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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 regards
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 stage
of developed external operations, to ‘reduce’ these operations prop-
erly and to transfer them to the oral plane, in short to build a capacity
‘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, in
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>

- Steve

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 behind.

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, "retired"
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 experience
and god knows, of practrical importance to lots of kids.
On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <jaylemke@umich.edu> wrote:
Yes, Mike and F.K., these are very disturbing issues. Both that what we think we want to teach seems to depend on deeper (e.g. 4000-year deep) knowledge than it's realistic to expect most people to learn (or want to learn), and that how we teach even the most practical bits of mathematics (like 15 minus 8) seems to have gone so wrong that it's hard to know where
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 admit that it's time to seriously re-think the whole of the what, why, and how of education. Math is a nice place to focus because at least some of it seems universally agreed to be useful by almost everyone, because professional
mathematicians and most people, including teachers and mathematics
educators, seem to hold radically different views about what the subject is, and because success in teaching it, measured in almost any way, is pretty
near the bottom of the heap.

Yes, we can find somewhat better ways to teach the same stuff, but maybe it's the stuff itself (the content of the curriculum, viewed not just as information, but as activity) that needs to be rethought? along with the
ethics and efficacy of who decides.

No matter how many times you multiply a minus by any number of pluses, you
still get a minus.


 Jay Lemke
Educational Studies
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! And your
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 learners."

Could we consider four years, just for whole numbers? Davydov starts with Algebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay and others believe his method of introducing youngesters to math has some extra power. As I understand it, others on xmca are dubious and look to other sources of difficulty. Karen Fuson, in her article on "developing mathematical power ins whole number operations" focuses on introducing number operations
through very simple, familiar, imaginable,
events where exchange is involved.

Its odd to me experiencing the cycle of time, the "coming back to the
beginning and recognizing it
for the first time" that is happening for me right now with arithmetic and
early algebra. The source
is quite practical with social significance: the unbridgable gap the
children I work with face between
what their teachers are teaching about (say) subtraction (2005-118 is my
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 task 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 by one,
with full compliance and even eagerness by the child, brings
her to 7.

Now suppose this phenomenon is ubiquitous, affects 100's of thousands of
children, and is heavily correlated with social class.

Then ....  ??? ....
I think my frustration is probably equivalent to yourse in intensity, but
the quality is of a somewhat different nature.

On Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong <lefouque@gmail.com> wrote:

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

for learners?

thus the huge gulf of understanding still persists.  that's why

as an educator, i feel so useless being unable to help other

people.   :-(


2009/6/4 Mike Cole <lchcmike@gmail.com>:

I am currently reading article by Fuson suggestion by Anna Sfard on whole

number operations. I also need to study Anna's paper with exactly this

example in it. Not sure what moment of despair at deeper understanding


me. Now that I am done teaching and have a whole day to communicate


are looking up!! Apologies for doubting I could have deep understanding


why minus x minus = plus and minus x plus = minus. At present my

understanding remains somewhat bifurcated. The former is negation of a

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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