teachers giving her
homework she doesn't understand.
She is now 15 and her maths homework is also beyond her father! :)
and the crisis of the transition from childhood to adulthood around
this age, makes it impossible for the father to get Marissa talk
about it to him, or engage Marissa in games of 20 Questions or
something to lead her to the joys of asking others. Discussion over
the dinner table is apparently also unconducive to her participation.
Does anyone have any ideas? I've run out of suggestions. I could
probably help if I was there, but I'm 1000 km away.
Mike Cole wrote:
SO glad you are interested in this, Jay.
I have just made contact with Karen Fuson who has, lucky for us,
from Northwestern and moved nearby. She is away for a week or so
but then we are getting together. This is a problem that just may be
tractable, theoretically interesting for sure, attractive of
and god knows, of practrical importance to lots of kids.
On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <email@example.com>
Yes, Mike and F.K., these are very disturbing issues. Both that
think we want to teach seems to depend on deeper (e.g. 4000-year
knowledge than it's realistic to expect most people to learn (or
learn), and that how we teach even the most practical bits of
(like 15 minus 8) seems to have gone so wrong that it's hard to
to start, especially for those we have most systematically failed.
We do indeed need to not give up. But we also need, I think, to
it's time to seriously re-think the whole of the what, why, and
education. Math is a nice place to focus because at least some of
universally agreed to be useful by almost everyone, because
mathematicians and most people, including teachers and mathematics
educators, seem to hold radically different views about what the
and because success in teaching it, measured in almost any way, is
near the bottom of the heap.
Yes, we can find somewhat better ways to teach the same stuff, but
it's the stuff itself (the content of the curriculum, viewed not
information, but as activity) that needs to be rethought? along
ethics and efficacy of who decides.
No matter how many times you multiply a minus by any number of
still get a minus.
University of Michigan
Ann Arbor, MI 48109
On Jun 6, 2009, at 6:12 PM, Mike Cole wrote:
Hi Foo Keong-- It is so generous of you to even try to explain!
question re math seems to me
relevant to other areas of knowledge as well when you ask, "Can we
condensefour thousand years of
human development into an easily digestible four minutes for
Could we consider four years, just for whole numbers? Davydov
Algebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay
believe his method of introducing youngesters to math has some
As I understand it, others on xmca are dubious and look to other
difficulty. Karen Fuson, in her article on "developing
ins whole number operations" focuses on introducing number
through very simple, familiar, imaginable,
events where exchange is involved.
Its odd to me experiencing the cycle of time, the "coming back to
beginning and recognizing it
for the first time" that is happening for me right now with
early algebra. The source
is quite practical with social significance: the unbridgable gap
children I work with face between
what their teachers are teaching about (say) subtraction (2005-118
current keystone example)
trying to get their kids to learn that the first step is to
subtract 8 from
15 and know enough to treat the
second zero as a 9. But the child, even understanding that the
teacher is focused on is
disabled because when asked 15-8 the answer =3 and only painstaking
attention to the problem set up with fingers and subtracting one
with full compliance and even eagerness by the child, brings
her to 7.
Now suppose this phenomenon is ubiquitous, affects 100's of
children, and is heavily correlated with social class.
Then .... ??? ....
I think my frustration is probably equivalent to yourse in
the quality is of a somewhat different nature.
On Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong <firstname.lastname@example.org>
I was trained in mathematics at the University of Cambridge (UK)
for my undergraduate studies, concentrating more on pure
mathematics (including algebra). I am able to roll out a
rigorous abstract proof of why "minus times minus" is a "plus",
using only the basic axioms of real numbers (actually you only
need a few of those axioms).
However, abstract proofs aren't likely to be useful for non-math
specialists and struggling neophyte learners of algebra. in
order to pull off such a proof, or even just to understand just
the few lines of proof, you almost need to be a mental masochist.
Who likes to go through mental torture?
Can we condense four thousand years of human development of
mathematical understanding into an easily digestible four minutes
thus the huge gulf of understanding still persists. that's why
as an educator, i feel so useless being unable to help other
2009/6/4 Mike Cole <email@example.com>:
I am currently reading article by Fuson suggestion by Anna Sfard
number operations. I also need to study Anna's paper with exactly
example in it. Not sure what moment of despair at deeper
me. Now that I am done teaching and have a whole day to communicate
are looking up!! Apologies for doubting I could have deep
why minus x minus = plus and minus x plus = minus. At present my
understanding remains somewhat bifurcated. The former is negation
negation as david kel long ago suggested, linking his suggestion to
approach. The second I think more of in terms of number line and
multiplication as repeated addition.
Perhaps the two will coalesce under your combined tutelage.
And member book links are coming in. Nice.
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